colour and the optical properties of materials: an exploration of the relationship between light,...
TRANSCRIPT
Colour and the OpticalProperties of Materials
An Exploration of the Relationship Between Light,the Optical Properties of Materials and Colour
PROFESSOR RICHARD J. D. TILLEY
Emeritus Professor, University of Cardiff, UK
Colour and the Optical Properties of Materials
Colour and the OpticalProperties of Materials
An Exploration of the Relationship Between Light,the Optical Properties of Materials and Colour
PROFESSOR RICHARD J. D. TILLEY
Emeritus Professor, University of Cardiff, UK
This edition first published 2011
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Library of Congress Cataloging-in-Publication Data
Tilley, R. J. D.
Colour and the optical properties of materials : an exploration of the relationship
between light, the optical properties of materials and colour / Richard J. D. Tilley.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-74696-7 (cloth) – ISBN 978-0-470-74695-0 (pbk.)
1. Light. 2. Optics. 3. Color. I. Title.
QC355.3.T55 2010
535.6–dc22
2010025108
A catalogue record for this book is available from the British Library.
ISBN 9780470746967 [HB]
ISBN 9780470746950 [PB]
Set in 10/12pt Times Roman by Thomson Digital, Noida, India
To Anne, for her continued help and support
Contents
Preface xv
1 Light and Colour 1
1.1 Colour and Light 1
1.2 Colour and Energy 3
1.3 Light Waves 5
1.4 Interference 7
1.5 Light Waves and Colour 9
1.6 Black-Body Radiation and Incandescence 10
1.7 The Colour of Incandescent Objects 13
1.8 Photons 14
1.9 Lamps and Lasers 16
1.9.1 Lamps 16
1.9.2 Emission and Absorption of Radiation 17
1.9.3 Energy-Level Populations 17
1.9.4 Rates of Absorption and Emission 18
1.9.5 Cavity Modes 21
1.10 Vision 23
1.11 Colour Perception 28
1.12 Additive Coloration 29
1.13 The Interaction of Light with a Material 33
1.14 Subtractive Coloration 37
1.15 Electronic ‘Paper’ 39
1.16 Appearance and Transparency 40
Appendix A1.1 Definitions, Units and Conversion Factors 43
A1.1.1 Constants, Conversion Factors and Energy 43
A1.1.2 Waves 43
A1.1.3 SI Units Associated with Radiation and Light 45
Further Reading 47
2 Colours Due to Refraction and Dispersion 49
2.1 Refraction and the Refractive Index of a Material 49
2.2 Total Internal Reflection 54
2.2.1 Refraction at an Interface 54
2.2.2 Evanescent Waves 54
2.3 Refractive Index and Polarisability 58
2.4 Refractive Index and Density 60
2.5 Invisible Animals, GRINs and Mirages 62
2.6 Dispersion and Colours Produced by Dispersion 65
2.7 Rainbows 68
2.8 Halos 75
2.9 Fibre Optics 75
2.9.1 Optical Communications 75
2.9.2 Optical Fibres 77
2.9.3 Attenuation in Glass Fibres 79
2.9.4 Chemical Impurities 80
2.9.5 Dispersion and Optical-Fibre Design 81
2.10 Negative Refractive Index Materials 84
2.10.1 Metamaterials 84
2.10.2 Superlenses 87
Further Reading 89
3 The Production of Colour by Reflection 91
3.1 Reflection from a Single Surface 92
3.1.1 Reflection from a Transparent Plate 92
3.1.2 Data Storage Using Reflection 94
3.2 Interference at a Single Thin Film in Air 94
3.2.1 Reflection Perpendicular to the Film 96
3.2.2 Variation with Viewing Angle 97
3.2.3 Transmitted Beams 98
3.3 The Colour of a Single Thin Film in Air 99
3.4 The Reflectivity of a Single Thin Film in Air 101
3.5 The Colour of a Single Thin Film on a Substrate 102
3.6 The Reflectivity of a Single Thin Film on a Substrate 104
3.7 Low-Reflection and High-Reflection Films 105
3.7.1 Antireflection Coatings 105
3.7.2 Antireflection Layers 106
3.7.3 Graded Index Antireflection Coatings 108
3.7.4 High-Reflectivity Surfaces 110
3.7.5 Interference-Modulated (IMOD) Displays 110
3.8 Multiple Thin Films 111
3.8.1 Dielectric Mirrors 111
3.8.2 Multilayer Stacks 113
3.8.3 Interference Filters and Distributed Bragg Reflectors 114
3.9 Fibre Bragg Gratings 115
3.10 ‘Smart’ Windows 119
3.10.1 Low-Emissivity Windows 119
3.10.2 Self-Cleaning Windows 121
3.11 Photonic Engineering in Nature 121
3.11.1 The Colour of Blue Butterflies 122
3.11.2 Shells 122
Contents viii
3.11.3 Labradorite 122
3.11.4 Mirror Eyes 125
Appendix A3.1 The Colour of a Thin Film in White Light 126
Further Reading 127
4 Polarisation and Crystals 129
4.1 Polarisation of Light 129
4.2 Polarisation by Reflection 131
4.3 Polars 135
4.4 Crystal Symmetry and Refractive Index 137
4.5 Double Refraction: Calcite as an Example 138
4.5.1 Double Refraction 138
4.5.2 Refractive Index and Crystal Structure 140
4.6 The Description of Double Refraction Effects 143
4.6.1 Uniaxial Crystals 143
4.6.2 Biaxial Crystals 144
4.7 Colour Produced by Polarisation and Birefringence 147
4.8 Dichroism and Pleochroism 149
4.9 Nonlinear Effects 151
4.9.1 Nonlinear Crystals 151
4.9.2 Second- and Third-Harmonic Generation 153
4.9.3 Frequency Mixing 155
4.9.4 Optical Parametric Amplifiers and Oscillators 156
4.10 Frequency Matching and Phase Matching 157
4.11 More on Second-Harmonic Generation 160
4.11.1 Polycrystalline Solids and Powders 160
4.11.2 Second-Harmonic Generation in Glass 160
4.11.3 Second-Harmonic and Sum-Frequency-Generation by
Organic Materials
161
4.11.4 Second-Harmonic Generation at Interfaces 162
4.11.5 Second-Harmonic Microscopy 162
4.12 Optical Activity 162
4.12.1 The Rotation of Polarised Light 162
4.12.2 Circular Birefringence and Dichroism 166
4.13 Liquid Crystals 168
4.13.1 Liquid-Crystal Mesophases 168
4.13.2 Liquid-Crystal Displays 169
Further Reading 173
5 Colour Due to Scattering 175
5.1 Scattering and Extinction 175
5.2 Tyndall Blue and Rayleigh Scattering 176
5.3 Blue Skies, Red Sunsets 178
5.4 Scattering and Polarisation 181
5.5 Mie Scattering 184
5.6 Blue Eyes, Blue Feathers and Blue Moons 187
5.7 Paints, Sunscreens and Related Matters 188
ix Contents
5.8 Multiple Scattering 190
5.9 Gold Sols and Ruby Glass 191
5.10 The Lycurgus Cup and Other Stained Glass 193
Further Reading 195
6 Colour Due to Diffraction 197
6.1 Diffraction and Colour Production by a Slit 198
6.2 Diffraction and Colour Production by a Rectangular Aperture 200
6.3 Diffraction and Colour Production by a Circular Aperture 202
6.4 The Diffraction Limit of Optical Instruments 203
6.5 Colour Production by Linear Diffraction Gratings 205
6.6 Two-Dimensional Gratings 208
6.7 Estimation of the Wavelength of Light by Diffraction 210
6.8 Diffraction by Crystals and Crystal-like Structures 211
6.8.1 Bragg’s Law 211
6.8.2 Opals 213
6.8.3 Artificial and Inverse Opals 218
6.8.4 The Effective Refractive Index of Inverse Opals 221
6.8.5 Photonic Crystals and Photonic Band Gaps 223
6.8.6 Dynamical Form of Bragg’s Law 224
6.9 Diffraction from Disordered Gratings 225
6.9.1 Random Specks and Droplets 225
6.9.2 Colour from Cholesteric Liquid Crystals 228
6.9.3 Disordered Two- and Three-Dimensional Gratings 230
6.10 Diffraction by Sub-Wavelength Structures 231
6.10.1 Diffraction by Moth-Eye Antireflection Structures 231
6.10.2 The Cornea of the Eye 233
6.10.3 Some Blue Feathers 234
6.11 Holograms 235
6.11.1 Holograms and Interference Patterns 235
6.11.2 Transmission Holograms 235
6.11.3 Reflection Holograms 237
6.11.4 Rainbow Holograms 239
6.11.5 Hologram Recording Media 240
6.11.6 Embossed Holograms 242
Further Reading 243
7 Colour from Atoms and Ions 247
7.1 The Spectra of Atoms and Ions 247
7.2 Terms and Levels 252
7.3 Atomic Spectra and Chemical Analysis 254
7.4 Fraunhofer Lines and Stellar Spectra 255
7.5 Neon Signs and Early Plasma Displays 256
7.6 The Helium Neon Laser 259
7.7 Sodium and Mercury Street Lights 262
7.8 Transition Metals and Crystal-Field Colours 264
7.9 Crystal Field Splitting, Energy Levels and Terms 270
Contents x
7.9.1 Configurations and Strong Field Energy Levels 270
7.9.2 Weak Fields and Term Splitting 271
7.9.3 Intermediate Fields 273
7.10 The Colour of Ruby 277
7.11 Transition-Metal-Ion Lasers 281
7.11.1 The Ruby Laser: A Three-Level Laser 281
7.11.2 The Titanium Sapphire Laser 282
7.12 Emerald, Alexandrite and Crystal-Field Strength 283
7.13 Crystal-Field Colours in Minerals and Gemstones 284
7.14 Colour as a Structural Probe 287
7.15 Colours from Lanthanoid Ions 288
7.16 The Neodymium (Nd3+) Solid-State Laser: A Four-Level Laser 290
7.17 Amplification of Optical-Fibre Signals 294
7.18 Transition Metal, Lanthanoid and Actinoid Pigments 295
7.19 Spectral-Hole Formation 297
Appendix A7.1 Electron Configurations 300
A7.1.1 Electron Configurations of the Lighter Atoms 300
A7.1.2 The 3d Transition Metals 301
A7.1.3 The Lanthanoid (Rare Earth) Elements 301
Appendix A7.2 Terms and Levels 302
A7.2.1 The Vector Model of the Atom 302
A7.2.2 Energy Levels and Terms of Many-Electron Atoms 304
A7.2.3 The Ground-State Term of an Atom 306
A7.2.4 Energy Levels of Many-Electron Atoms 306
Further Reading 307
8 Colour from Molecules 309
8.1 The Energy Levels of Molecules 309
8.2 The Colours Arising in Some Simple Inorganic Molecules 311
8.3 The Colour of Water 315
8.4 Chromophores, Chromogens and Auxochromes 316
8.5 Conjugated Bonds in Organic Molecules: The Carotenoids 317
8.6 Conjugated Bonds Circling Metal Atoms: Porphyrins and Phthalocyanines 319
8.7 Naturally Occurring Colorants: Flavonoid Pigments 323
8.7.1 Flavone-Related Colours: Yellows 323
8.7.2 Anthocyanin-Related Colours: Reds and Blues 324
8.7.3 The Colour of Red Wine 328
8.8 Autumn Leaves 332
8.9 Some Dyes and Pigments 333
8.9.1 Indigo, Tyrian Purple and Mauve 335
8.9.2 Tannins 337
8.9.3 Melanins 337
8.10 Charge-Transfer Colours 340
8.10.1 Charge-Transfer Processes 340
8.10.2 Cation-to-Cation (Intervalence) Charge Transfer 341
8.10.3 Anion-to-Cation Charge Transfer 345
8.10.4 Iron-Containing Minerals 346
xi Contents
8.10.5 Intra-Anion Charge Transfer 348
8.11 Colour-Change Sensors 349
8.11.1 The Detection of Metal Ions 349
8.11.2 Indicators 350
8.11.3 Colorimetric Sensor Films and Arrays 353
8.11.4 Markers 354
8.12 Dye Lasers 355
8.13 Photochromic Organic Molecules 358
Further Reading 361
9 Luminescence 363
9.1 Luminescence 363
9.2 Activators, Sensitisers and Fluorophores 365
9.3 Atomic Processes in Photoluminescence 368
9.3.1 Energy Absorption and Emission 368
9.3.2 Kinetic Factors 370
9.3.3 Quantum Yield and Reaction Rates 371
9.3.4 Structural Interactions 374
9.3.5 Quenching 374
9.4 Fluorescent Lamps 379
9.4.1 Halophosphate Lamps 379
9.4.2 Trichromatic Lamps 381
9.4.3 Other Fluorescent Lamps 382
9.5 Plasma Displays 383
9.6 Cathodoluminescence and Cathode Ray Tubes 385
9.6.1 Cathode Rays 385
9.6.2 Television Tubes 386
9.6.3 Other Applications of Cathodoluminescence 389
9.7 Field-Emission Displays 390
9.8 Phosphor Electroluminescent Displays 391
9.9 Up-Conversion 394
9.9.1 Ground-State Absorption and Excited-State Absorption 395
9.9.2 Energy Transfer 399
9.9.3 Other Up-Conversion Processes 401
9.10 Quantum Cutting 402
9.11 Fluorescent Molecules 405
9.11.1 Molecular Fluorescence 405
9.11.2 Fluorescent Proteins 407
9.11.3 Fluorescence Microscopy 409
9.11.4 Multiphoton Excitation Microscopy 410
9.12 Fluorescent Nanoparticles 411
9.13 Fluorescent Markers and Sensors 412
9.14 Chemiluminescence and Bioluminescence 413
9.15 Triboluminescence 416
9.16 Scintillators 416
Further Reading 418
Contents xii
10 Colour in Metals, Semiconductors and Insulators 419
10.1 The Colours of Insulators 420
10.2 Excitons 421
10.3 Impurity Colours in Insulators 424
10.4 Impurity Colours in Diamond 424
10.5 Colour Centres 429
10.5.1 The F Centre 429
10.5.2 Electron and Hole Centres 430
10.5.3 Surface Colour Centres 434
10.5.4 Complex Colour Centres: Laser Action 434
10.5.5 Photostimulable Phosphors 435
10.6 The Colours of Inorganic Semiconductors 436
10.6.1 Coloured Semiconductors 436
10.6.2 Transparent Conducting Oxides 437
10.7 The Colours of Semiconductor Alloys 440
10.8 Light Emitting Diodes 441
10.8.1 Direct and Indirect Band Gaps 441
10.8.2 Idealised Diode Structure 443
10.8.3 High-Brightness LEDs 445
10.8.4 Impurity Doping in LEDs 446
10.8.5 LED Displays and White Light Generation 446
10.9 Semiconductor Diode Lasers 448
10.10 Semiconductor Nanostructures 449
10.10.1 Nanostructures 449
10.10.2 Quantum Wells 451
10.10.3 Quantum Wires and Quantum Dots 454
10.11 Organic Semiconductors and Electroluminescence 457
10.11.1 Molecular Electroluminescence 457
10.11.2 Organic Light Emitting Diodes 459
10.12 Electrochromic Films 463
10.12.1 Tungsten Trioxide Electrochromic Films 465
10.12.2 Inorganic Electrochromic Materials 467
10.12.3 Electrochromic Molecules 468
10.12.4 Electrochromic Polymers 468
10.13 Photovoltaics 471
10.13.1 Photoconductivity and Photovoltaic Solar Cells 471
10.13.2 Dye-Sensitised Solar Cells 472
10.14 Digital Photography 474
10.14.1 Charge Coupled Devices 474
10.14.2 CCD Photography 476
10.15 The Colours of Metals 477
10.16 The Colours of Metal Nanoparticles 478
10.16.1 Plasmons 478
10.16.2 Surface Plasmons and Polaritons 479
10.16.3 Polychromic Glass 481
10.16.4 Photochromic Glass 482
10.16.5 Photographic Film 484
xiii Contents
10.16.6 Metal Nanoparticle Sensors and SERS 486
10.17 Extraordinary Light Transmission and Plasmonic Crystals 487
Further Reading 488
Index 491
Contents xiv
Preface
This book is concernedwith colour. It is not primarily a textbook on optics, but focuses attention upon theways
that colour can be produced and how these ways govern device applications. However it is not possible to
discuss colour without reference to numerous optical properties, so these, too, are explained throughout the
text. Colour, though, remains the dominant theme.
When writing about colour and colour production from a scientific point of view one is beset by a number of
language conflicts, arising from the historical importance of the subject. Much of this confusion is due to the
fact that the terminologyhas arisengradually, as a result of historical experiences that scientists of theday found
difficult to understand and interpret. Thus, diffraction, scattering, reflection and refraction can all be considered
to be scattering of photons, and the variety of terms in use only confuses the modern reader. Indeed, the nature
of light itself leads to problems. Is it a series of waves or a spray of bullet-like photons? A light wave can
apparently pass through holes in a metal foil that are far smaller than its wavelength. How can this be? Is the
light, instead, a series of photons that can do this, and if so, how big is a photon?
Other similar difficulties exist.Adecaying andglowing fungus exhibitingbioluminescencedoesnot produce
light by the same mechanism as a light-emitting diode (LED) using electroluminescence, although both are
termed luminescence. The termination ‘-chromic’ suffers from the same lack of precision. Thermochromic
molecules may or may not exhibit colour changes by the same mechanism as electrochromic thin films. The
names do not supply any information about this. The units used in the measurement of light are equally
confusing. This is because absolute measurements of energy, radiometric units, do not correspond to visual
perception, measured in photometric units.
Many of these questions are resolved in this book, particularly with respect to light and colour. The
explanations are taken at as simple a level that will allow an appreciation of the topic.
The book falls into three recognizable sections. Chapter 1 is introductory and covers ideas of light as rays,
waves or photons. The emission and absorption of radiation is described, as is the difference in light from an
incandescent source and light froma laser.Vision and the perception of colour (physiology or psychology), and
related aspects are described in outline, as is the technical measurement of colour. These are specialist topics,
and the information here is designed only to cover the need of subsequent chapters. Finally, the way in which
light can interact with a material is summarized, as a prologue to later chapters.
Chapters 2 6 explain optical phenomena mostly in terms of light waves. Colour is generated when light
waves comprising all colours (white light) are subdivided physically into a series of smallerwavelength ranges
(i.e. colours). Traditional divisions of the topic are retained, although there is little to choose, theoretically,
between labelling a process scattering, diffraction or reflection. Because of this, there is sometimes an
ambiguity as towhere a particular topic should be placed. For example, fibreBragg gratingsmight be treated as
multiple reflectors or as diffraction arrays. Mie scattering can be regarded as diffraction. The layout adopted
here is one that fits best with the explanations involved.
Chapters 7 10 require a photon explanation to account for colour production. Fundamentally, the absorption
and emission of light from atoms, ions and molecules forms the central theme, and for this a quantum
mechanical approach is needed. Many of these processes are widely exploited in displays. Although these are
technologically complex and require considerable engineering skills in production, the way in which they
produce colour is always based upon recognisable physical and chemical principles. Because of this, displays
are introduced throughout the text in terms of the appropriate colour-generating mechanism, rather than as a
separate section.
The topics covered encroach upon physics, chemistry, biology,materials science and engineering andmany
aspects of these intertwined subject areas are touched upon. Students of all of these disciplines should find this
book of relevance to some of their studies or interests. Readers who need more information can turn to the
Further Reading sections at the end of each chapter. These include selected references to the original literature
or substantial reviews andwill allow them to takematters further. In addition, thewebsite (www.wiley.com/go/
tilley colour) that accompanies this book contains exercises and numerical problems which have been
provided to illustrate and reinforce the concepts presented in the text. All readers are encouraged to attempt
them. There are also introductory questions that appear at the start of each chapter which are designed to
stimulate interest. The answers to these are found in the Chapter itself. In addition, the answers to these
introductory questions and all the other exercises and problems are to be found on the accompanying website.
Unfortunately, some important light-related topics have been omitted. These include the important
biological topicsof photoperiodism inanimals andplants andphotosynthesis.Althoughcolour is of importance
in these topics, the specialist knowledge here is biological rather than optical, and information in this field is
best reserved for biological texts.
It is a pleasure to acknowledge the considerable help and encouragement received in the preparation of this
edition. The editorial staff of JohnWiley& Sons have always given both assistance and encouragement in the
venture. I am indebted to Professor D. J. Brown, University of California, Irvine, USA; Mr A. Dulley, West
Glamorgan Archive Service, Swansea, Wales; Dr A. Eddington, Dr J. A. Findlay; Professor I. C. Freestone,
University of Cardiff,Wales; Spectrum Technologies plc, Bridgend,Wales; DrM. Sugdon, De La Rue Group;
DrR.D. Tilley, VictoriaUniversity ofWellington,NewZealand; ProfessorX. Zhang, University of California,
Berkeley,USA;Dr P.Vukusic, University of Exeter, England; DrG. I. N.Waterhouse, University ofAuckland,
New Zealand. All of them readily provided photographic material. To all of these I express my sincere thanks.
AllanCoughlin gave encouragement and advice, and themembers of staff of theTrevithickLibrary,University
of Cardiff, Wales, were indefatigable in answering my obscure queries.
Finally, my thanks, as always, are due to my wife Anne, who tolerated my hours reading or sat in front of a
computer without complaint, and made it possible to complete this work.
Richard J. D. TilleySouth Glamorgan
May 2010
Preface xvi
1
Light and Colour
. What is colour?
. Why do hot objects become red or white hot?
. How do e-books produce ‘printed’ words?
1.1 Colour and Light
Colour is defined as the subjective appearance of light as detected by the eye. It is necessary, therefore, to look
initially at how light is regarded. In fact, light has been a puzzle from earliest times and remains so today. In
elementary optics, light can usefully be considered to consist of light rays. These can be thought of as extremely
fine beams that travel in straight lines from the light source and thence, ultimately, to the eye. The majority of
optical instruments can be constructed within the framework of this idea. However, the ray concept breaks
down when the behaviour of light is critically tested, and the performance of optical instruments, as distinct
from their construction, cannot be explained in termsof light rays.Moreover, colour is not conveniently defined
in this way. For this, more complex ideas are needed.
The first testable theory of the nature of light was put forward by Newton (in 1704) in his book Optics, in
which it was suggested that light was composed of small particles or ‘corpuscles’. This idea was supported on
philosophical grounds byDescartes.Huygens, a contemporary, thought that lightwaswavelike, a point of view
also supported by Hooke. Young provided strong evidence for the wave theory of light by demonstrating the
interference of light beams (1803). Shortly afterwards, Fresnell andArago explained the polarisation of light in
terms of transverse light waves. However, none of these explanations was able to refute the particle hypothesis
completely. Nevertheless, the wave versus particle theories differed in one fundamental aspect that could be
tested.When light enters water it is refracted (Chapter 2). In terms of corpuscles, this implied a speeding up of
the light inwater relative to air. Thewave theorydemanded that the light shouldmovemore slowly inwater than
in air. The experiments were complicated by the enormous speed of light, which was known to be about
Colour and the Optical Properties of Materials Richard J. D. Tilley
� 2011 John Wiley & Sons, Ltd
3� 108m s 1, and it was not until April 1850 that Foucault first proved that lightmoved slower inwater than in
air, and seemingly killed the corpuscular theory then and there. Confirmation of the result by Fizeau a few
months later removed all doubt.
Over the years the wave theory became entrenched and was strengthened by the theoretical work of
physicists such as Fresnel, who first explained interference and diffraction (Chapter 6) using wave theory.
Polarisation (Chapter 4) is similarly explained on the assumption that light is a wave. Thewave theory of light
undoubtedly reached its peak when Maxwell developed his theory of electromagnetic radiation and showed
that lightwas only a small part of an electromagnetic spectrum. Lightwas then imagined as an electromagnetic
wave (Figure 1.1). Maxwell’s theory was confirmed experimentally by Hertz, whose experiments led directly
to radio.
The problem for the wave theory was that waves had to exist in something, and the ‘something’ was hard to
pin down. It became called the luminiferous aether and had the remarkable properties of pervading all space,
being of very small (or even zero) density and having extremely high rigidity. Attempts tomeasure the velocity
of the Earth relative to the luminiferous aether, the so-called aether drift, byMichelson andMorley, before the
end of the nineteenth century, proved negative. The difficulty was removed by Einstein’s theory of relativity,
and for a time it appeared that a theory of light as electromagnetic waves would finally explain all optical
phenomena.
This proved a false hope, and the corpuscular theory of light was revived early in the twentieth century,
principally by Einstein. Since 1895, it had been observed that when ultraviolet light was used to illuminate the
surfaces of certain metals, negative particles, later identified as electrons, were emitted. The details of the
experimental results were completely at oddswith thewave theory. The electrons, called photoelectrons, were
only observed if the frequency of the radiation exceeded a certain minimum value, which varied from
onematerial to another. The kinetic energy of the photoelectrons was linearly proportional to the frequency of
the illumination. The number of photoelectrons emitted increased as the intensity1 of the light increased, but
their energy remained constant for any particular light source. Very dim illumination still produced small
numbers of photoelectrons with the appropriate energy.
2101-210-410-610810-1010-1210-1410
radiowaves
micro-wavesinfrared
ultra-violetX-rays
gammarays
wavelength (m)
400 500 600 700 Wavelength / nm
vis ble
violet blue green yellow orange red
Figure 1.1 The electromagnetic spectrum. Historically, different regions have been given different names. Theboundaries betweeneach regionarenot sharply definedbut grade intoone another. Thevisible spectrumoccupiesonly a small part of the total spectrum
1 The imprecise expression ‘intensity’ has largely been replaced in the optical literature by well defined terms such as irradiance
(Appendix 1.1). The term intensity is retained here (in a qualitativeway to designate the amount of light) because of the historical context.
Colour and the Optical Properties of Materials 2
The explanation of this ‘photoelectric effect’ by Einstein in 1905was based upon the idea that light behaved
as small particles, now called photons. Each photon delivered the same amount of energy. If this was
sufficiently large, then the electron could be ejected from the surface. The energy of each photon, E, was
proportional to the frequencyof the illumination, so that the photoelectron could be ejectedwhen the frequency
passed a certain threshold, but not before that point was reached. Thereafter, increasing the frequency of the
illumination allowed the excess energy to be displayed as an increase in kinetic energy. The kinetic energy of
the photoelectrons ejected from a metal under this hail of photons could then be written as:
1
2mv2 ¼ E�f
where f is known as the work function of the metal and is simply the energy required to liberate the electron
from themetal surface. The intensity of the light simply indicated the number of photons arriving at the surface,
so that the number of photoelectrons emitted is a function of irradiance, but the energy of these electrons is a
function of the frequency of the radiation. Einstein thus rescued the wave theory from the dilemma of the
luminiferous aether and then seemingly wrecked the self-same theory via his explanation of the photoelectric
effect.
At present, all experiments show that light and its interaction with matter (i.e. atoms) is best described in
terms of photons.At its simplest level, the statistical behaviour of a large number of photons is then represented
verywell by an electromagnetic wave. That is to say, photons are the components of a light beam,whilst waves
are a mathematical description of a beam of light.
In this book, explanations are given in terms of the simplest approach that is in accord with the observations.
For large-scale phenomena, such as the operation of amagnifying glass, it is adequate to use the idea of a ray of
light.Whenobjects havingdimensions of the order of hundreds of nanometres are encountered it is necessary to
consider light tobeawave.Atomicprocesses require aphotonapproach. It needs tobe stressed that these are not
different fundamentally. All are contained within the theory of optics available today, generally described as
quantum optics or quantum electrodynamics.
Nomatter how it is described, light has no colour as such. Light simply leaves the generating source, possibly
interactswithmatter in the course of passage and then enters the eye. Colour, ormore accurately the perception
of colour, is the result of an eye brain combination that serves to discriminate between light of different
wavelengths or energies. In the following chapters, the production of light and its interaction with matter is
discussed from the point of view of colour that of the original light source, and how this is modified by
interaction with matter to generate new colours.
1.2 Colour and Energy
Colour is generated by interactions of light and matter atoms and molecules, or, more strictly, the electrons
associated with these. If light is considered as an electromagnetic wave, then the energy density of the wave,
which is the energy per unit volume of the space through which the light wave travels, is given by:
E ¼ e0ðE0Þ2
where e0 is the vacuum permittivity and E0 is the amplitude of the electric component of the wave. Classical
optics, the interaction of lightwith a transparent solid, in themain, is concernedwith scattering of the light. This
leads to the phenomena of reflection, refraction and so on. In these processes, colour is produced by interaction
between various light waves, and energy exchange considerations hardly matter. These aspects of colour
formation are covered in Chapters 2 6.
3 Light and Colour
When light is absorbed by or emitted from a material, say a gemstone such as ruby, energy changes are
paramount. In this case, light is best regarded as a streamofphotons; the energyof eachphotonbeingdefined as:
E ¼ hn ¼ hc
lð1:1Þ
where n is the frequency of the equivalent light wave, l is the wavelength of the equivalent light wave, h is
Planck’s constant and c is the velocity of light in vacuum.
The absorption of light by isolated atoms or molecules involves a change in energy of the electrons
surrounding the atomic nuclei. These occupy a series of atomic or molecular orbitals, each of which can be
assigned a precise energy. The energies of the orbitals form a sort of ladder (with variable rung spacing) from
low to high, each separated from the next by an energy gap. Electrons are fed into the orbitals from lowest to
highest energy until all of the electrons have been allocated, leaving the extra outer electron orbitals empty.
The total energy of all the electrons in the atom atmodest temperatures is represented by an energy level called
the ground state. The absorption of light will cause an electron tomove from the low-energy ground stateE0 to
an empty orbital at a higher energy. The new energy situation is represented by an energy level at energy E1
(Figure 1.2a). (These energy levels and how they are enumerated will be described in detail in later chapters.)
The relationshipbetween the energychangeDE and the frequencynor thewavelengthlof the light absorbed is:
E1�E0 ¼ DE ¼ hn ¼ hc
lð1:2Þ
where h is the Planck constant and c is the speed of light. When energy is lost from an isolated atom it moves
from the excited state back to the ground state. The simplest case is when the species passes directly fromE1 to
E0 (Figure 1.2a), with an energy output given by:
E1�E0 ¼ DE ¼ hn ¼ hc
l
identical to that of the absorbed radiation. However, the release of energy often takes place by more complex
mechanisms that will be explored in later chapters.
In both cases, if the frequency associated with the energy changeDE lies in the band that is registered by the
eye, then colour is perceived.
When atoms unite to form a solid (or a liquid) the precise energies of the orbitals are broadened out into
continuous bands of energy. The main energy landscape in a solid is the band structure which is the
geometrical form of the energy bands throughout the matrix. In a solid, electrons are allocated to the energy
bands, from the lowest energy up, until all have been allocated. The energy bands of highest energy are then
empty, similar to the orbitalswith highest energy in an atom. In the simplest depictions, the highest filled energy
band (the conduction band) is separated from the lowest empty energy band (the valence band) by a constant
band gap (Figure 1.2b). In real structures, the band architecture is more complex. Light absorption, emission
and colour generation in a solid cannot be discussed without consideration of the role of the band structure.
In this case, the energy difference DE which corresponds to colour registration might correspond to the
promotion of an electron from a full conduction band to an empty valence band. However, impurities and
defects can introduce further energy levels into the energy gap between the conduction and valence bands. In
these cases, energy transitions between these levels or between them and the energy bands of the solidmay then
be of an appropriate energy to act as important sources of colour (Figure 1.2b). Examples of these instances are
presented in Chapters 7 10.
Colour and the Optical Properties of Materials 4
1.3 Light Waves
In termsof thewave theory, lightwaves comprise a small segmentof the electromagnetic spectrum(Figure1.1).
Anypart of the electromagnetic spectrum is regarded as awave ofwavelengthlwith an electrical andmagnetic
component, each described by a vector and moving with a velocity, the ‘speed of light’, in a vacuum. The
electric field vector2 E is perpendicular to the magnetic vector, described in terms of the magnetic induction
B or the magnetic fieldH, and they are in phase, so that a peak in the electric field component coincides with a
peak in the magnetic field component. Moreover, these vectors lie in a plane perpendicular to the direction in
which thewave ismoving, described by the velocity vector v, or the propagation vector orwave vectork. Thus,
E and B both lie perpendicular to the direction of propagation, so that light is regarded as a transverse
electromagnetic (TEM) wave. The wave is a progressive wave, a travelling wave or a propagating wave, all
(a)
(b)
E0
E1
impurity energy levels
ΔE
valence band
conduction band
Eg
Figure 1.2 Energy transitions leading to colour production, shown as arrows: (a) transitions between energylevels in isolated atoms or molecules; (b) transitions between impurity levels and energy bands in a solid. Notethat each single energy level shown may actually be composed of several closely spaced energy levels in realsystems. Eg is the magnitude of the energy gap between the valence and conduction band
2 Vectors are given in bold throughout this book.
5 Light and Colour
terms being used more or less interchangeably. The electric field vector, the magnetic field vector and the
velocity vector can be represented by the three (right-handed) Cartesian axes (Figure 1.3a).
As far as the topics in this book are concerned, the electric field E can usually be considered in isolation. In
this case, aone-dimensional continuous electromagneticwavemoving in the þ xdirection canbeconveniently
depicted by the equation:
Ey ¼ E0 cos½ð2p=lÞðx�vtÞ� ð1:3Þ
distancex
crest
trough
y
z
x
E
H (B)
v (k)
(a)
(b)
ε
ε0
Figure 1.3 Lightwaves. (a) Light can be thought of as a TEMwave. The electric (E) andmagnetic (Hor B) vectorslie perpendicular to each other and to the vector representing the direction of travel of the wave (v or k).The shaded planes represent the positions of peaks in the electric and magnetic fields. (b) Part of a light wavetravelling along x. The curve represents the magnitude E of the electric field vector as a function of position. Thedistance between the crests or troughs is the wavelength l. Any point on the wave moves with a speed v. Ifthe electric field vector remains in the plane of the paper, as drawn, the light is linearly polarised. If the orientationof the electric field with respect to the plane of the page varies at random so that the curve continually adoptsdiffering angles with the plane of the paper, the light is unpolarised
Colour and the Optical Properties of Materials 6
Here, Ey is the magnitude of the electric field vector at position x and time t and v is the wave speed
(or velocity3). The term E0 is the amplitude of thewave (themaximumvalue that the electric field vector takes)
and is a constant. The speed v at which any point on thewave, say a peak or a trough, travels is called the phasespeed or phase velocity. The velocity of an electromagnetic wave in vacuum, denoted by the speed of light c, is
an important physical constant.
Taking t as fixed gives a snapshot of thewave at a single instant (Figure 1.3b). The spatial period of thewave,
which is the distance over which thewave subsequently repeats itself, is called thewavelength l. The peaks inthe wave are referred to as crests and the valleys as troughs. The term in square brackets, ½ð2p=lÞðx�vtÞ�, i.e.the argument of the cosine function, is called the phase of thewave, represented byf. The phase of thewave isusually quoted in radians, in the form (mp/n), i.e. 3p/4. Clearly, the phase of thewavevaries along its length andchanges by 2p in one wavelength. The phase of a light wave cannot be determined. However, the phase
difference between corresponding points on two different waves, say two equivalent crests, can be measured
with considerable precision.
Taking x as fixed will show that the magnitude of the electric field vector Ey will oscillate up and down
betweenvalues of�E0. The temporal period of thewave t, which is the time overwhich thewave subsequently
repeats itself, ismoreusually encountered as the reciprocal 1/tand is equal to the temporal frequencyn,which is
the number of waves that pass a point per second.
The speed of the wave v is related to the frequency n by:
v ¼ ln
(or in a vacuum by c ¼ ln).A beam of light is said to bemonochromaticwhen it is comprised of a very narrow range ofwavelengths and
it is coherent when all of the waves which make up the beam are completely in phase; that is, the crests and
troughs of all the waves are in step. The way in which the electric field vector is constrained describes the
polarisation of thewave. If the electric field vector remains in one plane, then the light is said to be linearly (or
plane) polarised. In general, the polarisation of the light wave must be considered when describing optical
phenomena.
Normal light, such as that from the sun, say, is not emitted in a continuous stream, but in short bursts lasting
about 10 8 s. Within each burst all of the light waves are in phase and linearly polarised. However, both the
phase and polarisation change fromburst to burst in a random fashion, so that the phase and polarisation of each
burst are unrelated to those in the preceding burst. Thismeans that the phase and the polarisation of a lightwave
fluctuate continuously and at random within a fraction of a second. Normal light is thus described as being
incoherent and unpolarised. Because of this, the interaction of daylight with objects can be interpreted (at least
as a good approximation) without considering polarisation. Light from lasers (Section 1.9) is, by and large,
coherent and polarised, and these aspects cannot usually be ignored.
1.4 Interference
One of the advantages of the wave description of light is that the interactions between two beams are easily
explained. If two light waves occupy the same region of space at the same time then they add together, or
interfere, to form a product wave. This idea, called the principle of superposition, was stated by Young some
3 Strictly speakingwe are discussingwave speed,which is a scalar quantity. Velocity is a vector quantity. However, it makes things simpler
to brush over this distinction in the present case.
7 Light and Colour
two centuries ago, in 1802. If two identical waves are exactly in step then they will add to produce a resultant
wavewith twice the amplitude (Figure 1.4a c) by the process of constructive interference. If the twowaves are
out of step, then the resultant amplitudewill be less, due todestructive interference. If thewaves are sufficiently
out of step that the crests of one correspond with the troughs of the other, then the resulting amplitude will be
zero (Figure 1.4d f).
Interference can occur between light waves with different relative frequencies, amplitudes and phases.
However, for this to be observed the phase difference between the beams must remain constant. That is, the
waves must be coherent. Many of the difficulties inherent in observing interference effects using normal light
stem from the incoherent nature of thewave trains used, and effortsmust bemade to ensure that the incoherence
does not destroy any visible interference patterns that may be generated. The use of laser light makes the
observation of interference much simpler.
a0
a0
(a)
(b)
(c)
2a0
resultant
Figure 1.4 Interference of light waves. (a)–(c) The addition of twowaves in phase, (a), (b), will produce awave oftwice the amplitude of the original wave, (c). (d)–(f) The addition of two waves out of phase by l/2, (d), (e), willproduce a wave with zero amplitude, (f)
Colour and the Optical Properties of Materials 8
The effects of interference can be assessed analytically using algebraic methods. An intuitive feeling for the
phenomenon is best gained by adding waves represented by formulae such as Equation 1.3 using a computer
and displaying the results graphically.
1.5 Light Waves and Colour
Our eyes can detect only a small part of the whole electromagnetic spectrum, called the visible spectrum
(Figure 1.1). The amount of light that the eye records in any situation,which can loosely be called the brightness
or intensity of the light, is not the amplitude of the wave but is the irradiance I, which is proportional to the
square of the amplitude:
I ¼ KðE0Þ2
a0
a0
(d)
(e)
(f)
resultant
zero
Figure 1.4 (Continued)
9 Light and Colour
where the value of the constant of proportionalityK depends upon the properties of the medium containing the
wave. (See Appendix A1.1 for information on units.)
The extent of the visible spectrum is defined in terms of the wavelength or frequency of the light waves
involved.Perception of the differentwavelengths is called colour. The precisemeasurement of colour involves
a determination of the energy present at each wavelength in the light using a spectrometer.
Thewavelength range that an eye can perceive varies from individual to individual. In general, it is assumed
that the shortest wavelength of light that an average person can detect corresponds to the colour violet, with a
wavelength near to 400 nm. Similarly, the longest wavelength of light registered by an average observer
corresponds to the colour red, with a wavelength close to 700 nm. Between these two limits the other
wavelengths of the spectrumare associatedwith the colour sequence from red toorange, green, blue, indigo and
finally to violet (Figure 1.1 and Table 1.1). The divisions between these colours are, of course, artificial, and
each colour blends into its neighbours. (Note that these colours are simply approximate labels for the
wavelength. The perceived colour of an object is a function of a number of factors (Section 1.10).) It is known
that the sensitivity of the eyes of animals is different than those of humans. Many insects, for example, can
detect wavelengths shorter than humans but do not see so far into the red.
Radiation with wavelengths shorter than violet falls in the ultraviolet region of the spectrum. Ultraviolet A
(UVA) is closest to the violet region and is taken to have a wavelength range of 400 320 nm. This radiation is
largely responsible for suntan. Ultraviolet B (UVB), with an approximatewavelength range of 320 280 nm, is
more damaging and causes sunburn.Ultraviolet radiationwith shorterwavelengths is called the far ultraviolet,
(280 200 nm) and vacuum ultraviolet (below 200 nm). UVB and shorter wavelengths are able to damage
biological cells severely, and excessive exposure leads to the occurrence of skin diseases. Radiation with
wavelengths longer than red is referred to as infrared radiation.Although not visible, the longerwavelengths of
infrared radiation, called thermal infrared, are detectable as the feeling of warmth on the skin.
1.6 Black-Body Radiation and Incandescence
There are many ways in which light can be generated, but the action normally tales place at an atomic level.
Individual atoms (ormolecules) lose energy, which is given out as radiation. These processes generally need to
be discussed in terms of photons rather than waves. In this section, just one example is given, the generation of
light by a hot body. This was the first light-generating process to be understood at a fundamental level, and led
directly to the photon concept as well as to an appreciation of our idea of the make-up of white light.
Table 1.1 The visible spectrum
Colour l/nm 10�14n/Hz 10�15o/rad s�1 1019� Energy/J Energy/eV
Infrared 750 4.00 2.51 2.65 1.65Deep red 700 4.28 2.69 2.84 1.77Orange red 650 4.61 2.90 3.06 1.91Orange 600 5.00 3.14 3.31 2.07Yellow 580 5.17 3.25 3.43 2.14Yellow green 550 5.45 3.42 3.61 2.25Green 525 5.71 3.59 3.78 2.36Blue green 500 6.00 3.77 3.98 2.48Blue 450 6.66 4.19 4.42 2.75Violet 400 7.50 4.71 4.97 3.10Ultraviolet 350 8.57 5.38 5.68 3.54
Colour and the Optical Properties of Materials 10
Incandescence is the emission of light by a hot body. The sun and tungsten-lamp filaments provide
commonplace examples, and both are regarded as producing (more or less)white light. The light characterising
the upper part of a candleflame also arises from incandescence. In this case, small particles of carbon are heated
to high temperatures in the flame and emit light which is perceived as yellow in colour. When light from an
incandescent object is spread out according towavelength by a prism (Chapter 2) the result is a continuous fan
of colours following the sequence listed inTable 1.1 and called a continuous spectrum. The radiation emitted is
both incoherent and unpolarised.
Incandescence comes about in the following way. At absolute zero all atoms and molecules making up the
solid are in the lowest possible energy state. As the temperature increases they absorb energy and are promoted
to higher energies and, at the same time, atoms and molecules which have already absorbed energy lose some
and they fall back to lower energies. (The energy levels involved in this processwill be described inmore detail
in later chapters.) The radiation emitted in thisway effectively extends over a continuous range of energies. For
a solid a little above room temperature all thewavelengths of the emitted energy lie in the infrared; although the
radiation is invisible, it is detectable as a sensation of warmth. At a temperature of about 700 �C the shortest
wavelengths emitted creep into the red end of the visible spectrum. The colour of the emitter is seen as red and
the object is said to become red hot. At higher temperatures the wavelengths of the radiation given out extend
increasingly into thevisible region and the colour observed changes from red to orange and thence to yellow, as
in the example of a candle flame,mentioned above.When the temperature of the emitting object reaches about
2500 �C all visible wavelengths are present and the body is said to be white hot. The sun provides a perfect
example, and the ‘colour’ white as applied to light is a combination of energies or wavelengths that spans the
visible spectrum with the same composition as that of the radiation from the sun.
These qualitative colour changes canbe understood in termsof the radiation emitted byablackbody.Ablack
body is an idealized objectwhich absorbs and emits allwavelengths perfectly.A reasonable approximation to a
source of black-body radiation would be a small pinhole in the wall of a hot furnace. If the irradiance of the
radiation issuing from the pinhole is measured as a function of wavelength, a characteristic curve is obtained
called ablack-body spectrum (Figure 1.5). The shape of the curve is dependent only upon the temperature of the
body, and the maximum in the curve moves to shorter wavelengths as the temperature of the black body
increases. The curve also mirrors the energy distribution inside the black body when in thermal equilibrium.
The explanation of the form that this curve takes played a significant role in the physics of the twentieth
century. Despite many attempts, the form of the black-body spectrum could not be explained by the classical
wave theory of electromagnetic radiation. The successful theoretical description of this curve by Planck in
1901, now known as the Planck law of black-body radiation or Planck’s radiation law, signalled the start of
the quantum theory. The equations describing the spectral radiance of all the radiation components within a
black body at equilibrium at temperature T in the frequency range n to n þ dn or the wavelength range l to
l þ dl are:
Ln ¼ 2hn3
c2½expðhn=kBTÞ�1� units: Wm 2 sr 1 Hz 1 ð1:4aÞ
or
Ll ¼ 2hc2
l5½expðhc=lkBTÞ�1� units: Wm 3 sr 1 ð1:4bÞ
In these equations, h is a constant that is now called Planck’s constant, c is the speed of light, l is the
wavelength,kB isBoltzmann’s constant andT (K) the temperatureof thebody.These equationsare often seen in
11 Light and Colour
the form describing the spectral irradiance (if the energy falls upon a surface) or the spectral exitance (if the
energy is observed after leaving a black body via a pinhole not large enough to disturb the thermal equilibrium
within), In in the frequency range n to n þ dn or Il in the wavelength range l to l þ dl as a function of the
wavelength l for a black body at a temperature T:
In ¼ 2phn3
c2½expðhn=kBTÞ�1� units: Wm 2 Hz 1 ð1:5aÞ
or
Il ¼ 2phc2
l5½expðhc=lkBTÞ�1� units: Wm 3 ð1:5bÞ
or as the corresponding spectral energy density un in the frequency range n to n þ dn or ul in the range l to
l þ dl as a function of the wavelength l for a black body at a temperature T:
un ¼ 8phn3
c3½expðhn=kBTÞ�1� units: J m 3 Hz 1 ð1:6aÞ
500 1000 1500 2000
1x1014
2x1014
3x1014
4x1014
5x1014
Spe
ctra
l irr
adia
nce
/ W m
-3
Wavelength / nm
8000 K
5000 K
4000 K
vis ble spectrum
Figure1.5 The radiation emitted fromablackbodyas a functionofwavelength.As the temperature of thebody isincreased, themaximumof the curve both increases andmoves towards shorter wavelengths (higher energy). Thespectrum emitted by the sun is similar to that for a black body at 6000 K and that from a red-hot object is similar tothe curve for a black body at 1000 K
Colour and the Optical Properties of Materials 12
or
ul ¼ 8phc
l5½expðhc=lkBTÞ�1� units: J m 4 ð1:6bÞ
The revolutionary concept that Planck employed in the derivation of these equations to successfully
reproduce the black-body curve was that the energy absorbed or given out by the atoms and molecules (the
‘oscillators’ in Planck’s time) in the black body could not take any value from a continuous spread of energies,
but had to be deliveredonly in packets orquanta q0, 2q0, 3q0 and so on.The relationship between the energyof a
quantum E and the frequency of the radiation n was given by what has since become one of the most famous
equations of science:
E ¼ hn ð1:1Þ
The constant h, Planck’s constant, is one of the important fundamental physical constants.
More recently, in the mid-twentieth century, it was realized that the cosmos was filled with some sort of
background electromagnetic radiation. The peak of the radiation lies in the microwave part of the electro-
magnetic spectrum. Naturally, it is invisible to optical instruments andwas first mapped using radio telescopes
and latterly by satellites. The spectrum of this radiation fits that of a black body; and indeed, this radiation,
called the cosmic microwave background radiation, is possibly the most accurately measured black-body
radiation curve available. It is interpreted as lending strong support to the ‘Big Bang’ theory of the origin of the
universe.
1.7 The Colour of Incandescent Objects
From the point of view of the colour of incandescent objects, one of the most important attributes of the
emission curve is the variation in the position of the maximum as the temperature of the black body increases.
(This was derived before the Planck radiation law and represents the final success of classical electromagnetic
theory.) The relationship, known as the Wien displacement law, is:
lmaxT ¼ constant
where T (K) is the temperature of the body and the constant has a value of 0.002 898mK. It can be derived from
Equations 1.5a and 1.5b by differentiatingwith respect to l and setting the result equal to zero. The colour of anincandescent object is then controlled by the maximum of the black body curve (or an approximation to it), as
mentioned below. The second factor of importance is the spread of the spectrum.A cool bodywill be perceived
as initially showing a colour when the peak of the curve is close enough to the visible range that some radiant
energy creeps into the low-energy (red) end of the spectrum. As the temperature of the incandescent
object increases, the peak moves to higher energies, following the displacement law, and the spread moves
further across the visible spectrum, resulting in the colour sequence of dull red, red hot to white hot to
blue white.
The colour of an incandescent object is described by its colour temperature if the spectrum resembles that of
a blackbody closely.Most solids behave like black bodies over some range of temperature andwavelength, and
stars are a close approximation over the whole of the wavelength range. If the match is approximate, the term
used is correlated colour temperature and this expression is used for light sources that are not incandescent,
13 Light and Colour
such as fluorescent lighting (Table 1.2). Colour photographs taken on film designed to be used in daylight
(colour temperature of about 5 400K) will show incorrect tones when used to photograph objects illuminated
with tungsten lights (colour temperature of about 3 400K) or fluorescent lights (colour temperature of about
3 000K) unless correcting filters are used.
Themost important incandescent object for us is the sun,which is the ultimate sourceof energyonEarth. The
solar spectrum has a form quite similar to a black-body curve corresponding to a solar temperature of about
5 780 �C (about 6 000K), which has a maximum near 480 nm. The form of the spectrum when it reaches the
surface of the Earth is a function of a number of variables, including the elevation of the sun, the amount of
scattering material in the atmosphere and so on. Light is perceived as white if it has a make-up like that of the
solar spectrum from an overhead sun on a clear day. Stars which are cooler than the sun give a redder colour,
whilst those which are hotter are perceived as whiter. The effective temperature of a star is the temperature
calculated as if it were a black body radiating with the same energy over the same wavelength ranges
(Table 1.3). The effective temperature is generally a good approximation to the surface temperature of a star.
The hottest visible stars are the Bellatrix type, with blue white colour and an effective temperature of
approximately 25 000K, whilst the reddest naked-eye star is m-Cephei, the Garnet Star, with a temperature of
approximately 2 600K.
1.8 Photons
The quantization of radiation proposed by Planck in the derivation of the radiation law was not seized upon
instantly. After a lapse of some years it was exploited by Einstein in his explanation of the photoelectric effect
Table 1.2 Colour temperature of incandescent sources
Light source Correlated colour temperature/K
Mean noon sunlight 5 400Electronic flash �7 000Blue flash bulb �6 000Tungsten filament photographic lamps �3 400Tubular triphosphor fluorescent lamp, 36W 3000Household tungsten filament light bulb, 100W 2850Standard candle 1 930
Table 1.3 Effective star temperatures
Star colour and example Effective temperature/K
Blue white, Bellatrix 25 000White, Sirius 11 000Yellow white, Sirius Solar 7 500Solar, the Sun 6000Orange yellow, Arcturus 4 200Orange, Antares 3 000Deep orange red, m Cephei 2 600
Colour and the Optical Properties of Materials 14
in 1905 (Section 1.1). He proposed that the quantization of radiation contained in Planck’s formula for black-
body absorption and emission of energy, i.e.:
E ¼ hn
wherenwas the frequencyof the radiation andh is Planck’s constant, could be applied to the radiation itself, not
just to the energyexchangewith atomsormolecules.That is to say, lightwas tobe regardednot as awavebutas a
hail of bullet-like objects (which are now called photons), each of which had an energy hn. Each photon
delivered the same amount of energy. If this was sufficiently large then the electron could be ejected from the
surface. The energy of each photon was proportional to the frequency of the illumination, so that when the
frequency passed a certain threshold, the photoelectron could be ejected, but not before that point was reached.
Thereafter, increasing the frequency of the illumination allowed the excess energy to be displayed as an
increase in kinetic energy. The kinetic energy of the photoelectrons ejected from a metal under this hail of
photons could then be written as:
1
2mv2 ¼ hv�f
where f is known as the work function of the metal and is the energy required to liberate the electron from
themetal surface. The irradiance of the light indicated the number of photons arriving at the surface, so that the
number of photoelectrons emitted is a function of irradiance, but the energy of these electrons is a function of
the frequency of the radiation.
A description of light in terms of photons is mandatory when dealing with events at an atomic scale. The
energy E of a photon is given by Equation 1.1:
E ¼ hn ¼ hc
lð1:1Þ
where n is the frequency of the equivalent light wave, l is the wavelength of the equivalent light wave, h is
Planck’s constant and c is the velocity of light in vacuum.
This conjunctionof the particle andwavedescriptions, calledwave particle duality, is evident in the fact that
n is the frequency and l is the wavelength of the wave-like properties associated with the photon. In fact, allparticles exhibit wave-like properties. The momentum p of a particle (such as an electron, say), is given by:
p ¼ ðE2�m2c4Þ1=2c
where E is the energy, m the particle mass and c the speed of light. For a photon, m¼ 0, so that:
p ¼ E
c
The wavelength of a particle is:
l ¼ h
p¼ hc
ðE2�m2c4Þ1=2
15 Light and Colour
For a photon, m¼ 0, so that:
l ¼ hc
E
The velocity of a particle is:
v ¼ pc2
E¼ c 1� m2c4
E2
� �� �1=2
For a photon, m¼ 0, so that:
v ¼ c
(For particles such as electrons, m is not zero.)
For many purposes the wave and particle aspects of light can be used interchangeably, as dictated by
experiment.Thewaveaspectof lightexpresses thefact that thephotonsdonotobeydeterministic lawsofmotion,
but laws of probability. The waves associated with light photons are a way of describing these probabilities.
1.9 Lamps and Lasers
1.9.1 Lamps
Until the end of the nineteenth century artificial illumination was via incandescence either firelight, candles,
oil lamps or gas light. At the end of this period, new light sources began to be invented in parallel with the
generation and availability of electricity. In 1897 Nernst invented the ‘glower’. This lamp consisted of a bar of
electrically conducting ceramicmade from amixture of lanthanide oxides that became incandescent under the
action of an electric current. Although Nernst glowers were widely used and were more efficient than
the competing incandescent carbon-filament electric light bulbs developed by Edison, they fell into disuse
following the successful introduction of tungsten-filament lamps after the invention of the Coolidge process
for the production of ductile tungsten wires for the fabrication of lamp filaments. Throughout the twentieth
century, tungsten-filament lamps dominated the lighting market.
Although incandescence was the most widespread source of artificial light, other lighting was well known.
Neon signs (Chapter 7) and various forms of luminescence (Chapter 9) were used in specialist light-generating
ways, such as, in the case of neon signs, for advertising. These latter mechanisms relied directly upon atomic
transitions in a way that was obscured in the complex incandescence reactions. In addition, instead of
generating a continuous ‘white light’ spectrum, these new light sources tended to give out coloured light, the
wavelengths produced depending upon the actual atoms emitting the photons.
All of these light sources, however, were similar to each other in oneway the light emitted was incoherent
and usually unpolarised. Towards the middle of the twentieth century, advances in communications technolo-
gies reinforced the utility of using light directly to carry signals. This necessitated the use of coherent radiation.
Initially the push came from radio, as radio waves are normally emitted as a coherent wave train, not as
incoherentwaves. Thewavelength of thewavesused for carrying signals continually decreased via longwaves,
mediumwaves and short waves. At the same time, the engineering skills required to encode greater and greater
information on thesewaves increased to an amazing extent, making television and stereo broadcasting a norm.
Unfortunately, the production of coherent radiation seemed to be stuck somewhere in the microwave region.
The idea of using lowerwavelengths, though, especially optical wavelengths, was enormously attractive, and a
Colour and the Optical Properties of Materials 16
great deal of effort was invested into breaking into this wavelength range. Success came in the 1960s, with the
invention of the laser. Lasers are a completely new sort of lamp compared with those already described.
The word laser is an acronym for the expression Light Amplification by Stimulated Emission of Radiation.
The first laser to be madewas the ruby laser, and the first laser light emitted was on 15May 1960. Since then a
vast number of lasers have been produced, including solid-state lasers, gas lasers, semiconductor diode lasers
and dye lasers. From an exotic beginning lasers have become ubiquitous inmodern life, being used as pointers,
at check-outs in supermarkets, in surveying and measurement, in micromachining, microsurgery and so on.
Here, the general principles of laser actionwill be outlined. Examples which illustrate particular facets of laser
light generation will be discussed throughout the text.
1.9.2 Emission and absorption of radiation
When a photon of energy hn is absorbed by an atom or molecule it passes from the normally occupied lower
energy state, often called the ground state, to an upper or excited state, as described above. The transition will
take place if the frequency of the photon n, is given exactly by:
E1�E0 ¼ DE ¼ hn ¼ hc
lð1:2Þ
where E0 is the energy of the ground state, E1 is the energy of the excited state and h is Planck’s constant. If the
atom is in the excited stateE1 andmakes a transition to theground stateE0, energywill be emittedwith the same
frequency, given by the same equation.
In this description the actual emissionmechanism is ignored. In 1917Einstein suggested that there should be
two possible types of emission process (Figure 1.6):
1. An atom in an excited state can randomly change to the ground state, by a process called spontaneous
emission.
2. A photon having an energy equal to the energy difference between the two levels (i.e. E1�E0) can interact
with the atom in the excited state, causing it to fall to the lower state and emit a photon at the same time, a
process called stimulated emission.
The light emission from ‘ordinary’, i.e. non-laser, sources is the result of spontaneous emission. Lasers are
concerned with stimulated emission. In spontaneous emission, the light photons all have the same frequency
but possess random phases and polarisation so that the light is incoherent. In stimulated emission the photon
produced has the same frequency, phase and polarisation, as the onewhich caused the emission so that the light
is coherent. It is these important features of stimulated emission on which the special properties of laser light
depend.
1.9.3 Energy-level populations
Under conditions of thermal equilibrium the relativepopulations of a series of energy levelswill begivenby the
Boltzmann law, which for two energy levels can be written as:
N1
N0
¼ exp�ðE1�E0Þ
kBT
� �
where kB is Boltzmann’s constant, T is the absolute temperature,E1 andE0 are the energies of the excited state
and the ground state respectively and N1 and N0 are the numbers of atoms (the populations) in each of these
energy levels. For ordinary atoms, in a gas, liquid or solid at ordinary temperatures, the fraction N1/N0 will be
17 Light and Colour
negligible for energy levels which are sufficiently separated to give rise to visible light. Atoms can be assumed
to be in the ground state as far as visible light emission is concerned.
When a photon of the appropriate energy interacts with an atom in the ground state it will be absorbed and
shortly afterwards re-released by spontaneous emission (Figure 1.7a). This will be repeated at each atom in the
ground state. There will be no amplification and we may well see a net absorption of energy. To obtain laser
amplification one needs to ensure that stimulated emission is the dominant process occurring. This means that
there are more atoms in the excited state of energy E1 than in the ground state E0. In this instance, a photon
interacting with an excited atom can cause energy to be released by stimulated emission and two photons
emerge. Ifmost atomsare in the excited state then amplificationmayoccur (Figure1.7b).The situation inwhich
more atoms are in the excited state than in the ground state is called a population inversion.
From the Boltzmann equation it is obvious that an increase in temperature cannot achieve this objective.
Even an infinite temperature will only result in equal numbers of atoms in E0 and E1. To obtain a population
inversion, therefore, a nonequilibrium statemust be achieved. The crux of laser action is how to create such a
nonequilibrium situation in a material and then exploit it to produce the desired amplification. Examples of
practical ways in which this is achieved are given later in the text (i.e. see Chapters 7 and 10).
1.9.4 Rates of absorption and emission
In the previous section itwas implicitly implied that the rate of spontaneous emissionwas fast. This aspectmust
be looked at in more detail to obtain a better understanding of laser action. When equilibrium between
absorption and emission holds, the rate of depopulation of an upper level (�dN1/dt) by spontaneous emission
E1 E1
E0E0
E0
E0
light absorbed
stimulated emission
spontaneous emission
hν
hν
hνhν
hν
(a)
(c)
(b)
Figure 1.6 Light absorption and emission. (a) Light absorption occurs when a photon excites an atom (ormolecule) from the ground state E0 to an excited state E1. (b) During spontaneous emission, the atoms lose energyand release photons at random. (c) During stimulated emission, an atom in an excited state is triggered to loseenergy by interaction with a photon of energy (E1 E0 )
Colour and the Optical Properties of Materials 18
will be given by a first-order rate law:
� dN1
dt¼ A10N1
where the negative sign denotes that the number N1 of atoms in the upper state E1 (per cubic metre, say) is
decreasingwith time. The rate is proportional to the number of atomsN1 in the state. The rate constant, denoted
here asA10, is called theEinstein coefficient for spontaneous emission, where the suffix ‘10’ means that we are
considering a transition from the excited state E1 to the ground state E0. The number of downward transitions
due to spontaneous emission, per second, will be given by:
A10N1
Similar rate laws can be written for the cases of stimulated emission and for absorption, but in this case the
rates are proportional to the numbers of atoms in the relevant state and, in addition, the number of photons
present. The reactions can be taken to be first order with respect to both of these quantities.
The rate at which atoms in state E0 are excited to state E1 is then given by:
� dN0
dt¼ B01rðn01ÞN0
whereN0 is the number of atoms in stateE0 (per cubicmetre, say),r(n01) is the radiation density responsible forabsorption, which is the number of quanta per cubic metre incident per second at the correct excitation
frequency n01, andB01 is theEinstein coefficient for absorption of radiation. Similarly, the rate of depopulation
E1
E0
E1
E0
hνhν
hν
(a)
(b)
Figure 1.7 Amplification. (a) When most atoms are in the ground state the absorption of a photon and thesubsequent spontaneous re-emission will not lead to amplification. (b) Whenmost atoms are in the excited state,stimulated emission can lead to amplification
19 Light and Colour
of state E1 by stimulated emission is given by:
� dN1
dt¼ B10rðn10ÞN1
where N1 is the number of atoms in state E1 (per cubic metre), r(n10) is the radiation density responsible fordepopulation, which is the number of quanta per cubic metre incident per second at the correct frequency n10,
and B10 is the Einstein coefficient for stimulated emission of radiation. Now, the correct frequency for
excitation will be the same as that for depopulation, so that n10¼ n01, which we can simply write as n, and the
radiation density will be the same in each case, so that we can write:
rðn10Þ ¼ rðn01Þ ¼ rðnÞ
The number of stimulated downward transitions per second will be given by:
N1B10rðnÞ
while the total number of upward transitions in the same time will be given by:
N0B01rðnÞ
At equilibrium, the total number of transitions in each direction must be equal; hence:
N0B01rðnÞ ¼ N1A10 þN1B10rðnÞ
so
rðnÞ ¼ N1A10
N0B01�N1B10
In addition, at equilibrium the Boltzmann distribution applies; thus:
N1
N0
¼ exp�hn
kBT
� �
and by making this substitution we have:
rðnÞ ¼ A10
expðhn=kBTÞB01�B10
This expression represents the radiation density at frequency n. At thermal equilibrium, this should be identical
to Planck’s equation, Equation 1.6a:
rðnÞ ¼ 8phn3
c3½expðhn=kBTÞ�1�
Colour and the Optical Properties of Materials 20
which leads to the conclusion that:
B01 ¼ B10 ¼ B
and:
A10
B¼ 8phn3
c3
The ratio of the rate of spontaneous emission to stimulated emission under conditions of thermal equilibrium is
given by:
R ¼ A10
rðnÞB ¼ exphn
kBT
� ��1
This is an extremely interesting result. At 300K, at visible wavelengths, R � 1. This shows that, for light,
stimulated emission will be negligible comparedwith spontaneous emission and reinforces the idea that it will
be impossible to make a laser under equilibrium conditions. On the other hand, if the wavelength increases
beyond the infrared into the microwave and radio-wave regions of the electromagnetic spectrum, R becomes
much less than unity and all emission will be stimulated. Hence, radio waves and microwaves arise almost
entirely from stimulated emission and are always coherent. This is one of the main reasons that commu-
nications in the early part of the twentieth century used radio waves.
Perhaps because of this equation, and the towering reputation of Einstein, it seems that for the first part of the
twentieth century it was felt that lasers were not feasible. In the middle of the century, scientists started to
explore stimulated emission at microwave frequencies, developing themaser. This soon led to the first lasers,
the ruby laser and then theHe Negas laser, produced in 1960with these early devices oftenbeing calledoptical
masers. Once the way to overcome the production of laser light was understood, laser development became
prolific. Later sections show how the equilibrium problem has been bypassed and how the difficulty of
achieving stimulated emission at optical wavelengths has been overcome.
1.9.5 Cavity modes
Supposing that a population inversion is obtained between energy levels thatwould give rise tovisible light, it is
still necessary to design the equipment so that amplification of the signal takes place. The losses from the laser
must be less than the total emission for amplification to be achieved. Losses in oscillating systems are often
defined in terms of a quality factor Q, a term borrowed from radio technology. In effect, a high value of Q is
needed to ensure amplification.
One of the most important of these design features is the shape of the cavity that the laser medium
occupies. Suppose that this is simply a crystal rod. The population is an unstable state and after a short time
some spontaneous emission will occur from E1. Naturally, these photons will rapidly leave the crystal rod;
and although in so doing a few other atoms might lose energy via stimulated emission, no amplification will
occur. It is necessary to prevent the photons from leaving the crystal in order to increase the chances of
stimulated emission occurring. The simplest way to achieve this is to coat the ends of the crystal rod with a
highly reflecting mirror. In this case the photons are reflected to and fro, causing stimulated emission from
the other populated E1 levels. Once started, the stimulated emission rapidly depopulates these levels in an
avalanche. In order to permit some light to emerge, one of the mirrors is not perfect and allows a small
21 Light and Colour
amount of light to pass. There will then be a burst light emerging from the cavity which is not only coherent
but also shows amplification. Thus, the simplest cavity geometry is simply cylindrical with one end fitted
with a completely reflecting mirror and the other with an almost perfect mirror, appropriate to the
wavelength of the light generated by the stimulated emission.
There are several consequences of this simple geometry which are easiest to explain if the light trapped in
the cavity is regarded as a wave. Taking the cavity as a rod with reflecting end faces, it is clear that initially
all photons will be emitted at random, but only those that are emitted more or less parallel to the long axis of
the cavity will bounce to an fro and so cause the stimulated emission avalanche. In terms of wave optics, the
photons form a series of standing waves in the cavity, which is described as resonance. The standing waves
form only if there is a node at each reflecting surface. The allowed waves are called longitudinal cavity
modes and are given by the condition that a complete number of half wavelengths must fit into the length l of
the cavity, i.e.:
mc ¼ l
l=2¼ 2l
l
wheremc is an integer, l is the cavity length and l is the wavelength of the mode. The frequency of a mode is
given by:
nm ¼ mcv
2l
where v is thevelocity of the lightwaves in the cavity, givenbynml, andnm is the frequencyof themodemc. The
separation of the modes is given by:
nm�nm 1 ¼ Dn ¼ v
2l
The velocity of light in the cavity is given by:
v ¼ c
n
where c is the velocity of light in a vacuum and n is the refractive index of the cavity medium (Chapter 2),
so that:
nm�nm 1 ¼ Dn ¼ c
2ln
Howdoes thisworkout in practice?The emission from theupper to the lower energy level has beenwritten as
a single energy with a negligible width. In the case of real materials, atoms and molecules are in continuous
motion, vibration in solids, translation in gases, and the sharp energy levels idealized in Figures 1.6 and 1.7 give
rise to a spread of energies (or of frequencies or wavelengths) called the transition bandwidth (Figure 1.8a).
Only that part of this output that fulfils the longitudinalmode criterionwill be allowed to grow.The output from
the cavitywill thenbe composed of a set ofmodes (Figure 1.8b). Thesemodeswill depend upon the shape of the
initial emission pulse and the overall power of the excitation process.
Colour and the Optical Properties of Materials 22
By extension, it is apparent that, in general, there will be transverse modes as well as longitudinal modes in
the laser emission. These must be taken into account when the optics of the laser beam are considered. Laser
cavity design is, therefore, of considerable importance in practice.
1.10 Vision
As stated earlier, light has no colour as such. Light radiation leaves the source, possibly interacts withmatter in
the course of passage and then enters the eye. Light is perceived by the eye brain combination, and colour is a
description of this perception. The colour that the observer is conscious of is thus a combination of many
factors, including the energy spread of the source light, the addition or subtraction of energy during any
interactionswith othermaterials and the sensitivity of the eye. For example, the blue skycontains all the colours
of the spectrum, as can be demonstrated by passing this light through a prism (Chapter 2). Blue is the colour
attributed to the sky when all the factors mentioned above are taken into account.
Frequency
Frequency
Irra
dian
ceIr
radi
ance
/ 2ln
emission
(a)
(b)
cavity modes
Figure 1.8 (a) The emission from an excited state E1 to the ground state E0 is not sharp, but consists of a range offrequencies dependent upon temperature and other factors. (b) In a laser cavity, only certain frequencies, thecavity modes, are allowed to propagate
23 Light and Colour
The physiological response of the eye brain combination arises when light waves fall upon the light-
sensitive retina, which makes up the inner surface of the eye. In 1876 Boll reported that the red purple
pigment found in this part of the eyes of animals bleached in the presence of light to a colourless form. The
change was found to be reversible, and in the dark the purple colour was regenerated. This important
photochromic reaction is the source of vision. The compound involved became known as ‘visual purple’ and
is now called rhodopsin.
Vision in humans and other animals involves a complex set of reactions which take place in two types of
photoreceptor cells located in the retinaof the eye: rods and cones. There are about 108 rods and4� 106 cones in
an eye. In humans the rod cells, of about 0.002mm diameter, are about four times as sensitive as cones and are
responsible for vision at low light intensities. Although they detect light all across the visible, the peak
sensitivity is at �500 nm. The light not absorbed, red and blue/violet, gives rise to the purple colour of the
membrane. The rod cells are not sensitive to colour and give rise to a monochrome image. Moreover, they
saturate in high light levels, making them unresponsive under these conditions. The cone cells, approximately
0.006mm diameter, are sensitive to bright light and form the daylight colour detection system. They exist in
three varieties with peak sensitivities in three different regions of the visible: L cones, most sensitive to red,
l(peak) �560 nm, M cones, most sensitive to green, l(peak) �530 nm, and S cones, most sensitive to blue
l(peak) �420 nm (Figure 1.9a). The human eye is optimally sensitive to green light and is noticeably less
sensitive to red and especially blue light (Figure 1.9b). The sensitivity of the eye to colour depends not only
upon the amount of light, but also upon which area of the retina is being stimulated. Themost sensitive region,
called the fovea, is almost directly behind the lens of the eye and predominantly contains cone cells. The
maximum sensitivity of a normal eye to bright white light focused on the fovea, which is the sum of the
contributions of the cone and rod cells, is for a wavelength close to 555 nm (Figure 1.9c). Colour blindness
results from a fault or deficiency in one or more varieties of the cone cells or in the way in which these cells
communicate with the brain.4
Human vision is said to be trichromatic. There is considerable variation across the human population in the
sensitivity ranges of the cone cells, giving rise to avariation in colour vision.Trichromaticity is commonamong
primates, butmost nonprimate animals can only detect two colours and are referred to as dichromats. However,
some birds, fish and reptiles have four different cone cell receptors and can detect ultraviolet light with l(peak)as low as 360 380 nm in addition to three ‘normal’ colours.
When light photons impinge on both rod and cone cells they are absorbed by stacks of photoreceptor
moleculeswhich are bleached in the process. This sends a nerve impulse to the brain. The system is remarkably
sensitive and there is considerable evidence to suggest that in the rod cells just onephoton is enough to stimulate
the nerve. The light-absorbing pigments consist of a protein, an opsin, bound to a light-absorbing molecule,
retinal. The receptor in the rod cells is called rhodopsin,while those in the cone cells are called cone opsins. The
opsin part of the receptor, consisting of 364 amino acid residues in humans, is arranged in the form of seven
helices, which penetrate the cell wall and enclose the retinal, which is bound to the amino acid lysine 296
(Figure 1.10). The opsin proteins differ from one cone cell to another and from rhodopsin in the rods, and it is
these differences that confer the differing sensitivities to the receptors. However, the differences are rather
small. For example, the amino acid sequences in the green (M) and red (L) cone receptors in humans differ in
only three of the amino acid residues in 364.
4 The existence of colour blindness itself was first recorded as such by John Dalton, who realized that his own perception of colours was
different than themajority of his friends (but the same as his brother’s), and formany years the conditionwas known asDaltonism. Amore
recent study of his careful observations suggests that hewas unable to distinguish the colour red. It is of interest to learn that Dalton himself
felt that he possessed some visual advantages over his friends because of the nature of the abnormal sensitivity of his eyesight. He did not
find that he was at a disadvantage at all. (Also see Figure 1.15.)
Colour and the Optical Properties of Materials 24
564 nm534 nm420 nm
(a)
Wavelength / nm
Abs
orba
nce
(nor
mal
ised
)
400 450 500 550 600 650 700
Wavelength / nm
Rel
ativ
e se
nsiti
vity
red
green
blue
400 450 500 550 600 650 700
(b)
Wavelength / nm
Spe
ctra
l lum
inou
s ef
ficie
ncy
400 450 500 550 600 650 700
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0(c)
Figure 1.9 The sensitivity of the eye to light, schematic. (a) Sensitivity of the cone cells in a normal eye to light asa function of the wavelength. (b) Visual sensitivity of a normal eye to red, green and blue light. (c) Overall visualsensitivity of a normal eye to light; the photopic spectral luminous efficiency function. Themaximum sensitivity isfor a wavelength close to 555 nm
25 Light and Colour
In humans, retinal is derived from the compound b-carotene (Section 8.5), an orange pigment found in
carrots. This is transformed intovitaminA in the liver,which then forms retinal. Thevisual pigments in animals
then consist of retinal plus opsin. (There are two formsof vitaminA:A1,which gives retinal1 (11-cis-retinal, the
aldehyde of vitaminA1), andA2,which gives retinal2 (3-dehydro-retinal). Retinal1 is used by allmammals and
birds and will just be referred to as retinal in what follows.)
The framework of the processes triggering vision is well established. It is described here with respect to rod
cells, which have been studied in most detail. The chromophore (light absorbing part) of rhodopsin is the cis-
form of the molecule retinal, 11-cis-retinal (Figure 1.11a). This cis-retinal molecule is bound to the opsin via
the amino acid lysine, to form rhodopsin (Figure 1.11b). The cis-retinal by itself is not coloured and has an
absorption maximum between 370 and 380 nm. However, when joined to the opsin the absorption maximum
moves to about 500 nm. Molecules which can cause the deepening of the colour of a chromophore are called
bathochromes and the resultant movement of the absorption maximum is referred to as a bathochromic shift.
The bathochromic shift comes about because of the particular conformation of the cis-retinal molecule in
conjunction with the protein. The bonding and slight differences in the various forms of the opsin molecules
produce different bathochromic shifts and, hence, make the cones sensitive to the different wavelengths of
red, green and blue light.
The molecular mechanism leading to the nerve impulse hinges on the fact that retinal can exist in two
isomeric forms, the cis-formalreadydescribed anda trans-form, calledall-trans-retinal.Under the influenceof
a photon the cis-retinal molecule changes to all-trans-retinal rhodopsin (Figure 1.11c). Absorption of light by
rhodopsin drives the molecule through several intermediates to the bleached state, which can consist of a
number of different molecules (metarhodopsin I, metarhodopsin II and so on), depending upon the conditions
experienced. Thereafter the reaction reverses, again passing through a number of intermediates, so that the
trans-retinal readopts the cis-conformation and reforms rhodopsin (Figure 1.11d). Another photon can trigger
the cycle again. Each cycle takes only a fraction of a second and can repeat indefinitely in normal light
conditions so as to send a stream of nerve impulses to the brain. These impulses end when the light is
extinguished and all molecules revert to rhodopsin.
It is worth commenting on the enormous complexity of vision. The description of the cycle occurring in rod
cells and presumed to occur in the cone cells described above is only true atmoderate light intensities. At lower
light intensities the trans-retinal molecule in rod cells is released completely from the opsin. Two processes
then operate, dependent upon the weakness of the light signal. At the ‘higher’ of these lower intensities the
trans-retinal is transformed back to the cis-conformation by the action of enzymes in the eye itself, whereupon
(a)
(b)opsin helices
retinal
Figure 1.10 (a) The schematic structure of an opsin protein in the cell wall of a photorectetor. (b) The opsinproteinmolecule is in the form of seven helices arranged to enclose a retinalmolecule. [(a) is adapted fromhttp://en.wikipedia.org/wiki/Rhodopsin]
Colour and the Optical Properties of Materials 26
CH3 CH3
CH3
CH3 CH3
H3C
H3C
CH3
CH3
CH3
CH3 CH3
H3C
H C3
H C3
7
7
7
8
8
8
9
9
9
10
10
10
11
11
11
12
12
12
13
13
13
14
14
15
15
1415
CHO
11-cis-retinal
rhodopsin
(a)
(b)
(c)
N+
N+
H
H
opsin
opsin
light photon
all-trans-retinal rhodopsin
rhodopsinall-trans
rhodopsin
light photon
intermediatesintermediates
metarhodopsins(d)
Figure 1.11 The structures of (a) 11-cis-retinal, (b) rhodopsin and (c) all-trans-retinal rhodopsin, produced bythe action of light on (b); (d) cycle of chemical changes producing vision. In normal illumination this process isrepeated many times a second. Each cycle results in the transmission of a signal along the optic nerve to the brain
27 Light and Colour
themolecule is reattached to the opsin.At the lowest light intensities, the trans-molecules actually leave the eye
completely, enter the bloodstream and are reprocessed to the cis-form in the liver, an occurrence which
contributes to the length of time that it takes to become fully ‘dark adapted’.
Rhodopsin has another role to play in the broader picture of life. It has been found that some purple
halobacteria, bacteria which inhabit very salty environments, are coloured purple by a version of rhodopsin
called bacteriorhodopsin. This consists of 247 amino acid residues, arranged in seven helices, with the
photoactive retinal attached to lysine 216. It is, however, not used for vision, but in an analogous fashion to
chlorophyll in plants. Absorption of light by chlorophyll initiates a chain of electron transfer reactions which
eventually provide the energy for plant growth. In the purple halobacteria, the rhodopsin converts sunlight into
energy for the metabolism of the bacterium. In essence it appears that the cis trans change acts as a proton
pump, and the resulting electrochemical potential created initiates the energy building steps.
1.11 Colour Perception
Recognition of colour is a function not only of the physical make-up of the light falling on the eye and
physiological factors, but also of psychological biases. The ‘colour’ of an object in this sense is changed by
factors such as surface roughness or texture. Subsurface scattering,which returns some incident light on a body
to the exterior, is of importance in the appearance of skin, cosmetics and paint. Because of this interplay it is
possible to distinguish a hard red plastic surface from a red velvet surface even though in terms of physics the
colours of both may be identical, originating in the same dye or pigment. It is clear that when describing the
appearanceof anobject in colour terms it is necessary to consider specular (mirror-like) reflection, diffuse (non-
mirror-like) reflection and subsurface scattering, as well as the make-up of the light which is reflected or
scattered. Moreover, human eyes vary in colour-interpreting ability. It appears that an average person can
distinguishmore than amilliondifferent colours.All of these aspects are impliedwhen the colour of anobject or
a light source is mentioned in a colloquial way. Because of this, colour is difficult to quantify.
Despite the complexity inherent in the concept of colour and its perception, it has been found that all colours
can be precisely specified by three parameters. Colours can then conveniently be represented by points in a
three-dimensional coordinate system. There are many diagrammatic ways of representing the three attributes,
and these are called colour spaces. The way in which the coordinates of any colour in the colour space are
derived is called a colourmodel. There aremany colourmodels, ofwhich only threewill be described briefly in
this book. (More information can be found in Section 1.17.)
Onewidely used colour model takes as initial parameters the three attributes hue, saturation and brightness
to give the HSB model. These characteristics are generally taken to be:
1. Hue,which corresponds to thewavelengthor frequencyof the radiation.The hue is givena colour name such
as red or yellow.
2. Saturation or chroma, which corresponds to the amount ofwhite lightmixed inwith the hue and allows pale
‘washed out’ colours to be described.
3. Brightness, lightness, luminance, or value, which describes the intensity of the colour, the number of
photons reaching the eye.
This model is also given the acronyms HVC (hue, chroma, value), HSL (hue, saturation, luminance), HIS
(hue, intensity, saturation) andHCL (hue, chroma, luminance). Oneway of building a colour space in terms of
this colourmodel is to arrange the hue around the periphery of a disc with the degree of saturation of the colour
represented by the distance from the centre of the disc along the radius. Brightness is defined by an axis
perpendicular to the centre of the disc (Figure 1.12a). This arrangement has been quantified in constructions
such as the Munsell colour cylinder or Munsell colour solid (Figure 1.12b).
Colour and the Optical Properties of Materials 28
1.12 Additive Coloration
Additive colourmixing occurswhen two ormore beams of differently coloured light combine (i.e. overlap on a
perfectly white surface, or arrive at the eye simultaneously).
Colours on television screens are produced by additive coloration, as the screen is composed of small dots of
three different phosphors each of which shines with one of three primary colours when activated. Additive
coloration is also used in the painting technique known as pointillism. In this method of painting, the image is
built up by placing small dots of relatively saturated colour onto the canvas, making sure that they do not
overlap. When viewed from a distance of a few metres such pictures appear bright and dynamic.
The colour patterns on thewings ofmany butterflies andmoths are produced in a similar way. Thewings are
tiled with a fine mosaic of scales, each of which reflects only one colour. The colour perceived by the eye is an
brightness / white
brightness / black
green
yellow
red
blue
purple
hue
saturation
white
(a)
(b) black
Figure 1.12 The representation of colours on a cylindrical colour space in the HSB colour model. (a) The hue isgiven by a point on the circumference of a planar disc, the saturation by the distance along the radius from thecentre of the disc and the lightness by the vertical axis of the system. (b) The solid representation of the coloursforms a colour cylinder, thebest knownof these being theMunsell colour cylinder [adapted from theEpsonOnlinePrinter Guide]
29 Light and Colour
additive colour arising from the numerous closely spaced scales. The range of colours which can be produced
by rather a few basic pigments is remarkable. For example, some perceived purples arise from mixtures of
black, white and red scales, while some greens arise from mixtures of yellow and black scales.
It has been found that the majority of additive colours can be produced by mixing just three additive
primary colours, red, green and blue. (Strictly speaking, any fairly monochromatic light near to these
colours will suffice). Moreover, mixing equal quantities of these three primary colour lights will produce
white light. There are a number of ways of quantifying the amounts of each primary colour light present,
which can represented by the values, r of the red component, g of the green component and b of the blue
component; thus:
colour ¼ rþ gþ b
Use of these three additive primaries is called the RGB colour model.
A simple colour space can be constructed by using Cartesian axes to represent the amount of the three
primary colours, red, green and blue, while the diagonal represents the transformation from black to white
(Figure 1.13a). Sections through this colour space allow one to represent colours by a planar figure. Such
representations are called chromaticity diagrams. A simple example is given by taking the triangular sheet
running diagonally through the cube normal to the black white diagonal and cutting the corners of the cube that
represent pure red, green and blue. This produces a colour triangle (Figure 1.13b). Other colours can be
Figure 1.13 Colour spaces and chromaticity diagrams. (a) RGB colours represented by Cartesian axes, withblack to white along the body diagonal. (b) A colour triangle, a section of (a) taken normal to the body diagonalpassing through red, green andblue corners of the cube.A combinationof the three primary colours at the verticesof the triangle will yield grey, but is shown white here. Other colours within the triangle (the gamut) can berepresented by a point in the plane of the triangular system
Colour and the Optical Properties of Materials 30
specifiedbycoordinates in theplaneof the colour triangle.The locationgivenby the coordinates corresponds to
the amounts r, g and bmaking up the colour. The coordinates which specify the case when the three primary
colours are mixed in equal amounts will correspond to a shade of grey, but is usually represented by the colour
white. The range of available colours that can be obtainedbymixing lights corresponding to the threevertices is
thegamut of colours available.Chromaticity diagramsgenerally represent hue and saturation, but not lightness
(i.e. the grey tone), which must still be added as a third axis perpendicular to the chromaticity diagram if this
information has to be displayed.
The study of lightmixing has been quantified by theCommission Internationale de l’Eclairage (CIE), which
has, onanumberofoccasions, refined the rather simple colour triangle concept so as toallowcolourperceptions
to be more accurately characterised. A colour is specified by a pair of x- and y-coordinates, which are derived
from the r, g and b values noted above by the application of a standardized set of equations. In this
representation, the triangular shape has been distorted into an outline something like a parabola, depending
upon the way in which the x- and y-axes are plotted. A commonly encountered form of the CIE chromaticity
diagram is that first proposed in 1931 (Figure 1.14a). The spectral colours are arranged around the outer edge of
the shape and colours not seen in the spectrum, the purples and browns, are found to lie between the red and
violet ends of the curve. The colours are fully saturated along the outer edge of the curve and become less and
less saturated as the centre of the diagram is approached. Standard daylightwhite is represented by a point close
to the coordinates x¼ y¼ 0.33, shown as W in Figures 1.14a, b.
If a straight line is drawn through the pointWand extended to the boundaries of the curve, the pair of colours
reached, when mixed, will givewhite light. For example (see Figure 1.14b), a line connects the colours red, of
wavelength 700 nm and blue green, of wavelength 492 nm and passes through the pointW. The proportions of
the end colours red and blue green light needed to producewhite light is given by the lever rule (Figure 1.14c):
amount of red light ¼ r=ðrþ cÞamount of blue-green light ¼ c=ðrþ cÞ
Measurement shows thatmixing redofwavelength 700 nmandblue green light ofwavelength492 nm in the
proportions 39% red to 61%bluewill producewhite light. The colours at the ends of a line through the pointW
are called a complementary pairof colours. If one of these colours is subtracted fromwhite light then the colour
remaining is called the complementary colour to the first.
As with the colour triangle, all planar chromaticity diagrams represent hue and saturation, but not the exact
value of lightness, which must still be added as a third axis perpendicular to the chromaticity diagram if this
information has to be displayed. In general terms, therefore, the white region on the chromaticity diagram
should be represented by grey, with white and black being extremes on the vertical axis perpendicular to the
plane of the figure.
Theaccurate renditionofadditivecoloration isofprime importance indisplays, suchas televisionscreensand
computer monitors. Additive coloration and the interconvertion between various colour models is most easily
explored using a computer which has photography or drawing editing software installed. On most of these
packages, sevenor eight or so different colourmodels are available, includingRGBand at least oneCIEmodel.
The coordinates of any colour are given and comparisons between several systems are rapidly made. The
instructions and help facilities give full information upon these options and how they affect colour rendition.
The confusion that colour blindness can cause is easily understood in terms of a chromaticity diagram. For
example, Dalton had a lack of red receptors (Footnote 4). The CIE 1931 chromaticity diagram can be used to
illustrate this. Any colour formed by mixing red with another colour, C, around the periphery of the curvewill
not be differentiated from any other colour along the line joining red to C. These lines show the loci of colour
confusion (Figure 1.15). Other types of colour blindness will lead to other loci of colour confusion.
31 Light and Colour
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
(a)
400
480
490
500
510
700
600
580
560
540
520
W red
orange
yellow
green
cyan
violet
x
y
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
400
480
490
500
510
700
600
580
560
540
520
W red
orange
yellow
green
cyan
violet
x
y
(b)
cyan492 nm
red700 nm
white
r(c) c
Figure 1.14 The CIE 1931 chromaticity diagram. (a) The colours of the spectrumare arranged around a curved lineand nonspectral colours fall on the line joining violet (400nm) and red (700nm). The figures marked around the outeredgeof the curve denote thewavelengthof the colour. Pointswithin the area of the diagram represent colours formedbythe additive mixing of light and can be specified by the appropriate x- and y-values. The point W represents white light.(b) A straight line through W links two complementary colours on the periphery of the diagram, in this examplered and cyan. (c) The lever rule gives the proportions of complementary colourswhich are needed to createwhitelight. In this example, the amount of red light is given by r/(r þ c) and the amount of cyan light by c/(r þ c)
Colour and the Optical Properties of Materials 32
1.13 The Interaction of Light with a Material
Colour is inherent in the light that leaves an emitting source; butmost often before it reaches the eye it interacts
with matter of many types: gases, liquids and solids. The colour observed is thus a function of both the source
radiation and the interactions that have occurred.
The way that light interacts with a material can be described in terms of scattering or absorption. To a first
approximation, scattering is well treated by assuming that the light behaves as an electromagnetic wave,
while absorption is best treated in terms of photons. If the energy of the scattered wave/photon is the same as
that of the incident wave/photon then the scattering is called elastic scattering, and otherwise inelastic
scattering.
For historical reasons, the term scattering itself, especially elastic scattering, is usually reserved for the
interaction of light with randomly distributed small particles. Elastic scattering from a surface is normally
called reflection, and elastic scattering into a transparent solid is called refraction. Scattering from ordered
collections of small particles, or fromsmall detail on largerobjects, is calleddiffraction. For the purposes of this
book these terms are retained as they stand, although all are simply different aspects of scattering. All of these
processes are wavelength dependent, and so can result in the production of coloured light from white light.
Inelastic scattering arises when energy is transferred from the light photons to an absorption centre.
Absorption is generally the termreserved for usewhen someor almost all of the incident radiation is takenupby
thematerial and inelastic scatteringwhen the changes are rather small. During absorption the energy is used to
excite the component atomsormolecules that constitute the absorption centres into higher energy levels.Often,
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
400
480
490
500
510
700
600
580
560
540
520
Wred
orange
yellow
green
cyan
violet
x
y
Figure 1.15 The dashed lines represent the loci of colour confusion for a personwith red-defective vision plottedon the CIE 1931 chromaticity diagram. Because of a fault in the red perception, all colours on each line appearsimilar to the colour at the low wavelength extremity
33 Light and Colour
the absorbed energy is manifested as a rise in temperature of the body. On occasion, some of this energymight
be re-emitted as light, giving rise to fluorescence and related features. A material that does not absorb
significantly is said to be transparent. Absorptionmay beminimal and transparencymaximal for high-quality
optical components over the visible spectrum, but no material is transparent over all wavelength ranges.
Silicon, for example, appears ‘metallic’ over the visible spectrum but is transparent to infrared wavelengths.
Absorption is wavelength dependent and an important source of colour production. It is often difficult
experimentally to separate the relative roles that absorption and scattering play in the interaction of light with a
material.
As a beam of light passes through a material it gradually loses intensity, a process generally called
attenuation (formerly extinction). Attenuation is due to the interaction of light with a material in two basic
ways: scattering or absorption (Figure 1.16).When attenuation takes place in a homogeneous solid the amount
of light transmitted by a semitransparent plate of thickness x is given by:
Ix ¼ Io expð�aexÞ ð1:7Þ
where Ix is the irradiance leaving the plate,5 Io is the incident irradiance and ae (m1) is the (Napierian)
linear attenuation coefficient (formerly extinction coefficient). Equation 1.7 is known as Lambert’s law or
Beer’s law, although it was first clearly set out by Bouguer and should, by rights, be called Bouguer’s law. The
incident light
reflected light
scattered light
fluorescence
transmitted light
absorption centre
scattering centre
fluorescence centre
Figure 1.16 The interaction of light with a transparent material. The light can be reflected, absorbed orscattered. Some absorption centres are able to re-emit light as fluorescence or luminescence. All of the processeslabelled are wavelength dependent and can lead to colour production
5 The symbol I is used for irradiance instead of E to avoid confusion with the use of E for energy throughout this book. See also
Appendix 1.1.
Colour and the Optical Properties of Materials 34
attenuation length is defined as 1/ae. The amount of light removed from the beam is thus:
Irem ¼ Io�Ix ¼ Io�Io expð�aexÞ ¼ Io½1�expð�aexÞ�
If the attenuation of the beam is solely due to absorption, then the attenuation coefficient is replaced by the
(Napierian) linear absorption coefficient aa. Similarly, if the attenuation is solely due to scattering, then the
attenuation coefficient is replaced by the (Napierian) linear scattering coefficient as. For nonhomogeneous
solids these coefficients may vary with direction. Note that the degree of attenuation will vary significantly
across the spectrum and the attenuation coefficient is not a constant.
It is sometimes convenient, as when discussing the absorption of X-rays, to define a mass absorption
coefficientm,whichdescribes thedecrease in transmitted irradiance throughahomogeneousmaterial of density
r and thickness x:
Ix ¼ Io expð�mrxÞ
In this case:
m ¼ aer
where m has units m2 kg 1 (in older literature cm2 g 1).
Attenuation is often associated with the presence of chemical or physical ‘centres’, which may be atoms,
molecules or larger particles, distributed throughout the bulk of a material. In the case of the mass absorption
coefficient described above these are the totality of the atoms thatmake up thematerial itself. In this case, if the
atoms in the material are supposed to absorb radiation independently of each other, then the mass absorption
coefficient of the phase is simply related to the weight fraction of each atom species present. Thus, the mass
absorption coefficient of a material M with a formula AxByCz is:
mM ¼ ðwt fraction AÞ � mA þðwt fraction BÞ � mB þðwt fraction CÞ � mC
The weight fraction of each species is given by:
wt fraction A ¼ mass of A present
total mass¼ xðmAÞ
xðmAÞþ yðmBÞþ zðmCÞ
and so on,wheremA is themolarmass of species A,mB is themolarmass of species B andmC is themolarmass
of species C.
More often, extinction is associated with a dilute concentration of centres distributed throughout the bulk
phase. In this case, the degree of extinction is often taken to be a function of the concentration of these centres.
This is taken into account in the Beer Lambert or Beer Lambert Bouguer law:
logIx
Io
� �¼ �ecx
35 Light and Colour
where Ix is the irradiance after passage through a length of sample x, Io is the incident irradiance and c is the
molar concentration (mol L 1, i.e.mol dm 3) of the active centres or species. Thequantity e is called themolar
(decadic) attenuation coefficient and has units6 of m2mol 1. The attenuation coefficient has units of area and
can, therefore, be regarded as an attenuation cross-section. In practical terms the units employed are often
Lmol 1m 1 (i.e. dm3mol 1m 1).Writing 1 Las 0.001m3, themolar attenuation coefficient can be expressed
as 0.001m2mol 1 or 1m2mmol 1.
The dimensionless product A¼ ecx is called the absorbance (sometimes the optical density) and the ratio
Ix/Io is the transmittance or transmissivity T. Thus, we can write:
logT ¼ �A
TheBeer Lambert law finds use in themeasurement of concentrations. For example, the clarity or otherwise
of polluted air is oftenmeasured by comparing the irradiance of light at a certain timewith the irradiance on a
fine day.
These interactions with a material can be expressed thus:
Io ¼ Ir þ Is þ Ia þ It
where Io is the incident irradiance, Ir is the amount reflected, Is is the amount scattered, Ia is the amount absorbed
and It is the amount transmitted, or as:
1 ¼ Rþ SþAþ T
whereR is the fraction of light reflected, S is the fraction of light scattered, A is the fraction of light absorbed
and T is the fraction of light transmitted and the quantities measured are the appropriate irradiance values.
In good-quality optical materials the amount of light scattered and absorbed is small and it is often adequate
to write:
Io ¼ Ir þ It
or
1 ¼ Rþ T
In a pure liquid the Beer Lambert law is often written in the form:
logIx
Io
� �¼ �ax
wherea (m 1)¼ ec is themolar (decadic) attenuation (or absorption) coefficient. The absorptionwill be due to
molecular or atomic processes taking place in the pure medium.
6 Chemists frequently use the termmolarity for concentration inmol L 1, given the symbolM. Thus, e is given the unitsM 1m 1, ormore
often M 1 cm 1. To convert values of e in M 1 cm 1 to M 1m 1, multiply the value by 100.
Colour and the Optical Properties of Materials 36
1.14 Subtractive Coloration
Absorption has been used formanycenturies to produce colour. For example, the colour of stainedglass and the
colours seen in ordinary colour filters are examples of colour production in this way (Figure 1.17). The colours
perceivedby the eye inwhich absorptionand selective reflectionor transmission are important are said tobedue
to subtractive colour mixing. For example, the photosensitive pigments in green leaves preferentially absorb
red and blue light and reflect more of the green component of the incident white light. Similarly, colour filters
absorb some wavelengths strongly and transmit the remainder. Figure 1.18a shows the fraction of light
transmitted as a function ofwavelength for a commercial glass colour filter. The range of thevisible spectrum is
indicated above the transmittance curve. The filter absorbs red light strongly and transmits violet and blue
green light (Figure1.18b). If thefilter is heldup to the light itwill lookblue green.When it is viewed in reflected
light it appears dark, as red light is absorbed and blue green passes through the film. This is the reason why
stained glass windows in medieval churches look impressive when viewed inside the building, with light
transmitted through the glass, yet often look dull when viewed from outside the building in reflected light.
By analogy with additive coloration, one would expect to be able to combine three subtractive primary
colours to produce thewhole range of subtractive colours. These subtractive primary colours are: cyan, which
absorbs red and transmits blue andgreen;magenta,which absorbsgreenand transmits blue and red; and yellow,
which absorbs blue and transmits green and red. If the three subtractive primaries are mixed in equal amounts
we obtain black, as one primary will absorb red, onewill absorb green and onewill absorb blue, thus removing
the whole of the visible spectrum. Colour construction using these three subtractive primary colours is
described as employing the CMY model, where the letters simply represent the initial letters of the colorants.
If the wavelength range of light absorbed is rather small, then the colour remaining is called the
complementary colour to that absorbed (Table 1.4). It is seen that the additive and subtractive primary
colours are complementary colours.
Colour printers use cyan, yellowandmagenta dyes to produce the coloured images. These dyes are deposited
upon white paper and absorb the appropriate subtractive primary colour.White light reflected from the dyes is
depleted in these colours and yields the appropriate toned image by subtractive coloration. Although the
Figure 1.17 Mediaeval stained glass window in Gloucester Cathedral, viewed from inside the building.[Reproduced with permission from Gloucester Cathedral www.gloucestercathedral.org.uk]
37 Light and Colour
300 500 700
Wavelength / nm
Fra
ctio
nal t
rans
mitt
ance
1.0
0.5
visible
white lightblue-green light
filter
blue red
(a)
(b)
Figure 1.18 (a) The fractional transmittanceof a commercial blue colourfilter. About three-quarters of the bluelight incident on the filter is transmitted, but most red light is absorbed. (b) When the filter is viewed intransmitted white light it will appear blue–green
Table 1.4 Complementary colours
Wavelength/nm Colour absorbed Complementary colour
400 435 Violet Yellow green435 480 Bluea Yellowb
480 490 Blue green Orange490 500 Green blue Red500 560 Greena Magentab
560 580 Yellow green Violet580 595 Yellow Blue595 605 Orange Blue green605 700 Reda Cyanb
aAdditive primary colours.b Subtractive primary colours.
Colour and the Optical Properties of Materials 38
overlap of cyan, yellow and magenta produces black, this tone is often not dark enough for many representa-
tions. Printers, therefore, often add black to the trio. This system of colour production is known as the CYMK
model of colour formation, where the letter K stands for the black component. Although these four colours are
satisfactory formany colour printing applications,more hues, intermediate between theCYMKset, are used to
obtain more accurate colour rendition, in, for example, high-quality art reproductions.
1.15 Electronic ‘Paper’
Paper is an extremely convenientway of displaying information using subtractive coloration, but once a page is
printed it is permanent. Electronic paper, with the advantages of a printed page, but the flexibility of electronic
erase and rewrite has been pursued for over 30 years. As of 2000, e-book readers, which are rigid units
displaying one paper-like page at a time, have been increasingly available.
There are two aspects to electronic paper. In the first, electronic ‘ink’ must be developed that will retain the
display indefinitely but is erasable at will. At least for black-and-white displays this has been accomplished.
The second is the production of a flexible page that can support the electronic circuitry needed to drive the
display. In this section the characteristics of the ‘ink’ are the main focus of attention, as this is the aspect that
impinges upon the topic of colour.
The first electronic paper, using the Gyricon process, consisted of small polyethylene spheres of approxi-
mately 90 mmdiameter, colouredwhite onone hemisphere andblack on the other.Thewhite part held a positive
charge and the black portion a negative charge, due to additives to the polymers used. These spheres were
embedded in a transparent siliconefilmand the sheetswere immersed in clearoil.This penetrated the sheets and
coated the beads, so that they were effectively encapsulated in a bubble of oil. The application of a negative
charge to an electrode on the surfacewill attract the positively chargedwhite side facing one side of the ‘page’.
In this way pixels of the display could be made black or white at will (Figure 1.19a). Rearranging the applied
voltage allows the image to be erased and rewritten.
The e-ink process is rather similar but uses themovement of charged particles in an electric field, the process
of electrophoresis. Once again, small polymer capsules containing submicrometre particles of white titanium
dioxide, TiO2, holding a negative charge due to appropriate surfactants, and black particles holding a positive
charge are central to the system.Themicrospheres also contain a nonviscous liquid and are embedded in a clear
plastic film. A charge applied to surface electrodes will attract white or black particles depending upon the
polarity of the electrodes. Reversal of the charge on the electrodes reverses the particles that are attracted and
the area will swap colour (Figure 1.19b). Erasure and rewriting is carried out as before.
Naturally, the use of polymer spheres to contain the black and white particles is not mandatory, and any cell
structure could be used. The device also becomes simpler if the black particles are replaced by a dark-coloured
fluid. The white particles are then the only active species present. When attracted to a surface the appropriate
pixel looks white and when not attracted the dark fluid is seen.
The colour of the pixels is due to absorption and scattering. Titanium dioxide is a well-known white
scatterer (Section 5.7) and the dark colour is simply absorption of the incident light by the dye present.
The system can be made into a colour display by putting red-, green- and blue-coloured filters in front of
the electrodes (Figure 1.19c). A white pixel will now become a coloured subpixel corresponding to one
of the colours.
The electrodes used to control the display can be a simple passive array of vertical strips on one face of the
device and horizontal strips on the reverse face. Application of charges to appropriate columns and rows
ensures that pixels can bemade black or white as required. The activematrixmethod of control, as used in flat-
panel television, inwhich a transistor controls each pixel, is alsowidely used. An advantage of these displays is
that once the page has been created, no further electrical input is needed until the page is rewritten. Of course,
39 Light and Colour
the requirements are less demanding for a rigid e-book than for a portable and flexible sheet-like page, which
has still to achieve widespread commercialization.
1.16 Appearance and Transparency
Scattering and absorption give rise to the world of colour around us (Figure 1.20). Even small changes in the
relative amounts of each wavelength band present in a light beam will contribute significantly to colour and
appearance. A striking example of this is the blue sky. Blue sky is so coloured because of light scattering (see
Chapter 5). However, blue sky contains all of the wavelengths of the spectrum something easily proved by
passing the light through a prism. The sky appears blue because the balance in the various colours has been
tipped slightly in favour of the blue end of the visible spectrum.
The appearance of an objectwill depend on a number of factors, especially on roughness and surface texture.
Thesewill alter the reflectivity of the surface considerably. If the surface is smooth then the reflection is said to
be specular, while if the surface is rough then the reflection is diffuse (Figure 1.21a). The diffuse reflection
component increases with surface roughness at the expense of the specular component, so that a finely ground
powder shows only diffuse reflection. The gloss of a surface is a measure of the relative amounts of diffuse to
specular reflections. Glossy surfaces have a large specular component. Aswell as diffuse reflection, subsurface
scattering is of considerable importance inmodifying the appearance of a surface. This is particularly sowhen
the surface is composed of layers with different optical properties, such as skin. Controlling these forms of
scattering and reflection are of great importance to the cosmetics industry, and imitating them is vital to both
artists and personnel involved in the representation of skin tones in computer-generated images.
_ _ _+ + + +
+ + +
+
+– – –
–
+ +
+
+ + +
+
+
+
_
_
_
_
_ _ _
_
_
–
(a)
(b)
(c)
colour filter layer
––
Figure 1.19 Electronic paper displays: (a) the rotating sphere Gyracon system; (b) the electrophoretic e-inksystem; (c) coloured filters allow for a full colour display to be achieved
Colour and the Optical Properties of Materials 40
Closely related to this is the property of transparency or invisibility in animals. Invisibility confers obvious
advantages to both predator and prey in the living world. It is not surprising, therefore, that many marine
animals almost achieve this object. To attain invisibility, the interactions of light with a material described
abovemust be bypassed. That is to say, reflection and refraction at surfaces and scattering and absorption from
internal centres need to be suppressed.
Reflection and refraction at the surface of the animal’s body can be substantially reduced by making the
refractive indices on both sides of the boundary the same (Chapters 2 and 3). For many marine animals,
including numerous species of zooplankton, jelly fish and similar creatures, the inner body fluid is essentially
watery, and reflection and refraction are virtually eliminated. This alone serves to make the animals virtually
invisible.7However, any inhomogeneities in the tissues andmembraneswill act so as to scatter light and render
the animal visible to a greater or lesser degree. Air pockets are particularly problematic. For instance, small
bubbles of air in water are easily visible in ordinary light and shine like silver spheres. Pigments, which colour
by absorption, cannot be totally avoided. The photoreceptors of the eye are pigments which absorb visible
radiation (Section 1.10). Similarly, the prey of the animal, once consumed, will be visible in the gut, unless this
matches the surroundings in both transparency and refractive index. Thus, although manymarine animals can
be extremely difficult to spot, and may be termed invisible for all practical purposes, some traces will remain
visible.
Solids and liquids cannot be manipulated so that their refractive index matches that of air, but can be made
with a refractive index that matches that of a liquid. A transparent solid immersed in a liquid of the same
refractive index will be invisible. For many solids, internal surfaces are a major cause of loss of transparency.
Figure 1.20 A moorland scene displaying colours due to scattering (blue sky), reflection (the blue stream) andabsorption (the green–browns of grass and soil)
7 This is the basis of the famous story The Invisible Man by H. G. Wells, published in 1897.
41 Light and Colour
Glass, the best known of transparent solids, is, in effect, in the liquid state, and no internal boundaries occur.
However, it is relatively simple to cause a glass to crystallize and the ensuing tiny crystallites act as scattering
centres. The resulting scattering, which can contain diffuse and specular components, renders the material
nontransparent although the solid transmits a certain amount of the incident light. Such materials are termed
translucent. The light emerging from a translucentmaterialwill also contain a diffuse and specular component
(Figure 1.21b). Translucency is a desirable property of fine porcelain, which consists of crystallites of mullite
(�Al6Si2O13) dispersed throughout a glassymatrix.More opaqueglasses, such asopalglasses, are deliberately
made with large numbers of scattering centres present. The resultant scattering renders the material white
because the scattering affects all wavelengths of the incident light equally.
Similarly, most plastics as fabricated are noncrystalline and have no internal boundaries, rendering them
transparent. If these contain impurities, inhomogeneities or polymer crystallites they become translucent and
take on a slightly milky appearance.
Non-glassy solids are mainly composed of polycrystalline aggregates or ‘grains’. The grain boundaries
between each crystallite scatter light, and any impurity phases that exist in the matrix or the grain boundary
regions enhance this effect, so that polycrystalline solids are invariably opaque. However, it is of considerable
benefit to make such materials transparent. This can be achieved by careful processing that achieves a high
incident beam
specularreflection
diffusereflection
rough surface
speculartransmissiondiffuse
transmission
opalglass
incident beam
specularreflection
diffusereflection
(a)
(b)
Figure 1.21 (a) Reflection of light from a rough surface consists of two components, diffuse reflection andspecular reflection. The ratio of diffuse reflection to specular reflection increases as the surface roughnessincreases. The ratio is an indication of surface gloss. (b) The passage of light through a translucent materialcontaining many scattering centres gives rise to both surface reflection and transmitted light with diffuse andspecular components
Colour and the Optical Properties of Materials 42
density, so that internal pores andbubbles of gas are eliminated, andproduces a solid composedof small, evenly
sized crystallites with no impurity grain-boundary phases present. In this way, transparent refractory ceramics
such as alumina (Al2O3), aluminium oxynitride (�Al23O27N5) and SiAlONs (materials occurring in the SiO2
Al2O3 Si4N3 AlN system) have been produced. These and similar materials have uses as lamp housings and
windows which need to be stable in air to temperatures of 2000 �C or more. In addition, these are hard and
durable ceramics and are favoured for applications such as specialist optical windows and domes where
resistance to abrasion and erosion are important selection criteria.
Appendix A1.1 Definitions, Units and Conversion Factors
A1.1.1 Constants, conversion factors and energy
Constants
The important constants for light are:
velocity of light in vacuum c 2.99792� 108m s 1
Planck constant h 6.62608� 10 34 J s
Boltzmann constant kB 1.38066� 10 23 JK 1
Conversion Factors
E (J)¼E (eV)� 1.60219� 10 19
E (J)¼E (cm 1)� 1.98645� 10 23
E (eV)¼E (cm 1)� 1.23987� 10 4
l (A�)¼ l (nm)� 10
l (nm)¼ l (mm)� 1000
l (nm)¼ 1239.9/E (eV)
l (nm)¼ 198 645� 10 21/E (J)
l ðnmÞ ¼ 107=n ðcm 1Þ
Energy
TheSI unit of energy is the joule (J).Awidevariety of energyunits are used in the literature connectedwith light
apart from the joule. A common nonstandard unit of energy in atomic work is the electron volt (eV).
Spectroscopy often uses energy values given in cm 1. These are not energy values at all really, butE/hc values.
To convert ‘energy’ in cm 1 to joules, multiply the value in cm 1 by h (J s) and c (cm s 1); see Conversion
Factors above.
A1.1.2 Waves
Waves
The wave equation, Equation 1.2, is a one-dimensional continuous harmonic wave that represents the electric
field vector E:
E ¼ E0 cos½ð2p=lÞðx�vtÞ� ðA1:1Þ
43 Light and Colour
E is the magnitude of the electric field vector at position x and time t, E0 is the amplitude of the wave, l is thewavelength of thewave, ½ð2p=lÞðx�vtÞ� is the phase of thewave (radians), v is the speed at which any point onthe wave, say a peak or a trough, travels in the positive x direction, and is called the phase speed or phase
velocity. The velocity of an electromagnetic wave in vacuum (the speed of light) has the symbol c.
The relationships described below allow the equation to be written in other equivalent forms. Those most
frequently met are:
1. a standing (non-travelling) wave:
E ¼ E0 cos½ð2p=lÞx�
2. a wave travelling in the negative x direction:
E ¼ E0 cos½ð2p=lÞðxþ vtÞ�
3. a wave travelling in the positive x direction, where o is the angular frequency:
E ¼ E0 cos½ð2px=lÞ�ot�
4. a wave travelling in the negative x direction, where o is the angular frequency:
E ¼ E0 cos½ð2px=lÞþot�
Frequency
The temporal frequency n of light (the number of waves that pass a point per second) has units of cycles per
second, hertz (Hz) or s 1. It is usually just called the frequency. The reciprocal of the temporal frequency, 1/n, is
the temporal period t, which is the amount of time for a completewave oscillation to pass a stationary observer
at a fixed value of x.
The angular (temporal) frequency o of a wave is given by:
o ¼ 2pt
¼ 2pn units: rad s 1
Using the relationship c¼ ln gives o¼ 2pc/l.
Wavelength
The wavelength l. is the spatial period of the wave the distance over which the wave subsequently repeats
itself. Inwavelength designations concerning light, nanometre (nm) is the preferred unit, but a commonly used
nonstandard unit, especially in X-ray diffraction, is the a�ngstr€om (A
�), 10 10m. To convert between units, see
Conversion factors above.
Wavelength and Energy
Planck’s law (E¼ hn¼ hc/l¼ ho/2p¼ ho) relates energy to wavelength. To convert between units, see
Conversion factors above.
Colour and the Optical Properties of Materials 44
Wavenumber
Thewavenumber is the reciprocal of the spatial periodof thewave (the number ofwaves per unit length) and so
is the reciprocal of the wavelength, 1/l. The wavenumber is given the symbol s when the light traverses a
transparent medium or n when in a vacuum. In spectroscopy it is often given units of cm 1.
sðnÞ ¼ 1
l
Using Planck’s law, in a vacuum:
E ¼ hn ¼ hc
l¼ hcn
E
hc¼ n
Similar equations can be written for light in a substance, by replacing c by the velocity v in the medium and
replacing n with s.Spectroscopy often uses wavenumbers (units: cm 1). To convert these to wavelength (units: nm):
l ðnmÞ ¼ 107
n ðcm 1Þ
In physics, the magnitude of the propagation vector or wave vector k is called the propagation number or
wavenumber, given by k¼ 2p/l. (By analogy with temporal frequency and temporal angular frequency, it
might be better to call k¼ 2p/l the angular spatial frequency to avoid confusion.) Additionally, physics alsouses k¼ 1/l for thewavenumber, omitting the factor 2p. To avoid confusion, thewave vector will bewritten as2p/l or 1/l rather than k.
A1.1.3 SI units associated with radiation and light
There are twoparallel sets of units in use for themeasurement of radiation and light.Photometricunitsmeasure
theperceptionof a light as it appears to the eyeof anaverageobserver.Radiometric unitsmeasure theamountof
electromagnetic radiation, including light, in terms of absolute quantities,without any reference to the eye.The
difference can be understood by considering the light output of four small light-emitting diodes (LEDs), one
infrared, one deep red, with an emission at 670 nm, one green with an emission at 555 nm and one bluewith an
emission at 490 nm. These may all emit exactly the same absolute power (measured in radiometric units, say
5mW), but the green light will appear ‘brighter’ than the other two visible LEDs because the eye is more
sensitive to green than red or blue. Calculation shows that the visible outputs will be: green, approximately
3.4 lm; blue, approximately 0.75 lm; red, approximately 0.1 lm; infrared, 0 lm. The green LED will appear
about 31 times brighter than the red LED, and the blue LED about seven times brighter than the red LED. The
infrared-emitting LED will be invisible to the eye and will not register at all in terms of photometric units,
although it still emits the same amount of power as the visible ones.
Clearly, photometric units are of importance in the design of displays and lighting,whereas radiometric units
are ofmore importancewhen comparing the energy requirements of the same structures. Although the two sets
of units are analogous, as set out in Table A1.1, because theymeasure different aspects of light, they cannot be
trivially interchanged in this regime.
45 Light and Colour
Table A1.1 Units used in radiometry and photometry
Radiometry Photometry
Name, symbol Comments Units Name, symbol Comments Units
Radiant power,radiant flux,F, P
Rate of flow of energyemitted by a source
W Luminous power,luminous flux,F, Fv
Rate of flow of luminous energyemitted by a source
lumen (lm)(cd sr)
Radiant intensityI¼ dF/d�
The power of an emittingsource per unit solid angle
Wsr�1 Luminousintensity, Iv
Light emitted from a source per unitsolid angle; SI base unit
candela(cd)¼ lm sr�1
Radiance, L¼d2F/(dA d�)
Radiant power per unit areaper unit solid angle.Radiant intensity of aradiating source per unitsurface area.
Wm�2 sr�1 Luminance, Lv A measure of ‘brightness’; luminousintensity of a light emitting sourceper unit area of source; may varyover the source surface
cd m�2 (nit!)
Irradiance E (I)¼dF/dA
Radiant power incidentupon a unit area of asurface.
Wm�2 Illuminance, Ev A measure of illumination; theluminous flux falling on a surfaceper unit area
lux (lmm�2)
(Radiant) ExitanceM¼ dF/dA
Radiant power emitted by asurface per unit area
Wm�2 Luminousexitance, Mv
Luminous flux emitted from a surface lux (lmm�2)
Flux is the amount of something flowing through a specified surface per unit time.
Luminous flux or luminous power F, unit lumen (lm): 1 lm is the amount of luminous flux passing in 1 s through a unit solid angle emitted by a point source of 1 cd. The total luminous flux of such a
point source is 4p lumens.
Luminous intensity Iv, unit candela (cd): 1 cd is the photometricmeasurement of luminous intensity in a givendirection of a source that emitsmonochromatic radiation of frequency 540� 1012Hz
and that has a radiant intensity in that direction of (1/683) watts per steradian. One square metre of a black body at 2042 K emits 600 000 cd.
Radiance L, unitsWm 2 sr 1: the radiance is the incoming radiation collected from a small angle of surroundings (measured in steradians) as if, for example, the detector is at the bottomof a tube.
The units of radiance are energy per unit area per unit solid angle, Wm 2 sr 1. The radiance is direction sensitive – the value recorded depends upon the direction in which the tube is pointing.
Irradiance Eor I, unitWm 2: the radiometric term irradiance is the total energy that a detector ‘sees’ fromahemisphere of surroundings. The preferred symbol for irradiance is E, but because of the
use of E for energy, and of E for the amplitude of an electromagnetic wave, it is less confusing here to use the symbol I.
Illuminance Ev, unit lux (lx): this is the photometric analogue of irradiance, being the total luminous flux incident upon unit area of a surface, with units of lux¼ lmm 2. The photometric term
illuminance has replaced the term brightness.
Radiant exitance or radiant emittance M, unit Wm 2: the amount of electromagnetic radiation leaving a surface is described by the radiometric term (radiant) exitance. The exitance is the
opposite of the irradiance, as it measures the total energy emitted by a surface into a hemisphere of the surroundings. The exitance has the same units as irradiance.
Luminous exitance or luminous emittance Mv, unit lux (lx): this is the photometric analogue of the radiometric radiant exitance.
Spectral units. These give the distribution of the quantity under discussion with respect to the wavelength or frequency of the radiation. For example, the spectral irradiance takes the form
irradiance per unit wavelength, written El, or irradiance per unit frequency En. The units of spectral quantities must contain the units of wavelength or frequency as appropriate. Thus, the units of
spectral irradiance are Wm 2m 1¼Wm 3.
Colouran
dtheOptical
Properties
ofMaterials
46
Further Reading
The followingfivebookscontain avast amount ofmaterial of relevance to thewholeof thisbook.The twobooks
by Bohren present material in a nonmathematical format and will repay repeated reading.
E. Hecht, Optics, 4th edition, Addison-Wesley, San Francisco, 2002.
K. Nassau, The Physics and Chemistry of Color, 2nd edition,Wiley-Interscience, NewYork, 2001, Chapters 1
and 2 and Appendix A.
C. F. Bohren,Clouds in aGlass of Beer, Dover, NewYork, 2001 (originally published by JohnWiley and Sons,
Inc., New York, 1987).
C. F. Bohren,What Light Through Yonder Window Breaks? Dover, New York, 2006 (originally published by
John Wiley and Sons, Inc., New York, 1991).
B. E. E. Saleh, M. C. Teich, Fundamentals of Photonics, John Wiley and Sons, Inc., New York, 1991.
Colour, from the point of view of artist’s pigments, is the subject of
V. Findlay, Colour; Travels through the Paintbox, Folio, London, 2009.
Goethe’s Theory of Colour, written 1808, published 1810, gives an interesting historical view of colour. It is
included in
D. Miller, (ed. and trans.), Goethe: Scientific Studies, Suhrkamp, New York, 1988 (Goethe Edition Vol. XII).
A clear discussion of radiometric and photometric units is given by J. M. Palmer (2003) to be found at http://
www.optics.arizona.edu/Palmer/rpfaq/rpfaq.htm. Also see
C. F. Bohren, E. C. Clothiaux, Fundamentals of Atmospheric Radiation, Wiley VCH, Weinheim, 2006,
Chapter 4.
The electromagnetic theory of radiation is clearly set out in
M. Kotlarchyk, Electromagnetic radiation and interactions with matter, in Encyclopedia of Imaging Science
and Technology, J. P. Hornak (ed.), Wiley-Interscience, 2002.
N. Braithwaite (ed.), Electromagnetism, Book 3, Electromagnetic Waves, The Open University, Milton
Keynes, 2006.
Light described in terms of quantum electrodynamics is explained lucidly and nonmathematically in
R. P. Feynman, QED: The Strange Theory of Light and Matter, Princeton University Press, Princeton, 1985.
The fascinating history of the theories of light is given by
G. N. Cantor, Optics after Newton, Manchester University Press, Manchester, 1983.
A comparison between thewave and particle explanation of the photoelectron effect and profound discussions
of the relationship between particle and wave theories of atomic physics are given by
D. Bohm, Quantum Theory, Prentice-Hall, Englewood Cliffs, NJ, 1951.
A detailed discussion of the solar spectrum and related topics is given by
C. F. Bohren, E. E. Clothiaux, Fundamentals of Atmospheric Radiation, Wiley VCH, Weinheim, 2006,
Chapter 1.
D. K. Lynch, W. Livingston, Color in Light and Nature, Cambridge University Press, Cambridge, 1995,
Chapters 2 and 7.
The (coincidental) relationship between the sensitivity of the human eye and the solar spectrum is discussed by
B. H. Stoffer, D. K. Lynch, Am. J. Phys. 67, 946 958 (1999).
For discussions of the molecular basis of vision, see
D. M. Hunt, L. S. Carvalho, J. A. Cowing, W. L. Davies, Phil. Trans. R. Soc. Lond. Ser. B 364, 2941 2945
(2009).
47 Light and Colour
For a discussion of bacteriorhodopsin, see
J. Whitford, Proteins, Structure and Function, John Wiley and Sons, Ltd, Chichester, 2005, pp. 114 119.
The evolution of primate colour vision is detailed by
G. H. Jacobs, J. Nathans, Sci. Am. 300 (April), 40 47 (2009).
The complexity of the retina and much information about vision in specialist circumstances is to be found by
consulting
S. Temple, N. S. Hart, N. J. Marshall, S. Collin, Proc. R. Soc. Lond. Ser. B 277, 2607 2615 (2010).
Many aspects of vision and the interpretation of visual images, including optical illusions, are detailed in the
series of articles by
J. C. Russ, Seeing the scientific image, Proc. R. Microscop. Soc. 39 (2004); Part 1: 97 114; Part 2: 179 193;
Part 3: 267 281.
The complexities of analysing colour and descriptions of the construction and use of chromaticity diagrams are
detailed in the following sources.A largenumberof articles concerning colour, colour theory, colour systems
and colour spaces will be found on Wikipedia (http://en.wikipedia.org/wiki/). Up-to-date details of colour
and colour reproduction will be found in the Instructions and Help functions for computer drawing and
photograph editing software, many of which are available: typically as in manuals for Nikon Coolscan,
Coreldraw, Photoshop and so on. Other sources are
http://www.efg2.com (this site has programs for the display and representation of chromaticity diagrams,
colour mixing and many other topics of relevance to the material in this chapter).
R.McDonald,Colour Physics for Industry, 2nd edition, Society of Dyers and Colourists, Bradford, UK, 1997.
F. Grum, C. J. Bartleson, Colour Measurement, Academic Press, New York, 1980.
R. Jackson, L. MacDonald, K. Freeman,Computer Generated Colour, JohnWiley and Sons, Ltd, Chichester,
1994.
Other interesting sources on colour and colour perception are as follows:
Animal colour patterns
J. A. Endler, J. Linn. Soc. 41, 315 352 (1990).
Reviews of transparency in biological tissues
S. Johnsen, E. A. Widder, J. Theor. Biol. 199, 181 198 (1999).
S. Johnsen, Sci. Am. 282 (February), 62 71 (2000).
Texture and computer modelling of surfaces
J. Dorsey, P. Hanrahan, Sci. Am. 282 (February), 46 53 (2000) and references cited therein.
There are a number of demonstrations of relevance to this chapter, including diffuse versus specular reflection,
available at http://demonstrations.wolfram.com/index.html.
Colour and the Optical Properties of Materials 48
2
Colours Due to Refraction and Dispersion
. Why are images of objects in water displaced?
. How do rainbows form?
. What is a negative index material (NIM)?
Many of the most beautiful colours arise simply by the passage of a beam of ordinary white light through a
transparent material. Rainbows provide a familiar example. These effects are due to refraction and dispersion
by thematerial. In this chapter, these rather straightforward features are described.The effect of thepolarisation
of the light is ignored for the present, although this feature is of importance, particularly for refraction within
crystalline solids or when the beam is especially powerful. These complexities are considered later.
2.1 Refraction and the Refractive Index of a Material
When light travelling through the air enters a transparentmedium, say a sheet of glass or a pool of clearwater, it
appears to bend. This is called refraction. The effect of refraction is familiar to most. A stick will appear bent
towards the surface when dipped into water (Figure 2.1a and b). Similarly, the bottom of a swimming pool
always seems closer to the surface than it really is. Kingfishers and other birds that catch fish by diving into
rivers must allow for this effect and aim below the object that they apparently see in order to hit the target
(Figure 2.1c). An equally complex problem is faced by Archer fish. These animals capture insect prey by
spitting a jet of water onto overhanging vegetation when an insect is present, knocking the prey into the water
below. The eyes of the Archer fish remain below the water during a capture attempt, and refraction as well as
associated colour changes (Section 2.6) must all be included in the ‘equation’ for the water jet trajectory.
The magnitude of the deviation as a light ray passes across the boundary into a transparent substance is
quantified by the refractive index (or index of refraction) of the material. For all practical purposes involving
Colour and the Optical Properties of Materials Richard J. D. Tilley
� 2011 John Wiley & Sons, Ltd
air
water
kingfisher
B
C
air
water
observer
A
B
C
(b)
(c)
Figure 2.1 Refraction at an air–water surface. (a)Ahalf-immersedpencil seems tobendupwards; (b) the causeofthe apparent displacement is that rays of light reflected fromBappear to come fromC; (c) anobject suchas afish ina pool appears to be nearer the surface than it is because light rays reflected from the fish at B appear to originatefrom C
Colour and the Optical Properties of Materials 50
light, the refractive index of a transparent solid is positive.1 To treat the interaction of light with awide range of
materials, not just transparent ones but also those that are semitransparent or opaque, it is necessary to consider
the refractive index as a complex number.2 In this case, the complex refractive index N is written in the form:
N ¼ nþ ik
where n is the refractive index and k is the absorption index or extinction coefficient.3 (Note that the complex
refractive index is alsowrittenN¼ n� ik. The formalismchosen depends upon theway inwhich the lightwave
is described mathematically.) The pair of terms n and k are called the optical constants of a material. (Both of
these terms vary considerably with wavelength and, in practice, are by no means constant!)
The real part of the complex refractive index, n, accounts for the interaction of light with the nonabsorbing
part of the medium and the imaginary part of the complex refractive index, k, represents the absorptive
properties of themedium throughwhich the light travels. For a completely nonabsorbingmaterial (at least over
thewavelength range of interest), the absorption index k is zero. The behaviour of these materials is expressed
simply in terms of n, the ‘ordinary’ refractive index. In cases where absorption is present, the absorption
coefficient aa is related to k by:
aa ¼ 4pkl
ð1:7Þ
Materials that absorb over part of the electromagnetic spectrum are frequently transparent in other parts.
Silicon, for example, appears to be metallic in visible light but is transparent in the infrared. Water strongly
absorbs over much of the electromagnetic spectrum, making it a potent ‘greenhouse gas’, although it is
transparent in the visible. The refractive index of the oxide ceria, CeO2, is close to 2.35 over the visible
spectrum. It is a transparent phase with k¼ 0. At the onset of the ultraviolet, this changes rapidly and the
value of k reaches approximately 1.4 at 310 nm. This property makes ceria a possible component of
sunscreen creams.
The magnitude of the ray deviation when light enters a transparent medium is related to the index of
refraction by:
n ¼ sin�1sin�2
ð2:1Þ
�1 being called the angle of incidence and �2 the angle of refraction (Figure 2.2). This equation is known as
Snel’s law.4 The plane of incidence is the plane containing the incident ray and the normal to the surface. The
1 There is no theoretical reasonwhy the refractive index should always be positive, and research into negative refractive indexmaterials is
important. This is discussed in Section 2.10.2 Complex numbers are simply an ordered pair of numbers. One part is called the ‘real’ part and the other the ‘imaginary’ part. This
terminology frequently gives rise to confusion or a lack of understanding. In fact, one part could easily be called the ‘red’ part and the other
the ‘blue’ part instead of the ‘real’ part and the ‘imaginary’ part. In the present case, the real part gives the interaction with the transparent
aspect of the medium and the imaginary part describes the interaction with the absorptive part of the medium. The use of the complex
number formalism allows both aspects of the phenomenon to be analysed mathematically simultaneously.3 Note that the symbol k is used both in the complex refractive index, for the wavenumber of an electromagnetic wave (see Appendix
A1.1), for the Boltzmann constant and for the refractive coefficient (Section 2.4). In this book, the use of k is restricted to the absorption
index. The wavenumber will be written 1/l or 2p/l as appropriate, the Boltzmann constant as kB and the refractive coefficient as kr.4 The originator of this ‘law’ was Willibrord Snel van Royen (1591 1626). The spelling ‘Snell’ is incorrect, but well established.
51 Colours Due to Refraction and Dispersion
above equation is a special case of the more general relation:
sin�1sin�2
¼ n2
n1ð2:2Þ
for light passing from a medium of refractive index n1 to one of refractive index n2.
In effect, the refractive index of a transparent material is a manifestation of the fact that the light wave is
slowed down on entering a transparent material. This is due to the interaction of the light with the electrons
around the atoms which make up the solid. The absolute refractive index is given by:
n ¼ c
vð2:3Þ
where c is the velocity of light in a vacuum and v the velocity of light in the medium.
The frequency of the light does not alterwhen it enters a transparentmedium, and because of the relationship
between the velocity v and frequency n, that is:
nl ¼ v
it is possible to write:
n ¼ c
v¼ l0
lsð2:4Þ
where l0 is thewavelength of the light wave in a vacuum (which is close to that for air) and ls is thewavelengthin the transparent substance. It is thus seen that light has a smaller wavelength in a transparent material than in
air or a vacuum (Figure 2.3a).
glass
air
θ 1
θ 2
Figure 2.2 The refraction of a beamof light as it enters a block of a transparentmedium, such as glass,with a highrefractive index fromamediumof low refractive index, such as air. The light beamappears to bend at the interfaceby an amount given by Snel’s law
Colour and the Optical Properties of Materials 52
This can introduce confusion when the path of light rays through different materials has to be compared. To
overcome the difficulty it is useful to define the optical path or optical thickness [d] and distinguish it from the
real or physical thickness of a material d. The relationship is given by:
½d� ¼ nd ð2:5Þ
A slab of thickness d and refractive index 2n (optical thickness 2nd) will ‘contain’ twice as manywavelengths
as a slab of thicknessd and refractive indexn (optical thicknessnd) (Figure 2.3a). In effect, the optical thickness
is a measure of dimensions in terms of the wavelength of the light passing through it. Two different materials
with the same optical thickness contain the same number of wavelengths of the light which traverses each of
n = 1.0
(b)
(a)
(c)
n = 2.0 n = 1.0
d d d
n1
n2
dout of step
n1
d1 d2 d3 d4 d5
n1n2 n2n3
Figure 2.3 The effect of the refractive index on the wavelength of light. (a) The wavelength in a medium ofrefractive index2.0 is half that in amediumof refractive index 1.0. (b)Awavedivided into twoandpassing throughmaterials of refractive index n1 andn2 and of thickness dwill be out of step on emergence. (c) The optical thicknessof a series of slabs is the sum of the optical thickness, n1d1, etc. of the optical thickness of all of the slabs
53 Colours Due to Refraction and Dispersion
them. Similarly, if a beam of light is divided so that one part enters a slab of thickness d and refractive index n1while the other part enters a slab of thickness d and refractive index n2, on emerging from the slabs any point on
wave 1, say a crest, will have travelled n1d while the equivalent crest on wave 2 will have travelled n2d. In
general, the two waves will be out of step by (jn1� n2j)d, where only the absolute value of the difference in
refractive indices is important (Figure 2.3b). For several transparent materials traversed in sequence
(Figure 2.3c):
½d� ¼ n1d1 þ n2d2 þ n3d3 þ � � � ¼Xni¼1
nidi ð2:6Þ
In many crystalline materials the index of refraction varies with the direction of the beam of light. This is
taken further in Chapter 4. In the present chapter only those materials where the index of refraction is
independent of direction are considered. These are said to be optically isotropic and are typified by gases,
liquids, glasses and crystals with cubic symmetry. The refractive index can also vary locally in intense light
beams, as when a laser beam enters a transparent solid. This gives rise to a nonlinear refractive index, which is
most conveniently discussed in the context of crystal optics (Chapter 4).
2.2 Total Internal Reflection
2.2.1 Refraction at an interface
When light passes fromahigher refractive indexmaterial, such as glass, tooneof lower refractive index, suchas
air, the refraction causes the emerging ray to bend towards the interface (Figure 2.4a). As the angle of incidence
�1 at which the ray approaches the surface increases, the angle of the emerging ray �2 increases and the ray
approaches the surface (Figure 2.4b). At the critical angle �c the emerging ray actually travels exactly along the
surface (Figure 2.4c). If �c is exceeded, then no light escapes and it behaves as if it were reflected from the
undersurface (Figure 2.4d). This effect is called total internal reflection. The light is trapped in the high
refractive index medium.
The critical angle is given by:
sin�c ¼ n2
n1¼ n ðlowÞ
n ðhighÞ ð2:7Þ
which follows from Equation 2.2 when �2¼ 90�.Total internal reflection is not an ‘all-or-nothing’ process. When the incident light falls onto the interface at
normal incidence it is completely transmitted.As the angle of incidence approaches the critical angle,more and
more light is reflected back into the medium of higher refractive index and less and less is transmitted. At the
critical angle, this shift is complete and everything is reflected (but see the following section).
Total internal reflection from the glass air interface is used in prismatic binoculars and single-lens-reflex
cameras to ‘reflect’ light and to channel light in optical fibres (Section 2.9). It is the reason why a swimmer
underwater will see the air surface as a bright ‘hole’ in a surrounding dark continuum.
2.2.2 Evanescent waves
When waves encounter a barrier they often get around it in one way or another. In the present context, a light
wave that is totally internally reflected at an interface can ‘leak’ across it, although such an occurrence is
Colour and the Optical Properties of Materials 54
low refractiveindex n2
high refractiveindex n1
θ1
θ2
(a)
n2
n1
θ1
θ2
(b)
n2
n1
θ1 = θc
θ2
(c)
n2
n1θ1 θ1
(d)
Figure 2.4 (a, b)When light passes from amediumof high refractive index (such as glass) to one of low refractiveindex (such as air) the transmitted portion will be refracted into a path which lies closer to the interface betweenthe materials while the remainder is reflected below the interface. The amount transmitted gradually decreasesand the amount of reflected light gradually increases as the angle of incidence increases. (c) When the angleof incidence reaches the critical angle uc the vanishinglyweak transmitted raywill emerge along the surface itself.(d) For angles of incidence greater than the critical angle, total internal reflection will occur and all the light isreflected
55 Colours Due to Refraction and Dispersion
n2
n2
n1
n1
n1
θ1
θ1
θ1
θ1
θ1
evanescent wave
evanescent wave
(a)
(b)
coupling region
input signaloutput signal 1
output signal 2
input signal
output signal 1
output signal 2
thin layer of low refractive index cement
(c)
(d)
Figure 2.5 Evanescent waves. (a) An evanescent wave exists in the medium of lower refractive index when totalinternal reflection occurs. (b) Frustrated total (internal) reflection across a narrow gap. (c) Fused fibre couplers(schematic); signal transfer is by frustrated total reflection. (d) Cubic beam splitter (schematic); one signal isgenerated by total internal reflection and one by frustrated total reflection
Colour and the Optical Properties of Materials 56
forbidden in termsof rayoptics.The leakyportionof thewavedecays rapidly and is calledan evanescent5wave,
in contrast to the travelling, progressive or propagating waves described in Chapter 1.
The existence of an evanescent wave is discovered if the reflected and transmitted wave amplitudes
of the incident electromagnetic wave are determined by solution of the electromagnetic wave equations
for the appropriate boundary between two dielectric phases. The solution surprisingly shows that, when
total internal reflection occurs, the waves not only have amplitude within the dielectric of high
refractive index (corresponding to the total internal reflection) but also there is wave amplitude within
the dielectric of low refractive index. This belongs to the evanescent wave. The evanescent wave is
found to be a periodic wave, with the same period as the incident beam, but it decays exponentially on
moving away from the interface between the dielectric phases. In general, the decay is quite abrupt
and falls to negligible values within a couple of wavelengths, say approximately 1000 1500 nm
(Figure 2.5a).
Despite the fleeting nature of this wave, it has important properties that have been exploited in recent
years. First, the evanescent wave can leak back into another dielectric as long as it is close enough to be
within range of the rapidly decaying amplitude. Once inside this phase it will generate a propagating wave
similar to the one that originally produced the evanescent wave itself. In the case where the two dielectric
phases have the same refractive index, the new propagating wave is parallel to the original (Figure 2.5b). In
this sense the original propagating wave has jumped the gap between the two dielectric phases and so has
escaped total internal reflection a phenomenon called frustrated total (internal) reflection. This is of value
in several areas of photonics, especially optical data processing, as it allows information carried by a laser
beam to be introduced (coupled) into an optical fibre. There are a number of ways of building such a coupler.
The simplest, conceptually, is simply to place a short section of the information-carrying fibre in close
proximity to a section of the receiving fibre a fused-fibre coupler (Figure 2.5c). The evanescent wave
transfers the data from the source to the receiver. This technique can also be used as a nondestructive sensor
for signals in a fibre-optic waveguide. An information stream can also be divided into two streams with a
beam splitter (Figure 2.5d). In this technology (as an example), a cube of glass is divided along a diagonal
and the two halves are cemented together with a clear cement of low refractive index. Part of the signal beam
is totally internally reflected at the diagonal prism face while part is transferred across the cement gap into
the second prism.
This phenomenon is not just of research interest. Coupling using evanescent radio-frequency waves lies at
the heart of contactless charging of electronic devices, including batteries in heart pacemakers.
Evanescent waves have another important property. Waves, in general, carry information. A normal
light wave only carries information at a scale equal to or greater than the wavelength of the light itself.
This imposes severe restrictions on the performance of optical instruments and essentially means that the
resolution of an optical instrument, such as a microscope, is never more than the wavelength of the light used
(Section 6.4). It is one reasonwhy, in order to packmore electronic components onto silicon chips, lithographic
techniques are continuously trying to use shorter and shorter wavelength radiation. Similarly, DVDs have a
higher density of information content because they use shorter-wavelength red light, whereas CDs use longer
wavelength infrared light. Evanescent waves, it turns out, carry information at a scale less than thewavelength
of the light involved.Thismeans that, if suchwaves canbeused to forman image, the resolutionwill be less than
the wavelength of light. Such lenses have now been fabricated with resolution a fraction of the wavelength of
light (Section 2.10).
5 ‘Evanescent’ means ‘that which quickly passes away’ (OED).
57 Colours Due to Refraction and Dispersion
2.3 Refractive Index and Polarisability
To understand the relationship between refractive index and the atomic ormolecular structure of amaterial it is
necessary to recall that light can be treated as a varying electric field. If a static electric field is applied to an
insulating material, the internal components which carry a charge will try to line up with the field and the
material is said to become polarised. Thismeans that any positively and negatively charged species present are
rearranged slightly, with the positive charges moving in the field direction and the negative changes against it.
Hence, some parts of the material take on a slightly positive charge while an equal number of parts become
negative. The extent of this separation ismeasured as the relative permittivity (formerly dielectric constant) of
the substance. The magnitude of the relative permittivity is found to be closely related to that of the refractive
index.
The most important of the internal components that contribute to the relative permittivity are (i) the
permanent molecular dipoles present, (ii) the positive and negative ions present and (iii) the electrons
present. In a static electric field, existing molecular dipoles will reorient themselves in the field as much as
the surroundingswill allow (Figure 2.6a). Similarly, a static electric fieldwill cause the ions tomove slightly so
as to produce a net dipole moment (Figure 2.6b). The lightest component, the negatively charged
electron cloud surrounding the atomic nucleus, is also deformed by an external field to create a dipole
(Figure 2.6c).
If the electric field is not static, but consists of an alternating field, the dipoles, ions and electrons will try
to follow the changes in the field direction and move to and fro. (This effect is utilized in microwave ovens,
which bombard the contents with radiation at frequencies of about 1010Hz. As the applied electric field
changes direction, the dipoles, especially those associated with water molecules, reorient to and fro. This
continuous motion heats the food in the oven.) Motion is restricted for molecular dipoles, and when the
frequency of the applied electric field becomes much higher than that of microwaves (of the order of
1011Hz) any contribution of the dipoles is lost as the electric field is now changing too rapidly for them to
keep up. The magnitude of the polarisability thus falls to a lower plateau (Figure 2.7). When the frequency
of the field reaches that of near-infrared radiation (approximately 1014Hz) even the lightest ions can no
longer move to and fro quickly enough and their contribution to the polarisability is now lost. The magnitude
of the polarisability then falls to a lower plateau (Figure 2.7). The electrons, however, can follow the
oscillations of a varying electrical field even at visible and ultraviolet frequencies, and it is these which are
most important in colour production. This response of the electrons to an applied alternating electric field is
called the electronic polarisability.
Classical electromagnetic theory considers that the oscillating electronswhich are drivenby the electric field
of the light wave emit, in turn, an electromagnetic wave of the same frequency but at a reduced velocity
compared with that in a vacuum, the difference being measured as the refractive index. (This, though, is true
only for relatively low field strengths; see Chapter 3.) Theory relates the relative permittivity to the refractive
index thus:
n2 ¼ er ð2:8Þ
where er is the relative permittivity of the material and n is the refractive index. Strictly speaking, Equation 2.8
applies when the relative permittivity is measured at optical frequencies. Recorded values of relative
permittivity are often measured at frequencies far from those appropriate for light waves, and so the
other contributions to the relative permittivity may be important. In such cases, the relationship given
Colour and the Optical Properties of Materials 58
in Equation 2.8 does not hold well especially for solids. A more useful relationship is given by the Lorentz
Lorenz equation:
n2�1
n2 þ 2¼ Nae
3eoð2:9Þ
where N is the number of polarisable units in the material and ae is the electronic polarisability of each
(identical) unit. This equation is only applicable to homogeneous isotropic materials that do not contain
permanent dipoles or dipolar molecules. However, it is often taken to be approximately true for crystals of low
E=0E
+ + +
+
+
+
+ + + +
+
_
_ _ _
_
_
__ _
(a)
(b)
(c)
Figure 2.6 The effects of an external electric field E on the components of a solid. (a) Molecules with permanentdipoles (such as water, H2O) will align in the field as much as the structure will allow. (b) Ions which are evenlyspaced (such as in rocksalt, NaCl) tend to be displaced in the field so as to create a net dipole moment. (c). Auniform electron cloud around an atom or an ion (such as lead, Pb) tends to distort so as to produce a dipole. Thedisplacements have all been grossly exaggerated for clarity.Dipoles are indicated as arrowswith the arrowhead atthe positive end
59 Colours Due to Refraction and Dispersion
symmetry, provided that they do not contain permanent dipolar molecules. In this case, because the different
constituentswill showdifferent polarisabilities, these termsmust be summed over the i different units to get the
appropriate refractive index thus:
n2�1
n2 þ 2¼
XNiai3eo
Ingeneral, stronglyboundelectrons, trapped at atomicnuclei or in strong chemical bonds, have a lowelectronic
polarisability and this leads to a low refractive index. Loosely bound electrons, such as outer electrons on large
atoms or lone pair electrons, are highly polarisable and so will yield materials with a larger refractive index.
This effect is well known. For example, lead oxide, PbO, contains large ions with a highly polarisable lone
pair on each Pb2þ ion. When lead oxide is added to ordinary glass the highly polarisable Pb2þ ions (which
occupypositions between the chains of SiO4 tetrahedramakingup the structure) have the effect of considerably
increasing the refractive index of the glass. Flint glass (which contains significant amounts of lead oxide),
therefore, is prized and used as ‘cut glass’ and lead ‘crystal’ because the higher refractive index gives a more
attractive appearance to the articles. Table 2.1 shows the effect of added PbO on the refractive index of three
different flint glasses.
Equation 2.9 is valid formost ordinary electric field strengths, and that applies to the electric field component
of light. However, at high field strengths, such as those generated by intense laser light, it may not hold. In this
case the refractive index of the material will be changed by the field. Solids in which this change can be made
permanent are called photorefractive materials.
2.4 Refractive Index and Density
As the previous section demonstrated, electronic polarisability and the number of polarisable atoms present are
important factors that contribute to themagnitude of the refractive index.Gases, because of their low densities,
dipolar contr bution
ionic contribution
electroniccontribution
Pol
ariz
abili
ty
Frequency / Hz
far mid nearinfrared
microwave
9 10 11 12 13 14 15 16 17 1810 10 10 10 10 10 10 10 10 10
dipoles + ions + electrons
ions + electrons
visible
electrons
Figure 2.7 A simplified schematic illustration of the contribution of permanent dipoles, ions and electrons to thetotal polarisability of a material as the frequency of the applied field is increased. The contribution due topermanent molecular dipoles is lost when the field frequency reaches themicrowave region and the contributionof the ions is lost at near-infrared frequencies. Only the effect of electronic polarisability occurs at opticalfrequencies
Colour and the Optical Properties of Materials 60
have refractive indices close to unity.However, although small, thevariation of the density of air as a function of
temperature is the source of mirages and related visual effects (Section 2.5).
Densely packed arrays of atoms in liquids and solids have a higher refractive index than gases. The refractive
index increases with density, as can be confirmed from the Lorentz Lorenz equation, (Equation 2.9). The
number of scattering centres per unit volume can be expressed as a density, so that the equation can bewritten:
n2�1
n2 þ 2¼ rrs
where n is the refractive index,r is the densitymeasured at the same temperature as the refractive index and rs is
a constant the specific refraction. Themolar refractionRmof acompound is defined as rs times themolarmass,
so that:
ðn2�1ÞVm
n2 þ 2¼ Rm
Although the refractive indices of most simple compounds are known, it is sometimes useful to estimate the
refractive indexofmore complex or hypotheticalmaterials. One of themost successfulways of doing this is via
the Gladstone Dale equation, which combines density and, indirectly, polarisability terms. It is especially
useful for complex oxides, for which the Gladstone Dale formula can be written:
n ¼ 1þ rð p1kr1 þ p2kr2 þ p3kr3 þ � � � Þor
n ¼ 1þ rX
pikri ð2:10Þwhere r is the density of the complex oxide and the terms pi and kri are defined below. The assumption
underlying the formula is that the refractive index of a complex oxide is made up by adding together the
contributions from a collection of simple oxides, oxide 1, oxide 2 and so on, for which optical data are known.
The polarisability is taken into account by allocating to each of the simple oxide components a factor kr called
the refractive coefficient, an empirically determined constant. The amount of each oxide is taken into account
Table 2.1 Refractive indices
Substance Refractive indexa n Substance Refractive indexa n
Vacuum 1.0 (definition) Dry air, 1 atm 15 �C 1.000 27Water 1.3324 Na3AlF6 (cryolite) 1.338b
MgF2 1.382b Fused silica (SiO2) 1.460KCl (sylvite) 1.490 Crown glass 1.522Extra light flint glassc 1.543 NaCl (halite) 1.544Flint glassc 1.607 MgO (periclase) 1.735Dense flint glassc 1.746 Al2O3 (corundum) 1.765b
ZrO2 (zirconia) 2.160b C (diamond) 2.418CaTiO3 (perovskite) 2.740 TiO2 (rutile) 2.755b
aA value appropriate to the yellow light emitted by sodium atoms (the sodium D lines; Chapter 7), with an average wavelength 589.3 nm, is given.bThe refractive index varies with direction; the average value is given.c The flint glasses contain significant amounts of lead oxide, PbO, as follows: extra light flint, 24mass% PbO; flint, 44mass% PbO; dense flint,
62mass% PbO.
61 Colours Due to Refraction and Dispersion
bymultiplying the refractive coefficient by its weight fraction p in the compound. A number of values of kr for
use in the Gladstone Dale formula are given in Table 2.2.
The ruleworkswell and usually gives answerswithin about 5%.Note, however, that the value obtained is an
average refractive index. Many oxides have refractive indices which vary according to crystallographic
direction. The Gladstone Dale relationship ignores this feature.
The equation can also be used to determine a value of either density or average refractive index for unknown
polymorphs of simple oxides. For example, the Gladstone Dale equation for the polymorphs of SiO2 is:
n ¼ 1þ 0:208r
and for the polymorphs of TiO2 it is:
n ¼ 1þ 0:393r
Provided the density of each polymorph is known, its average refractive index can be found and vice versa.
2.5 Invisible Animals, GRINs and Mirages
As mentioned earlier (Section 1.16), the initial premise of the H.G. Wells tale The Invisible Man is that if the
refractive index of the body matches that of air, the body would become invisible, and this principle is widely
used by aquatic animals to avoid predators. These creatures, typified by jellyfish, have a gelatinous bodywhich
has a refractive index very close to that of water. This renders them more or less invisible. Indeed, some
creatures succeed so well at this that they cannot be detected until the observer is only centimetres away.
The relationship between density and refractive index can be exploited quite simply tomakematerialswith a
lower thannormal refractive index.Apractical use of this idea, conceivedmore than50years ago, is to fabricate
the material in the form of foam. Provided that the air bubbles are smaller than thewavelength of light they are
not resolved and the light encounters a medium in which the effective refractive index lies between that of air
and that of the foam matrix. For example, silica containing pores of the about 4 nm in diameter has been
fabricated with refractive index of 1.23, compared with the refractive index of a nonporous film, 1.457. Such
structures are used in antireflection coatings (Section 3.7).
The refractive index of porous materials depends upon the pore shape and distribution, as well as the phase
that fills the pore. The polarisation and wavelength of the light are also important variables. To a first
approximation, the refractive index of the whole, nt, can be assessed as that of a simple mixture:
nt ¼ nmVm þ npVp ð2:11Þ
where nm is the refractive index of the material that can be regarded as the matrix, Vm is the volume fraction of
the matrix, np is the refractive index of the material filling the pore and Vp is the volume fraction of the pore
phase. The volume fractions are given by:
Vm ¼ volume of matrix phase
total volume
and
Vp ¼ volume of pore material
total volume
Colour and the Optical Properties of Materials 62
Table 2.2 Refractive coefficients for some oxidesa
Oxide krb Oxide kr Oxide kr Oxide kr Oxide kr Oxide kr Oxide kr Oxide kr Oxide kr
H2O 0.340Li2O 0.307 BeO 0.240 B2O3 0.215 CO2 0.221 N2O5 0.242Na2O 0.190 MgO 0.200 Al2O3 0.207 SiO2 0.208 P2O5 0.183K2O 0.196 CaO 0.210 TiO2 0.393 Ga2O3 0.170 GeO2 0.167 As2O5 0.162Rb2O 0.128 SrO 0.145 Y2O3 0.170 ZrO2 0.211 Nb2O5 0.268 MoO3 0.237 In2O3 0.130 SnO2 0.143 Sb2O5 0.153Cs2O 0.119 BaO 0.128 La2O3 0.148 Ta2O5 0.151 WO3 0.171 PbO 0.133 Bi2O3 0.139
a Data from: J. A. Mandarino, Can. Mineral., 14, 498–502 (1976); 16, 19–174 (1978); 17, 71–76 (1979); 19, 441–450 (1981).b These values give correct results if the density is in g cm 3. To use density in kgm 3, multiply the values of kr by 10 3.
63
Colours
Dueto
Refractio
nan
dDisp
ersion
that is:
Vm þVp ¼ 1
If there are several different types of pore material present then the equations can be extended; for example:
nt ¼ nmVm þ np1Vp1 þ np2Vp2 þ � � �
where
Vm þVp1 þVp2 þ � � � ¼ 1
The equations can be written in terms of the weight fraction of the componentsW1, etc. by substituting via the
density equation:
V1 ¼ W1
r1
Although the refractive index of most isotropic materials is uniform, transparent solids with a varying
refractive index are manufactured for a number of purposes. These are called graded-index (GRIN) materials.
AGRIN solid can bemade by arranging a nonuniformdistribution of dopants throughout the bulk. Thus,GRIN
optical fibres (Section 2.9) are made by the diffusion of GeO2 into SiO2. The optical path of a ray in a GRIN
material will follow a curve, the form of which depends upon the refractive index distribution in the material
(Figure 2.8a). Cylinders of materials with an appropriate refractive index variation can then act as a focusing
lens (Figure 2.8b). The optical path length [d ] in amaterialwith a smoothly varying refractive index is obtained
by replacing the summation sign in Equation 2.6 by an integral, which for a path between points a and b is:
½d� ¼ðb
a
nsð pÞ dp
where ns( p) is the refractive index in the substance at position p and dp is a small element of path (Figure 2.8c).
GRINsolids are not uncommon innature. The lens of the human eye is an example. It is built up of layerswith
a refractive index which varies from about 1.41 at the centre to 1.39 at the outer layers. The atmosphere also
has a continuouslyvarying refractive index, from1.0 in space to approximately1.000 292 for dry air and light of
wavelength 589.3 nm (the mean of the sodium D lines, Chapter 7) at 0 �C and 1 atm pressure. The refractive
indexwill varywith pressure, temperature and the content of other gases, especiallywater vapour. In particular,
local density fluctuations can have a considerable effect on the refractive index and give rise to a variety of
meteorological phenomena, such asmirages. In general, a temperaturegradient in the air changes the refractive
indexof the air and sets upan ‘air lens’.Because the lens is imperfect, the images reaching the eyeare imprecise.
For this reason, the human imagination has constructed a variety of fanciful explanations for the apparitions
observed, including the familiar water pools and more arcane Atlantis myths.
The idea of using GRIN optics was evolved in some night-flying insects some millions of years ago. A
number of animals, notablymoths, have eyeswell adapted to night vision. The surface structure of these ‘moth
eyes’ is bumpy and acts like a GRIN layer with a refractive index between that of the surroundingmedium and
the substrate. The net result is to cut down or eliminate surface reflection (see Section 3.7.3).
Colour and the Optical Properties of Materials 64
2.6 Dispersion and Colours Produced by Dispersion
As mentioned above, the refractive index is far from constant. The variation of the refractive index of a
transparent material with wavelength is known as dispersion (Figure 2.9). Dispersion can be formally defined
as the slope of the refractive index n versus the wavelength l curve, dn/dl. In general, the index of refractionincreases as the wavelength decreases, so that the refractive index of red light in a material is less than that of
violet light. This situation is referred to as normal dispersion. Although the normal dispersion of many
materials is rather small, it is important to include itwhen calculating the optical properties of lenses and similar
high-quality optical components. Anomalous dispersion is found in the region of absorption bands in the
material, when transparency is lost. These absorption bands are associated with transitions from one energy
configuration (often the ground state) to higher energy levels.
Formany transparent materials a good representation of the variation of refractive indexwith wavelength in
the visible region is given by Cauchy’s equation:
n ¼ Aþ B
l2þ C
l4
dp
p
n
(a)
(b)
(c)
Figure 2.8 GRIN materials. (a) The path of a ray in a GRIN is generally curved. (b) Suitable refractive indexvariation can produce a focusing effect. (c) The path length in a GRIN material is specified by the integral of therefractive index ns(p) at point p over the length of the path dp
65 Colours Due to Refraction and Dispersion
whereA,B andC are empirically determined parameters. For lens design Cauchy’s equation is not sufficiently
precise, and amore accurate formula, which gives the refractive index of glasses in thewavelength range 365
2300 nm to high degree of fidelity, is the Sellmeier equation:
n ¼ 1þ B1l2
l2�C1
þ B2l2
l2�C2
þ B3l2
l2�C3
� �1=2
where the wavelength l is in micrometres and B1 B3 andC1 C3 are the Sellmeier constants appropriate to the
glass. The Sellmeier equation can also be applied to transparent crystals. For those that are not isotropic,
different equations must be obtained for each of the crystallographically independent directions.
normal dispersion
anomalousdispersion
absorptionband
Wavelength
Ref
ract
ive
inde
x
(a)
1.50
1.45
Wavelength / nm Wavelength / nm
Ref
ract
ive
inde
x
(b)
fused silicaglass
400 500 600 700
1.80
1.85
Refr
act
ive in
dex
(c)
400 500 600 700
aluminium oxide (corundum,sapphire)
Figure 2.9 The variation of refractive index with wavelength. (a) Schematic dispersion curve for a transparentmaterial. Anomalous dispersion occurs close to energy transitions from a lower to a higher energy level.(b) Dispersion curve for fused silica glass. (c) Dispersion curve for corundum, Al2O3. In the case of corundumthe refractive index depends upon direction and average values are plotted
Colour and the Optical Properties of Materials 66
TheAbb�eV-value, orAbb�enumber,writtenVd, awidelyusedmeasureof thedispersionof a transparent solid,
is given by:
Vd ¼ nd � 1
nF � nC
where nd is the refractive index of the material at a (yellow) wavelength, 587.56 nm (the helium d-line; see
Chapter 7),nF is the refractive indexof thematerial at a (blue)wavelength of 486.1 nm(thehydrogenF line) and
nC is the refractive indexof thematerial at a (red)wavelength of 656.3 nm (the hydrogenC line). The reciprocal
of the Abb�e number is often called the dispersive power.
The dispersion of refractive index is the cause of the formation of a spectrum when white light is passed
through a glass prism (Figure 2.10a). Snel’s law tells us that, for a given angle of incidence �i, sin �r is inversely
white light silica glass prism
θiθr
whitered
violet
violet
redwhite
white white
lens
red
violet
α
δwhite
(a)
(b)
(c)
Figure 2.10 (a) The refraction of white light by a silica glass prism. For silica glass, the refractive index for violetlight is greater than for red light,whichdisperses the light to forma spectrum. (b) The edgeof a simple lens acts as aprism and so causes chromatic aberration. (c) The deviation of light by a thin prism provides a usefulmodel for thedispersion from a thin lens. Each colour will be deviated by a different amount d¼ (nl 1)a
67 Colours Due to Refraction and Dispersion
proportional to the refractive indexn, so that asn increases�r decreases and the raydeviatesmore.Red light then
tends to be the least deviated and violet light themost. The higher the dispersion, thewiderwill be the spectrum.
Exactly the same effect is found in simple lenses. The edge of the lens is approximately prism shaped and
dispersion causes the image to become coloured at the periphery of the field of view, (Figure 2.10b). This effect
is known as chromatic aberration. If the lens is considered to be a thin prism, the angle of deviation of the rays dwill be given by:
d ¼ ðnl�1Þa
where nl is the refractive index appropriate to the colour and a is the small angle at the top of the prism
(measured in radians) (Figure 2.10c). Chromatic aberration is avoided in expensive lenses by using
combinations of glasses chosen so as to compensate for the effects of dispersion in each component. Such
compound lenses are called achromats.
Dispersion is responsible for the flashes of colour, called fire, that are such an important feature of diamonds.
The production of such fine colours is due to the combination of very high refractive index and high dispersion.
The stones are ‘cut’ (actually cleaved) so as to producemany facets, each of which can act as a tiny prism, thus
greatly enhancing the display of fire as the gem moves (Figure 2.11).
2.7 Rainbows
The rainbow is one of the most beautiful examples of colour produced by refraction (Figure 2.12). Most
frequently seen, when the observer’s back is to the sun, is a single bright arc called the primary rainbow
(Figure 2.13a). The colour violet is always innermost, at an angle of 41� to the incident beam. The colours
proceed through indigo, blue, green, yellow, orange to red on the outside of the arc at an angle of 43� to the
incident beam. The locus of the various angles generates the arc seen and the observer appears to be at the apex
of a cone with an average semi-vertical angle of about 42� (Figure 2.13b).A careful examination of the sky near a rainbow will often, but not always, show many other features,
including a fainter secondary rainbow at an angle of about 50� and various supernumerary arcs inside the
primarybow.The secondarybow ishigher in the sky than theprimarybow, ismuch less intense than the primary
and the colour sequence is reversed with respect to the primary bow. Also, though not so easily seen, is the fact
whiteviolet red
cut diamond
Figure 2.11 The combination of high refractive index and high dispersion in cut diamonds gives these gemstonesthe ability to produce spectral colours, known as fire
Colour and the Optical Properties of Materials 68
that the skybetween the twobows is noticeably darker than the skybelowor above the bows. This darker region
is known as Alexander’s dark band (Figure 2.14).
Although a complete description of all of these features is complex, the (first-order) explanation of the
primary rainbow is relatively simple.6 It is producedby a single reflection from inside a raindrop (Figure 2.15a).
The point where the incident light beam falls on the drop can be defined by the impact parameter IP, which lies
between 0 and 1.0 and is expressed as a fraction of the drop radius R (Figure 2.15b). From the geometry of the
refraction and internal reflection it is seen that the deviation d of a ray (Figure 2.15c) is given by:
d ¼ 180þ 2i�4r
sin i ¼ IP
sin r ¼ IP
n
where i is the angle of incidence, r is the angle of refraction and n is the refractive index ofwater. It is possible to
calculate values of d as a function of IP (Table 2.3).
Figure 2.12 Primary and secondary rainbows. The secondarybow is higher in the sky than the primarybowand isnoticeably fainter. The colour sequence in the secondary bow is reversed comparedwith that in themuch brighterprimary bow. The region of sky between the two bows is noticeably darker than the sky above and below the twobows. This is Alexander’s dark band
6 The discussion here is termed ‘first order’ because raindrops are not spherical as they fall through the air. They have aflattenedbottomand
have a shape more like a ladybird than a ball.
69 Colours Due to Refraction and Dispersion
It is seen that the deviation falls gradually and reaches a minimum at about 138�. In fact, there is a
considerable bunching of the rays in the region of thisminimumdeviation,which is the reasonwhy the rainbow
appears to be bright.
The exact minimum can be determined analytically by the differentiation of the formula for the deviation of
the ray. The result is found to be:
IP ¼ 0:862 38; d ¼ 137:6�
where the refractive index of water has been taken to be 1.33.
The refractive index of natural water depends upon any dissolved impurities, the temperature and the
wavelength of the light. For accurate results, precise values for the refractive index of water are required.
Reasonable values to take are:
n ðred; 20 �CÞ 1:3310; n ðviolet; 20 �CÞ 1:3440
red
violet
primaryrainbowrain
4143
observer
sunlight
~138°sunlight
raindrop
rainbow
(a)
(b)
Figure 2.13 The geometry of a primary rainbow. (a) To observe a rainbow, sunlight must come from behind theobserver. (b) In the main or primary bow, each raindrop generates a cone of refracted and reflected light; theviolet light appears to come from a cone of semi-vertical angle 41� and the red light from a cone of semi-verticalangle 43�
Colour and the Optical Properties of Materials 70
Using these, it is found that the minimum deviation of red light is 137.63� and violet light is 139.35�, giving anangular width of 1.72� for the bow. As the different colours diverge on leaving a drop, they do not all enter theeye. In fact, each colour observed comes from a different raindrop, appropriately positionedwith respect to the
observer.
Normally, all of a drop is illuminated in sunlight and so every impact parameter occurs over every angle in the
vertical plane, so that, as described above, the refracted and reflected light is deviated into a cone with a semi-
vertical angle of approximately 42� andwithmost intensity concentrated into the outer surface.Thismeans that
each of the colours that enters the eye from a rainbow originates in a separate arc of water drops. For the same
reason, no two observers ever see exactly the same rainbow. Each person sees only a unique part of the raindrop
curtain that subtends the correct angles with respect to the observing eye.
The secondary bow is caused by two internal reflections (Figure 2.16a and b). The geometry drawn in this
figure allows one to conclude that:
d ¼ 360þ 2i�6r
sin i ¼ IP
sin r ¼ IP
n
red
red
violet
violet
primaryrainbow
secondaryrainbowrain
4251
observer
Alexander’sDark Band
138°
129°
sunlight
Figure 2.14 The positions of the primary and secondary rainbows. The region between them is Alexander’s darkband
71 Colours Due to Refraction and Dispersion
where d is the deviation of a ray, i is the angle of incidence, r is the angle of refraction, IP is the impact parameter
and n is the refractive index of water (Table 2.4). To compare this with the primary bow, the incident ray to be
considered needs to enter the drop below the centre line. The relevant angle of deviation for comparison with
that of the primary bow is (360� d )� (Figure 2.16c and d).
Table 2.4 shows that the deviated rays cluster, this time with a minimum deviation of close to 230� (or amaximum value of (360� d ) of 129�). The angle that the bow subtends to the observer is close to 51�, whichindicates that the secondary bow will lie above the primary bow (Figures 2.12 and 2.13).
sunlight
violetred
43°41°water
drop
observer
(a)
(b)i
r
rr
r
i
i
IP
d(c)
Figure 2.15 The reflection and refraction that a ray of light undergoes in forming a primary rainbow. (a) Theprimary bow is produced by a single reflection within each raindrop combined with dispersion of the light due tothe variation of refractive indexwithwavelength. (b) The refraction and reflectionwithin a rain drop. (c) The angleof deviation of the reflected ray
Colour and the Optical Properties of Materials 72
Table 2.3 The deviation of a ray which forms a primary rainbowa
IP i/deg r/deg d/deg
0.1 5.74 4.31 174.20.2 11.54 8.65 168.50.3 17.46 13.04 162.80.4 23.58 17.50 157.20.5 30.00 22.08 151.70.6 36.87 26.82 146.50.7 44.43 31.76 141.80.8 53.13 37.00 138.30.9 64.16 42.59 138.00.95 71.81 45.58 141.3
aCalculated using n¼ 1.33.
i
r
rr
rr
ri
IP
d
(a)
(b)
Figure 2.16 The reflection and refraction that a ray of light undergoes in forming a secondary rainbow. (a) Thesecondary bow is produced by a double reflectionwithin each raindrop combinedwith dispersion of the light dueto the variation of refractive index with wavelength. (b) The angle of deviation of the reflected ray. (c) As (a) withan impact parameter below the centre of the drop. (d) The angle of deviation of the reflected ray in (c)
73 Colours Due to Refraction and Dispersion
i
r
rr
rr
ri
IP
360 - d
(c)
(d)
Figure 2.16 (Continued)
Table 2.4 The deviation of a ray which forms a secondary rainbowa
IP i/deg r/deg d/deg (360 d )/deg
0.1 5.74 4.31 345.6 14.30.2 11.54 8.65 331.2 28.20.3 17.46 13.04 316.7 43.30.4 23.58 17.50 302.2 57.80.5 30.00 22.08 287.5 72.50.6 36.87 26.82 272.8 87.20.7 44.43 31.76 258.3 101.70.8 53.13 37.00 244.2 115.70.9 64.16 42.59 232.8 127.20.925 67.67 44.07 230.9 129.10.95 71.81 45.58 230.1 129.90.975 77.16 47.15 231.4 128.6
aRefractive index of water taken as 1.33.
Colour and the Optical Properties of Materials 74
As in the case of the primary bow, the value of the maximum deviation can be determined precisely be
differentiation. The result is found to be:
IP ¼ 0:950 73; 360� d ¼ 129:9�
where the refractive index of water has been taken to be 1.33.
Usingmore precisevalues for the refractive indexofwater,wefind that themaximumdeviation of red light is
129.63� and for violet it is 126.52�, giving an angularwidth of 3.11� for the bow. It also confirms that the colours
in the secondary boware reversedwith respect to theprimarybow. In addition, because a largeamount of light is
not reflected internally, each additional internal reflection decreases the intensity of the bow considerably. A
secondary bow is, therefore, rarely very intense, being only about 43% as bright as the primary.
The calculations shown in the tables indicate that light rays have a minimum deviation from 180� to about138� for the primarybowandamaximumdeviation from0� to about 129� for the secondarybow.No light entersthe space lying between 129� and 138�, the region between the primary and secondary bows. Observation will
show that this region is indeed darker than the sky below the primary bow and above the secondary bow. It is
referred to as Alexander’s dark band (Figures 2.12 and 2.13)
Other rainbows are produced by more internal reflections within the water drops.
Although higher order rainbows are too faint to see in the sky, they can be observed in the laboratory. Three
internal reflections produce a ternary bow and so on, and up to a dozen orders can be seen indoors. The formula
for the deviation d of a light ray after m internal reflections, generating the mth order rainbow, is:
d ¼ 2ði� rÞþmð180� 2rÞThe maximum or minimum deviation of the rays is given by:
½ðmþ 1Þ2 � 1�ðIPÞ2 ¼ ðmþ 1Þ2 � n2
where m is the order of the bow, n is the refractive index of water and IP is the impact parameter.
Rainbows are found to be polarised even though sunlight, the cause of the bow, is unpolarised. This is
explained in Chapter 3.
2.8 Halos
Ahalo is a rather pale diffuse ring of colour, often red on the inner side and blue on the outer side, seen around a
bright object such as the sun or moonwhen partially obscured by thin, high cloud (Figure 2.17a). Halos are not
as spectacular as rainbows and can often bemissed during casual observation. The total angular width of a halo
is 44�. Once again, refraction and dispersion are responsible for the colour, but in this instance the refractionoccurs within randomly oriented hexagonal ice crystals in the upper atmosphere. The commonest halo, the 22�
halo, is a ring subtending an angle of 22� to the observer’s eye (Figure 2.17b). Dispersion of the blue
wavelengths ismore than that of the reds, and so thehalo is red internally andblue violet externally.Figure2.18
shows a halo-like arc of colour due to refraction and dispersion in ice crystals in stratospheric clouds.
2.9 Fibre Optics
2.9.1 Optical communications
In the early years of the twentieth century, data transmission was mainly by way of electrical impulses
sent along copper wires, by radio waves or by manual transport of written records. In the early years of the
75 Colours Due to Refraction and Dispersion
(a)
(b)
red
violet
whitelight
60°
ice crystal
~22°
violet redsun
Figure 2.17 (a) A halo is sometimes seen around a bright object such as the sun when partially obscured by highcloud. The total angularwidth of a halo is 44�. (b) The halo is formed by refraction of light through a random arrayof hexagonal ice crystals. The average deviation of the light in each crystal is 22�, with the deviation of the red rayabout 1.5� less than that of the blue (The angles are exaggerated for clarity)
Figure 2.18 A halo-like arc of colour produced by reflection and dispersion when sunlight falls upon prismaticice crystals in the upper atmosphere
Colour and the Optical Properties of Materials 76
twenty-first century, optical data transmission along glass fibres is normal. Tyndall, in 1870, first showed that
light could be transmitted along a jet ofwater even if the pathwas curved. The reason for the transmission is that
total internal reflection at the water air interface prevents the light from escaping. Shortly after this, the
transmission of light within a glass rod was also demonstrated. As glass can be easily dawn into fine fibres it
soon became clear that bundles of fibres could be used for the remote illumination and viewing of inaccessible
or dangerous areas. The subject of light transmission along thin fibres of glass, plastic or other transparent
materials is referred to as fibre optics.
Until themid 1950s fibre optics remained something of a curiosity. This was because the transparency of the
glass was poor, due to the presence of impurities.Moreover, different colours tended to separate because of the
dispersion of the refractive index of the glass, resulting in strong chromatic aberration and producing
unsatisfactory images. Despite these problems, use was made of short lengths of glass fibres for decorative
purposes and lighting.However, during theseyears themain use of glass fibre bundleswas inmedicine,making
the examination of internal organs possible without surgery.
The situation changed in the mid 1960s. The impetus was for rapid communication of large amounts of data
along secure lines, and for this glassfibresweredeemed ideal.Asa spin-off fromdevelopments incommunication
technology, uses of fibre bundles as long-distance light guides, for remote viewing of inaccessible objects or
dangerous devices, in medical imaging and in many applications of laser technology has also burgeoned. The
developmentswhich led to these changes are described,with emphasis onmaterials properties. Information upon
engineering aspects will be found in the sources listed in this chapter’s Further Reading.
2.9.2 Optical fibres
Data is carried in optical communications by a series of pulses of light encoded so that information can be
stored and retrieved. The transparent optical wave carrier used for communications is silica (SiO2) glass.
The light pulses launched into the fibre are constrained to stay within the fibre by total internal reflection.
Thus, the core of the fibre, along which light travels, must possess a higher refractive index than the outer
surface of the fibre. Moreover, a glass surface at which the total internal reflection is to occur is easily
damaged and needs protection. Both of these objectives are met by providing a surface cladding of lower
refractive index glass, compared with the core of the fibre. The core and the cladding make up a single glass
fibre (Figure 2.19). The cladding should not be confused with a plastic protective covering, which has no
optical role to play.
The starting point for a fibre is a high-purity silica tube containing only a few parts per million of hydroxyl
ions. In order to create a fibre with the correct refractive index profile, the silica tube is rotated and heated
while a gas consisting of various amounts of silicon tetrachloride (SiCl4), germanium tetrachloride (GeCl4),
phosphorus oxychloride (POCl2), Freon (a chlorofluorohydrocarbon, typified by CF2Cl2) and oxygen is
plastic cover
claddingn ~ 1.48
coren ~ 1.5
opticalfibre
Figure 2.19 The structure of a silica optical fibre. The core has a higher refractive index than the cladding, whichserves to confine light rays by total internal reflection. The fibre is covered with a protective plastic coating
77 Colours Due to Refraction and Dispersion
allowed to flow through its centre (Figure 2.20a). At the temperatures of the tube, 700 900 �C, the gases
decompose:
SiCl4 þO2 ! SiO2 þ 2Cl2
GeCl4 þO2 !GeO2 þ 2Cl2
4POCl2 þ 3O2 ! 2P2O5 þ 4Cl2
The result is that a ‘soot’ consisting of a mixture of silicon, germanium and phosphorus oxides forms on the
inside of the tube. As the heating zone traverses the tube the ‘soot’ merges with the tube to form a glass inner
coating about 10 mm thick.
The refractive index variation is achieved in two stages. Of the order of 12 to 32 layers are deposited initially
to form an inner coating inside the tubewhich will become the cladding on the fibre. This has a composition of
SiO2 containing 0.1 to 1% P, F and Ge, the fluorine incorporation arising from the Freon gas present. The
refractive index is close to that of pure SiO2. Following this stage the material which will ultimately form the
core is deposited. This is achieved by depositing 4 to 10 layers of material with an overall composition
somewhere between the limits (Ge,P,F)0.06Si0.94O2 to (Ge,P,F)0.3Si0.7O2, depending upon final use. The
replacement of Si by the heavier Ge and to a lesser extent P increases the refractive index over that of the first
layers laid down.
When sufficient layers have been formed the temperature is raised enough to cause the tube to collapse under
surface tension. The result is a solid rodwith a centre of higher refractive index glass surrounded by a region of
lower refractive index glass enveloped in the original silica of the tube. This solid rod is called a preform
(Figure 2.20b).
To transform the preform into a fibre by a process called fibre drawing, the end of a preform rod is softened to
near to its melting point. Under these conditions glass has the property that it can be pulled out and will form a
long fibre. Surprisingly, the refractive index profile of the preform is preserved exactly in the fibre even though
the preform diameter of 15 100mm is drawn down to approximately 0.1mm.
reactants:SiCl4GeCl4POCl2FreonO2 rotate and heat
(Ge, P, F) (Si) O1-x x 2
silica tube
collapse
preform rod
(a)
(b)
core
cladding
Figure 2.20 The formation of a preform rod for silica fibre production. (a) Reactive gases are passed through thecentre of a hot, rotating silica tube, where they decompose to form layers of (Ge,P,F)xSi1�xO2. (b) After reaction,increased heating causes the tube to collapse to the perform with the cladding (lower dopant content) and core(higher dopant content) at the centre
Colour and the Optical Properties of Materials 78
2.9.3 Attenuation in glass fibres
Attenuation describes the loss of light intensity as the signal is transmitted along the fibre. This is of major
concern, as any degradation of the signal must be minimized. The loss is defined as:
loss ðdBÞ ¼ �10log10½PðxÞ=Pð0Þ�
where P(0) is the power input, at x¼ 0, and P(x) is the power at a remote point x. The attenuation is defined as
the loss per kilometre; thus:
attenuation ¼ �10log10½PðxÞ=Pð0Þ�x
The units of attenuation7 are decibels per kilometre. Ordinary window glass has an attenuation of about
100000 dB km 1.Attenuation, like dispersion, varieswithwavelength. The spectral response of a fibre defines
the way in which the fibre attenuation changes with the frequency of the radiation being transmitted.
Attenuation is caused by a combination of absorption and scatteringwithin the glass.Extrinsic attenuation
is due to poor processing or fabrication techniques, and may be due to artefacts such as bubbles, particles,
impurities and variable fibre dimensions. These problems have been eliminated in modern optical fibre
manufacture. Intrinsic attenuation is a property of the pure material itself, and cannot be removed by
processing. It is the ultimate limit on the performance of the fibre and mainly arises from two factors:
Rayleigh scattering (Section 5.2) and lattice vibrations.
Rayleigh scattering is due to small density fluctuations in the glass. This variation is an inevitable feature of
the noncrystalline state and cannot be removed by processing. As Rayleigh scattering is proportional to l 4,
where l is thewavelength of the optical pulse, the effect is more important for short-wavelength radiation. For
anyparticularglass,most of the factors affectingRayleigh scattering are constant and cannot be easily changed.
However, materials with a low refractive index and glass transition temperature tend to exhibit low Rayleigh
scattering.
Absorption due to lattice vibrations, referred to as phonon absorption, occurs when the lattice vibrations of
the solidmatch the energy of the radiation. This occurs for infraredwavelengths, and converts the signal energy
into heat. It is a function of the mass of the atoms in the glass and the strength of the chemical bonds between
them and results in a decrease in the transparency of the glass at long wavelengths.
Absorption due to electronic transitions (Chapter 7), mostly at high energies and associated with ultraviolet
wavelengths, does not figure significantly in present-day applications, but may become important if shorter
signal wavelengths are to be used in the future. The dependence on wavelength of absorption due to electronic
transitions can often be expressed by a formula of the type:
Electronic absorption ¼ B1expB2
l
� �
where B1 and B2 are constants relating to the glass used and l is the wavelength of the radiation.
By1979, the best silica fibres showedonly intrinsic attenuation andhad a loss of about 0.2 dB km 1at 1.5mmwavelength. The current industry standard is slightly less than this, at about 0.16 dB km 1.
Despite this achievement, new fibre materials are constantly being explored. The absorption maxima
caused by lattice vibrations can be manipulated both by changing the strength of the chemical bonds
between the components and by changing the mass of the atoms linked. For example, silica, with rather light
7 The unit of loss is the decibel, dB; the base unit, the bel, is almost never used.
79 Colours Due to Refraction and Dispersion
atoms and strong bonds, transmits satisfactorily only to about 2.5mm and strongly absorbs radiation in the
8 15mm wavelength range. One group of materials with great promise are glasses primarily made from
fluorides of zirconium (ZrF4), barium (BaF2) and lanthanum (LaF3), called ZBLAN glasses. Although the
chemical bonding in fluorides is regarded as being as strong as in oxides, the heavy atoms move the
absorption maximum to wavelengths of 17 25mm. In addition, they are found to have an attenuation which
is only one-hundredth of that of silica. They are, therefore, enormously attractive for long-distance high data
density communications.
Desirable characteristics are also found in the arsenic triselenide (AsSe3) glasses, composed of very heavy
atoms and linked by weak bonds. These do not absorb strongly until 44 46 mm and so show great potential for
the transmission of infrared radiation. Unfortunately, the chemical difficulties associated withmaking fluoride
and selenide glasses have not yet been solved and they are not currently used in long-distance commercial
applications.
2.9.4 Chemical impurities
The preparation of high-purity glass was one of the most important advances needed to allow fibre-optic
communications to become a reality, and enormous strides in improvement of glass purity have been made
since the earliest times (Figure 2.21). In the original glass fibres, transitionmetal impurities caused difficulties
because they absorb strongly in the visible. The gravest problem was iron, present as Fe2þ , and it is this ionwhich giveswindowglass its greenish tint (Section 7.7). Even as low a concentration as 1 ppmof iron can result
1
10
100
310
410
510
610
710
500 800 1200 1600 1800 1900 1960 1970 1980 1990
drying
chemical vapour deposition
high purity melting
optical fibres
highest quality optical glass
high quality lenses
ordinary “window glass”
1000BC BC AD
Year
Atte
nuat
ion
/ (dB
/ km
)
Figure 2.21 The quality of glass through the ages. Recent improvements have been in response to the needs ofoptical fibre manufacturers. Currently, silica fibres are routinely made with an attenuation of less than0.2 dB km�1
Colour and the Optical Properties of Materials 80
in an attenuation of 15 dB km 1. The presence of transition metal cations was overcome by the preparation of
silica usingveryhighpurity chemicalsmade available by the semiconductor industry.Atpresent it is possible to
purchase silica with no significant transition metal ion impurities present.
The most important impurity in silica fibres today remains hydroxyl ( OH) (Figure 2.22). Hydroxyl arises
from water or hydrogen incorporation into the glass during fabrication. Flames used to melt silica are rich in
both of these impurities, and any silica melted in a gas flame will be heavily contaminated by hydroxyl. The
main absorption peaks are at 950, 1240 and 1390 nm and an impurity OH level of 1 ppm can give an
attenuation of the order of 102 dB km 1 at 1.4 mmsignal wavelength. It is clear, therefore, that silica for optical
fibre use must be melted in electric furnaces in a dry atmosphere to eliminate hydroxyl as much as possible.
Despite careful processing, fibres currently in production still contain significant amounts of hydroxyl, which
remains an important source of attenuation.
When hydroxyl absorption is superimposed upon the intrinsic attenuation of a pure silica glass, of which
Rayleigh scattering is themain contributor at shorter wavelengths and phonon absorption (infrared absorption)
at longer wavelengths, it is seen that the best window for signals is close to 1500 nm (Figure 2.23).
2.9.5 Dispersion and optical-fibre design
A short pulse of light launched into a fibre will tend to spread out, due to dispersion (Figure 2.24a c). When
dispersion was discussed above it was defined in terms of the change of refractive index with wavelength. In
optical fibres, the dispersion is defined as the delay between the arrival time of the start of a light pulse and its
finish time relative to that of the initial pulse. It is measured at half peak amplitude (Figure 2.24d). If the initial
pulse has a spread of t0 seconds at 50% amplitude and the final pulse a spread of tx seconds at 50% amplitude
after having travelled x kilometres, the dispersion is given by:
dispersion ¼ tx�t0
x
The units of dispersion in optical fibres are nanoseconds per kilometre.
700 800 900 1000 1100 1200 1300 1400
Wavelength / nm
0.01
0.10
1.0
10.0
-1-1
Atte
nuat
ion
/ [dB
km
ppm
(O
H-)
]
Figure 2.22 The attenuation introduced into a silica fibre by the presence of OH�
81 Colours Due to Refraction and Dispersion
Dispersionwill obviously arise if the light used is notmonochromatic.An initially sharp pulse consisting of a
group of wavelengths will spread out as it travels down the fibre because the refractive index depends on
wavelength, and thus the light of different wavelengths will travel at different speeds. This effect is known as
wavelength dispersion.
The first optical-fibre communications installations used LEDs (Chapter 10). These had a spectral width of
about 35 nmcentreduponawavelengthof0.82 mm,which sets a severe limit on the rateofdata transmissionand
forced the introduction of lasers with a spectral width of 2 nm or less for signal sources.
Unfortunately, even with completely monochromatic light, pulse spreading can still occur, due to the
fact that the radiation can take various modes (paths) through the fibre. A ray that travels along the axis of a
fibre will travel less far than one which is reflected many times on its journey (Figure 2.25). (In fact, the
dispersion that results cannot be properly understood in terms of the transmission of light rays, and the
various modes are better described in terms of the allowed wave patterns that can travel down the fibre.)
The resultant pulse broadening, due to the various modes present, is called modal (or intermodal )
dispersion.
Inorder toovercomemodal dispersionanumberofdifferentfibre typeshaveevolved.Theearliest fibreswere
called stepped-index multimode fibres. These fibres have a large core region, allowing many modes to
propagate (Figure 2.26a). The ray labelled H in Figure 2.26a is known as a high-order mode and the ray L is a
low-ordermode. Stepped-indexmultimode fibres are easy tomake and join, but have a lower performance than
those described below. Stepped-index fibres are adequate for short-distance communications but not for
medium- or long-distance links.
The first advance on stepped-index fibres was theGRIN fibre. In this design, the refractive index of the fibre
varies smoothly from high at the centre to low at the periphery of the core region. The refractive index gradient
means that light travels faster and faster as it approaches the edge regions of the fibre. The velocity of mode A
will be fairly constant,while thevelocityofmodeBwill vary smoothly from lowest at thefibre centre togreatest
near to the fibre edge (Figure 2.26b). The differences in path length between high-order and low-order modes
1.0
0.5
0.3
0.2
0.11.1 1.2 1.3 1.4 1.5 1.7 2.0
Wavelength / μm
-1Lo
ss /
dB k
m
intrinsic loss
hydroxylabsorption
Rayleighscattering
silica infraredabsorption
Figure 2.23 The attenuation in a silica fibre due to intrinsic attenuation and OH� impurities
Colour and the Optical Properties of Materials 82
Am
plitu
de
Distance
0 0
0.50.5
1.0
1.0
t0 tx
(a)
(b)
(c)
(d)
Figure 2.24 The gradual dispersion (or spreading) of a series of initially sharp light pulses (a), as theymove alongan optical fibre (b, c). (d) The dispersion of a light pulse is given by (tx t0); the 50% amplitude peak widths afterthe pulse have travelled 1 km
ray 1 (mode 1)ray 2 (mode 2)
core
cladding
Figure 2.25 The allowed paths that light can take through an optical fibre are called modes; although drawn asray paths they are really alternative light wave patterns
83 Colours Due to Refraction and Dispersion
are thus minimized by this velocity variation. GRIN fibres reduce modal dispersion by a factor of about 25 to
about 1 ns km 1.
Even this improvement is insufficient for long-distance communications. For best resultsmonomode fibres
are required (Figure 2.26c). The number of possible modes is reduced simply by reducing the diameter of the
core. When the core diameter reaches 10mm or less, only one mode can propagate and, in principle, modal
dispersion is zero for these fibres.Monomode fibres, therefore, have a high performance but are harder tomake
and join.
Althoughfibres incommercial usearemadeof silicaglass, it is notperfect.Thedispersion is lowest at 1.3 mm,
buttheminimumattenuationoccursat1.5mm,leadingtosomesacrificeofperformanceirrespectiveof thesignal
wavelengthchosen.Thesearchfornewmaterials toresolvethisconflictcontinuesinmanyresearch laboratories.
2.10 Negative Refractive Index Materials
2.10.1 Metamaterials
For all practical purposes involving light, the refractive indexof a transparent solid is positive.However, there is
no theoretical reason why this should always be so, and much current interest centres upon materials which
display a negative refractive index. Thesematerials are often called negative-indexmaterials (NIMs). In fact, it
~125 μm
~125 μm
~140 μm
~100 μm
~65 μm
~10 μm
cladding
cladding
core
core
claddingcore
Refractiveindex n
H
L
AB
(a)
(b)
(c)
Figure 2.26 Types of optical fibre: (a) stepped-index fibre; (b) GRIN fibre; (c) monomode fibre
Colour and the Optical Properties of Materials 84
has been known for any years that the refractive index of many materials is negative for X-rays. However, the
effect with X-rays isminute and the current interest is inmaterials that show an appreciable negative refractive
index at frequencies near to those of the visible spectrum. A good deal of research is at present underway in
order to create structures that will lead to a negative refractive index.
The current awareness of the potential of negative refractive index materials started with an exploration of
their use in the construction of perfect lenses (see Pendry in this chapter’s Further Reading). The problem in
creating a NIM is implicit in Equation 2.8, Section 2.3:
n2 ¼ er ð2:8Þ
Now this is an approximation valid for transparent phases at optical frequencies. The more correct
equation is:
n2 ¼ ermr
where er is the relative permittivity and mr the relative permeability of the material. Roughly speaking, this can
be interpreted by saying that both the electric and magnetic dipoles in the material contribute to the refractive
index. For all ordinary transparentmaterials er is positive andmr is approximately unity, leading toEquation2.8.
This is because, at optical frequencies, the light electrons are able to keep up with the electric field component
(Section 2.3), while the magnetic dipole contribution, which arises with unpaired electron spins and electron
orbital moments, is unable to react to the rapidly changingmagnetic field component. This results in a value of
approximately unity for mr and leads to the contention in Chapter 1 that only the electric field component of the
electromagneticwaveneeds to be consideredwhen optical properties are paramount. Tomake a negative index
phase for any particular frequency range, it is ideally necessary to obtain a material in which both er and mr arenegative.
It turns out that it is easy to obtain a material with a negative value of er. Metals containing free
electrons, especially the noble metals, copper, silver and gold, show this, as do some ferroelectric
compounds over some ranges of the electromagnetic spectrum. The problem is to obtain a matching
negative value of mr.This is not possible in a single natural material, but has been possible in artificially created composite
structures. These combinations are known as metamaterials. A metamaterial is a periodic structure (like a
crystal) with artificially designed component ‘atoms’ that give the material the desired properties. The
important structural features must be smaller than the wavelength of the electromagnetic radiation that the
material is designed to influence, and so it is not surprising that the first negative index metamaterials were
designed to act on microwaves with wavelengths of the order of centimetres. An early metamaterial ‘crystal’
was composed of copper wires in a cubic grid, together with open copper rings (split-ring resonators) at the
nodes (Figure 2.27). In this design, the copper wires provide the negative permittivity component and the split
rings the negativepermeability component. Thewavelength of radiation thatwill respond to this negative index
structure is approximately the same as the spacing of the pairs of split-ring resonators. Many other designs of
metamaterial are currently being explored, including ‘fishnet’ structures. In this design, a periodic array in
which thin films of silver sandwich a thin layer of an insulator such as alumina (Al2O3) provides a negative
permeability (Figure 2.28a). When two sets of these arrays are arranged perpendicular to each other a fishnet
structure is formed (Figure 2.28b). One part of the net provides the negative permeability ‘atoms’ and the
continuous threads of the net provide the negative permittivity wires. These structures can be fabricated with
spacings such that they respond to optical-frequency radiation. Photonic crystals (Chapter 6), structures made
up of ordered ‘crystal-like’ arrays of pores or similar ‘defects’, can also be designed to show a negative
refractive index at optical frequencies.
85 Colours Due to Refraction and Dispersion
(a)
silver
insulator
silver
substrate
(b)
Figure 2.28 (a) An array of silver strips separated by an insulating layer is able to form a negative refractive indexsolid. (b) Two arrays as in (a) make up a ‘fishnet’ metamaterial
split ring resonator
copper wire
Figure 2.27 Schematic diagram of a metamaterial made up of pairs of metallic split rings mounted on copperwires to form a crystal-like structure that shows negative refractive index for radiation with a wavelengthapproximately equal to the spacing of the rings
Colour and the Optical Properties of Materials 86
2.10.2 Superlenses
Negative refractive index materials are being explored in a number of contexts, including ‘invisibility’ or
cloaking screens,which act so as todivert incident radiation in such awayas to render theobject ‘invisible’ to an
observer. However, the application ofmost relevance to the subjectmatter of this book concerns superlenses or
hyperlenses. Normal lenses, those in telescopes, microscopes, cameras and other optical instruments, have a
resolution limited to approximately thewavelengthof the imaging light, due to diffraction (Section 6.4). That is
to say, any feature smaller than thewavelengthof light cannot be imagedoptically. (Image formation canalsobe
treated in terms of information theory, in which case the information content of the image is limited to detail of
the order of thewavelength of light or greater.) This has always been an impediment for scientists whowish to
understand physical, chemical or biological processes at the molecular or cellular level and led directly to the
invention and use of, among other techniques, electron microscopy, because the wavelength of an electron
beam is considerably less than that of a light beam. The limitation on imaging detail is also of considerable
importance with respect to data storage and transmission, as the desire to make ever smaller components on
silicon chips and related circuitry has continually been hampered by this imaging constraint.
The use of negative refractive indexmaterials has allowed these barriers to be overcome, at least in part, and
lenses able to image details considerably smaller than thewavelength of the incident light have been produced.
Lenses that can bypass the diffraction limit are called ‘superlenses’ or ‘hyperlenses’.
The first remarkable result with respect to this phenomenon is that a plane slab of negative refractive index
material can actually form an image. This is because Snel’s law continues to operate, but the refracted beam
now lies on the same side of the normal as the incident beam rather than the opposite side, which can be seen by
applying Equation 2.2:
n1sin�1 ¼ n2sin�2
When n2 is negative, �2 will be negative (Figure 2.29a and b). This in turn implies that a slab of negative
refractive index material can form an image lens shapes are not needed (Figure 2.29c).
The second remarkable point is that use of a negative refractive index slab can form an imagewith a superior
resolution to that of a conventional lens. A conventional lens forms an image using progressive waves, the
normal light waves. The detail that these images contain is limited by diffraction and is roughly of the order of
the wavelength of the light used (Chapter 6). Now, evanescent waves carry more detailed information and if
they can also contribute to image formation then the resultant ‘information content’ or resolution should be
better than that of progressivewaves alone, and be able to reveal detail that is smaller than thewavelength of the
light.
This is possible and comes about in the following way. A slab of NIM acts so as to increase the amplitude
of an evanescent wave in an exponential fashion, which is opposite behaviour to that of a normal material
(Figure 2.29d). This means that the image formed by the slab, as described above, can also include
information provided by the evanescent wave and thus may have a superior resolution to that shown by an
ordinary lens.
The question, therefore, is can a slab ofNIMbe used in this conceptually simpleway? The answer is yes, and
this (third) remarkable feature of NIMs is that, despite the limitations on the fabrication of these substances, a
simple layer of silver metal will act as a superlens. This comes about in the following way. It is found that the
equations for the reflection and transmission of light are independent of the permeability of the solid for light
that is polarised in the plane of incidence (the p-wave or transverse magnetic wave; Chapter 4). As pointed out
above, silver is a metal that shows a negative refractive index in the optical region that arises from the
permittivity component of the refractive index. Hence, use of the correctly polarised incident light should give
the required sub-wavelength resolution, a feat that has been successfully achieved.
87 Colours Due to Refraction and Dispersion
n positive n negative
air air
θ1 θ1
θ2 θ2
(a) (b)
(c)
n1(+ve) n2 (-ve)
ray path if n2 is positive
(d)n1 (+ve) n2 (-ve)
evanescent wave if is +ven2
Figure 2.29 (a) Snel’s law for a normal material. (b) Snel’s law for a NIM. (c) Image formation by a slab of NIM.(c) Image formation by a slab of NIM. (d) The amplitude of an evanescent wave in a NIM
Cr layer
Silica
Hyperlens:16 (Ag / Al2O3) layersobject
magnified image
illumination (365 nm)
microscope(a)
Figure 2.30 (a) Magnifying hyperlens composed of 16Ag/Al2O3 cylindrical layers; schematic. The object isinscribeduponaCr layer and themagnified image can be viewed andphotographedusing amicroscope. (b)A sub-diffraction limit image obtained with the lens [(a) and (b) from Science, Far-Field Optical Hyperlens MagnifyingSub-Diffraction-Limited Objects by Z. Liu, et al., 315, 5819, 1686 Copyright (2007). Reprinted with permissionfrom AAAS.]
Colour and the Optical Properties of Materials 88
There is a drawback to this technique. The image is formed close to the foil, in a region where the
evanescent wave is appreciable. This limits usefulness considerably, but can be of value in the fabrication of
nano-scale devices using contact methods of detail transfer so that the distance between the object and image
is minimal.
This limitation can be bypassed using more complex metamaterial design. A curved metamaterial,
composed of alternating cylindrical layers of alumina and silver, acts as a magnifying lens (Figure 2.30a).
The image is still formed close to the exit surface of the lens, but the image is magnified by the material
curvature. Provided that the magnification is greater than the diffraction limit of an ordinary optical
microscope, it can then be magnified optically at will. This technique has allowed images of features about
1/10 the wavelength of the imaging illumination to be photographed directly (Figure 2.30b). This rapidly
advancing field is at a very exciting stage.
Further Reading
The effect of refraction on vision in water, especially from the point of view of a fish or of a fisherman, is
described by
J. D. Walker, Sci. Am. 250 (March), 108 113 (1984).
Details of archer fish vision are given by
S. Temple, N. S. Hart, N. J. Marshall, S. Collin, Proc. R. Soc. Lond. Ser. B 277, 2607 2615 (2010).
Figure 2.30 (Continued)
89 Colours Due to Refraction and Dispersion
Evanescent waves are introduced clearly in
N. Braithwaite (ed.), Electromagnetism, Book 3, Electromagnetic Waves, The Open University, Milton
Keynes, 2006.
See also
W. Knoll, Mater. Res. Soc. Bull. 16 (July), 29 39 (1991).
The relationship between refractive index, polarisability and the Gladstone Dale (and other) equations is
given in
F. D. Bloss, Crystallography and Crystal Chemistry, Holt Rinehart and Winston, New York, (1971),
Chapter 11.
R. E. Newnham, Structure Property Relations, Springer, Berlin, 1975, Chapter 5.
J.A.Mandarino,Can.Mineral.14, 498 502 (1976);16, 19 174 (1978);17, 71 76 (1979);19, 441 450 (1981).
Animal eyes and the relevance of graded-index optics are described in
L. P. Lee, R. Szema, Science 310, 1148 1150 (2005).
The moth-eye antireflection surface structure is described by
P. Vukusic, J. R. Sambles, Nature 424, 852 855 (2003).
The physics of the rainbow is given by
C.B.Boyer,TheRainbow:FromMyth toMathematics,MacMillanEducationLtd, Basingstoke. (1987). (This
gives a very full history of the rainbow and its explanations, from ancient times. The first edition (1959) only
has black-and-white photographs. The 1987 edition has colour illustrations.)
V. Khare, H. M. Nussenzveig, Phys. Rev. Lett. 33, 976 980 (1974).
H. M. Nussenzveig, Sci. Am. 236 (April), 116 127 (1977).
J. D. Walker, Am. J. Phys. 44, 421 433 (1976); Sci. Am. 237 (July), 138 144 (1977).
A discussion of rainbows, halos and other dispersion colours found in nature is given by
D.K.Lynch,W.Livingston,ColorandLight inNature,CambridgeUniversityPress,Cambridge,1995,Chapter4.
Fibre optics is covered in detail by
J. Hecht, Understanding Fibre Optics, 3rd edition, Prentice Hall, Upper Saddle River, NJ, 1999.
The evolution of fibre-optic communications can be appreciated by reading the following series of articles:
W. S. Boyle, Sci. Am. 237 (August), 40 48 (1977).
A. Yariv, Sci. Am. 240 (January), 54 62 (1979).
M. G. Drexhage, C. T. Moynihan, Sci. Am. 259 (November), 76 81 (1988).
E. Desurvire, Sci. Am. 266 (January), 96 103 (1992).
G. Stix, Sci. Am. 284 (January), 68 73 (2001).
D. J. Bishop, C. R. Giles, S. R. Das, Sci. Am. 284 (January), 74 79 (2001).
D. J. Blumenthal, Sci. Am. 284 (January), 80 83 (2001).
Negative refractive index materials, superlenses and hyperlenses can be reviewed by consulting
J. B. Pendry, Phys. Rev. Lett. 85, 3966 3969 (2000).
J. B. Pendry, D. R. Smith, Sci. Am. 295 (July), 43 49 (2006).
Various authors in Mater. Res. Soc. Bull. 33 (October), (2008).
Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, Science 315, 1686 (2007).
There are a number of demonstrations of relevance to this chapter, including refraction by prisms, water
droplets and NIMs, available at http://|demonstrations.wolfram.com/index.html.
Colour and the Optical Properties of Materials 90
3
The Production of Colour by Reflection
. Why are soap bubbles coloured?
. How do antireflection coatings on lenses function?
. How can perfect mirrors be made from transparent
materials?
Reflection is a commonplace phenomenon and the appearance of a solid is often dominated by reflection. As
well as modifying the perceived colour of a body in terms of surface gloss, reflection as such can give rise to
a surprising range of colours. The most vivid of these are associated with the presence of reflection by thin
transparent films. Bright colours are often seen in soap bubbles, and close examination of transparent insect
wings shows that these can show areas which are beautifully coloured. Casual observation also reveals that the
colours haveametallic aspect (due to a considerable specular component) and seem tovarywith thedirectionof
viewing and with the thickness of the film. They are said to be iridescent. In this chapter the origin of these and
other colours due to reflection is explored. Attention is confined to reflection by more or less transparent
insulating solids (dielectrics in older literature). Metals are considered in a later chapter. However, recall that
the refractive index varies with wavelength, and metals are transparent at some wavelengths and in these
circumstances the conclusions of this chapter will then apply.
It is necessary tomention that polarisation of light is important in reflection. In this chapterwe are concerned
mainly with the colours produced by unpolarised sunlight, and the refinements needed to account for the
polarisation of thewaves are considered inChapter 4.This objective is aidedbymainly considering light falling
perpendicularly onto surfaces, for which the polarisation direction of the wave becomes redundant.
Colour and the Optical Properties of Materials Richard J. D. Tilley
� 2011 John Wiley & Sons, Ltd
3.1 Reflection from a Single Surface
3.1.1 Reflection from a transparent plate
When light falls onto a smooth, thick transparent plate such as a slab of glass, so that the lower surface can be
ignored, someof itwill be reflected and some transmitted (Figure 3.1). For a smoothmetal plate almost all of the
lightwill be reflected, as the amount of transmitted light is negligible. Forbothmaterials, thewell-known lawof
reflection is:
�1 ¼ �3
where �1 (also �i or i) is the angle of incidence and �3 (also �r or r) the angle of reflection. The plane of incidencecontains the incident ray, the reflected ray and the normal to the reflecting surface. In Figure 3.1 this is the plane
of the page.
The amount of light reflected from such a surface depends upon the polarisation of the light (Chapter 4). For
a polished, thick plate and light at normal incidence (i.e. perpendicular to the surface) the polarisation can be
ignored and the coefficient of reflection r, defined such that if awave of amplitude E0 falls upon the surface thenthe amplitude of the reflected wave is rE0, is given by:
r ¼ n0�ns
n0 þ ns
where n0 and ns are the refractive indices of themedia on the two sides of the boundary in the direction inwhich
the light travels (Figure 3.2). The eye detects irradiance changes rather than amplitude changes, and so it is the
more convenient to work with the reflectivity or reflectance R:
R ¼ r2 ¼ n0�ns
n0 þ ns
� �2
This is because the irradiance I0 is proportional to the square of the amplitude E20. The reflected irradianceR(I0)
(Figure 3.2) is then proportional to r2E20.
θ1 θ3
θ2
air
glass
Figure 3.1 Light falling on a transparent plate such as a slab of glass will be partly reflected and partly refracted.The angle of incidence u1 will be equal to the angle of reflection u3
Colour and the Optical Properties of Materials 92
Remember that becausendependsuponwavelength, the coefficient of reflectionand the reflectivitywill vary
across the spectrum.
When light travelling through a medium of low refractive index (such as air) is reflected at the surface of
a substance of higher refractive index (such as glass), r is negative. This signifies a phase change ofp radians onreflection, which means, in terms of a light wave, that a peak turns into a trough, and vice versa, whenever the
refractive indices at the interface are in the sequence low/high (Figure 3.3).
The reflectivity R for a transparent plate of refractive index ns in air is:
R ¼ 1�ns
1þ ns
� �2
¼ ns�1
ns þ 1
� �2
ð3:1Þ
ε0 I0 R I0r ε0
n0
ns
Figure 3.2 Reflection of a beam of light perpendicular to a transparent surface. The amplitude of the incidentbeam is E0 and incident irradiance is I0. The reflected amplitude will be given by rE0 and the reflected intensity byRI0. The angles have been exaggerated for clarity
incident ray
ray after reflection
phasechange
n0 n1
Figure 3.3 A phase change of p radians is introduced in a ray reflected at a surface of higher refractive index,which means that a peak changes to a trough and vice versa
93 The Production of Colour by Reflection
For ordinary glass, with a refractive index of approximately 1.5, R is about 0.03 0.04; that is, about 3 4 %.
Although this may seem to be a rather insignificant amount, it is noticeable in everyday life. Reflections from
windows and from the glass over a painting are frequently annoying. Moreover, it is too high for specialist
purposes, such as high-performance lenses and optical components, so these are given antireflection coatings,
discussed below. In addition, this small degree of reflectivity turns out to be an essential component in the
production of colour through interference by thin films.
3.1.2 Data storage using reflection
In essence, the active data storage layer (or layers) on CDs, DVDs, HD-DVDs and Blu-Ray discs is reflective.
Data is stored by making small dots on the recording layer that have a different reflectivity to the background.
The writing process involves decreasing the reflectivity and the reading process involves detecting these
reflectivity differences.
The discs that contain permanent read-only data, such as those sold in stores, are producedwith physical pits
in the surface. The pitwill have a different reflectivity to the higher surrounding plateau. The smaller the pit, the
more data can be stored on the disc. Pit size is more or less controlled by the wavelength of the light used to
record the data. The CD, introduced in 1983, used infrared wavelengths (780 nm) generated by semiconductor
lasers (Section 10.9). The pit size minimum was 0.83mm and the track pitch (the separation between lines of
pits) was 1.6mm. Great effort was expended in moving towards red light, wavelength 650 nm, which allowed
for smaller pit dimensions. Thus, the DVD, introduced in 1996, using this wavelength, has a minimum pit size
of 0.4mm and a track pitch of 0.74 mm. Recently, there has been a move to blue light, wavelength 405 nm,
allowing for smaller pits and greater data storage, resulting in the transition to HD-DVD and Blu-ray
technology.
Recordable CDs (CD-R) use similar reflectivity differentials to record data. In this instance, a writing laser
in the computer marks small spots in a layer of polymer containing a dye. These spots are detected as
changes in reflectivity when scanned by the reading laser. Rewritable CDs (CD-RW) make marks in a layer
of a crystalline Ag In Sb Te alloy. The exact composition of the alloy is carefully chosen so as to yield
optimum recording and erasing facilities under the influence of the lasers normally present in home
computers. The alloy is crystalline as prepared. When heated by a pulse from the writing laser beam, to
a temperature of approximately 700 �C, the alloy melts locally and cools too rapidly to crystallise, thus
ending in an amorphous (glass-like) state. This has a lower reflectivity than the surrounding crystalline
surface. Erasure is carried out by heating the amorphous data spot with a laser pulse to about 200 �C, whichallows solid-state crystallisation to occur, restoring the original surface reflectivity. The writing process can
then be carried out once more.
The marks on a CD or DVD are comparable in size to the wavelength of light. This means that light
rays can interfere and produce iridescent colours. These are discussed in terms of diffraction in
Chapter 6.
3.2 Interference at a Single Thin Film in Air
Part of a monochromatic beam of light incident upon the top surface of a homogeneous thin film of refractive
index nwill be reflected. The remainder will enter the film and be repeatedly reflected from the bottom surface
and the underside of the top surface. At each reflection some of the light will escape to produce additional
reflected and transmitted rays (Figure 3.4a). The consequence of this repeated reflection and transmission is to
produce the bright colours seen on thin films of many types.
Colour and the Optical Properties of Materials 94
1 2 3
air
air
n
nn
1 2 3
d
(a)
(b)
ray 1 (incident ray )
ray 2 (reflected)
ray 3 (refractedand reflected)
out of step =retardation
top surface(c)
Figure 3.4 The reflection and transmission of a ray of light incident on a transparent film in air. (a) A number ofreflected and transmitted beams occur due to repeated reflection at the top and bottom faces of the film. (b) Atnormal incidence (the angles of incidence and reflection have been changed from 90� for clarity), ray 3 will havetravelled further than ray 2 by apath difference of2nd. (c) Thewavesmaking up rays 2 and 3will be out of step dueto the combined effects of a phase change on reflection and the path difference
95 The Production of Colour by Reflection
3.2.1 Reflection perpendicular to the film
When the light beam is perpendicular to the surface the complexity of dealing with polarisation is avoided, so
that the analysis of the phenomenon is simplified. In this case, part of the light seenbyanobserver above thefilm
will have been reflected at the top surface (ray 2). In addition, somewill have travelled through thefilmandbeen
reflected from the bottom surface before reaching the observer (ray 3) (Figure 3.4b). As the reflectivity is rather
small, about 4% for a glass surface, the first reflected ray and the first ray transmitted through the glass and then
reflected from the lower surface are ofmost importance. For the present, the other transmitted and reflected rays
will be ignored.
Because of the difference in the paths taken by the two rays, thewaveswill be out of step. In addition, because
ray 2 is travelling through a medium of low refractive index and is reflected at a low high refractive index
interface awavepeakwill turn into a trough andviceversa.Thiswill not happen to ray3because it is reflected at
a high low refractive index surface (Figure 3.4c).
On leaving the thin film thewavesmaking up rays 2 and 3 can now interfere,whichwill cause the film to look
either dark or bright. The effect is easily understood. Ray 3 will have travelled further than ray 2 by a twice the
film thickness, 2d. However, the physical thickness does not give the mismatch between the wave crests and
peaks of these two rays. This is givenby the additional optical path travelled by ray3, in this case 2nd,wheren is
the refractive indexof thefilmandd is the physical thickness. Thepathdifference (or retardation) between rays
2 and 3 is equal to the optical path difference between the two rays, so that:
p ¼ 2nd
The appearance of a thin film when viewed by reflection at normal incidence will depend upon this path
difference. If the path difference p considered in isolation is equal to an integral number of wavelengths
then the waves will be exactly in step as they travel away from the surface. Adding in the phase change of
half a wavelength for ray 2 will make it out of step with ray 3 by this amount as they leave the surface.
The consequence is that destructive interference will occur between ray 2 and ray 3. The film will,
therefore, appear dark. In general, the film will appear dark under irradiation with light of wavelength l0 inair when:
p ¼ 2nd ¼ ml0 ðm ¼ 1; 2; 3; . . .Þ minimum ðdarkÞ
In a similar way, a path difference p between ray 2 and ray 3 equal to a half-integral number of wavelengths
will cause the two rays to be exactly out of step.Adding in the half-wavelength phase change for ray 2willmake
them exactly in step. The filmwill then appear bright, because constructive interferencewill occur. In general,
the film will appear bright when:
p ¼ 2nd ¼ ðmþ 1
2Þl0 ðm ¼ 1; 2; 3; . . .Þ maximum ðbrightÞ
At other path differences the film will appear to have an intermediate tone, depending upon the exact phase
difference between ray 2 and ray 3.
When a tapered or wedge-shaped film is viewed by monochromatic light at normal incidence, some
thicknesses will be appropriate for constructive interference and some for destructive interference. The film
will then appear to be crossed by a series of bright and dark bands (Figure 3.5a and b).When the thickness of the
film is considerablybelow12l0 thefilmwill appear dark, because thepathdifference in thefilm,p¼ 2nd,will not
be sufficient to counteract the change of phase of the ray reflected from the upper surface, and destructive
interference occurs. As the film thickness increases, it will eventually reach the stage where the destructive
Colour and the Optical Properties of Materials 96
interference is replaced by constructive interference, and a bright band appears, centred at p ¼ 1
2l0. Thereafter,
bright and dark bands will succeed each other, each at an interval of p ¼ 1
2l0.
Precisely the same effect will be obtained for an air wedge between two inclined transparent plates
(Figure 3.5c). Bright fringes will be observed as at intervals of:
p ¼ 2nd ¼ ðmþ 1
2Þl0 ðm ¼ 1; 2; 3; . . .Þ maximum ðbrightÞ
In cases where the wedge or air gap is not uniform, the fringes follow contours of equal thickness. This gives
a dynamic impression of surface contours.
3.2.2 Variation with viewing angle
When the light beam is at anangle to the surface, polarisation effects become important. These arenot severe for
angles of incidence of up to approximately 25� to the surface (see Figure 4.4) andwill be ignored here. The pathdifference p between rays 2 and 3 becomes (Figure 3.6):
p ¼ 2nd cos�2
The analysis now follows that given in the previous section. If p is equal to awhole number ofwavelengths then
thefilmwill appear dark, due to the combined effect of pathdifference and change of phase of ray2on reflection
at the surface. Thus:
0
λ/2
1λ
3λ/2
2λ
5λ/2
3λ
7λ/2
4λ
dark
dark
dark
dark
dark
bright
bright
bright
bright
(a) (b)
air gap(c)
Figure 3.5 Interference in a wedge shaped film: (a) film profile; (b) bright and dark reflection bands resultingfrom the interference ofmonochromatic light viewednormal to thewedge fromabove; (c) an air gap between twotransparent plates behaves in a similar way to a wedge
97 The Production of Colour by Reflection
p ¼ 2nd cos�2 ¼ ml0 minimum ðdarkÞ
For the same reason, if the path difference turns out to be a half wavelength, reinforcement will occur and
we find:
p ¼ 2nd cos�2 ¼ ðmþ 1
2Þl0 maximum ðbrightÞ
When viewed in monochromatic light, the regions of the film which look bright or dark will change with the
viewing angle. If the light is normal to thefilm, then cos�2 ¼ cosð0Þ ¼ 1and theequations reduce to thosegiven
in the previous section. Naturally, the same is true of wedges and wedge-shaped air gaps.
3.2.3 Transmitted beams
The analysis given in the preceding section for interference between the reflected beams can be repeated for the
transmitted beams (Figure 3.7). The path difference between beams 4 and 5 is:
p ¼ 2nd cos�2
In this case, there is no extra phase change, as all reflections take place at the high low refractive index
interfaces. Thus, when:
p ¼ 2nd cos�2 ¼ ml0 maximum ðbrightÞ
there will be constructive interference and a maximum in transmitted intensity. When:
p ¼ 2nd cos�2 ¼ ðmþ 1
2Þl0 minimum ðdarkÞ
1
air
n d
2 3
air
4
θ1
θ2
θ2
Figure 3.6 The reflection of a ray of light incident on a transparent film in air. The path difference between rays 2and 3 is 2nd cosu2
Colour and the Optical Properties of Materials 98
therewill be destructive interference and aminimum in the transmitted intensity. This is converse to the case of
reflection, so that a dark reflection band corresponds to a bright transmission band. The two patterns are
complementary.
Note, however, the appearances of the patterns are somewhat different. In the case of reflection, the
intensities of rays 2 and 3 are similar. In the case of transmission, ray 4 will have approximately 96 % of the
incident intensity, while ray 5, which has suffered two reflections, will have about 1/1000 of this value.
3.3 The Colour of a Single Thin Film in Air
Although adiscussionofmonochromatic light is helpful so as to understand the physical processes taking place
on reflection at a thin film, we are really much more interested in what the appearance of the film will be in
daylight. When the film is viewed in white light, the same reflection and interference discussed above will
occur, except that the effects of all of the different wavelengths present must be added. These interference
effects lead to intense colours, familiar in soap films, oil films on water puddles and thin flakes of minerals
which can glint with bright colours in sunlight (Figure 3.8).
For example, violet light with a wavelength of 400 nm will reflect a maximum of intensity when the film
produces a path length difference or retardation p (¼2nd) of ðmþ 12Þl, i.e. 200 nm, 600 nm, 1000 nm and so on
(Figure 3.9). Of course, these values of the path difference will not give a maximum for the other wavelengths
present in white light. In fact, for a wavelength of 600 nm, there will be a minimum of intensity for the same
retardation of 600 nm (Figure 3.9).
In order to determine the reflected colour of a thin filmwhenviewed inwhite light it is necessary to add up all
of these contributions over all of the values of thewavelength present. It is seen that there is a large contribution
from the 400 nm wave. The contribution from succeeding waves decreases until at a wavelength of 600 nm
there is no contribution at all. Thereafter, a small contribution is obtained fromwavelengths of 650 and 700 nm.
(In reality, a continuumofwavelengths occurs between400and700 nmof course.Here, just sevenwavelengths
are used as an illustration.) The overall colour perceived will be the sum of all of these. Because of the
dominance of the 400 nm contribution the filmwill appear to be a violet blue colour. Thereafter, the perceived
colour will vary as the film thickness increases or decreases, as certain colours are either reinforced or
cancelled. The sequence of colours seen will repeat in a cyclical fashion as the film thickness changes
(Figure 3.10). Each sequence of spectral colours is called an order, which starts with the first order for the
1 2 3
air
n
d
4 5
θ2
Figure 3.7 The transmission of a ray of light incident on a transparent film in air. The path difference betweenrays 4 and 5 is 2nd cosu2
99 The Production of Colour by Reflection
thinnest of films. A new order begins every 550 nm of retardation (¼2nd). The colour of a thin film viewed by
reflection in white light is given in Appendix A3.1.
Ultimately, strong interference effects will be lost. This is because ordinary white light is emitted in bursts
which undergo a sudden change of phase every 10 8 s or so. When the film is very thin the two rays which
interfere come from within the same burst and interference effects are noticeable. With thicker films,
interference takes place between different bursts of light and the interference effects are weaker. This results
in films showing mainly pale pinks and greens in the fourth and fifth orders. With even thicker films all
interference effects are smoothed out and colours are no longer apparent to the eye.
Since the fraction of incident white light which is reflected is coloured, it follows that the transmitted light
will be depleted in this colour and the colour seen will, therefore, be the additive complementary colour to that
strongly reflected. These are listed in Appendix A3.1.
If the angle of viewing is not perpendicular to the film then the retardation changes slightly. The correct
expression for this is:
p ¼ 2nd cos�2
as described before. This formula indicates that, as the viewing angle moves away from perpendicular to the
film, the colour observed will move towards lower retardation. Thus, for example, second-order orange red
will change towards green and blue (Figure 3.10). (But note that, at all angles except for perpendicular viewing,
polarisation will also occur and be important.)
This discussion explains the familiar colours of soap films seen in air. These are best seen if the film is viewed
against a black background, which prevents the effects being masked by other reflections. As the thickness of
the films varies, due to water flow within the films themselves, the colours change in a dramatic and beautiful
way. A draining film has a number of possible equilibrium thicknesses. The thinnest, with a thickness of about
6 nm, gives a black film called Newton’s black film. The ways in which a thinning film produces rivers and
streams of black in a coloured surrounding film are legion, and no two casually produced films seem to drain in
Figure 3.8 Interference colours in thin films: (a) soap bubbles; (b) flakes of molybdite (molybdenum trioxide,MoO3). Both films are viewed in reflected white light
Colour and the Optical Properties of Materials 100
the same way. If the films are formed on a wire frame, then the transmitted and reflected colours can be
compared.
3.4 The Reflectivity of a Single Thin Film in Air
Interference and colour, as just discussed, should be differentiated from reflectivity. It could be that a certain
colour, say red, is produced by interference effects in a film, but whether the colour is readily seen will depend
upon the reflectivityof thefilm for thiswavelength.The reflectivity of a thinfilm inairwill bedifferent from that
for a thick plate (Equation 3.1), as interference effects from the bottom surface also need to be considered.
However, the polarisation of the light will be important and can only be neglected when the light is incident
perpendicularly to the surface of the film.
400 800 1200Retardation / nm
1400
g
b
c
d
e
f
a
Inte
nsity
λ = 400 nm
450 nm
500 nm
550 nm
600 nm
650 nm
700 nm
Figure 3.9 The intensity (in arbitrary units) of light reflected from a single thin film in air at various wavelengthsplotted as a function of the retardation – the optical path difference between the two interfering beams
101 The Production of Colour by Reflection
For light of a singlewavelength at normal incidence the reflectivity of a homogeneous transparent thin film is
given by:
R ¼ 2r21�2r21cos2d1�2r21 cos2dþ r41
where
r1 ¼ n0�nf
n0 þ nf
n0 is the refractive index of the surrounding medium (often air, with n0¼ 1.0) and nf is the refractive index of
the film:
d ¼ 2p½d�l
¼ 2pnf dl
where [d] is the optical thickness of the film and d is the physical thickness. The reflectivity is found to vary in
a cyclic fashion from zero for values of [d] equal to 0, l/2, l, etc. to a maximum (of approximately 0.24 for
nf¼ 1.7) for values of [d] given by l/4, 3l/4 and so on. Because the refractive index of the film is a function of
wavelength, the reflectivity will also vary across the spectrum.
3.5 The Colour of a Single Thin Film on a Substrate
The behaviour of a single thin film on a substrate is similar to that discussed for the case of a single thin film in
air. Thus, a thin transparent film on a substrate would be coloured when viewed in white light. To analyse this
situation, it is necessary to take into account any change of phase that might occur on reflection at the back
surface of the film. The actual hue perceived will be found by a summation of all of the reflected intensities, as
was discussed earlier.
Inte
nsity
100 500 1000
1st order 2nd order
Retardation / nm
grey
whi
teye
llow
yello
w
oran
ge
viol
et-b
lue
viol
et-b
lue
blue gr
een
oran
ge-r
ed
Figure 3.10 The total intensity (in arbitrary units) reflected from a thin film in air illuminated by white light asa function of the retardation (the optical path difference) of the film. The approximate colours observed by eye areindicated
Colour and the Optical Properties of Materials 102
If the substrate has a lower refractive index than the film on the surface then the treatmentwill be identical to
that for a thin film in air, from the point of view of interference effects. The reflected colours observed when
the film is viewed at normal incidence in white light will be the same as those listed as ‘colour reflected’ in
Appendix A3.1. (The transmitted colours are normally absorbed by the substrate.)
If the refractive index of the substrate is greater than that of the film then a phase changewill be introduced at
both the air film interface and the film substrate interface. In this case the reflected colour seen at normal
incidencewhen viewed in white light will be the complementary colour to that just described, listed as ‘colour
transmitted’ in Appendix A3.1. For example, if it is necessary to estimate the thickness of a film of SiO2 grown
on the surface of a single crystal of carborundum (SiC, silicon carbide) use the ‘colour transmitted’ list, because
the refractive index of silicon carbide is greater than that of silica. In fact, silicon carbide is strongly absorbing
over the visible spectrum and when first grown the crystals of carborundum are a shiny black. However, they
soon take on a wide variety of attractive iridescent colours because of surface oxidation, which produces thin
silicon dioxide surface films in a wide variety of thicknesses (Figure 3.11). These films are protective and
preserve the underlyingmaterial from further oxidation, so that the colours only change slowly over the course
of time.
Similar colours are seen on the surfaces of somemetals due to oxidation. The film is usually of a transparent
oxide, Al2O3 onAl, TiO2 on Ti, Ta2O5 on Ta and so on.When they form as a result of themetal being used as an
electrode in an electrochemical cell themetal is said to beanodized. These films, if of the appropriate thickness,
will be brightly coloured. (Note that some anodized films are made especially thick to protect the underlying
Figure 3.11 Colours due to white light interference in a thin transparent film of silicon dioxide (SiO2) oncarborundum (SiC, silicon carbide). The colour variation is due to changes in film thickness
103 The Production of Colour by Reflection
metal. They are frequently coloured by the incorporation of dyes for decorative purposes. These colours are not
thin film effects and arise from the dye molecules (Chapter 8).)
Thin-film colours are also frequently seen when an oil layer covers a puddle of water on a road, where the
refractive index of the oil is usually greater than that of the underlyingwater. These colours are enhanced to the
eyeby theblack road surface,which absorbs all light not reflectedby thefilm.As the thicknessof theoil changes
in response to wind or water movement, the colours vary considerably.
In fact, the eye is able to detect minute changes in surface appearance when a thin film is deposited upon
a substrate. As surprising as this may seem, the fastest and simplest way to detect single atomic layers of
graphene is to use reflected white-light optical microscopy. Graphene, which is being actively studied
because of its remarkable electronic properties, is composed of a single layer of carbon atoms linked in a
hexagonal array a single sheet from a graphite crystal, in fact. Graphene can be prepared by rubbing
a graphite crystal over a smooth surface, a process called mechanical defoliation. A graphene layer on a
such a substrate can be readily detected by eye because of the additional path difference introduced by the
graphene layer, even though this is of only one atom in thickness, coupled with the fact that graphene
absorbs a little of the incident light. Graphene sheets on silicon dioxide, for example, look transparent pale
purple.
3.6 The Reflectivity of a Single Thin Film on a Substrate
The reflectivity of a single thin film deposited on a substrate, like that of a single thin film in air, depends upon
the polarisation of the light, the film thickness and direction of the incident radiation. In the case of
monochromatic illumination perpendicular to a homogeneous nonabsorbing thin film:
R ¼ 2r21 þ 2r1r2 cos2dþ r221þ 2r1r2 cos2dþ r21r
22
where
r1 ¼ n0�nf
n0 þ nf
r2 ¼ nf�ns
nf þ ns
n0 is the refractive index of the surroundingmedium, nf is the refractive index of the film and ns is the refractive
index of the substrate. The expression for d is:
d ¼ 2p½d�l
¼ 2pnfdl
where [d] is the optical thickness of the film and d is the physical thickness of the film. For values of [d] given by
l/2, l, 3l/2, etc. the equation reduces to:
R ¼ ðn0�nsÞ2ðn0 þ nsÞ2
Colour and the Optical Properties of Materials 104
This is identical to the equation for an uncoated surface. Thus, a layer of optical thickness l/2, etc. can be
considered to be optically absent and the surface has normal uncoated reflectivity. This is an intriguing and
useful result. It means that if a delicate surface is coated with a l/2 layer of a hard transparent material the
surface will be protected without any effect on optical properties.
For values of [d] given by l/4, 3l/4, etc. the reflectivity is given by:
R ¼ n2f�n0ns
n2f þ n0ns
� �2
ð3:2Þ
and the reflectancewill be eitheramaximumor aminimum. Thiswill dependuponwhether thefilmhas ahigher
refractive index than the substrate or a lower refractive index than the substrate. When the refractive index of
the film is between that of the surrounding medium and the substrate (n0 < nf < ns), the reflectivity will be
aminimum. When the film has a higher refractive index than both the substrate and the surrounding medium
(n0 < nf > ns), the reflectivity will be a maximum.
As with a thin film in air, the value of the reflectivity will cycle with film thickness between a lower value at
[d] equal to 0, l/2, l, etc. to a maximum for values of [d] equal to l/4, 3l/4 and so on. Because the refractiveindices are a function of wavelength, the reflectivity will also vary across the spectrum.
3.7 Low-Reflection and High-Reflection Films
3.7.1 Antireflection coatings
Wecaneasily use the above equations to see how thinfilmsmodify the reflectivity of a surface. Suppose that it is
desired to make a nonreflective coating on a glass surface in air. (Such coatings are called antireflection (AR)
coatings.) Equation 3.2 shows that if thevalue of nf lies between that of air and theglass then the reflectivitywill
be aminimum for al/4 film. PuttingR¼ 0 inEquation3.2 yields avalue of the refractive indexof thefilmwhich
will give no reflection at all:
nf ¼ nsp ð3:3Þ
For glass, ns is about 1.5, so the antireflecting film must have a refractive index:
nf ¼ 1:5p
¼ 1:225
Very few solids have such a low index of refraction, and a compromise material often used is magnesium
fluoride,MgF2, for which n in the middle of the visible is 1.370 at 500 nm.1 This is not perfect, but does reduce
the reflectivity from about 4 % down to about 1 % (Figure 3.12). The coating will actually be maximally
antireflective for the designwavelength, which is thewavelength forwhich theAR coating is optimised and the
amount of light reflected will increase for wavelengths on either side of the design wavelength and also for
oblique angles of incidence. For camera lenses, which commonly use AR coatings, the design wavelength is
usually near the middle of the visible spectrum, say 550 nm. Such films reflect violet and red more than yellow
or green; an effect readily observed when a good coated camera lens is examined.
1 Note that MgF2 is not isotropic and the refractive index depends upon crystal direction (see Chapter 4). In this and similar cases, the
coatings are made by evaporation and generally have a single effective refractive index.
105 The Production of Colour by Reflection
3.7.2 Antireflection layers
Apart from their utility as surface coatings, AR layers are also important in a variety of applications. For
example, the fabrication of an integrated circuit on a silicon chip involves one or more steps in which the
material is exposed to light through a pattern called amask. Themask is used to selectively illuminate areas on
the chip which, after further processing, build into the array of transistors which manipulate data. The light
actually interacts with a layer of substance called a photoresist. After illumination the photoresist employed is
weakened in those areas which were exposed to light and these are subsequently dissolved away so as to reveal
the underlying silicon, which can then be selectively doped or otherwise treated. The length of time of the
exposure of the photoresist to light is critical to the success of the process.
The desire to pack more and more transistors onto a chip has led to the drawing of ever finer detail onto the
mask and the use of increasingly shorter wavelength light in the illumination steps. At present, the use of
ultraviolet radiation is commonplace. The sharpness of the pattern produced on the silicon, and hence the
number of transistors which can be placed onto the chip, is limited by diffraction (Chapter 6) and multiple
reflections within the photoresist. Themultiple reflections expose parts of the photoresist which should remain
unexposed (Figure 3.13a). This has the effect of reducing the sharpness of the projected pattern and can also
introduce spurious detail or defects.
In order to combat this difficulty an AR coating can be applied between the silicon substrate and the
photoresist (Figure 3.13b). The aim is to introduce a film of the correct thickness to ensure that successive rays
reflected at the bottomsurface of thephotoresist are out of phasebyl/2 so that destructive interference occurs inthe photoresist layer. In terms of the AR layers previously discussed, the photoresist becomes the surrounding
medium, refractive index n0, the new layer is the AR layer, refractive index nf, and the substrate remains as
silicon, refractive index ns.
Although the idea is conceptually simple, the thickness of theAR layer is rather difficult to determine. There
are two main reasons for this. As the layer is interposed between the silicon and the photoresist the simple
400 500 600 700
Ref
lect
ivity
(%
)
1
2
3
Wavelength / nm
Figure 3.12 The reflectivity of a quarter-wave film of amagnesiumfluoride (MgF2) AR coating on a glass surfacewith a refractive index of 1.52. The designwavelength of the film is 550 nmand the beam is taken as perpendicularto the surface
Colour and the Optical Properties of Materials 106
formula in Equation 3.3 cannot be applied and rather complex calculations of the reflectivity must be made.
Second, the simple refractive index term n of the layer must be replaced by the complex refractive index N,
because at wavelengths in the ultraviolet region many materials which are transparent at visible wavelengths
absorb strongly.
One suitable material that has been used in AR layers is silicon oxynitride, SiOxNy, often written as SiON.
The compound has an advantage in that a change of composition alters the optical properties of the film
(Table 3.1). The material is laid down as a thin film by passing a mixture of silane (SiH4), nitrous oxide (N2O)
and nitrogen (N2) over the silicon wafer. The various proportions of the gases control the composition and,
hence, allow thewavelength atwhich thefilm is optimally antireflective to bevaried atwill. The SiOxNy layer is
thus said to be a tuneable AR layer.
silicon
silicon
photoresist
(a)
(b)
photoresist
antireflectionlayer
rays λ /2 out of phase reflections cancel
multiplereflections
Figure 3.13 (a) Multiple light reflections in a film of photoresist on a silicon surface. (b) The deposition of an ARlayer between the photoresist and the silicon results in cancellationof reflectedbeamsbydestructive interference,thus increasing the precision of the process
Table 3.1 Optical properties of Si–O–Nfilms at 248 nmwavelength as a function of composition
Composition Refractive index n Extinction coefficient k
SiO0.86N0.24 1.8948 0.4558SiO0.71N0.27 1.9682 0.5253SiO0.54N0.59 2.0821 0.5004SiO0.47N0.49 2.2127 0.6030
107 The Production of Colour by Reflection
3.7.3 Graded index antireflection coatings
Theproblemof forming a single-filmARcoating on a surface is contained inEquation 3.3. In the case of glass it
has not proved possible to find a filmmaterialwith a refractive index that fits this equation exactly.One solution
to the problem is to make a film in which the refractive index varies gradually from that of the surrounding
medium, usually air, to that of the substrate; that is, a GRIN material (Section 2.5).
The first practical use of this idea, conceivedmore than 50 years ago, was to fashion a surface foam. The idea
is to have a high concentration of air bubbles at the outer surface that gradually falls to zero at the inner
(substrate) surface (Figure 3.14). Provided that the air bubbles are smaller than thewavelength of light, they are
not resolved and the light encounters amedium inwhich the effective refractive index gradually increases from
that of air to that of the substrate. Porous coatings of this type can bemade from a silica gel. If a glass surface is
dipped into the gel and then heated to form a porous glass layer, an antireflective surface coating can be formed
which fits Equation 3.3 precisely.
The refractive index of the film will depend upon the volume and size distribution of the pores, the
polarisation of the incident light and will be wavelength dependent. To a first approximation, the assumption
that thematerial behaves as a simplemixture (Section 2.5) canbe employed.The average refractive indexof the
whole film is then given by Equation 2.11, which is also written in the form:
nf ¼ n0 þFðns�n0Þ ð3:4Þ
where n0 is the refractive index of the surrounding medium, which also fills the pores, usually air, with n0¼ 1,
and ns is the refractive index of the substrate; the amount of solid in the antireflecting layer is called the filling
factor F (identical to the volume fraction of substrate Vs), which runs from 0 % at the surface to 100 % at the
substrate.
The idea of usingGRIN optics inAR coatings, albeit in a slightly different form, was evolved in night-flying
insects somemillionsof years ago. It is clearly of advantage tooptimize the amount of light that the eyesof these
night-flying insects receive and anARcoatingon the eyeshelps in this. TheARcoating derives from thenormal
surface architecture of the insect eye. Adult insects use compound eyes, each of which is formed of many
separate imaging units called ommatidia. The ommatidia form a hexagonal pattern of facets on the surface of
the eye, each facet corresponding to the outer surface of an ommatidium. The eye facets of most day-flying
insects, such as bees anddragonflies, are smooth, but certainnight-flyingmoths and a fewbutterflies have facets
that are covered with tiny bumps (Figure 3.15). The dimensions of the bumps are about half the wavelength of
light, being about 200 nm at the base and 200 nm high. These bumps form an effectiveGRIN layer that endows
the surface of each facet with marked AR properties.
nf 1
nf 2
nf 3
nf 4
ns
Figure 3.14 A surface foam can act as an AR coating. The reflectivity of the coating is determined by dividing upthe surface into a large number of parallel layers and assessing the refractive index of each slice. The overallrefractive index of the surface layersmust equal the square root of the refractive index of the substrate for perfectAR behaviour
Colour and the Optical Properties of Materials 108
In order for the moth-eye surface to act as an AR layer, the protuberances must be small enough not to be
resolved by light, whichmeans in practice that the bumpsmust be separated by half thewavelength of the light
or less. If this is not so, the array will act as a diffraction grating (Chapter 6). The surface architecture must also
not be confused with surface roughness, which will increase diffuse reflection compared with specular
reflection, butnot decrease the total amount of light returned towards the source.The antireflectiveproperties of
moth-eye surfaces can be determined by dividing the bumpy surface into slices parallel to the substrate surface,
estimating the average refractive index and reflectivity of each layer and then summing over the slices to
determine the reflectivity of thewhole structure (Figure 3.16). This is a complex calculation, as the polarisation
of the lightmust be taken into account.However, an approximate estimationof the effective refractive indexcan
bemade using Equation 3.4. (An alternative approach via a diffraction problem is given in Section 6.2.) These
bumpy surfaces are nowbeing reproduced artificially for use asARcoatings.As the effectwas first recorded on
the surface of the eyes of somemoths, these types of surface AR coating are calledmoth-eye AR coatings. The
more formal name for a moth-eye AR coating is ultrahigh spatial-frequency surface relief grating.
Nanoparticles can be used in a similar fashion. The refractive index of a layer of nanorods depends upon the
constituents of the rods, the spacing between them and the angle at which they lie on the surface. The use of
Figure 3.15 The antireflective GRIN structure on the surface of the eye of Morpho butterfly. [Reprinted bypermission from Macmillan Publishers Ltd: NATURE, Photonic structures in biology, Pete Vukusic and J. RoySambles, 424, 852–855, copyright (2003)]
nf 1
nf 2
nf 3
nf 4
ns
Figure 3.16 Amoth-eye surface structure can act as an AR coating. The reflectivity of the coating is determinedby dividing up the surface structure into many thin layers and assessing the refractive index of each slice. Theoverall refractive index of the surface layers must equal the square root of the refractive index of the substrate forperfect AR behaviour
109 The Production of Colour by Reflection
several layers of rods can thusmake aGRIN surface layer. The best AR coating (2009) has beenmade fromfive
100 nm layers of silica and titanium dioxide nanorods deposited at 45� to the surface of aluminium nitride
plates, amaterial of use in LEDs and semiconductor lasers. The final layer, with an effective refractive index of
1.05, gives a reflectivity of 0.01 % (see this chapter’s Further Reading).
3.7.4 High-reflectivity surfaces
Thin-filmcoatings can also be used so as to optimize the reflectivity; that is,make thevalue ofR as close to unity
as possible. A film of thickness l/4 will achieve this provided that the refractive index of the film nf is greater
than both n0 and ns. Two materials frequently used are SiOx, with x approximately equal to 1.0 (n� 2.0), and
TiO2 (n� 2.5 2.8). ATiO2 film of thickness l/4 on glass will have a reflectivity of about 0.40 (40%). As R for
a single glass surface in air is about 0.04 (4 %), 40 % represents a tenfold improvement. The effect is used in
costume jewellery. Rhinestones consist of a glass object with refractive index close to 1.52 coated with an
approximately l/4 thickness film of TiO2. Variations in film thickness and viewing angle give these objects
a wide variety of fleeting colours which are meant to simulate the fire of diamonds. Sparkling paints and nail
varnish alsomake use of an approximately quarter-wavelength thickness of TiO2 deposited onto flakes ofmica
which are subsequently dispersed in the product. The various colours seen are created in a similar way to the
colours on rhinestones.
3.7.5 Interference-modulated (IMOD) displays
Thin film interference is able to generate bright remarkably colours. This is exploited in a display technology
aimed at mobile phone screens. The idea is based on the interference of white light falling on a pair of parallel
reflecting surfaces, so that colour is essentially developed in a thin air film. The arrangement is similar to that of
a Fabry P�erot �etalon. This device, which is an interferometer, consists of a semitransparent film separated
froma fully reflectingfilmby anarrowair gap. Light fromabroad source falling on the top surface is repeatedly
reflected from thebottomsurface and leaks from the top surface as a reflected beamand from thebottomsurface
as a transmitted beam. In a classic Fabry P�erot�etalon the transmitted beam is exploited and the reflected beam
is suppressed. In an IMOD display the reflected beam is exploited.
The arrangement of a single pixel consists of a pair of mirrors separated by a narrow air gap (Figure 3.17a).
The principle of operation is as given in Sections 3.2 and 3.3. The pixel will reflect light of a wavelength lbrightly for incident light falling normal to the surface when:
2d ¼ ðmþ 1
2Þl
whered is the separationof themirrors andm is an integer.As theviewermovesaway fromnormal incidence the
colour will appear tomove to shorter wavelengths; that is, red tends tomove towards blue. The actual colour of
thepixelwill not be a singlewavelength, of course, butwill dependupon the interferenceof thewhole spectrum,
as described above (Figure 3.10).
The device can operate in an interactive fashion if the separation of the air space between themirrors is varied
according to a controlled input. In current displays this is achieved by using electrostatic attraction between
the top (fixed) and lower (moveable) film. Piezoelectric movement, used to control electrodes in a number of
devices, including surface tunnelling microscopes, which are able to reveal atomic features on a surface, can
also be utilized. As the separation varies, so the colour of the pixel changes (Figure 3.17b). These displays are
currently beingwidelyexplored formobile telephone screens.Theyhaveanadvantage in that althoughpower is
needed to change the colour of a pixel, once that colour is set, no power is needed to maintain it. In competing
Colour and the Optical Properties of Materials 110
technologies, such as liquid crystal displays or organic light emitting diode displays, the power must be
supplied continuously to maintain colour and brightness. Moreover, these pixels are easily visible in bright
daylight, which is a drawback of some present displays.
3.8 Multiple Thin Films
3.8.1 Dielectric mirrors
Traditionally,mirrors have beenmade frommetals. The bestmetallicmirrors aremade of a thick layer of silver,
which has a reflectivity of about 0.96 in the visible. (The reflectivity of metals is considered in more detail in
Section 10.15.) Surprisingly, multiple thin films of transparent materials can be laid down one on top of the
other in such away as to form perfectmirrors. These are often called dielectricmirrors. The fabrication of such
devices forms part of the subject area known as photonic or thin-film engineering. Awide variety ofmultilayer
mirrors are now manufactured, mainly from oxides and fluorides. These are all stable in air and have the
additional advantage over metallic mirrors of not degrading in normal use.
The simplest formula for the reflectance of such amirror refers to the specific case inwhich all layers are l/4thick and of alternating high (H) and low (L) refractive indices, nH and nL, illuminated by light falling
perpendicular to the surface. The arrangement (Figure 3.18) is called a quarter-wave stack. The maximum
reflectance of a quarter-wave stack deposited on a substrate in the sequence:
substrate; L; H; L; H; L; H; . . .L; H; air
air
mirror
semi-transparent mirror
daylight
daylight
(a)
(b)
no reflection (black)
Figure 3.17 The principle of operation of an IMOD display: (a) reflection of light from a pair of parallel mirrors(a Fabry–P�erot etalon); (b) variation of mirror separation gives rise to different coloured pixels
111 The Production of Colour by Reflection
is given by the formula:
R ¼ ns f2N�n0
ns f 2N þ n0
� �2
where f is equal to (nH/nL), n0 is the refractive index of the surroundingmedium, usually air (n0¼ 1.0), ns is the
refractive indexof the substrate, usually glass (ns� 1.5) andN is the number of (LH) pairs of layers in the stack.
For a stack in air this equation is equivalent to:
R ¼ ns f2N�1
ns f 2N þ 1
� �2
¼ ns�ðnL=nHÞ2Nns þðnL=nHÞ2N
" #2
Computation shows that, as the number of pairs of layers increases,R rapidly approaches 1.0, implying perfect
reflectivity.
The form of the reflectivity as a function of wavelength for light falling on the stack at normal incidence has
a typical structure consisting of a central plateau together with small side maxima distributed about the design
wavelength l0 (Figure 3.19). In general, the central plateau becomes squarer and higher as the number of layers
increases until a reflectivity of unity is reached. The width of the central plateau is given by:
Dl ¼ 4lparcsin
1�f
1þ f
� �
where f¼ (nH/nL).
Different formulaemust be used if the stack terminateswith an L-layer, if there are not complete sets of pairs
of layers or for oblique illumination. When the beam is at an oblique angle of incidence the polarisation of the
beam must also be taken into account.
ns
n0
nH
nL
nL
nL
nH
nH
(glass)
(air)
1 pair
λ /4
λ /4
Figure 3.18 A stack of thin films, eachof optical thickness l/4, called a quarter-wave stack, can act as an effectivedielectric mirror. The reflectivity increases with the number of pairs of layers and rapidly approaches 1.0
Colour and the Optical Properties of Materials 112
3.8.2 Multilayer stacks
The difficulty of making calculations of the reflectivity and transmissivity of thin-film multilayers prevented
large-scale use of this technology in the first half of the twentieth century. The mathematical formulation of
the problem, though, was solved at this time, and the optical properties of a stack can be described by
a methodology that allows the contribution of each layer to be represented by a matrix. The total optical effect
of the stack is obtained by multiplying the matrices together. Suitable computational software is now readily
available (see this chapter’s Further Reading).
The general approach used tomake amultilayer optical component is to lay down a stack of thin filmswhich
have alternately higher and lower refractive indices using vacuum evaporation of the materials. Manipulation
of the thicknesses and the refractive indices of the layers in the stack, in accordance with computation, allows
for themodification of the optical properties at will. This technology is thus equally suitable for the production
of multilayered AR coatings. For example, Figure 3.20a d shows the variation in reflectivity of a stack of four
thin films as the thickness of just one of the layers is changed. The four thin films are deposited on a glass
substrate and alternate between high refractive index (H) and low refractive index (L), ending with air. The
arrangement of the layers is: air (n¼ 1.0); (1) L, 93 nm, n¼ 1.48; (2) H, 120 nm, n¼ 2.30; (3) L, 37 nm,
n¼ 1.48; (4)H, variable thickness, 30 nm, 24 nm, 18 nm, 12 nm, n¼ 2.30; substrate, glass, n¼ 1.52. The single
layer to be changed was that next to the glass substrate, and then only from a thickness of 30 nm to 12 nm.
The curves are all evaluated for a design wavelength of 550 nm and for light normally incident upon the stack.
The final curve (Figure 3.20d) makes an almost optimal AR coating.
In general, when amultilayer stack is tilted the reflectivitymust take into account polarisation. Although this
has only a small effect on the total reflectivity, the wavelength which is strongly reflected or transmitted shifts
towards lower values, but it does somuchmore slowly than for a single thinfilm.Thus, thefilmwill lookbluer as
the stack is tilted.
It is found that the centralwavelengthwill decrease froml0 for normal incidence tol�when the stack is tiltedthrough small angles � (than 20�) given by the expression:
l� ¼ l�½1�ð�2=2n2f Þ�
Ref
lect
ance
Wavelength
λ 0
Figure 3.19 General form of the reflection from a quarter-wave stack as a function of thewavelength for light atnormal incidence
113 The Production of Colour by Reflection
where nf is the effective refractive index of the stack and � is in radians. Thismeans that thesemultilayer stacks
are tunable over small degrees of rotation. For accurate work the stack must be aligned precisely to ensure
wavelength-specific performance.
If the layers are uneven in thickness, or to some extent disordered, a wide variety of wavelengths will be
reflected. Thesewill be perceived aswhite or silver, depending upon the smoothness of the surfaces. This is the
reasonwhy a roll of thin plastic filmused for foodwrap looks silver.Many insects show silvermarkings that are
similarly made up of thin layers of transparent material of varying spacings (Figure 3.21).
3.8.3 Interference filters and distributed Bragg reflectors
The same technique of multiple dielectric layer deposition can be used to make interference filters. The form
of the reflection curve of a multilayer stack (Figure 3.19) shows that wavelengths to either side of the central
plateau will be transmitted and those within the plateau will be reflected. By using multiple thin films the
regions that transmit or reflect can be precisely manipulated to make optical filters. These fall into three
different categories. Shortpass filters transmit visible wavelengths and cut out infrared radiation
(Figure 3.22a). They are often used in surveillance cameras to eliminate heat radiation. Longpass filters
block ultraviolet radiation and transmit the visible (Figure 3.22b). Other filters, called bandpass filters, pass
only a limited section (or band) of the electromagnetic spectrum (Figure 3.22c). (These thin-film interference
filters generally give a much sharper transmittance than the type of filter made from dye molecules
distributed in a gelatine matrix; the type of filter illustrated in Figure 1.18). As the filters are made of
transparent layers, the wavelengths not transmitted are reflected. Bandpass filters, therefore, act as mirrors
for the complementary colour of the transmitted band. Because of this effect, these filters are often vividly
coloured (Figure 3.23).
Whenmultilayer reflectors are included in an optical device such as awaveguide or some types of laser they
are called distributed Bragg reflectors. They are typically made from layers of TiO2 and SiO2. The reflectivity
of such amultilayer is computed in the sameway as anymultilayer stack, taking into account the surroundings
1
3
5
7
500 600 700550 6500
2
4
6
8
750450
Wavelength / nm
Ref
lect
ance
/ %
a
b
c
d
Figure 3.20 The reflectivity of a stack of four thin films on a glass substrate in air. The thickness of each layer isconstant except for the one adjacent to the glass, which takes values of (a) 30 nm, (b) 24 nm, (c) 18 nm, (d) 12 nm.The stack (d) shows almost perfect AR behaviour. Computations were made using ‘Filmstar’ software (see thischapter’s Further Reading)
Colour and the Optical Properties of Materials 114
and the substrate onwhich the Bragg reflector is deposited. Thewavelength interval that is reflected from such
a reflector is called the photonic stopband. The approximate width of the stopband is given by the formula:
Dl ¼ 4lparcsin
1�f
1þ f
� �
where f¼ nH/nL.
3.9 Fibre Bragg Gratings
Multilayer interference filters can also be produced in the cores of optical fibres. These are called fibre Bragg
gratings (FBGs) and are used for controlling light signals as they travel along the fibre. FBGs are formedwithin
the core of an optical fibre (usually amonomode fibre) (Section 2.9) in which the refractive index is modulated
in a periodic way with a repeat spacing d of about the wavelength of light.
The formation of an FBG was first observed in 1978, more or less by accident, rather like the initial
observation of the formation of frequency-doubled light in a fibre (Section 4.11). Light from an argon-ion laser
was focused into a length of germanium dioxide (GeO2)-doped silica (SiO2) fibre and, surprisingly, more and
more lightwas reflected back along thefibre as time passed. Itwas concluded that a refractive indexgratingwas
being created in the fibre by interference between the incidentwave and awave reflected from the far end of the
fibre. The two waves formed an interference pattern in the fibre, which produced the refractive index change.
Although initially treated as a bizarre phenomenon, it has since been found that anyGeO2-dopedSiO2fibrewill
behave in a similar fashion. The refractive index gratings that form in this way are called Hill gratings. Hill
gratings are limited to the wavelength of the radiation producing the effect.
Figure 3.21 The Silver-washed Fritillary butterfly, Argynnis paphia. The silver ‘wash’ on the wings is caused byreflection from a disordered thin-film multilayer stacks
115 The Production of Colour by Reflection
400 700
700
1000
50
50
100
100
Wavelength / nm
% T
rans
mis
sion
Wavelength / nm
% T
rans
mis
sion
(b)
(a)
400
400 700 1000
50
100
Wavelength / nm
(c)
% T
rans
mis
sion
Figure 3.22 Transmission profiles of multilayer dielectric filters: (a) a shortpass filter; (b) a longpass filter; (c) abandpass filter
Colour and the Optical Properties of Materials 116
The simplest (conceptual) modulation is a step repeat (Figure 3.24). This type of grating is described as
a uniform grating. If the spacing of the refractive index modulation is not constant, but varies in spacing in
a uniform way along the length of the modulation from d1 to d2, the grating is said to be chirped. All can be
described as a form of distributed Bragg reflector (Section 3.8).
Theway inwhichFBGs influence light pulses passingdown the core of thefibre canbeunderstood in termsof
multiple thin-filmoptics. The lightwill be reflectedback from thegrating if thewavelength of the light l and thespacing of the grating d are given by:
l ¼ 2navd
(a)
(b)
cladding
core
n1
n2 n3
Refractive index
Positiond
n2
n3
Figure 3.24 FBGs. (a) Periodic modulation of the refractive index in the core of an optical fibre. (b) The simpleststepmodulation of refractive index. The refractive indices of the cladding, core andmodified core are n1, n2 andn3respectively
Figure 3.23 Multilayer interference filters. The bright reflected colours are complementary to the colourstransmitted by the filters and absorbed by the black backing
117 The Production of Colour by Reflection
where nav is the average of the refractive index of the core and themodulated region. The value of the refractive
index change Dn between the modulated and unmodulated regions is of the order of 10 3 to 10 4 on the core
refractive index, which is close to 1.5. It is clear, therefore, that a single modulation would hardly cause any
change in a pulse. However, FBGs are often of the order of 40 000modulations in length (which only occupies
200 nmoffibre for blue green light of 500 nmwavelength) and, hence, considerable intensity can be reflected.
A pulse of white light introduced into a fibre containing such a grating would reflect back a pulse of
monochromatic blue green light of 500 nm wavelength. The other wavelengths would be transmitted. The
grating is thus able to act as a filter or as a mirror, as in the case of the other multiple thin-film devices.
There are a number of ways of fabricating FBGs. The simplest is to use the interference of two ultraviolet
laser beams shone onto the fibre from the side. The peaks and troughs of the interference pattern of two beams
focused on the fibre create the refractive index changes required (Figure 3.25a). Because the spatial frequency
of the interference pattern is readily changed by altering the angle at which the beamsmeet, awide range in the
spacing of the refractive indexmodulations can be imposedupon the core.Gratings can also be created by using
a mask, which, because of the dimensions involved, acts as a diffraction grating to create a pattern of maxima
and minima in the fibre core (Figure 3.25b). If the part of the fibre which lies in the interference pattern is bent
into a curve, chirped gratings can be made.
As described above, FBGs form under the influence of external radiation, most often of ultraviolet
frequencies. However, not all fibres are susceptible to the formation of refractive index gratings. The glass
must be photosensitive; that is, they react to light in a specified way (see also Sections 10.17 and 10.18).
Although GeO2-doped SiO2 glass is satisfactory, much better gratings form in fibres which also contain boron
(a)
ultravioletlight
ultravioletlight
ultravioletlight
(b)
mask
Figure 3.25 Fabrication of FBGs: (a) interference of two beams of ultraviolet light; (b) diffraction patternfrom a mask
Colour and the Optical Properties of Materials 118
trioxide (B2O3) or tin dioxide (SnO2) as co-dopants. Additionally, fibres can be transformed into a
photosensitive state by forcing hydrogen into the structure.
There is some uncertainty about the mechanism by which the change in refractive index is produced. It is
agreed, though, that defects in the structure are involved in some way. From a number of possibilities,
local density variation, the formation of colour centres involving GeO or the formation of centres involving
a germanium hydrogen (GeH) pair seem to be the most likely candidates at present.
There is considerable interest in FBGs because they have numerous applications in fibre-optic commu-
nications. Clearly, each different wavelength that passes down a cable can carry data. As colour signals do not
become mixed, the more wavelengths that can be crammed into a fibre the more data that it can carry per unit
time. The technique of putting large numbers of different wavelengths down a fibre is called dense wavelength
divisionmultiplexing (DWDM). In this context, FBGs can be used for adding or removing signals from a fibre,
necessary in wavelength multiplexing of optical communication systems (Figure 3.26).
3.10 ‘Smart’ Windows
Smart windows are those that respond to changes in the external and internal environment. There are a number
of different types under active investigation. Here, just two are mentioned, both of which rely on thin-film
reflectivity for the active function, low-emissivity windows and self-cleaning windows. For other approaches
see Section 10.12.
3.10.1 Low-emissivity windows
Windows in buildings are targets for improved energy efficiency. The reason for this is that normal window
glass is an extremely good absorber and emitter of thermal energy. The black-body equations (Section 1.6)
show that a room with a temperature of 21 �C has approximately 94 % of the thermal energy in the range 5
40mm, with a peak at about 10 mm. Glass absorbs and re-emits about 80% of this energy, making windows an
appreciable gateway for loss of heat. Windows which address this problem are known as low-emissivity
windows.
The details depend upon the place of use. In colder regions it is not only necessary tominimize heat loss to the
outside, but also to guarantee that solar energy penetrates the glass and acts as a passive heating agent. In desert
regions it might be more desirable to make reflection of external solar energy the priority.
All the systems in use rely on coating the inside of one or both panes of a double sheet of glasswith a thin film
ofmaterialwhich, in simple terms, is transparent tovisiblewavelengths andopaque to infrared.Thepositioning
of the coating dependsupon the use forwhich thewindow is designed.Toprevent heat loss from rooms in cooler
climates the coating is frequently upon the inside of the inner pane (Figure 3.27a).
One commonly used substance is tin dioxide (SnO2) doped with fluoride ions (F ), with a refractive index
of approximately 2.0. This material is transparent to visible wavelengths but strongly absorbent to the
wavelengths that characterize the thermal energy from the room, thus capturing the energy. These films have a
droppedfrom signal
addedto signal
Figure 3.26 The addition and removal of a signal from a fibre using an FBG; schematic
119 The Production of Colour by Reflection
low emissivity for thesewavelengths and, hence, cannot lose the energy by radiation. The energy is conducted
back through the glass and returned into the room by radiation from the uncoated surfaces. The useful
performance of the film is limited by its thickness. As films become thicker, the emissivity increases, so it is
important to keep film thickness low.
Unfortunately, the ideal thickness for SnO2 films is exactly that which produces a green colour due to
interference andgivesgreen-tintedwindows.Thegreen reflection from thedopedSnO2 layer canbe suppressed
by coating the glass with a thin layer of transparent material with a lower refractive index than the SnO2 before
theSnO2 is applied (Figure 3.27b).Colour suppression occurs via the sameprinciples outlined inSection 3.7.2.
Theaim is tocause reflections from the topandbottomsurfaces of thedopedSnO2 layer tobeout of phase and so
interfere destructively, hence eliminating colour production via interference. If the film is viewed from inside
the room, to a first approximation it is convenient to call the glass pane the surrounding medium, refractive
indexn0, and the dopedSnO2 coating as the substrate.Using the formula for al/4AR layer,Equation3.2, shows
that the ideal film refractive index nf is given by:
nf ¼ n0nsp ð3:5Þ
where n0 is the refractive index of the doped SnO2 layer and ns is the refractive index of the glass (�1.5). This
suggests that a thin film with a refractive index which is of the order of 1.75 might form a suitable colour
glassn ≈ 1.5
glassn ≈ 1.5
SnO2/F-
n ≈ 2.0
room
room
outside
outside
air gap
colour suppressionlayer, n ≈ 1.75
(a)
(b)
Figure 3.27 Low-emissivity coatings: (a) the coating is applied to the inside of a double glass unit and on the sidenearest to the room; (b) another thin film with a lower refractive index is often applied to reduce reflections andact as a colour suppression layer
Colour and the Optical Properties of Materials 120
suppression layer for someone inside the room. Exact multiple thin-film calculations are needed to refine the
thin-film characteristics, both in reflection and transmission.
3.10.2 Self-cleaning windows
Self-cleaning windows need to destroy organic molecules and bacteria that stick to the glass. An additional
desirable property is that the surface should be hydrophilic so that water flows over it readily and allows debris
to be washed away with rain. Titanium dioxide (TiO2) is an oxidation photocatalyst. It is able to decompose
organic molecules and disrupt the surfaces of bacteria when irradiated with ultraviolet light. This has made it
a promising surface coating for ‘self-cleaning’ widows, as normal sunlight contains sufficient ultraviolet to
effect the removal of organic deposits on window surfaces over the course of the day. Moreover, thin films of
TiO2 pick up hydroxyl (OH ) groups on the surface, making it hydrophilic. Thus, self-cleaning windows use
external coatings of TiO2.
Aswith low-emissivitywindows, the thinfilmcausesunwanted interference effects. In this case, thepresence
of a l/4 thin film of TiO2 on the window increases surface reflectivity greatly. This is given by Equation (3.2):
R ¼ n2f�n0ns
n2f þ n0ns
� �2
ð3:2Þ
wherenf is the refractive indexof theTiO2film,n0 is the refractive indexof air (1.0) andns the refractive indexof
the window glass (�1.5). There are two common forms of TiO2: anatase, with an effective refractive index in
thin film form of approximately 2.52, and rutile, with an effective refractive index in thin film form of
approximately 2.76. Substituting these into Equation 3.2 shows that the reflectivity of the surface will lie
between approximatevalues of 38 and 45%.Both of these are too high for convenient use in ordinarywindows.
It is possible to try to suppress this high reflectivity by the inclusion of an AR coating between the TiO2 film
and the glass. However, this faces the same problem as described above for low-emissivity windows, and it is
not easy to find a film that suppresses high reflectivity when viewed from both sides of the glass. GRIN
techniques can help. Self-cleaning windows fabricated with a surface coating of porous silica about 120 nm
thick containing nanoparticles of TiO2 are able to combine both the self-cleaning andAR properties in one. As
describedabove, al/4AR layer onglass should ideally possess a refractive indexof about 1.225 (Section3.7.1).
Porous silica can give a lower value than this, which is increased by the presence of the TiO2 nanoparticles. The
refractive index of the film can be calculated using the methods in Section 2.5, that is:
nf ¼ n1V1 þ n2V2 þ n3V3 þ � � �where n1 represents the refractive index of component 1, etc. and V1 represents the volume fraction of the
material 1, etc.:
V1 þV2 þV3 þ � � � ¼ 1
In practice one would use computer software to evaluate the ideal thicknesses of the TiO2 and SiO2 (or other)
AR layers so as to optimize the transparency of the window.
3.11 Photonic Engineering in Nature
The application ofmultiple thin films in nature iswidespread, and a volume could easily bewritten on this topic
alone. If thefilm thicknesses are fairly uniform, then abright colourwill be reflected. Such colours aregenerally
121 The Production of Colour by Reflection
referred to as iridescent, meaning that the colour has a metallic appearance and the tone changes with viewing
angle. If the layers are uneven in thickness, or to some extent disordered, a wide variety of wavelengths will be
reflected. These will be perceived as white or silver, depending upon the smoothness of the surfaces (see
Figure 3.21). All of these are referred to as structural colours to differentiate them from colours produced by
pigments. Here, just a few examples from a legion are touched upon.Manymorewill be found if the references
(see this chapter’s Further Reading) are consulted.
3.11.1 The colour of blue butterflies
An example of the vivid blue colouring seen in butterflies is provided by the Common Blue, Polyommatus
icarus (Figure 3.28a). The colour arises in tiny scales that cloak thewings (Figure 3.28b). The colour perceived
is built up by a mosaic of these tiny scales and is similar in result to that used by pointillist painters such as
Seurat. The blue scales of this butterfly are made up of sheets of transparent multilayers running parallel to the
scale base (Figure 3.29a). There are four layers of transparent material with a thickness of about 50 nm and
a refractive index of about 1.57 separated by air layers of approximately twice this dimension. In addition, the
layers of transparent material are be perforated into a ‘pepper-pot’ structure (Figure 3.29b). Calculation
confirms that this arrangement is highly reflective for violet blue wavelengths.
In nature, there aremany similar species of blue butterfly, eachofwhich is characterized by adifferent tone of
blue and which can be recognized one from another by these subtle differences. It is easy to appreciate that the
colour of the reflective scales can be tuned by small changes in the multilayer thickness, spacing and degree of
perforation. This latter attribute is equivalent to a GRIN layer that has a refractive index somewhere between
that of air and 1.57.
3.11.2 Shells
Many shells have amultilayer construction, as this affords the desirable combination of strength and lightness.
Occasionally this feature gives rise to structural iridescent colours. The colours are more often visible on the
inside of a shell, as the outsides tend to be camouflaged or otherwise coloured to aid concealment. In many
species these colours are pale greens and pinks and are known as mother-of-pearl or nacre. However, the New
Zealand paua, Haliotis iris, has a very marked iridescence and displays intense colours that change with
viewing direction (Figure 3.30). The colours exhibited aremanyvivid blues and greens. Themultilayers giving
rise to this spectacular effect are derived from alternating organic and inorganic layers. The colour effect is
enhanced by dark-pigmented underlying material that absorbs any light that has not been reflected. The shells
are used for decoration and jewellery.
3.11.3 Labradorite
Minerals can develop as multilayer structures in a number of ways. An example is provided by the mineral
labradorite. This material exhibits flashing rainbow-like colours which vary as the angle of observation
changes in a typically iridescent fashion (Figure 3.31). The phenomenon is also known as schiller and
labradorescencewhen applied to the mineral. Most commonly the colours exhibited are violets and blues, but
greens, yellow and orange colours can also be seen in some specimens.
Geologically, labradorite is a plagioclase feldspar; feldspars being minerals constructed from a strong
framework of corner-sharing (SiO4)4 groups with alkali or alkaline earth cations contained in the cages
present. It has a composition lying between the parent compounds anorthite (CaAl2Si2O8) and albite
(NaAlSi3O8), both of which are also feldspars. Labradorite consists of between 50 % and 70 % anorthite,
so that its formula can be written as Ca0.5–0.7Na0.5–0.3(Al,Si)AlSi2O8.
Colour and the Optical Properties of Materials 122
Figure 3.28 (a) TheCommonBlue butterfly P. icarus. (b) Scales from thewingofP. icarus.Only some scales havea blue reflecting microstructure. The yellow–brown scales are coloured by melanin-related pigments. [Figure (a)reproduced with kind permission of Dr J.A. Findlay]
123 The Production of Colour by Reflection
It is believed that, during the formation of the parent rocks, the feldspar leading to labradorite had a
homogeneous composition in which the various cations were distributed at random over the possible sites
available. This is known to happen at high temperatures, and a complete solid solution is said to form between
the parent phases anorthite and albite. However, at low temperatures this homogeneous solid is thermody-
namically unstable and over geological time scales the sodium and calcium ions segregate to form alternating
lamellae which are sodium rich and calcium rich. This also necessitates the diffusion and subsequent ordering
of aluminium and silicon cations at the same time. The result is that adjacent layers possess differing refractive
indices. In rare circumstances the segregation can result in lamellae which have the appropriate thickness and
degree ofordering to reflect visible light and amultiple thin-filmstructure results. For example, an investigation
of the microstructures of labradorite giving rise to a blue schiller had stacks of alternating lamellae of
dimensions 72.5 nm and 65.1 nm, whereas materials showing a red schiller had lamellae of 176.6 nm
alternating with lamellae of thickness 87.4 nm. As expected, the colours observed will depend upon the
relative thickness of the lamellae and the angle of illumination and observation, and the refractive indices of the
component lamellae. As these are subject to many variables, no two samples of labradorite from different
locations are truly identical.
Figure 3.29 Electron micrographs of a blue scale from the wing of the butterfly P. icarus. (a) Transmissionelectron microscope transverse section. (b) Scanning electron micrograph of a fractured blue scale. The multipleinternal layers with a perforated structure that give rise to the blue reflectivity are revealed
Colour and the Optical Properties of Materials 124
3.11.4 Mirror eyes
The design of eye most familiar to readers is the ‘camera’ type, which uses a lens to focus light onto a light-
sensitivemembrane the retina.However, a number of different eyedesigns are found in nature, someofwhich
usemirrors rather than lenses. Just one frommany examples is provided by scallops of the genusPecten, which
Figure 3.30 New Zealand paua (H. iris) shells showing characteristic iridescent colours that are noticeablyangle dependent
Figure 3.31 A specimen of labradorite from Madagascar. The colours displayed (labradoresence or schiller)change with viewing angle
125 The Production of Colour by Reflection
have focusing elements made up from multilayers of cytoplasm, with a refractive index of 1.34, and guanine
crystals, with a refractive index of 1.83. These layers form amirror to bring light to a focus. The total thickness
of the mirror is about 6mm and contains 60 or so layers. The multilayer mirror is hemispherical in shape and
forms the interior rear surface of the eye so that rays of light entering the eye fall more or less perpendicularly
upon the stack. As a rough estimate, each layer has an optical thickness of about l/4, corresponding to strongreflection ofl¼ 600 nm.However, the layers of themirror are not evenly spaced, and for this reason the eyewill
focus a range of wavelengths.
Appendix A3.1 The Colour of a Thin Film in White Light
Retardationa/nm Colour reflectedb Colour transmittedc
Start of first order
0 black bright white
40 iron grey white
97 lavender grey yellowish white
158 grey blue brownish white
218 grey brownish yellow
234 green white brown
259 white bright red
267 yellow white carmine red
281 straw yellow deep violet
306 bright yellow indigo
332 yellow blue
430 yellow brown grey blue
505 orange red blue green
536 red green
551 deep red yellow green
555 End of first order; start of second order
565 magenta purple bright green
575 violet green yellow
589 indigo gold
609 dark blue yellow
664 sky blue orange
680 blue orange brown
728 blue green brown orange
747 green carmine red
826 bright green purple red
843 yellow green violet purple
866 green yellow violet
910 yellow indigo
948 orange dark blue
998 orange red green blue
1050 crimson violet yellow green
1100 dark violet red green
Colour and the Optical Properties of Materials 126
1110 End of second order; start of third order
1128 blue violet yellow green
1151 indigo off yellow
1258 blue green pink
1314 emerald green red
1334 sea green brown red
1350 green purple violet
1376 dull green violet
1400 yellow green violet grey
1426 green yellow grey blue
1450 yellow indigo
1495 rose pink sea green
1534 carmine red green
1621 dull purple dull sea green
1650 violet grey yellow green
1665 End of third order; start of fourth order
1682 blue grey green yellow
1710 dull sea green yellow grey
1750 blue green lilac
1800 green brown purple red
1811 green carmine
1900 pale green red
1927 greenish grey grey red
2000 pale grey blue grey
2100 carmine red green
2220 End of fourth order; start of fifth order
�2500 green
�2700 pink
Beyond this point, orders overlap and the film colour is generally pale pink or pale green in reflection.
a The retardation is equal to the path difference p between the interfering rays. For a single film, p¼ 2nd, where n is the refractive index of the film and
d (nm) is the physical thickness. For a birefringent crystal, p¼d(|n1 n2|),where d is the thickness of the slice of crystal and n1 and n2 are the effective
refractive indices of the slice for light of two perpendicular polarisation directions. For a uniaxial crystal this ismaximally d(|n0 ne|) (see Chapter 4).b This colour is seen in reflection from a thin film in air when illuminated by white light at normal incidence. It is the same colour as that shown in
transmission by a thin transparent plate of an anisotropic crystal viewed at normal incidence in white light between crossed polars (see Chapter 4).c This colour is the complementary colour to that reflected and is the same as that shown in transmission by a thin film in air when illuminated by
white light at normal incidence. It is the same as that shown in transmission by a thin transparent plate of an anisotropic crystal viewed in white light
between parallel polars (see Chapter 4). In addition, these colours are seen in reflection when a thin transparent film on a substrate with a greater
refractive index is viewed at normal incidence in white light.
Further Reading
Much of this chapter is concerned with thin-film optical engineering. An introduction to the topic is given by
E. Hecht, Optics, 4th edition, Addison-Wesley, San Francisco, 2002.
B. E. E. Saleh, M. C. Teich, Fundamentals of Photonics, John Wiley and Sons, Inc., New York, 1991.
Appendix A3.1 (Continued)
Retardationa/nm Colour reflectedb Colour transmittedc
127 The Production of Colour by Reflection
An interesting historical perspective of the evolution of CDs and DVDs is
A. E. Bell, Sci. Am. 275 (July), 28 32 (1996).
R. L. Gunshor, A. V. Nurmikko, Sci. Am. 275 (July), 34 37 (1996).
G. Zorpette, Sci. Am. 283 (August), 19 20 (2000).
S. Nakamura, M. Riordan, Sci. Am. 300 (April), 54 59 (2009).
The colours produced by soap films are described and explained in
C. Isenberg, The Science of Soap Films and Soap Bubbles, Tieto, Clevedon (UK), 1978. Reprinted by Dover,
New York, 1992.
The use of nanorods as an antireflective surface is given in
J.-Q. Xi et al., Nat. Photonics 1, 176 179 (2007).
Complete coverage of the theory of single and multiple thin films is in
H. A. McLeod, Thin-Film Optical Filters, 3rd edition, Institute of Physics, London, 2001.
For background on the history of thin-film computation, see
O. S. Heavens, Rev. Prog. Phys. 23, 1 65 (1960).
For IMOD displays, see
M. M. Waldrop, Sci. Am. 297 (November), 68 71 (2007).
Free versions of the software ‘Filmstar’, for the computation of thin-film optics (and used to compute Figure
3.20), are available from Dr F. T. Goldstein, FTG Software Associates, PO Box 597, Princeton, NJ 08542,
USA (www.ftgsoftware.com).
Full information on FBGs will be found in
R. Kashyap, Fiber Bragg Gratings, Academic Press, London, 1999.
The topic of colour in nature is described from an evolutionary perspective,with examples of thin-film colours,
mirror eyes, etc. by
A. R. Parker, In the Blink of an Eye, Free Press, London, 2003.
Structural colour is reviewed by
P.Vukusic, Structural color, inDekker Encyclopedia of Nanoscience andNanotechnology, J. A. Schwarz, C. I.
Contescu, K. Putyera, (eds), Vol. 5, Marcel Dekker, New York, 2004, pp. 3713 3722.
Many aspects of structural colour, including butterfly scales, moth-eye AR surfaces and mirror eyes, will be
found in the following papers and the references cited therein:
P. Vukusic, R. J. Wooton, J. R. Sambles, Proc. R. Soc. Lond. Ser. B 271, 595 601 (2004).
P. Vukusic, J. R. Sambles, C. R. Lawrence, R. J. Wooton, Proc. R. Soc. Lond. Ser. B 269, 7 14 (2002).
A. R. Parker, D. R. McKenzie, M. C. J. Large, J. Exp. Biol. 201, 1307 1303 (1998).
A. R. Parker, Z. Hegedus, R. A. Watts, Proc. Roy. Soc. Lond. Ser. B 265, 811 815 (1998).
A. R. Parker, Am. Sci. 87, 248 255, (1999).
H. Ghiradella, Appl. Opt. 30, 3492 3500 (1991).
D.-E. Nilsson, Nature 332, 76 78 (1988).
A. A. Fincham, Nature 287, 729 731 (1980).
M. F. Land, Sci. Am. 239 (December), 88 99 (1978).
Colour and the Optical Properties of Materials 128
4
Polarisation and Crystals
. Why do some crystals produce double images?
. How can infrared radiation be changed into green light?
. How do liquid crystal displays form images?
The interaction of crystals and light has long produced fascinating and puzzling experimental results. In the
preceding chapters, the polarisation of light has not figured prominently. However, when the interplay of light
and crystal symmetry is considered the polarisation of light can no longer be ignored. This chapter describes
these and related interactions and shows how they lead to colour generation.
4.1 Polarisation of Light
Light can be regarded as awave of wavelength lwith electrical andmagnetic components lying at right angles
to one another, each described by a vector (Chapter 1). For the majority of optical processes only the electric
vectorE is important and the light can be represented by a sinusoidal wave that describes the amplitude ofE as
a function of position and time (Figure 4.1a). The vector E is always perpendicular to the direction of
propagationof the light but can adopt anyangle otherwise, similar to thepositions available to ahandonaclock.
For ordinary light, such as that from the sun, the orientation of the electric vector changes in a random fashion
every 10 8 s or so, as if the seconds-hand of a clock jumped unpredictably from position to position without
rotating in a steady manner. The position of the electric vector defines the polarisation of the light wave.
Ordinary light is said to be unpolarised.
Colour and the Optical Properties of Materials Richard J. D. Tilley
� 2011 John Wiley & Sons, Ltd
Light is described as linearly or plane polarisedwhen the electric vectorEwhich describes the light wave is
forced to vibrate in a single plane.1 This is analogous to the seconds-hand on a clock being stuck permanently in
one position. The plane that constrains the electric field vector can lie at any angle � to the propagation direction
electric fieldvector E
L R
(a)
z
y
x
y y(b) (c)
x x
θ
(d) (e) yy
xx
θ
Figure 4.1 (a) A snapshot of a light wave moving from left to right with the electric field vector E in the plane ofthe paper at that instant. When viewed along the ray the tip of the electric field vector may: (b) oscillate along aline at a constant angle u to forma linearly polarised beam; (c) trace out an ellipse to forman elliptically polarisedbeam; (d) trace out a circle to forma circularly polarised beam. Theposition of the electric field vector is shownatthree different instants in (c) and (d). (e) The electric field vector at any instant can be resolved into twomutuallyperpendicular components
1 In this book, the shorthand term ‘polarised light’ will be taken to mean linearly polarised light. Other forms of polarisation will be
described explicitly.
Colour and the Optical Properties of Materials 130
(Figure 4.1b). On looking into the beam, if the tip of theE vectorwerevisible, it would appear to oscillate along
a fixed line at an angle �.In the most general representation of polarised light, the tip of the vector E traces out an ellipse, called the
vibration ellipse, and the light is said to be elliptically polarised (Figure 4.1c). If the tip of the electric field
vector traces out a circle the light is said to be circularly polarised (Figure 4.1d). Taking time into account, the
true path of the tip of the vector forms a helix in both cases. Elliptically or circularly polarised light can be right
polarised, so that the tip of the E vector traces out a path like that of a screw thread a right-handed helix.
Left circularly polarised light has the tip of the E vector travelling in the opposite way as a left-handed helix.
Both circularly and linearly polarised light are special forms of elliptically polarised light, in the first casewhen
the major and minor axes of the ellipse are equal and in the second case when one of the axes is zero in length.
A linearly polarised light beam can also be considered to be composed of a superposition of right and left
circularly polarised light beams with the same frequency and amplitude. This approach is useful in the
discussion of optical activity (Section 4.12).
It is often convenient, when discussing the effects of polarised light on amaterial, to resolve the electric field
vector into two components along (any) mutually perpendicular axes, x and y (Figure 4.1e). The amplitude of
the component directed along the y-axis varies from a maximum value þ y to a minimum value �y through
zero, while the amplitude of the component directed along the x-axis varies from amaximum value of þ x to a
minimum value of �x through zero. Each resolved component is linearly polarised. These two parts may or
may not be in phase with each other initially; that is, the maximum þ y may or may not coincide with the
maximum þ x value. After interaction with a nonisotropic material, the phase between the two components
will be changed. The resultant beam can be obtained by recombination of the two components in the reverse
procedure to that used to divide the initial beam.The shape of thevibration ellipse depends upon the amplitudes
of thex- and y-components and the phase difference between them.Thus, a general beamof polarised lightwill
emerge from an isotropic material, such as a glass or a cubic crystal, with unchanged polarisation but from a
nonisotropic material with a different polarisation.
Many light beams canbe considered to be composedof two fractions, onepolarised andoneunpolarised.The
relative amount of each is expressed as the degree of polarisation of the light. While unpolarised light can
interfere freely, it is important to note that two light beams polarised perpendicular to one another do not
interfere or form interference patterns.
In the case of linearly polarised light, it is often helpful to resolve the polarisation into two components
perpendicular and parallel to the plane of incidence of the beam the planewhich contains the incident ray, the
normal to the surface and the reflected ray. The component of the lightwavepolarised such that the electric field
vector lies in the plane of incidence is called the p-wave, or transversemagnetic (TM)wave. The component of
the light wave polarised such that the electric field vector lies perpendicular to the plane of incidence is called
the s-wave, or transverse electric (TE) wave (Figure 4.2). In general, s- and p-waves differ in the way they are
reflected and refracted.
Whilst the human eye is unable to detect polarisation direction, many animals have this ability. A number
of examples will be given later in this book.
4.2 Polarisation by Reflection
When light is incident upon the surface of a transparent dielectric such as glass, part will be reflected and part
refracted. The p-wave (TMwave) is reflected to a different extent than the s-wave (TEwave). The difference is
dependent upon the angle of incidence �1 (Figure4.3). Formanyanglesof incidence the reflection of thep-wave
is somewhat suppressed relative to that of the s-wave. This causes the reflected light to be noticeably polarised.
When this occurs, the refracted part of the incident light will also be partly polarised.
131 Polarisation and Crystals
With reference to Figure 4.3, the reflectivity or reflectance of the surface is given by Fresnel’s laws:
Rs ¼ sinð�1��2Þsinð�1 þ �2Þ
� �2
and:
Rp ¼ tanð�1��2Þtanð�1 þ �2Þ
� �2
for the s-wave and p-wave respectively. (Remember that �1 is equal to �3 for reflection.) These equations showthatwhen light passing through air falls perpendicularly onto the surface of thematerial,with refractive indexn,
refracted (transmitted) s- (TE) wave
reflected s- (TE) waveincident s- (TE) wave
B
E kEr
kr
Br
(b)
refracted (transmitted)p- (TM) wave
reflected p- (TM) waveincident p- (TM) wave
n1
n2
n1
n2
θ1 θ3
θ2
θ1 θ2
θ3
B
E
kEr
Et
kr
kt
Br
Bt
Et
ktBt
(a)
Figure 4.2 Geometry of linearly polarised light with respect to the plane of incidence of the ray (the plane of thepage): (a) p (TM) wave with E in the plane of incidence; (b) s (TE) wave with E perpendicular to the plane ofincidence
Colour and the Optical Properties of Materials 132
the angle of incidence, �1 is zero and the reflectivity of the p-wave equals that of the s-wave. The reflectivityof the s-wave remains almost the same as that of the p-wave up to an angle of incidence �1 of about 20
�.Thereafter, the reflectivity of the s-wave smoothly increases to 100 % at grazing incidence (�1¼ 90�). Thereflectivity of the p-wave diverges from that of the s-wave and decreases as the angle of incidence increases
until, at a particular angle, it becomes zero (Figure 4.4). At this point the reflected beam is polarised to its
maximum extent. This optimal angle of incidence is given by Brewster’s law:
0 60 9030
oAngle of incidence /
Ref
lect
ance
/ %
100
50
0
s-wavep-wave
Figure 4.4 Reflection at a glass surface (n¼ 1.52) in air showing the p-wave and s-wave components. Thereflectivity of the p-wave component is zero at the Brewster angle, �57� for a glass–air interface
refracted raycontainings- and p- components
reflected raycontainings- and p- components
incident ray
n1
n2
θ1 θ3
θ2
s- (TE) wave
p- (TM) wave
Figure 4.3 The geometry of reflection that leads to the production of polarised light. Note that u1¼ u3 and boththe reflected and refracted rays generally contain s-wave and p-wave components. Polarisation perpendicular tothe plane of incidence (the plane of the page) is represented by filled circles along the ray and polarisation normalto this direction as double-headed arrows. When both polarisation modes are present the symbols aresuperimposed
133 Polarisation and Crystals
tan�1 ¼ tan�3 ¼ n2
n1
where the angles are given in Figure 4.3, n1 is the refractive index of the initial medium that the light ray
traverses andn2 is the refractive indexof themediumcausing reflection. Forglasswith a refractive indexof 1.52
in air (so that n1¼ 1 and n2¼ 1.52), Brewster’s angle will be 56.7� (Figure 4.5). As the angle of incidenceincreases past Brewster’s angle the reflectivity of the surface for the p-wavewill increase smoothly to 100% at
grazing incidence (�1¼ 90�). Thus, at both perpendicular and grazing incidence the s-wave and p-wave behaveidentically and all of the light is reflected.
Unpolarised light shone on a sheet of high-quality optical glass arranged at the Brewster angle will reflect
100 % s-wave polarised light and transmit about 42 % s-wave and 58 % p-wave light. A stack of transparent
glass plates aligned at the Brewster angle make up a Brewster window, which transmits almost 100 % p-wave
polarised light and reflects 100 % s-wave polarised light.
When total internal reflection is considered, the reflectivity of the s-wave andp-wave components of the light
beamwill be angle dependent in the sameway. The reflectivity of the p-wavewill become zero at the ‘internal’
Brewster angle of (90� 56.7)� for glass, i.e. 33.3�. This has important consequences for the long-distance
performance of optical fibres.
Polarisation caused by reflection is present in many natural phenomena. For example, a primary rainbow is
polarised in a direction perpendicular to the arc. The reason for this is because polarisation is introduced at the
reflection inside the raindrop. The formof the curve is similar to that shown in Figure 4.4, but 100% reflectance
occurs at the critical angle for water, �49� rather than 90� (Figure 4.6). The Brewster angle �B at which
the reflectance of the p-wave falls to zero for reflection at an internal water air surface is given by:
≈57° ≈57°
90°
s-wave
s- + p-wave
air
glass
unpolarised incidentlight
completely polarisedreflected light
partly polarisedrefracted light
Figure 4.5 Unpolarised lightonreflection fromaglassplateat theBrewsterangle (�57�)willproduceacompletelylinearlypolariseds-waveandapartiallypolarisedrefractedwave.Polarisationperpendiculartotheplaneofincidence(theplaneof thepage) is representedbyfilledcirclesalongtherayandpolarisationnormal tothisdirectionasdouble-headed arrows. When both polarisation modes are present the symbols are superimposed
Colour and the Optical Properties of Materials 134
tan�1 ¼ tan�3 ¼ tan�B ¼ nair
nwater¼ 1
1:33
�B ¼ 36:9�
The angle of reflection in the drop is close to 38� for the important rays which suffer aminimum deviation of
about 138� (see Table 2.3). Thus, the reflected light is almost 100% polarised, with the s-wave being themajor
component present.
Although the human eye is unable to detect polarisation direction, many animals have this ability. When
discussing invisibility in animals, a point notmade, but of considerable importance, is that numerous predators
can detect polarisation differences. Thus, although a jellyfish, say,may appear invisible to humans, it maywell
be visible to a predator because of surface reflection, as this will generate considerable degrees of polarisation
that will change as the jellyfish moves.
4.3 Polars
Polars are devices which transmit light vibrating (mainly) in a single plane. This plane is referred to as the
vibration direction or the allowed direction of the polar. Light can thus be made into a linearly polarised wave
by passing it through a polar. Polars can be made in a variety of ways. A stack of thin films arranged at the
Brewster angle to form a Brewster window, as described above, is one such method. Polarised light can also
be produced using prisms of certain crystals, such as calcite, described below.Asmight be anticipated, a grid of
conductingmetallic wires can also polarise electromagnetic radiation. The component of theE vector parallel
to thewirewill be absorbed, as it can excite the free electrons in thewire readily, while the component of theEvector perpendicular to thewires passes largely unhindered.Naturally, the grid ofwiresmust be closely spaced
compared with the wavelength of the radiation in order to function efficiently, which makes these devices
particularly useful in the infrared and microwave regions of the electromagnetic spectrum (�20mm�1 cm).
0 60 9030oAngle of incidence /
Ref
lect
ance
/ %
100
50
0
criticalangle
Brewster’sangle
p-waves-wave
Figure 4.6 The internal reflection of light in a water drop (n¼ 1.33) in air showing the p-wave and s-wavecomponents. The reflectivity of the p-wave component is zero at the Brewster angle, �37� for a water–airinterface
135 Polarisation and Crystals
Many organic molecules are able to interact with light in a similar way to a wire. Organic molecules are
essentially composed of chains of carbon atoms, each linked to its neighbours by strong chemical bonds
(see Chapter 8). Themolecular feature that allows useful polarising behaviour is a long andmore or less linear
molecular skeleton.The electricfield of the incident lightwhich is parallel to the longmolecular axis is strongly
absorbed, while the electric field that is perpendicular to the long axis is only weakly absorbed. This is because
the electrons forming the chemical bonds can readily distort along the molecular axis but are much more
constrained perpendicular to the axis. Naturally, some bonding patterns reinforce this trend, and these are the
molecules that are of greatest importance as polars. For similar reasons, it would be anticipated that carbon
nanotubes would also show strongly anisotropic behaviour with respect to polarisation of light.
Should suchmolecules be arranged at random, thennooverall polarisationwill be recorded, but if theycanbe
aligned then light will be emerge from the system strongly polarised perpendicular to the molecular axis.
Inexpensive sheets of polarising material are made in this way, with all molecules aligned parallel to one
another. The first of these to be widely available was Polaroid, invented and developed by Land in the years
following 1927. A common molecule employed for polarising films is polyvinyl alcohol (PVA), the
polymerized form of vinyl alcohol (CH2¼CHOH), first used in Polaroid sheets by Land in 1938. These
long molecules are embedded in a sheet of an inert polymer, which is treated with iodine and then stretched.
The stretching aligns the polymer molecules and the iodine enhances light absorption. Such materials are
known as dichroic sheet polarisers (Section 4.8). These polarising sheets generally absorb almost 100 % of
the light component with the E vector parallel to the polymer axis and transmit about 65 % of light with the E
vector perpendicular to the polymer axis. While not perfect, large-area polarising sheets are inexpensive and
widely available.
Polarising films are put to practical use in ‘Polaroid’ sunglasses, where the film is arranged so as to endow it
with a vertical vibration direction. Reflected light contains a considerable proportion of light polarised parallel
to the reflecting surface, the s-wave component. The Polaroid sunglasses eliminate this horizontal component
and so considerably reduce glare. Similar devices, polarisation filters, are used in photography to reduce the
glare caused by reflection at water surfaces and clouds, and polarisation due to scattering (Chapter 5). Exactly
the same mechanism operates in nematic liquid crystals (Section 4.13).
If two polars are arranged in tandem so that light passes through both (Figure 4.7), then the first polar
encountered by the light beam is called the polariser (that is, the object that introduces the polarisation) and the
analyserpolariser
unpolarisedincident light
polarisedtransmitted light
light polarised invertical plane
light polarised inplane at θ to vertical
I0I0cos2θ
Figure 4.7 Normal light of irradiance I0 transmitted by a polar (at left) will emerge linearly polarised parallel tothe vibrationdirectionof the polar,marked as a double-headed arrow.When twopolars are arranged in sequence,the first polar is called the polariser (at left) and the second the analyser (at right). In such a case a beamtransmitted by both polariser and analyser will have a linear polarisation parallel to the vibration direction of theanalyser. The irradiance will be given by I0 cos
2u. If the vibration directions of the polariser and analyser areperpendicular to each other then no light will be transmitted by the pair
Colour and the Optical Properties of Materials 136
second the analyser (that is, the object that determines the resultant polarisation). The light irradiance
transmitted by a pair of polars was first investigated some 200 years ago by Malus. When the vibration
directions of polariser and analyser are parallel the light transmitted will consist of a linearly polarised wave
with irradiance equal to that of the incident radiation. If the analyser is now rotatedwith respect to the polariser
the transmitted irradiance will diminish according to the law of Malus:
I ¼ I0 cos2�
where I0 is the incident irradiance and � is the angle between the vibration directions of the polariser and
analyser. The emergent wave will be linearly polarised in a plane corresponding to the vibration direction of
the analyser (Figure 4.7). No light will be transmitted when the vibration directions of the two polars are
perpendicular to each other. In this orientation the polars are said to be crossed.
Apart from polars that transmit or absorb linearly polarised light, polars made of materials that transmit or
absorb circularly or elliptically polarised light are commonplace. The polarised-light-absorbing filters used in
photography to enhance contrast by reducing glare or reflection are mostly circularly polarised films.
4.4 Crystal Symmetry and Refractive Index
Gases, most liquids and some solids, such as glasses, are isotropicwith respect to their refractive index. That is
to say, the refractive index is the same irrespective of the direction taken by the light beam.This is not generally
true for crystallinematerials, which aremore often anisotropic. The optical behaviour is found to depend upon
the symmetryof the crystal.Note that here it is the internal symmetrywhich is important, not the external shape,
called themorphology or habit. Symmetry is defined in terms of symmetry operators, which apply reflections,
rotations and so on to the atomic and molecular components making up the crystal. From among the various
symmetry operators, the presence of a centre of symmetry is of considerable significance from the point of view
of optical properties. A centre of symmetry, at (0, 0, 0) transforms any point (x, y, z) to (�x, �y, �z). Both
crystals and molecules which do not have a centre of symmetry are termed non-centrosymmetric.
The unit cell of a crystal is the smallest convenient volume of crystal which displays the symmetry of the
crystal and, if extended in three directions (like building up a cube or pyramid from bricks), will produce the
macroscopic crystal. It is characterized by three axes, labelled a (of length a),b (of length b) and c (of length c),
and the anglesbetween them,a,b andg,wherea lies betweenband c,b lies betweena and candgbetweena andb. For historical reasons, the classification of external symmetry led to the derivation of six crystal families,
which later was refined into seven crystal systems (Table 4.1).
Table 4.1 The crystal systems
Crystal family Crystal system Unit cell Example
Isometric Cubic a¼ b¼ c, a¼ b¼ g¼ 90� Rock salt, NaClTetragonal Tetragonal a¼ b 6¼ c, a¼ b¼ g¼ 90� Rutile, TiO2
Orthorhombic Orthorhombic a 6¼ b 6¼ c, a¼ b¼ g¼ 90� Stibnite, Sb2S3Monoclinic Monoclinic a 6¼ b 6¼ c, a¼ g¼ 90�, b 6¼ 90� Tungsten trioxide, WO3
Anorthic Triclinic a 6¼ b 6¼ c, a 6¼ b 6¼ g 6¼ 90� Copper sulfate, CuSO4�5H2OHexagonal Hexagonal a¼ b 6¼ c, a¼ b¼ 90�, g¼ 120� Zincite, ZnO
Trigonal or rhombohedrala a¼ b¼ c, a¼ b¼ g 6¼ 90� or Calcite, CaCO3
a0 ¼ b0 6¼ c0, a¼ b¼ 90�, g¼ 120� Dolomite, CaMg(CO3)2
a Trigonal (rhombohedral) crystals are often described in terms of an alternative hexagonal unit cell given in the second line of this box.
137 Polarisation and Crystals
Cubic (isometric) crystals like commonsalt (halite or rock salt) are isotropicmaterials.Theyexhibit the same
refractive index in all directions and behave in the same way as a glass with respect to light. In all of the other
classes crystals are anisotropic. Tetragonal, trigonal and hexagonal crystals have identical refractive indices
along the a- and b-axes and a different refractive index along the c-axis. These crystals display two principal
values of the refractive index, or two principal (refractive) indices. In orthorhombic, monoclinic and triclinic
crystals there are three principal values of the refractive index, related to three mutually perpendicular axes.
These axes may coincide with crystallographic axes for orthorhombic crystals but not for monoclinic or
triclinic crystals, which are characterized by nonorthogonal axes.
The numerical difference between the highest and lowest values of the principal refractive indices is
called the birefringence of the crystal. The refractive index encountered by a beam of light entering a
crystal in an arbitrary direction depends upon the polarisation of the light and lies between the values of the
highest and lowest principal refractive indices. Because of the reciprocal relationship between
the refractive index and the velocity of light in a material (Equation 2.3), the direction with the
lowest refractive index is often called the fast direction or the fast axis, while the direction along the
highest refractive index is the slow direction or slow axis. The relationship between crystal structure and
refractive index is described in greater detail in Section 4.6.
This variation of refractive index with crystal direction is unsurprising. The refractive index depends upon
the density of atoms in a crystal (Section 2.4). In cubic crystals the atom density averages to be the same in all
directions, while in crystals of lower symmetry some directions contain more atoms than others. For example,
in the tetragonal rutile structure of TiO2, chains of TiO6 octahedra run along the c-axis. This structural feature
results in the atoms in the crystal beingmuch less densely packed along the a- and b-axes than along the c-axis
chains. The refractive index along a and b is 2.609, while along c it is 2.900.
4.5 Double Refraction: Calcite as an Example
4.5.1 Double refraction
Although a variation in refractive index with direction may not be surprising, the way in which crystals with
structures other than cubic interact with light is certainly so. This is well illustrated by the mineral calcite.
Calcite is a mineral form of calcium carbonate (CaCO3). The unit cell is trigonal with a¼ 0.641 nm and
a¼ 101.9�, but it is sometimes more convenient to refer to a hexagonal unit cell in which a¼ 0.499 nm and
c¼ 1.71 nm. The form of optical interest is called Iceland spar, and is a particularly clear form of the material.
Iceland spar crystals are easily cleaved into rhombohedra. If such a rhombohedron is placed over a line or
mark, double images will form when the crystal is in particular orientations (Figure 4.8). This can be
demonstrated with greater precision by examination of a black spot on a sheet of paper through such a crystal.
In general, two spotswill be seen on looking fromabove through the crystal (Figure 4.9a). The spots also appear
to be at different heights within the crystal itself. One spot will appear to be undeviated in positionwith respect
to the spot on the paper.Theundeviated spot is formedby lightmoving through the crystal as if itwereglass, and
the ray producing this effect is variously called the ordinary ray, theO-ray or o-ray. If the crystal is then rotated
the ‘ordinary’ spotwill remain inplacewhile theotherwill rotate in a circle about thefixedspot (Figure4.9b d).
The ray causing this behaviour is called the extraordinary ray,E-ray or e-ray. The crystal is displaying the fact
that it has two indices of refraction and the feature is called double refraction.
If a (linearly polarised) polar is placed over the crystal and rotated, at first one dot disappears and then the
other (Figure 4.9e and f ). If the crystal is picked up and tilted, then the separation of the two dots will change;
and if it is possible to look down the diagonals of the rhombohedron, in one case only one dot will be seen, that
formed by the o-ray, no matter how the crystal is rotated about this diagonal. This direction is called the optic
Colour and the Optical Properties of Materials 138
axis. In a normal cleaved rhobohedron of calcite the optic axis lies along the body diagonal which passes
through the ‘bluntest’ pair of corners. These occur at the two corners where the faces which meet all show
obtuse angles.
These results suggest that the beam of light entering a calcite crystal in a general direction is split into
two refracted rays which are mutually polarised perpendicular to each other. The crystal thus resolves an
incident beam of unpolarised light into two linearly polarised components with vibration directions
perpendicular to each other; the s-wave and p-wave components. When the beam enters the crystal, each
of the two linearly polarised components experiences its own refractive index and produces two refracted
rays (Figure 4.10). The o-beam (the s-wave) is undeviated and consists of light linearly polarised with a
horizontal vibration direction parallel to the base of the crystal rhomb. The e-beam (p-wave) is deviated
and consists of light linearly polarised with the vertical vibration direction perpendicular to that in the
o-beam. No light is absorbed (in a perfectly clear crystal) and half of the incident intensity is found in
each of the beams.
A prism of glass (or any isotropic substance) will produce a spectrum when a beam of white light falls onto
one of the faces (Section 2.6). If the prism is made of a uniaxial material such as calcite, two spectra can form
(Figure 4.11). In general, an incident beam of unpolarised light will be split into two, an ordinary and
extraordinary beam, each of which will produce a spectrum due to dispersion. Unless the two rays are widely
separated, the violet portion of the upper spectrum, due to the e-ray, will overlap the red portion of the lower
spectrum, due to the o-ray. In this case the observed spectrum will appear as if a white band has appeared in
the centre of an otherwise abnormally elongated spectrum. If the prism is cut so that the beam travels along the
optic axis then only one normal spectrum will form.
Figure 4.8 Double refraction by an ordinary unpolished rhombohedron of Iceland spar. If the crystal is rotated,then the separation of the two pairs of lines visible through the crystal will alter and in some orientations only asingle line will appear
139 Polarisation and Crystals
The compound eyes of the now extinct ammonites were composed of calcite crystals, with each facet of the
eye made from a single crystal. (There are several thousand facets making up each eye.) To allow for image
formation without double diffraction effects, the optic axis of the calcite (the crystallographic c-axis) was
aligned along the long axis of the facet.
4.5.2 Refractive index and crystal structure
More information on double refraction can be gained by investigating the refractive indices experienced by the
rays. As Iceland spar is trigonal it will possess two principal refractive indices, related to the crystallographic
direction of rotation
o
o
o
o
e
e
e
ee
projection ofc-axis (optic axis)
o
(a) (b)
(c) (d)
(e) (f)
Figure 4.9 Schematic representation of the appearance of double refraction by a crystal of calcite placed overa black spot on a sheet of paper. (a)–(d). As the crystal is rotated, one image (due to the o-ray) remains stationaryand one (due to the e-ray) rotates. (e), (f). A sheet of polariser placed with its vibration direction, indicated asa double-headed arrow, perpendicular to the projection of the c-axis (the optic axis) of the calcite causes the e-rayto disappear, while the same polariser rotated by 90� causes the o-ray to disappear
Colour and the Optical Properties of Materials 140
axes. It is convenient to use hexagonal axes, in which case the hexagonal c-axis is the optic axis. When
unpolarised light is transmitted along the c-axis (optic axis) only one spot is seen and both polarisation
components experience the same refractive index, no¼ 1.658. When the unpolarised light is transmitted
perpendicular to the c-axis (optic axis) it is resolved into two beams, one with a vibration direction
perpendicular to c, the o-ray, experiencing a refractive index n0, and one with a vibration direction parallel
to c, the e-ray, experiencing a refractive index ne¼ 1.486. The values of no and ne represent the principal
refractive indices of the crystal.2 The birefringence (that is, the numerical difference between the
principal indices for calcite) is given by no (1.658) minus ne (1.486), that is 0.172. The direction with the
highest refractive index, parallel to the c-axis, is the slow direction.
Whenunpolarised light is transmitted in anyother direction, the crystalwill show two refractive indices, each
ofwhichwill apply tooneof the polarisation components of the incident light.Oneof thesewill always be equal
to no but the other onewill depend upon the direction of the light ray and is variable, written n0e.When the light
beam travels parallel to the c-axis n0e is equal to no and there is only one effective refractive index
for the material. When the light beam travels perpendicular to c the value of n0e is equal to ne.
To understand this difference it is necessary to turn to the crystal structure of calcite (Figure 4.12). The
structure can be thought of in terms of planar (CO3)2 ions and Ca2þ ions. The (CO3)
2 ions are arranged
in sheets perpendicular to the optic (c-) axis. When the light beam travels down the optic axis (the
projection ofc-axis (optic axis)
o-raye-ray
109°71°
spot
observer
Figure 4.10 The passage of a monochromatic beam of light through a cleaved prism of calcite. Normal lightfalling perpendicularly upon the bottom face of the prism (as in Figure 4.9) is split into two components withdifferent polarisation. The o-ray, with a vibration direction in a plane perpendicular to the c-axis, indicated byfilled circles along the ray, is undeviated. The e-ray, with a vibration direction in a planewhich includes the c-axis,indicated by double-headed arrows, is deviated by about 6�. The top and bottom cleavage faces of the prismare (101) planes with respect to the hexagonal unit cell, and the c-axis (the optic axis) is a body diagonal of thecleavage rhombohedron
2 There are a number of conventions in use: no is also written as o, no, No, O, nO, NO, No; ne is also written as e, ne, Ne, E, nE, NE, Ne.
141 Polarisation and Crystals
optic axis (c-axis)
(CO3)2–
Figure4.12 The structure of calcite (schematic). Theplanar (CO3 )2� ions are arranged in layers perpendicular to
the crystallographic c-axis (the optic axis). The direction of the groups alternates from one layer to the next. TheCa2þ ions are omitted for clarity
optic axis
red
violet
ordinaryray
red
violet
extraordinaryray
optic axis
red
violet
(a)
(b)
unpolarized
incident beam
incident beam
Figure4.11 (a)Aprismmadeof adoubly refractingmaterial suchas calcitewill produce two spectrawithnormalwhite lightwhen the optic axis is perpendicular to the beam. The e-ray is polarised parallel to the optic axis and theo-ray is polarised perpendicular to the optic axis. (b)Whennormal light is propagated along the optic axis only onespectrum forms, as the o- and e-rays are not separated
Colour and the Optical Properties of Materials 142
crystallographic c-axis) the vibration directions of the E vectors lie parallel to the planes containing the
(CO3)2 groups. Interaction with the planar groups is strong. The light beam is slowed; that is, the
refractive index is high and corresponds to no. When the light beam travels in a direction perpendicular
to the optic axis, one E vector component vibrates parallel to the planes containing the (CO3)2 groups and
so experiences the same refractive index as before: no. The other E vector component vibrates normal to
the planes containing the (CO3)2 groups, is less impeded and is not slowed so much; that is, the refractive
index is low and corresponds to ne.
4.6 The Description of Double Refraction Effects
4.6.1 Uniaxial crystals
In the case of amorphous materials, such as air, water or glass, and cubic (isometric) crystals the refractive
indices experienced by the horizontally and vertically polarised components of the light are identical. This
means that they behave identically. The two beams cannot be separated and it is simplest to say that only one
refracted beam is present.
This is not true with all tetragonal, hexagonal and trigonal crystals, but unless the two refractive indices are
quite different then the double refraction observed is too small to be noticed casually. All crystals in these
systemswill have one optic axis, the crystallographic c-axis, and they are described asuniaxial.Abeamof light
entering such a crystal splits into two beams. One polarisation component experiences a refractive index noand the other a refractive index that has a magnitude n0e lying between no and ne.
A convenient way to visualize this interaction and to determine the refractive indices encountered by the
horizontally and vertically polarised components of a light beam is by way of a construction called the optical
indicatrix. This is an ellipsoid with the dimensions of the mutually perpendicular axes determined by the
principal refractive indices of the crystal. The optical indicatrix for tetragonal, hexagonal and trigonal crystals
is drawnwith thevaluene taken as parallel to the c-axis of the crystals andno as perpendicular to it (Figure 4.13).
If ne is greater than no then the crystal is termed optically positive, and if ne is less than no then it is optically
negative. (As the refractive index of a cubic crystal is the same in all directions, the optical indicatrix is a
sphere.) The fast axis for uniaxial negative crystals is along the optic axis and perpendicular to it for uniaxial
positive crystals.
In order to determine the refractive indices experienced by the polarised components of a light ray, the beam
is projected onto the indicatrix. The polarisation of the incident beam is resolved into two perpendicular
components, normal to the beam direction, which form the major and minor semiaxes of the elliptical cross-
section of the incident beam projected onto the indicatrix (Figure 4.14). Within this elliptical section the
polarisation directions can be chosen to be parallel to the no axis of the indicatrix and perpendicular to this.
Thus, a beam travelling down the optic axis has both vibration directions of the polarisation lying parallel to the
no axes in the indicatrix, which indicates that both polarisation terms will see only a single refractive index, no,
and sowill not be separated. A beam travelling perpendicular to the optic axis will have thevibration directions
resolved along one of the no axes, and along the ne axis. The incident beam will split into two, as described
above, each component polarised perpendicular to the other. The vibration directions of a beam at an arbitrary
angle to the indicatrix are resolved parallel to no and in a perpendicular direction to this. The refractive indices
encountered by the two polarisation forms can be read from the lengths of the semi-major and semi-minor axes
of the ellipse so formed. It is seen that, no matter what angle � to the optic axis that the incident beam makes,
it generates an elliptical cross-section inwhich one semi-axis is always no. The other semi-axis is n0e, which hasa value between no and ne. The relationship between themagnitude of n0e and the angle � that the raymakeswith
the optic axis is:
143 Polarisation and Crystals
1
ðn0eÞ2¼ cos2�
n2oþ sin2�
n2e
When the light beam travels parallel to c the value of � is zero and n0e is equal to no and there is only one effectiverefractive index for the material. When the light beam travels perpendicular to c the value of � is 90� and n0eis equal to ne.
4.6.2 Biaxial crystals
Similar effects to those just describedwill be seenwith crystals belonging to the orthorhombic,monoclinic and
triclinic systems. In these cases, crystals exhibit three principle refractive indices, na (which has the smallest
value), nb and ng (which is the greatest value). The crystals have two optic axes and are referred to as biaxial.
The horizontally and vertically polarised components of a beam of light entering such a crystal encounter
different refractive indices, with magnitudes lying between the lowest, na, and the highest, ng. However, the
refractive index encountered by both polarisation components of a light beam directed along either optic axis
is nb. There is not usually an intuitive relationship between the optic axes and the crystallographic axes.
(a)
(b)
ne
nono
optic axis (c-axis)
ne
nono
optic axis (c-axis)
Figure 4.13 The optical indicatrix for a uniaxial (tetragonal, hexagonal, trigonal) crystal: (a) uniaxial positive,ne > no ; (b) uniaxial negative, ne < no. In both cases the optic axis coincides with the crystallographic c-axis. Thecross-section shaded is circular, with a radius no
Colour and the Optical Properties of Materials 144
Aswith uniaxial crystals, a convenientway to visualize the interaction of lightwith crystals and to determine
the refractive indices encountered by the horizontally and vertically polarised components of a light beam is by
way of the construction if the optical indicatrix (Figure 4.15). This is an ellipsoid with the dimensions of the
mutually perpendicular axes determined by the principle refractive indices of the crystal. In order to determine
ne
no
incidentray
n′e
optic axis (c-axis)
Figure 4.14 A beam of light incident on a uniaxial indicatrix experiences two refractive indices given by themajor andminor semiaxes of the elliptical cross-section of the perpendicular to the beamdirection.One of these isalways no and the other is n0e. For a beam directed down the optic axis both refractive indices are no, while for abeam perpendicular to the optic axis one is no and the other ne
nα
nγ
nβ
Figure 4.15 The general form of the optical indicatrix for biaxial (orthorhombic, monoclinic and triclinic)crystals
145 Polarisation and Crystals
the refractive indices experienced by the polarised components of a light ray, the indicatrix is sectioned
perpendicular to the beamdirection and the refractive indices read from the lengths of the semi-major and semi-
minor axes of the ellipse so formed. In the case of a biaxial crystal it is seen that the incident ray, in one
orientation, will generate a section which is circular and with a semi-axis equal to nb (Figure 4.16). The
direction of the light, at an angle V to the ng axis in this case, defines one optic axis. Clearly, there will
be another optic axis at an equal angle V (or Vg if it is necessary to stress that the angle is with respect to the
ng axis). The crystal is defined as optically positivewhen the angle between the two optic axes, 2Vg, is less than
optic axisoptic axisnγ
nβ
nβ
nα
(b)
2Vγ
nγ
nβ
nβ
nα
(a)
incidentray along oneoptic axis
Figure 4.16 The optical indicatrix of a biaxial crystal. (a) A ray of light incident upon a biaxial crystal can give riseto a circular cross-section with radius nb because nb lies between na and ng. This direction defines one optic axis,which is perpendicular to the cross-section. (b) The second optic axis makes an equal angle to the ng axis and isconstructed in a similar fashion
Colour and the Optical Properties of Materials 146
90� and optically negative when 2Vg is greater than 90�. As with uniaxial crystals, the maximum values of
the refractive indices are called the principal indices of the crystal and the difference between the principal
indices, ng� na for a biaxial crystal, is called the birefringence of the crystal.
To summarise, in all crystals of symmetry lower than cubic the refractive index depends upon the direction
of vibration of the light ray. Any ray not passing down an optic axis is resolved into two rays linearly polarised
in two mutually perpendicular directions.
4.7 Colour Produced by Polarisation and Birefringence
Birefringence, as such, does not normally result in colour production.A strongly birefringent crystal of Iceland
spar is clear when viewed in ordinary daylight. However, this changes when polarised light is involved, and
many beautiful colours can be seen in thin plates of anisotropic crystals when examined using polarised light.
A good example is provided by a sheet of mica3 placed between two polars and viewed in transmission by
holding the sandwich up to a white light. Suppose that the polars are crossed. As the mica sheet is rotated with
respect to the polars, four positions, at 90�, will be found at which the mica sheet becomes dark. These are the
extinction positions. When the mica sheet is midway between these positions it will seem to be brightly
coloured. The colour seen is very sensitive to the viewing angle, but if care is taken to look at the foil without
any change of viewing angle then the colour will be seen to remain unchanged as the mica sheet is rotated. It is
only the overall intensity which changes. The colour observed will also be found to depend upon the thickness
of the mica sheet, although the overall pattern of variation of intensity will be the same as that just described
as the mica foil is rotated.
The colours produced by birefringent films are explained in the following way. The beam leaving the
polariser is linearly polarised. On entering the crystal this beam will be split into two, the ordinary and
extraordinary rays. Because of the difference in refractive index experienced by the ordinary and extraordinary
rays, each will move at a different velocity in the crystal. The result is a phase difference between the two rays
called the relative retardation, which will be different for each wavelength. The optical path length of each
beam is given by:
ordinary ray ½d�o ¼ dno
extraordinary ray ½d�e ¼ dn0e
where d is the (real) thickness of the plate. The relative path difference p between these rays is given by:
p ¼ djno�n0ej
where only the positive numerical difference between the refractive indices is important.
The relative phase difference D� between the ordinary and extraordinary rays is:
D� ¼ p2pl0
¼ 2pl0
djno�n0ej
where l0 is the vacuum wavelength of the light. The maximum phase difference is for rays travelling along or
perpendicular to the optic axis, in which case:
3 Mica is the name applied to a group of structurally relatedminerals that are generallymonoclinic and so biaxial in nature. Here, the exact
species of mica is irrelevant.
147 Polarisation and Crystals
D� ¼ 2pl0
djno�nej
The retardation between the two rays means that the light is now elliptically polarised, not plane polarised.
On traversing the analyser, the elliptical polarisation will be resolved along two mutually perpendicular
directions: one parallel to the vibration direction of the analyser and one perpendicular to this. Some light will
now be transmitted, the amount depending upon the wavelength.
The resultant colour production is an interference effect. Although two light beams polarised perpen-
dicular to one another do not interfere or form interference patterns, two beams with parallel polarisation
can. On meeting the analyser, only the electric field components of the ordinary and extraordinary rays
parallel to the allowed direction will pass. The phase difference will result in interference because the
resultant electric field vectors are now parallel in each ray, which fulfils the interference condition. This
causes the image to take on a colour because some of the wavelengths of the white light spectrum will
interfere constructively and so be enhanced, while some wavelengths will interfere destructively and so be
diminished. The colour perceived will be the sum of the effects over the visible spectrum. The colour
observed as a function of retardation is given in Appendix A3.1. If the crossed polars are now rotated to be
in the parallel position without changing the orientation of the mica, the complementary colour will be seen
(Appendix A3.1).
The colour will vary as a function of the thickness of the plate because the retardation is a function of the
distance travelled by the two rays. It will also vary as the orientation of the beams change with respect to the
optic axis for the same reason.
This same effect can be exploited to reveal stress and strain4 in an isotropic material. When a material is
stressed the density will change slightly. If the stress is directional then the density will vary in a pattern
which mirrors this. Thus, an isotropic material under stress can contain optically anisotropic regions. In the
case of molecular materials, including polymer films, the molecules can also become partly oriented parallel
to each other during stretching, which enhances the effect. If the material is observed between crossed polars,
coloured fringes will reveal the stressed areas. In effect, the stress encodes information on the linearly
polarised incident beam which is decoded by the analyser. The effect is easily seen. Take a piece of plastic
film and look at it between crossed polars. Generally, nothing of interest will be seen. If you now stretch the
film (technically subject it to a uniaxial tensile stress) brightly coloured areas will appear (Figure 4.17). The
birefringence so produced in the now anisotropic film is called stress birefringence. This feature is widely
used in glass blowing to make sure that residual strain is not present. A glass workpiece is viewed between
crossed polars and the strained regions are revealed. If necessary, the piece can then be annealed (reheated
at a moderate temperature) to allow the glass to flow slightly and so relieve the strain present. Before the
advent of high-speed computers the strain in complex engineering components could be analysed by
building them of clear plastic and viewing the stress and strain fields present using crossed polars. Regions of
the structure containing high levels of stress show coloured fringes, the spacing of which indicates the stress
gradients present.
This phenomenon is also well known to car drivers who wear Polaroid sunglasses. The windscreens of
cars are stressed in a predetermined way so as to avoid catastrophic failure if hit by a flying stone or similar
object. Light reflected from a hot road will be partly polarised, as explained above. The Polaroid sunglasses
act as an analyser and coloured fringes delineating the strained areas are clearly visible over the
windscreen.
4 The result of a stress (a force or load applied to a material) is a strain (a deformation).
Colour and the Optical Properties of Materials 148
4.8 Dichroism and Pleochroism
If uniaxial or biaxial crystals are viewed by transmitted linearly polarised white light, many will be seen to
change colour on rotation. Uniaxial crystals may display two colours (dichroism) and biaxial crystals three
colours (trichroism). The term pleochroism (more than the usual number of colours, many coloured) is
frequently used generically instead of either dichromism or trichromism. Note that dichroism (pleochroism)
is due to the fact that theabsorptionof linearlypolarised light is a functionof the polarisation direction,whereas
colour due to birefringence (double refraction) is due to the retardation introduced by the refractive indices
encountered by linearly polarised light.
The trigonalmineral tourmaline provides a good example of pleochroism.Tourmaline is the nameof a group
of hexagonal minerals of complex general formula. The best-known tourmalines are the gemstone elbaite,
Na(Li,Al)3Al6Si6O18(BO3)3(OH,F)4 and the iron-rich form, schorl, NaFe3(Fe,Al)6Si6O18(BO3)3(OH,F)4. The
best samples from the point of viewof dichroism are prepared from schorl, which is a negative uniaxialmineral
inwhich the value of no varies from approximately 1.66 to about 1.672, ne from1.633 to 1.64 and birefringence
from 0.027 to 0.032.
Plates of schorl about 1mm thick and containing the c-axis will transmit most of the incident light with a
vibration direction parallel to the c-axis (the e-ray) and absorb most of the incident light with a vibration
direction perpendicular to the c-axis (the o-ray). If the crystal is illuminated with polarised light and rotated
through 90� it will become alternately dark and light (Figure 4.18). The use of linearly polarised light ensures
that the beam of incident radiation is, in effect, solely made up of the o-ray component when the vibration
direction is perpendicular to the c-axis or the e-ray component when the vibration direction is parallel to the
Figure 4.17 A thin piece of polymer film used to wrap food stretched and viewed between crossed polars. Thebright colours in the normally transparent film reveal regions of high strain in the film
149 Polarisation and Crystals
c-axis. The o-ray is strongly absorbed, leading to a dark-brown grey or even black appearance, while the e-ray
is weakly absorbed, producing a light brown or grey colour.
The origin of this dichroism lies in the presence of the transition metal ions Fe2þ and Fe3þ present and
intervalence charge-transfer between these ions in which an electron is transferred from an Fe2þ ion onto a
neighbouring Fe3þ ion (see Section 8.10). The linearly polarised light is strongly absorbedwhen the vibration
direction, that is the electric field, coincides with the orientation of the ion pairs involved in the charge transfer.
The electric field can then be seen to aid the electron transfer considerably. When the vibration direction is
perpendicular to this orientation the efficiency of the electric field in aiding the electron transfer is minimal.
The light is not absorbed and the crystal remains clear. Because the effect is viewed in transmitted light, rather
low concentrations of the transition metal ions are required for best effects.
Ruby (aluminium oxide (Al2O3) containing about 0.5 % chromium oxide (Cr2O3), approximate formula
Cr0.005Al0.995O3) belongs to the hexagonal system and is dichroic. If viewed in linearly polarised white light
with the plane of vibration parallel to the c-axis (the optic axis) the crystal appears orange red.When rotated by
90� so that the plane of vibration is perpendicular to the optic axis the colour seen is purple red. Thismineral is
c-axis
direction ofvibration of incident light
direction ofvibration of incident light
(a)
(b)
(c)
e
ounpolarised light
Figure 4.18 Dichroism in tourmaline. (a) A plate of the mineral tourmaline, cut so as to contain thecrystallographic c-axis (the optic axis), transmits linearly polarised light differently depending upon the directionof vibration. The relative transmission factors are shown by the orthogonal pair of double-headed arrows. (b). Anobserver positioned above the crystal which is illuminated from belowwith linearly polarised light will see eithera dark crystal (b) or a clear crystal (c), depending upon the orientation of the slab with respect to the direction ofvibration of the light
Colour and the Optical Properties of Materials 150
negative uniaxial with no¼ 1.768, ne¼ 1.760 and birefringence 0.008; this latter value underlines the fact that
themagnitudeof thebirefringencedoesnot directly control theobservationofpleochroism. (The colour of ruby
is examined in more detail in Section 7.10.)
The reason for pleochromism is that the absorption of light is dependent both upon its direction in the crystal
with respect to the optic axes and its state of polarisation. Thus, all uniaxial and biaxial crystals show
pleochroism, as in these materials the absorption spectrum of a crystal with light polarised along one crystal
axis is different to the absorption spectrum when light is polarised along the other axes. The magnitude of the
effect does not depend upon the refractive indices or the birefringence of the crystal, but very strongly upon
crystal thickness. Because the effect is that of absorption, thin plates of crystal will usually show little change
in colour on rotation even for strongly pleochroic materials, although there are exceptions. One group of
strongly dichroicmaterials are sheet polarsmade from aligned arrays of organicmolecules. These show strong
absorption changes even in thin films. However, because all visible wavelengths are absorbed, the change is
from a light grey to dark grey/black and spectral colours are not seen. In fact, many polymer films possess quite
large degrees of birefringence because the manufacturing process tends to align the long molecules in one
favoured direction. This can be readily shown by examination between polars.
4.9 Nonlinear Effects
4.9.1 Nonlinear crystals
Imagine a bright green beam of light emerging from a 1 cm cube of a perfectly transparent crystal, with no
apparent electrical or other connections to it. This is certainly one of the most spectacular ways of colour
production and always impresses on first sight. What is happening is that a beam of invisible laser radiation in
the infrared is being converted by the crystal into a beam of green light. There are a number of ways that this
can come about. In this chapter, only pure undoped materials are considered and the phenomenon is called
frequencydoublingor secondharmonic generation (SHG). It is oftendescribed in termsof adding two identical
photons of frequencyn together, to produce a single photonwith double the frequency, 2n; that is, half the initial
wavelength. However, this description will be avoided so as to differentiate the process from up-conversion,
another way of adding two photons together so as to produce a photon of doubled frequency (Section 9.9). In
up-conversion an impurity ion acts so as to achieve the optical transformation.5 Frequency doubling utilizes
only the pure matrix and impurity dopants are not involved. Frequency doubling is a nonlinear effect.
There are two vital ingredients needed for the manifestation of nonlinear effects in crystals: a high electric
field and a matrix of the correct symmetry. Consider the electric field initially. A time-varying electric field,
such as that of a light wave, causes electronic polarisation in the crystal (Section 2.3). For electric fields of
normal intensity (in sunlight, E0� 102Vm 1), the bulk polarisation of a material P is a linear function of the
electric field E:
P ¼ e0wE ð4:1Þ
where e0 is the permittivity of free space and w is the dielectric susceptibility of the material. (To relate the
equation to quantities applicable at optical frequencies, note that the dielectric susceptibility is given by:
w ¼ er�1 ¼ n2�1
5 The language of the literature is often less than precise and frequency doubling is often called up conversion.
151 Polarisation and Crystals
where er is the relative permittivity and n the refractive index of a transparent phase.) This is easily understood
in a qualitative fashion. A transparent solid has strongly bound electrons. The electric field displaces these
slightly and the resulting displacement (i.e. polarisation) is a linear function of the field strength. Although this
serves perfectly well for ordinary light sources, it is only a first approximation. More exactly, the polarisation
can be written as a series:
P ¼ e0wð1ÞEþ e0wð2ÞE� Eþ e0wð3ÞE� E� Eþ � � � ð4:2aÞ
where w(1) is the linear dielectric susceptibility, w(2) the second-order dielectric susceptibility, w(3) the third-order dielectric susceptibility and so on.6 The polarisation is no longer a simple linear function of the electric
field. To a first approximation the vector complexities of Equation 4.2a can be ignored and it can be written in
scalar form as:
P ¼ e0wð1ÞE þ e0wð2ÞE2 þ e0wð3ÞE3 þ � � � ð4:2bÞ
where E is the magnitude of the electric field.
The first term is the ‘linear’ term and is the only term of relevance in traditional optics. The succeeding
‘nonlinear’ terms are important when light from ordinary sources is replaced by laser light pulses in which the
electric field E0 can reach a value above 108Vm 1. In such pulses, the electron cloud surrounding the atoms
in thematrix is considerably distorted. If the electron cloud deformation can be approximated as varying as the
square of the field, then the second term becomes appreciable. If the variation is best expressed in terms of a
cubic variation, then the third termmust be considered. At low field strengths, all give an approximately linear
response (Figure 4.19).
Electric field strength E
Pol
aris
atio
n P
linear, P ∝ E
2parabolic, P ∝ E
3cubic, P E∝
Figure 4.19 The variation of the polarisation with electric field strength for linear, quadratic and cubicdependence. At low fields all approximate to linear behaviour
6 Equation 3.2 is a vector equation that is most easily manipulated mathematically by tensor methods in which the terms w(1), w(2) and w(3)
are arrays of coefficients. For our purposes, the vector quantitiesP andEwill be treated as scalars and the terms w(1), w(2) and w(3) as singlevalued numbers.
Colour and the Optical Properties of Materials 152
Although the electric field strength is important in the observation of nonlinear properties of solids, the
observed polarisation of a crystal is also strongly influenced by crystal symmetry. In a centrosymmetric unit
cell (one that possesses a centre of symmetry), electronic polarisation in one part of the unit cell is equal and
opposite to that in another part of the unit cell. In such materials only the odd-order w terms w(1) and w(3) havenonzero values. In non-centrosymmetric crystals (those lacking a centre of symmetry), the second-order term
w(2) has a nonzero value and all terms are relevant. It is these latter types ofmaterial that are generally known as
nonlinear optical materials.
Nonlinear effects havemany implications for optical properties and are beingwidely exploredwith a view to
applications in numerous branches of photonics and optical engineering. Here, only the importance of
nonlinear effects in colour production is considered. The major effects in this category are summarized in
Scheme 4.1. The commonest nonlinear crystals for these uses are often labelled by acronyms rather than
chemical formula: ADP (ammonium dihydrogen phosphate, (NH4)H2PO4), KDP (potassium dihydrogen
phosphate, KH2PO4), BBO (beta barium borate, b-BaB2O5), LBO (lithium triborate, LiB3O5), AGS (silver
gallium sulfide, AgGaS2), AGSe (silver gallium selenide, AgGaSe2). Less-common nonlinear crystals are
usually called by their scientific names: lithium niobate (LiNbO3) and lithium iodate (LiIO3).
4.9.2 Second- and third-harmonic generation
The nonlinear terms in the polarisation equation allow photons to be added and subtracted in certain specific
ways to generate light frequencies not available from existing sources. Commonly, nonlinearity is used to
generate light of double the frequency, second-harmonic generation (SHG), of the inputwave, but light of triple
the frequency, third-harmonic generation (THG), has also been achievedusing the same technique. Frequency
doubling and tripling (SHG and THG) comes about in this way. The electric field associated with a light beam
P = ++ε0χ(1)E ε0χ(2)E2 ε0χ(3)E3
1 Dispersion 1 SHG: ω, 2ω
2 mixing of
ω 3 → ω1+ω 2
ω1-ω 2
ω1+ω 2,ω1, ω 2,
SFG: DFG:
1 THG ω, 3ω
3. OPO, OPA:
4. THG via SHG + SFG
Nonlinear colour production
SHG: Second harmonic generation
SFG: Sum frequency generation
DFG: Difference frequency generation
THG: Third harmonic generation
OPO: Optical parametric oscillator
OPA: Optical parametric amplifier
Scheme 4.1 Colour production using nonlinear effects
153 Polarisation and Crystals
is not steady, but varies sinusoidally. This variation can be expressed in terms of the angular frequency (see
Appendix A1.1) as:
E ¼ E0 cosðotÞ
where E is the magnitude of the electric field vector, E0 is the amplitude of the electric field vector, o is the
angular frequency of the oscillation and t is the time. If this is substituted into a scalar formofEquation 4.2b the
magnitude of the polarisation P is:
P ¼ e0wð1ÞE0 cosotþ e0wð2ÞðE0 cosotÞ2 þ e0wð3ÞðE0 cosotÞ3 þ � � � ð4:3Þ
The polarisation, thus, oscillates in a complex way, depending upon how many terms are of importance in
Equation 4.3. This equation can be rewritten as:
P ¼ AþBcosotþC cos2otþDcos3otþ � � � ð4:4Þ
An oscillating charge gives rise to an electromagnetic wave, and so each of the terms in Equation 4.4 can
then be thought of as the source of such a wave. The first term, surprisingly, implies that a static electric field
will form in a nonlinear material, when illuminated with a suitably intense laser beam. The second term gives
rise to a wave of the same frequency as the initial wave (that is, o) and is the normal interaction dealt with in
traditional optics. The main contribution to the constant B is w(1). The third-term constant contains w(2) andgives rise to a wave of double that frequency, 2o, which corresponds to SHG (Figure 4.20a). If w(2) is zero, asin all centrosymmetric crystals, C is zero and no SHG wave can form. The forth-term constant contains w(3)
and gives rise to a wave of tripled frequency, 3o, triple-harmonic generation (THG) (Figure 4.20b). (Note
that, from a practical point of view, frequency tripling is usually carried out rather differently, as described
below.)
The irradianceof thesewaves dependsupon themeasuredvalues of the dielectric susceptibilities.When light
froma suitable laser passes through a crystalwith appreciable second-order dielectric susceptibility, twobeams
may emerge, with frequencies ofo and 2o. (In fact the production of a second harmonic from a crystal when
illuminated by a laser is usually taken as a good test for the lack of a centre of symmetry.) One of the first
nonlinear crystals to be utilized for this purpose was potassium dihydrogen phosphate (KDP), which can
non-linear crystal
non-linear crystal
optic axis
optic axis
input wave
input wave
output waves
output waves
ω
ω
2ω
3ω
ω
ω
(a)
(b)
Figure 4.20 The (schematic) generation of (a) frequency-doubled (SHG) and (b) frequency-tripled (THG) outputusing a laser input and suitable nonlinear crystals
Colour and the Optical Properties of Materials 154
convert red 694 nm output from a ruby laser into ultraviolet light of wavelength 347 nm. The crystals used in
semiconductor lasers (Section 10.9), such as gallium aluminium arsenide (GaAlAs2), are nonlinear materials,
and the laser output can consist of botho and 2owaves in certain circumstances. Crystals ofBiB3O6 have been
used to obtain pure frequency tripling, as in this phase the centrosymmetric term w(2) is zero.
4.9.3 Frequency mixing
More complicated processes can also take placewhenmore than one beam irradiates a nonlinear crystal. If two
beams characterised by angular frequencies o1 and o2 are used, not only 2o1, 2o2 (the second- harmonic
frequencies from each beam), and 3o1, 3o2 (the third-harmonic frequencies from each beam), but also
o1 þ o2 ando1�o2 (the sum and difference frequencies) can all be produced. The production of the sum and
difference frequencies (Figure 4.21a) is known as frequency mixing sum frequency mixing (SFM) or sum
frequency generation (SFG) and difference frequency mixing (DFM) or difference frequency generation
(DFG). It is analogous to the formation of ‘beats’ easily heard when two sound waves mix.
non-linear crystal
optic axis
input waves output waves filter
2ω1ω1
ω22ω2
ω1 + ω2
ω1 − ω2
(b)
(b)
SHG
SFG
DFG
SHG
SHG: Second harmonic generation
SFG: Sum frequency generation
DFG: Difference frequency generation
non-linear crystal
optic axis
input wave
input waves
output waves
output wavesω
2ω3ω
ω mirror
Figure 4.21 (a) The (schematic) generation of sum and difference frequencies as well as doubled (SHG) andtripled (THG) output using a laser input and a suitable nonlinear crystal. (b) The (schematic) formation of THGusing SHG followed by SFG
155 Polarisation and Crystals
Frequencymixing can be understood in the following way. Suppose the crystal is irradiated with two beams
simultaneously so that:
E1 ¼ E01 cosðo1tÞE2 ¼ E02 cosðo2tÞ
The electric field in the sample is then:
E1 þE2 ¼ E01 cosðo1tÞþ E02 cosðo2tÞ
Substituting this into Equation 4.3 will produce a series for P which will contain terms containing the sum
and difference of the frequencies, cos(o1 þ o2)t and cos(o1�o2)t. These are the formal source of the two
outputwaves, one of frequency (o1 þ o2) and one of frequency (o1�o2). SFG iswidely used to obtainwaves
that are not readily produced by existing lasers. For example, infrared waves from a CO2 laser (l¼ 10.6mm)
and a Nd3þ :YAG laser (l¼ 1.06mm) can be combined in an Ag3AsS3 crystal, which is opaque in the visible
but transmits infrared, to give an output wave with l¼ 0.96mm.
THG is often accomplished by the use of SHG in tandemwith SFG (Figure 4.21b). Thefirst process produces
an output frequency 2o1, and the addition of 2o1 plus the unconverted beam of frequencyo1 using SFG gives
anoutputwaveof frequency3o1.A typical application involves thegeneration of ultraviolet light from infrared
laser output which is then used in LIDAR7 equipment for atmospheric surveying and the measurement of
atmospheric properties such as ozone content. For these purposes, Nd:YAG (yttrium aluminium garnet) or
Nd:YLF (yttrium lanthanum fluoride) lasers produce an initial output in the infrared wavelength range
(1300 925 nm) which is frequency doubled by an LBO (b-BaB2O3) crystal to a wavelength range of
750 463 nm and then tripled by a following LBO crystal to give an output of 433 310 nm.
4.9.4 Optical parametric amplifiers and oscillators
Optical parametric amplifiers and optical parametric oscillators are devices which use a nonlinear crystal to
produce andamplify awave fromaspecific input. From theprevious discussion it is seen that the introductionof
apair ofwavesof frequencieso1 ando2 into a suitablenonlinear crystal generates the sumfrequencyoutputo3.
That is:
o1 þo2 ¼ o3
In this process, two input frequencies unite to give a single output frequency. There is no inherent reasonwhy
this should not operate ‘backwards’, so that a sufficiently powerful initialpumpbeamof frequencyop produces
output waves of frequencies os, which is the desired lower energy output wave (called the signal) and a
redundant lower energy outputoi (called the idlerwave).On traversing the crystal, the powerfulop input pump
wave is gradually decomposed into two waves with angular frequencies os and oi as it crosses the crystal.
Conservation of energy implies:
op ¼ os þoi
The amount of light converted in a single pass is usually small, but is improved by repeated reflectionwithin the
nonlinear crystal in an oscillator design (Figure 4.22a). In an amplifier, a signal wave of angular frequency os,
7 LIDAR is the optical equivalent of radar and is an acronym of light detection and ranging.
Colour and the Optical Properties of Materials 156
which has a low irradiance, is passed through a nonlinear crystal in conjunction with the powerful pump
beam, op. The decomposition of the pump wave into an idler wave of frequency oi and signal wave of
frequency os is identical to oscillator operation. The frequency of the newly generated signal wave, with a
frequency os that matches the frequency of the input signal wave, adds to, and so amplifies, the signal wave
(Figure 4.22b).
In a suitable nonlinear crystal the refractive index encountered by the pump, signal and idler waves is a
continuous function of the angle of the waves to the optic axis. This means that there are a range of allowed
combinations of frequency generation that can occur.
The actual frequencies generated, os and oi, will, therefore, vary as the angle of the incident beam on the
nonlinear crystal is changed. This is known as tuning. The effect is considerable. For example, a commercial
oscillator using a b-BaB2O4 (BBO) crystal using frequency-doubled Nd3þ :YAG laser output of 532 nm pump
radiation can produce a signal wave varying in wavelength from approximately 650 to 1060 nm and an idler
wave varying in wavelength from approximately 1060 to 3000 nm by rotation of the crystal over an angle
of just 2�.
4.10 Frequency Matching and Phase Matching
In principle, any non-centrosymmetric crystal can be used for the generation of other colours using harmonic
and sum and difference methods. As the incident beams traverse the crystal they are gradually converted from
one angular frequency to the others, such that:
o1 þo2 ¼ o3 ð4:5Þ
This equation sets out what is known as the frequency matching condition, which must be fulfilled in all
cases. However, the newly created waves are generally out of phase with each other and the incident beam.
Beams with a phase difference will interfere with each other. A result of this interference, the intensity of the
new rays emerging from the crystal is very low due to destructive interference. Destructive interference can
only be prevented if all of the beams remain in phase. In the crystals thatwe are speaking of, this canbe achieved
by making the refractive indices and angular frequencies agree with the equation:
non-linear crystal
non-linear crystal
optic axis
optic axis
signal wave ω s
(a)
(b)
pump wave ω p
pump wave ω p
residual pump wave ω p amplified signal wave ω s
signal wave ω s residual pump wave ω p
idler wave ω i
idler wave ω i
mirrormirror
Figure 4.22 (a) The (schematic) operation of an optical parametric oscillator; an intense pump wave isconverted into a signal wave and an idler wave. Repeated reflection increases the degree of conversion achieved.(b) The (schematic) operation of an optical parametric amplifier; input consists of a strong pumpwave and aweaksignal wave. Conversion of the pump into two waves, one of which matches the signal wave, results inamplification of the latter
157 Polarisation and Crystals
n1o1 þ n2o2 ¼ n3o3 ð4:6Þ
This is known as the phase matching condition, which can be compared with the frequency matching
condition given by Equation 4.5. As an illustration, consider SHG. In this case:
o1 ¼ o2 ¼ o o3 ¼ 2o
n1 ¼ n2 ¼ n
To satisfy Equation 4.6:
noþ no ¼ n32o
which implies that the refractive index of the crystal for light of frequencyomustmatch the refractive index of
the crystal with respect to output with a frequency 2o. In an ordinary material, the refractive index decreases
with increasing wavelength (Section 2.6). As:
o ¼ 2pcl
the refractive index at the frequency 2o will, therefore, be greater than that at o. Although obtaining phase
matchingwould appear to be a tall order, it can be achieved in someuniaxial and biaxial crystals. Recall that if a
beam of light is passed into a uniaxial or biaxial crystal it splits into two parts, the ordinary and extraordinary
rays, each of which encounters its own unique refractive index (Sections 4.5 and 4.6). This provides a solution
to the problem. Take a hypothetical example. In a uniaxial positive crystal the refractive index n0e encounteredby the extraordinary ray is greater than the refractive index of the ordinary ray no. It is conceivable, therefore,
that the refractive index encountered by the ordinary ray at wavelength l (angular frequency 2o) could be
identical to the refractive index encountered by the extraordinary ray, with wavelength 2l at an angular
frequencyo (Figure 4.23a). It is then necessary to find a crystal direction inwhich the refractive indices for the
fundamental and the frequency-doubled beam match the phase matching angle. The same strategy can be
applied in the case of a uniaxial negative crystal, remembering that in this case n0e is less than no (Figure 4.23b).Thus, the general principle is to use a uniaxial or biaxial crystal and to set the crystal at an angle to the incident
wavesuch that the ordinary and extraordinarywaves (that is, theo and2owaves) encounter the same refractive
index. Whether the input wave is taken as the ordinary or extraordinary component will depend upon the
refractive indices of the frequency-doubling crystal. This strategy will not work with all crystals, but it is
possible in some.
The phase matching angle for second-harmonic-generated waves is given by:
uniaxial positive crystals; ne > no
sin2�m ¼ neð2lÞnoðlÞ
� �2noðlÞ2�noð2lÞ2neð2lÞ2�noð2lÞ2
uniaxial negative crystals; ne < no
sin2�m ¼ neðlÞnoð2lÞ
� �2noðlÞ2�noð2lÞ2noðlÞ2�neðlÞ2
Colour and the Optical Properties of Materials 158
where �m is the phasematching angle, no is the refractive index of the ordinary ray at wavelengths l and 2l andne is the refractive index for the extraordinary ray at wavelengths l and 2l. (Specifically, these formulae are for
‘Type I’ phase matching, which is the condition that optimizes the birefringence of the nonlinear medium;
see below.)
Because the refractive index is temperature sensitive, crystals have to be placed in temperature-controlled
cells to achieve reasonable amounts of conversion. This allows for an alternative method of tuning the output.
Instead of rotating the crystal small amounts so as to achieve perfect phase matching, the temperature can be
varied to obtain the same goal. Which of these alternatives is preferred will depend upon a number of factors.
In some situations temperature variation is preferred to orientation variation.
An important point has been glossed over in the description above the polarisation of the beams. The
ordinary ray and the extraordinary ray passing through a low-symmetry crystal are polarised at right angles to
each other and the polarisation direction is related to the direction of the optical axis (Sections 4.5 and 4.6). It is
clear, therefore, that in all nonlinear optical devices (not just in SHG), not only must the beam directions with
respect to the optic axis be precise, but the polarisationmust also be correct. This leads to a number of different
phase-matching schemes which quantify the relative polarisation of the pump, signal and idler waves with
respect to the optical axis of the nonlinear crystal. (The designation of Type I phase matching above gives
specific information on the relative polarisation directions of the input and output waves.)
A second point of importance also needs to be mentioned. The paths of the ordinary and extraordinary rays
in a nonlinear crystal are not parallel, but diverge (Figure 4.10). This effect is calledwalk off, and is quantified
by the angle between the two rays, thewalk-off angle. This serves as ameans of separating the two rays, but also
will drastically lower efficiency unless compensated.
Ref
ract
ive
inde
xR
efra
ctiv
e in
dex
no ne
WavelengthAngular frequency
λ2ω
2λω
uniaxial positivene > no
ne no
WavelengthAngular frequency
λ2ω
2λω
ne > no
uniaxial negative
(a)
(b)
Figure 4.23 Phase matching. (a) Uniaxial positive crystal; the refractive index of the e-ray of wavelength 2l canbe equal to the refractive index of the o-ray of wavelength l. (b) Uniaxial negative crystal; the refractive index ofthe e-ray of wavelength l can be equal to the refractive index of the o-ray of wavelength 2l
159 Polarisation and Crystals
A final point to note is that there are other ways of obtaining phase matching. Strictly speaking, the method
described above is called birefringence phase matching. Clearly, it is limited both to birefringent crystals and
to the subgroup of these that allow the refractive indices to be matched.
4.11 More on Second-Harmonic Generation
4.11.1 Polycrystalline solids and powders
The drawback of the method of SHG for the generation of colours so far described is that expensive large
crystals are needed and precise phase-matching angles or temperaturesmust bemaintained for intense second-
wave output. Large crystals are needed because the output wave is generated successively as the input
wave travels across the crystal. However, if a very high intensity output wave is not required then non-
centrosymmetric power (polycrystalline) samples will generate second-harmonic radiation.
The observed intensity will depend upon the crystal size. Small crystallites will not suffer from the
destructive interference that necessitates phase matching. This makes materials that cannot be phase matched
using birefringence available both for study and application. As the crystallite size increases, so the second-
harmonic output will increase, but at some stage destructive interference will begin. This point is generally
specifiedby thecoherence length8Lc,which is thedistanceoverwhich theoand2owavesbecomeout ofphase
by half a wavelength (p radians), given by:
Lc ¼ lo4ðn2o�noÞ
where l is the wavelength of the fundamental wave and n2o and no are the relevant refractive indices for the
fundamental and frequency-doubled waves.
Use of polycrystalline materials for SHG is widespread. Pure powders can be pressed into thin layers or
fabricated as thin films by a variety of techniques. Alternatively, particles can be mechanically distributed in
glasses or other amorphousmaterials such as aerogels, or formed by the partial recrystallisation of glasses. This
approach yields materials which have the formability of glass yet maintain SHG potential. A solid composed
ofmany small grains (i.e. crystallites) will also give appreciable SHGoutput. This approach has been usedwith
the important group of non-centrosymmetric III V and II VI semiconductors that includes GaAs and ZnSe.
These arewidelyusedoptoelectronicmaterialswith highvaluesofw(2), andalthough large crystals are available,they cannot be phase matched using birefringence. The use of a solid composed of partially oriented small
grains gives rise to output waves that are all roughly (but not perfectly) in phase, resulting in a useful output.
The generation of frequency-doubled visible light, say green light from the 1064 nm output of a Nd:YAG
laser, makes optical microscopy possible. This technique has been used to image crystallites in glass and other
amorphous materials. Naturally, SFG that leads to visible output can also be used.
4.11.2 Second-harmonic generation in glass
Glasses are centrosymmetric and the value of w(2) for any glass is zero. A glass, therefore, should not give rise
to SHG. However, it is found that intense infrared laser light pulses with a wavelength of 1064 nm from a
Nd3þ :YAG laser sent down an ordinary commercial optical fibre eventually produces SHG. After an hour or
so a green light with a wavelength of 532 nm starts to appear along with the input infrared. As time goes by the
intensity of the green light increases, and after 10 h or so is quite prominent.
8 This terminology is unfortunate. Traditionally, in optics the coherence length refers to the length of the wave train emitted by a light
source in which the waves are all in step; in effect the length over which the wave can be considered to be a single sinusoid.
Colour and the Optical Properties of Materials 160
How is it that a nonzero second-order dielectric susceptibility term has evolved during the infrared
irradiation? The major sequence of steps occurring seems to be these. The intense electric field of the
infrared radiation is strong enough to cause ions in the glass tomigrate in a process akin to ionic conductivity.
In the region near to the core cladding interface (see Section 2.9) the ionic displacements give rise to a
permanent charge separation. The resulting permanent electric field Edc that builds up in the boundary regionhas been found to be as large as 108 Vm 1. It is this intense field that is partly responsible for the SHG. In
addition, species migration results in the formation of defects in the glass. These contribute to the electric
field and also result in a loss of symmetry. The two effects result in a nonzero w(2) term which in turn allows
SHG to occur.
Since the original observation, SHG inmany glasses has been detected. For example, pure silica glass plates
heated to a temperature of between 250 and 325 �C and at the same time subjected to a static electric field of
about 3 kV, which is maintained while the glass cools (called thermal poling), also show SHG in a region of
about 3 mm close to the surface adjacent to the positive contact. This region has been found to be depleted in
ionic constituents, and it seems that ionicmovement in the applied voltage both establishes apermanent electric
field at the surface and creates defects in the phase. Heating the glass at a temperature of a few hundred degrees
in the absence of an applied voltage allows for ionic diffusion to re-occur, cancels the effect and returns theglass
to its original nonactive state.
Thermal poling is now frequently used to make SHG in homogeneous glass possible.
4.11.3 Second-harmonic and sum-frequency-generation by organic materials
Crystals that are built of organic molecules behave in exactly the same way as inorganic crystals and the
considerations given above apply. However, organic molecules themselves can display nonlinear effects. The
polarisability p of a molecule is written, in scalar form, as:
p ¼ p0 þ aE þ að2ÞE2 þ að3ÞE3 þ � � �where p0 is the permanent dipole (if any) on the molecule, E is the electric field, a is the molecular
polarisability, a(2) (or b) is the first hyperpolarisability, a(3) (or g) is the second hyperpolarisability and soon. Clearly, a(1), a(2) and a(3) are the molecular equivalents of the macroscopic terms w(1), w(2) and w(3). Notethat here E represents the field experienced by themolecule, themicroscopic field. In general, this will differ
from the external field applied to the collection of molecules, the macroscopic field, as it will include a
contribution from the neighbouring polarised molecules.
One great advantage ofmolecular nonlinearity is that thewell-known techniques of organic synthesis can be
used to modify the hyperpolarisability values at will. The addition or subtraction of polar groups, and their
placement relative to themain body of the molecule, can all be adjusted precisely. In a crystal, the requirement
for a non-centrosymmetric arrangement still applies for the case of SHG. However, molecules dispersed in
liquids can show considerable bulk nonlinear effects if the molecules are partly aligned by an external electric
field poling. The alignment need not be perfect to obtain appreciable bulk values of w(2). Similarly, nonlinear
polymers or copolymers containing nonlinear molecules can be fabricated by alignment in an electric field
during fabrication.As in the case of powders, below a certain thickness, phasematching becomes irrelevant for
polymer films or molecular solutions.
Theoverall polarisabilityPof themediumcontaining thenonlinearmolecules is, to afirst approximation, the
sumof the contributionsof the individual species present. For the second-order parameterw(2), for example, it is
possible to write:
wð2Þ ¼ N1hað2Þ1 iþN2hað2Þ2 iþN3hað2Þ3 iþ � � �
161 Polarisation and Crystals
whereN1, etc. are the number ofmolecules of type 1 (units m 3)withmolecular hyperpolarisability að2Þ1 and so
on. The angle brackets represent the average value of the hyperpolarisability. In a crystal this average will
simply be the value for a single molecule, but in a liquid the average will depend upon the temperature and the
amount of thermal jostling each molecule undergoes. The equation can also apply to surface species (see
below), in which case the units of N are m 2.
4.11.4 Second-harmonic generation at interfaces
Interfaces are non-centrosymmetric: one side of the interface differs considerably from the other. This means
that atoms or molecules situated in the interface can be used for SHG because an atom or a molecule in the
interface is exposed to quite different forces from one side comparedwith the other. This feature has long been
exploited. Thus, at the surface of a metal the nearly free electrons act as the SHG oscillators, at semiconductor
surfaces the incomplete bonds act in the sameway, while manymolecular species absorbed onto a surface can
generate waves by virtue of the arrangement of bonds along the molecular length. Moreover, the SHG signals
are fairly easy to detect because the bulk phases, be they solid, liquid or gas, generally are SHG inactive. Thus,
it is routinely possible to detect less than a monolayer covering of a surface using SHG signals.
There are many applications of the technique. Interfacial reactions, including absorption and desorption,
corrosion, and the dynamics of electrode processes in electrochemical cells, are all open to study using
interfacial SHG. The orientation of molecules absorbed onto a surface is also accessible with SHG. In
interfaces, including biological cells, at which ordering takes place, strong signals can be generated, making
opticalmicroscopy possible. SHG allows the degree of chirality (i.e. right or left handedness; see Section 4.12)
present to be determined.
4.11.5 Second-harmonic microscopy
The second-harmonic signal generated in a material can be used as the light source in optical microscopy,
provided that the harmonic lies within the visible region, if the eye is the detector used. The technique has been
mostly used in biologically oriented studies. Many molecules utilized in biological tissue are birefringent and
are arranged intomoreor lessorderedarrays, often at interfaces. These provide ideal environments for imaging.
One advantage of using SHG is that light is not absorbed by the tissues, and so tissue damage that might occur
with powerful illumination is avoided.Moreover, disordered or amorphousmaterials are not involved,making
for greater discrimination in suitable subjects. Thus, SHG microscopy has been used to form high-resolution
optical images of collagen and similar muscle tissue and the study of the retina in subjects suffering from the
blindness-causing disease glaucoma (Figure 4.24). (See this chapter’s Further Reading, formore information.)
4.12 Optical Activity
4.12.1 The rotation of polarised light
One of the most intriguing results obtained by scientists trying to unravel the physics and chemistry of natural
materials during the nineteenth century was the phenomenon of optical activity. For example, crystals of salts
of the two acids tartaric acid and racemic acid were well known even hundreds of years ago and could be
collected from old wine casks. The sodium salts of these two acids, sodium tartrate and sodium racemate,
seemed to be chemically and physically identical. However, if linearly polarised light was passed through a
solution of the tartaric acid salt the plane of polarisation rotated to the right as viewed by an observer looking
towards the light source (Figure 4.25). The amount of rotationwas as good a physical property of the compound
Colour and the Optical Properties of Materials 162
Figure 4.24 Second-harmonic-generated image of an eye. The top left image is of a single field of view from asingle section. The top right image is a composite of overlapping fields to constitute an entire section. Severalhundred sections were reconstructed using Almira software to produce the bottom image. The green light is SHGlight from collagen and the red light is from two-photon emission from elastin-enriched collagen. [Reproducedwith permission from Professor D. J. Brown, Eye Institute, School of Medicine, University of California, USA]
163 Polarisation and Crystals
as, for instance, the melting point, and it could be used for characterisation purposes. The puzzle was that the
corresponding salt of racemic acid was optically inactive and caused no rotation.
The resolution of the problem was glimpsed when Pasteur made a painstaking optical examination of
sodium racemate crystals. In 1848 he announced that these contained equal numbers of two forms, one ‘right-
handed’ and one ‘left-handed’,meaning that they had the same relationship to each other as a left-hand glove to
a right-hand glove or an object and its mirror image. Solutions of the two crystal types rotated the plane of
polarisation by equal amounts, but in opposite directions. The compound sodium racemate could be described
as a mixture of two forms of sodium tartrate, each of which rotated polarised light in equal and opposite
directions. One of these was identical to the natural material described above, while the other appeared not to
occur in isolation.
The process of dissolution separates the solid crystals intomolecules or ions. It was clear, therefore, that this
example of optical activity was a feature which needed to be explained at a molecular level and not entirely at
the crystallographic level. Since then it has long been established that anymolecule that can exist in twomirror-
image-related forms that cannot be superimposed one upon the other is optically active. They are referred to as
chiralmolecules. Themirror imagemolecules are called enantiomers and in organic chemistry are also known
as optical isomers. Enantiomers, therefore, differ in one physical property: they display optical activity. One
formof a chiral moleculewill rotate the plane of polarised light in one direction and its enantiomerwill rotate it
in the opposite direction. The form of molecule which rotates the plane of polarisation to the right is labelled
dextrorotatory. The formofmoleculewhich rotates the plane of polarisation to the left is called laevorotatory.
Enantiomers are efficient polarisation rotators. Mixtures of enantiomers in equal proportions will produce
no resultant rotation of the plane of linearly polarised light and are called racemicmixtures, after the ‘racemic
acid’ of Pasteur.
In organic compounds, optical isomers occur whenever four different groups are attached to a tetrahedrally
coordinated central carbon atom, making it a chiral carbon atom or chiral centre (Figure 4.26). Although it is
not easy to see from drawings that the two structures cannot be superimposed, the construction of a simple
model (putty and matchsticks) will convince you.
The amount by which the linearly polarised light is rotated by an optically active material depends upon
the number of chiral centres in the beam path, the wavelength of the light used to measure the effect and the
temperature. Optical activity is expressed in terms of specific rotation [a]tl measured under standard
conditions, which includes the wavelength l of the light used and the temperature t of the optically active
material.9 For solutions, the specific rotation is given by:
α
observersolution
l
polariser analyserlight source
Figure 4.25 Schematic diagram of the rotation of the plane of linearly polarised light by a solution of opticallyactive molecules. The instrument used for the accurate measurement of the rotation is a polarimeter
9 Specific rotation is also termed the rotatory power r, but this is a poor descriptor, as the units are angle per unit length, not those of power,Watts.
Colour and the Optical Properties of Materials 164
½a�tl ¼adc
ð4:7Þ
where a is the rotation observed over a path length d (dm) for a solution of concentration c (g cm 3). For a pure
liquid theconcentration c is replacedby thedensityr (g cm 3) inEquation4.7.For crystals, the specific rotation
ismeasured as the rotation permillimetre of crystal. If the plane of polarised light is rotated clockwisewhen the
observer looks towards the light source then the value of specific rotation is positive and when the rotation is
anticlockwise the specific rotation is negative.
Although a molecule with a single chiral centre exists as a left- or right-handed form, more complexity is
introduced inmolecules that contain several such centres. The results can then lead to increased optical activity
(i.e. increased specific rotation), reduced optical activity or no optical activity at all. Tartaric acid, Pasteur’s
crystals, is of thismore complex type because themolecules contain two chiral carbon atoms.These can ‘cancel
out’ internally in the molecule so that three molecular forms actually exist: the two optically active mirror-
image structures, which cannot be superimposed on each other (laevorotatory and dextrorotatory), and the
optically inactive form, calledmeso-tartaric acid,which canbe superimposedon itsmirror image (Figure 4.27).
The nomenclature is thus that if the optical activity is cancelled internally by the action ofmore than one chiral
centre then the form is labelledmeso-, as inmeso-tartaric acid. If optical activity is lost because equal numbers
of (þ ) and (�) optically active enantiomers are present then the term used is racemic or racemate, as in
racemic-tartaric acid.
Inorganic molecules with tetrahedral or octahedral bond geometry can also form enantiomeric pairs. In
addition, it should be noted that althoughmanyoptically active crystals contain optically activemolecules, this
is not mandatory. In fact, the first instance of optical activity was noted in quartz by Arago in 1811. Crystals of
quartz, one form of silicon dioxide (SiO2), occur in left- or right-handed forms although no molecules are
present. Quartz is hexagonal (i.e. uniaxial), with the optic axis parallel to the crystallographic c-axis. In this
material, the corner-shared [SiO4] tetrahedra that make up the crystal form helices (either right handed or
left handed) along the optic axis. The plane of polarisation of linearly polarised light directed through a slice
of crystal along the optic axis is rotated left or right, depending upon the handedness of the crystal. Similar light
directed normal to the optic axis shows no change.
(S )-(+)-alanine
occurs in nature
* chiral C atom
NH2 NH2
CH3 CH3C C
COOH COOH
m
* *H H
(a)(b)
Figure 4.26 The enantiomers of the amino acid alanine as examples of a chiral molecule. The chiral carbon atomin each molecule is marked C� and is coordinated to the other groups by tetrahedrally arranged chemical bonds.Only one form occurs naturally, that in (a), (S)-(þ )-2-aminopropionic acid. The form in (b), (R)-( )-2-aminopropionic acid, can be synthesised
165 Polarisation and Crystals
Enantiomers display identical physical and chemical properties except when they react with other chiral
molecules. This has profound effects for life, because many biologically important molecules are chiral.
Naturally occurring amino acids are ‘left handed’, while naturally occurring sugars are ‘right handed’. The
molecules important to life on Earth are thus described as homochiral. (It seems that this bias is not restricted
to life on Earth. Studies of theMurchison andMurray meteorites, reported from the late 1990s onwards, show
that they also contain a preponderance of left-handed amino acids.)
The differences in biological and pharmacological activity between two enantiomers can be pronounced.
Drugs and pharmaceuticals derived from natural products are often chiral and the two enantiomers differ
considerably in activity, one perhaps being beneficial and one being nonactive or even toxic. The sensation of
the odour of caraway, for example, is triggered by the left-handed enantiomer of limonene and that ofmandarin
oranges by the right-handed isomer. Vitamin C prevents the disease scurvy; the other enantiomer of this
substance is biologically inactive. Such a list could be extended indefinitely.
4.12.2 Circular birefringence and dichroism
The occurrence of optical activity is related to the polarisation of the incident wave. Taking the incident beam
of linearly polarised light as made up of two equal and oppositely circularly polarised beams (Section 4.1), in
a chiral material the electric field is given by:
E� ¼ ER þ e2�iEL
where R represents the right circularly polarised component andL the left circularly polarised component. The
relative phase between the two polarisation modes is 2� and the orientation of the composite linear polarised
*chiral C
COOH
* * *
* * *
H
laevorotatory dextrorotatory inactive
OH
mirror
Figure 4.27 The three forms of tartaric acid. Two of these, the laevorotatory and dextrorotatory forms, aremirror images and are optically active. The third form is inactive. The chiral carbon atoms in each molecule aremarked C�. The chemical bonds formed by these atoms have a tetrahedral geometry
Colour and the Optical Properties of Materials 166
beam is �. In an optically active phase the refractive index of the left circularly polarised ray nLwill be differentthan that of the right circularly polarised ray nR. The difference in refractive indices is given by:
Dn ¼ nL � nR
so that:
Dn is positive for nL > nR
Dn is negative for nL < nR
ThevalueofDn, the circular birefringence, is characteristic of thematerial and is related to the specific rotation.
The phase difference between the two polarisation modes, 2D�, after travelling a distance d in the chiral
medium is:
2D� ¼ 2pdDnl0
where l0 is the vacuum wavelength of the light. The rotation that the linearly polarised beam suffers is D�,given by:
D� ¼ pdDnl0
D� positive (nL > nR) represents dextrorotation (clockwisewhen the observer looks into the beam).D� negative(nL < nR) represents laevorotation (anticlockwise when the observer looks into the beam).
For a crystal, if d is measured in millimetres then the value of [D�/d] is equal to the specific rotation
[a]lt.
There will be a slight difference in the absorption coefficients of the incident right-hand and left-hand
circularly polarised light as it travels through an optically active crystal:
Dk ¼ kL � kR
where kL is the absorption coefficient for left-handed polarised light and kR is the absorption coefficient
for right-handed polarised light. By analogy with normal dichroism (Section 4.8), Dk is called circular
dichroism.
The specific rotation of a material is generally dependent upon wavelength. For manymaterials the specific
rotation dispersion can be given by Boltzmann’s equation:
½a�lt ¼A1
l2þ A2
l4
where A1 and A2 are experimentally determined constants.
Optically active compounds are not coloured by virtue of this property. However, the optical activity is
translated into colours when these compounds are viewed between polars. This is due to interference between
167 Polarisation and Crystals
the rotated and nonrotated components of the light beam after they have passed through the analyser, similar to
that described with respect to thin crystal plates (Section 4.7).
4.13 Liquid Crystals
4.13.1 Liquid-crystal mesophases
Early experiments established that normal liquids were isotropic and had no effect upon the polarisation of
light. Concurrent with this, it was observed that some organic crystals derived from cholesterol seemed to have
two ‘melting points’. For example, cholesteryl benzoate appeared to show a lower ‘melting point’ (when the
crystals turned into a cloudy liquid) and an upper ‘melting point’ (when the liquid became clear) separated by a
temperature interval of 33K.
The cloudy region was described as consisting of one or more mesomorphic phases or, more usually now,
mesophases.When studiedwith a polarisingmicroscope themesophase region, although certainly liquid-like,
had a noticeable effect upon polarised light and seemed to behave rather like a low-symmetry crystal. Because
of this, these curious materials were referred to as liquid crystals. The higher temperature clear liquid phase,
formed above the second ‘melting point’, containsmoleculeswhich are truly random in direction and the liquid
formed has no effect on polarised light.
A century ormore of investigation has shown that the liquid crystals first investigated aremade up of needle-
like or calamitic rigid molecules (from the Greek kalamos, reed). On raising the temperature a liquid crystal
disorders in a number of steps, rather than all at once as is normal for a solid liquid transition. The structures
which formwithin themesophase region are complex and depend upon intermolecular forces and temperature,
as well as on the geometry of the components. Above the first ‘melting point’ the molecules lose the strictly
ordered intermolecular spacing typical of a normal crystal but still retain a partial degree of order. One of
the commonest forms of disorder is found when the molecules retain a roughly parallel orientation but the
geometric centres of the molecules are arranged at random as in a conventional liquid (Figure 4.28a). The
preferred direction along which the molecules align is called the director. This structure defines a nematic
liquid crystal, (from the Greek nematos, thread), so called because, when viewed in polarised light, long dark
threads appear to occur throughout the bulk. These dark thread-like lines are optical effects caused by linear
defects calleddisclinationswhich run through the structure.Most nematic liquid-crystalmesophases haveonly
one optical axis and so are uniaxial. A considerable number of liquid-crystal phases retain the same roughly
parallel orientation of the linear molecules, but they aggregate into sheets. This mesophase structure is that of
a smectic liquid crystal (Figure 4.28b).
This behaviour exactly parallels the situation to be found in Polaroid (Section 4.3). The effect on light is due
to the presence long organic molecules, which act as uniaxial units and preferentially absorb one vibration
direction of light comprising the incident beam. When the molecules are more or less aligned, as in Polaroid
sheets or in liquid crystals, the effect of each molecule on an incident light beam is cumulative and the
transmitted light is, to a greater or lesser extent, depending upon the degree of order, polarised.When the same
molecules are randomly distributed the overall effect upon the polarisation of the incident light cancels and
no polarisation is observed.
More recently, liquid-crystal-like behaviour has been obtained frommaterials which are built up from disc-
shaped molecules. To differentiate them from the calamitic molecules described above, disc-like molecular
liquid crystals are called discotic. Although layers of molecules in a smectic arrangement are unknown to date
in discotic mesophases, disordered columns of discs can occur to form columnar liquid crystals.
Because nematic liquid crystals behave as uniaxial materials they can generate colours in polarised light in
the same way as uniaxial crystals described above. However, some forms of liquid crystals, cholesteric or
twisted nematic phases (described in Section 6.9), can produce colour directly by diffraction.
Colour and the Optical Properties of Materials 168
4.13.2 Liquid-crystal displays
The orientation of the molecules in a mesophase is easily influenced by external disturbances such as electric
fields. This has led to the most important use for liquid crystals in displays. These were first invented in the
1960s and subsequently developed intensively over the remainder of the twentieth century.
Liquid-crystal displays (LCDs) were originally introduced on portable calculators and digital watches as
black-on-grey images and are still widely used in clocks, watches, calculators and many other displays where
the primary purpose is to display figures (Figure 4.29). Liquid crystals are not, themselves, coloured and do not
emit light, so that external illumination is required. However, the molecules present in the liquid crystal have
two important and exploitable properties. The molecules can alter the orientation of the direction of
polarisation of the source light and the alignment of the molecules is easily changed by an externally applied
electric field. This means that an electric field can easily change the orientation of an entry beam of polarised
light.
These attributes apply to a liquid-crystal film which is at the heart of these displays. The film is sandwiched
between two glass sheets which are bounded by crossed polars (Figure 4.30a). White unpolarised background
lighting provides the illumination. The liquid-crystal film is divided into pixels by grids of transparent
conducting electrodes imprinted upon the glass sheets. When no voltage is applied to the electrodes the
molecules in the liquid-crystal layer remain in the orientation originally imposed during manufacture. The
polarised light beam from the entry polar is not rotated on passing through the liquid-crystal layer
(Figure 4.30b). In this case the light is blocked by the exit polariser and the pixel appears dark. Applying
a voltage to the electrodes causes the liquid-crystal molecules to rotate and in so doing to change the plane of
(a)
(b)
director
director
Figure 4.28 The structure of calmitic liquid-crystal mesophases (schematic): (a) nematic; (b) smectic
169 Polarisation and Crystals
polarisation of the linearly polarised light beamby90� (Figure 4.30c). The polarised light ‘follows’ themolecules
and the plane of polarisation is rotated. It is then passed by the exit polariser and so the pixel looks bright.
Use of colour filters extended this technology to include small-area colour displays on portable computers
and digital cameras (Figure 4.31a and b). The relatively poor quality of these earlier screen displays (see
figures) was less important than the fact that they were small, coloured and portable. Rapid improvements in
performance has meant that at present, LCD screens are the norm for small portable displays such as cameras
and phones, aswell as computer displays and television and have replaced cathode-ray tube technology almost
completely (Figure 4.31c). In colour displays each pixel is composed of three subpixels, each of which has a
colourfilter imposedbefore thefinal polariser.The fabricationof these colourfilters is intricate and involves the
dispersion of an organic pigment in a clear polymer substrate. In order to obtain a full-colour image three
different pigments need to be used, corresponding to the three primary colours. Each of these is allocated to its
appropriate subpixel. The colour seen by a viewer is additive and the primary colours used are red, green and
blue. The display is usually viewed against a black background which enhances the colour contrast.
There are two ways of controlling the output of the device. In a passive display, used for small screens and
especially black-and-white displays on clocks andwatches, the grid of electrodes is fixed and eachpixel is ‘open’
or ‘shut’ depending upon the state of the voltage applied to the electrode grids. For colour displays, especially
those showing moving images, such as television, the display is powered using an active matrix. In this
technology, the fixed grid of ruled electrodes is replaced with a grid of transistors connected to each electrode.
One transistor controls one pixel and each pixel can then be switched independently of any others.
The light source in thedisplay is of importance.As liquidcrystals donot generate light, the light source canbe
daylight or artificial.Whendaylight is the source, as it is formany simple clock,watch andhand-held calculator
displays, the liquid-crystal matrix is backed by a mirror. Daylight then traverses the unit and is reflected back
again to the viewer. This simple solution is unsuitable for active displays such as television. In these, a light is
provided behind the liquid-crystal matrix. The light then passes through the unit and is viewed directly. The
quality of the light source is important in controlling the perceived performance of the screen. Direct-view
Figure 4.29 Black-and-white LCD
Colour and the Optical Properties of Materials 170
incomingunpolarisedwhite light
polariser on glass sheetvertical electrodes
liquid crystal layerhorizontal electrodes
polariser on glass sheetoutgoing light
to viewercolour filter (optional)
(a)
(b)
no lighttransmitted
lighttransmitted
horizontallypolarised
light
liquid crystal molecules
vertical polariser
(c)
Figure 4.30 LCDs (schematic). (a) The arrangement of the components in an LCD. (b) In one state, the liquid-crystal film does not rotate the plane of polarisation of the light traversing it and no light is passed by the secondpolariser. (c) In a second state, the liquid-crystal film does rotate the plane of polarisation of the light traversingit and light is passed by the second polariser
171 Polarisation and Crystals
Figure 4.31 (a, b) The colour LCD on a digital 35mm camera (1999). The sharpness and colour rendition ofthe image is rather poor and pixels of the display are clearly visible. The LCD image size is approximately3.5 cm� 2.8 cm. (c) Liquid-crystal computer display (2004) showing far superior resolution and colour comparedto (a) and (b). The display dimensions are approximately 34 cm� 27 cm
Colour and the Optical Properties of Materials 172
television screens havefluorescentwhite light bulbs installed behind a light-diffusing screen, to give as uniform
white light-emitting background. More recently, light emitting diode (LED) illumination (Section 10.8),
behind the LCD, alsowith a light-diffusing screen, has been introduced. These give a wider colour gamut than
fluorescent back lighting, and also allow for a thinner screen.
Further Reading
Polarisation is described by
E. Hecht, Optics, 4th edition, Addison-Wesley, San Francisco, 2002, Chapter 8.
B.E.E. Saleh,M.C.Teich,Fundamentals ofPhotonics, JohnWileyandSons, Inc.,NewYork, 1991,Chapter 6.
Acollectionof classicpapersonpolarised light,which includes reprints of studiesbyHuygens andNewton, and
which makes fascinating reading is
W. Swindell (ed.), Polarised Light, Dowden, Hutchison and Ross, Pennsylvania, 1975 (distributed by John
Wiley and Sons).
The easiest route to further information about polarisers is to visit the websites of optical component
manufacturers. For example, enter ‘dichroic sheet polarisers’ into any search engine to find up-to-date
information. An extremely interesting (and well worth reading) account of the crystal structure of
herapathite, the active polarising material in Polaroid, together with historical references, is
B. Kahr, J. Freudenthal, S. Phillips, W. Kaminsky, Science 324, 1407 (2009).
Crystal structures are described by
J. V. Smith, Geometrical and Structural Crystallography, Wiley, New York, 1982.
R. J. D. Tilley, Crystals and Crystal Structures, Wiley, Chichester, 2006.
The relationship between crystal properties and light is summarized by
F.D. Bloss,Crystallography andCrystal Chemistry, Holt Rinehart andWinston, NewYork, 1971, Chapter 11.
Figure 4.31 (Continued )
173 Polarisation and Crystals
R.E. Stoiber, S. A.Morse,Crystal Identificationwith the PolarisingMicroscope, Chapman andHall, London,
1994.
E. E. Wahlstrom, Optical Crystallography, 5th edition, John Wiley and Sons, Inc., New York, 1975.
The information regarding ammonite eyes is given by
R. Fortey, Life, Folio, London, 2008, p. 95; (originally published by HarperCollins, UK, 1997, and A Knopf,
USA, 1998).
A. R. Parker, In the Blink of an Eye, Free Press, London, 2003, pp. 217 220.
The early studies of nonlinear optical properties of crystals are described in
M. E. Lines, A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials, Clarendon
Press, Oxford, 1977, Chapter 13. Reprinted in the series Oxford Classic Texts, 2001.
Recent information on nonlinear optics is given in
R. W. Boyd, Nonlinear Optics, 2nd edition, Academic Press, New York, 2003.
Various authors, J. Mater. Chem. 19 (40), (2009); a single-topic issue on (mostly molecular) nonlinear
materials.
The observation of SHG in glass fibres described in Section 4.11.2 is given in
U. Osterberg, W. Margulis, Opt. Lett. 11, 516 518 (1986).
SHG in surfaces is reviewed by
K. B. Eisenthal, Chem. Rev. 96, 1343 1360 (1996).
Y. R. Shen, Nature 337, 519 525 (1989).
The use of SHG in biological microscopy is surveyed by
W. Mohler, A. C. Millard, P. Campagnola, Methods 29, 97 109 (2003).
D. J. Brown, N. Morishige, A. Neekhra, D. S. Minckler, J. V. Jester, J. Biomed. Opt. 12, 024029 (2007).
Optical activity is discussed in all textbooks concerned with organic chemistry and many concerned with
inorganic chemistry. For example, see
J. McMurry, Organic Chemistry, 6th edition, Thomson Brooks/Cole, Belmont, CA, 2004, Chapter 9.
K. P. C. Vollhardt, N. E. Schore, Organic Chemistry, 3rd edition, W. H. Freeman, San Francisco, 1999.
D.F.Shriver, P.W.Atkins,C.H.Langford, InorganicChemistry, 2nd edition,OxfordUniversityPress,Oxford,
1994.
An overview of chirality is given by
G. H. Wagni�ere, On Chirality and the Universal Asymmetry, Wiley-VCH/VCA, Weinheim/Zurich, 2007.
The homochiral nature of the molecules important to life on the Earth is discussed by
A. Guijarro, M. Yus, The Origin of Chirality in the Molecules of Life, RSC Publishing, Cambridge, UK, 2008.
Liquid crystals are described by
P. J. Collins, Liquid Crystals: Nature’s Delicate Phase of Matter, Princeton University Press, Princeton, NJ,
1990.
A simple description of active matrix LCDs will be found in
S. Musa, Sci. Am. 277 (November), 87 (1997).
More technical information is given in
J. Hanna, I. Shimizu, Mater. Res. Soc. Bull. 23 (March), 35 38 (1996).
There are a number of demonstrations of relevance to this chapter, including polarised light and nonlinear
phenomena, available at http://demonstrations.wolfram.com/index.html.
Colour and the Optical Properties of Materials 174
5
Colour Due to Scattering
. Why is skylight polarised?
. Why are eyes blue at birth?
. How can yellow gold colour glass red?
Scattering, defined somewhat imprecisely, is the deviation of a beam of light from a straight path after
interaction with an object. In this sense, refraction and reflection (and, in fact, many other optical phenomena)
cancorrectly be regardedas scattering. In commonparlance, however, scattering tends to refer to the interaction
of light with small particles, often distributed at random in a continuous medium, such as small dust particles
in air.
Elastic scattering is a complex process that generally applies to the interaction of a light beam with specks
such as smoke, dust orwater droplets, inwhich little or no energy is exchanged. To a reasonable approximation,
elastic scattering simply involves the redirection of light from its original trajectory into another one. It is elastic
scatteringwhich causes sunbeams to becomevisible in dusty or smoky rooms. Inelastic scattering describes the
complex process occurring when there is significant energy exchange with the object, so that the scattered
radiation has (generally) a lower energy than the incoming radiation.
Intense colours can be produced by scattering. Some of theways in which this comes about are described in
this chapter.
5.1 Scattering and Extinction
As a beamof light passes through a transparentmedium, a solid, liquid or gas, it gradually loses intensity, due to
elastic scattering (Figure 5.1). The scattering particles might be the atoms or molecules that make up the
medium, or else impurities of one sort or another within the medium. The gradual loss of intensity is generally
Colour and the Optical Properties of Materials Richard J. D. Tilley
� 2011 John Wiley & Sons, Ltd
called extinction. Provided thatmultiple scatteringdoesnotoccur that is, eachphoton in the incident radiation
is scattered only once as it crosses the medium the attenuation of a beam of light which has traversed a plate
containing scattering centres follows the same exponential law given in Section 1.13):
I ¼ Io expð�asxÞ ð5:1Þ
where I is the irradiance leaving the plate, Io is the incident irradiance, x (m) is the thickness of the plate and
as (m1) is the (Napierian) linear scattering coefficient. The scattering length is defined as 1/aa.
Equation (5.1) assumes that each scattered photon is lost from the forward-propagating beam. In point of
fact, photons can be scattered in a forward direction as well as in any other direction. Forward scattering will
tend to diminish the measured attenuation, thus reducing the apparent value of the absorption coefficient.
The amount of beam attenuation obviously depends upon the number of scattering centres present. In
addition, the total extinction that occurs is found to depend upon:
1. The ratio of the particle size to the wavelength of the light. Broadly speaking, large particles scatter more
than small particles.
2. The ratio of the refractive indices of the particle and the surrounding medium. If the refractive index of the
particle is the same as that of the medium, then no scattered radiation is registered, as has been mentioned
previously (Sections 1.16 and 2.5).
3. The particle shape.Although calculations are difficult tomake for geometries other than spheres, spheroids,
rods and other shapes have been analysed. The degree of scattering depends upon the relative orientation of
the particlewith respect to the illumination and the formof the particle, that is a long and needle-like rodwill
scatter differently than a short, thick rod.
5.2 Tyndall Blue and Rayleigh Scattering
In order to understand how scattering can lead to colour production the variation of scattering withwavelength
must be investigated. The scattering of light by small particleswas studied, from this point of view, by a number
of scientists in the nineteenth century, but the most detailed experiments were made by Tyndall. He observed
that liquids containing suspensions of small droplets, such as water containing a little milk, looked sky blue
when illuminated with white light and viewed from the side. The beam of light responsible was also visible in
the liquid and the light emerging in the beam direction took on a red hue (Figure 5.2a). The fact that a beam
of light is visible in a suspension of small particles but invisible in a true solution is still the easiest way of
distinguishing one from the other (Figure 5.2b). Tyndall supposed (correctly) that blue lightwas scatteredmore
strongly than red light, and this blue scattering is still referred to as Tyndall blue.
Figure 5.1 A beam of light passing through a medium containing scattering centres will gradually diminish inintensity
Colour and the Optical Properties of Materials 176
Thefirst importantmathematical studyof scatteringwas carried out byRayleigh,who investigated scattering
by a small insulating (nonabsorbing) sphere with a diameter less than one-tenth of the wavelength of the
incident light. Scattering by such bodies is referred to as Rayleigh scattering. The classical model for this
scattering is that the incident electromagnetic wave (the light beam) causes the electrons associated with
the scatterer to oscillate at the same frequency as thewave. The oscillating electrons then emit a wavewith the
same frequency (i.e. colour) as the incident wave, but in a different direction. In the case where a beam of
unpolarised light of irradiance Io is scattered once only by a single scattering centre, the irradiance of the
scattered light Is is given by1:
Is ¼ Io9p2V2
2d2l4m2�1
m2 þ 2
� �2
ð1þ cos2�Þ ð5:2Þ
where the measurement is taken at a distance d from the scattering centre, V is the volume of the scattering
particle, l is the wavelength of the light incident upon the particle andm is the relative refractive index of the
particle:
m ¼ np
nm
In this case np is the refractive index of the particle and nm the refractive index of the surroundingmedium. For
air nm is 1.0. The angle � is the angle between the incident beam and the direction of the scattered beam
white
white
reddish
white
bluish
bluish
beam visible
beam invisible
(a)
(b)
Figure 5.2 (a) Small particles suspended in a liquidwill preferentially scatter blue light. The transmitted lightwilltake on a reddish colour and the beamwill be visible in the liquid. (b) In a true solution the amount of scattering isvery small and the beam remains invisible
1 There are a surprising number of expressions for Rayleigh scattering to be found in textbooks and elsewhere. Many of these look quite
different from one another. The expression in Equation 5.2 is the clearest for the present purposes.
177 Colour Due to Scattering
(Figure 5.3). Note that these refractive indices only contain real parts n (the absorption index k is zero) and that
the wavelength of the light impinging upon the particle should be that in the medium surrounding the sphere,
rather than the vacuumwavelength. This is not important for air, butmust be taken into account if scattering in
glass or water is considered. The wavelength in a medium of refractive index nm is given by:
lm ¼ lvacuumnm
For example, if the scatteringofyellow light of (vacuum)wavelength550 nmfromasmall particle embedded in
glass with refractive index 1.5 is considered, the correct wavelength to use in Equation 5.2 is 550/1.5 (that is,
367 nm), whilst if the same particle is suspended in water, refractive index 1.333, the correct wavelength to use
is 550/1.333, or 412 nm.
If there areNv particles per unit volume, then the scattered irradiance from this volume is simply Ismultiplied
by Nv, as each photon is only scattered once.
If the irradiance of the scattered light in a plane containing the incident beam, the scattering volume and the
observer, theplaneofobservation, is plotted, then acharacteristicRayleigh scatteringpatternorpolar diagram
is formed (Figure 5.3). It indicates that as much light is scattered backwards as forwards and that only half as
much is scattered normal to the beam direction. As the 1/l4 term in the equation shows, all wavelengths
scatter in this pattern, but the shorter wavelengths are more strongly scattered than the longer wavelengths.
The importance of the equation went beyond simply explaining scattering. A comparison of measurements
of scatteringwith the theorymade it clear thatmolecules alone could operate as scattering centres. That is, even
the purest gas would still show light scattering. Moreover, the formula allowed an estimate of molecular size
and the number of molecules present in a unit volume of a gas to be made. These values permitted scientists to
estimateAvogadro’s number and themolarmasses of gases. Such informationwas of great interest towards the
end of the nineteenth century, when the atomic theory of matter was still a topic of controversy.
5.3 Blue Skies, Red Sunsets
The blue colour of the sky has been a topic of interest since antiquity. Newtonmade the reasonable suggestion
that it arose by reflection from small water droplets in the atmosphere. Rayleigh showed that it was due to
unpolarisedincident beam
x
y
θ
Figure 5.3 The Rayleigh scattering pattern of unpolarised light from small particles. The lengths of the arrowsdiverging from the small scattering centre can each be thought of as defining the scattered irradiance at a distanced and at an angle u to the forward direction
Colour and the Optical Properties of Materials 178
scattering by gas molecules in the atmosphere. In the absence of an atmosphere the sky would appear black,
even when the sun is high above the horizon, as indeed it does on the moon.
The colour of the sky is the result of thewavelength differential inherent in Rayleigh scattering. Because this
is proportional to 1/l4, violet light is scattered far more than red light (Figure 5.4). However, it is important
to remember that all wavelengths are scattered in the Rayleigh pattern, as analysis with a prism will show
(Section 2.6, Figure 2.10a). This suggests that when we look at the sky in a direction which is not towards the
sun, the colour seen should be indigo or violet. In fact, the sky appears to be blue. This is for two reasons. First,
the solar energy reaching the ground has less intensity in the violet than at longer wavelengths such as yellow
and, second, the sensitivity of the eye to colour peaks in theyellow green region of the spectrumnear to 555 nm
(Figure 1.9). The result of these factors is that the sky away from the sun is perceived to be blue (Figure 5.5a).
Towards sunset, when it is possible to look in the direction of the sun through a thicker layer of atmosphere, the
scattering will remove blue light preferentially and the sun and sky will appear red (Figure 5.5b). The effect
will be enhancedwhen this light is reflected at a shallow angle from clouds or fine dust in the upper atmosphere,
as can occur after volcanic eruptions, when spectacular sunsets are often recorded (Figure 5.6).
This analysis suggests that the evocative phrase ‘blue remembered hills’2 requires further explanation. If the
hills are far away, surely light scattered from them and entering the eye should be diminished in blue and hence
look reddish. This ignores scattering of sunlight from the bodyof air that lies between the hills and the observer,
400300 500 600 700 800
Wavelength / nm
1.0
1.2
0.8
0.4
0.6
0.2
Rel
ativ
e sc
atte
red
irrad
ianc
e
visible
Figure 5.4 Rayleigh scattering of visible light as a function ofwavelength. Violet light is scattered approximatelynine times more than red light
2 The earliest use of this phrase that I have located is inA Shropshire Lad, byA. E. Houseman, published 1896, poemXL ‘What are those
blue remembered hills’. The hills are those in Shropshire, running along the Welsh Marches. The expression has also been used more
recently by Rosemary Sutcliff, as the title of a memoir of childhood, and by Dennis Potter (in 1979) as the title of a play.
179 Colour Due to Scattering
which is blue enriched. This scattered light is called airlight. When hills are close, the amount of airlight is
relatively small and the hills are normal in appearance. As the distance between the observer and the hills
increases, the airlight scattering becomes dominant and the hills take on a blue indistinct appearance. The
contribution of the airlight increases both as the sun rises in the sky and as the distance to the hills extends.
Further hills look bluer. However, a stage will come when the hills eventually become obscured or invisible
(Figure 5.7).
A total eclipse of the moon was observed across much of Europe during the night of 21 January 2000. The
moonwas seen to be a copper colour. This distinctive colour is also due to light scattering in the atmosphere, as
in the case of the red sky at sunset.A total eclipse of themoonoccurswhen themoonpasses into the shadow (the
umbra) of the Earth (Figure 5.8). The colour that the eclipsed disc takes on depends upon the light that is
refracted by the Earth’s atmosphere to reach the surface of the moon. When the relevant part of the
atmosphere is clear this can be extensive. The light reaching themoon is reddened due to Rayleigh scattering
on its passage through the atmosphere. When reflected it gives the full moon a copper appearance. In less
favourable conditions, when the atmosphere is cloudy, the amount of refracted light is low and the moon is
barely visible when in the umbra.
When themoonpasses through thepenumbra there is little reduction in light incidence andnocolour changes
are seen.
Finally, note that a blue moon is also caused by scattering, but not by air molecules (see
Section 5.6).
yellow-white light from sun
light from sun, depleted in blue,seen as red
scatteredlight, seen as blue
scatteredlight, seen as blue
observerin day time
observerat sunset
(a)
(b)
Figure 5.5 (a) An observer during the daywill see light from the sun as yellowwhite, and scattered light, in otherdirections, will give the sky a blue colour. (b) At sunset, light in the direction of the sunwill be depleted in blue andappear red, while the sky overhead will remain blue due to the scattered light
Colour and the Optical Properties of Materials 180
5.4 Scattering and Polarisation
A characteristic of Rayleigh scattering is that it produces strongly polarised light. Assume that an
unpolarised light beam is travelling along the positive x-direction and that observations of the scattered
light (at a distance d and angle � to the positive x-axis) are made in the x y plane, the plane of observation
(Figure 5.9a). The incident light beam can be resolved into two linearly polarised components: one with the
electric field vector lying parallel to the x y plane of observation and one with the electric field vector lying
Figure 5.6 Orange–red light reflected from clouds. The photograph was taken before sunrise on a clear wintermorning. The sky is starting to take on a blue hue due to light scattered from the upper atmosphere. The clouds arelower in the atmosphere and reflect light that has travelled a significant distance through the air, andwhich has, asa consequence, become reddened
Figure 5.7 The foothills of the Pyrenees, France. The nearest ground is normal in colour, further hills appear bluedue to scattered airlight and the furthest hills look indistinct and start to merge with the background
181 Colour Due to Scattering
SunEarth
Moon
umbra
penumbra
light refracted byEarth’s atmosphere
Figure 5.8 A full eclipse of the moon is due to the moon passing through the shadow (umbra) of the Earth. Theangles are greatly exaggerated here
(a)
(b)
total =perpendicular+parallel
electric field vector perpendicular to planeof observation
electric field vectorparallel to planeof observation
y
x
x
y
z
incident beamunpolarised
scattering centre
d
θ
Figure 5.9 (a) A beam of unpolarised light travelling along the positive x-direction can be resolved into twolinearly polarised components parallel andperpendicular to the x–y plane. This is taken as the planeof observationof the scattered light, which is recorded at a distance d and angle u to the positive x-direction. (b). The Rayleighscattering pattern ismadeupof the sumof light scatteredwith its electric field vector perpendicular andparallel tothe plane of observation
Colour and the Optical Properties of Materials 182
perpendicular to the plane of observation. Thewave polarised perpendicular to the plane of observation is found
to be scattered equally in all directions in the plane (Figure 5.9b). This wave contributes the term ‘1’ in the
ð1þ cos2�Þ factor of Equation 5.2. The scattering from the component with the electric field vector in the plane
of observation has a dumbbell shape (Figure 5.9b). This wave contributes the cos2� term in the ð1þ cos2�Þfactor in Equation 5.2. The total scattering curve is the sumof both of these contributions. Thus,within the plane
of observation, the scattered light is unpolarised in the beamdirection, completely polarisedperpendicular to the
beam direction and partially polarised between these two directions (Figure 5.10).
Sky light is polarised due to this differential scattering. The degree of polarisation is least (virtually zero) in
the direction of the sun.However, sky light in a planewhich includes the observer and is at 90� to the line joiningthe observer to the sun is strongly polarised (Figure 5.11). Theoretically, the light should be completely
polarised, but in reality it is found to be only about 75 85%polarised. The reason for the discrepancy is that the
actual polarisation observed at any point in the sky is a result ofmultiple scattering, the atmospheric conditions
scattering centre
unpolarisedincident beam
unpolarised
partly polarised
partly polarised
completely polarised
completely polarised
x
y
–y
–x
Figure 5.10 The polarisation of scattered light is zero in the beam direction and at a maximum in the directionperpendicular to the incident beam. In other directions in the plane of observation it is partially polarised
light from sun
horizon
observer
light maximally polarisedin this plane
Figure 5.11 Light scattered from smallmolecules in the air is optimally polarised in a plane at 90� to the directionof the incident radiation. It is not completely polarised in this plane due tomultiple scattering. Observation of thepolarisation of the light of the sky will allow the observer to estimate the position of the sun even on overcast days
183 Colour Due to Scattering
and the relative positions of the sun and the observer. In other directions the degree of polarisation lies between
zero and this value. In reality, the accurate evaluationof the polarisationof the sky light is complex, and it is only
in the second half of the twentieth century that accurate polarisation maps of the sky have been produced with
the aid of computers.
The polarisation of the sky can be observed using uniaxial or biaxial crystals. Cordierite (a magnesium
aluminosilicate,Mg2Al4Si5O18,with the beryl structure) is a biaxialmineral and absorption of polarised light is
strong alongonly one crystallographic axis. It is recounted thatVikings used this property of cordierite crystals,
called sunstones, to locate the sun (and so navigate) even when the sun was not visible. The sky is viewed
through a cordierite crystal which is rotated at the same time. If the direction of observation is in a plane
perpendicular to the direction of the sun a clear patch of skywill appear alternately darkened and brightened as
the crystal rotates. Viewing in the direction of the sun does not produce this effect, as the light is not very
polarised. Hence, the direction of the sun can be determined even on cloudy days. The effect is easily checked
with a piece of Polaroid film.
Humans are unable to detect polarised light,3 but bees and ants, and perhaps many other insects, possess this
ability. They use this skill to navigate to and from the hive or nest even under conditions when the sun is hidden
from them.The capability arises because themolecule responsible for photoreception in the eyes of all animals,
rhodopsin (see Section 1.10), is a dipolar molecule with an optic axis. These molecules absorb polarised light
energymaximallywhen the direction of polarisation is parallel to the optic axis of themolecule. In insects’ eyes
these molecules are aligned in a fixed direction, making them polarisation sensitive. In humans the molecules
are free to rotate, so that the orientation of the optic axis is random and polarisation perception is lost.
5.5 Mie Scattering
Rayleigh himself extended scattering theory to particles of any size and shape, provided that the relative
refractive index of the particle was small. Further work on this topic was carried out later by Debye and Gans,
and the result is generally called Rayleigh Gans theory. This produces approximate expressions for scattering
for a particle of arbitrary shape and size provided that the relative refractive index of the particle is small
(usually just greater than unity) and the diameter of the particle is larger (but not too large) than thewavelength
of the scattered light in themedium surrounding the particle.Although each element in the scattering particle is
treated as aRayleigh scatterer, the resulting angular distributionof scattered light differs considerably from that
given by the simpler Rayleigh formula, Equation 5.2. Rayleigh Gans theory can be used, for example, to study
the scattering of light by long polymeric molecules in solution.
Despite this body ofwork, themost important advancewas to apply electromagnetic theory to the scattering
and absorption of light by an isotropic absorbing homogeneous sphere of any size. The mathematics of the
problem is formidable and thefirst complete theorywaspublished in1908byG.Mie.4Onceagain, though, each
photon was presumed to be scattered only once. The theory is more general than that of Rayleigh, because it
includes absorbing (i.e. metallic) bodies as well as insulators. It includes Rayleigh scattering as a special case
for nonabsorbing small particles with a radius less than that of the incident light. Despite this universality, the
termMie scattering is sometimes reserved only for scattering by particles that are somewhat larger than those
for which Rayleigh scattering is valid, say about one-third the wavelength of light or more.
3 This is not altogether correct. The visual phenomenon called Heidinger’s brushes, a faint small hourglass shape centred in the field of
view, is caused by the detection of polarised light in the retina of the eye. It is not observed by everyone.4 In fact, considerable progresswas previouslymade on the problembyLorenz, andDebye published on this topic shortly afterMie, so that
the theory is also called the Lorenz Mie theory, or the Mie Debye theory.
Colour and the Optical Properties of Materials 184
Mie scattering theory allows for a complete solution to the scattering from a spherical particle provided that
the optical constants of the material, n and k, in the complex refractive index:
N ¼ nþ ik
are known (Section 2.1). Note that n and k are not constant but vary considerably with wavelength, making the
computations arduous without a computer. It is found that for small particles the scattering is proportional
to l p, with p¼ 4, as in Rayleigh theory. As the particle size approaches that of thewavelength of light, p takes
values between 4 and 0.2, while p¼ 0 for the largest particles. In the Rayleigh scattering limit, the forward
andbackward lobes of scattered irradianceare equal (Figure 5.12a).As the radius of theparticle approaches and
passes that of the wavelength of light, the forward scattering lobe becomes dominant and the backward
scattering lobe becomes negligible (Figure 5.12b). At larger particle sizes, forward scattering remains
dominant, but side bands develop representing maxima and minima of scattering at definite angles
(Figure 5.12c). The positions of these lobes depend upon the wavelength of the scattered light and so they
can be strongly coloured. These coloured bands, referred to as higher-order Tyndall spectra, are dependent
upon the particle size and so can be used for particle size determination. For the largest particle radii,
wavelength dependence is lost. That is to say, large droplets scatter all wavelengths equally (although the
scattering pattern still shows the strong forward-pointing lobe), which is the reason why fogs are white to the
eye. These can be compared with the calculated scattering patterns for nonabsorbing spherical particles with
a refractive index of 1.50, for a wavelength of light of 550 nm (Figure 5.13).
With large particles, white light becomes reflected (rather than scattered as discussed here) evenly in all
directions. This is the situation that holds in fogs and mists composed of fairly coarse droplets.
Although Mie theory provides an exact solution for light scattering from spherical particles, the scattered
irradiance pattern is a complex function of particle radius, the wavelength of the light and its polarisation, the
refractive index of the particle and the refractive index of the surrounding medium. The resulting function
describing the scattering is rarely written out in full, although approximate forms applicable to various special
cases are to be found in the literature. Indeed, the scattering function is ‘neither simple nor intuitive’.
(a)
(b)
(c)
incident beam
incident beam
incident beam
Figure 5.12 Light scattering by small particles (schematic). (a) Rayleigh scattering from particles much smallerthan the wavelength of light. (b) Particles approaching the wavelength of light; the scattering becomespronounced in the forward direction. (c) Particles larger than the wavelength of light; lobes appear whichare wavelength dependent and so give rise to colours at specific viewing angles
185 Colour Due to Scattering
On the other hand, the power of theMie theory is that it allows the scattering to be evaluated via a calculation
of the cross-sections for extinction Ce, scattering Cs and absorption Ca, where:
Ce ¼ Cs þCa
When absorption can be ignored:Ce ¼ Cs
These figures are often presented as efficiencies, or efficiency factors, Q:
Qe ¼ Ce
GQs ¼ Cs
GQa ¼ Ca
G
whereG is the geometrical cross-sectional area of the particle projected onto the beam direction. For spherical
particles, G¼ pa2, where a is the sphere radius.
The attenuation of a beam of light given by Equation 1.7:
Ix ¼ Io expð�aexÞ
can be expressed in terms of the extinction cross-section as:
Ix ¼ Io expð�NvCexÞ ð5:3Þ
and in the case where only scattering is important as:
Ix ¼ Io expð�NvCsxÞ
where Nv is the number of extinction (scattering, absorption) centres per unit volume. These equations can be
written in terms efficiency factors, which, for spherical particles, are:
Ix ¼ Io expð�Nvpr2QexÞIx ¼ Io expð�Nvpr2QsxÞ
ð5:4Þ
1.01.01000 nmparticle radius = 500 nm
(d) (e)
0.10.10.1100 nmparticle radius = 50 nm
(c)(b)(a)
10 nm
Figure 5.13 Mie scattering patterns for light of wavelength 550 nm from nonabsorbing particles of increasingradius with a refractive index of 1.5, in air. The scattered irradiance is proportional to the length of the arrows.Scale bars indicate the extent of scattering. Note that the scale in (d) and (e) is 10� that of the scale in (a)–(c)
Colour and the Optical Properties of Materials 186
In general, the scattering cross-section of a particle Cs is proportional to the wavelength:
Cs / l p
where p¼ 4 in the limit for small particles that fall into the Rayleigh scattering regime and lies between 4 and
0.2 for particles near toor larger than thewavelengthof the scattered radiation.Aplot of logCs versus log lhas aslope of �p and can be used to determine the scattering dependence.
Ideally, a plot of the extinction efficiency of a particle Qe varies as a damped sinusoidal function
(Figure 5.14). The principal and first maximum of the curve occur when the particle radius is about half
that of thewavelength of the light being scattered.Unfortunately, the exact curves are not smooth, as portrayed,
but have a pronounced ‘ripple structure’, which means that each wave of the sinusoid is made up of wavelets.
In fact, thesewavelets can be so severe as to obliterate the sinusoidal wave, and calculations are always needed
to obtain accurate efficiency factors.
5.6 Blue Eyes, Blue Feathers and Blue Moons
When trying to explain almost all nonpigmentary biological colours, such as, for instance, blue eyes, it is
difficult to tease apart the many optical effects that take place simultaneously. The tissues showing the colour
are generally more or less transparent and contain thin films and reflecting surfaces. In the cases in which
colour is believed to be primarily caused by scattering it is not a simple matter to describe the colour as due to
either Rayleigh or Mie scattering. (In fact, the two terms merge into one another for particles of appropriate
sizes and optical constants.) In such cases the term Tyndall scattering is a useful if rather imprecise expression
that can used to describe the preferential scattering of shorter wavelengths of light by small particles or optical
inhomogeneities in transparentmaterials that give rise to a visible blue colour. The hue so produced can then be
called Tyndall blue. The colours generated in this way are rather ‘soft’, lacking in saturation, of rather low
intensity and lacking the strong iridescence of multilayer colours (Sections 3.8 and 3.11).
2
0.3 0.5 0.7
1
3
4
5
0.1 0.2 0.4 0.6 0.8 0.9
Ext
inct
ion
effic
ienc
y, Q
e
Particle radius, / nm
Figure 5.14 Idealised sinusoidal curve for the extinction efficiency Qe for scattering of light of wavelength550 nm by a spherical particle of TiO2 (rutile), with refractive index 2.755, in a transparent oil of refractiveindex 1.500
187 Colour Due to Scattering
All eyes are blue at birth, and this is, in fact, a scattering effect. Of course, thewhole of the eye is not blue, and
the expression refers to the delicate colouring of the iris. This is essentially a transparent material consisting of
a composite of various tissues, small crystalline regions and air vesicles, each ofwhich have differing refractive
indices. This inhomogeneity gives rise to preferential scattering of blue light. As the light transmitted through
the lens and iris is absorbed in the underlying tissues, only the scattered light re-emerges, to give the impression
of blue irises. After someweeks a pigment is laid down inmany irises and it is this that changes the colour from
blue to green or brown.
The blue colours of many feathers from exotic birds are coloured blue in the sameway. The outer part of the
feather is a composite of several differentmore or less transparent proteins togetherwith small air vesicles. The
inhomogeneity causes preferential blue scattering. This is not easily seen against a background of reflected
light, but if the feathers are backed by a dark absorbent layer, the blue colour becomes easily visible.
Aswith feathers, blue scales on thewings of themale butterflies of the speciesPapilio zalmosis appear to be
coloured by scatteringwhich gives rise to Tyndall blue colour. The scattering is produced by a layer of air-filled
tubes (alveoli) that penetrate a more or less transparent medium making up part of the scale structure and it
appears that Tyndall blue coloration is the result.
A similar phenomenon that falls neatly between nominal Rayleigh andMie scattering domains results in the
formation of the rare blue moon (or blue sun). This effect is caused by scattering in the atmosphere. Small
particles, say of the order of 50 nm diameter, scatter violet wavelengths more strongly than red (Rayleigh).
Large particles, say of the order of 10 000 nmdiameter, scatter all wavelengths about equally (Mie). In between
these limits some particle sizes, say about 7000 and 8000 nm diameter, scatter red wavelengths more strongly
than violet. The light transmitted through a haze of these particles will then lose red light preferentially, giving
the object, moon or sun, a blue green cast. The size and optical properties of the scattering particles must fall
into a narrowandprecisely defined range for the effect to be observed.This occurs rarely, but has been known to
happenwhen forest fires inject uniformsmall oily droplets into thehigh atmosphere.At such times, a bluemoon
or sun may be spotted by a lucky observer.
5.7 Paints, Sunscreens and Related Matters
Many paints, plastics and glazes are made opaque by the addition of white pigment. Most frequently this is
titaniumdioxide (TiO2), but china clay and limestone are also commonly employed. Thesematerials are not, in
fact, white, but colourless. They give the appearance of whiteness when in powder form (in air) due to surface
reflection and scattering. When mixed within a transparent matrix opacity is mainly the result of scattering. A
similar opacity is found in opal glass, devitrified glass, glass ceramics and porcelain, which contain varying
amounts of precipitated crystalline phases in aglassymatrix.Both components are transparent in bulk formand
the opacity comes principally from scattering by the inclusions.
Acontrol of the scatteringpower is essential if a satisfactory degreeof opacity is to beobtained.The scattered
irradiance Is, which determines the opacity, is given by Equation (5.4):
Is ¼ Io½1�expð�Nvpr2QsxÞ�
whereNv is the number of scattering particles per unit volume and x is the thickness of the pigment-containing
layer. The idealized general form of the scattering efficiency curve plotted against size is a damped sinusoid
(Figure 5.14). This shows that themaximumextinction efficiency is givenby thefirst principalmaximumof the
curve. However, the calculation of scattering cross-sections or efficiencies is complicated, and, as the first
principal maximum is of most importance, an approximation valid for nonabsorbing spheres, derived from
Colour and the Optical Properties of Materials 188
geometric optics (see van de Hulst in Further Reading), is useful. The equation takes the form:
Qs ¼ Qe ¼ 2� 4
rsin rþ 4
r2ð1�cos rÞ ð5:5Þ
where r is given by:
r ¼ 4prðm�1Þl
r is the sphere radius,l is thewavelength of the light in themediumsurrounding the particle andm is the relative
refractive index of the particle:
m ¼ nparticle
nmedium
This equation holds whenm is close to one (and is reasonable up to m¼ 2) and when r is greater than l. Thecurve (Figure 5.15) is a useful approximation to curves obtainedwith exact computations, and ismuch simpler
to handle. Note, though, that exact calculations can show any inherent ripple structure present; the simpler
equation (Equation 5.5) does not reproduce this aspect of the scattering. The first and highest maximum is at
r� 4.0, implying that the ratio of the particle radius r to that of the scattered wavelength l for maximum
extinction is:
r � lpðm�1Þ
Substitutionof typical values shows that extinction is at amaximumfor aparticlediameter approximately equal
to the wavelength of the light scattered, but this depends upon the medium surrounding the particles.
The majority of ordinary ceramic materials are produced by firing a mixture of finely grained powders or
reactants that decompose to such ingredients. Much of the opacity of these bodies, which appear white to the
eye unless pigments are deliberately added, is due to light reflection and scattering from the boundaries that
remain between the crystallites of the final body.However, if the crystallites are of uniform size and are sintered
so that they are in contact along the grain boundaries that separate one crystallite from another, the material
regains transparency. This is the principle behind the fabrication of transparent polycrystalline ceramic bodies.
One of the important successes of this approach was the fabrication of transparent polycrystalline alumina
(Al2O3) tubes for sodium vapour lamps (Section 7.7). Sodium vapour is highly corrosive and reacts with silica
302010
1.5
2.0
2.5
3.0
Ext
inct
ion
effic
ienc
y, Q
e
ρ
Figure 5.15 Plot of Equation 5.5 for the extinction efficiency of spherical particles as a function of the parameterr. The first and most important maximum occurs at a value of r� 4.0
189 Colour Due to Scattering
glass, but does not attack alumina. Initial attempts to make transparent tubes were foiled because of the
presence of small air-filled pores between each crystallite, which scattered light and rendered the ceramics
opaque. The addition of magnesium oxide (MgO) as a sintering aid removed these pores and allowed
transparent tubes, trade named Lucalox, to bemanufactured and so allowed for thewidespread introduction of
high-pressure sodium street lighting.
Scattering is also of immense importance in biologicalmaterial, and transparencyof biologicalmaterials has
already been mentioned (Sections 1.16 and 2.5). The size of biological components spans the range from
Rayleigh scattering (proteins, ribosomes, etc.) through traditional Mie scattering (bacteria, small cells) to
reflection (large cells). The transparency or otherwise ofmuch biological tissue depends upon the scattering of
these components. Moreover, the appearance of skin and complexion, in both people and other animals, is
closely related to surface and subsurface light scattering from structures lying within and on the surface of the
skin, as cosmetics manufacturers are well aware.
Sunscreens are a case in point. Titaniumdioxide, aswell as being awhite pigment, is also a strong absorber of
ultraviolet radiation (Section 10.1). It is widely used in sunscreen creams and lotions. However, it is considered
undesirable that these products should be opaque. Fortunately, scattering drops towards zero as thevalue ofr, orthe ratio of particle radius to the wavelength of light, falls (Figure 5.15). Thus, the ultraviolet absorbing
characteristics of titanium dioxide can be utilized without a high opacity if particle sizes are decreased. This is
the region where Rayleigh scattering becomes dominant. Thus, small particles of titanium dioxide will not
register any significant extinction and so not render the surrounding medium opaque at all. It is found that a
particle diameter of about 20 nm is optimal. Particles of this size are invisible to the eye and provide best for the
balance between transparency and absorption. This compares with maximum opacity for particles of titanium
dioxide in paints and oils, which occurs when particles are approximately 200 nm diameter.
5.8 Multiple Scattering
Towhat extent can the examples in the previous sectionbe treated in termsofRayleigh orMie scattering?These
theories depend upon the assumption that each particle scatters only once, and this is surely not so in heavily
loaded paints or sunscreens. Counterintuitively, the effects of multiple scattering can lead to increased
transparency rather than increased extinction. For example, in experiments to make glasses containing
lanthanoid5 ions for up-conversion purposes (Section 9.9), some slightly crystallized glasses were found
to bemore transparent than the original glass. The glassesweremade from sodiumoxide, aluminiumoxide and
silica (Na2O Al2O3 SiO2) together with small amounts of lanthanum trifluoride (LaF3). The initial glass is
clear.Heat treatment causes crystals of LaF3 to form.Although some of thesematerialswere opaque, due to the
formation of large crystals in the matrix, others were more transparent than the parent glass according to the
measured linear absorption coefficient (Table 5.1). This was because the scattering centres were rather close
together. Multiple scattering, particularly in a forward direction, from randomly distributed closely spaced
scattering centres caused the diminution in the measured extinction.
There is a second way in which multiple scattering can lead to greater transparency, most obviously
displayed in the occurrence of transparency in inhomogeneous biological tissue. Perhaps of greatest
importance (to us) is the evolution of the transparent cornea and lens of the mammalian eye. These structures
aremade of layers of differentmaterials andwould be expected to be opaque, just like the similarly constructed
surrounding ‘white’ of the eye, the schlera.However, as the front part of the schlera is transparent, scatteringhas
somehow been negated in this volume. This comes about by arranging the scattering material so that the
scattered waves are out of phase with each other. The scattering is effectively suppressed and the material
remains transparent.
5 The term lanthanoid now replaces the older term lanthanide (see Chapter 7 Footnote 7).
Colour and the Optical Properties of Materials 190
It should be noted that scattering centres can be arranged so that the scattered waves are in phase with each
other and light intensity is reinforced. Thismay lead to strong colours.When scattering centres are arranged in
such a way as to give rise to strongly reinforced or suppressed scattering they are found to be fairly regularly
spaced. At this stage it is easiest to treat the problem byway of diffraction theory, which will be described later
(Chapter 6).
5.9 Gold Sols and Ruby Glass
Gold sols, first deliberately prepared by Faraday utilizing the chemical reduction of gold chloride solution,
are brightly coloured. The colour is due to microscopic crystallites of gold which are small enough to remain
suspended in aqueous solutions. Ruby-coloured glass has been known for much longer, apparently dating to
Roman times. It was produced regularly from the fifteenth century; the first published report on the production
appears to be that of J. Kunckel, in 1689. The process involved gold as the ‘magic’ colorant. Slight
modifications in the processing also allowed craftsmen to make blue or purple glass.
Ruby glass is made by dissolving of the order of 0.01 % of gold in molten glass. If the glass is cooled in
a normal way, which is fairly rapidly, then the glass remains clear, as isolated gold atoms are distributed evenly
throughout the material. Colour is developed by annealing (reheating) the glass to 650 �C for several hours.
At this temperature the gold atoms in the glass aggregate to produce gold crystals with diameters of between
30 and140 nm, distributed throughout theglassmatrix. The colour is causedby these crystallites. Control of the
crystallite size and, hence, colour by processing is difficult, which iswhy early glassworkerswho had perfected
recipes for the production of ruby glass guarded their knowledge jealously.
A precise explanation of the colour of dilute dispersions of gold was given in 1908 within the framework of
the theory derivedbyMie for this purpose. In the previous discussionswehave assumed that the particleswhich
are scattering light are nonabsorbing insulators, allowing calculations to be performed using the ordinary
refractive index n.6 However, metal particles are strongly absorbing, making the use of the complex refractive
index:
N ¼ nþ ik
mandatory (Section 2.1). For strongly absorbing collections of small particles it is found that, although the
amount of light scattered is still proportional to V2l 4 (the Rayleigh dependence), the absorption of light is
Table 5.1 The increase in transparency of glass ceramics containing LaF3 crystallitesa
Heating temperature/�C Crystallite size/nm Absorption coefficient/cm�1
0.075750 7.2 0.060775 12.4 0.070800 19.6 0.095825 33.3 0.101850 opaque
aData extracted from information given by M. J. Dejneka, Mater. Res. Soc. Bull. 23 (November), 57–62 (1998).
6 Although this may be reasonable over very restricted wavelength scales for some compounds, it is not even valid for oxides such as
titanium dioxide over a more extensive wavelength range.
191 Colour Due to Scattering
proportional toVl 1, where V is the volume of the particles which are interacting with the incident light. Now,
as V becomes smaller, the main interaction with light changes from scattering to absorption. Classical Mie
calculationswith spherical goldparticles reveal howabsorptionand scatteringchangewithparticle diameter. In
the case of gold crystals with a diameter of less than approximately 50 nm, absorption is dominant and the total
light removed by absorption plus scattering, the extinction, peaks at approximately 550 nm, in the green region
of the spectrum (Figure 5.16a). The transmitted colour lacks this wavelength and imparts a ruby red colour to
the glass.7 As the particle size increases, the scattered contribution becomes larger and eventually dominates
the absorption (Figure 5.16b d). Owing to this shift, the perceived transmission colour also moves from ruby
red towards purple and then blue. Further increase in size leads to the domination of scattering, and ultimately
reflection, over absorption.Theglass loses bright colour andbecomes opaque.Other colours canbeproduced in
glass by using other noble metals, notably silver for yellow and platinum for pink.
400
400
400
400
500
500
500
500
600
600
600
600
700
700
700
700
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
r = 25 nm
r = 75 nm
r = 50 nm
r = 100 nm
Qa
Qa
Qe
Qe
Qa
Qa
Qe
Qe
Qs Qs
Qs
Qs
Wavelength / nm
Wavelength / nm
Wavelength / nm
Wavelength / nm
Effi
cien
cy F
acto
r,Q
Effi
cien
cy F
acto
r,Q
Effi
cien
cy F
acto
r,Q
Effi
cien
cy F
acto
r,Q
Figure 5.16 Mie calculations for spherical gold particles of radius r: (a) r¼ 25 nm; (b) r¼ 50 nm; (c) r¼ 75 nm;(d) r¼ 100nm. The calculations were made using ‘Scatlab’ software (see this chapter’s Further Reading). [Theoptical constants for gold were taken from P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370–4379 (1972)]
7 The red colour of ruby gemstones and the synthetic ruby crystals used in the first lasers arises for a different reason, and is treated in
Chapter 7.
Colour and the Optical Properties of Materials 192
Note that the calculations for gold are for spherical particles and use the refractive index data for bulk
material. More refined calculations are now possible, for a variety of crystallite shapes (see this chapter’s
Further Reading). These confirm the broad accuracy of the details given above.
When the dimensions of themetallic particles fall below a diameter of 50 nmor so, absorption dominates the
colour effects observed. Although Mie theory documents these changes, it does not explain them and is
confined to spherical objects. The precise absorption characteristics of these small particles depend critically
on the shape and are not well explained in terms of spheres. Further discussion of these colours is postponed to
Section 10.12.
5.10 The Lycurgus Cup and Other Stained Glass
Ruby-coloured and similarly coloured glass was more or less reliably produced from medieval times using
a variety of known recipes. The Lycurgus cup, an artefact that dates from the late Roman period, is unique
and has not been duplicated. Like ruby glass, it is composed of glass coloured by metal nanoparticles but
is renowned for its unusual colouring, as the glass is dichroic. In reflected light the colour is jade green,
while in transmitted light it is a deep wine-red (Figure 5.17). The numerous small metal particles in the
Figure 5.17 The Lycurgus cup: (a) in reflected light; (b) in transmitted light. [Copyright � The Trustees of theBritish Museum]
193 Colour Due to Scattering
glassy matrix are polygonal with an approximate composition of 66.2 at.% silver, 31.2 at.% gold and 2.6
at.% copper. There are a wide range of particle sizes present, but most fall in the range 50 100 nm
diameter, with an average of approximately 70 nm diameter. The colour is supposed to arise chiefly by the
physical processes of scattering and absorption by the metallic particles embedded in the glass matrix
making up the body.
This suggestion can be evaluated to a first approximation by calculating the optical characteristics usingMie
theory for spherical particles. The important parameters are the absorption efficiency Qa, the scattering
efficiency Qs, the extinction efficiency Qe and the backscattering efficiency (also known as the radar
backscattering efficiency)Qr. The extinction efficiency should correspond with the colour seen by transmitted
light, while the backscattering efficiency should correspond to the colour seen in reflected light. Because the
metal particles are an alloy of unusual composition, the optical constants were obtained by using the known
optical constants of the puremetals silver, gold and copper, added in proportion to the alloy composition. Thus,
the optical constants for the alloy, at wavelength l, are computed as:
nalloyðlÞ ¼ 0:662nAg þ 0:312nAu þ 0:026nCu
kalloyðlÞ ¼ 0:662kAg þ 0:312kAu þ 0:026kCu
The computed values of the efficiencies Qext, Qabs, Qsca and Qb (¼Qr /4p) for metal particles of 50 nm
radius are a very close approximation to the experimental spectra taken from the cup (Figure 5.18). The results
can thus be interpreted as supporting the idea that the colour of the Lycurgus cup can be explained in terms of
Mie scattering from the alloy particles in the glass.
It is noteworthy that neither pure silver nor pure gold particles of any size are able to reproduce the colours
observed.
The difference in colour between a typical ruby glass and the Lycurgus cup can be explained in terms of the
relative amounts of scattering and absorption. The small particles which occur in gold sols, gold colloids and
ruby glass are of the dimensions which exhibit high absorption and low scattering. The colour produced on
transmission of daylight is white minus the colour absorbed by the particles, i.e. subtractive coloration. If the
glass is examined in reflected white light it will look dark, as light is absorbed on entering the solid and little is
returned to the eye. The surface may also be shiny, due to reflection from the glass matrix, and a slight ruby
colour may be discerned due to light traversing the glass and being reflected back to the observer from the rear
face of the object.
The colour effects of stained glass windows in churches are similar to ruby glass. These glasses are coloured
by subtractive coloration, often due to the incorporation of transition metal ions into the solid. Viewed in
reflected light the glass appears dark and slightly shiny, for the same reasons as ruby glass. For this reason,
stained glass seen from outside a church, for example, is always rather disappointing compared with the
remarkable effects when illuminated by strong sunlight and viewed inside the building.
As the particle size increases, scattering becomes more important and rapidly dominates the interaction of
the metal particles with incident white light. In the Lycugus cup, scattering has reached this level. The colour
scattered ismostly at the short (blue violet) end of the spectrum. The light traversing the glass will then appear
depleted in blue and appear an orange red colour. In reflected light, little is absorbed, and the glass will be
coloured by backscattered radiation from themetallic particles. This will be predominantly light from the blue
end of the spectrum.
The actual colours observed (in both ruby glass and the Lycurgus cup and related artefacts) depend strongly
upon the particle composition, size and shape and on the density of particles in the glass. Thus, many subtle
variations are to be expected in glassmadeby artisans using relatively irreproducible techniques. Itwould seem
Colour and the Optical Properties of Materials 194
that the craftsmen that fabricated the Lycurgus cup were both remarkably skilled and rather lucky on the day.
For further discussion of these colours, refer to Section 10.16.
Further Reading
Classical scattering theory is treated in detail by
H. C. van deHulse, Light Scattering by Small Particles, JohnWiley and Sons, Inc., NewYork, 1957 (reprinted
by Dover, New York, 1981).
C. F.Bohren,D.R.Huffman,Absorption andScattering of Light by SmallParticles, JohnWiley andSons, Inc.,
New York, 1983 (reissued by Wiley-VHC, Weinheim, 2004).
Atmospheric scattering, including the formation of blue moons, is in treated in
C. F. Bohren, E. E. Clothiaux, Fundamentals of Atmospheric Radiation, Wiley-VCH, Weinheim, 2006.
D. K. Lynch, W. Livingston, Color and Light in Nature, Cambridge University Press, Cambridge, 1995.
400 500 600 700
1
2
3
4
5
6
7
8
r = 25 nm
Qa
Qa
QeQe
QbQb
Qs
Qs
Wavelength / nm Wavelength / nm
Effi
cien
cy F
acto
r,Q
Effi
cien
cy F
acto
r,Q
400 500 600 700
1
2
3
4
5
6
7
8
r = 50 nm
400 400500 500600 600700 700
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
r = 75 nm r = 100 nm
QaQa
Qe
Qe
Qb Qb
Qs
Qs
Wavelength / nm Wavelength / nm
Effi
cien
cy F
acto
r,Q
Effi
cien
cy F
acto
r,Q
Figure 5.18 The extinction, scattering and absorption efficiencies (Qe, Qa, Qs) and the radar backscatteringefficiency/4p (Qb ), as a function of wavelength of the incident radiation, for particle radii in the range25–200 nm
195 Colour Due to Scattering
Related references of interest with respect to atmospheric phenomena are
J. Walker, Sci. Am. 260 (January), 84 87 (1989).
J. Walker, Sci. Am. 238 (January), 132 138 (1978).
The use of cordierite crystals for navigation, the navigation of insects using polarised light and information
about the polarisation of sky light are given in
A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers, Prentice-Hall, Englewood
Cliffs, 1976, p. 369.
J. Walker, Sci. Am. 238 (January), 132 138 (1978).
R. Wehner, Sci. Am. 235 (July), 106 115 (1976).
There are a considerable number of programs available to compute Mie scattering parameters. The original
routine used, Mie Tab (which was available at http://www.zianet.com/damila), seems not to be compatible
with newer operating systems. More recently I have used Scatlab, found at www.scatlab.com.
Mie scattering calculations by Scott Prahl can be found at Oregon Medical Laser Center, omic.ogi.edu/calc/
mie calc.html.
Others are available and can be quickly located via a Web browser.
The Tyndall blue colour of Papilio zalmosis is described by
J. Huxley, Proc. R. Soc. Lond. Ser. B 193, 441 453 (1976).
The invention and production of Lucalox transparent alumina ceramics is described by
J. E. Burke, Mater. Res. Soc. Bull. 21 (June), 61 68 (1996).
The discovery of enhanced transmission in LaF3 glass ceramics is reported by
M. J. Dejneka, Mater. Res. Soc. Bull. 23 (November), 57 62 (1998).
Opal glasses, polychromatic glass and other glassy materials are described from the inventor’s viewpoint by
S. D. Stookey, Explorations in Glass, The American Ceramic Society, Westerville, OH, 2000.
Scattering by biological tissues is described by
S. Johnsen, A. A. Widder, J. Theor. Biol. 199, 181 198 (1999).
The original paper of Mie on the colours of gold colloids is
G. Mie, Ann. Phys. 25, 377 445 (1908).
The microstructure of the Lycurgus cup is given by
D. J. Barber, I. C. Freestone, Archaeometry 32, 33 45 (1990).
Colour and the Optical Properties of Materials 196
6
Colour Due to Diffraction
. What causes the colours reflected from compact discs
(CDs) and digital versatile discs (DVDs)?. Why are opals coloured?. How do liquid-crystal thermometers work?
Diffraction is a particular form of light scattering. There is no hard and fast distinction between scattering and
diffraction, although the term scattering tends to be usedwhen discussing light interactionwith small randomly
distributed particleswhile the expressiondiffraction is associatedmorewith organized structures. In the case of
Rayleigh andMie scattering, the scattered waves have no implicit relationship to one another and the situation
is called incoherent scattering.When the scattering object ismade up of amore or less ordered arrangement of
scattering centres, the scattered waves have a close relationship with each other, defined, in part, by the
separation of the scattering centres.Under these circumstances the outgoingwaves can interfere constructively
or destructively and the phenomenon is called diffraction.
With this rather loose distinction preserved for convenience, one can note that the term diffraction also tends
to be limited to the effects that occurwhen lightwaves interactwith objects havingmore or less ordered features
of a size similar to the wavelength of the radiation. After this interaction the waves travelling away from the
diffracting feature, the diffracted waves, can interfere, thus giving rise to complex patterns of intensity. The
result of diffraction is then a set of bright and dark fringes, due to constructive and destructive interference,
called a diffraction pattern. When the separation between the scattering objects is less than the wavelength of
light, similar effects still occur, but the classical diffraction equations are very restricted in application. The
scattering is then often called coherent scattering rather than diffraction, although the two processes
are identical.
Colour and the Optical Properties of Materials Richard J. D. Tilley
� 2011 John Wiley & Sons, Ltd
Diffraction by a small pinhole was first described by Grimaldi in 1665. Since that time, many famous
scientists have studied the diffraction effects arising when light passes through small apertures in an otherwise
opaque screen. Themathematical analysis of the intensity patterns produced in this waywas not trivial, but the
solutions for apertures of various shapes, which agreed perfectly with observations, provided strong support
for the wave theory of light. Classically, two regimes have been explored in most detail: (i) diffraction quite
close to the object which interactswith the light, calledFresnel diffraction, and (ii) the effects of diffraction far
from the object which interacts with the light, called Fraunhofer diffraction.
An ordered collection of objects that diffract light, such as slits or circular apertures, atoms ormolecules, etc.
when arranged in a regular array, forms a diffraction grating and the mathematical analysis of diffraction
gratings is an important constituent of the field of optics. Broadly speaking, the interference effects can occur
after transmission by the collection, which then forms a transmission grating or after reflection from the
collection, in which case it is a reflection grating. A transmission diffraction grating made up of a series of
transparent apertures in anopaquematerial produces its effect by selectivelychanging theamplitudeof the light
passing through it and is knownas anamplitude object and specifically anamplitude grating. Suppose, instead,
that the grating is composed of adjacent strips of material which are transparent but of differing refractive
indices. In this case a phase difference will be selectively introduced between the beams traversing adjacent
regions, and the material is known as a phase object or phase grating. Differences in phase are not visible,
but can be transformed into visible intensity differences using wave recombination techniques such as
interference. Reflection gratings can also alter the amplitude or phase of the interfering beams to form
amplitude and phase objects.
It is diffraction that often sets a limit to the performance of optical instruments, including the eye, and the
topic is, therefore, of particular practical importance. As well as allowing a quantitative analysis of the
performance of optical instruments to bemade, themathematical analysis of diffraction led to the production of
diffraction gratings for use in spectroscopy and the understanding of crystal structures, thus providing the
foundations of much of modern science. More recently, diffraction studies have expanded into areas in which
disorder, partial order or sub-wavelength order are dominant. These have important consequences in
explaining the transparency of the cornea of the eye, for example
The diffraction patterns formed by diffracting centres are sensitive to the wavelength of the incident
light. When white light is involved, a multiplicity of such patterns form. When these are spatially separated,
intense colours can be observed. Commonplace examples of this abound. Diffraction effects contribute to the
shifting colours seen on many multicoloured wrapping papers and bags. The colours noticed in reflected light
from the surface of a CD, and in some reflected patterns on banknotes or security logos, are also the results of
diffraction.
6.1 Diffraction and Colour Production by a Slit
Tounderstand thediffraction patternproducedbya rectangular aperture (Section6.2) it is easiest tobeginwith a
long, narrow slit. If such an aperture is illuminated bymonochromatic light then some of the incoming wave is
scattered by the edges of the slit and some passes through the central open part. The resultingwaves interfere to
produce a diffraction pattern on the sideof the slit away from the source of illumination. Thepattern far from the
slit, theFraunhofer diffraction pattern, consists of a set of bright and dark fringes running parallel to the slit (if
the complexities caused by the ends of the slit are ignored) (Figure 6.1). These are generally called orders. The
first (straight through) bright fringe is called the zero-order fringe for constructive interference, because here
the interfering waves have a phase difference of zero. Thereafter, the bright fringes are labelled as first order
(phase difference l), second order (phase difference 2l) all for constructive interference. The same is true for
theminima,which represent placeswhere thewaves that interfere are out of phase by amultiple of l/2. Thefirst
Colour and the Optical Properties of Materials 198
minimum is called the first-order fringe for destructive interference, with a phase difference of l/2 between theinterfering waves, the second minimum is called the second-order fringe for destructive interference, with a
phase difference of 3l/2, and so on.
The irradiance pattern observed far from the slit (the Fraunhofer diffraction pattern) is given by the
expression:
Ix ¼ Iosinx
x
� �2
ð6:1Þ
x ¼ kwsin�
2¼ pwsin�
l
where k is the propagation number of thewave (2p/l),w is thewidth of the slit, � is the angular deviation fromthe ‘straight through’ position andl is thewavelengthof the light (Figure 6.2). Thepositions of theminima (that
is, the set of dark fringes) are given by:
x ¼ �p; �2p; �3p; . . . ¼ �mp
sin�min ¼ mlw
screen withnarrow slit
diffractedlight
diffractionangle θ
incidentlight
0th order
1st order
1st order
Figure 6.1 The fringes produced by diffraction of monochromatic light by a long narrow slit. Diffracted light isconcentrated into bands at various values of the angle �u to the undeviated beam
0 0.5 1.0 1.5-0.5-1.0-1.5
1.0Ix / I0
θ / rad
0.8
0.6
0.4
0.2
Figure 6.2 The relative irradiance profile (Ix/Io) for a single slit of width 3l, plotted over the range u¼�p/2
199 Colour Due to Diffraction
wherem takes values 1, 2, 3, etc. For �min to be appreciable, the slit widthwmust be similar to thewavelength of
the light l. Moreover, the formula shows that the spacing between the minima will be proportional to the
reciprocal of the slit width, so that the narrower the opening the wider will the fringe spacing be. The width of
theprincipal (central)maximumis twice that of theothers.Thepositionsof themaximabetween thedarkbands
are not givenby such a simple formula, but can be approximatedby assuming that they liemidway between the
minima.
The sine of the angle through which a ray is diffracted is related to its wavelength. This indicates that
each wavelength in white light will be diffracted through a slightly different angle, with red light
diffracted through the greatest angle and violet light diffracted through the least. In this way, white light
will produce a set of diffraction patterns, each belonging to a different wavelength (Figure 6.3). In the
principal maximum, the spread of the red (long wavelength) waves will be greater than the spread of the violet
waves. In the central part of the peak, all colours will overlap to give white. At the extreme edges, the red will
extend further to give the fringe a reddish hue. The effect in the other maxima will be different as there is
no overlap, as the different colours spread out. These patterns look like, and are called, spectra. They are
referred to as first-order, second-order (and so on) spectra as they are recorded further and further from the
undeviated beam.
The intensities of these spectra are very low comparedwith that of the undeviated central fringe. They can be
estimated by usingEquation 6.1. Taking the central peak as irradiance 1.0, the first-order bright fringes have an
irradiance of 0.0472. The other fringes are even weaker. (These spectra must not be confused with the intense
spectra produced by diffraction gratings; Section 6.5.)
Remarkably, the diffraction pattern of any object is identical to that of the complementary object that is, an
object which is opaque, where the first object is clear. Thus, the diffraction pattern of a thin wire is identical to
that of a thin slit. This is a statement of Babinet’s principle. It means, for example, that the central fringe in the
diffraction patterns of both a thin wire and a thin slit are bright!
6.2 Diffraction and Colour Production by a Rectangular Aperture
A rectangular aperture is formed, conceptually, by shortening the length of the corresponding slit. The bright
anddark ‘slit’ fringeswill now formparallel to both the longand short edges of the rectangle. Thesewill overlap
violet
violet
violet
violet
red
red
red
red
screen
whiteincidentbeam
2nd order
2nd order
1st order
1st order
Figure 6.3 The diffraction orders, resembling spectra, produced by diffraction of white light by a long narrowslit. The angle through which red light is deviated is greater than that by which violet light is deviated for eachorder of diffraction. These spectra are very weak compared with the central zeroth-order white fringe
Colour and the Optical Properties of Materials 200
at the corners so that in someplaces bright fringeswill coincide and in other places dark fringeswill coincide, to
reinforce the pattern. Elsewhere, bright anddark fringeswill overlap to give intermediate degrees of brightness.
The irradiance is given by an equation almost identical to Equation (6.1):
Ixy ¼ Iosinx
x
� �2siny
y
� �2
where all the terms have similar meanings to those in Equation 6.1, with y-coordinates substituted for
x-coordinates where necessary. This produces diffraction maxima in the form of small rectangular spots
running in two perpendicular directions (Figure 6.4a c). The intensity of the central rectangular spot is much
greater than that of the others. The spot spacing is inversely proportional to the dimensions of the slit. Avertical
narrowaperturewill give rise towidely spaced spots in ahorizontal direction (parallel to the aperturewidth) and
closely spaced spots in a vertical direction (parallel to the aperture length). If the rectangular aperture
is replaced by a square, a square array of spots is formed. White light will produce coloured spots, as the
diffraction angle is wavelength sensitive, as described above.
Babinet’s principle allowsone to state that thediffraction pattern of a rectangular speck is identical to that of a
rectangular hole.
(b)
(a)
w
h
(c)
Figure 6.4 Diffraction by a rectangular aperture: (a) aperture dimensions; (b) schematic diffraction pattern;(c) computed diffraction pattern for a slit of length 40l and width 30l. The spacing of the spots in (b) is inverselyproportional to the aperture width w (horizontally) and height h (vertically)
201 Colour Due to Diffraction
6.3 Diffraction and Colour Production by a Circular Aperture
The diffraction pattern of a circular aperture, a pinhole, is formed by the interference of light scattered from the
periphery of the hole. The form of the diffraction pattern produced can be inferred by reference to a square
aperture. In this latter case, a set of bright patches are formed parallel to the edges of the square. If the square is
converted into an octagon, by cutting off the corners, it can be surmised that a set of bright patches will again
form parallel to the straight edges. As the number of sides increases, so does the number of sets of bright spots.
The diffraction pattern of a circular aperture takes this extrapolation to the limit. The pattern will consist of a
series of bright and dark fringes concentric with the original aperture (Figure 6.5). The spacing of the maxima
(b)
(a)
d
(c)
Figure 6.5 The diffraction pattern from a circular aperture: (a) aperture dimensions; (b) schematic pattern,consisting of a bright central disc (Airy’s disc) surrounded by a set of circular light and dark rings (Airy’s rings);(c) computed image of the diffraction pattern. The radii of the rings are inversely proportional to the aperturediameter d
Colour and the Optical Properties of Materials 202
and minima is given by:
sin� ¼ mld
where � is the angle between the directly transmitted ray and the diffraction ring, l is thewavelength of the lightand d the diameter of the aperture. The computation of m requires rather sophisticated mathematics, first
completed by Airy in 1835. The results show that m takes the values 0 (central bright spot), 1.220 (first dark
ring), 1.635 (first bright ring), 2.333 (second dark ring), 2.679 (second bright ring) and 3.238 (third dark ring).
As before, the intensity of the central spot will be considerably greater than the intensities of the surrounding
rings. The angular separation from the centre of the pattern to the first dark ring is given by:
sin� � D� ¼ 1:220ld
ð6:2Þ
The angular spread of the pattern increases as the pinhole gets smaller.
The driving force forAiry’sworkwas the interpretation of the image of a star in a good telescope, Airy being
Astronomer Royal at the time. Under ideal conditions the star will appear as a small point-like disc of light
surrounded by diffraction rings. (These are much fainter than the disc and can often be more easily
distinguished if the eyepiece is pulled in or out by a short distance so as to defocus the image slightly.)
The central bright region is knownasAiry’s disc and the surrounding circles asAiry’s rings. Theperformanceof
a telescope can be estimated by the appearance of these images. A distortion ofAiry’s rings, when atmospheric
conditions are good, is indicative of poor optics. Exactly the same effect will be seen if a point of light is
observed in anopticalmicroscope. Slight defocusingwill reveal an expanding set ofAiry’s rings, theperfection
of which mirrors the perfection of the lenses.
6.4 The Diffraction Limit of Optical Instruments
When an object is imaged in an optical system, a telescope or amicroscope, for example, it can be considered to
bemade up of innumerable point sources. Each of thesewill be imaged not as a point, but, in instrumentswith a
circular limiting aperture, as a set of Airy discs. The idealized case is when the object consists of two small
points, giving rise to two separated Airy discs. As the object points approach each other the image discs will
begin to overlap. The resolution of the optical instrument can, therefore, be equated with the separation of the
pair of adjacent Airy discs just before they appear to merge into one. There are a number of ways in which this
can be translated into a number. Rayleigh suggested, for convenience’s sake, that this limit was taken as
equivalent to the separation of point images when the Airy disc of one fell on the first dark ring of the second.
That is, the angular limit of resolution is:
D� ¼ 1:220ld
where D� (radians) is the angular separation of the image pair, d is the diameter of the limiting aperture in the
optical system and l is the wavelength of the imaging radiation.1 This equation (or similar) expresses the idea
1 This assumes that the optical components are perfect and ignores the resolution of the detector. In digital cameras the image resolution is
generally limited by the resolution of the detector (quoted in megapixels) rather than by diffraction.
203 Colour Due to Diffraction
that the ultimate quality of an image is limited by diffraction, the so-called diffraction limit of optical
instruments.
The form that this equation takes when applied to a particular instrument varies. For telescopes and
binoculars, Equation 6.2 is taken as it stands. The value of d corresponds to the diameter of the objective lens or
mirror and the value of D� is usually quoted as an angle. Star catalogues, for example, list the separation of
double stars in terms of their apparent angular separation. Clearly, the resolution limit of these instruments can
be increased (nominallywithout limit) by increasing thevalue of the objective lens ormirror diameterd.Hence,
the drive to larger and larger telescopes, including, in recent years, the construction of telescope arrays with
effective apertures of many kilometres in diameter. The great Mt Palomar telescope has a mirror of
approximately 5m diameter, giving the instrument an ideal diffraction-limited resolution of 2.8� 10 2
seconds of arc.
Cameras for photography of scenes far from the lens, and eyes, are similarly constrained. In these
instruments, resolution is limited by the apparent diameter of the lens. For an eye, this equates to the diameter
of thepupil.Takingapupil diameterof 3mmgivesadiffraction-limited resolutionof approximately46 seconds
of arc for light of 550 nm. Thus, the ideal eye cannot separate objects that have an angular separation less than
this amount. Real eye performance is poorer than this. For example, the middle star of the ‘handle’ of Ursa
major (the Big Dipper) star constellation, zeta Ursae majoris (Mizar), has a fainter companion, Alcor, at a
separation of 11.8minutes of arc, roughly 10� the diffraction limit of a perfect eye.This pair of stars, the ‘horse
and rider’, is a ‘naked-eye double’, and the ability to separate them is considered a good test of eyesight (and of
local atmospheric conditions).
The optics of the microscope requires a different interpretation of Equation 6.2. In this instrument the
diameter of the objective and the closeness of approach to the object are of importance.Moreover, the distance
separation of the object points is of importance, rather than the angular separation. Keeping the same Rayleigh
criterion that is, theAiry disc ofone imagepoint falls on thefirst dark ringof the second leads to the equation:
s ¼ 1:220l2sini
where s is the minimum separation of two self-luminous object points that can be resolved and i is the semi-
angle subtended at the objective lens by one of the points.
Note that the real situation in amicroscope ismore complicated, as the object points are rarely self-luminous.
Abbe considered the problem in detail and produced the formula generally used for microscope resolution,
which drops the factor 1.220 and takes into account the refractive index of the medium surrounding the points:
s ¼ l2nsini
wherel is thewavelength of the light used,n is the refractive indexof themediumsurrounding the points and i is
the semi-angle subtended at the objective lens by one of the points. The product ðnsiniÞ is the Abbe numerical
aperture value of the objective, with values of 1.5 or better for good lenses.
Ineither case, detail less thanabout half thewavelengthof the light used forobservation is not recovered.This
wavelength connection accounts for the drive towards the use of short-wavelength ultraviolet light for the
preparation of integrated circuits on silicon chips to increase the density of packing of circuit elements. The
recent development of ‘superlenses’ (Section 2.10) allows the diffraction limitation to be bypassed.
Just as with the slit, the dependence of the diffraction angle upon wavelength means that a circular aperture
illuminated with white light will produce a central white spot edged with red and a concentric set of coloured
rings, rather likeminiature circular rainbows. The formula indicates that each ringwill have a violet inner edge
and a red outer edge.
Colour and the Optical Properties of Materials 204
The same effect is seen when a beam of white light is scattered by a small mote of dust. Babinet’s principle
indicates that this scatteringwill take the same formas that by a small pinhole. The central disc of the diffraction
pattern will be bright in both cases and the colour sequencewhen illuminatedwith white light will be the same.
The propagation of a beamof light is similarly affected bydiffraction.Abeamof laser light, for instance,will
spread because of diffraction as it leaves the laser. This effect, although small, can be important, as when, for
example, lasers beams are used to prepare optical masks or gratings (Section 6.5). Recently, in the same way
that thediffraction limit for lenseshas beenbypassed, thediffraction spreadingof laser beamshasbeennullified
by patterning the emerging light to resemble the intensity profile found in the Airy diffraction pattern from a
circular aperture, emerging as an ‘Airy wavefront’ (see this chapter’s Further Reading).
6.5 Colour Production by Linear Diffraction Gratings
The simplest diffraction grating to visualize consists of a sheet of material inscribed with a set of regularly
spaced parallel lines with a repeat period similar to that of the wavelength of light. Originally, gratings were
made by carefully ruling lines on a metal sheet. This ‘master grating’ was replicated by making copies of the
surface using a suitable dimensionally stable polymer film. These can act as reflection gratings when coated
with a reflectivemetal such as aluminium. Amechanically simpler method of making a linear grating is to use
the interference pattern formed by two laser beams. If these interact in a film of photoresist,2 then a sinusoidal
interference pattern is formed which is transformed into a set of sinusoidal groves when the photoresist is
processed.An advantage of thismethod is that the grating spacing is easy to control, because it is simply altered
by changing the angle of the interfering laser beams and thewavelength of the laser light, both of which can be
adjusted precisely.
Light transmitted or reflected from such a linear diffraction grating forms a pattern of intense maxima
separated by much weaker intensity oscillations. The positions of the maxima, called principal maxima,
produced by a thin planar transmission or reflection grating in air are given by the grating equation:
dðsin�i þ sin�mÞ ¼ ml ð6:3Þ
where d is thegrating spacing, �i is the angle of incidence and �m is the angle of diffraction of themth-order line,
taking values of 0,�1,�2,�3, etc. (Figure 6.6). (The convention when using this formula is that the angles of
incidence and diffraction are considered to be positivewhen the incident and diffracted beams are on the same
side of the normal to the grating and the angle of diffraction is negative when the diffracted beam lies on the
opposite side to the grating normal to that of the incident beam.)3 Each value of m corresponds to a different
diffraction maximum, called an order. For the zero-order diffracted beam, �i¼��m, which is identical to
straight through transmission or ordinary (specular) reflection. It is important to be aware that these orders are
not the same as the weak orders from a single scattering object (as depicted in Figure 6.3, for example) but are
new, intense peaks formed by the periodic grating. The most intense orders (after m¼ 0, which is the most
intense), are them¼�1, thenm¼�2. For a particular value of � the orders form a row of diffracted maxima
with spacing proportional to 1/d. Because each line on the grating acts as a contributing slit they are of much
2 A photoresist is a polymeric material that is altered by exposure to radiation, (usually light). After illumination, the photoresist is
weakened or strengthened in those areas which were exposed to light. Theweakened areas can be selectively removed by dissolution so as
to reveal the underlying material, which can then be further manipulated.3 The grating equation iswritten in a number ofways, differing in the sign of the sin � terms. These alternatives simply reflect different sign
conventions for the angles of incidence and diffraction and which side of the grating normal is allocated to þm and m. All forms of the
equation give identical results if used with the correct sign convention.
205 Colour Due to Diffraction
greater intensity than those formed from a single slit. These principal maxima become sharper as the spatial
frequency of the grating (the number of lines per unit length) increases. Quite ordinary gratings, with in excess
of 1000 lines per millimetre, produce very narrow orders which are, in effect, imaged as lines.
There are two particularly useful formulations of the grating equation. The positions of the diffraction
maxima for a transmissionor reflection gratingwhen illuminatedbymonochromatic light normal to the surface
are given by the formula:
dsin�m ¼ ml
(a)
transmission gratingincidentbeam
d
order m
order m = 0(straight through)
θmθ i
(b) incidentbeam
reflection grating
d
order m
order m = 0(specular reflection)
θm
θ i
Figure 6.6 Diffraction grating: (a) diffraction from apertures in an amplitude transmission grating; (b)diffraction from reflection in an amplitude reflection grating
Colour and the Optical Properties of Materials 206
This is derived fromEquation6.3 byputting�i¼ 0 (Figure 6.7a andb).When light falls on a reflection grating at
close to grazing incidence, �i¼ 90� and the formula for the positions of the diffraction maxima is:
dð1�sin�mÞ ¼ ml
or
dð1�cos�Þ ¼ ml
and � is the angle that the diffracted beammakeswith the reflecting grating surface,which is the complement of
�m (Figure 6.7c).
transmission grating
reflection grating
incidentbeam
incidentbeam
incidentbeam
reflection grating
(a)
d
d
d
order m
order m
order m = 0
order m = 0
order m
order m = 0
θm
θm
θm θ
(b)
(c)
Figure 6.7 Diffraction at normal incidence: (a) amplitude transmission grating; (b) amplitude reflection grating.Diffraction at grazing incidence: (c) amplitude reflection grating
207 Colour Due to Diffraction
When a grating is illuminated by white light, each wavelength will be diffracted through a slightly different
angle so that each order will consist of a spectrum except for the m¼ 0 order, in which all of the different
wavelengths overlap to give white. The resolution of a grating that is, the difference in wavelength of two
adjacent lines that can be separated is a function of the spatial frequency (number of lines per unit length) of
thegrating.Thegreater the spatial frequency, thegreater is the resolvingpower of thegrating. For spectroscopic
purposes, especially when examining the light from faint objects such as distant stars, it is important to
concentrate as much light as possible into the intensem¼�1 orders. For the simple gratings discussed, most
light falls into the spectroscopically uselessm¼ 0 order. If the reflecting units of a reflection grating are cut at a
slight angle to the plane of the grating, to produce a blazed grating, then themaximum intensity can be directed
into any chosen spectral region. Gratings for specialist use are invariably of blazed construction.
Diffraction gratings can give rise to very intense colours. Although accurately ruled gratings are expensive,
plastic replicas are inexpensive and readily available. These can be used to show the spectra of many light
sources, including those from torches using incandescent light bulbs (Figure 6.8) and from street lights (see
Chapter 7).
6.6 Two-Dimensional Gratings
Adiffraction grating consisting of a set of ruled lines is, in effect, a one-dimensional grating. The simplest two-
dimensional grating is formed by two sets of ruled lines at right angles to one another. This is equivalent to an
array of apertures arranged in an ordered pattern in an opaque screen. The diffraction pattern from such an
arrangement consists of an array of intense spots arrangedon agridwith a symmetry thatmatches the symmetry
of thegratingpattern. For example, agratingconsistingof a rectangular arrayof apertureswill produceapattern
consisting of a zero-order central maximum of greatest intensity surrounded by a rectangular array of bright
spots. If the grating repetition is characterized by perpendicular spacings a and b then the rectangular grid of
diffracted maxima will be spaced proportional to 1/a and 1/b. As before, it is important to be aware that the
maxima surrounding the central peak are not the same as theweaker subsidiary maxima described above for a
Figure 6.8 The continuous first-order spectrum formed by passing the light from a pen torch through a lineartransmission grating of 1000 lines/mm. The disc to the right is the zero-order spot
Colour and the Optical Properties of Materials 208
single rectangular aperture (Section 6.2 and Figure 6.4), but are new intense orders produced by the grid of the
diffraction grating, as are the principal maxima in the line grating described above.
For example, a net curtain acts as a (not perfectly aligned) array of square or rectangular apertures. If a far-off
sodium street lamp is viewed through fairly closely woven net curtains, a square or rectangular grid of yellow
spots will be seen centred upon the image of the light itself (Figure 6.9a). If the light is white, such as a beam of
strongly reflected sunlight, the diffracted orders, except for the central (zeroth) spot, will be coloured, although
the separationof the spectrawill not begreat (Figure6.9b).The intensityof thepatterns is seen tobest advantage
when viewing a small, distant bright light through a closely woven black fabric such as an opened umbrella,
which absorbs superfluous reflection and scattering.
The array of surface pits on a compact disc makes a good reflection grating. The pits that record the data are
arranged along tracks of a constant spacing (Section 3.1) on a surface that is subsequently covered with a
Figure 6.9 Two-dimensional transmission gratings: (a) a sodium streetlight viewed through a white net curtain;(b) bright sunlight reflected from parked cars viewed through a white net curtain. The rectangular array ofdiffraction orders mirrors the symmetry of the curtain mesh
209 Colour Due to Diffraction
reflecting layer. The surface, therefore, forms a curved line grating and the image of a white light viewed by
reflection from the surfacewill showseveral orders of diffraction, seen as continuous spectra, as the disc is tilted
(Figure 6.10a). Because of the curvature of the tracks these spectra take a complicated form. Similar
complications arise, and are put to good effect, in reflection gratings used in decorative coatings. In these,
the plastic grating replicas are incorporated into the decorativepattern. Theobserved colours varywith viewing
angle and the degree of distortion of thematerial (Figure 6.10b). Colour effects are often enhanced by covering
the reflection grating with other coloured transparent layers.
6.7 Estimation of the Wavelength of Light by Diffraction
It is surprisingly easy to estimate the wavelength of light using a digital versatile disc (DVD), compact disc
(CD), hi-fi gramophone (phonograph) record or a steel rule in conjunctionwith the phenomenon of diffraction.
The method uses these objects as reflection diffraction gratings, which, for a narrow beam of light, can all be
Figure 6.10 Reflection grating colours: (a) colours formed by the reflection grating on the surface of a CD;(b) colours formed by the reflection grating on the surface of a gift bag
Colour and the Optical Properties of Materials 210
considered to be linear gratings. The spread of the diffracted spectra depends upon the spacing of the grating.
The principal grating ondiscs is formedby the tracks,which are 1.6mmapart on aCDand0.74 mmonaDVD.A
hi-fi gramophone record has a grating spacing of about 0.1mm and a steel rule is graduated down to 0.5mm.
Several orders of diffraction can be seen if a narrow beam of white light from a pen torch is reflected at near to
grazing incidence to a disc surface. The orders diffracted from a hi-fi record are harder to see, and those from a
steel rule are the most difficult to detect. Much more accuracy can be obtained using a laser pointer. Because
laser light is coherent (Section 1.9), diffraction effects are pronounced.
If a laser pointer is shone on aCDat close to grazing incidence several diffracted orderswill be easily visible as
brightspotsonanearbywallor screen(Figure6.11). (Ashort lineofclosely spacedsubsidiarymaximawillappear
to either side of these bright spots. They can be ignored for the present purposes, as only the most intense spot
of thisgroupis theprincipalmaximumoftheorder.) Inorder toestimate theanglesofincidenceanddiffraction,use
the fact that tan(90� �i) is given by s0/D for the zero order (m¼ 0, reflected) beam. The angle of diffraction of
the order-m beam, given by tan(90� �m), is given by sm/D. These values are substituted into the grating equationto obtain the laser wavelength, remembering to pay attention to the sign convention used (Section 6.5).
In a home experiment, a helium neon laser pointer was attached to the top of a camera tripod by tape and the
beam directed at a shallow angle onto a CD. The positions of the diffraction maxima could be measured to
within 1mm by allowing the beams to fall onto a sheet of graph paper. In a quick trial the distances were
s0¼ 6.3 cm, s1¼ 23.5 cm, D¼ 15.0 cm (Figure 6.11). Hence:
tanð90��iÞ ¼ 6:3=15:0 �i ¼ 67:2�
tanð90��mÞ ¼ 23:5=15:0 �m ¼ �32:6�
Substitution inEquation6.3withm¼ 1givesl¼ 613 nm.The red laser light has awavelengthof632.8 nm. It
is surprising that such an accurate value can be obtained so easily. Careful experimentationwill give an answer
much closer to the known wavelength.
6.8 Diffraction by Crystals and Crystal-like Structures
6.8.1 Bragg’s law
The atoms in a crystal are arranged in ordered arrays and form a three-dimensional grating. The separation
of atoms in crystals is similar to the wavelength of X-rays, and the diffraction of X-rays from these
red laser pointer
θ i
θm
θ i
d
D
s0
s1
first order
zeroorder
screen
Figure 6.11 The arrangement to measure the wavelength of light with a simple reflection grating such as a CD
211 Colour Due to Diffraction
three-dimensional gratings has been used for the elucidation of crystal structures since the early years of the
twentieth century. Electrons or neutrons can also be diffracted by crystals, and both these techniques are also
widely used in structure analysis. The resulting diffraction pattern is analogous to that produced by a two-
dimensional grating.A grating consisting of an array of atoms placed at the corners of a lattice built up by stacking
brick-like units of sides a, b and c will produce a pattern consisting of a zero-order central maximum of greatest
intensity surrounded by a three-dimensional array of bright spots also arranged on a brick-like lattice. The grid of
diffracted maxima will be spaced proportional to 1/a, 1/b and 1/c. To be able to determine a crystal structure
precisely it is necessary to measure the positions and intensities of the diffracted beams. However, even the
position alone of a diffracted beam will give information about the spacing of the planes of atoms responsible.
This comesabout in the followingway.Thepositionsof thediffractedbeams froma lineof atomsaregivenby
Equation 6.3. Focusing attention on the zero-order reflection, for conciseness, a strong diffracted beam (the
‘reflected’ beam) will occur at an angle dependent upon (but not equal to) the direction that the incident beam
makeswith the line of atom scatters.When the atoms are arranged on aplane, a strong diffracted beamwill only
occur when the diffraction maxima from each of the rows of atoms in the plane are in phase. This imposes
restrictions such that a strong zero-order diffracted beam is only producedwhen the plane is treated as amirror
and the angle of incidence of the beam falling onto the plane is equal to the angle of ‘reflection’ of the diffracted
beam off the plane. There is, though, no restriction on the angle of incidence itself. When the atom planes are
stacked up to form a three-dimensional grating (i.e. a crystal), there are further limitations to the diffraction.
Once again, a strong zero-order diffracted beamonly occurs for ‘reflection’, i.e. when the angle of incidence on
the stackofplanes is equal to the angle of ‘reflection’ off the stackof planes, but in addition, only certain specific
values of the spacingbetween the planes give rise to anysignificant intensity.Thismeans that a strongdiffracted
beam only occurs at a few specific angles of incidence. Similar arguments apply to other orders of diffraction.
Thus, when a crystal is bathed in a beam of X-rays, no diffracted maxima will, in general, be observed. As the
crystal is tilted and rotated, sometimes a set of atomic planes in the crystal will be in just the right orientation for
‘reflection’, and the spacing of the planeswill be just right for the production of a strong diffracted beam, so that
an intense ‘reflection’ flashes out from the crystal, quickly being extinguished as the crystal is rotated or tilted
further. The well-known formula relating the planar spacing and the occurrence of a strong diffracted beam is
known as Bragg’s law.
Consider the diffraction (for the reasons given above, often called ‘reflection’) of a beam, 1, of monochro-
matic X-rays from a plane of atoms in a crystal (Figure 6.12). Each atom acts as a point scattering centre for the
d
A
B
C D
2
1
θB
θB
θB
Figure 6.12 The geometry of the diffraction of X-rays from a crystal lattice needed to derive Bragg’s law. Twobeams, 1 and 2, are ‘reflected’ from two adjacent planes of atoms spaced a distance d apart and reinforce eachother when the path difference between them is a whole number of wavelengths
Colour and the Optical Properties of Materials 212
X-rays and the maximum diffracted intensity will lie at the same angle �B, the Bragg angle, with respect to theatom layer as the incident beam. (Care here the traditional angle of incidence used in optics, i.e. �i, is thecomplement of the Bragg angle �B). If another beam, 2, is reflected from a parallel layer of atoms a distance d
below the first layer it will travel further than beam 1. For beams 1 and 2 to reinforce each other theymust be in
phase on leaving the crystal. (Because the process occurring with both beams is identical we can ignore any
change of phase that might occur on diffraction.) This means that the path difference between beams 1 and 2
must be a whole number of wavelengths. Using the information in the figure it is seen that beam 2 has a longer
path than beam 1 by CB þ BD.
For reinforcement:
CBþBD ¼ ml
where m is an integer and l is the wavelength of the X-rays. However:
CB ¼ BD ¼ dsin�B
Hence:
ml ¼ 2dsin�B ð6:4Þ
where d is the separation of the planes of atoms which are responsible for the diffraction, l is the X-ray
wavelength, m is the order of the diffracted beam and �B is the Bragg angle between the X-ray beam and the
atom planes. This relationship is Bragg’s law. It was first applied in 1913 to determine the spacing of the lattice
planes in a crystal of sodium chloride.
Crystals do not scatter X-rays very strongly. Only a very small proportion of the incident beam is diffracted,
and implicit in the Bragg equation is the notion that anyX-ray photon is only scattered once. (This also applies
to biological material, and it is for this reason that X-rays can be used inmedical diagnosis.) It is not unusual to
use exposure times of hours in order to obtain X-ray diffraction patterns from small crystals. The theory
describing this diffraction is called the kinematical theory of X-ray diffraction.
In contrast to this, electrons, which are also diffracted by crystals, interact very strongly with the atoms in a
crystal. Thus, it is easy to obtain an electron diffraction pattern of a crystal in a fraction of a second, using an
electronmicroscope (Figure 6.13). The diffraction pattern consists of an array of bright spots, each of which is
derived from a plane in the crystal and conforms to Bragg’s law in position. Because the electron beam passes
through the crystal, it gives information about the grating formed by atom planes parallel to the electron beam.
However, very thin crystalsmust beused toobtain thesepatterns, because eachelectron is scatteredmany times.
If the crystal is thicker than a few nanometres the electrons are completely absorbed by the crystal and no
diffraction effects are recorded. The theory describing electron diffraction for suchmultiple scattering is more
complex than the kinematical theory and is called the dynamical theory. Moreover, because of the multiple
scattering, it is not easy to relate the intensity of the scattered beams to the atomic structure of the crystal, and
electron microscopy and electron diffraction are mainly used to determine crystal structures when methods
such as X-ray diffraction cannot be employed.
The dynamical theory ofX-ray diffraction reduces to the kinematical theorywhen the scattering ofX-rays is
weak.Bragg’s law,derivedfromthekinematical theory, is anapproximation that suffices formanyapplications.
6.8.2 Opals
Thegemstone precious opal is an example of a naturalmaterial that diffracts light in the sameway that ordinary
crystals diffract X-rays. Common opal (potch opal) has a milky appearance and is the origin of the adjective
213 Colour Due to Diffraction
opalescent. Precious opal shows flashes of colour from within the stone, blazing out brilliantly over small
angles as the stone is tilted. In the rarest opals the colours flash out from a black background. Figure 6.14 shows
veins of opal in ironstone fromAustralia. The colours in the veins changes if the fragment is tilted only slightly.
(In reality the figure does not do justice to the colours seen with an optical microscope, which reveals fleeting
reds greens and blues, all of which are angle dependent and seemingly buried within the veins.)
The colour of precious opal is due to the diffraction of white light. The regions producing the colours are
madeupof anorderedpackingof spheresof silica (SiO2)which are embedded in amorphous silica or amatrix of
Figure 6.13 Electron diffraction pattern from a single crystal of the oxide WNb12O33. The pattern is a planesection through the three-dimensional diffraction pattern. The spacingof thediffracted spots gives informationonthe dimensions of the crystal structure
Figure 6.14 Veins of opal in ironstone
Colour and the Optical Properties of Materials 214
disordered spheres (Figure 6.15). These small volumes resemble small crystallites. They interact with light
because the spacing of the ordered regions of silica spheres is similar to that of the wavelength of light. These
flashing colours arise from regions where the spheres of silica are ordered, whereas the pale milky colour of
potch opal arises in regions containing disordered spheres.
1. The conditions underwhich diffraction takes place are the same as those discussedwith respect to theBragg
equation. However, the Bragg equation, which was derived for X-ray diffraction, must be modified in the
following two ways.The incident light is not travelling in a vacuum, but in a matrix of silica, so that the
diffraction conditions will relate to the wavelength of the light in the solid. Thus, use:
lðopalÞ ¼ l0=ns
where ns is the refractive index of the opal matrix, approximately that of silica in opal, about 1.45, and l0 isthe vacuum wavelength of the light.
2. Refraction of the light beamwill take place at the opal surface and the diffraction angle �B in the opalwill notbe the same as the angle that the beammakeswith the external surface.Writing �1 for the angle of incidenceand �2 for the angle of refraction (Figure 6.16a):
�B ¼ ð90��2Þ
ordered silicasphere “crystallites”
disordered spheres and amorphous silica
Figure 6.15 The structure of precious opal consists of regions where spheres of silica pack together into ordered‘crystallites’ surrounded by a disordered matrix of silica spheres and amorphous silica. The ordered ‘crystallites’vary from one to another in orientation and in the diameters of the spheres
215 Colour Due to Diffraction
The form of the Bragg equation becomes:
ml0 ¼ 2nsdsin�B
¼ 2nsdsinð90��2Þ¼ 2nsdcos�2
¼ 2nsdð1�sin2�2Þ1=2
Using Snel’s law:
sin�2 ¼ sin�1ns
d
colour(wavelength λ)
θ1
θ2
θB
white light(a)
red
yellow-green
orangeyellow
(b) whitelight
array of spheres
surfaceof opal
θ1
θB θB
θc
Figure 6.16 Diffraction from precious opal. (a) Diffraction from an ordered array of silica spheres in anamorphous silicamatrix. (b) Not all colours will be able to escape from the opal, due to total internal reflection. Ifred light is observed normal to the diffracting layers, yellow–green light will just escape along the surface. Allshorter wavelengths will remain within the opal
Colour and the Optical Properties of Materials 216
and just writing l for the wavelength of the light observed in air:
ml ¼ 2nsd 1� sin2�1n2s
� �1=2
¼ 2dðn2s � sin2�1Þ1=2ð6:5Þ
(For precious opal this formula is adequate, but in circumstances in which the volume of the voids becomes
comparable to the volume of the solid it is better to use the effective refractive index; Section 6.8.4.)
When an opal is illuminated with white light some regions strongly diffract red, some green and so on,
dependent upon the incident angle �1 of the incident light, the sphere diameter and the order of diffractionm.
The colour seen is given by:
l ¼ 2dðn2s � sin2�1Þ1=2
m
The colour of a single grain will change with viewing angle because of the sin2� term in the equation. As the
diffracting grain is tilted, the colour noted by afixedobserverwillmove fromamaximumvalue to lower values;
that is, red colours will shift towards violet.
The longest wavelength observable lmax will occur at normal incidence, when sin�1 ¼ 0 andm¼ 1. In this
case the wavelength diffracted back to the viewer will be:
lmax ¼ 2nsd
At a certain angle total internal reflection will prevent the light from escaping (Figure 6.16b). The actual range
of colour play will thus be less than that suggested by the Bragg equation. If the opal has a flat surface and is
surrounded by air, the critical angle �c (see Chapter 2) will be given by:
sin�c ¼ 1:0
ns
The diffraction angle �B is given by (90� �c)� (Figure 6.16b). Thus, it is possible to write:
lmax=lc ¼ ð2 ns dÞ=ð2ns dsin �BÞ ¼ 1=sin�B ¼ 1=cos �c
where lc is thewavelength of the colour diffracted just at the critical angle. No light of shorter wavelength willescape.
The equation for natural opal, Equation 6.5, can be generalized for light passing from amediumof refractive
index n1 into a medium of refractive index n2 thus:
ml ¼ 2n2dsinð90� �2Þ¼ 2n2dcos�2
¼ 2n2dð1� sin2�2Þ1=2
217 Colour Due to Diffraction
Using Snel’s law, sin�2 ¼ n1sin�1=n2 (Chapter 2):
ml ¼ 2n2d 1� n21sin2�1
n22
� �1=2
¼ 2dðn22 � n21sin2�1Þ1=2
If the surrounding medium is air, n1¼ 1.00, and the medium of the opal is silica, n2¼ ns:
ml ¼ 2dðn2s � sin2�1Þ1=2
as above.
6.8.3 Artificial and inverse opals
There is considerable interest in the use of artificial opals and related structures for the diffraction of light, with
the ultimate aimof employing thesematerials for optical datamanipulation and computing. Artificial opals are
generally prepared by forming polymer (frequently polystyrene or poly-methyl methacrylate) spheres in
suspension and then allowing these to aggregate in controlled conditions, often aided by centrifuging. The
chemistry of formation is carefully controlled so that only a small range of sizes is produced amonodisperse
suspension and the solid aggregates then adopt structures analogous to those of puremetals such as copper or
gold. The products are called colloidal (photonic) crystals or colloidal opals.
The initial step in the formation of a colloidal crystal is the deposition of a close-packed hexagonal array of
spheres onto aplanar substrate (Figure 6.17a). Successive sheets of sphereswith the samegeometry formon top
of the first, fitting into the dimples on the preceding layer. The commonest structure formed corresponds to a
three-layer repeat stacking4 (Figure 6.17b).Theunit cell of this arrangement is, in fact, cubic and representative
of the face-centred cubic structure (Figure 6.17c). The sheets that are laid down in this way correspond to
crystallographic (111) planes and the direction normal to the sheets is the [111] direction.5 In terms of the
conventional crystallographic cubic unit cell, (111) planes are cell diagonal planes and the [111] direction is the
cell body diagonal. For an array of cubic closest packed spheres the fraction of the volume occupied is 0.7405
and the relationship between the sphere radius and the spacing of the (111) planes is:
d111 ¼ 2 2p
r
3p � 1:633r
It is of interest to discover that a mixture of spheres of two sizes will often aggregate to form superlattices, the
structures of which are analogous to those of alloys such as brass, an alloy of copper and zinc.
Inverse opals are fabricated from colloidal crystals (Figure 6.18a) by infiltrating the spaces between the
polymer spheres with a suitable inorganic precursor in solution. The precursor is transformed to a solid by
drying or heating. The polymer spheres that form the crystalline template are removed by solution or heating in
air. The end result is a crystalline array of hollow spheres, the shells of which are made of the inorganic solid
chosen (Figure 6.18b). The shells may be crystalline or amorphous, depending upon preparation methods.
4 This arrangement is called cubic closest packing. The two simplest ways of stacking the layers one on top of another are a two layer
repeat called hexagonal closest packing and the three layer repeat called cubic closest packing.Manymore complex packing arrangements
can be devised.5 The designation of planes in a crystal is byMiller indices (hkl). The indices h, k and l specify the fractions of the unit cell edges a, b and c
intercepted by the plane. Directions [uvw] are perpendicular to (hkl) cubic crystals. Note that the type of brackets used, (xxx) or [xxx], are
part of the nomenclature.
Colour and the Optical Properties of Materials 218
(a)
layer A
layer Blayer C
(b)
[111](c)
Figure 6.17 Close packing of spheres: (a) a single close-packed array of spheres; (b) cubic closest packing ofspheres, with all layers identical to that in (a); (c) the cubic unit cell of the packing in (b). Each layer in (b) liesperpendicular to the unit cell diagonal, [111]
219 Colour Due to Diffraction
Figure 6.18 Scanning electron micrographs of: (a) a poly-methyl methacrylate colloidal crystal; (b) an inverseopal formed of CeO2 fabricated from (a). [Reprinted with permission from Chemistry of Materials, Physical andOptical Propoerties of InverseOpal CeO2 Photonic Crystals byGeoffrey I. N.Waterhouse et al., 20, 3, 1183-1190Copyright 2008]
Colour and the Optical Properties of Materials 220
When these artificial structures are illuminated with white light they will strongly diffract colours
of wavelength l in a similar manner to natural opals. It is convenient to replace ns in the equation for
natural opal, Equation 6.5, with the effective refractive index of the opal or inverse opal phase ne (see
Section 6.8.4 below) to give:
ml ¼ 2dðn2e�sin2�1Þ1=2
wherem is the order of the reflection, d is the spacing between the layers of spheres or voids that make up the
diffracting plane of the crystal and �1 is the (conventional) angle of incidence of thewhite light inmedium1.As
the angle of incidence increases, so thewavelength diffractedwill decrease; that is, a red reflection at �¼ 0�willmove towards green and blue, as described for natural opals above.
The maximum value of l is given bym¼ 1, due to diffraction from the (111) planes of spheres, which have
the greatest value of d (see Section 6.8.5). Because of shrinkage during processing it is useful to replace the
sphere radius with the more easily measured average distance between the sphere centres D:
d111 � 1:633r � 0:8165D
The wavelength diffracted by the (111) planes is then given by:
l111 � 1:633Dðn2e�sin2�1Þ1=2
At normal incidence:
l111 � 1:633Dne
6.8.4 The effective refractive index of inverse opals
The effective refractive index ne of the opal or inverse opal can be estimated experimentally by plotting the
square of the wavelength diffracted from the (111) planes versus sin2�1:
l2111 � ð1:633DÞ2ðn2e�sin2�1Þ¼ ð1:633DÞ2n2e�ð1:633DÞ2sin2�1
The slope and intercept of the linear graph (Figure 6.19a) are given by:
slope ¼ �ð1:633DÞ2
intercept ¼ ð1:633DÞ2n2e
The effective refractive index can be related to the refractive indices of the components of the inverse opal in
several ways. The most widely used is to employ the volume fractions (Section 2.5) thus:
ne ¼ n1V1 þ n2V2 þ n3V3 þ � � �V1 þV2 þV3 þ � � � ¼ 1
221 Colour Due to Diffraction
where ni is the refractive index of the ith component and Vi is the fraction of the total volume of the solid
occupied by the ith component. For a two-component system, such as that composed of spheres and air or
spherical shells and air:
V2 ¼ 1�V1
If the air-filled voids in an inverse opal in air are filled with a liquid, the effective refractive index change will
result in a change in the colours diffracted.Generally speaking, thewavelength diffracted increases, so that, for
example, blue green diffracted beams become red on adding liquid. For simplicity, consider an inverse opal
in air diffracting light from the (111) planes. At normal incidence:
λ2111
sin2 θ1
intercept(1.633 D)2 ne
slope– (1.633 D)2
λ111
n2
intercept(1.633 D) n1 V1 slope
(1.633 D) n2 (1 – V1)
(a)
(b)
Figure 6.19 Schematic plots of (a) l2111 versus sin2u1; (b) l111 versus n2 for an inverse opal structure; l111is the wavelength of light diffracted by the (111) planes of the array, u1 is the angle between the normal to thesurface and the incident beam of white light and n2 is the refractive index of the fluid filling the voids in thestructure
Colour and the Optical Properties of Materials 222
l111 ¼ 1:633Dne
ne ¼ n1V1 þ n2ð1�V1Þ
where subscript ‘1’ refers to the walls and subscript ‘2’ to the voids. Thus:
l111 ¼ 1:633D½n1V1 þ n2ð1�V1Þ�¼ 1:633Dn1V1 þ 1:633Dn2ð1�V1Þ
where n2 is the refractive index of the liquid within the void. Using a series of different liquids, a plot of l111versus n2 (Figure 6.19b) will be a straight line with the parameters:
slope ¼ 1:633Dn2ð1�V1Þintercept ¼ 1:633Dn1V1
Once this graph has been constructed, the inverse opal can be used as a refractive index meter for the
determination of an unknown refractive index.
6.8.5 Photonic crystals and photonic band gaps
Photonic crystals are artificial structures that have unit cells with dimensions approximately equal to the
wavelength of light and so give rise to intense diffraction colours. The ‘crystal’ can be made up of arrays of
particles (as in opals), voids (as in inverse opals) or any other structures (tubes, layers and so on) provided that
the structural repeat distance is similar to the wavelength of light. In addition to structures fabricated in the
laboratory, many beautiful colours in nature are produced by natural photonic crystals created by living
organisms. Probably best known are some of the spectacular colours of certain butterflies or beetles, but lesser
known creatures, such as the sea mouse, have vividly coloured spines due to a photonic crystal type of
microstructure (see this chapter’s Further Reading).
There is considerable interest in the fabrication of photonic crystals, including artificially mimicking the
natural photonic crystals that appear in living organisms, because these provide a compactway ofmanipulating
photonswithout additional energy requirements. Because of this research perspective, the terminology used in
electronics has been used to describe some aspects of the physical processes that occur on reflection and
diffraction. Thus, when a colour is strongly reflected by an opal or an opal-like array it will not pass through the
solid. In the jargon of photonics, this state of affairs corresponds to a photonic band gap (PBG) or a stop band.
Thus, a PBG or stop band occurs when a range of frequencies will not propagate through the crystal. The terms
reflection,Bragg reflection, stop band and PBGare frequently used synonymously.More explicitly, a complete
PBG occurs when the propagation of a range of frequencies is forbidden for every state of polarisation and
propagation direction.
The rules regarding the existence of PBGs are identical to those that determine if an X-ray beam will be
diffracted and are documented in crystallography texts (see this chapter’s Further Reading). The positions of the
band gaps, corresponding to the strongly diffracted wavelengths, are readily computed via Bragg’s law, which
holds for any ‘crystalline’ array no matter the size of the constituent ‘atoms’ and whether man-made or not.
As an example, the situation in the type of colloidal crystal described abovewill be outlined. These colloidal
crystals and inverse opals are laid down in sheets of (111) planes described in terms of a cubic unit cell. The
strongly diffracted wavelengths are given by Bragg’s law, and it remains to calculate the interplanar spacing
223 Colour Due to Diffraction
(d values) of the various planes of spheres or voids making up the solid. For the close-packed arrangement
described, the interplanar spacing is given by:
dhkl ¼ a
h2 þ k2 þ l2p
where a is the cubic lattice parameter. The relationship between the measured average distance between the
sphere centres D and the cubic lattice parameter a is:
a ¼ 2p
D
Hence:
dhkl ¼ 2p
D
h2 þ k2 þ l2p
There is one other factor to take into account. Not every plane in the structure will give rise to a strongly
diffracted beam.This is because, for some specific values of (hkl), interference effects not described previously
cancel out the beams diffracted from adjacent planes to give zero diffracted intensity. For example, there is no
Bragg reflection from a (100) plane in a face-centred array of the same arrangement as found in artificial opals.
The lowest order reflecting planes (hkl) for this structure in order of interplanar spacing are (111), (200), (220),
(311) and (222).6 PBGswill then occur for each of these planes. Thewavelength and angle dependence is given
by the equations above, simply by substituting the appropriate value of dhkl; that is:
ml ¼ 2dhkl n2e�sin2�1
q
¼ 2 2p
D
h2 þ k2 þ l2p n2e�sin2�1
q
For illumination perpendicular to the planes:
ml ¼ 2 2p
Dne
h2 þ k2 þ l2p
If the photonic crystal is thin, then the transmitted colour will be complementary to the reflected colour.
The production of these different reflected and transmitted colours is sometimes called optical filtering.
(For more information on the enormous topic of photonic crystals, see this chapter’s Further Reading.)
6.8.6 Dynamical form of Bragg’s law
In the foregoing section the kinematical (single scattering event) theory of scattering is used. However, light
photons are strongly diffracted by gratings and very little light would penetrate a stack of gratings. This means
6 Note that reflection from (200) corresponds to the second orderm 2 reflection from (100), the reflection from (220) corresponds to the
second order reflection from (110) and reflection from (222) corresponds to the second order reflection from (111). X ray crystallography
has adopted the system of keepingm 1 and changing the interplanar spacing as in the equation for dhkl given. In optics it ismore common
to keep the diffraction plane as constant and vary the order m. In either case the numerical results from both approaches are exactly the
same.
Colour and the Optical Properties of Materials 224
that for a precise understanding of the scattering by opals and similar arrays the dynamical (multiple scattering)
theory is required. For normal incidence, the dynamical theory gives Bragg’s law as:
ld ¼ lB 1þ j2
� �
where ld is the wavelength of the diffraction peak maximum computed by dynamical theory, lB is the Bragg
wavelength andj is related to the ratio of the refractive indices of thewall nwall and voids nvoid in the following
way:
j ¼ 3Vwall
n2rel�1
n2rel�2
nrel ¼ nwall
nvoid
For the instance of diffraction normal to (111) planes in air:
ld ¼ 1:633DnwallVwall þ 1:633Dnvoidð1�VwallÞ½ � 1þ j2
� �
And in general, for normal incidence:
mld ¼ 2dn2e
¼ 2d nwallVwall þ nvoidð1�VwallÞ½ � 1þ j2
� �
These corrections are needed in the most precise work.
6.9 Diffraction from Disordered Gratings
6.9.1 Random specks and droplets
Randomly sited copies of a single object will produce a diffraction pattern which is a brighter version of that of
the isolated object. Thus, the diffraction pattern of a random collection of circular apertures or rectangles will
consist of the same patterns as described above, but with an increased intensity. This has interesting
consequences for pattern recognition. For example, suppose that the object consists of an array of pairs of
circular apertures arranged so that the axes of the pairs are parallel but the position of the pairs is at randomover
a plane. To the eye this will resemble a random collection of single circular apertures. The diffraction pattern
will not, however, look like anAiry pattern, butwill be a fringe pattern consistent with that from a single pair of
apertures with the fringes perpendicular to the axis of the aperture pair (Figure 6.20a and b). If the pairs are
arranged randomly bothwith respect to position and the orientation, the single fringe patternwill be duplicated
at every angle that the pairs of points display in the object.With enoughpoint pairs the unidirectional fringes are
transformed into a diffuse ring pattern (Figure 6.20c and d). This provides a very simpleway of distinguishing
between a random array of objects and an array that appears to be random but does include some hidden order.
For example, the diffraction pattern of an amorphous material, such as a glass or a film of evaporated carbon,
usually consists of a few very diffuse rings (Figure 6.21), indicating that a certain degree of order is present,
although invisible to the eye. If the atomswere truly independent of each other, the diffraction patternwould be
225 Colour Due to Diffraction
anAiry pattern equivalent to that from a randomcollection of points. In a silicate glass, for example, the diffuse
rings reflect the occurrence ofmany Si O bonds, each of which have a similar length but which are arranged at
randomwith respect to the incident illumination. In carbonfilms theC Cbondplays the same role.The average
bond length in the amorphous state can be estimated from the ring diameter and comparedwith bond distances
in crystalline materials.
Diffraction patterns from random droplets or specks can be seen frequently. Because of the wavelength
sensitivity of the diffraction, the effects give rise to colours. One of the commonest of these phenomena is the
corona around the sun or moon, seen through high, thin clouds.7 They lie close to the disc of the object and are
much narrower than the halos described earlier (Section 2.8). The pattern is the Airy ring (Fraunhofer)
diffraction pattern from the collection of randomly distributed droplets. These add together and an observer, in
reality, sees fragments of the diffraction patterns frommanydroplets or specks, each ofwhich contributes to the
overall effect. When the clouds consist of similarly sized droplets or specks, the effect will be strong. At their
best, the coronae show multicoloured rings surrounding the central disc of the sun or moon. Usually only the
first ring is easily seen, and has a colour sequencewith violet on the inside and red on the outside. If more than
one ring is possible then the same colour sequence, violet inside and red outside, is seen. When the drops are
variable in size the effect is diminished and then just a pale ring can be made out.
For a similar reason, a multicoloured ring can sometimes be seen to surround a narrow beam of white light
which has passed through a pane of glass covered with a fine powder or with fine drops of moisture. Each
particle diffracts as a small circular aperture. The eye intercepts many of these diffracted rays and a coloured
d
(a)
1/d
(b)
d
(c)
1/d
(d)
Figure 6.20 Diffraction patterns of arrays of points (schematic). (a) A randomarray of pairs of points (one pair inbox, separation d). (b) Diffraction pattern of (a) consists of fringes of separation 1/d. (c) A randomarray of pairs ofpoints as in (a), but with random orientation. (d) Diffraction pattern of (c) consists of a set of diffuse rings ofspacing 1/d. By eye, neither (a) nor (c) indicate the presence of any internal order, but the diffraction patternsshow this clearly
7 This is different from the outer atmosphere of the sun, which is also called the corona.
Colour and the Optical Properties of Materials 226
ring is seenwhich is composed of fragments of colour frommanydifferent dust particles.As before, violet is on
the inside and red is on the outside of the circle. The same effect can sometimes be seen around the image of a
small light in a dustymirror. Again, each dust particle acts so as to diffract the light, which is reflected from the
mirror surface back towards the observer and is the Fraunhofer diffraction pattern of the specks.
Another series of coloured fringes can also arise from dust on a mirror surface. In this case the ring pattern,
which is also multicoloured, is called the Whewell Qu�etalet pattern. The ring structure is again caused by
diffraction from light scattered from the particles reflected from themirror and then returned to the eye, but the
interference paths of the rays are different from those that give rise to the Fraunhofer pattern.
Figure 6.21 Amorphous carbonfilm: (a) high resolution electronmicrograph; (b) diffraction pattern. Thediffuserings indicate that some order is present in the apparently randomfilm. [Reprintedwith permission fromGeoffreyI. N. Waterhouse et al., Physical and Optical Properties of Inverse Opal CeO2 Photonic Crystals, 20, 3, 2008,American Chemical Society]
227 Colour Due to Diffraction
6.9.2 Colour from cholesteric liquid crystals
The structure of a liquid crystal made up of longmolecules (a nematic liquid crystal) was described in Section
4.13. In thesemesophases, the director, which is the average direction taken by the long axes of themolecules,
falls along just one singledirection. In cholestericphases,whichare alsocalled twistednematicphases or,more
correctly, chiral nematic phases, the director rotates steadily as one travels along a direction perpendicular to
the sheets ofmolecules (Figure 6.22). The result is the generation of a helical structurewithin thematerial. The
physical reason for the rotation is that the molecules in each layer are asymmetric. When suchmolecules pack
together the interactions areminimised ifmolecules in one layer rotate slightly comparedwith those in the layer
below. Because these interactions are the same from one layer to another the same twist occurs between each
layer. In this way the uniform helical structure results.
The helix so formed can be right or left handed, and this influences the way in which a beam of white light
interactswith themesophase.Unpolarised light canbe regardedas twobeamsof oppositely circularly polarised
light (Section 4.1). The beam with the same handedness as the helical arrangement of the mesophase passes
straight through an ordered or partly ordered array, while the opposite beam may interact with the array and a
coloured diffracted beam can then appear.
This colour arises by diffractionwhen the pitch of the helices in the cholestericmesophase (that is, the repeat
distance along each helix) is similar to thewavelength of light. Scattered light can then interfere constructively.
axis
director
Figure 6.22 The cholesteric liquid-crystal structure. The average orientation of the molecules (the director) ineach layer rotates in a regular fashion as one moves along the axis to create a helical structure
Colour and the Optical Properties of Materials 228
The conditions for this to occur follow Bragg’s law:
ml ¼ 2nmdsin�B
where nm is the average refractive index of the mesophase, d is the helical repeat distance and �B the angle
between the layer and the light beam (Figure 6.23). (Corrections for refraction at the surface of the cholesteric
phase can also be made, if important.) For light normally incident on the film:
l ¼ 2nmd
When illuminated with white light, any wavelength satisfying the Bragg relationship will be diffracted
strongly and give a colour in reflection. If the liquid crystal is backed by a dark background then the colour will
appear quite bright. This is because the transmitted light will be absorbed. On viewing the film normally and
thenmoving towards grazing incidence the colourwill appear to change towards shorterwavelengths due to the
sin� term in the Bragg equation. Because of the uniaxial nature of the molecules, the refractive index of the
d
θ B
whitelight
Figure 6.23 Diffraction of light from a cholesteric chiral nematic array. The pitch d of the array determines thewavelength of light that is strongly diffracted
229 Colour Due to Diffraction
medium will be different along and perpendicular to the molecular axis. Thewavelength range diffracted will
be given by:
Dl ¼ 2Dnd
whereDn is the birefringence of themolecules (the difference between the refractive indices along the two axes
of the molecule).
The pitch of the helical structure can be engineered by both temperature and impurities. The end result is
widely seen in liquid-crystal thermometers. A commonly encountered form of these inexpensive devices
consists of a cardwith a black strip of plastic running across it. In the black band a coloured numberwill be seen
corresponding to the temperature. A different number lights up at a different point on the band as the
temperature varies.
These devices operate in the following way. A series of spots of a cholesteric material are arranged in a row.
Theyare chosen so that the periodicity of themolecular helix in each spotwill diffract visible light at a precisely
defined temperature. An increment of the temperature dT of the order of 1 �C is engineered between each
successive spot by additives or by slightly modifying the cholesteric molecule. Within the design temperature
range, each spot will diffract light only when the mesophase is at the correct temperature and not do so
otherwise. This causes the appropriate temperature value to ‘light up’. Moreover, on approaching the
temperature, each spot will run through a spectrum of colours as the pitch of the helix varies. The effect
of temperature on colour, of which this is an example, is called thermochromism.
These effects have already been anticipated bynature, and a number of beetles show iridescent colours due to
acholesteric arrangement of layers offibres in the outer integument of the body.Thefibredirection in each layer
is slightly different from that on either side and ahelical layered structure is built up. If thepitch of the spiral is of
similar dimensions to the wavelength of light then they give rise to intense ‘metallic’ colours when viewed in
white light.
One of the puzzles surrounding early research on cholesteric liquid-crystal phases centred upon the fact that
these often showed a transient bright blue colour on cooling. The effect was confined to a narrow temperature
range close to the upper melting point of the phase, where the true liquid forms. This transient state contains
several different forms and they are known as the cholesteric blue phases. The blue colour arises by diffraction
of white light from a superstructure within the mesophase. These superstructures consist of ordering of the
cholesteric helices into supercells with a dimension of the same order of magnitude as blue light. They act as
crystals and diffract blue light in accordancewith Bragg’s law. The ordering is only stable over a narrow range
of conditions, which accounts for the fleeting nature of the feature.
6.9.3 Disordered two- and three-dimensional gratings
The situationdescribed for a randomarrayof specksor droplets applies equallywell to randomarrays of two- or
three-dimensional gratings. A scattering object which is made up of a random collection of two- or three-
dimensional gratings will give a diffraction pattern which is a brighter version of that of the isolated object.
Here, there are two extravariables to consider besides the randomspatial position: the relativeorientation of the
grating fragments (that is, the relative rotation about an axis parallel to the illuminating light beam) and the
physical extent of the grating. Both of these modify the diffraction pattern observed.
The formation of iridescent suspensions of polymer spheres provides an example. During the preparation of
colloidal crystallites, monodisperse suspensions of polymer can order whilst still in the fluid, well before
solidificationoccurs. The extent of theorderingwill dependupon the concentration of spheres in the suspension
and the interaction between them. A frequent form of ordering involves the spheres aggregating into
hexagonally packed layers within the liquid phase. These disordered two-dimensional gratings may diffract
Colour and the Optical Properties of Materials 230
white light and give rise to intense colours. When the concentrations and interactions are favourable, three-
dimensional ordering can also occur over small volumes of the suspension, giving rise to diffracted colours
when the sphere spacing is appropriate.
Diffraction by disordered three-dimensional structures is most commonly described with respect to the
X-ray diffraction pattern from a large number of randomly oriented crystallites the ‘powdermethod’ ofX-ray
diffraction. As described above, a small crystal will give rise to a strong diffracted beam when the interplanar
spacing and the angle of incidence of the X-ray beam agree with Bragg’s law. The diffraction pattern from a
single crystal will consist of a three-dimensional array of spots with a spacing and symmetry that matches the
dimensions of the atomic grating that makes up the crystal. If the crystallites are randomly oriented, the
diffraction pattern from each one will also be randomly oriented. Each spot in a single crystal pattern can now
formanywhere on the surface of a sphere and the spot pattern becomes a set of concentric shells.Aplane section
through the pattern yields a series of rings, typifying a powder X-ray diffraction photograph. If the number of
crystallites included in the incident beam is rather small, the rings are broken up into ‘spotty’ rings
(Figure 6.24).
The effect of limited crystallite size is to broaden the extent of each diffraction spot. In the case of a ring
pattern, each ring will become broadened. This is analogous to the effect described earlier, in which a small
aperture produces a greater spread of diffracted intensity than a wide aperture. The spread of the rings can be
used as a measure of crystallite size.
6.10 Diffraction by Sub-Wavelength Structures
6.10.1 Diffraction by moth-eye antireflection structures
Night-flying insects need to optimize the amount of light that reaches the receptor cells in the eye.This has been
achieved by covering the surface of each of the components of the insect compound eye (the ommatidia) with a
large number of tiny bumps that are somewhat smaller than about half the wavelength of the incident light,
being about 200 nm at the base and 200 nm high (Figure 3.15). This creates an AR coating and these types of
Figure 6.24 A ‘spotty’ electron diffraction pattern from a polycrystalline sample of titanium dioxide (TiO2)
231 Colour Due to Diffraction
surfaceARcoatings are also calledmoth-eyeARcoatings. The description of the optical consequences of these
surface features can bemade in termsofGRINeffects (Section 3.7) or in terms of a two-dimensional diffraction
grating, i.e. an ultrahigh spatial-frequency surface relief grating. This latter aspect is described here.
Themoth-eye surface grating is a reflection grating. To optimize night vision, it is important tominimize all
reflected light. This means that the diffracted orders must be suppressed. Although the surface grating is two-
dimensional, it is possible to gain an idea of the action of the small surface bumps on the surface of themoth eye
by using the one-dimensional diffraction grating equations given above. Consider, initially, light prevented
from entering the eye by diffraction from the reflection grating on the surface. For light falling on the surface at
normal incidence:
dsin�m ¼ ml
where d is thegrating spacing,m is the order of the principalmaximum,l is thewavelength of the light and �m is
the diffraction angle of themth-order maximum. The equation shows that, as the value of d approaches l, thediffracted orders make larger angles with the surface normal. The limiting cases occur when sin�m is equal to
�1 andm¼ 1. Inserting these values into the equation shows that the limit is reachedwhen l/d¼ 1. That is, the
grating spacing corresponds to thewavelength of light. All diffracted beams except the zero (m¼ 0) order will
be suppressed when d is slightly greater than l, because that would correspond to a value of sin�m > 1.
At the other extreme, consider the light which hits the eye surface at grazing incidence. The equation for
diffraction is now:
dð1�cos�Þ ¼ ml
As before, the limiting cases occur when cos� is equal to�1 andm¼ 1. Inserting these values into the equation
shows that the limit is reached when l/d¼ 2. Thus, in either case, if the grating spacing is less than l/2, alldiffracted orders will be suppressed except the zero (m¼ 0) order.
The light entering the eyewill be maximised if the surface has a grating with a repeat of about l/2. For bluelight this corresponds to approximately 200 nm. In order to determine the intensity of this zero-order reflection
for amoth-eye structure it is necessary to calculate the intensity as a function of the surface profile (whichmay
be square, sinusoidal or irregular), the depth of the grooves, the angle of incidence and the polarisation of the
light. Generally, the calculations are numerical and no analytic solutions are available (see this chapter’s
Further Reading).
The diffraction equation can be generalized for application in other similar situations; for example, a fish
eye adapted to dim light conditions, which must also optimise light gathering. When light passes from a
medium with refractive index n1 into a medium with a refractive index n2 (Figure 6.25), the grating
equation is:
ml ¼ dðn1sin�i þ n2sin�mÞ
where n1 is the refractive index of the medium containing the incident beam (above the grating) and n2 is the
refractive index of the medium containing the diffracted beam of orderm (below the grating). In the case of a
reflection grating, n1¼ n2 and the grating equation is:
ml ¼ dðn1sin�i þ n1sin�mÞ¼ n1dðsin�i þ sin�mÞ
When the initialmediumis air,with a refractive indexofunity, the equation reduces toEquation6.3.Clearly, the
same reasoning used for moth eyes applies, except now the factor n1 must be taken into account. The limiting
Colour and the Optical Properties of Materials 232
spacing is thenl/n1 for normal incidence andl/2n1 forgrazing incidence. That is, the critical spacing is reducedby a factor of n1.
6.10.2 The cornea of the eye
The cornea of the eye provides a second example of diffraction by structures with a repeat dimension less than
the averagewavelength of visible light. The cornea is the outer surface of the eyedirectly in front of the lens and
iris and has a thickness of approximately 0.6mm.Naturally, it is vital to survival for the cornea to be completely
transparent. Now the cornea ismade up of long collagen fibrils embedded in a gel-like transparent protein. The
refractive index of the fibrils is close to 2.17, while the surrounding matrix has a refractive index of 1.81. This
arrangement seems ideal for giving rise to considerable incoherent Mie or Rayleigh scattering, thus rendering
the cornea opaque like the rest of the ‘white’ of the eye. Transparency is obtained by making use of coherent
scattering, that is diffraction, fromacarefully constructed arrangement of layers, to forma transmission grating
that is able to eliminate all diffracted orders except the directly transmitted beam.
Apart from an exterior and interior film, the epithelium and endothelium, the cornea consists of about 250
lamellae stacked one on top of the other (Figure 6.26). Each lamella contains long collagen fibrils of a diameter
of approximately 30 nm arranged parallel to each other and stacked up in an ordered array with about 50 nm
between centres, into a lamella 2400 nm thick. The long collagen molecules are birefringent and so each
lamella is also birefringent. However, the direction of the collagen fibrils changes abruptly from one lamella to
the next. In fact, the lamellae are distributed so that the collagenmolecules are arranged in a cross-ply fashion,
so that over the 250 or so lamellae the effect is averaged and the cornea is not able to detect polarised light.
incident beam
m = –1
m = 0m = 1
m = 0m = 1
m = –1
reflection
transmission
refractive index n2
refractive index n1
d
θ i
θr1
θ t1
Figure 6.25 Diffraction at the surface between two transparentmedia of refractive index n1 and n2. The angle ofincidence is ui, the angle of diffraction for the first-order reflected ray is ur1 and the angle of diffraction of the first-order transmitted ray is ut1
233 Colour Due to Diffraction
The arrangement of the fibrils on a regular grid of spacing approximately 50 nm will give rise to sub-
wavelength diffraction. According to the grating equation, Equation 6.3, diffraction will occur when:
dsin�m ¼ ml
In the present case,d is of the order of 50 nm, i.e. about 0.1l, and the only real solution iswhenm and �m are both
equal to zero. In essence, the scattering is coherent and reinforcement occurs for the ‘straight through’ beam
while all other scattering is suppressed by destructive interference.
6.10.3 Some blue feathers
As has been apparent throughout this chapter, many of the intense colours found in nature are produced by
diffraction. Sub-wavelength diffraction has also been found to be important here. A recent example is given by
the iridescent blue colour of the Plum-throated Cotinga, Cotinga maynana. The colour arises in the feather
barbs. Rather like the cornea, the barbs aremade of amixture of substances, in this case keratin separated by air
spaces. Keratin is a tough fibrous protein widely distributed throughout the animal kingdom, occurring, for
example, in hair, hooves, feathers and fingernails.
Again, this structure would be expected to scatter light strongly. This is averted because the tissue is not a
random arrangement, but consists of a disordered diffraction grating (Section 6.9.3). The (computed)
light
light
epithelium endothelium
stroma~250 lamellae~2.4 μm thick
1 lamella~2.4 μm
fibril
Figure 6.26 The cornea of the eye (schematic). (a) Themain part of the cornea, the stroma, is composed of about250 lamellae. (b) Three lamellae, each containing collagen fibrils approximately 30 nm apart in a quite well-ordered array. Those at the left and right are roughly parallel to the page and those in the middle are roughlyperpendicular to the page
Colour and the Optical Properties of Materials 234
diffraction pattern confirms this. Thegrating spacing, averaging about 165 nm, is below thewavelength of light
and ‘normal’ diffraction is not possible. Bragg’s law, Equation 6.4:
ml ¼ 2d sin�B
suggests that, for normal incidence and m¼ 1, the colour strongly diffracted would be 330 nm, in the
ultraviolet. This estimate is too simplistic by far, but exact calculations give a reflectance peak in the
500 520 nm range, in good agreement with measurement. Once again, the structural colour is produced by
coherent scattering from a partly ordered matrix with sub-wavelength repetition. The desirable coloured
reflected light is enhanced by constructive interference and other colours are lost by destructive interference,
arising in the ordering of the keratin air barb microstructure.
6.11 Holograms
6.11.1 Holograms and interference patterns
Objects are perceivedwhen a train of light waves enters the eye and the resulting nerve impulses are processed
by the brain. Strictly speaking, the source of the waves entering the eye is unimportant for the perception to
occur. A hologram is a permanent store of the information needed to create these light waves in such detail that
the observer is given the impression that the real object is being observed even though it is, in fact, an illusion.
There are many forms of holograms. In this section, only those that really involve colour production are
described. This chapter’s Further Reading will give more information on broader aspects of the subject.
The information stored is an interference pattern formedbetween light scattered fromanobject and light that
has not been scattered. The most important requirement is that the object should be illuminated by coherent
light and the interference pattern is created by an overlapping of the reflected light with the unchanged incident
beam. To do this, amonochromatic laser beamwith awavelength somewhere in the visible is divided into two.
One part illuminates an object and some of the light is reflected from the object to create the object beam or
signal beam. The other part of the beam traverses an identical distance but without having encountered the
object, to form the reference beam. The object and reference beams are arranged to intersect and in this region
the two reunited beams will interfere with each other and an interference pattern of variable irradiance and
phase will be present. Holograms are recorded versions of these two- or three-dimensional interference
patterns.
6.11.2 Transmission holograms
A transmission hologram is formedwhen the object beam and the reference beam enter the recordingmedium
from the same side (Figure 6.27a). In order for the wave pattern to record the image details accurately, all
vibrations must be eliminated. Any disturbance at all will introduce additional ‘information’ into the
interference pattern, which amounts to a degradation of the record.
To view the holographic image, a procedure known as image reconstruction, the recorded hologram is
illuminatedwith the referencebeamalone (Figure6.27b).Thebeampasses through thehologram,which iswhy
these holograms are called transmission holograms. If the viewer looks back along the reference beam a
reconstructed virtual image of the object is seen, as if looking through awindow, which is the hologram frame.
As the viewer moves, the image remains fixed in space, but the aspect of the image that is seen and its position
within the hologram frame changes. The image appears to show all of the three-dimensional properties of the
original object. (Exactly the same impression is created when a real object, a tree, say, is viewed through a
235 Colour Due to Diffraction
window.As onemoves around the room, the tree stays in the same place, but the extent of the view of the tree is
curtailed by the window frame.)
In point of fact, the reconstruction results in two images, a real image formed in front of the hologram (i.e.
between theviewer and thehologram)andavirtual image, just described, formedbehind thehologram.The real
image is rather difficult to locate and the image most often viewed is the virtual image seen when looking into
the hologram.
The way in which the holographic record is formed, by the two interfering wave fronts approaching the
recording medium from the same side, results in the interference pattern being restricted to a thin layer. This
laser beam
objectspatial filter
spatialfilter
beam splitter mirror
mirror
hologramrecorded here
(a)
laser beam
(virtual)image
spatialfilter
beam splitter mirror
hologram “window”
(b)
Figure 6.27 Transmission holograms: (a) recording the hologram; (b) reconstruction of the image;(c) reconstruction with a different wavelength giving rise to a displaced and distorted image
Colour and the Optical Properties of Materials 236
type of hologram is called a thin or planar hologram. If such a hologram is illuminated with laser light of a
different colour, then an imagewill still be formed.However, the imagewill be distorted anddisplaced, so that if
the laser light is red, when green light is used to create the hologram, the image will appear to the left of
the original because red light is diffracted more than green (Figure 6.27c). What is more, the image will move
around disconcertingly as the viewpoint is changed. Similarly, if violet light is used, the imagewill form to the
right of the original image, as violet light is diffracted less than green, and the reconstruction will be distorted
andmove as the viewer’s viewpoint changes. If white light is used to reconstruct the hologram the result will be
a blurred and distorted shape that may not be recognizable. This type of hologram, therefore, is not suited to
white-light reconstruction. The behaviour of a thin hologram can thus be seen to parallel that of a thin
diffractiongrating.On irradiationwithwhite light a thinhologramgives rise to several orders of diffracted light,
of decreasing intensity as the diffraction angle increases.
6.11.3 Reflection holograms
A reflectionhologram is formedwhen the referencebeamand theobject beamenter the recordingmediumfrom
opposite sides (Figure 6.28a). Although vibrationsmust be eliminated to record a good hologram, the stringent
requirements required for a transmission hologram can be considerably reduced in a reflection hologram by
placing the object more or less in contact with the recording medium.
Image reconstruction is similar to that for a transmission hologram. However, in this case the beam is
reflected from the hologram and does not pass though it (Figure 6.28b). Because the reflection holographic
record is formed by the two interferingwave fronts approaching the recordingmedium fromopposite sides, the
interference pattern extends for a considerable distance through the recodingmedium.This type of hologram is
called a thick or volume hologram.
There is a significantdifferencebetween reconstructionusinga reflection (volume)hologramcomparedwith
using a transmission (planar) hologram. In this case of a volume hologram, if the hologram is illuminated with
laser beam
(virtual)image
spatialfilter
beam splitter mirror
hologram
c
Figure 6.27 (Continued)
237 Colour Due to Diffraction
white light, the interference pattern in the hologram is able to interact selectively with light of the appropriate
wavelength and so a coloured reconstructed image is still visible. The difference between a planar and volume
hologram is, in point of fact, analogous to the difference between a planar diffraction gating and a three-
dimensionalgrating, suchasa crystal (Sections6.5, 6.6 and6.8).The selectivelycoloured reconstruction froma
volume hologram is described as arising from ‘Bragg planes’ in the holographic record. The comparison is
quite accurate.Whenabeamof ‘white’X-rays (that is, anX-raybeamwitha spreadofwavelengths) illuminates
a crystal, only those wavelengths that fit the Bragg condition will be diffracted by the three-dimensional
distribution of electron density within the crystal. In the sameway, when a beam of white light interacts with a
thick hologram, only those wavelengths that fit with the ‘Bragg condition’ imposed by the variation in
interference fringes within the volume of the hologramwill be diffracted and so contribute to the reconstructed
image. In comparison with a thin hologram and a thin diffraction grating, a volume hologram gives rise to one
diffraction order (in addition to the zeroth order), which is intense in a direction satisfying Bragg’s law.
This leads to another point of interest, but one which is not restricted to volume holograms alone, but also
applies to planar holograms. Each fragment of the hologram contains sufficient information to reconstruct the
laser beam
objectspatial filter
spatialfilter
beam splitter mirror
mirror
mirror
hologramrecorded here
(a)
(virtual)image
hologram
(b)
laser beamor white light
Figure 6.28 Reflection holograms: (a) recording the hologram; (b) reconstruction of the image with the samelaser light or white light gives a coloured image
Colour and the Optical Properties of Materials 238
image. That is, if a hologram is smashed into fragments, each will serve to reconstruct the image. This is the
same as inX-ray diffraction, where a large crystal can be continually subdivided but the diffraction pattern still
contains all the information to deduce the structure of the diffracting crystal. Of course there are limits. As the
crystal volume decreases beyond a certain point, information is lost and the crystal structure so deducedmay be
incomplete. Similarly, if the hologram fragment becomes too small, information is lost and the reconstructed
image degrades.
6.11.4 Rainbow holograms
Rainbow holograms are the brightly coloured holograms seen on credit cards, security documents and so on.
The first point to be clear about is that rainbow holograms are transmission holograms, even though they give
the appearance of being reflection holograms. In reality, the white viewing light passes through the hologram
and is reflected from a backing layer before reaching the observer. The image seen and the colours visible
depend upon both the viewing angle and the viewing distance, so that amultiplicity of colours and patterns can
be picked out as the hologram is tilted andmoved.Theywere inventedbyBenton in 1968, and are also knownas
Benton holograms. Technically they are described aswhite-light transfer transmission holograms. In order to
understand this terminology it is necessary to consider transmission hologram construction.
The shortcomings which prevent transmission holograms being viewed satisfactorily in white light were
overcome by two innovations. Take the problem of colour spread and distortion first. The degree of distortion
andmovement of an image formed by light of a different wavelength to that of the original laser depends upon
the distance d between the hologramplane and the object (Figure 6.29a). If d is reduced to zero, some distortion
will still exist, but the different coloured reconstructions will overlap and, as the viewpoint is moved, image
movement will be minimal. Naturally, the object cannot be placed right up against the recording medium
because it would block the reference beam. This is surmounted bymaking a hologram of the reconstruction. In
this case it is necessary to use the real image formed, not the virtual image normally viewed, and this is
positioned in the recording plane of the second hologram (Figure 6.29b). The original hologram is called the
master, or H1, and the second hologram the transfer hologram, or H2. When H2 is viewed in white light the
multicoloured (virtual image) reconstructionsnowoverlap anddonotmovemuchwhen theobservationpoint is
moved (Figure 6.29c). The reconstruction may be variously coloured or it may appear as black and white,
depending upon the colour overlap and the nature of the illumination.
The amount of the reconstructionvisible to the observer is, however, constrained.The transfer hologram,H2,
is a hologram of a hologram, and so the observer would see the reconstructed image as if it were viewed
through the same ‘window’ or ‘porthole’ that circumscribed the original hologram H1 (Figure 6.29d). If
theviewpoint is changed toomuch, parts of the imagewill seem to fall behind this aperture and sobeblocked for
the observer. Moreover, each wavelength of light used in the reconstruction will create its own window, with
red corresponding to themost diffracted light andviolet the least.Thus, anobserverona levelwith thehologram
will see a green reconstruction, whereas if the observer moves up the image colour will move through
yellow and orange to red, or if the observer moves down to green, then blue, indigo and violet (Figure 6.29e).
There will still be considerable overlap of colours in parts of the image, dependent upon the viewing distance
and angle.
This latter problem is overcome bymaking the viewingwindow a narrow strip. H1, therefore, takes the form
of a narrow strip rather than a rectangular or square frame. This time, as the observer moves up, the part of the
reconstruction visible through the narrow window changes colour from green to yellow, orange and red.
Similarly, on moving down, the part of the reconstruction visible to the observer changes colour sequentially
from green to blue, indigo and violet (Figure 6.29f). The result is a rainbow hologram.
There is a price to pay for this, because the strong sensation of depth in a ‘vertical’ direction, normal to the slit
length, is lost. This is not so along the strip, in a ‘horizontal’ direction, andhere, as theobservermoves to and fro,
239 Colour Due to Diffraction
it is possible to ‘see behind’ the image, as expected. The loss invertical parallax is due to the fact that the narrow
slit excludes information upon this vertical direction from the holographic record. This is of little consequence
when rainbow holograms are used for security-related purposes.
6.11.5 Hologram recording media
The interference pattern that is the information content of a hologram is stored in a photosensitivematerial. To
some extent, the type of photoresponsivemediumused depends on the purpose of the hologram. For artwork or
security labels a thin hologram may suffice; for data storage and retrieval a thick hologram, able to store
multiple ‘sheets’ of data in the same volume may be necessary. The photosensitive response can involve a
change of optical absorption, refractive index, optical anisotropy or thickness. The information can then be
imprinted on the beam used for reconstruction by these changes.
objecthologram H1
(a)
d
laser beam
H1 H2
(b)
laser beamlaser beam
real image
H2
(c) white light
Figure 6.29 Transfer transmission holograms. (a) The distortion and movement of white light images in aconventional transmission hologramH1 increases as the object distance d increases. (b) A transfer hologramH2 ismade from themasterH1 transmission hologramwith the realH1 image in the plane ofH2. (c) Illumination ofH2withwhite light reconstructs coloured virtual images that overlap (exaggerated here) and showminimal distortionandmovement. (d) The image seen will still appear as if viewed through the H1 ‘window’. (e) Each wavelength oflightwill generate its ownwindow(three colours only shownhere for clarity). (f)A rainbowhologram ismadewitheach H1 window reduced to a narrow slit (three colours only shown here for clarity)
Colour and the Optical Properties of Materials 240
For example, if the hologram is stored by a change in the optical absorption of the recording medium, the
amplitudeof the beamused for reconstruction ismodulated.By analogywith diffraction gratingnomenclature,
this type of hologram is called an amplitude hologram. In cases where the refractive index or the thickness of
the recording film ismodulated, the information is impressed upon the reconstruction by changes in phase, and
these holograms are called phase holograms. The use of optical anisotropy, particularly of refractive index,
which will affect the polarisation of the reconstruction beam, gives rise to polarisation holograms.
The first holographic recording medium used was fine-grain photographic emulsion deposited on a glass
plate formaximumdimensional stability or else on a stable polymer base. Exposure to the object and reference
beams causes silver crystallites to form in the emulsion (see Section 10.19). The emulsion, after development,
contains blackened lines corresponding to the peaks in the interference pattern and transparent areas
corresponding to the troughs. In this form the processed emulsion acts as an amplitude hologram. These
primary holograms can be ‘bleached’ to remove the silver grains. After this process the refractive index of the
emulsion varies in a mirror of the amplitude variation. In this form the emulsion acts as a phase hologram. It
red H1 window
violet H1 window
(e) white light
H1 window
(d) laser beam
(f) white light
rainbowhologram
Figure 6.29 (Continued)
241 Colour Due to Diffraction
should be noted that commercially available photographic emulsions and films are becoming difficult to obtain
(2010) and many hologram makers now prepare photographic emulsions themselves.
Dichromated gelatine (DCG) deposited upon a glass or stable polymer film is awidely used phase hologram
recording medium. The material consists of gelatine, a form of the natural product collagen, a widely
distributed fibrous protein found in bones and connective tissues. The gelatine needs to be sensitised with
dichromate, Cr2O72 . On exposure to light, particularly violet or near ultraviolet, the dichromate causes the
gelatinemolecules to cross-link, giving the region a higher refractive index than the unexposed volumes. After
appropriate treatment the interference pattern is fixed to give a phase hologram. Thismaterial is one of a family
of photopolymers that record information in a similarway.The problemswith this group centre upon long-term
stability of the irradiated volumes or dimensional changes caused by shrinkage.
There are a number of inorganic materials that are also being explored for recording holograms that use
refractive index changes. Most studies have been carried out on ferroelectric crystals, such as lithium niobate
LiNbO3. These are insulating oxides. Irradiation with light can excite electrons within the structure, usually
fromdeliberately added dopant impurities such as Fe2þ . (Fordetails of theseprocesses, seeChapters 7 and10).Once the electrons have been formed they are free tomove through the conduction band of the crystal until they
come to another impurity atom or ion which is able to trap them. In effect, electrons shy away from the
illuminated volumes and congregate in the dark regions. This charge distribution pattern not only mimics
the irradiance distribution of the interference pattern, but also causes refractive index changes throughout the
crystal volume. These record the information as a phase hologram.
Anumber of inorganic glasses, especially of the chalogenides (S, Se andTe), are also utilised to record phase
holograms. In these solids, light of sufficiently high irradiance is used to break some of the bonds in the glass.
This leads to refractive index changes and, hence, to a phase hologram.
Polarisation holograms are formed by embedding linear molecules with a dipole moment in a polymer
matrix. A widely explored group of molecules considered for this application are derived from azobenzene
modified by adding side chains to the benzene rings.When such amaterial is irradiated with linearly polarised
light, the side chains orient parallel or perpendicular to the electric field vector of the light, causing the initially
isotropic polymer matrix to become birefringent with an accompanying large change in refractive index.
Irradiation with circularly polarised light can erase the hologram.
6.11.6 Embossed holograms
Embossed holograms are the physical holograms that are used as security labels. They are thin rainbow
holograms made in the following way. The original master hologram H1 is a slit transmission hologrammade
as described above (Figure 6.30a). The H1master is used to make a rainbow transmission hologramH2which
is made in a photoresist (Figure 6.30b). This is a polymeric material that reacts to light in one of two ways:
the photoresist becomes insoluble (a negative photoresist) or soluble (a positive photoresist) on illumination
with ultraviolet light. This photoresist layer is of the order of 1mm thick and is mounted on a glass plate.
The interference pattern making up the hologram causes the photoresist to react. Processing of the
photoresist dissolves parts of the upper layers, which then leaves H2 as a grooved and ridged surface with
either peaks or troughs corresponding to the peaks of the interference pattern (Figure 6.30c). The result is a
phase hologram.
Thismatrix is sprayedwith a thin layer of silverpaint tomake it electrically conducting and thennickel plated
to give a thin nickel foil, the master or mother shim, which is the negative of the photoresist surface
(Figure 6.30d and e). This is detached from the photoresist and used to make further copies, child or stamper
shims, by further electroplating onto the preparedmother shim (Figure 6.30f). These are positive copies and are
detatched from the mother shim for use. As many child shims as are needed are formed in this way. The child
shims are used in conventional embossingmachines (Figure 6.30g). These press the shim, under an appropriate
Colour and the Optical Properties of Materials 242
temperature and pressure regime, into the surface of a thermoplastic sheet so that the surface relief is now
imposed upon the polymer film. The film is backedwith a reflective layer, which can be aluminium, zinc oxide,
titanium dioxide or another plastic film, depending upon the ultimate use, and a layer of adhesive if needed.
Naturally, the embossingmachines operate on rolls of plastic and so can producemany thousands of holograms
quickly.
The initial preparation of the H1 slit master is costly and extremely difficult to duplicate. However, once in
production the stamped copies cost just a few pence each. The combination of these two features, coupledwith
the unique appearance of these holograms,makes themubiquitous both for securitymarking and for decorative
purposes.
Further Reading
The classical theory of optical diffraction is described by
E. Hecht, Optics, 4th edition, Addison-Wesley, San Francisco, CA, 2002.
The prevention of diffraction spreading of a light beam is described by
K. Dholakia, Nature 451, 413 (2008) and references cited therein.
Theway to obtain thewavelength of light using a steel rule or similar grating and some informative background
to the method is given in
W. P. Trower (ed.), Discovering Alvarez, University of Chicago Press, Chicago, IL, 1987, p. 1.
An introduction to crystal structures and X-ray diffraction is
R. J. D. Tilley, Crystals and Crystal Structures, John Wiley and Sons, Ltd, Chichester, 2006.
The corona around the sunormoonand thepatterns formedbydroplets or dust onmirrors or otherglass surfaces
is described by
object
hologram H1
(a)slit
cylindrical lens
reference beam
H1 H2 on photoresist
(b)
develop
(c)
electroplate mother shim
(d) (e)
plastic film
embossing shim(g)(f)
mother shim
child shim
Figure 6.30 Embossed holograms (schematic): (a) preparation of slit hologram H1; (b) preparation of H2 onphotoresist; (c) etched photoresist, forming a phase grating; (d) electroplate photoresist; (e) separate mothershim; (f) electroplate mother shim give a chid shim; (g) emboss plastic film with child shim
243 Colour Due to Diffraction
J. Walker, Sci. Am. 245 (August), 116 120 (1981).
D. K. Lynch, W. Livingston, Color in Light and Nature, Cambridge University Press, Cambridge, 1995,
Chapter 4.
C. F. Bohren,What Light Through Yonder Window Breaks? Dover, New York, 2006 (originally published by
John Wiley and Sons, Inc., New York, 1991), Chapter 2.
The microstructure of precious opal is described in
J. V. Sanders, Acta Crystallogr. Sect. A 24, 427 434 (1968).
J. V. Sanders, P. J. Darragh, Mineral. Rec. 2 (6), (1971).
P. J. Darragh, A. J. Gaskin, J. V. Sanders, Sci. Am. 238 (April), 87 95 (1978).
There are currently large numbers of research papers concerned with inverse opals, colloidal crystals and
photonic crystals. Some starting points for further study are
G. I. N. Waterhouse, J. B. Metson, H. Idriss, D. Sun-Waterhouse, Chem. Mater. 20, 1183 1190 (2008).
S. John, Nature 460, 337 (2009).
K. Ishizaki, S. Noda, Nature 460, 367 370 (2009).
A. S. Iyer, L. A. Lyon, Angew. Chem. Int. Ed. 48, 4562 4566 (2009).
For further information on photonic crystals and related materials, especially those that generate beautiful
natural colours, see
A. van Blaaderen, Mater. Res. Soc. Bull. 23 (October), 36 43 (1998).
A. Parker, Proc. R. Soc. Lond. Ser. B 262, 349 355 (1995).
A. Parker, R. C. McPhedran, D. R. McKenzie, L. C. Botten, N.-A. P. Nicorovici, Nature 409, 36 37
(2001).
P. Vukusic, Structural colour, in Dekker Encyclopedia of Nanoscience and Technology, Volume 5, J. A.
Schwarz, C. I. Contescu, K. Putyera (eds), Marcel Dekker, New York, 2004, pp. 3713 3722.
P. Vukusic, J. R. Sambles, Nature 424, 852 855 (2003).
Various authors, Mater. Res. Soc. Bull. 26, 608 646 (2001).
The topic of colour in nature is described from an evolutionary perspective, with examples of diffraction
colours, by
A. R. Parker, In the Blink of an Eye, Free Press, London, 2003.
For further information on diffraction by moth-eye structures and high spatial-frequency surface gratings, see
T. K. Gaylord, W. E. Baird, M. G. Moharam, Appl. Opt. 25, 4562 4567 (1986).
T. K. Gaylord, E. N. Glytsis, M. G. Moharam, Appl. Opt. 26, 3123 3135 (1987).
A. R. Parker, Am. Sci. 87, 248 255 (1999).
Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, Appl. Opt. 26, 1142 1146 (1987).
A very interesting description of an AR surface grating on the eye of a 45 million-year-old fly preserved in
amber is given by
A. R. Parker, Z. Hegedus, R. A. Watts, Proc. R. Soc. Lond. Ser. B 265, 811 815 (1998).
The cholesteric colours of certain beetles and how thesemay be proved to come from twisted layered structures
is given by
A. C. Neville, S. Caveney, Biol. Rev. 44, 531 562 (1969).
The cholesteric blue phases are described by
P. H. Keyes, Mater. Res. Soc. Bull. 16 (January), 32 37 (1991).
Colour and the Optical Properties of Materials 244
The structure of the cornea is given by
J. P. Giraud, Y. Pouliquen, G. Offret, P. Payrau, Exp. Eye Res. 21, 221 229 (1975).
The structure and colour of the Plum-throated Cotinga is described by
R. O. Prum, R. H. Torres, S, Williamson, J, Dyke, Nature 396, 28 29 (1998).
Simple descriptions of making holograms, interesting from a historical point of view, are
J. Walker, Sci. Am. 242 (February), 124 128 (1980).
J. Walker, Sci. Am. 260 (May), 100 103 (1989).
See also
F. Unterseher, J. Hansen, B. Schlesinger, Holography Handbook, Ross Books, 1996.
P. Hariharan, Basics of Holography, Cambridge University Press, 2002.
An extremely clear explanation of rainbow holograms is found in the tutorial by K. Bazargan, at
http://holographer.org (2004).
245 Colour Due to Diffraction
7
Colour from Atoms and Ions
. How can the chemical composition of the sun and other
stars be determined?. How do sodium street lamps produce yellow light?. Why are transition metal compounds coloured?
In the preceding chapters, colour generation has been described in terms of the wave theory of light. This no
longer suffices, and inmuchof the remainingmaterial, startingwith this chapter, photons andquantum ideas are
necessary.
7.1 The Spectra of Atoms and Ions
The spectrum of electromagnetic radiation emitted by an incandescent solid (Sections 1.6 and 1.7) is
continuous and contains all wavelengths. The spectrum of light emitted by a rarefied gas of atoms or ions
consists of a sequence of bright lines. For example, if the light frommercury or sodium street lamps is dispersed
by an inexpensive diffraction grating positioned in front of a camera lens, the image will consist of a set of
individual coloured copies of the light source, each of which arises from a particular line in the spectrum
(Figure 7.1a). Narrowing the source image to a slit, as in a spectrometer,1 will yield a series of sharp lines when
an excited gas is the source (Figure 7.1b). This latter pattern is called a line spectrum. If a continuous spectrum
Colour and the Optical Properties of Materials Richard J. D. Tilley
� 2011 John Wiley & Sons, Ltd
1 Devices for the display of spectra are called spectroscopes, spectrographs or spectrometers. They use prisms or diffraction gratings and
the resultant spectra are recorded and displayed electronically.
from an incandescent source is passed through a gas of the same atoms then the bright line spectrum will be
shown as a negative and appear as a set of dark lines on the continuous bright background (Section 7.4). In this
chapter, the origin of these lines is described.
For most chemical purposes an atom or an ion can be considered to consist of a dense minute nucleus
surrounded by electrons which are said to occupy a series of orbitals. The electron configuration of an atom or
an iondescribes theway inwhich these electrons are allocated to these orbitals (AppendixA7.1 andFigure 7.2).
The spectral lines emitted or absorbed (the source of colour of atoms) arise when electrons jump from one
orbital to another. The energies of the orbitals in isolated atoms and ions are precise and the total energy of all of
the electrons is then represented as a sharp energy level. Energy is absorbed when electrons are excited from a
lower energy level to a higher level and exactly the same energy is releasedwhen the electron drops back to the
same lower level again.
From this point of view, the spectral lines shouldbe infinitelynarrow,butbecause of atommotion the lines are
broadened due to the Doppler effect. The broadening amounts to v /c, where v is the velocity of the atoms and c
the velocity of light. Doppler broadening is thus greatest for light atoms at high temperatures. Other
interactions, notably electron electron and electron nucleus, which are described by quantum electrodynam-
ics, also place limits upon the sharpness of the lines, and even in ideal circumstances the lines have a width
termed the natural linewidth.
These refinements explain the shape of spectral lines, but the main point remains that the energy of the line
is centred upon the energy separating the final and initial states that the electron occupies. When the energy of
the radiation absorbed or emitted falls in the visible these transitions give rise to colours. For transitions giving
404.7433.6434.7435.8
546.1577.0579.1
690.8Wavelength / nm
(a)
(b)
Figure 7.1 (a) The image of a mercury-vapour street light taken with an inexpensive transmission diffractiongrating over the lens. Each coloured lamp image arises from one line in the first-order diffraction spectrum ofmercury. Thewhite lampat the right is the zero-order image and thedeepblue image at the far left is thebeginningof the second-order series. (b) The line spectrum ofmercury vapour. The bright lines correspond to the images in(a) as shown
Colour and the Optical Properties of Materials 248
rise to lines in the visible, 400 700 nm, the width of a spectral line at ordinary temperatures is about
�0.0005 nm.
A light photon can only interactwith an electron if it has the exact amount of energy to allowanelectron to pass
fromoneprecise energy level to another. Thus,whenaphotonof energyhn isabsorbedbyanatomor ion it passes
from a lower energy state, often the lowest available state in the system, called the ground state E0, to an upper
one E1 (Figure 7.3a). The transition will only take place if the frequency n of the photon is given exactly by:
n ¼ E1�E0
h¼ DE
h
whereDE is the energy separation of the two energy levels and h is Planck’s constant. If the atom is in the upper
state E1 and makes a transition to the lower state E0, the same quantity of energy, DE, will be emitted
(Figure 7.3b). This will have the same frequency, given by the same equation:
n ¼ E1�E0
h¼ DE
h
Each transition gives rise to a line in the spectrum (Figure 7.3c).
The line spectra ofmost atoms are complex and contain large numbers of lines, but hydrogen has a relatively
simple visible spectrum, which contains four strong lines (Figure 7.4a) and because of this has played a
Period
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Group
IA IIA IIIB IVB VB VIB VIIB VIIIB IB IIB IIIA IVA VA VIA VIIA VIII
Figure 7.2 The Periodic Table of the elements, giving the outer electron configuration (below element symbol)and ground-state level (upper right) of the atoms. The transitionmetals are coloured orange, the lanthanoids blueand the actinoids mauve. Note that the terms "lanthanoid" and "actinoid" are now preferred by IUPAC to"lanthanide" and "actinide" (see footnote to Appendix A7.1.3)
249 Colour from Atoms and Ions
prominent role in twentieth-century physics. The lines form part of a series, the Balmer series, with positions
given by the formula:
n ¼ 1
l¼ RH
1
22� 1
n2
� �
where n is the wavenumber (usually measured in cm 1; Appendix A1.1), RH is the Rydberg constant
(1.097� 107m 1¼ 1.097� 105 cm 1) and n takes values of 3, 4, 5, . . .). As the formula shows, the lines
gradually approach each other and reach a series limit as n approaches infinity, given by RH/22 (Table 7.1).
This series was explained theoretically by Bohr in his celebrated theory of the hydrogen atom. The lines
are caused by transitions of an excited electron from the n¼ 3, 4, 5, . . . shells to the n¼ 2 shell of the atom
(Figure 7.4b). Because the ns, np, nd, etc. orbitals of hydrogen all have the same energy, it does not matter
which exact orbital the electron is excited to or falls from, and this is the reason for the apparent simplicity of
the spectrum. (In fact, the lines all are multiplets and the explanation of these features required the profound
(a) (b)
(c)
light absorbed light emitted
hν hν
E0
E1
E0
E1
ν Frequency
Inte
nsity
Figure 7.3 The absorption and emission of radiation by isolated atoms or ions. (a) Light is absorbed if the energyof the photon exactly matches the energy gap between the lower level and the upper level. (b) When energy isreleased spontaneously the energy of the photon is again exactly equal to the difference in energy between theupper and lower levels. (c) The emission fromacollection of atoms in a gas at lowpressurewill consist of a narrowline. The frequency at which the line occurs is the same as that of the photons involved in absorption and emissionin (a) and (b)
Table 7.1 The Balmer series of visible spectral lines of atomichydrogen
n Designation l/nm l�1/cm�1
3 H a 656.3 15 2374 H b 486.1 20 5725 H g 434.0 23 0416 H d 410.2 24 378¥ 364.7 27 420
Spectral lines are often specified in terms of n ¼ 1=l given in units of cm 1 (see Appendix A1.1).
Colour and the Optical Properties of Materials 250
development of quantum theory; see this chapter’s Further Reading.) Other series form as electrons fall from
excited levels to the n¼ 1 (Lyman), n¼ 3 (Paschen), n¼ 4 (Bracket), n¼ 5 (Pfund) orbitals, but these lie
outside the visible. The spectra of atoms containing many electrons, even of ‘one-electron’ atoms such as
lithium and sodium, are far more complex, and the explanation of these spectra was one of the great
triumphs of twentieth-century science.
Each spectral line gives precise information about the difference in energy between the two energy levels
involved in the electron transition, and this can be indicated by using a formula for the frequency or
wavenumber containing the difference between these two quantities. Thus, the Balmer series can be written
in the form:
n ¼ 1
l¼ T1�T
Tl andT are called terms,withTl representing the series limitRH/22.All other spectral lines canbedescribed in a
similar way, as the difference between two terms. How these terms are derived follows.
Ene
rgy
0
13.6 eV
n = 1
n = 2
n = 3
n = 4
n = ∞
Balmerseries
H-α H-β H-γ H-δ
656.
3 nm
486.
1 nm
434.
0 nm
410.
2 nm
(b)
656 3486.1434.1410 2
(a)
Figure 7.4 (a) The visible spectrum of atomic hydrogen. The wavelength of each line is noted in nanometres.At high resolution these lines are seen to split into more complex groups of lines. (b) The transitions to then¼ 2 level of the hydrogen atom give rise to the visible spectral lines in the Balmer series
251 Colour from Atoms and Ions
7.2 Terms and Levels
The electron configurations of atoms or ions (Appendix A7.1, Figure 7.2) are a first-level approximation as far
as energies are concerned. They do not even take electron electron repulsion into account and are worked out
by assuming that there is only one electron circling a nucleus surrounded by a negative cloud made up of all of
the other electrons present.Because only one electron is involved in the computations the quantumnumbers are
called hydrogen-like or one-electron symbols and they are given lower case letter labels.
Atomic or ionic spectra, on the other hand, consist of a series of lines which give information on the exact
energy difference between two energy levels in the species under investigation. A measurement of atomic
spectra thus allows the real energy levels of atoms tobe assessed.The energy levels of an isolated atomaregiven
labels called term symbols.A term is a set of states or energy levelswhich arevery similar in energy.Transitions
between these terms, or, more precisely, the energy levels that make up the set of states specified by the term,
give rise to the observed line spectrum of an atom.
Term symbols are derived by taking into account electron interactions. The most important of these, for
an understanding of spectra, are electron electron repulsion and the combining, or coupling, of the orbital
and spin angular momenta. There are a number of ways of carrying out this coupling. The best known
method, which is mainly applicable to light atoms, is called Russell Saunders or LS coupling (Appendix
A7.2). In this designation, each term is written as 2Sþ 1L, where L is a many-electron quantum number
describing the total orbital angular momentum of all of the electrons surrounding the atomic nucleus and S
is a many-electron quantum number representing the total electron spin. Sometimes the term symbol has an
initial value n, when all of the electrons outside the core (that is, those involved in transitions) come from
the same shell. The superscript 2Sþ 1 is called the multiplicity of the term. Upper case letters are used to
make it clear that all electrons are included and to differentiate them from the hydrogen-like configurations.
Terms, therefore, apply to the overall energy state of the atom or ion as a whole. The total angular
momentum quantum number L is replaced by a letter symbol similar to that used for the single electron
quantum number l. The correspondence is set out in Table 7.2. After L¼ 3, F, the sequence of letters is
alphabetic, omitting J. Be aware that the symbol S (italic) means the value of total spin while S (roman)
gives the value of L.
As an example, the alkali metal atoms in their lowest energy state all have a single ns electron outside of a
closed shell. All core electrons can be ignored, so that the only electron to consider when constructing the term
is the outer s electron. Thevalue ofLmust be equal to thevalue of l for the electron, i.e. zero, so that the state is S.
The spinon the electron is 12, so that the total spinquantumnumberS (donot confusewith that just used forL¼ 0)
is 12. The multiplicity is then (2S þ 1)¼ 2. The lowest energy term for all of the alkali metals is thus 2S.
Appendix A7.2 sets out in detail the way in which the terms of an atom can be derived.
Table 7.2 The correspondence of L valuesand letter symbols
L Symbol
0 S1 P2 D3 F4 G5 H
Colour and the Optical Properties of Materials 252
Even the term symbol does not account for the true complexity found in most atoms. This arises from the
interaction between the spin and the orbital momentum (spin orbit coupling) that is ignored in Russell
Saunders coupling. For this the quantum number J is needed. It is given by:
J ¼ ðLþ SÞ; ðLþ S�1Þ; . . . ; jL�Sj
where |L� S| is the modulus (absolute value, irrespective of whether þ or�) of the quantity L� S. The new
quantum number is incorporated as a subscript to the term, now written 2Sþ 1LJ and this is no longer called a
term, but a level. Eachvalueof J represents a different energy level.Thus, the term for the alkalimetals 2S canbe
expanded bynoting that thevalue of the quantumnumber J is givenbyLþ S ¼ 12. Theground-state energy level
term for all of the alkali metals is thus 2S1=2. To specify the ground state of a sodium atom, for example, the fact
that the outer electron is in the 3s orbital would allow the level to be written 32S1=2.
It is found that a singlet term always gives rise to one level, a doublet to two, a triplet to three and so on. The
progression from the electron configuration of an atom to a set of energy levels thus involves a number of steps,
shown schematically for a 3d2 ion such as Ti2þ or V3þ in Figure 7.5. At the far left of the figure, the electron
configuration is shown and it is assumed that the ion can be represented by a single energy level. This is useful
chemically, but is unable to account for the spectra of the atom. Russell Saunders coupling is a reasonable
approximation to use for the 3d-metals, and the terms that arise from this are given to the right of the
configuration. In Russell Saunders coupling the electron electron repulsion is considered to dominate the
interactions. The ion is now allocated five energy levels, the lowest being represented by a term 3F. The terms
are split further if spin orbit coupling (j j coupling) is introduced. The number of levels each term forms is the
same as the multiplicity of the term, 2S þ 1, and this leads to nine energy levels in total.
The procedure for dealing with atoms in general is given in Appendix A7.2 and the ground-state level of all
atoms can be found in Figure 7.2.
3d2
1S0
1S
1G4
1G
3P2
3P13P 3P0
1D2
1D
3F4
3F3
3F2
3F
Nointeraction
(one-electronconfiguration)
Electron–electronrepulsion (Russell-Saunders terms)
Spin–orbitinteraction( j–j levels)
Terms LevelsOrbitals
Figure 7.5 The schematic development of the energy levels of a free d2 ion, such as Ti2þ or V3þ
253 Colour from Atoms and Ions
In a heavy atom it might be preferable to derive the energy levels by proceeding from the electron
configuration to levels derived by j� j coupling and then add on a smaller effect due to electron electron
repulsion. In real atoms, the energy levels determined experimentally are often best described by an
intermediate model between the two extremes of Russell Saunders and j� j coupling, and for these atoms
alternativecoupling schemesmaybepreferred.The splittingof terms into levelsdue to spin orbit coupling is of
considerable importance for the lanthanoids.
In addition, note that in the presence of a magnetic field these spin orbit levels are split further, so that the
spectra of atoms and ions in magnetic fields are more complex than that already discussed. This has relevance
not only for laboratory work, but also for the interpretation of stellar spectra, where extremely strongmagnetic
fields can occur. The same is true of static electric fields. In both cases, atoms or ions in a gas or free space will
show an average effect because of themotion of the particles. However, in a crystal, the atom and ion positions
aremore or less fixed and the application of eithermagnetic or electric fields along certain symmetry directions
will, in general, cause different degrees of splitting of the levels than the same fields applied along different
symmetry directions.
7.3 Atomic Spectra and Chemical Analysis
Although the terms and levels give a picture of the energy levels availablewithin an atom, the spectra cannot be
explained simply byworking out all of the possible transitions between them.Apart from the energy restriction
mentioned above, light can only interact with electrons if the wave functions specifying the initial and final
states fulfil certain conditions. This latter restriction leads to a number of selection rules which allow one to
determine whether the transition is probable or improbable. These selection rules depend upon how the light
interacts with the electrons. The quantum mechanical interactions that the wave functions describe can be
specified in terms of an interaction of the electric field of the light wavewith the electrons, a situation known as
an electric dipole transition, or with the magnetic field of the light wave, resulting in a magnetic dipole
transition. (Other less frequently observed transitions involving electric and magnetic quadrupoles and more
complex configurations are also possible but become increasingly rare.)
The selection rules which set out which transitions are allowed andwhich are not allowed actually represent
the probabilities of the transition occurring. Thus, allowed transitions have a very high probability of taking
place, whereas forbidden transitions have a very low probability of occurring, but are not absolutely forbidden.
Electric dipole transitions have the highest probability, giving rise to intense lines in the spectrum of an atom
and so are of primary importance in colour production. The other types of transition have lower probabilities of
occurring, and at best give rise to weak lines in the spectrum.
The selection rules applicable to Russell Saunders terms and levels leading to the absorption or emission of
light due to electric-dipole transitions are
n no restriction
DS cannot change
DL can change by 0, �1, but L¼ 0 to L¼ 0 is forbidden
DJ can change by 0, �1, but J¼ 0 to J¼ 0 is forbidden.
In addition, theLaporte selection rule, sometimes called theparity selection rule, limits transitionsonly to those
forwhich the symmetryof thewavefunctions specifying the start andend states are of opposite parity. Itmeans that
transitions between orbitals of the type s to s, p to p, d to d and f to f are all forbidden. This has considerable
importance for the operation of lasers, as forbidden transitions (i.e. transitions with a low probability) are
Colour and the Optical Properties of Materials 254
associated with energy states with long lifetimes. These are necessary to obtain the population inversions needed
for laser action (Section 1.9).
The exact arrangement of the energy levels in an atom is very sensitive to the electron configuration. When
this constraint is coupled to the selection rules operating, it emerges that the line spectrum of each chemical
element is unique.Thus, the spectrumbecomes a powerful analytical tool. Each atomor ion can be thought of as
having a line spectrum fingerprint which can be used as a diagnostic test for the element.
At the simplest level this is made use of in inorganic chemistry as a ‘flame test’. A small quantity of the
material beingexamined is placeduponaplatinumwire andheated tohigh temperature in aflame.Thecolour of
the flame is a guide to the atoms present. This method works well with the alkali metals and alkaline earth
metals, which produce clearly identifiable colours (Table 7.3).
At this point it is well to be aware that the colours produced in a flame, or in fireworks, which are similar,
are the result of complex interactions and frequently arise from molecular species rather than isolated atoms
or ions. Thus, the scarlet colour arising from lithium compounds is due to radiation from LiOH molecules
rather than isolated Li or Liþ . Green colours from barium compounds are derived from the molecular
species BaClþ and BaOHþ , and the red of strontium compounds derives from SrOHþ and SrClþ rather
than Sr or Sr2þ . The same could be said about other flame and firework colours. Colour from molecules is
described in Chapter 8.
Muchmore information about the colours givenout by the flame can be obtained by allowing the light to pass
through anarrowslit and viewing itwith an inexpensiveplastic diffraction grating.Thegrating spreads the light
out into a series of spectra which in the case of atoms or ions consist mainly of lines and which in the case of
molecules also contains bands (see Chapter 8). In fact, such an arrangement is a simple spectroscope. The
technique can yield more information if the intensities and positions of the lines in the spectrum can be
recorded, and it is this technique which allows one to determine that molecular species are important in flame
colours rather than isolated atoms or ions. Comparison of the intensities of lines with those from standard
solutions of ions allows quantitative analysis of even very small quantities of impurities to be made. The
technique is called atomic absorption analysis. It is routinely used to detect quantities of metal impurities at
concentrations of parts per million.
7.4 Fraunhofer Lines and Stellar Spectra
In 1814, Fraunhofer, bymaking better spectrographs than any others available at that time, discovered that the
solar spectrumwas interspersed with a number of dark lines, now calledFraunhofer lines. Themost important
visible Fraunhofer lines are illustrated in Figure 7.6 and listed in Table 7.4. These features are actually
absorption spectra and consist of both sharp lines and wider bands. The lines are due to absorption by single
isolated atoms and ions, whereas the bands arise from molecules that lie between the source of light and the
observer. (The reasons why molecules often give rise to absorption bands is described in Chapter 8.)
Table 7.3 Flame test colours
Atom Colour Atom Colour Atom Colour
lithium scarletsodium yellow calcium orange red copper bluepotassium violet strontium crimsonrubidium red violet barium greencaesium blue
255 Colour from Atoms and Ions
The Fraunhofer lines arise in two ways. One set of absorption lines and bands, the telluric lines, are due to
components of the Earth’s atmosphere. These absorb incoming solar radiation and give rise to dark lines or
bands in theotherwisecontinuousspectrumfromthesun.Theprincipalcontributionsare fromoxygenandwater
vapour.However, another set of lines ariseswhen light from the sun is absorbedbyatomsor ions in the relatively
cool outer solar regions.Among themost prominentof these are lines fromhydrogen, sodium, calciumand iron,
which could all be identified by comparisonwith spectra from standards available in the laboratory. Significant
among the information, a Fraunhofer line at awavelength 587.6 nm, discovered in 1868, could not be attributed
to any known element. The new line was taken as an indication of the presence of a new element in the solar
atmosphere. The element was subsequently named helium, from the Greek word for the sun, helios. Almost
30 years were to pass before the gas was discovered on Earth, by Ramsay, who isolated it in 1895.
Nowadays, the presence of metallic atoms and ions in the outer atmospheres of stars or even far-off galaxies
is generally confirmed by recording the spectrum of the star and examining the dark absorption Fraunhofer
lines found.
7.5 Neon Signs and Early Plasma Displays
Faraday, in 1835, first discovered that gases at low pressure could conduct electricity and at the same time give
out light. The complex processes taking placewere investigated in depth byGeissler in the 1860s and at the end
of the 1890s by Crookes.
400 500 600 700Wavelength / nm
G F E2 D3 D2 D1 Cα B
Figure 7.6 The main visible Fraunhofer lines and bands, visible as dark lines within the continuous spectrum ofthe sun. The atoms or molecules responsible for these features are listed in Table 7.4
Table 7.4 Some Fraunhofer lines
Designation Origin Wavelength/nm
B O2 molecules 687.7 688.4 (band)C hydrogen, H a 656.2a, a O2 molecules 627.6 628.7 (band)D1 sodium 589.6D2 sodium 589.0D3 or d helium 587.6E2 iron 527.0F hydrogen, H b 486.1G0 hydrogen, H g 434.0G iron, calcium 430.8h hydrogen, H d 410.2
Colour and the Optical Properties of Materials 256
The experimental observations are easy to report. If a gas is contained in a tube and is at atmospheric
pressure it will not conduct electricity unless it is subjected to an extremely high voltage, as when lightning
strikes through the atmosphere. If the pressure is reduced and the gas is subjected to a voltage of the order
of kilovolts it begins to show electrical conductivity and at the same time it starts to emit light. The colour
of the light depends upon the gas in the tube. This is the basis for the operation of neon signs, invented in
1910, and of sodium- and mercury-vapour street lighting. Ultimately, as the pressure falls, the number of
spectral lines diminishes, until only the so-called ‘persistent lines’ can be observed. Finally, all light
emission ceases and the tube no longer glows. During the first half of the twentieth century the gas pressure
in scientific vacuum equipment, including electron microscopes, was estimated by using the light emitted
when an electric field was imposed on the residual gases in a tube (a Geissler tube), which connected with
the main body of the equipment. When the glow had been totally extinguished the vacuum was usually
as good as could be obtained with the then available vacuum pumps and was colloquially called a
‘black vacuum’.
In Geissler tubes, neon signs and similar devices, electrons are emitted from the cathode (the negative
electrode) and are accelerated in a high electric field across the gas. These electrons collidewith gasmolecules
and excite them to higher energies. Some molecules are ionized and these ions in turn are accelerated in the
electric field and cause further ionization and excitation. The light emitted is due to the excited atoms and ions
losing energy by releasing photons as they return to lower energy states. Lamps that emit light by this
mechanism are generally termed gas discharge lamps.
‘Neon’ signs are gas discharge lamps thatmake use of atoms of the inert gases as theworkingmedium. These
elements exist as monatomic gases at normal temperatures and all can be used in what are now collectively
known as neon signs, neon being the first to be used. To make a neon sign, a glass tube is evacuated and filled
with a low pressure of one of the inert gases. The gas is subjected to a voltage of about 10 kV via electrodes at
opposite ends of the tube, which causes it to glow with a characteristic colour (Table 7.5). In some displays a
mixture of gases is used to change the apparent colour of the glow.
Gas discharge tubes using xenon (Xe) as the working medium are also widespread. Under low pressures
and fairly low current densities, the light takes on a blue hue and is high in ultraviolet radiation. However,
when the gas is present at rather higher pressures, and when the current density becomes high, the tubes emit
light that is perceived to be white. This is made use of in two common ways. Xenon arc lamps, operating at
high pressures (up to 300 atm) and temperatures give out continuous light. They are notably used in cinema
projectors. Flashlamps or flashtubes are used to give a high-intensity white-light output over an extremely
short interval. These contain low pressures of xenon (up to 0.1 atm). The flash is achieved by sending a large
pulse of current through the gas via a charged capacitor. Xenon flashtubes are commonly found as the flash
unit in cameras.
Table 7.5 Colours produced by the inert gases
Gas Colour
Helium yellowNeon pink redNeon þ argon redArgon pale blueArgon þ mercury blueKrypton lavenderXenon blue
257 Colour from Atoms and Ions
The energy level diagrams of the inert gases are complex because the simplest excited state, [np5 (n þ 1)s],
consists of two unfilled shells. Each configuration of the excited atom thus gives rise to a considerable number
of levels.These aremostoftendisplayed in the formof aGrotriandiagram, inwhich the termsare set out as a set
of ladders of increasing energy. The termswith a givenmultiplicity all form a single ladder. Spectral transitions
are indicated by lines between relevant terms: absorption represented byanupward transition and emissionbya
downward transition. A simplified version of such a diagram for neon, which shows electron configurations
rather than terms (Figure 7.7), reveals that the main lines contributing to the red colour are due to transitions
from the excited 2s2 2p5 3p configuration, comprising ten levels in total, to the 2s2 2p5 3s excited state. These
include the twomost intense lines, at 692.9 nmand 703.2 nm. The ground state, with configuration 2s2 2p6with
a single level, 1S0, does not figure in transitions giving rise to visible radiation.
In the case of xenon, the spectrumconsists of relatively few lines at lowexcitations, giving rise to the blue tint
of the output light. However, there are so many transitions possible that can produce light when the excitation
energy is high that the output appears to bewhite, although the emission is still in the form of lines and is not a
continuous spectrum. The observed colour changes from blue to white as the discharge energy increases.
Similar colour changes occur with the other gases mentioned.
15
16
17
18
19
20
3s
2s22p5 ns: 2s22p5 np: 2s22p5 nd:4 levels ineach box
3p
10 levels ineach box
4p
5p
3d
12 levels ineach box
4d
5d 6d
4s
5s
6s 7s21
22
130000
140000
150000
160000
170000
Ene
rgy
/ eV
Energy / cm
–1
703 2, 692.9, 650.6, 640 2
478 9
488 5, 482.7
576.4, 534.1
471 0, 470.4, 453.8
471.5
(a)
(b)
Figure 7.7 (a) Line spectrum of neon. (b) Schematic Grotrian energy-level diagram for neon with configura-tions rather than terms displayed. The ground state, at energy zero, is not shown. Each excited state gives rise to anumber of levels which are not drawn individually but represented schematically by a box. The transitions givingrise to emission colours are drawn as downward-pointing arrows. The wavelengths marked (in nanometres) arejust a fewof themore intense lines in the visible. (Conversionof the units eVandcm�1 to theSI unit joule (J) is givenin Appendix A1.1)
Colour and the Optical Properties of Materials 258
The state of matter giving rise to the colours is a form of plasma, in which the high electrical field imposed
across the tube strips the enclosed atoms of some electrons and creates a fluid consisting of positive and
negative entities. The plasma usually forms close to the cathode and is localized in a small volume giving out
high-intensity light. In some lamp designs, plasmas can form at both electrodes, thus increasing the output.
Similar plasma colours can occur in the atmosphere, although potentials of the order of 30 000V cm 1 are
needed. The best known natural display of this type is St Elmo’s fire. This a deep blue or violet glow
appearing around tall and generally sharp objects, especially, in historic times, the masts of sailing ships.2 It
occurs sporadically and is particularly observed when the weather is heavy and thundery. It is believed that
high static electric fields, enhanced in the neighbourhood of sharp points, are intense enough to break apart
the molecules in the surrounding air, mostly nitrogen and oxygen, to form a local plasma of ionised
fragments. The release of energy as the excited fragments regain the ground state gives rise to colours in an
analogous way to those in neon lights. Note that, as in the case of flame tests described above, the actual
constituents involved, which are likely to contain molecular fragments as well as ionized single atoms, are
not well understood.
With the advent of portable computers a need arose for a lightweight flat display screen. Among the first of
thesewas themonochrome gas plasma display,which, having been developed someyears previously, operated
on the principles just outlined. Ionised inert gases (mainly neon)were employed to produce the illumination. A
photograph of a display of this type, froma computer available in 1989, is shown in Figure 7.8. This technology
rapidly gave way to full-colour plasma displays, which are described in Section 9.5.
7.6 The Helium–Neon Laser
For laser action, two objectives have to be fulfilled (Section 1.9). It is necessary to obtain a population inversion
between two energy levels and then ensure that the higher energy level is depopulated by stimulated emission,
not by spontaneous emission. This was first achieved in the ruby laser by Maiman in 1960 (Section 7.11).
However, not long after this, at the end of 1960, Javan constructed the helium neon (He Ne) laser. This laser is
the ubiquitous red laser common in supermarket check-out counters and laser pointers, although colours other
than red can also be produced by the helium neon combination; in fact, the first laserwavelength producedwas
at 1.15 mm. The laser consists of a low pressure (10 2 to 10 3 atm) of helium mixed with about 10 % neon,
enclosed in a narrow glass tube.
The laser uses excited helium to transfer energy to neon and so obtain a population inversion. The helium is
excited by a high voltage, just as in a neon sign. High-energy electrons are produced by subjecting the cathode
(thenegative terminal) to ahighvoltage.These energetic electrons, e�, collidewith theheliumatoms toproduce
an excited state, He�:
He ð1s2Þþ e*!He* ð1s1 2s1Þþ e
2 A good description of St Elmo’s fire is given by Darwin at the start of Chapter III in his Journal of Researches . . ., better known as the
Voyage of theBeagle. The incident described takes place in the estuary of theRiver Plate. ‘On a second nightwewitnessed a splendid scene
of natural fireworks; the mast head and yard arm ends shonewith St. Elmo’s light; and the form of the vane could almost be traced, as if it
had been rubbed with phosphorus. The sea was so highly luminous, that the tracks of the penguins were marked by fiery wakes, and lastly
the skywasmomentarily illuminated by themost vivid lightning.’ Later in the same chapterDarwin remarks that ‘the neighbourhood of the
Rio Plata seems particularly subject to electric phenomena’. St Elmowas the patron saint of sailors, and the presence of St Elmo’s firewas
believed to give protection against storms.
259 Colour from Atoms and Ions
The 1s1 2s1 configuration gives rise to two energy levels 21S0 and 23S1 (Figure 7.9). The excitedHe
� can passits energy over to a neon atom during a collision to produce an excited neon atom, Ne�. This can happen
because, quite by chance, the energy to be transferred is almost exactly the same as two excitation energy
transitions of Ne:
He* ð1s1 2s1; 21S0ÞþNe ð2s2 2p6Þ!HeþNe* ð2s2 2p5 5s1Þ
He* ð1s1 2s1; 23S1ÞþNe ð2s2 2p6Þ!HeþNe* ð2s2 2p5 4s1ÞThe neon energy levels derived from these two configurations, each consist of four energy levels. In addition,
there are two other sets of 10 energy levels present on the neon atoms, derived from the configurations
2s2 2p5 3p1 and 2s2 2p5 4p1. (All of the energy levels on the neon atoms are complex and laser workers use a
labelling system of s and p designations (Table 7.6). Unfortunately, this mimics the chemical configuration
symbols, butwithout the same implications, which leads to unnecessary confusion. In order to relate Figure 7.9
with the labels found in laser texts, the correspondence in nomenclature is given in Table 7.6.) The collisions
between neon and excited helium atoms (He�) produces a population of excited neon atoms (Ne�) in which
several series of occupied and empty energy levels exist in close conjunction. These excited Ne� atoms can
release energy by stimulated emission, thereby dropping to many of the empty levels, and about 100 or more
output wavelengths can appear. Themain transition, however, is from the 2s2 2p5 5s1 set of levels to the 2s2 2p5
3p1 set of levels:
Ne* ð2s2 2p5 5s1Þ!Ne* ð2s2 2p5 3p1Þþ hn
Figure 7.8 A gas plasma flat-screen display on a portable computer of 1989
Colour and the Optical Properties of Materials 260
The transition produces thewell-known red laser output with awavelength of 632.8 nm. Transitions to some of
the other levels in the samemanifold give rise to the other coloured output frequencies,which include 543.5 nm
(green), 594.1 nm (yellow) and 612.0 nm (orange).
The still-energized neon atom thereafter rapidly decays to the ground state, 2s2 2p6, in two steps:
Ne* ð2s2 2p5 3p1Þ!Ne* ð2s2 2p5 3s1Þ!Ne ð2s2 2p6Þ
The 2s2 2p5 3p1 to 2s2 2p5 3s1 transition is fast and helps to maintain a population inversion between the
2s2 2p5 3p1 level and those above it. The final transition is radiationless and energy is often lost to the walls of
15
16
17
18
19
20
3s
3p
4p
5p
4s
5s
6s21
22
130000
140000
150000
160000
170000
Ene
rgy
/ eV
Energy / cm
–13391 nm
1152 nm
633-543
21S0
23S1
pump
He Ne
rapid decay
wall collisions
1s1 2s1
2s2 2p5 ns: 2s2 2p5 np:4 levels ineach box
10 levels ineach box
Figure 7.9 Schematic processes operating in a helium–neon laser. Helium (He) atoms are excited from the 1s2
ground state into two 1s1 2s1 (He�) states and subsequently transfer energy to neon (Ne) atoms to excite them tothe 2s2 2p5 4s and2s2 2p5 5s groups of energy levels. Themain laser transition is from the2s2 2p5 5smanifold to the2s2 2p5 3p group of levels, from which the Ne atoms return to the ground state in two steps
Table 7.6 Energy levels in neon
Electron configuration Laser terminology
2s2 2p5 3s1 1s2 1s5 (4 energy levels)2s2 2p5 3p1 2p1 2p10 (10 energy levels)2s2 2p5 4s1 2s2 2s5 (4 energy levels)2s2 2p5 4p1 3p1 3p10 (10 energy levels)2s2 2p5 5s1 3s2 3s5 (4 energy levels)
261 Colour from Atoms and Ions
the laser tube in transitions which do not give out light. The renewed population of ground-state He and Ne
atoms allows the process to begin all over again.
7.7 Sodium and Mercury Street Lights
Sodium street lights give out a characteristic yellow colour, which arises from excitedNa atoms andNaþ ions.
As sodium is a solid at normal temperatures, the initial discharge is through a lowpressure of neonwhich is also
contained in the lamp tube. This ‘neon lamp’ is first activated, which is the reasonwhy sodium lamps glowwith
a pink red colour when they are warming up. After a short time the energy supplied to the neon generates
enough heat for the sodium to evaporate. At this stage, collisions between electrons accelerated by the electric
field and sodiumatoms excite these latter to higher energy levels.On falling back to the ground state this energy
is released.
A partial energy level Grotrian diagram of sodium atoms is given in Figure 7.10. This shows that, for neutral
sodium atoms, each return path traverses the closely spaced pair of levels 32P1=2 and 32P3=2 that arise from the
0
1
2
3
4
5
3s
4s
2s2 2p6 ns:
1 level2s2 2p6
np:2 levels
2s2 2p6 nd:
2 levels
4p
5p6p
4d
5d
5s
6s7s
6
10,000
20,000
30,000
40,000
50,000
Ene
rgy
/ eV
Energy / cm
–1
589.0, 589.6 (D)
615.4, 616.1
514.9, 515 3474 8, 475.2
568 3, 568.8
497 8, 498.3
2S1/22P1/2, 2P3/2
2D1/2, 2D3/2
3p
3d
0
(b)
(a)
D
Figure 7.10 (a) The line spectrum of sodium. The D-line doublet is not resolved in this image. (b) SchematicGrotrian energy-level diagram for sodium atoms. The transitions giving rise to emission colours are drawn asdownward-pointing arrows. Thewavelengths emitted are given beside each transition (in nanometres). By far themost intense transitions are those at 589.0 and589.6 nmthat produce abright yellowdoublet – the sodiumDlines.The term symbols of all the levels in each column are the same, and the terms 2P and 2D each consist of a pair ofclosely spaced levels which are not resolved on the energy scale used. (Conversion of the units eV and cm�1 to theSI unit joule (J) is given in Appendix A1.1)
Colour and the Optical Properties of Materials 262
configuration 1s2 2s2 2p6 3p before returning to the ground state 32S1=2 arising from the ground-state
configuration of 1s2 2s2 2p6 3s. These two transitions are the only significant lines in the visible spectrum
andgiveout the familiar yellowsodium light,whichmakesupapproximately 90%of thevisible emissions.The
two 32P levels differ slightly in energy so that the emission consists of two wavelengths, 588.995 and
589.592 nm. These constitute the bright yellow sodium D lines, widely used in spectroscopy and as standard
wavelengths at which to record optical properties such as refractive index.
Sodium lamps operated at relatively low sodium pressures give a light output dominated by the sodium D
lines. This means that objects illuminated by these lamps and observed by reflected light do not show the
colour that would be experienced when illuminated by daylight (see Chapter 1). Such lamps are said to have
poor colour rendition and have a limited usefulness. To overcome this, high-pressure sodium lamps are now
more commonly used in applications such as street lighting, where colour perception is important. The high
pressure broadens the lines emitted and other materials, notably mercury (see below), add further emission
lines to the spectrum, balancing the D-line emission and giving a light that is perceived as ‘whiter’
(Figure 7.11).
Mercury-vapour lights operate in a similar fashion to sodium lights. A high voltage is imposed across a tube
containingmercuryvapour.Collisions betweenelectrons andmercuryatomsexcite them tohigher energies and
these same atoms emit light as they lose energy again. When the light is first switched on, the low mercury
pressure means that only the persistent lines of the spectrum appear and the light has a deep blue colour. As the
lamp warms, the pressure increases, more spectral lines appear and these also broaden to give a more white
colour, but still with a noticeable blue green aura.
The ground-state configuration of mercury, [Xe] 4f14 5d10 6s2, shows that there are a pair of 6s electrons
outside of filled inner shells, giving a ground-state level 1S0. Themost important transitions as far as the visible
spectrumare concernedare from the excited6s7s configuration to theexcited6s6pconfiguration (Figure7.12).
These have wavelengths in the blue green (404.7, 453.8 and 546.1 nm), which gives these lamps their rather
eerie coloration. The lack of any reds means that faces viewed by reflected mercury light have a pallid
appearance.Topartly correct this,mercury tubes are often coatedwith afluorescentmaterialwhich converts the
Figure 7.11 The emission spectrum of a high-pressure sodium street lamp obtained using an inexpensivetransmission diffraction grating
263 Colour from Atoms and Ions
strong ultraviolet emission with a wavelength of 253.65 nm arising from a transition between the 63P1 level to
the ground state 61S0 (not shown in Figure 7.12) into visible light (see Chapter 9).
The spectra of street lights are, in fact, easy to observe or photograph using an inexpensive plastic
transmission diffraction grating (see this chapter’s Further Reading).
7.8 Transition Metals and Crystal-Field Colours
The majority of atoms or ions in solids or solutions do not give rise to pronounced colours because the
energy difference between the normally occupied ground state and the nearest excited states is generally
outside that equivalent to the visible spectrum. The main exceptions to this rule are the enigmatic
transition metal ions, which are often described as ‘coloured’. The most important transition metals from
the point of view of colour are the 3d transition metals, listed in Appendix A7.1. As an example of these
colours, Figure 7.13 shows aqueous solutions of green Ni(H2O)62þ and blue Cu(H2O)6
2þ and crystalline
examples of green nickel nitrate Ni(NO3)2�6H2O and blue copper nitrate Cu(NO3)2�3H2O. In both
solutions and crystals, six water molecules are arranged so that the oxygen atoms form an octahedral
coordination polyhedron around a central cation. The colours can be quantified by recording the absorption
spectra of the solutions (Figure 7.14a and b). These show that the nickel-containing solution absorbs in
both the violet and red regions of the spectrum, whereas the copper-containing solution absorbs only in the
yellow to red region.
4
5
6
7
8
9
6s ns:1 level
6s np:3 levels
6s np:1 level
6s nd:1 level
6s nd:3 levels
6s 6d 6s 6d
6s 7d6s 7d
6s 7s
10
40,000
30,000
50,000
60,000
70,000
80,000
Ene
rgy
/ eV E
nergy / cm–1
546.1
435.8
433 9
577.0
434.7
579.1
404.7
3S1
3P23P13P0
3P2
3P13P0
1P1
1D2
1D23D1,2,3
3D1,2,31D2
3P 1P 3D
6s 6p
6s 7p
6s 6p
6s 7p
Figure 7.12 Partial Grotrian diagram for mercury atoms. The transitions giving rise to colours are drawn asdownward-pointing arrows. Thewavelengths emitted are given besides each transition. Themost intense line is at435.8 nm in the blue region of the spectrum. (Conversion of the units eV and cm�1 to the SI unit joule (J) is given inAppendix A1.1)
Colour and the Optical Properties of Materials 264
In these and the other 3d transition metal ions the five 3d orbitals contain one or more electrons and electron
transitions between the various d orbitals are associated with the colours observed. However, a glance at the
free-ion terms shows that these do not provide an explanation no suitable energy intervals exist close to the
ground state.Thus, the introductionof these cations into solids or liquidsmust change theenergy levels in sucha
way that transitions that give rise to colours become possible. Theway in which this comes about involves the
shapes of the d orbitals, which can be described as pointing along or between a set of x-, y- and z-axes
Figure 7.13 Crystal-field colours of transition metal ions: (a) green Ni2þ (H2O)6 and (b) Cu2þ (H2O)6, both inwater solution; (c) green nickel nitrate Ni(NO3)2�6H2O and (d) blue copper nitrate Cu(NO3)2�3H2O crystals.The transition metal ions are in similar environments in both solution and crystals
265 Colour from Atoms and Ions
(Figure7.15).Theorbitals directedbetween theaxesare thedxy, dyzanddxz set and thosepointingalong theaxes
are the dx2 y2 and dz2 pair.
In a free ion or atom, all of these orbitals have the same energy. However, this is not true when the atom
or ion is placed into a crystal because of the interaction (most easily imagined as repulsion) between the
electrons on the surrounding atoms and the d orbitals. If these surrounding electrons were distributed
evenly over the surface of a sphere the five d orbitals would still have the same energy as each other,
although higher than in the isolated state by an amount E0. If the surrounding electrons are arranged
differently, the energy of some of the d orbitals might be different than the others, so that the energies of
the orbitals become split (Figure 7.16). This is called crystal-field splitting or ligand-field splitting.3 The
extent of the splitting depends upon the symmetry of the surrounding ions and the strength of the local
crystal field.
The two most important geometries to consider, especially for oxide pigments and ceramics, are
octahedral and tetrahedral coordination (Figure 7.17). When an ion is surrounded by an octahedron of
400
400
500
500
600
600
700
700
5
5
10
10
Wavelength / nm
Wavelength / nm
Abs
orpt
ion
(arb
itrar
y un
its)
Abs
orpt
ion
(arb
itrar
y un
its)
Ni(H2O)62+
Cu(H2O)62+
(a)
(b)
Figure 7.14 The absorption spectra of aqueous solutions containing (a) Ni(H2O)62þ and (b) Cu(H2O)6
2þ . Theabsorption scales are arbitrary
3 The difference between these two labels reflects the method of calculation of the splitting. If the material is treated as ionic and the
surrounding charges represented as points (the simplest model), the expression crystal field splitting is appropriate, whereas if molecular
orbital theory is used, ligand field splitting is utilized. The terms are usually employed interchangeably and for convenience only the
expression crystal field theory will be adopted here.
Colour and the Optical Properties of Materials 266
negative O2 ions the d orbitals pointing directly towards the oxygen ions, dx2 y2 and dz2 (the eg pair), will be
strongly repelled and so raised in energy compared with those pointing between the oxygen ions, the dxy, dxzand dyz (the t2g group)
4 (Figure 7.18). The labels e and t refer to the degeneracy of the group. A set of orbitals
labelled ‘e’ is doubly degenerate; that is, two orbitals with the same energy form the e set. In the same way,
y x
x x
z y
y z
dxydxz
dyz dx 2-y 2
x
z
dz2
Figure 7.15 The shapes of the five d orbitals superimposed upon a set of orthogonal axes. The lobes of electrondensity in the group dxy, dxz and dyz lie between the axes. The lobes of the pair of orbitals dx2�y2 and dx2 lie along theaxes
4 The subscript g relates to the symmetry of the atomic or molecular orbitals under discussion. As far as this book is concerned, the
subscript g is added when the cation is situated at a centre of an octahedron of surrounding anions, whereas it is omitted when the cation is
situated at the centre of a tetrahedron of surrounding anions.
267 Colour from Atoms and Ions
free iond orbitals
sphericallysymmetricalfield
crystal fieldsplitting in a field ofrhombic symmetry
E0
(5)
(1)
(1)
(1)(1)(1)
Figure7.16 Schematic crystal-field splittingof theenergiesof thefivedorbitals. Ina free ion theenergiesof eachofthe five orbitals are degenerate (i.e. equal). In a crystal field of spherical symmetry the energies remain degeneratebut are increased over that in the free ion by an amount E0. In a field of lower symmetry, the energies of the orbitalssplit. That shown is the splitting in a field of rhombic symmetry, which totally removes the degeneracy of the set
(a)
(b)
x
y
z
x
y
z
Figure 7.17 Acation(largesphere)surroundedby: (a) six oxygen anions (small spheres) arranged as an octahedralanion coordination polyhedron; (b) four oxygen anions arranged as a tetrahedral anion coordination polyhedron.The cation-centred cubic outline indicates that in (a) the anions are located at the cube face centres and in (b) atcube vertices, so that the cation–anion distance is greater in (b) than in (a), leading to a smaller crystal field
Colour and the Optical Properties of Materials 268
the label ‘t’ indicates a triply degenerate set, in which case three orbitals with the same energy comprise
the group. (Although not relevant to the present situation, it should be mentioned that orbital groups
designated ‘a’ consist of a single orbital only.) The crystal-field splitting generates an energy gap between
the lower t2g group of orbitals and the upper eg group which is written D or 10 Dq. The distribution of
the two sets is unsymmetrical about E0, with the t2g group at �4 Dq¼�2/5D and the eg group at
þ 6 Dq¼ þ 4/5D.When a transition metal ion is surrounded by a tetrahedron of oxygen ions the crystal-field splitting is
reversed. In this case thedxy, dxz anddyzorbitals (the t2 group) are raised in energy relative todx2 y2 anddz2 (the e
pair). The distribution of the two sets is again unsymmetrical about the energy level for a spherically
symmetrical distribution of chargeE0with the t2 set at þ 4Dq¼ þ 2/5D and the e set being at�6Dq¼�4/5D(Figure 7.18). The magnitude of the splitting for ions in a tetrahedron will be less than that for ions in an
octahedron, and calculations give the result that the tetrahedral crystal field splitting is 4/9 of the octahedral
splitting.
Note that lower case symbols (t2g, etc.) are used to describe these crystal-field orbitals, as they ignore all
electron electron interactions and so behave as ‘one-electron’ states.
The colour of a transition metal ion is then supposed to be due to d electrons moving across the relatively
small energy gap created by the crystal-field splitting. The magnitude of the crystal-field splitting will depend
on the geometry of the surrounding ions and how close they are to the cation. In a strong crystal field, produced
when the surrounding anions are close to the cation, the crystal-field splitting is large. This means that the
transition energywill be large and any absorption peakwill be in theviolet or ultraviolet regionof the spectrum.
In a weak crystal field, produced when the surrounding anions are further away from the cation, the splitting is
t2(3 levels)
eg (2 levels)
t2g(3 levels)
e (2 levels)
tetrahedral spherical octahedral
Δ(tet) = (4/9) Δ(oct)
10 Dq =Δ(oct)
6Dq(oct) = 3/5 Δ(oct)
4Dq(oct) = 2/5 Δ(oct)
10Dq= Δ(tet)
4Dq(tet)= 2/5 Δ(tet)
6Dq(tet)= 3/5 Δ(tet)
E0E0
Figure 7.18 Crystal-field splitting in a field of cubic symmetry. In an octahedral field the t2g set has a lowerenergy and the eg set a higher energy, while in a tetrahedral field the situation is reversed. The separation ofthe upper and lower energy levels, the crystal-field splitting, isD(oct) in the case of an octahedral crystal field and(4/9)D(oct) for a tetrahedral crystal field
269 Colour from Atoms and Ions
smaller and any energy peakwill be in the red or infrared. This variation accounts for the fact that anyparticular
transitionmetal cationmay exhibit different colours in different compounds, as explained for ruby and emerald
below.
This neat solution to the question of colour in transition metal ions runs into a problem when the selection
rules that apply to transitions are consulted (Section 7.3). It is clearly stated that transitions of the type d to d
were forbidden by the Laporte selection rule. However, this rule breaks down for atoms or ions in compounds.
Themain reason for this iswhenan ion isnot located at a centre of symmetryadegreeofmixingbetweenvarious
orbitals, such as p and d orbitals, can occur.As p to d transitions are allowed, the transitions giving rise to colour
are also allowed, to a degree corresponding to the amount of orbital mixing achieved. Thus, ions situated in
tetrahedral coordination are not at a centre of symmetry and show quite strong colours. Ions at the centre of a
perfect octahedron are at a centre of symmetry, but, in most solids and liquids, thermal agitation of the
surroundings and crystal distortions remove the precise symmetry,making transitions possible, though they are
often less intense than those from similar ions in tetrahedral sites.
Another selection rule is also important. In a free ion, transitions are only allowed between states of the same
multiplicity. This rule does not change as an ion is introduced into a crystal, which means that allowed
transitions are between stateswith the samemultiplicity. These are called spin-allowed transitions. Transitions
between states of differing multiplicity can be weakly allowed, but in general these do not give rise to strong
colours.
The strength of the crystal field interaction is dependent upon the distance between the surrounding ligands
and the central ion. This will vary with temperature, and so the perceived colour of the material will change
as the temperature changes, an example of thermochromism. This effect is usually too small to notice over
the temperature ranges occurring close to room temperature, but in oxides used as pigments, or in oxide
gemstones, the colour at a temperature of several hundred degrees may be quite different to that at lower
temperatures. Thermochromism can also come about if temperature causes a change in the geometry of the
surrounding ligands. For example, a low-temperature distorted octahedron may transform to a regular
octahedral form as the temperature increases and a thermochromic colour change will be registered. Other
transformations, from tetrahedral to square planar, for instance, can have a similar thermochromic effect.
Having described the source of the colour of transition metal ions, as in the case of free atoms and ions, it is
now necessary to construct energy-level diagrams and explain theway in which these vary with the strength of
the crystal field in order to explain spectra correctly.
7.9 Crystal Field Splitting, Energy Levels and Terms
7.9.1 Configurations and strong field energy levels
The simplestway to get an idea of the energy levels available to a transitionmetal ion is towork out the electron
configurations assuming that there are no electron electron interactions. This is called the strong field
approach, and basically assumes that the crystal field is so strong that it dominates all other interactions, such as
electron electron repulsion. Electrons are then simply allocated to the split orbitals, filling the lower energy
orbitals first.
Takingoctahedral geometry as an example, suppose that the ionhasnd electrons in total, ofwhich (n� p) are
in the lower t2g set and p are in the upper eg set. The energy of the ion E (with respect to the spherically
symmetrical charge distribution E0) is computed by noting that the energy of the t2g orbitals set is�4Dq andthat of the eg set is þ 6Dq. The energy of an ion is then:
E ¼ ½ðn�pÞð�4Þþ ð pÞðþ 6Þ�Dq
Colour and the Optical Properties of Materials 270
To illustrate this, the energy levels of a 3d3 ion in an octahedral site are obtained in the following way. The
lowest energy for the ion will be when all three d electrons are in the lowest energy levels, so that this
configuration can be written (t2g)3 (eg)
0, and the energy is �12Dq. The next lowest energy for the ion will
correspond to twoelectrons in the t2g orbitals andone in the egpair, (t2g)2 (eg)
1, giving anenergy�2Dq.Anotherenergy levelwill correspond to the electron distribution, (t2g)
1 (eg)2, lying at þ 8Dq.Finally, the highest energy
will correspond to all electrons in the upper level, (t2g)0 (eg)
3, at þ 18Dq. The 3d3 configuration will then give
rise to four strong-field energy levels.
Energy levels for other d-electron populations or for ions in a tetrahedral crystal field can be calculated in a
similar fashion.
On the basis of thismodel, the spectrumof a transitionmetal ion in an octahedral crystal fieldwill consist of a
set of peaks, with energies given by multiples of 10Dq. The spectrum of Ni(H2O)62þ (Figure 7.14a) does not
agree with this simple prediction. The three peaks in the spectrum are at wavelengths of 395 nm, 680 nm
(centre) and 1176 nm. These are at energies of 3.14 eV (5.03� 10 19 J), 1.82 eV (2.92� 10 19 J) and 1.05 eV
(1.69� 10 19 J). Taking 10Dq as 1.05 eV, although the highest of the energy values (3.14 eV) is close to
3� 1.05 eV, the middle value does not fit at all. This indicates that the simple model needs refinement. If some
degree of electron electron interaction is introduced (i.e. the crystal field becomes weaker), then these levels
will split, just as a configuration such as 3d2 splits into a number of terms of varying energy when electron
electron repulsion is considered (Figure 7.5).
7.9.2 Weak fields and term splitting
At the other extreme, it is possible to assume that the crystal field is very weak. Electron electron repulsion
is then most important. In fact, in the case of a zero-strength crystal field, the dominant set of energy levels
are described by term symbols (Section 7.2). As the strength of the crystal field increases, this will modify
the energies represented by the terms in a predictable fashion, first set out by Bethe in 1929. The result is
that for ions in a field of cubic symmetry, which is applicable to octahedral and tetrahedral crystal fields, S
and P terms remain as single energy states, a D term splits into two energy states, an F term into three and a
G term into four (Table 7.7). The multiplicity of the term is carried over onto the new states. The energy
states in the crystal are given labels that parallel those for single-electron orbitals mentioned in the previous
section and describe the degeneracy of the orbitals. Thus, an S term, which is singly degenerate, is labelled
A in a crystal field, a P term, which is triply degenerate, is labelled T, meaning triply degenerate, a D term
splits into two, a doubly degenerate E term and a triply degenerate T term, and so on. Different
configurations of sets of orbitals with the same degeneracy are labelled with a subscript 1, 2 and so
on. Thus, a P term gives rise to a T1 term, while a D term gives rise to a T2 term. The subscript ‘g’ means a
Table 7.7 Splitting of terms in fields of cubic symmetry
Free ion termTerms in tetrahedral
crystal fieldTerms in octahedral
crystal field
S A1 A1g
P T1 T1gD E, T2 Eg, T2gF A2, T1, T2 A2g, T1g, T2gG A1, E, T1, T2 A1g, Eg, T1g, T2g
271 Colour from Atoms and Ions
centre of orbital symmetry exists, as described in the previous section. (Further information is given in the
Further Reading.)
The splitting of the ground-state terms of the 3d transition metal ions in an octahedral crystal field can
now be drawn (Figure 7.19a). It is seen that the arrangement of the new energy states is symmetrical about
the d5 6S term. This comes about because orbitals with n electrons can be treated as mirrors of those
containing n holes. Thus, the term splitting of a d1 ion (an empty shell plus one electron) is equivalent to that
of a d9 ion because the latter configuration can be regarded as equivalent to a filled shell plus one hole.
However, the order of the states is reversed. The energies of the states can also be calculated for a weak
crystal field. These give results similar to those in a strong field, but with significant differences for some
terms (Table 7.8).
It is also found that if the order of the split terms is inverted then the arrangement appropriate to ions in a
tetrahedral field is obtained (Figure 7.19b). Thus, if an F state for a d2 ion splits in an octahedral field into A2g,
T2g and T1g terms (in decending order), then in a tetrahedral field it will split into T1, T2 and A2 terms (in
decending order). The energy of the splitting will remain the same, in units of Dq, but the magnitude of the
3A2g
4A2g
6A1g
4A2g
3A2g
2Eg
5D
5Eg
5Eg
2Eg
3F
3F
4F
4F
4T1g
3T1g
4T2g
6S
2T2g
5T2g
4T1g
3T2g
4T2g3T2g
3T1g
5T2g
2T2g
d 6
d 1 d 2 d 5d 4d 3
d 7 d 8 d 9
2D
5D 2D
(a)
Figure 7.19 Splitting of the ground-state terms of d1 to d9 ions in a field of cubic symmetry (schematic):(a) octahedral field; (b) tetrahedral field. The magnitude of the splitting in an octahedral field will beapproximately double that in a tetrahedral field
Colour and the Optical Properties of Materials 272
splittingwill be smaller for tetrahedral fields comparedwithoctahedralfieldsbecauseof thedifferencebetween
Dq(octahedral) and Dq(tetrahedral).
7.9.3 Intermediate fields
The separation of each of the split levels described in the previous section will increase as the magnitude of
the crystal field increases. Ultimately these must link up with the energy levels described in the strong-field
case. In most real crystals the energy levels are then somewhere between the weak- and strong-field extremes.
The connection can be described schematically as:
dn! 2Sþ 1L! 2Sþ 1A; E; T t2gn peg
p dn
With this information it is now possible to interpret the colour of green Ni(H2O)62þ and blue Cu(H2O)6
2þ .Take the copper casefirst. The zero-fieldground state of the d9Cu2þ ion is 2D.When the ion is introduced into a
crystal field, the energy of the 2D term will be greater than the equivalent energy of an ion in a vacuum by the
interaction energy E0 (Figure 7.16) but will still remain as a precise 2D term if the crystal-field energy is set at
d 6
d 1 d 2 d 5d 4d 3
d 7 d 8 d 9
3A2
4A1
6A1
4A2
3A2
2E
5D
5E
5E
2E
3F
3F
4F
4F
4T1
3T1
4T2
6S
2T2
5T2 4T1
3T2
4T23T2
3T1
5T2
2T2
2D
5D 2D
(b)
Figure 7.19 (Continued)
273 Colour from Atoms and Ions
zero. As the crystal field increases, the 2D termwill divide into 2Eg and2T2g levels in the octahedral field of the
surrounding water molecules; more precisely, that of the surrounding oxygen ions. The divergence is a linear
function of Dq, with slopes of �6Dq (2Eg) and þ 4Dq (2T2g) (Figure 7.20). This representation is called an
Orgel diagram. At very strong fields the ground-state 2Eg level is associated with the ‘one-electron’
configuration t2g6 eg
3 obtained by placing six of the nine d electrons into the lowest energy t2g orbitals first
and then allocating the remaining three to the eg levels. In the excited state, one electron is promoted from the
lower to the upper set to give the configuration t2g5 eg
4 associated with the 2T2g state. This indicates that the
spectrum should contain one absorption peak. The maximum is found to occur at 780 nm, in the near infrared,
corresponding to the transition5 between the upper and lower energy level, 2T2g 2Eg (Figure 7.21). The
energy separation should be equal to 10Dq,with avalue of 1.6 eV (2.55� 10 19 J).However, aword of caution
is needed. The peak itself is very broad and suggests that it is necessary to take into account interactions not
included so far. (In fact, it is found that the coordination polyhedron is not a regular octahedron, but is distorted
so that the symmetry changes to tetragonal, which accounts for this feature.)
The colour of Ni(H2O)62þ can be explained in an analogous fashion. The zero-field ground state of the d8
Ni2þ ion is 3F. As the crystal field increases, the 3F term divides into 3A2g,3T2g and
3T1g in the octahedral
field of the surrounding water molecules. These should show a linear dependence upon Dq, with slopes of
Table 7.8 Energies of terms in an octahedral crystal field
Free ion electronconfiguration
Free ionground state term
Term in anoctahedral field Energy/Dq
d1 2D 2T2g 42Eg þ 6
d2 3F 3T1g 63T2g þ 23A2g þ 12
d3 4F 4A2g 124T2g 24T1g þ 6
d4 5D 5Eg 65T2g þ 4
d5 6S 6A1g 0d6 5D 5T2g 4
5Eg þ 6d7 4F 4T1g 6
4T2g þ 24A2g þ 12
d8 3F 3A2g 123T2g 23T1g þ 6
d9 2D 2Eg 62T2g þ 4
5 A spectroscopic transition is conventionally written with the higher energy state (H) first and the lower energy state (L) second. This
means that an absorption of energy is written H L, and an emission of energy is written H ! L. This is adopted when the equations
describing the transition are not written in a ‘chemical equation’ format.
Colour and the Optical Properties of Materials 274
Dq
2T2g
2D
2Eg
10 Dq
Ene
rgy
Figure 7.20 Orgel diagram (schematic) for a d9 ion such asCu2þ in an octahedral field. The separation of the twoenergy levels is equal to 10Dq
780 nm: infrared absorption
0
1
2
3
3T2g
2Eg
blue-green transmission
Ene
rgy
/ eV
Absorption
400 nm: violet
700 nm: red
Figure 7.21 The electronic transition responsible for the absorption spectrumofCu2þ ions inwater solution andblue hydrate crystals, Cu2þ (H2O)6
275 Colour from Atoms and Ions
�12 (3A2g),�2 (3T2g) and þ 6 (3T1g). However, the energy-level diagram for the free ion shows that the terms1D, 3P and 1G are quite close in energy to the 3F term, and thesemust also be considered. As optical transitions
are only expected between the triplet terms, 1D and 1G can be omitted in the first instance. (Note that this is not
always so. The operation of the ruby laser depends upon a spin-disallowed transition; Section 7.11.) However,
the 3P term cannot be ignored and transforms into a 3T1g term in the octahedral crystal fieldwith the slope of the
linear dependence equal to zero. This means that, at moderate field strengths, the upper 3T1g level from the 3F
termwill cross it (Figure 7.22a). Now the non-crossing rule of quantummechanics states that two energy states
with the same symmetry arising from a single ion never cross. Thus, the Orgel diagram for a d8 ion shows these
two straight lines becoming curved, as if repelling each other (Figure 7.22b).
Dq
Dq
Ene
rgy
Ene
rgy
8 Dq
8 Dq - interaction energy
10 Dq
10 Dq
3T2g
3T2g
3T1g
3T1g
3P
3F
3P
3F
3T1g
3T1g
3A2g
3A2g
(a)
(b)
Figure 7.22 Orgel diagram (schematic) for a d8 ion such as Ni2þ in an octahedral field: (a) without terminteractions between the two 3T1g levels (crossing allowed); (b) with interaction between the two 3T1g levels(non-crossing rule applied). The separation of the two lower energy levels ( 3A1g and
3T2g ) is equal to 10Dq inbothcases
Colour and the Optical Properties of Materials 276
The spectrum should contain three peaks that correspond to the transitions and energies
1. 3T2g 3A2g 1176 nm (1.05 eV, 1.69� 10 19 J, 8503 cm 1)¼ 10Dq
2. 3T1g (from3F) 3A2g 680 nm (1.82 eV, 2.92� 10 19 J, 14 706 cm 1)¼ 18Dq interaction energy
3. 3T1g (from3P) 3A2g 395 nm (3.14 eV, 5.03� 10 19 J, 25 316 cm 1).
Only two of these are close to the visible, one at each end of the spectrum, in good agreement with the
absorption data.
The energy separation for the lowest energy transition, in the infrared, is seen to be equal to 10Dq (¼D) forthis chemical environment. The energy separation for the next lowest transition ideally corresponds to 18Dq.
However, the non-crossing rule decreases this by an amount of energy called the interaction energy or
configuration interaction energy. The third absorption peak is not directly related, in this simple model, to the
crystal-field splitting, Dq, and so provides no new information on this parameter. The colour of Ni(H2O)62þ is
due to the two transitions:
3T1g (from3F) 3A2g 680 nm
3T1g (from3P) 3A2g 395 nm
The absorption of energy in the near ultraviolet and in the far red remove blue and red from the transmission
spectrum, resulting in the green colour perceived (Figure 7.23).
However, it is obvious that the central peak of the spectrum is split into two components, again indicating that
a further refinement of the simple theory so far presented is needed. (In fact, it is necessary to take the coupling
between the spins and the orbital angular momentum, or spin orbit coupling, into account for this.)
7.10 The Colour of Ruby
Ruby consists of single crystals of aluminium oxide (a-Al2O3) containing about 0.5 % Cr as an impurity. The
Cr3þ impurity ions in ruby are distributed at random over some of the positions normally reserved for Al3þ in
the oxide structure. The formula of the gemstone can be written (CrxAl1 x)2O3. When x¼ 0 (pure Al2O3) the
stone is colourless and found as the mineral corundum. At very small values of x close to 0.005 the crystal is
coloured a rich ruby red. As the Cr3þ concentration increases, so the colour becomes grey and then dull green,
which is the colour of pure Cr2O3, used as the pigment chrome green.
The fact that ruby is coloured while Al2O3 is colourless indicates that it is the Cr3þ in the structure that is of
paramount importance. The colour changes can be explained in terms of crystal-field splitting of the energy
levels of the Cr3þ ion. The oxides Al2O3 and Cr2O3 are isostructural and the positions occupied by Cr3þ all
across the composition range are at the centres of slightly distorted octahedral sites formed by six nearest
neighbour oxygen ions.
The outer electron configuration of the Cr3þ ion is 3d3 and the ground-state term is 4F. This splits in an
octahedral crystal field to give the states 4A2g at�12Dq, 4T2g at�2Dq and 4T1g at þ 6Dq. A higher 4P state
gives rise to a further 4T1g level. Apart from the multiplicity, the diagram is similar to that for the d8 ion Ni2þ
(Figure 7.22) and three absorption bands are expected. These are
1. 4T2g 4A2g 556 nm (2.23 eV, 3.58� 10 19 J, 18 000 cm 1)¼ 10Dq
2. 4T1g (from4F) 4A2g 400 nm (3.10 eV, 4.97� 10 19 J, 25 000 cm 1)¼ 18Dq interaction energy
3. 4T1g (from4P) 4A2g 270 nm (4.59 eV, 7.35� 10 19 J, 37 000 cm 1).
277 Colour from Atoms and Ions
The first two of these transitions contribute to the colour of the gemstone. The absorption at 556 nm removes
green yellow and the absorption at 400 nm removes violet. Between the absorption curves there is a relatively
small blue transmission window at 680 nm and a red transmission window is present at wavelengths greater
than about 650 nm. This means that the colour transmitted by the ruby will be red with something of a blue
purple undertone.
The colour of ruby, however, is richer than this explanation suggests, andmore detail needs to be added. Two
additional factors need to be taken into account: spin-forbidden transitions and the crystal symmetry. The
transitions that give rise to colour, such as those detailed above, are forbidden in terms of parity but are spin
allowed because themultiplicity of all the terms is identical and equal to four. The free ion terms of Cr3þ show
that a 2G term is found only slightly higher in energy than the 4P term (Figure 7.24). AG term splits into four in
an octahedral crystal field (Table 7.7) and in ruby two of these new levels fall within the spread of the levels
derived from the 4F ground state (Figure 7.24).
Themost important of these, from the point of viewof colour, is the lowest energy 2Eg level.Direct excitation
from the ground-state 4A2g to the 2Eg level is forbidden by both the parity and multiplicity selection rules.
However, a different circumstance operates for the excited states. At the same time as excitation is occurring,
395 nm: violet absorption
680 nm: red absorption
0
1
2
33T1g
3T1g
3T2g
3A2g
green transmission
Ene
rgy
/ eV
Absorption
Figure 7.23 The electronic transitions responsible for the absorption spectrum of Ni2þ ions in water solution,Ni2þ (H2O)6
Colour and the Optical Properties of Materials 278
manyof thehigher energyCr3þ ions return to theground state byemitting exactly the sameamount of energyas
was absorbed, so as to drop back to the ground state from either 4T1g or4T2g. However, as these transitions are
forbidden by the parity selection rule, they are not fast. Some ions lose energy instead to the crystal structure,
warming it slightly, dropping back only to the 2E energy level. This is not an optical transition, but involves heat
energy, phonon exchange, and so is not bound by the selection rules given earlier for optical transitions. It is
described as a radiationless or phonon-assisted transition. (However, the selection rules make it clear that a
direct (optical) transition from the ground state to the 2E state by absorbing energy is low, and so 2E only
becomes filled by this roundabout process.) The ions in the 2E state also slowly lose energy and return to the
ground state. This transition gives rise to red light emission, which features as a narrow band, called the
intercombination band J1 orR, close to 693 nm, in the ruby spectrum.This fluorescent radiation (seeChapter 9)
enhances the colour of the best rubies. At compositions close to Cr0.005Al0.995O3 it can be made to dominate
light emission, and the result is laser action (Section 7.11).
The second feature that adds to the colour of ruby stems from the symmetry of ruby crystals. The crystal
structure of ruby is trigonal (but usually referred to hexagonal axes), and aswith all crystals of symmetry lower
than cubic, the absorption spectrum depends upon the polarisation of the light used for the illumination. Ruby
crystals are dichroic (Section 4.8). In ruby, two absorption spectra arise: one for light polarised parallel to the
crystallographic c-axis and one for light polarised perpendicular to the c-axis (Figure 7.25). Although these
spectra are very similar to each other, noticeable differences in colour are apparent when ruby crystals in
4T1g
4T1g
4T2g
2T2g
2T1g2Eg
2A1g
2A2g
4F
4P
2G
Free ionterm
Octahedral crystalfield
Figure 7.24 Energy levels of the 3d3 ion Cr3þ due to splitting of free ion terms 4F,4P and 2G in an octahedralcrystal field (schematic)
279 Colour from Atoms and Ions
differing orientations are observed in polarised light. In ruby, when the plane of polarisation of the light is
perpendicular to the c-axis the crystal is perceived as ruby coloured, but when it lies parallel to the c-axis the
crystal takes on a more orange hue.
In terms of crystal-field theory, the excited 4T2g and 4T1g (from 4F) are split due to the change of local
symmetry from cubic (in an ideal octahedron) to trigonal (point group C3v) in the crystal (Figure 7.25) as
follows:
4T2g! 4A2 and4E
4T1g (from4F) ! 4A1 and
4E
The selection rules nowmean that light polarised parallel to the c-axis (the e-ray) interacts with only the 4A
levels and light polarised perpendicular to the c-axis (the o-ray) with the 4E levels. The separation of the new
0
1
2
3
4
Ene
rgy
/ eV
polarisation parallel to c-axis(e-ray): orange-red colour
polarisation perpendicular toc-axis (o-ray): purple-red colour
Absorption
yellow/green/orangeabsorption ~ 556 nm
violetabsorption~ 400 nm
deep red fluorescence
4T1g
4T2g
2Eg
4A1
4A2
3A2g
4E
4E
Figure 7.25 The electronic transitions that give rise to dichroism in ruby. The new energy levels are produced bythe decrease in symmetry from octahedral (Figure 7.24) to rhombic in crystals, which causes a splitting of the 4Tlevels. Light polarised perpendicular to the optic axis (the o-ray) gives the gemstone a purple–red colour, whilelight polarised parallel to the optic axis (the e-ray) gives an orange–red colour. The 2Eg level also splits, but thescale of the diagram is too small to show this
Colour and the Optical Properties of Materials 280
levels is small, of the order of 0.06 eV (500 cm 1), but does give a colour change that is readily detected by eye.
When illuminated by light polarised perpendicular to the c-axis (the o-ray) the ruby is purple red (ruby)
coloured, while when illuminated by light polarised parallel to the c-axis (the e-ray), the colour is perceived as
orange red.
In addition to these effects, the 2E level giving rise to the intercombination band J1 (R) is also found to be split
into two components due to the same symmetry change, so that the red line at 693 nm is resolved into two
narrow lines, R1 at 693.5 nm and R2 at 692.3 nm (not differentiated at the scale of Figure 7.25).
7.11 Transition-Metal-Ion Lasers
7.11.1 The ruby laser: a three-level laser
The first laser constructed was the ruby laser, built by Maiman in 1960. It consisted of a ruby crystal
(�Cr0.005Al1.995O3) about 7 cm long. One end facewas silvered to give total reflection while the other end was
partially silvered so as to release any stimulated emission. The population inversion was created by a bright
flash of light from a xenon flash tube. The whole was surrounded by a reflecting shield (Figure 7.26).
Laser operation makes use of the electronic energy levels of Cr3þ in a crystal field described above. The
electron transitions which lead to colour in rubies are due to the transitions
1. 4T2g 4A2g 556 nm, absorbs yellow green
2. 4T1g 4A2g 400 nm, absorbs violet.
These transitions are forbidden in terms of parity but are spin allowed. In addition, account must be taken of
the 2E term.Direct excitation from the ground-state 4A2g term to the 2E term is forbidden by both the parity and
multiplicity selection rules.As described in the previous section, this level is filled by a roundabout process that
involves the shedding of energy to the crystal structure as a radiationless transition. The rates of the transitions
between these states are
1. 4T2g! 4A2g: 3� 105 s 1
2. 4T2g! 2E: 2� 107 s 1.
The secondof these two transitions is about 100 times faster than thefirst. The rates of the transitions from the4T1g energy level to
2E and 4A2g are of a similar magnitude. This means that, on irradiating the rubywith white
light, a significant number of atoms end up in the 2E state. For the samequantummechanical reasons that forbid
reflector
Xenon flash tube
ruby crystallaser output
Figure 7.26 The original ruby laser (Maiman, 1960) (schematic)
281 Colour from Atoms and Ions
the direct transition from theground state to the 2E state, the transition from the 2E level back to theground state
is also forbidden, and so atoms in the 2E state have a long lifetime. Thus, it is possible to build a population
inversion between the 2E and 4A2g levels.
Laser operation takes place in the following way. An intense flash of white light is directed onto the crystal.
This process is called optical pumping. This excites theCr3þ ions into the 4T2g and4T1g states. These then lose
energy by radiationless transitions and ‘flow over’ into the energy level 2E. If the initial flash is intense enough
itwill cause apopulation inversion between 2Eand 4A2g.About 0.5ms after the start of the pumpingflash, some
spontaneous emissionwill occur from 2E. In order to prevent these first photons from escaping from the crystal
without causing stimulated emission from the other excited ions, one end is coated with a mirror and the other
with apartly reflectingmirror. In this case the photons are reflected to and fro, causing stimulated emission from
the other populated 2E levels. Once started, the stimulated emission rapidly depopulates these levels in an
avalanche. There will be a burst of red laser light of wavelength 694.3 nm which emerges from the partly
reflecting surface. In the original laser, silver mirrors were used. However, silver absorbs as much light as it
transmits. This resulted in overheating, which caused crystals to deformor crack.Now themirrors are thin-film
dielectric mirrors, one of which transmits about 1% of the incident photons.
Following the light burst, the upper levels will be empty and the process can be repeated. The ruby laser
generally operatesbyemittingenergy in short bursts, eachofwhich lasts about 1ms.This is referred toaspulsed
operation.
The ruby laser is called a three-level laser, becausebasically three energy levels are involved in theoperation.
These are the ground state (4A2g), an excited state reached by optical absorption or pumping (4T2g or4T1g) and
an intermediate state of long lifetime (2E) reached by radiationless transfer from the optically accessible state
and from which stimulated emission (laser emission) occurs to the ground state. It is energetically costly to
obtain a population inversion in a three-level laser because onemust pumpmore than half the population of the
ground state to themiddle level.Moreover, very little of the electrical energy supplied to the flash lamp ends up
pumping photons, and carefully designed reflectors are essential. Finally, the energy lost in the transitions from4T2g and
4T1g to2E ends up as lattice vibrations, which cause the crystal to heat up considerably. Tomake sure
that the ruby does not overheat and shatter, it is necessary to cool the crystal and to space the pulses to allow the
heat to dissipate.
7.11.2 The titanium–sapphire laser
The titanium sapphire laser is, at present, the laser chosen for the generation of femtosecond pulses.
Chemically, it is very similar to the ruby laser, in that the laser medium is a crystal of corundum doped
with a small amount of Ti3þ . This ion has a 3d1 configuration and the impurities are located in octahedral sites
in the corundum structure, totally analogous to Cr3þ in ruby. In an octahedral crystal field the 2D free-ion state
splits into a 2T2g ground state and a2Eg excited state (Figure 7.27). The absorption spectrum of Ti3þ ions in
corundum should then consist of a single maximum corresponding to the transition 2Eg 2T2g and the
emission spectrum, due to the transition 2Eg! 2T2g, should be similar. This transition is generally regarded as
the operative laser transition.
In reality, the situation is slightlymore complex.The symmetry of theTi3þ sites in corundum is not perfectly
octahedral, causing both the ground-state 2T2g energy level and the upper2Eg level to split (Figure 7.27). This
results in the absorption spectrum showing two overlapping peaks, at wavelengths of approximately 475 and
550 nm. The laser transition, from the lower of these levels to the ground-state levels is at a wavelength of
approximately 800 nm. (Note, however, that the transition is still labelled as 2Eg! 2T2g in most laser
literature.)
These are normal crystal-field-generated energy levels, and the excited and ground states have the same
multiplicity. This means that the lifetime of the excited state is short (approximately 3.2ms) and spontaneous
Colour and the Optical Properties of Materials 282
emission is the expectedmechanismof energy loss. To obtain a population inversion it is necessary to pump the
crystal with an intense laser beam.Dye lasers (Section 8.12), frequency-doubledNd:YAG lasers (Section 7.16)
and other lasers have been used for this excitation.
7.12 Emerald, Alexandrite and Crystal-Field Strength
The effect of the strength of the crystal field on colour is demonstrated by comparing ‘ruby-red’ ruby with the
gemstone emerald, with a characteristic ‘emerald green’ hue. Emeralds possess the hexagonal beryl
(Be3Al2Si6O18) structure, which, when pure, is a colourless mineral. The structural framework is composed
of Si6O18 rings forming tunnels parallel to the c-axis linked by Be-centred oxygen tetrahedra and Al-centred
octahedra. As with ruby, a trace of Cr3þ substitutes for some Al3þ . The source of colour in both gemstones is
thus Cr3þ in octahedral sites. The energy-level diagram for ruby is therefore relevant (Figure 7.24). However,
in beryl the octahedra surrounding the Cr3þ ions are slightly larger than in corundum and so the crystal field
experienced by the Cr3þ in emerald is weaker than in ruby. The energy-level diagram remains essentially the
same, but there is a shift in the energy levels 4T1g and4T2g towards the ground-state level
4A2g. This causes the
two main absorption bands to move towards the lower energy red end of the spectrum. The band that absorbs
yellow/green in ruby (�556 nm) now absorbs yellow/red with a peak at 650 nm. The violet-absorbing band in
ruby (400 nm) now absorbs more blue, with a peak at 450 nm. Between the absorption curves there is a blue
green transmission window at 500 nm. Emeralds, therefore, absorb red and some blue and transmit green with
some residue of blue to give the typical emerald colour.
The crystal structure of emerald is hexagonal, and just as with ruby, crystals are dichroic; the colour
depending upon the direction of the polarised light which irradiates them. In addition, the 2E state gives a red
fluorescence just as with ruby. However, neither of these effects is as noticeable as in the case of ruby itself.
Alexandrite is an extremely raremineral. In daylight or the light fromafluorescent tube the stone looks blue
green, while in incandescent light from an ordinary tungsten-filament lamp or candlelight the stone appears
deep red. The gemstone is a form of the mineral chrysoberyl, BeAl2O4. The crystal structure is orthorhombic,
2D
free ion octahedralfield
corundum
10 Dq
~475 nm
~550 nm ~800 nm
2E2g
2T2g
Figure 7.27 Schematic energy-level diagram for Ti3þ (3d1) ions in corundum (Al2O3). To a first approximationthe free-ion term splits into two, separated by 10Dq. In corundum, distortion of the coordination polyhedrasurrounding the cations splits both the ground and excited states, approximately as shown
283 Colour from Atoms and Ions
of the olivine structure type. The oxygen atoms in the structure are in approximately hexagonal close packing,
with the Al atoms occupying octahedral positions and the Be atoms in tetrahedral positions.
Aswith ruby and emerald, the colour of alexandrite is due to a small amount of Cr3þ impurity ions replacing
Al3þ ions in the octahedral sites of the structure. The colour, therefore, is produced in the same way as in
emerald and ruby, by the crystal-field splitting of the energy levels arising from the 3d3 configuration of the
Cr3þ impurity ions and the energy-level diagram inFigure 7.24 remains valid. In alexandrite, themagnitude of
the crystal field is about halfway between that of ruby and emerald. The two important absorption peaks lie
midwaybetween those found in these latter gemstones.However, to obtain the true alexandrite colour effect the
concentration of Cr3þ must be such that the two absorption bands, one in the blue region and one in the yellow/
red regionof the spectrum, are equal; and just as importantly, thewindowsof transmissionof red andgreen light
are also comparable. In these especial circumstances the colour of thegemstone that is perceivedby the eye then
depends upon the spectral characteristic of the light falling on the stone. In daylight or slightly blue-rich light, to
the eye,which ismore sensitive to green than to red, the stone takes on the appearance of emerald. On the other
hand, if this incident light is rich in red, as in the case of light from an incandescent source, little green light is
returned to theeyeand the stone looks a rubycolour.That is to say, the colour noted isdue to theperceptionof the
relative amounts of red and blue green light reaching the eye, not due to any changes in the crystal field under
different types of illumination.
For the alexandrite effect to be seen, the impurity content and the crystal fieldmust be finelymatched. It is for
this reason that natural alexandrite is rare. However, corundum (aluminium oxide, Al2O3) can be doped to
produce synthetic pseudo-alexandrites, which are sometimes sold as real alexandrite by unscrupulous dealers.
Note that because alexandrite (and chrysoberyl) possesses an orthorhombic crystal structure it has two optic
axes and is a biaxial material. The crystals exhibit strong trichroism when observed in transmitted linearly
polarised light. The colours vary from purple red to orange and green, depending upon the relative orientation
of the crystal and theplaneofpolarisationof the light.This is not related to thealexandrite effect and is shownby
colourless chrysoberyl as well as by alexandrite itself.
7.13 Crystal-Field Colours in Minerals and Gemstones
The colour produced by a transition metal ion will depend on the local crystal field, as the previous sections,
pertaining to Cr3þ in octahedral sites, make clear. A further example of this variation is given when this
latter ion is incorporated into chrome alum. The colour perceived is purple, again due to crystal-field
changes, as the Cr3þ is now surrounded by six water molecules. When absorption bands due to crystal-field
splitting occur well within the visible spectrum, then even small crystallographic changes lead to sufficiently
large changes in the local crystal field at the transition metal ion that the perceived colours change
enormously.
Despite this, many transition metal ions are thought of as showing a typical colour. Copper compounds, for
example, are usually green-blue and Fe2þ imparts a pale watery green colour to oxides and hydrates. Taking
copper as an example, in copper oxides, hydroxides or hydrates the Cu2þ ions are usually coordinated by six
oxygen atoms in a distorted octahedral coordination polyhedron. Although the crystal field varies from one
compound to another, the absorption peak usually lies in the infrared. The perceivedcolour, blue or blue green,
is due to the intrusion of this absorption band into the red end of the visible spectrum. The exact position of
the peak is thus largely irrelevant, with colour changes being due to the small differences in encroachment into
the visible. Thus, these compounds will always appear to be blue green. The exact tone will depend upon
both the crystal-field strength and the consequent encroachment, which gives ample room for subtle colour
variation. Figure 7.28 shows a sample of the mineral malachite, Cu2(OH)2(CO3), with a green colour. This is
mixedwith the relatedmineral azurite,Cu3(OH)2(CO3)2,which is bright blue.Although the chemical formulae
Colour and the Optical Properties of Materials 284
of these twominerals are similar, the crystal field in azurite is sufficiently different to that inmalachite that quite
different colours are perceived: blue versus green.
Many ions can occur inmore than one type of coordination and important variations of colour can occur. An
ion ina tetrahedral sitewillgenerallyexperienceacrystalfieldabout4/9of thatexperiencedbythesameioninan
octahedral field. Thus, absorption peaks will move from the violet towards red. In the case of octahedrally
coordinatedCu2þ , which has typically blue colours due to absorption in the red, an ion in a tetrahedral sitewillnow have an absorption peak well into the infrared, and hence tetrahedral Cu2þ will not show the typical blue
colour.
A similar change in coordination leads to the colour change exhibited bymanymoisture indicators or drying
agents. These are materials which often contain Co2þ ions. In the dry state these ions are in tetrahedral
coordination and appear deep blue the same colour as cobalt in glass.When thesematerials pick upwater, the
coordination changes to octahedral and Co2þ (H2O)6 units form. In these, the oxygen atoms in the water
molecules surround the Co2þ ions in an octahedral configuration. The colour is now pink.
Despite the variations in crystal-field colours, it is useful to list some of themore characteristic colours of the
3d transition metal compounds. These are given in Table 7.9.
As was highlighted by ruby and emerald, many gemstones are actually coloured by the presence of small
amounts of transition metal impurities in what would otherwise be colourless crystals. A short list of some
gemstones coloured by transition metal ion impurities is given in Table 7.10.
Figure 7.28 The differing crystal field colours of Cu2þ ions in malachite (Cu2(OH)2(CO3), green) and azurite(Cu(OH)2(CO3)2, blue) intermingled in a mineral sample
285 Colour from Atoms and Ions
Table 7.9 Typical colours shown by 3d transition metal compounds
IonNumber of d
electrons ColourApproximatesymmetry Examples
Ti3þ 1 purple tetrahedral doped glassa
V4þ 1 red tetrahedral doped glassV3þ 2 green tetrahedral doped glass
blue octahedral Al2O3
Cr3þ 3 green tetrahedral doped glassgreen octahedral emerald (Be3Al2Si6O18)red octahedral Al2O3 (ruby), doped TiO2
violet purple octahedral chrome alum (KCr(SO4)3�12H2O)Mn3þ 4 purple tetrahedral doped glassMn2þ 5 yellow tetrahedral doped glass
red octahedral MnCO3
green octahedral MnOpink octahedral MnSiO3
Fe3þ 5 yellow green tetrahedral doped glassred brown octahedral Fe2O3, rust
Fe2þ 6 blue green octahedral Fe(H2O)62þ in solution and hydrates
Co2þ 7 blue tetrahedral CoAl2O4, glasspink octahedral Co(H2O)6
2þ in solution and hydratesNi2þ 8 green octahedral Ni(H2O)6
2þ in solution and hydrates, NiOyellow octahedral doped Al2O3, NiCl2
Cu2þ 9 green octahedral Cu2(OH)2(CO3); malachiteblue octahedral Cu(H2O)6
2þ in solution and hydrates
a Glass refers to silicate glass.
Table 7.10 The colours of some gemstones
Gemstone General formulaStructuretype Colour
Origin of colour andcation replaced
Garnet Ca3Al2Si3O12 garnet red Fe2þ in cubic (8 coordinate)Ca2þ site
Peridot Mg2SiO4 olivine yellowgreen
Fe2þ in octahedral Mg2þ site
Topaz Al2SiO4(OH) topaz yellow Fe3þ in octahedral Al3þ siteEmerald Be3Al2Si6O18 beryl green Cr3þ in octahedral Al3þ siteAlexandrite BeAl2O4 olivine red/green Cr3þ in octahedral Al3þ siteRuby Al2O3 corundum red Cr3þ in octahedral Al3þ siteRubellite CaLi2Al7(OH)4(BO3)3Si6O18 tourmaline pink red Mn2þ in octahedral Al3þ siteIndicolite CaLi2Al7(OH)4(BO3)3Si6O18 tourmaline blue Fe2þ in octahedral Al3þ siteTurquoise CuAl6(PO4)4(OH)8�4H2O turquoise blue
greenCu2þ in octahedral (Cu2þ ) site
Colour and the Optical Properties of Materials 286
7.14 Colour as a Structural Probe
The spectrumof a transitionmetal ion in a solid can give information about the local position of the ion because
the colour depends upon the crystal field. Consider as an example the problem of cation distribution in oxide
spinels.
The spinel structure is adopted by many compounds with a formula AB2X4, where A and B are medium-
sized cations and X represents an anion, most often O2 . In this structure the anions are in a close-packed
array and the cations sit in octahedral and tetrahedral sites. The absorption spectra and the colour of
transition metal ions are quite different for these geometries, and so the site occupation can be easily and
unambiguously determined. The spinel NiAl2O4 is a case in point. The absorption spectrum of this material
reveals that the Ni2þ ions are found in both positions. Moreover, spectra taken in the earliest stages of the
formation reaction:
NiOþAl2O3!NiAl2O4
show that the Ni2þ occupies both sites from the very start of the reaction. In the related spinel NiGa2O4, the
Ni2þ ions exclusively occupy octahedral sites.
Because the structure of glasses cannot be solved by X-ray crystallography it is difficult to obtain structural
information at an atomic level, especially concerning the cation coordination in glasses. However, it is often
possible to incorporate a small amount of a transition metal into the structure as a probe of local geometry. For
example,Figure7.29 showsaglass bottle incorporating a small quantity ofCo2þ . Thebluecolour is consideredto be typical of tetrahedrally coordinated Co2þ in oxide matrices and indicates that the medium-sized Co2þ
ions replace the small Si4þ ions in tetrahedral sites in the glass network.
The tetrahedral network structure of silicate glass is well known, but new glasses often have quite unknown
structures. As an illustration of the use of transition metal ions as structural probes we can consider an
exploration of the structure of a ZnCl2 glass. A small amount of Mn2þ incorporated into an (Mn,Zn)Cl2 glass
imparts a yellow colour and yields a spectrum typical of tetrahedrally coordinatedMn2þ . As with the silicateglasses, small additions ofCo2þ in (Co,Zn)Cl2 gives a bluematerial characteristic of tetrahedrally coordinated
Co2þ ions. These results give strength to the argument that the amorphousZnCl2 glass is formed froma random
network of linked ZnCl4 tetrahedral units and that the added ions replace Zn2þ in the network. The suggestion
is further strengthened by noting that incorporation of Fe2þ in (Fe,Zn)Cl2 glass yields an absorption spectrum
expected from tetrahedrally coordinated Fe2þ ions.
This measurement of absorption spectra can also be used to determine the oxidation states of transition
metal ions and indirectly to yield information upon the local conditions prevailing during formation
reactions. To illustrate this, consider the fabrication of heavy metal fluoride glasses for potential optical fibre
use (Section 2.9). In order to gain some insight into the conditions occurring during reaction, a small amount
of vanadium was incorporated into a glass composed mainly of ZrF4, BaF2 and NaF. When the glass was
made in a nitrogen atmosphere, the colour was yellow green and spectral analysis showed it to contain V3þ
in octahedral sites. When a partial pressure of oxygen of about 0.1 atm was introduced into the nitrogen,
V5þ formed, which is colourless and so the glass loses its yellow green hue. Surprisingly, there was no trace
found of the stable ion V4þ under any processing condition. This information allowed the reaction
mechanisms occurring within the glass during fabrication to be determined, a task of considerable difficulty
by other methods.
These examples show that the colour of a transition metal ion and its careful measurement using absorption
spectra can give useful structural results in a variety of situations and often over a range of temperatures which
remain inaccessible to other experimental techniques.
287 Colour from Atoms and Ions
7.15 Colours from Lanthanoid Ions
The lanthanoids (also called the rare earths) have electrons in partly filled 4f orbitals (AppendixA7.1.3).Many
lanthanoids showcolours due to electron transitions involving the 4f orbitals, and these transitions are similarly
forbidden in terms of parity, leading to rather weak coloration (Table 7.11).
There is a considerable difference between the lanthanoids and the 3d transitionmetal ions. The 4f electrons
in the lanthanoids are well shielded beneath an outer electron configuration (5s2 5p6 6s2) and so are little
influenced by the crystal surroundings. This means that the important optical (and magnetic) properties
attributed to the 4f electrons on anyparticular lanthanoid ion donot depend significantly upon the host structure
and the colours do not arise from crystal-field splitting of the f-orbital energies. For this reason the transitions
Figure 7.29 A bottle coloured blue by the addition of Co2þ ions, which occupy tetrahedral sites in the glassmatrix
Colour and the Optical Properties of Materials 288
can be usefully labelled with atomic term symbols (Section 7.2). In addition, the transitions from one 4f
configuration to another are far narrower than those influenced by crystal-field splitting, as can be judged by
comparison of the absorption spectra of Ni(H2O)62þ and Cu(H2O)6
2þ (Figure 7.14) and of ruby (Figure 7.25)
with that ofNd3þ ions doped into a sodiumoxyfluorideglass (Figure7.30). (The spectral linewidths emittedby
lanthanoid ions in good crystalline matrices are narrower than those in more disordered structures, such as in
glasses.) Lanthanoid elements, thus, find use in phosphors (Chapter 9), lasers (Section 7.16) and other light-
emitting solids, where a host lattice can be chosen with respect to processing conditions without changing the
desirable colour properties of the ion greatly.
Table 7.11 Colours characteristic of lanthanoids ions
Ion Electron configuration Characteristic colour
Ce3þ 4f1 yellowPr3þ 4f2 greenNd3þ 4f3 lilac/violetPm3þ 4f4 pinkSm3þ 4f5 pale yellowSm2þ 4f6 red/greenEu3þ 4f6 pinkEu2þ 4f7 brownTb3þ 4f8 pinkDy3þ 4f9 pale yellowDy2þ 4f10 brownHo3þ 4f10 yellowEr3þ 4f11 pinkTm3þ 4f12 green
400 500 600 700 8000
0.5
1.0
Wavelength /nm
Abs
orba
nce
(arb
itrar
y un
its)
2 P1/
2 +
2 D5/
2
4 G5/
2 +
2 G7/
2
4 F7/
2 +
4 S3/
2
4 F5/
2 +
2 H9/
2
4 G9/
2 4 G7/
2
Figure 7.30 Themain peaks in the absorption spectrum of a glass doped with Nd3þ ions. The peaks are labelledwith the free-ion terms of the ion. [Data extracted fromB. Kartikeyan, S.Mohan,Mater. Res. Bull. 39, 1507–1515(2004)]
289 Colour from Atoms and Ions
The simplest lanthanoid ion is Ce3þ , with configuration [Xe] 4f1 5d1 6s2, the lowest energy levels arisingfrom the single f electron being 2F7=2 and
2F5=2. The next higher energy state for Ce3þ is the 5d level. Owing to
interactionof themoreexposed5delectronswith the surrounding crystal structure, this is broadened into aband
of energies, which also may overlap with another broadened band of energies derived from the 6s energy level
(Figure 7.31a). The transitions between the 5d/6s band and the 4f levels are allowed, and the colours produced
by transitions of this type are intense. However, the transition does not consist of a sharp line. Instead, any light
falling on the crystal will not be absorbed if the energy of the light is less than the energy difference between the2F7=2 and
2F5=2 levels and the upper band of energies.When the incident photons have energies greater than this
energy gap they will be absorbed. Thus, there will be a sharp change in absorption from low to high
(Figure 7.31b). The change in absorption the absorption edge lies towards the violet end of the spectrum.
The exact position of the absorption edge is a function of the surrounding matrix and will not be sharp due to
lattice vibrations, defects and other factors. In the case of CeO2, which is a good absorber of near-ultraviolet
light, the edge of the absorption creeps into the visible. Blue and green are strongly absorbed and to the eye the
oxide is perceived as pale yellow.
The exact position of the upper energy band is influenced by the surrounding crystal structure. Forming a
solid solution with another oxide with a different lattice parameter will move the absorption edge slightly. An
example is given by the solid solution formed by reacting yellow CeO2 with Y2O3. The change in structure
moves the absorption band further into the ultraviolet and so renders the CeO2 solid solution colourless. These
materials transmit the visible spectrum well but strongly absorb ultraviolet and have found use as ultraviolet-
absorbing transparent coatings. (Also see Section 10.1.)
As thenumberof f electrons increases, the energy-leveldiagramsbecome increasingly complex (seeChapter
9). However, Eu2þ , with a configuration 4f7, is an exception. In this case the higher energy state is obtained bytransferring an f electron to the outer 5d orbitals, which are lower than all of the other 4f energy levels. As the 5d
orbital is exposed to the crystal lattice, this 4f6 5d configuration forms a band of energies (Figure 7.32). As in
the case of Ce3þ , transitions from the ground state to the upper energy band are allowed. The energy gap is
slightly smaller than in the case of Ce3þ , and so the absorptionmoves slightly deeper into thevisible spectrum.
Because of this, the colour of the oxide EuO is a red brown rather than yellow.
The situation in Ce3þ - and Eu2þ -containing materials, therefore, is rather similar, which means that these
ions are very useful in providing a high-intensity luminescence (see Chapter 9). However, the 5d band is not
a continuum of energies, as represented schematically in Figures 7.31 and 7.32, but is usually split into sub-
bands that reflect the crystal-field splitting of the 5d orbitals. Absorption spectra that extend into the
ultraviolet thus show a variable number of absorption peaks which can affect emission spectra in this
wavelength region.
7.16 The Neodymium (Nd3þ ) Solid-State Laser: A Four-Level Laser
Although the ruby laser (Section 7.11) was the first laser made, three-level operation makes it inefficient,
because more than half of the ground state must be excited before a population inversion is possible. A more
energy-efficient device can be made employing a four-level energy-level scheme (Figure 7.33). Lasers using
this type of energy-level arrangement are referred to as four-level lasers.
Laser operation takes place in the following sequence of steps.
1. Atoms in the ground state E0 are excited to a rather high energy level E1 by optical pumping. This process
needs to be fast and efficient.
2. Atoms inE1 lose energy again byway of a fast and efficient radiationless process to an intermediate state I1.
Once in I1, atoms should have a long lifetime and not lose energy quickly.
Colour and the Optical Properties of Materials 290
5d energy band
2F7/22F5/2
~ 400 nm
1
2
3
4
5
Ene
rgy
0
Ce3+
5000
10000
15000
20000
25000
30000
35000
40000
cm–1eV(a)
300 400 500 600 700
Wavelength / nm
Abs
orba
nce
/ %
20
40
60
80
(b)
Figure 7.31 (a) Schematic energy-level diagram of Ce3þ in a typical oxide structure. The emission from thelower edge of the upper energy band to the ground state is close to the far-violet region of the spectrum. The exactposition depends upon the host structure. (b) The diffuse absorbance spectrum of CeO2. [Reprinted from J. SolidState Chem, 181, F. Tessier et al., Powder preparation and UV absorption properties of selected compositions inthe CeO2–Y2O3 system, 1204–1212, Copyright (2008), with permission from Elsevier]
291 Colour from Atoms and Ions
3. It is essential that another intermediate state, I0, is present and also sufficiently high above theground state to
be effectively empty. In this case, a small population in I1 gives a population inversion between I1 and I0.
4. Ultimately, a few photonswill be released by spontaneous emission as some atoms drop from I1 to I0. These
can promote stimulated emission between I1 and I0, allowing laser action to take place.
5. Atoms return from I0 to E0 by a step which needs to be rapid and radiationless.
6. If the energy corresponding to the transitions from E1 to I1 and I0 to E0 can be easily dissipated, continuous
operation rather than pulsed operation is possible.
Themost important four-level solid-state laseruses neodymium(Nd3þ ions) as theactivecentres.These ions
canbe introduced into awidevariety ofhost latticeswith little effect onoptical properties because the important
4f orbitals are shielded from the crystal surroundings as described above. Themost common host materials are
glass, yttrium aluminium garnet (YAG) and calcium tungstate (CaWO4).
The important transitions taking place inNd3þ -ion lasers can be understood in terms of a simplified energy-
level diagram (Figure 7.34). The f-electron levels are rather sharp. Above these lie bands of considerablewidth
derived from the interaction of the 5d and 6s orbitals. Optical pumping excites the ions from the ground state to
thesewide bands. This process is very efficient because broadbands allowawide range ofwavelengths to pump
the laser and because the transitions are allowed in terms of quantum theory. In addition, loss of energy from the
excited state down to the f-electron energy levels is fast. The energy loss halts at the pair of 4F levels. The
~ 420 nm
1
2
3
4
5
Ene
rgy
0
4f 6-5d band
8S7/2
Eu2+
5000
10000
15000
20000
25000
30000
35000
40000
cm–1eV
Figure 7.32 Schematic energy-level diagram of Eu2þ in a typical oxide structure. The emission from the loweredge of the upper energy band to the ground state is in the violet region of the spectrum. The exact positiondepends upon the host structure
Colour and the Optical Properties of Materials 292
4F
4I15/2
4I13/24I11/24I9/2
5d-6s band
0
1
2
3
4
5
1×10–19
2×10–19
3×10–19
4×10–19
5×10–19
6×10–19
7×10–19
8×10–19
Energy
J eV
0
Figure 7.34 Energy levels of most importance in the neodymium laser. The pump transition is from the groundstate to the broad 5d–6s band. The main laser transition is between the 4F and 4I11/2 levels. Internal radiationlesstransitions are marked with dashed lines
Laser emissionPump
E0
E1
I1
I0
Figure 7.33 Schematic arrangement of energy levels and transitions in a four-level laser. Internal (radiationless)transitions are marked with dashed lines
293 Colour from Atoms and Ions
principal laser transition is from these 4F levels to 4I11=2. The emission is at approximately 1060 nm in the
infrared.
The laser medium contains about 1 % Nd3þ and can have quite high power outputs. These lasers can be
operated continuously or pulsed. At higher Nd3þ concentrations the lifetime of the 4F upper state drops from
about 200ms in a typically 1%dopedmaterial to about 5ms at higher dopant concentrations. This is due toNdNd interactions and associated changes in lattice vibration characteristics. Under these conditions, laser
operation is no longer possible.
7.17 Amplification of Optical-Fibre Signals
The amplification of signals in fibre-optic transmission systems is of great importance, as the input signal
degrades with distance due to attenuation and dispersion. Originally, amplification used costly repeaters,
which transformed the optical pulses into electronic signals, amplified these electronically and then
recreated optical pulses. Operating systems are now available which use a section of optical fibre doped
with erbium (Er3þ ) as the activator. Erbium-fibre amplifiers using 1.48 and 0.98mm pump radiation were
perfected 1989. The amplifying section consists of about 30m of monomode fibre core containing just a few
hundred parts per million of Er3þ (Figure 7.35a). This section of the fibre is illuminated by a semiconductor
diode laser (Section 10.9) at the frequency of the carrier signal. The commonest wavelengths used are 980,
1480 and 1550 nm. The erbium ions transfer energy from the laser to the signal pulses as they traverse this
section of fibre.
The energy transfer comes about in the followingway. Illumination of the erbium-containing section of fibre
with energy ofwavelength 980 nm excites the ions from the ground state (4I15=2) to the upper state (4I11=2) from
980
nm
1480
nm
Energy passed to signal
4I11/2
4I13/2
4I15/2 ground state
(b)
incomingsignal
outgoingsignal
Er3+ doped length of fibre(a)
Figure 7.35 Signal amplification in an Er3þ -doped section of optical fibre. (a) A weak incoming signal issubstantially amplified on traversing the section. (b) Schematic energy-level diagram of Er3þ in SiO2. Pumpwavelengths at approximately 980 and 1480 nm populate the 4I13/2 level, which passes this energy to the signal
Colour and the Optical Properties of Materials 294
whence they rapidly decay to the 4I13=2 level (Figure 7.35b). This process is referred to as pumping and the laser
involved as the pump. The use of radiation of 1480 nm wavelength excites the Er3þ ions directly from the
ground state to the 4I13=2 level.This state hasquite a long lifetime.Apassing light pulse,with awavelength close
to 1480 nm, empties the Er3þ excited state via stimulated emission (Section 1.9). In effect, the pump energy is
transferred to the signal pulses over the course of the erbium-doped stretch of fibre. This achieves signal
amplification while retaining the coherence of the pulse constituting the signal.
The Er3þ -doped sections of fibre are made in a similar way to that described in Section 2.9. The gas stream
which is used to lay down what will become the core region of the fibre is modified by the addition of erbium
chloride and aluminium chloride. The aluminium chloride is added as a co-dopant because it has been found
that the presence ofAl3þ ions in the glass greatly increases the number of Er3þ ionswhich can be incorporated
before clustering starts to occur. The chlorides decompose in the same way as the chlorides of silicon and
germanium, to formsoot containing thedesired concentration ofEr3þ ions. Subsequent heating and collapseof
the tube produces a preform with an erbium-doped core.
Signal amplification is also used in theNational Ignition Facility for fusion research in theUSA.This aims to
use laser beams to ignite fusion in a deuterium tritium pellet. The output from 192 lasers is used, but as these
beams are nowhere near powerful enough, they are repeatedly passed through glass light guides containing
Nd3þ inorder tobeamplifiedasdescribedabove. In this facility, theNd3þ is pumpedbyXeflash lamps, like the
first ruby lasers (Section 7.11), and at each pass the pump energy is added to the laser beam energy until
sufficient power has been reached, at which stage all of the beams are focused onto the target (see this chapter’s
Further Reading).
7.18 Transition Metal, Lanthanoid and Actinoid Pigments6
Inorganic pigments are colorants used to enhance the appearance of an object. Pigments are incorporated as
finely groundpowders and are often applied to surfaces as paints and inks.Although organic pigments (Chapter
8) are usually brighter than inorganic pigments, they are not stable atmoderate or high temperatures. This poses
amajor problem in the fabrication of decorative ceramics and glasses, as high temperatures are essential during
manufacture (Figure 7.36). The use of transition metal and lanthanoid (or, more rarely, actinoid) compounds,
which are added in small quantities to the batch, overcomes this difficulty. The colours generated are often due
to the d d or f f transitions described above.
The actinoids have partly filled 5f orbitals and behave in a similar way to lanthanoid. These are not usually
associated with colour production because of the scarcity and radioactive nature of the heavy atoms. The most
commonly utilized examples are uranium compounds, which are used to colour glass and ceramics a yellow
green colour. Thematerial used in glasses is usually the yellow trioxide, UO3. Small quantities ofUO3 dissolve
completely in many glasses to yield a coloured yellow green transparent material (Figure 7.37). It is also used
in larger quantities as a green yellow pigment for ceramics. The other commonly utilized uranium compound
is uranyl nitrate, UO2(NO3)2�6H2O, often called uranium nitrate. It forms yellow crystals which are readily
soluble in water.
Large numbers of inorganic transition metal oxides are used as pigments. Chromic oxide (Cr2O3), known
as chrome green, is typical of a number of simple oxides used the green colour arising in the crystal-field
splitting of the energy of the d orbitals on octahedrally coordinated Cr3þ ions. Many complex oxides are also
used. For example, calcium chromium silicate (Ca3Cr2Si3O12), with the garnet structure, shows a green colour
attributed to the octahedrally coordinated Cr3þ ions, but because of differences in the crystal-field splitting of
the d-orbital energy levels in the two compounds the colours are perceived as different. Cobalt aluminate
6 The terms ‘lanthanoid’ and ‘actinoid’ are now recommended by IUPAC. See footnote to Appendix A7.1.3.
295 Colour from Atoms and Ions
(CoAl2O4), with the spinel structure, is a blue pigment; the colour arises from tetrahedrally coordinated Co2þ
ions in the crystal. Similarly, cobalt silicate (Co2SiO4), with the olivine structure, is a blue pigment, which also
relies upon tetrahedrally coordinatedCo2þ ions as the colour producer.Differences in the crystal-field splitting
of theCo2þ energy levels in these two crystals give each a unique tone.Cobalt chromite (CoCr2O4),which also
Figure 7.36 An enamel trinket box lid. The enamel is a glass-basedmaterialwhich has been fused to ametal base.The colours are derived from oxide pigments dissolved or dispersed in the glass
Figure 7.37 Yellow–green uranium (U6þ )-doped glass
Colour and the Optical Properties of Materials 296
adopts the spinel structure, contains tetrahedrally coordinatedCo2þ , imparting a blue colour, and octahedrally
coordinated Cr3þ ions in a similar geometry to that in Cr2O3 to give a green colour. The resulting compound
combines both of these tones to yield a blue green pigment.
The use of such colorants is hardly new. Several thousand years ago the Egyptians synthesized blue
objects using a colorant now known as ‘Egyptian blue’ and the Chinese synthesized both blue and purple
artefacts using ‘Han blue’ and ‘Han purple’. The mode of production of Egyptian blue is typical of the
techniques used. Artisans heated a mixture of lime, copper oxide and quartz, in approximate ratios of 1:1:4,
at high temperatures in a kiln. This produced a polycrystalline/glassy blue solid which was ground to make a
blue pigment which could be used in paints. All these ancient blue pigments have been shown to be complex
copper silicates. The formulae are CaCuSi4O10 for Egyptian blue, BaCuSi4O10 for Han blue and BaCuSi2O6
for Han purple. The compounds themselves are ring silicates in which the colour is derived from crystal-
field splitting of the Cu2þ d-orbitals in a square planar environment. Bearing in mind the fact that the
alkaline earth copper silicon oxygen systems are complex and contain a bewildering variety of both
coloured and noncoloured crystalline and glassy phases, the technological expertise of the craftsmen was
considerable.
In the past, the desires for bright colours led to the use of pigments which were dangerous and which would
not be allowed today. For example, Scheele’s green, a bright green compound precipitated from ‘arsenious
acid’ with copper sulfate solution, which has been assigned an approximate formula HCuAsO3, and Paris
green, a mixed copper arsenic acetate of approximate formula 3Cu(AsO2)2�Cu(CH3COO)2, nominally copper
acetoarsenite, were both widely used to colour much sought after green wallpaper in the nineteenth century.
These compounds, however, are rather unstable and release toxic arsenic-containing vapours in moist air.
Indeed, the death ofNapoleon, on the islandofStHelena, in1821, is attributed to arsenic poisoningarising from
the decomposition of Paris green pigments in the wallpapers of his accommodation.
In many of these examples, the agent causing the colour is a substituted transition metal or lanthanoid ion.
The degree of coloration can be adjusted by changing the amount of dopant or by adjusting co-dopants to
change the dimensions of the surrounding crystal structure. Thus, a colourless crystal can be made to appear
black by doping with two substituents, one of which absorbs radiation in the low-energy yellow part of the
visible spectrum and another that absorbs in the high-energy blue region. The black colour of the pigment
CoxZn7 xSb2O12 is attributed to such double absorption. In this material, which adopts the inverse spinel
structure, Co2þ ions occupy both octahedral and tetrahedral sites. The colour is due to the Co2þ , whichreplacesZn2þ to formsubstitutional defects.TheabsorptionofCo2þ in octahedral sites centres on red yellow,
and of Co2þ in tetrahedral sites on blue, giving an overall black material.
Despite this long history, there is considerable current research concerned with ceramic pigment
formulation.
7.19 Spectral-Hole Formation
The widths of the d d absorption bands in the spectra of transition metal ions in solids are generally
considerable, due to the strong interactions of the surrounding crystal matrix with the exposed d orbitals
on the cations. In the case of lanthanoid ions, the widths of the f f bands are considerably smaller, but still
appreciable due to thermal vibrations. These can be eliminated if the crystal host structure is cooled to 10Kor
less. Nevertheless, the narrow lines still have somewidth. These have been explored both to study fundamental
materials properties and for data storage. For this latter purpose, multiple bits are stored at a single location in
the host crystal, using the f f transition as themeans to this end. The technique is called spectral-hole burning.
Themethod uses crystals dopedwith lanthanoids, subsequently cooled to close to absolute zero.Under these
circumstances, the linewidth of a peak in the spectrum of the ion will, in an ideal case, consist of a single
297 Colour from Atoms and Ions
excitation frequencycalled thehomogeneous linewidthGh (Figure 7.38a).However, inmost crystals, not all the
cation sites are exactly identical. Thismeans that the active centres in the crystal give rise to a narrow spectrum
of different absorption frequencies. Of course, the line is still very sharp in spectroscopic terms, but is wider
than the single frequency absorption of an individual ion. The total absorption width of a collection of such
centres is called the inhomogeneous linewidth Gi (Figure 7.38b). The homogeneous linewidth of typical
lanthanoid ions such as Pr3þ , Sm3þ , Eu3þ and Er3þ is approximately 10 kHz, while the inhomogeneous
linewidth is of the order of 10GHz, a factor of 106.
(a)
(b)
(c)
(d)
Figure 7.38 Homogeneous and inhomogeneous absorption of ions in a crystal: (a) homogeneous linewidth of asmall subset of ions; (b) inhomogeneous linewidth of the whole set of ions; (c) single spectral hole in theinhomogeneous linewidth; (d) several spectral holes in the inhomogeneous linewidth
Colour and the Optical Properties of Materials 298
Irradiation of a small volume of crystal with a well-defined beam of laser light of the appropriate frequency
will excite only a subset of the atoms that contribute to the homogeneous linewidth. This causes a dip to be
recorded in the homogeneous profile at exactly the excitation energy of the subset of ions involved
(Figure 7.38c). A change in the irradiation frequency can excite a second subset of cations, and so on.
Hence, the homogeneous linewidth becomes pitted (Figure 7.38d). These dips in the profile can be used for
storage of several single bits of data at the same location in the crystal the tinyvolume irradiated by the laser. In
theory, the number of bits that can be accommodated at the location is equal to the ratio ofGi/Gh. In the case of
one system that has been studied, Er3þ doped into Y2SiO5,Gi¼ 0.6GHz andGh¼ 50 kHz, so that the number
of bits than can be accommodated is 0.6� 109/50� 103¼ 12 000.
In order to make a memory store, it is essential that the spectral holes have a reasonable lifetime. In
general, a transition between two f f levels, known as two-level hole burning (Figure 7.39a) gives a hole
lifetime near to 10 6 s, although hole lifetimes of up to 10 ms have been observed. A memory involving such
transitions would need to be refreshed continually, and would not be suitable for long term data storage. The
lifetime of a spectral hole can be increased by using different transitions. One method is analogous to
the transition from a 4T state to a 2E state in ruby, and involves an intermediate level with a much longer
lifetime than two-level hole burning. The mechanism, called metastable trapping (Figure 7.39b), increases
the spectral hole lifetime significantly. A third mechanism involves making use of transitions to the broad
band of energies that lies above the f f levels. In this, a laser beam initially excites an ion from the ground
state to an excited f level. The excited state is then ionised and an electron is promoted into the conduction
band of the solid (see Chapter 10) using ‘gated’ photo-ionisation (Figure 7.39c). This results in the creation,
for instance, of an Ln3þ ion from a dopant population of Ln2þ . The process involves irradiation of the crystalvolumewith the hole-burning wavelength lb and simultaneously with the ‘gating’ wavelength lg. The lifetime
of the hole now depends on the rate at which the electron drops from the conduction band to reform the
groundstate
f levelmetastablef level
conductionband
(a) (b) (c)
photoionisation
Figure 7.39 Mechanisms of spectral hole burning: (a) two level; (b) metastable trapping; (c) gated photo-ionisation
299 Colour from Atoms and Ions
Ln2þ state. At low temperatures, the lifetime of these spectral holes is considerable and the memory storage is
regarded as permanent.
Appendix A7.1 Electron Configurations
A7.1.1 Electron configurations of the lighter atoms
Formost chemical purposes anatomor an ioncanbeconsidered toconsist of adenseminutenucleus surrounded
by electrons which are said to occupy a series of orbitals. The electron configuration of an atom or an ion
describes theway inwhich the electrons are allocated to theseorbitals.The simplest approximationwhichgives
the occupancy of the orbitals is the independent particle model (the orbital approximation), in which each
electron is supposed tobe isolated andmoving inafieldcomprising that arising in thenucleus andall of theother
electrons combined. In this approach, each electron is assigned a set of four unique quantum numbers which
correspond to the atomic orbital that the electron occupies. The atomic orbitals form a set of shells which are
filled from the lowest energy upwards. The Pauli exclusion principle demands that only two electrons, with
opposed spins, can occupy an orbital. If this were not so, all electronswould end up in the lowest energy orbital.
The lowest energy shell is characterized by a principal quantum number n¼ 1 and contains only one atomic
orbital called an s-orbital. This, like any atomic orbital, can contain either one or two electrons. The two atoms
that these two alternatives correspond to are hydrogen (H) and helium (He). The electron configurations of
these two atoms are written
H 1s1
He 1s2
where the principal quantum number (1) is written first, the orbital (s) follows and then the number of electrons
in the orbital as a superscript.
The next lowest energy shell is characterized by a principal quantumnumbern¼ 2 and contains one s-orbital
and three p-orbitals, px py and pz, all of which have the same energy. The s-orbital can contain up to two
electrons, as above, and the three p-orbitals can contain a maximum of six electrons. The electron config-
urations of the atoms which make up the second shell, from lowest to highest energy, are
Li 1s2 2s1 or [He] 2s1
Be 1s2 2s2 or [He] 2s2
B 1s2 2s2 2p1 or [He] 2s2 2p1
C 1s2 2s2 2p2 or [He] 2s2 2p2
N 1s2 2s2 2p3 or [He] 2s2 2p3
O 1s2 2s2 2p4 or [He] 2s2 2p4
F 1s2 2s2 2p5 or [He] 2s2 2p5
Ne 1s2 2s2 2p6 or [He] 2s2 2p6
The second shell is now full. Note that in order towrite the configuration in a compact form the inner filled shell
is represented by the symbol of the atom with that configuration, which is He in this case.
The next energy shell is characterized by a principal quantum number n¼ 3 and contains one s-orbital,
three p-orbitals px, py and pz, and five d-orbitals, dxy, dxz, dyz, dx2 y2 and dz2 . The s-orbital can contain up
to two electrons and the three p-orbitals can contain a maximum of six electrons, as before. The five
d-orbitals can contain up to 10 electrons. Atoms with partly filled d orbitals are called transition
metals. The electron configurations of the atoms which make up the third shell, from lowest to highest
energy, are:
Colour and the Optical Properties of Materials 300
Na 1s2 2s2 2p6 3s1 or [Ne] 3s1
Mg 1s2 2s2 2p6 3s2 or [Ne] 3s2
Al 1s2 2s2 2p6 3s2 3p1 or [Ne] 3s2 3p1
Si 1s2 2s2 2p6 3s2 3p2 or [Ne] 3s2 3p2
P 1s2 2s2 2p6 3s2 3p3 or [Ne] 3s2 3p3
S 1s2 2s2 2p6 3s2 3p4 or [Ne] 3s2 3p4
Cl 1s2 2s2 2p6 3s2 3p5 or [Ne] 3s2 3p5
Ar 1s2 2s2 2p6 3s2 3p6 or [Ne] 3s2 3p6
The energy of the 4s-orbital is close to that of the 3d-orbitals and is usually filled before the 3d group. The
electron configuration of the 3d transition metals is given in Appendix A7.1.2.
The filling of the fourth, fifth and subsequent shells follows along the same lines as above. In heavier atoms
there is often someuncertainty in the order inwhich the orbitals are filled. Thiswill be observed, for example, in
someof the atoms listed inAppendixA7.1.3. The completely filled ns2np6 configurations (which correspond to
the inert gases), used to write the electron configurations in a compact form, are
He 1s2
Ne [He] 2s2 2p6
Ar [Ne] 3s2 3p6
Kr [Ar] 3d10 4s2 4p6
Xe [Kr] 4d10 5s2 5p6
The outer electron configuration of the atoms is given in Figure 7.2. The electron configuration of ions is
written in an identical fashion. Cationic configurations can usually be derived from that of the parent atoms by
removing a small number of electrons from the atomic orbitals last filled and anionic configurations by adding
electrons to these same orbitals.
A7.1.2 The 3d transition metals
The ten 3d transition metal elements are found in Period 4 of the Periodic Table: K, Ca, Sc,Ti,V,Cr,Mn, Fe,
Co, Ni, Cu, Zn, Ga, Ge, As, Se, Br, Kr. They are characterized by having partly filled 3d atomic orbitals
(Table A7.1. Cuprous (Cuþ ) and zinc (Zn2þ ) ions do not behave as typical transition metal ions as they have
completely filled 3d-orbitals.
A7.1.3 The lanthanoid (rare earth) elements
The 15 lanthanoid or rare earth elements7 are found in Period 6 of the Periodic Table: Cs, Ba, (La),Ce,Pr,Nd,Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, (Lu), Hf, Ta, W... .
The electron configuration of some of these atoms is uncertain and neither lanthanumnor lutetium behave as
typical lanthanoid, although both are frequently included in the group (Table A7.2).
7 Though ‘lanthanide’ is still in widespread use, the IUPAC recommendation IR 3.5 is that the term ‘lanthanoid’ be used for this group of
elements. The reasoning is that ‘ oid’means ‘having the formof, like, similar to’,whereas ‘ ide’ is normally indicative of negative ions. For
the same reason, ‘actinoid’ is now recommended over use of ‘actinide’.
301 Colour from Atoms and Ions
Appendix A7.2 Terms and Levels
A7.2.1 The vector model of the atom
The energy levels associated with the electron configurations of an atom (Appendix A7.1) are derived by using
the vector model of an atom. In thismodel, classical ideas are grafted onto the quantummechanics of the atom.
The quantum number l is associated with the angular momentum of the electron around the nucleus. It is
represented by an angular momentum vector l. Similarly, the spin quantum number of the electron s is
associated with a spin angular momentum vector s. (Vectors in the following text are specified in bold type and
quantum numbers in italic type.) The scalar values of s and l are writtenms andml. In the vector model of the
atom, the two angularmomentum vectors are added together to get a total angularmomentum for the atom as a
whole. This is then related to the electron energy levels of the atom.
There are twomainwaysof tackling this task.Thefirst of thesemakes theapproximation that the electrostatic
repulsion between electrons is themost important energy term. In this approximation, calledRussell Saunders
coupling, all of the individual s vectors of the electrons are summed vectorially to yield a total spin angular
momentumvectorS. Similarly, all of the individual lvectors for the electrons present are summedvectorially to
give a total orbital angular momentum vector L. The vectors S andL can also be summed vectorially to give a
total angular momentum vector J. Note that the convention is to use lower case letters for a single electron and
upper case for many electrons.
One alternative approach to Russell Saunders coupling is to assume that the interaction between the orbital
angularmomentum and the spin angularmomentum is themost important. This interaction is called spin orbit
coupling. In this case, the s and l vectors for an individual electron are added vectorially to give a total angular
Table A7.1 The 3d transition metals
Name SymbolElectron configuration
of atoma Iond electron
configuration of ion
Scandium Sc [Ar] 3d1 4s2 Sc3þ d0
Titanium Ti [Ar] 3d2 4s2 Ti4þ d0
Ti3þ d1
Ti2þ d2
Vanadium V [Ar] 3d3 4s2 V5þ d0
V4þ d1
V3þ d2
V2þ d3
Chromium Cr [Ar] 3d5 4s1 Cr3þ d3
Manganese Mn [Ar] 3d5 4s2 Mn4þ d3
Mn3þ d4
Mn2þ d5
Iron Fe [Ar] 3d6 4s2 Fe3þ d5
Fe2þ d6
Cobalt Co [Ar] 3d7 4s2 Co4þ d5
Co3þ d6
Co2þ d7
Nickel Ni [Ar] 3d8 4s2 Ni2þ d8
Copper Cu [Ar] 3d10 4s1 Cu2þ d9
Cuþ d10
Zinc Zn [Ar] 3d10 4s2 Zn2þ d10
a [Ar]¼1s2 2s2 2p6 3s2 3p6.
Colour and the Optical Properties of Materials 302
momentum vector j for a single electron. These values of j are then added vectorially to give the total angular
momentum vector J, for the whole atom. The technique of adding j values to obtain energy levels is called j j
coupling.
Broadly speaking, Russell Saunders coupling works well for lighter atoms and j j coupling for heavier
atoms. Other coupling schemes have also been worked out, and these find use in medium and heavy
atoms. In reality, the energy levels derived from each scheme represent approximations to those found by
experiment.
For almost all purposes, the Russell Saunders coupling scheme is adequate for the specification of the
energy levels of an isolatedmany-electron atom. In general, it is not necessary towork directlywith the vectors
S,L and J. Instead, many-electron quantum numbers (not vectors) S, L and J are used to label the energy levels
in a simpleway.Themethodof derivation is set out inAppendixA7.2.2.ThevalueofS is not useddirectly, but is
replacedby the spinmultiplicity, 2S þ 1.Similarly, the total angularmomentumquantumnumberL is replaced
by a letter symbol similar to that used for the single electron quantumnumber l (TableA7.3). After L¼ 3, F, the
sequence of letters is alphabetic, omitting J. Be aware that the symbol ‘S’ has two interpretations: S (roman) is
the value of L and S (italic) as the value of total spin.
The combinations are written in the following form:
2Sþ 1L
This is called a term symbol. It represents a set of energy levels, called a term in spectroscopic parlance. States
with amultiplicity of one are called singlet states, stateswith amultiplicity of two are called doublet states,with
Table A7.2 The lanthanoid elements
Name SymbolElectron configurationof atoma Ion
f electronconfiguration of ion
Lanthanum La [Xe] 5d1 6s2 or [Xe] 4f1 6s2 La3þ 4f0
Cerium Ce [Xe] 4f1 5d1 6s2 or [Xe] 4f2 6s2 Ce4þ 4f0
Ce3þ 4f1
Praseodymium Pr [Xe] 4f3 6s2 Pr4þ 4f1
Pr3þ 4f2
Neodymium Nd [Xe] 4f4 6s2 Nd3þ 4f3
Promethium Pm [Xe] 4f5 6s2 Pm3þ 4f4
Samarium Sm [Xe] 4f6 6s2 Sm3þ 4f5
Sm2þ 4f6
Europium Eu [Xe] 4f7 6s2 Eu3þ 4f6
Eu2þ 4f7
Gadolinium Gd [Xe] 4f7 5d1 6s2 Gd3þ 4f7
Terbium Tb [Xe] 4f9 6s2 Tb4þ 4f7
Tb3þ 4f8
Dysprosium Dy [Xe] 4f10 6s2 Dy3þ 4f9
Holmium Ho [Xe] 4f11 6s2 Ho3þ 4f10
Erbium Er [Xe] 4f12 6s2 Er3þ 4f11
Thulium Tm [Xe] 4f13 6s2 Tm3þ 4f12
Ytterbium Yb [Xe] 4f14 6s2 Yb3þ 4f13
Yb2þ 4f14
Lutetium Lu [Xe] 4f14 5d1 6s2 Lu3þ 4f14
a[Xe]¼ 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6.
303 Colour from Atoms and Ions
multiplicity of three, triplets, with multiplicity four, quartets and so on. Hence, 1S is called singlet S and 3P is
called triplet P.
A7.2.2 Energy levels and terms of many-electron atoms
The Russell Saunders terms of an atom are derived by adding the individual spin quantum numbers of the
electrons to yield a total spin quantumnumber S and adding the individual orbital angularmomentum quantum
numbers of the electrons to give a total orbital angular momentum quantum number L. For example, the total
spin angular momentum quantum number S(2) for two electrons is given by adding the individual quantum
numbers thus:
Sð2Þ ¼ ðs1þ s2Þ; ðs1þ s2�1Þ; . . . ; js1�s2j:
As s1 and s2 are both equal to 12:
Sð2Þ ¼ 1 or 0
(Note that this is amaximumnumber of values. If two electrons are spin paired then only the value zero applies.
If the electrons are in different orbitals, say an s and a p orbital, then they can have parallel or antiparallel spins,
making both one and zero possible.)
In order to obtain the value of S for three electrons S(3), the value for two electrons S(2) is added to the spin
quantum number of the third electron s3 thus:
Sð3Þ ¼ ðSð2Þþ s3Þ; ðSð2Þþ s3�1Þ; . . . ; jSð2Þ�s3j
Both of the values for S(2) are permitted, so we obtain:
Sð2Þ ¼ 1; Sð3Þ ¼ 1þ 12; 1þ 1
2�1 ¼ 3
2; 1
2
Sð2Þ ¼ 0; Sð3Þ ¼ 0þ 12
Sð3Þ ¼ 32; 1
2
This procedure is called theClebsch Gordon rule. It is used to obtain the S values for increasing numbers of
spins. It will be found that for an even number of electrons, S values are integers, and for an odd number of
electrons, S values are half-integers.
Table A7.3 The correspondence of L valuesand letter symbols
L Symbol
0 S1 P2 D3 F4 G5 H
Colour and the Optical Properties of Materials 304
The total angular momentum quantum number L is obtained in a similar fashion. For two electrons with
individual angularmomentumquantumnumbers l1 and l2, the total angularmomentumquantumnumberL(2) is:
Lð2Þ ¼ ðl1þ l2Þ; ðl1þ l2� 1Þ; . . . ; jl1� l2jIn the case of three electrons, the Clebsch Gordon rule is applied thus:
Lð3Þ ¼ ðLð2Þþ l3Þ; ðLð2Þþ l3� 1Þ; . . . ; jLð2Þ�l3j
using every value of L(2) obtained previously. Values of the quantum number L are given letter symbols as
described.
For example, the terms arising from the two p electrons on carbon (C), with l1¼ l2¼ 1, are obtained in the
following way:
S ¼ 12þ 1
2; 1
2� 1
2¼ 1; 0
2Sþ 1 ¼ 3 or 1
L ¼ 1þ 1; 1þ 1�1; 1�1 ¼ 2; 1; 0 ðD; P; SÞ
The total number of possible terms for the two p electrons is given by combining these values. The possible
terms for two p electrons are therefore:
3D; 3P; 3S; 1D; 1P; 1S
Not all of these possibilities are allowed for any particular configuration, because the Pauli exclusion
principle limits the number of electrons in each orbital to two with opposed spins. When this is taken into
account, the allowed terms are:
3P; 1D; 1S
Similarly, an atom with two d electrons, with a configuration of, say, 3d2, will again have:
S ¼ 12þ 1
2; 1
2� 1
2¼ 1; 0
The possible values of L are obtained by using the values l1¼ l2¼ 2, to give:
L ¼ 2þ 2; ð2þ 2�1Þ; ð2þ 2�2Þ; ð2þ 2�3Þ; ð2þ 2�4Þ ¼ 4; 3; 2; 1; 0 ðG; F; D; P; SÞ
Combining all of these gives:
3G; 3F; 3D; 3P; 3S; 1G; 1F; 1D; 1P; 1S
Taking into account forbidden configurations gives the allowed terms as:
3F; 3P; 1G; 1D; 1S
305 Colour from Atoms and Ions
The energies of the terms are difficult to obtain simply, and theymust be calculated using quantummechanical
procedures.However, the lowest energy term, theground-state term, is easily foundusing themethoddescribed
in Appendix A7.2.3.
A7.2.3 The ground-state term of an atom
The lowest energy term, the ground-state term, can be found using Hund’s first and second rules:
1. The term with the lowest energy has the highest multiplicity, equivalent to the highest total spin quantum
number S.
2. For terms with the same value of multiplicity, the term with the highest value of L is lowest in energy.
There is a simple method of determining the ground state of any atom or ion. The procedure is as follows:
1. Draw a set of boxes corresponding to the number of orbitals available. For a p electron, this is three
(Figure A7.1).
2. Label each box with the value of ml, highest on the left and lowest on the right.
3. Fill the boxeswith unpaired electrons, from left to right.When each box contains one electron, start again at
the left.
4. Sum the ms values of each electron, þ 12or �1
2. This is equal to the maximum value of S.
5. Sum the ml values of each electron to give a maximum value of L.
6. Write the ground term 2Sþ 1L.
Using this technique, set out in Figure A7.1, the ground term of the 2p2 and 2p4 configurations is 3P.
A7.2.4 Energy levels of many-electron atoms
The term symbol does not account for the true complexity found inmost atoms. This arises from the interaction
between the spin and theorbitalmomentum(spin orbit coupling) that is ignored inRussell Saunders coupling.
For this the quantum number J described above is needed. It is given by:
J ¼ ðLþ SÞ; ðLþ S�1Þ; . . . ; jL� Sj
ml
ml
1
1
0
0
–1
–1
p2
p4
S = ½ + ½ = 12S+1 = 3L = 1 + 0 = 1
term scheme 3P
term scheme 3P
S = ½ + ½ + ½ – ½ = 12S+1 = 3L = 1 + 0 + –1 + 1= 1
Figure A7.1 Determination of a ground state term
Colour and the Optical Properties of Materials 306
where |L� S| is the modulus (absolute value, regardless of whether þ or �). Thus, the term 3P, has J values
given by:
J ¼ ð1þ 1Þ; ð1þ 1�1Þ; . . . ; j1�1j ¼ 2; 1; 0
The new quantum number is incorporated as a subscript to the term, nowwritten 2Sþ 1Lj and this is no longer
called a termsymbol, but a level. Eachvalueof J represents a different energy level. It is found that a singlet term
alwaysgivesoneenergy level, a doublet two, a triplet three and soon.Thus, ground-state term 3P is composedof
three levels 3P0,3P1 and 3P2. The separation of these energy levels is controlled by the magnitude of the
interaction between L and S. Hund’s third rule (see Appendix A7.2.3 for rules 1 and 2) allows the values of J to
be sorted in order of energy. The levelwith the lowest energy is that with lowest J value if thevalence shell is up
to half full and that with the highest J value if the valence shell is more than half full.
The nomenclature just described is not adequate to describe either molecular energy levels or the
energy levels of atoms in crystal fields. In these cases a terminology based upon symmetry is most often
encountered.
Further reading
The electron configuration of atoms and ions at an introductory level is explained clearly in
P. W. Atkins, L. Jones, Chemistry, 3rd edition, W. H. Freeman, New York, 1997, Chapter 7.
A comprehensive tabulation of atomic spectra is given by
Y. Ralchenko, A. E. Kramida, J. Reader, NIST ASD Team (2008). NIST Atomic Database (version 3.1.5),
[Online]. Available at http://physics.nist.gov/asd3 [10 October 2008]. National Institute of Standards and
Technology, Gaithersburg, MD, USA.
Accounts of the history of the understanding of the spectrum of hydrogen and of the role of Rydberg are to be
found in
M. Sutton, Chem. World 1 (July), 38 41 (2004) and references cited therein.
T.W.H€ansch, A. L. Schawlow, G.W. Series, Sci. Am. 240 (March), 72 86 (1979) and references cited therein.
A simple and inexpensiveway of observing line spectra from flames and street lamps using plastic diffraction
gratings is described by
J. Walker, Sci. Am. 250 (January), 112 117 (1984).
For a description of crystal- and ligand-field theory with respect to colour, see
D.W. Smith, Ligand field theory and spectra, inWiley Encyclopedia of InorganicChemistry, 2nd edition, R. B.
King (ed.), Wiley, Chichester, 2005 and the many references cited therein.
The original group theoretical paper concerning the splitting of terms in a crystal field is
H. Bethe, Ann. Phys. 395, 133 208 (1929).
An introduction to group theory of relevance to colour is given by
S. B. Piepho, P. N. Schatz, Group Theory in Spectroscopy, Wiley, New York, 1983.
The use of Nd3þ laser amplifiers at the National Ignition Facility is described by
M. Moyer, Sci. Am. 302 (March), 35 41 (2010).
307 Colour from Atoms and Ions
The pigments Egyptian blue, Han blue and Han purple are described in
S.Colinart,M.Menu (eds),LaCouleur dans laPeinture et L’Emaillage de l’Egypte Ancienne, Edipuglia, Bari,
1998. The paper by H. G. Wiedemann, G. Bayer, A. Reller, p. 195, is of especial relevance.
Inorganic pigments, from the point of view of an artist, are the subject of
V. Findlay, Colour; Travels through the Paintbox, Folio, London, 2009.
For spectral holes, a good first source is
E. S. Maniloff, A. E. Johnson, T. W. Mossberg, Mater. Res. Soc. Bull. 24 (September), 46 50 (1999).
Colour and the Optical Properties of Materials 308
8
Colour from Molecules
. Why is deep water tinted blue?
. What colours roses red and cornflowers blue?
. What is a blueprint?
The subject matter encompassed by this chapter is enormous and the topics are only covered in outline. Some
books that cover these topics in detail are listed in this chapter’s Further Reading.
8.1 The Energy Levels of Molecules
Whereas a gas of atoms emits light at precise wavelengths to give a series of sharp lines, molecules may emit
sharp lines and extended bands. Each band in a molecular spectrum generally has one sharp side and a diffuse,
gradually fading side to it. Under high resolution the bands are seen to be made up of closely spaced series of
lines. Thus, the spectrum of even a simple molecule such as O2 will be vastly more complex than the line
spectrum of an isolated oxygen atom. However, the individual lines in a molecular spectrum, whether isolated
or as part of a band, still represent the energy difference between two energy levels:
DE ¼ E1�E0 ¼ hn ¼ hc
lð8:1Þ
The origin of the transitions can be broken down into three components. The electrons in themolecule can be
excited to higher energies involving an energy change DEel. Here, we can note that the outer electrons in
particular do not occupy orbitals centred upon atomic cores, like the atomic orbitals of Chapter 7, but occupy
molecular orbitals that extend over thewhole of themolecule and can be considered to be derived fromoverlap
Colour and the Optical Properties of Materials Richard J. D. Tilley
� 2011 John Wiley & Sons, Ltd
of atomic orbitals. Electronic transitions can then be considered to be analogous to those in atoms, but now the
electrons are switched from lower energymolecular orbitals to higher energymolecular orbitals andviceversa.
As in the case of atoms, these transitions are governed by selection rules, and energy states can be described by
molecular term schemes (see this chapter’s Further Reading). The electronic energy levels are separated by
energies of the order of 6� 10 19 J (360 kJmol 1, 3.7 eV, 30 000 cm 1), and these produce spectral lines in the
visible and ultraviolet region.
A molecule may also vibrate, and each single electronic energy level is accompanied by one or more sets
of energy levels that correspond to vibrational transitions. The energy increment of the vibrational levels
DEvib is about a tenth of that between the electronic energy levels, i.e. 6� 10 20 J (36 kJmol 1, 0.37 eV,
3000 cm 1) and transitions between these levels give rise to absorption and emission at infrared wave-
lengths. Finally, rotation gives rise to further energy level increments DErot which are added to the
vibrational levels. The energy step of the rotational levels is about a hundred times smaller again, at
approximately 6� 10 22 J (4 kJmol 1, 0.0037 eV, 30 cm 1) and transitions between these levels give rise to
microwave absorption and emission. Thus, each single line corresponding to an electronic transition in a
single atom is transformed into a closely spaced set of lines (a band) in amolecule (Figure 8.1). The equation
for energy exchange now becomes:
DE ¼ ðEel þEvib þErotÞ1�ðEel þEvib þErotÞ0 ¼ hn ¼ hc
l
The theoryunderlying the electronenergy levelsofmolecules is, in principle, but a littlemorecomplex than that
of atoms, and the calculations, using molecular orbital theory, can be carried out routinely. However, in
practice, the bewildering complexity of many molecules makes the work feasible only for simpler structures.
Fortunately, for our purposes, the colours arising in molecules can be understood by ignoring almost all of the
molecular orbitals and focusing attention upon just two. These are the molecular orbital of highest energy that
contains electrons and the first molecular orbital above it in energy that is empty of electrons (Figure 8.2). In
a shorthand notation this pair of orbitals is often referred to as the highest occupied molecular orbital, or
HOMO, and the lowest unoccupiedmolecular orbital, orLUMO.These are alsoknownas frontier orbitals, and
as well as of importance for colour, they also influence the outcome of chemical reactions between molecules.
Molecular orbitals are also labelled according to their effect upon the stability of a molecule. An electron-
containing orbital is abondingorbitalwhen the electronswithin it contribute to the chemical bondsbetween the
atomsof themolecule and so stabilize themolecule.Anorbital is anantibondingorbitalwhen its occupation by
electrons destabilizes themolecule.Antibonding orbitals are usually labelledwith an asterisk. Somemolecular
orbitals, which are neutral as far as molecular stability is concerned, are called nonbonding orbitals. These
frequently house d electrons or lone-pair electrons, more or less located on a single atom. The HOMO and
LUMO may be any of these types.
In the case of the intensely coloured organic molecules that are of most interest in this chapter, the highest
molecular orbital containing electrons, the HOMO, is often a pmolecular orbital, derived by overlap of atomic
p orbitals on the atomsmaking up themolecular skeleton. Similarly, the LUMO is often a p� molecular orbital,
derived from the same atomic orbital type, the asterisk indicating antibonding status.One of themost important
electronic energy transitions for colour production in these complex molecules is electron excitation from a
p-typeHOMOto thep�-typeLUMO, called ap top� transition. Such transitions give rise to intense absorptionbands with high absorption coefficients and are found in molecules containing conjugated single and double
bonds (Section 8.5 and elsewhere). Transitions from a nonbonding HOMO to a p� LUMO, n to p� transitions,are also possible. Thesegive rise to less-intense absorption bands than thep top� transitions but are nonethelessimportant. They occur, for example, inmolecules containing a (>C¼O) group (the ketones) and are the source
of colour in a variety of dyes.
Colour and the Optical Properties of Materials 310
Although the vibrational and rotational energy-level separation is too small to give rise to colours, these
additional increments of energy can significantly modify the tone of the gross colour due to the electronic
HOMO LUMO transition.
8.2 The Colours Arising in Some Simple Inorganic Molecules
Theelectronic transitions ofmany simplemolecules lie in the ultraviolet and sodonot lead to significant colour.
An exception is provided by the vapours of the halogens Cl2, Br2 and I2, which respectively exhibit colours of
yellow-green, red-brown andpurple-violet. Themolecular orbitals of interest for colour production are derived
from overlap of the outer p orbitals on the two halogen atoms. The HOMO is a filled pair of p *g orbitals.1
electronic
electronic
vibrational
vibrational
rotational
rotational
Ene
rgy
Figure 8.1 Electronic, vibrational and rotational energy levels of amolecule (schematic). Each electronic energylevel has additional associated energy levels due tomolecular vibration and rotation. The electronic energy levelsare separated by approximately 6� 10�19 J (3.7 eV), the vibrational levels by approximately 6� 10�20 J (0.37 eV)and the rotational levels by approximately 6� 10�22 J (0.0037 eV)
1 The labels g and u give information about the symmetry of the molecular orbitals and the electronic transitions that are possible.
311 Colour from Molecules
The LUMO is an unoccupied s *u orbital (Figure 8.3a). The colour of the gases is derived from an electronic
transition from the p *g HOMO to the s *
u LUMO. As one moves from Cl2 towards I2, the HOMO LUMO gap
decreases and the absorptionmaximummoves towards the red end of the spectrum,modifying the colour from
yellow-green to purple-violet.
An atomic electronic absorption peak is generally a simple narrow bell shape. In contrast to this, the
absorption spectrum for a molecule will consist of a series of bands that can be thought of as approximately
occupying the envelope of the corresponding electronic transition (Figure 8.3b). This is because the excited
state can be one of a number of vibrational levels associated with LUMO, and additionally can be one of
a number of rotational levels associated with each vibrational state. In the spectrum of Br2, for example, which
lies between the approximate limits of 500 850 nm, there are of the order of 80 000 transitions associated with
approximately 150 bands. Thus, although the vibrational and rotational energy-level separation is too small to
LUMO
HOMO
Ene
rgy molecular orbitals
electron pairs
hν in hν out
ground state
excited state ground state
(a)
(b)
Figure 8.2 The molecular energy levels of importance in producing colour in many compounds (schematic):(a) the highest occupiedmolecular orbital (HOMO) and the lowest unoccupiedmolecular orbital (LUMO); (b) themajor colour-producing electronic transitions, dotted arrows, are often between the HOMO and the LUMO
Colour and the Optical Properties of Materials 312
give rise to colours directly, these additional increments of energy can significantlymodify thegross colour due
to the electronic HOMO LUMO transition. In this way, the actual spectra of the halogens are much more
complex than the simple description given above suggests, and the tint of the colours displayedmay be almost
entirely attributable to the influence of vibrational and rotational energy levels.
Colours due to molecular transitions (often mixed in with atomic transitions) are seen in the upper
atmosphere as the spectacular displays aurora borealis or aurora australis. The auroras form at heights of
100 to 1000 km above the polar regions. Very energetic particles, mostly electrons together with some protons,
mainly originating in the sun, spiral along the Earth’s magnetic field lines towards the poles. When they reach
the tenuous outer limits of the atmosphere they collide with and excite the atoms and molecules encountered.
These excited species lose energy by radiating in part in the visible and give rise to the remarkable shifting
curtains of colour seen in far northern and southern latitudes.
The major components contributing to the colours are nitrogen molecules (N2) and oxygen atoms (O).
Nitrogen molecules can become ionized to Nþ2 which then can recapture an electron to leave an excited
nitrogen molecule2 (N*2). This species then decays to the ground state, giving out light in the process:
N2 þ e ðfrom spaceÞ!N þ2 þ 2e
N þ2 þ e !N *
2 þ violet and blue light
N *2 !N2 þ pink light
p5 p5
σ*u
σg
π*g
πu
HOMO
LUMO
Inte
nsity
electronic + vibrational + rotational (bands)
(a)
(b)
envelope of electronic transition
Figure 8.3 (a) Themolecular orbitals (schematic) of the halogensCl2, Br2 and I2 that arise from the overlap of theoutermost p5 orbitals on the atoms. (b) The envelope of the electronic transition, arrowed in (a), which isresponsible for the gross colour of the gases. The (highly simplified) band spectrum of a halogen molecule isapproximately bounded by the envelope of the electronic transition and is responsible for the perceived colour ofthe gases
2 Note that here the asterisk means an atomic or molecular excited state. This use of the asterisk symbol for both antibonding orbital and
excited state is commonplace.
313 Colour from Molecules
Oxygen atoms (O), which are more common than oxygenmolecules in the outer regions of the atmosphere,
are formed by photodissociation of O2 under intense ultraviolet irradiation in these near-space conditions.
These are excited by electron bombardment to form excited O� species which return to the ground state by theemission of whitish-green and crimson light:
Oþ e !O*þ e ðwith a lower energyÞO*!Oþwhitish green and crimson light
Colour from molecular transitions is also seen in the blue region around a candle flame (Figure 8.4). The
chemistry of this region is complex and a large number of molecular fragments occur when the candle wax is
vaporized. The blue colour is mainly produced by excitations of the two unstable molecular fragments C2 and
CH. The strongest CH band is at 432 nm in the blue region of the spectrum, while C2 has a strong band in the
green with less-intense bands in the blue and violet regions. Themain part of the flame appears orange yellow
due to incandescent carbon particleswhich are deposited as sootwhen striking a cold surface. (The spectrumof
a candle, measured with an inexpensive diffraction grating, will show a continuous spectrum from the heated
carbon particles. More-sophisticated equipment is needed to analyse the spectrum from the blue part of the
flame.)
yellow outer flame
dark unburnt gases
blue margins
(a)
Wavelength / nm
(b)
400 500 600 700
CH CH CH C2
Figure 8.4 A candle flame: (a) the blue colour of the outer sheath at the base of the flame arises from transitionswithin CH and C2 molecules; (b) the positions of the main emission bands from these species
Colour and the Optical Properties of Materials 314
Flame colours and St Elmo’s fire have been mentioned in the Sections 7.3 and 7.5, where it was pointed out
that many of the characteristic colours arise frommolecular transitions rather than atomic or ionic transitions.
The same is true of the colours produced in firework displays. The combustible part of the firework raises the
temperature of the colouring agent and excited atomic and molecular fragments then release energy, much as
visible colours, as they cool and recombine. The species present are complex and include metal atoms, metal
ions, and chloride, oxide and hydroxide fragments. For example, green colours are generated by barium salts,
such as BaCO3, Ba(NO3)2 and BaSO4. Themain colour-emitting fragments are believed to be BaOH (487 and
512 nm) and BaO (549, 564, 604 and 649 nm) Red is produced by the inclusion of strontium salts, especially
SrCO3, Sr(NO3)2 and SrSO4. The main colour-emitting fragments are believed to be SrOH (506 and 722 nm)
and SrCl (618, 636 and 661 nm). Naturally, great skill is required to blend these chemicals with the other
components of the firework to obtain reproducible effects.
Sonoluminescence, light generated when a high-intensity ultrasound wave is passed through a liquid, is
a related phenomenon. The intensity of the light emitted can be high and easily visible in daylight. The effect of
the ultrasound waves is to cause bubbles of vapour to grow within the body of the liquid which ultimately
collapse. During this cycle there is intense compressional heating taking place. The light emitted, the
sonoluminescence, is interpreted as the emission spectra of excited molecules and molecular fragments
that occur within the bubbles. These include OH , commonplace in water solutions, organic fragments
including C2 and inert gas atoms, especially when these are introduced into solution as markers. The
temperatures reached in the collapsing bubbles can reach the order of 10 000K, which is more than sufficient
to produce intense emission spectra. The characteristic colours can be used to determine the atomic and
molecular fragments present and estimate the temperatures within bubbles.
8.3 The Colour of Water
Water is a deceptively simple compound indeed, if a person knows but one chemical formula it is most likely
to be that of water, H2O. One aspect of water that seems to be of never-ending fascination is its colour. Many
famous scientists at the endof the nineteenth and early years of the twentieth centuries put forward explanations
for the colour, but it is only inmore recent times that a consensus has started to appear on this topic. The reasons
are not hard tofind for this apparently curious fact.The colour ofwater bodies innaturedepends upon reflection,
scattering, impurities, the aspect of the sky and so on. Here, the colour of pure water is discussed.
The colour of pure water in transmission is blue because red light is more strongly absorbed than blue.
Passing from the surface to greater depths in clear sea will render the light that penetrates a deeper and deeper
blue. Absorption is due to transitions between the various energy levels described above, in particular between
the vibrational energy levels. Thewater molecule is angular with a bendingmode of vibration n2, which, in thegas phase, absorbs energy in the infrared, at a wavelength of 6273 nm. In addition, two stretching modes, in
which the bonds in themolecule lengthenand shorten, alsooccur.Oneof these, inwhich the bonds lengthen and
shorten together, the symmetricalmode n1, absorbs energy at 2738 nm in the gas phase. The other, inwhich one
bond lengthens as the other shortens, the antisymmetricalmode n3, absorbs energy at 2662 nm in the gas phase.
These absorption wavelengths are far from the visible and, as is well known, water molecules in the vapour
phase are colourless.
Although the three absorption peaks formolecules ofwater do not directly produce colour, they can combine
to produce overtones, which are harmonics, and combinatorial tones, which are sums, of the fundamental
frequencies. For example, if we set the frequencies of the absorptionmaxima as n1, n2, and n3, the overtones areof the form 2n1 and the combinatorial tones are of the form 2n1 þ n3. The existence of these terms extends the
spectrumofwatermoleculesmuch closer to thevisible; close enough, in fact, to present a sensation of colour to
the eye.
315 Colour from Molecules
The weak absorption of light in the red region of the spectrum of both water and ice is due to a peak in the
absorption spectrum at 760 nm in the infrared, the tail of which extends into the visible. There are also weaker
peaks at 660and605 nm in the orange redpart of the spectrumwhich contribute to the removal of the redpart of
the spectrum. It is difficult to assign these peaks to specific overtones and combinatorial tones because of the
multiplicity of possible arrangements available. However, it has been suggested that the peak at 760 nm is due
to overtones of the fundamental n1, O�H stretching vibration, in particular 3n1 þ n3 and n1 þ 3n3. Althoughthe absorption due to these transitions is very weak, it is enough to remove a small fraction of red and orange,
which is sufficient to give sizeable bodies of pure water or ice a pale (watery!) blue colour.
There are twoways inwhich infrared absorption bands can bemoved. Themass of the atoms in the bonds can
be increased and the bonds can be made stronger. (Both of these attributes are of importance in the improved
optical transmission of heavy-metal fluoride-glass optical fibreswith respect to silica fibres (Section 2.9). In the
case of ordinary water, the atoms are of fixedmass, but in the liquid and solid states the interatomic bonding is
altered compared with the gas phase. The change comes about because of hydrogen bonding, which links the
molecules together by additional liaisons. Although hydrogen bonding is weak, with a bond energy of
approximately 20 kJmol 1, comparedwith theH�Obond strength of 463 kJmol 1, it is of significance. In the
case of water, the change in bonding is enough to shift the absorption spectrum to longer wavelengths; that is,
further into the infrared.Hydrogen bonding is stronger in solid ice than liquidwater and so the absorption bands
in ice are slightly red-shifted compared with those found in the liquid. This alters the colour of ice slightly,
compared with water, making it more blue green.
The effects of the mass of the atoms can be investigated by a study of heavy water (D2O). The vibrational
absorptionbands are considerably shifted to longerwavelengths inD2O.Forexample, the band at about 760 nm
in water is found at 1000 nm in D2O. These bands are now well into the infrared, and D2O will be ‘white’
compared with blue water.
8.4 Chromophores, Chromogens and Auxochromes
The earliest studies in organic chemistry showed that colours in organic molecules could be manipulated
experimentally. For example, it was found that many coloured organicmaterials were turned colourless by the
addition of hydrogen and returned to their original colours by the removal of hydrogen. To try to rationalize the
experimental observations theGerman chemistWitt suggested, in 1876, long before quantum theory andX-ray
structural studies, a series of guidelines relating to the colour of organic molecules. The source of the colour in
a molecule was supposed to be one or more ‘colour-bearing’ small groups of atoms with multiple bond
configurations, called chromophores.3 Some important chromophores are listed in Table 8.1.A compound that
Table 8.1 Some chromophores
Group name Formula Group name Formula
Nitro NO2 Azoxy N¼N OCarbonyl ¼CO Nitroso NOAzo N¼N Azoamine N¼N NHThiocarbonyl ¼CS Ene >C¼C<
3 This terminology is now not restricted to organic molecules and is often used in inorganic chemistry to denote a group of atoms or ions
which cause colour. For example, the Cr3þ centres in ruby are sometimes called chromophores.
Colour and the Optical Properties of Materials 316
could be made coloured by the addition of chromophores was called a chromogen. The depth of colour of the
chromogenwas proportional to the number of chromophores present. It was recognized that some groups in an
organic molecule, called auxochromes, also played a role. Although auxochromes did not produce colour
themselves they had the effect of intensifying the colour of a molecule if a chromophore was present. The
important auxochromes are hydroxyl (�OH), keto groups (>C¼O) and groups of atoms containing nitrogen.
The changes in colours produced by chromophores and auxochromeswere described as bathochromic if the
wavelength shifted to longer wavelengths (i.e. blue to red) and hypsochromic if the reverse occurred (i.e. red to
blue). The change in the depth of colour was described as hyperchromic if the absorbance increased and
hypochromic if the absorbance decreased.
Theoretical calculations show that the presence of chromophores decreases the energy between the HOMO
and the LUMO. Themore chromophores there are in amolecule, the greater is the decrease in energy. Thus, in
cases where the main absorption band of a parent molecule lies in the ultraviolet, the absorption band of a
daughter molecule containing one or more chromophores is moved towards the visible. In suitable cases the
result is the transformation of a colourless parent compound into an intensely coloured daughter molecule.
Much of the remainder of this chapter is concerned, in one way or another, with the way in which a HOMO
and LUMO energy separation that gives rise to an absorption in the ultraviolet is reduced in magnitude so that
the absorption is brought into the visible. In the carotenoids, which follow, this comes about by joining double
and single carbon bonds in a line. In the porphyrins, a metal cation bonds to several molecules so as to produce
the same double single bond effect. In sensors for the detection of metal ions, the colour change that serves to
indicate the presence of the cation is due to a change in HOMO LUMO separation induced by the cations
themselves, either by forming new molecular arrangements or by shifting the existing ultraviolet absorption
maximum. The same theme will be spotted throughout all the later sections.
8.5 Conjugated Bonds in Organic Molecules: The Carotenoids
The>C¼C< double bond (-ene) arrangement linking two carbon atoms is formed by the overlap of p orbitals
on the two adjacent carbon atoms to give a p HOMO and a p� LUMO. Although the>C¼C< double bond is
regarded as a chromophore, an isolated>C¼C< group has a p to p� absorption band centred at a wavelengthnear to 160 nm in the far ultraviolet and so does not lead to colour in a molecule. However, a dramatic change
occurs when a number of these units are arranged in an alternating single-bond double-bond arrangement, to
form a sequence of conjugated double bonds and the p to p� absorption band approaches the visible. For
example, whereas the absorption maximum of ethene (CH2¼CH2) is at 162.5 nm, that of the compound
CH3�CH¼CH�CH¼CH�CH¼CH�CH¼CH�CH3 (or CH3�(CH¼CH)4�CH3), with four conjugated
double bonds, has an absorption maximum at approximately 300 nm. Colour is first found in the molecule
containing six conjugated double bonds, CH3�(CH¼CH)6�CH3, in which the absorption maximum
encroaches into the blue end of the spectrum, causing the molecule to appear yellow. In this sense it is the
conjugated set of >C¼C< double bonds that is the chromophore.
(It must be remembered that the ultraviolet spectra of these molecules will be complex; far more so than
the inorganic molecules described above. Thus, although the HOMO LUMO transition might dominate the
spectrum, it will appear as a broad peak or band rather than a single sharp line.)
Two of the more important conjugated molecules are a- and b-carotene (Figure 8.5a and b). These
substances, when pure, form deep purple red orange crystals with a strong absorption maximum at
approximately 450 nm (indigo). These compounds are so named because they were first isolated from the
cultivated carrot, Daucus carota, although they are found in many orange and yellow flowers. Lycopene
(Figure 8.5c) has an absorption peak further into the visible than b-carotene, nearer to 475 nm (blue) and gives
a red colour to fruit and flowers. It is found in tomatoes, which it endowswith thewell-known bright red colour.
317 Colour from Molecules
H C3
H C3
CH3 CH3 CH3
CH3 CH3 CH3CH3
CH3
(a)
(b)
H C3
H C3
CH3 CH3 CH3
CH3 CH3 CH3CH3
CH3
α-carotene
β-carotene
CH3 CH3 CH3 CH3
CH3 CH3 CH3 CH3
(c)
H C3
H C3CH3
CH3
CH3 CH3
CH3 CH3
CH3
CH3
HO
OH(d)
anthophyll (lutein)
lycopene
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
COOH
(e)
(f)
HOOC
O
RO
O
RO
crocetin
crocin
Figure 8.5 The structures of conjugated molecules: (a) a-carotene, purple–red; (b) b-carotene, orange–red;(c) lycopene, deep red; (d) xanthophyll (lutein), yellow–orange; (e) crocetin, yellow; (f) crocin, yellow–orange; (g) (S,S)-astaxanthin. In these and succeeding figures, carbon (C) and hydrogen (H) atoms are omittedfrom the main skeleton of the molecule and only the carbon–carbon single and double bonds are depicted, assingle anddouble lines respectively. At the periphery of themolecule the atoms are indicated. Apart fromCandH,O represents oxygen and R symbolizes a general organic group of bonded atoms
Colour and the Optical Properties of Materials 318
Underripe tomatoes and thosewhich have been developed to show other colours have no ormodified lycopene
present.
Closely related to these are thepigments xanthophyll (also called lutein), crocin and crocetin (Figure8.5d f).
These are an orange yellowcolour.Xanthophyll occurs in the petals ofmanyflowers and also colours eggyolk.
Crocetin is a brick-red compound and crocin, yellower, is familiar as saffron, which is derived from crocus
pollen. The structures of these compounds clearly show the conjugated backbone of the molecules.4
The group of pigments structurally related to those shown are known generally as the carotenoids. All owe
their colours to the conjugated double bond configuration in the molecules. There are two main groups of
carotenoids as far as colour is concerned: the carotenes, which are hydrocarbons, and the related oxygen-
containing compounds comprising the alcohols, ketones, aldehydes, ethers and carotenol esters. These latter
are collectively also knownasxanthophylls. Themoleculea-carotin is a carotene,while crocetin and crocin arexanthophylls. Many xanthophylls can be recognized by the name ending -xanthin; for example, taraxanthin,
which is found in dandelions (Taraxacum species) and lycoxanthin (C40H56O), which is the mono-alcohol
derivative of the carotene lycopene (C40H56). Xanthophylls cannot bemade by animals; although important (in
vision for instance, Section 1.10), theymust be ingested fromplantmaterial. Similarly, theyellowcolour of egg
yolks is derived from ingested xanthophylls.
The structural chemistry of these molecules is complex and many puzzles regarding the colours perceived
remain to be solved. An example is given by the colour change experienced when lobsters, shrimps, crabs and
related crustaceans are boiled.When living, these animals appear in a variety of slate-blue tones. This is due to
the presence of a-crustacyanin, a complex molecule containing 16 protein chains bound to 16 astaxanthin
molecules.When the animal is boiled the protein chains denature and the colour turns orange/red, the colour of
isolated astaxanthin molecules. Astaxanthin itself exists as three optical isomers, each of which is closely
related in structure to b-carotene (Figure 8.5g). All of these are orange/red. The reason for the change in huefrom orange/red to slate bluewhen thesemolecules are incorporated into a-crustacyanin is still not completely
understood. It appears that the astaxanthinmolecules bound to the proteins are held in a flattened form that has
the effect ofmoving theperceivedcolour towardsblue, butnot sufficiently to account for the total colour change
observed. The full explanation of this colour change is still being sought.
8.6 Conjugated Bonds Circling Metal Atoms: Porphyrins and Phthalocyanines
Systems of conjugated bonds which circle a metal atom can give rise to rich colours and many molecules
containing this arrangement are important to life processes. The twomain classes of compounds in this context
H C3
H C3
CH3 CH3 CH3
CH3 CH3 CH3CH3
CH3
(S,S)-astaxanthin
O
HO
O
OH
Figure 8.5 (Continued)
4 An examination of Figure 8.4 will show that the sequence of double and single bonds can be drawn in more than one way. This is a
shortcoming of the stick like depiction of the bonds; the various forms are called resonance hybrids. In reality, the bonding is best thought
of in terms of molecular orbitals which extend over much or all of the structure.
319 Colour from Molecules
are called porphyrins and phthalocyanines. Of these, the porphyrin chlorophyll, the source of the green colour
of plants, is surely among the most important molecules known.
Chlorophyll is found in four forms, called chlorophyll a, chlorophyll b, chlorophyll c and chlorophyll
d. Higher plants and green algae contain chlorophyll a and b in a ratio of about 3:1. Red algae contain mainly
chlorophyll a and some chlorophyll d. Chlorophyll c (together with some chlorophyll a) is found in many
marine algae. The central core of all chlorophyll molecules is a magnesium atom surrounded by a sequence of
alternating double and single bonds (Figure 8.6). The molecule can be considered to be derived from a group
of four pyrrole rings, a molecule that contains extended molecular orbitals that encompass the ring structure,
and which, in chlorophyll, extend over thewhole of the central region. Chlorophyll molecules absorb strongly
in the blue and red parts of the spectrum. The colour reflected by leaves corresponds to those wavelengths not
strongly absorbed, which are the greens, providing a good example of subtractive coloration (Figure 8.7).
The core of a similar vital porphyrin molecule, haem (heme), is iron rather than magnesium (Figure 8.8a).
Haem is a planar structurewhich is responsible for oxygen transport in the bloodstream. Like chlorophyll, it is
alsomade up from four pyrrole rings and contains at its centre an Fe2þ ion. This in itself is remarkable, because
the stable form of Fe ions in the presence of oxygen is Fe3þ . It forms the central feature of the molecule
haemoglobin, which transports oxygen to and fro in the cells of the body. It is also responsible for the colour of
N
N
N
N
Mg
CH R
OCOOCH3
CH2COOR′
CH3
R = CH3 (chlorophyll a)
CH3
CH3
CH3
CH2CH3
CH2
H3C
H2C
( )3
R′ =
R = CHO (chlorophyll b)
NH
pyrrole
Figure 8.6 The structure of chlorophyll. The molecule is built around a central magnesium (Mg) atom, linked tofour nitrogen (N) atoms. This arrangement forms a typical porphyrin ring structure. The structure of the group Rvaries from one type of chlorophyll to another
Colour and the Optical Properties of Materials 320
Figure 8.7 Chlorophyll leaf colours: (a) strawberry tree (Arbutus unedo) ; (b) cyclamen; (c) rosemary; (d) sage.Although the green colours are produced by chlorophyll in each case, the appearance of the leaves differs greatly,due to shape and surface coatings
321 Colour from Molecules
blood, as the group has strong absorption maxima in the green-yellow part of the spectrum. Reds and purples,
therefore, are reflected to produce the coloration of fresh blood.
In blood, haem is unitedwith the colourless protein globin to formhaemoglobin. In adults, the principal form
of haemoglobin contains four protein chains, two a-chains and two b-chains, to form a roughly spherical
molecule of about 5.5 nmdiameter. A haemgroup is embedded in each of these chains in such away that the Fe
ions are bound to another nitrogen atom to one side of the plane of the haem centre, so that the Fe is surrounded
by a square pyramid ofN atoms. This geometry ensures that there is limited access to the Fe ion, which is to the
one side not bonded to N, and O2 molecules (as well as CO2 and CO) link here to complete octahedral
coordination about the cation.
This restricted access,which is due to thevery specific folding of the protein chains,makes the binding of the
oxygen and carbon dioxidemolecules reversible and also stabilizes the Fe2þ state over the Fe3þ state. It is also
responsible for the relatively weak binding of carbon monoxide, although this is still strong enough to cause
death when inhaled in sufficient quantities.
The mineral Fe2O3, called haematite, was so named because its red colour was reminiscent of fresh blood.
In Fe2O3 the colour arises from transitions involving the 3d electron levels on the Fe3þ ions (Section 8.10.4).
In haemoglobin, despite the fact that an iron atom is present, the colour arises from p to p� transitions, not fromthe iron at all!
The phthalocyanines, discovered early in the twentieth century but not characterized structurally until the
1930s, are rather similar metal-centred molecules related to porphyrins. The metal-free form of this series,
which is blue, was first synthesized in 1907, and metal-containing derivatives, incorporating typically Cu, Fe,
Al, Ni, Co, Zn, etc., soon followed. These mainly show colours in the blues and green/blues. Copper
phthalocyanine, a widely available blue compound, Pigment Blue 15, is manufactured in large quantities
and used in inks, paints and plastics (Figure 8.8b). The colour of thematerial is only slightly changedwhen the
Cu central atom is replaced by an alternative metal, revealing that the colour is not directly due to the copper.
The blue colour arises from p to p� transitions in the phthalocyanine ring system. This can bemodified to some
extent by changing the atoms linked to the ring system, andCl and F substitutions are used to this end. A typical
inorganic chemical catalogue will list several dozen of these derivatives, all of which offer slightly different
properties to the colour industry.
Although copper phthalocyanine is not found in nature, rather similar blue compounds do occur. They are
found in the blue blood of hermit crabs and related crustaceans.Theblue colour arises fromp top� transitions in
Figure 8.7 (Continued)
Colour and the Optical Properties of Materials 322
copper-containing haemocyanin molecules, which transport oxygen and play an analogous role to the
haemoglobins in mammalian blood. Thus, we find, as in the case of haem, that the colour of the molecule
is similar to the crystal field colour of the central cation, Cu2þ , but arises from a quite different mechanism.
8.7 Naturally Occurring Colorants: Flavonoid Pigments
8.7.1 Flavone-related colours: yellows
The flavonoids are an enormous group of diverse and colourful pigments named after the compound flavone,
first isolated from the Fairy primrose Primula malacoides. (Note though, that flavone itself is colourless.) The
group includes the chalcones (yellow orange), the flavones (ivory cream), the flavonols (yellows) and the
anthocyanins (pink violet). They are mostly derived from a phenylpropane-related precursor by a number of
metabolic pathways within the developing plant (Figure 8.9).
The wealth of flower colours derives from a limited number of basic molecules by the substitution of some
of the hydrogen atoms by a range of other groups. For example, the influence of increasing the number of
auxochrome (�OH) groups on colour is well illustrated in the sequence of compounds flavone, which is
colourless, flavonol, which is pale yellow, kaempferol, which is deep yellow, and quercetin, which is orange
(Figure 8.10).
H C3
H C3 CH3
CH3
CH
CH
CH2
CH2
CH2CH2
CH COOH2CH COOH2
N
Cu
Fe
N
N
N
N
N
N N
N
N
N
N
(a)
(b)
Figure 8.8 The structures of (a) red haem, a porphyrin; (b) blue copper phthalocyanine. The colour is producedwithin the organic structure, not by the transition metal cation
323 Colour from Molecules
Althoughmostly associatedwith plant colours, the flavones and flavonols also appear in animals, where they
are assimilated fromplants.This has been studied in anumber ofbutterfly species, including theMarbledWhite
(Melanargia galathea) and the Common Blue (Polyommatus icarus) (Figure 8.11a and b). In the Marbled
White, the presence of a number of flavonoids, including quercetin and kaempferol, are associated with
a yellow brown colouration. In the Common Blue, the presence of kaempferol glucoside is particularly
associated with the orange lunules on the underwings. As these flavonoids absorb strongly in the ultraviolet
region of the spectrum, it has been suggested that the distribution of flavonoid species in the wings might
produce patterns that are visible to butterflies but not to us.
The flavones react readily with ammonia (NH3) to produce much deeper yellow colours. This provides an
easy test for the presence of flavones in nature. For example, thewhite areas on thewings of theMarbledWhite
butterfly (Figure 8.11a) turn a deep yellowwhen exposed to ammonia vapour. This change is an example of an
auxochromic shift, caused by the fact that the nitrogen-containing ammonia, when bound to the flavone,
increases the electron delocalisation in the molecules. In butterflies the reaction is reversible, so that the deep
yellow colour returns to the original white tone when the ammonia fumes are removed.
8.7.2 Anthocyanin-related colours: reds and blues
Many of the blues and reds of flowers are derived from a group of flavonoid-related compounds called
anthocyanins. Thenamederives fromcyanin (¼blue) as thecompoundwasfirst isolated fromblue cornflowers,
Centaurea cyanus. All the anthocyanins absorb strongly in the green region of the spectrum, thus allowing the
flowers to reflect varying proportions of reds and blues. The colour range of flowers and fruits using
anthocyanins spans the range from salmon pink through to blue and violet (Figure 8.12).
The diversity of this group of plant pigments is considerable. The anthocyanins are composed of an
anthocyanidin plus one or more sugar molecules. The anthocyanins are glycosides of anthocyanidins, and the
anthocyanidins themselves are the aglycons of anthocyanins.5 There are about 30 anthocyanidins known,
which yield about 1000 anthocyanin pigments when the various sugar substitutions are taken into account.
Table 8.2 gives information for some of the most widely distributed pigments and the flowers in which they
occur.
phenyl propane
quinones(reds and browns)
flavonoids
chalcones(orange - yellow)
flavones(ivory - cream)
flavonols(yellow)
anthocyanidins-------anthocyanins(pink - scarlet - blue - violet)
absorption moves from ultraviolet towards red
Figure 8.9 Schematic relationships between various flavonoids. (Note that the quinones are not flavonoids)
5 An aglycon is the non sugar compound remaining after replacement of the glycosyl group from a glycoside by an H atom.
Colour and the Optical Properties of Materials 324
The generic structure of the anthocyanidins (Figure 8.13a) is transformed into specific pigments by the
substitution of other groups forR1 andR2.Note that the anthocyanidin unit is a cation, called a flavylium cation,
and is usually associated with a corresponding anion. For example, cyanidin is often isolated as the chloride
(Figure 8.13b). These anthocyanidins are transformed to anthocyanins by addition of sugars, usually at the
CH2
CH
CH3
C
phenyl propaneCH2
CH
O
chalcone
O
O
O
O
O
O
flavone
flavonol
kaempferol
OH
OH
HO
OH
OH
(a)
(b)
(c)
(d)
(e)
O
O
quercetin
OH
OHHO
OH
OH
(f)
Figure 8.10 The structures of some flavonoid-related molecules: (a) phenylpropane; (b) chalcone; (c) flavone;(d) flavonol; (e) kaempferol; (f) quercetin
325 Colour from Molecules
oxygen atomsO3 andO5. For example, cyanidin,when linked to twoglucose units at these positions, forms the
3,5 diglucoside called cyanin (Figure 8.13c).
The colour of the pigment produced in a flower depends upon R1 and R2 and the sugars attached to the
molecule. Although the absorption spectra of all of these derivatives are rather similar, slight changes in
the absorption maxima make significant changes to the hue perceived by the viewer.
Having identified the pigment is only a part of the story and it does not suffice to explain flower colours in
detail. The first observation on this wasmade in 1913 byWillst€ater and co-workers. They observed that cyaninoccurred in blue cornflowers (the origin of the name cyanin, as mentioned above) and in red rose petals.
Experiments showed that the colour of the cyanin molecule was red in acid solution, pale violet in neutral
solution and blue in alkaline solution. This lead to the pH theory of flower colours, in which different shades
were associated with differences in the pH of the sap or other cell fluids present in the organelle containing the
pigment molecules. However, the theory does not account for all colours, as alkaline plant fluids are not at all
usual. Moreover, these colours fade rapidly under normal conditions, leading to questions concerning the
stability of the colours in nature.
Figure 8.11 Flavonoid-containing butterflies: (a) Marbled White (Melanargia galathea); (b) Common Blue(Polyommatus icarus). In (a) the white colours turn bright yellow in ammonia fumes. In (b) the flavonoids areconcentrated in the orange lunules. In both insects the pigments areobtained fromplants eatenby the caterpillars.[Photographs provided by Dr J. A. Findlay]
Colour and the Optical Properties of Materials 326
Analternative theory, put forward shortly after the pH theory, is that the pigmentmight complexwith ametal
cation to bring about colour changes. This seems reasonable in the light of the previous two sections. Part of the
difficulty in assigning colour to a single molecule or molecule cation complex lies in the fact that the cyanin
molecules exist in a number of forms, all of which are in dynamic equilibrium and all of which depend upon
the pHof the surrounding liquidmedium (Figure 8.14).However, even taking this into account does not explain
the colours or stability of pigments in the natural state, and now a number of other ideas are current. Although
changing pH and adding metal cations are well-known horticultural recipes for changing plant colour
hydrangeas, for example, are treated with aluminium solutions and the soil is made acid to preserve their blue
colour and the soil is made alkaline to turn the colours pink the details of flower colour are more complex.
It is now clear that plants use metal complexes to stabilise colour, but these are often large molecules
made up of six anthocyanin molecules and six flavone molecules linked to two metal cations to form
Figure 8.12 Anthocyanin colours: (a) scarlet and blue fuchsia flowers; (b) pink rose; (c) geranium; (d) apples,showing red skin colours
327 Colour from Molecules
ametalloanthocyanin. Another strategy used by plants is the interleaving of molecules of cyanins with other
aromatic units to forma stable stack. (See this chapter’s FurtherReading, formore information.)Apart from the
intrinsic interest in such questions, there is also a certain amount of commercial relevance. For instance, much
effort is directed towardsmanipulating colour so as to breed blue roses, carnations and chrysanthemums; a task
that has been on the ‘verge of success’ for quite a few years now.
8.7.3 The colour of red wine
The difference between the colour of red and white wines rests with the presence or absence of rather complex
anthocyanin-relatedmaterials, includingmalvin (malvidin 3-glucoside) (Figure 8.15a). These are found in the
Figure 8.12 (Continued)
Colour and the Optical Properties of Materials 328
Table 8.2 Some anthocyanins and anthocyanidins found in flowersa
Anthocyanidin(aglycon)
Anthocyanin(glycoside) Source Colourb R1 R2
Absorptionmaximumc (nm)
Cyanidin cyanin cornflowers blue OH H 535Pelargonidin pelargonin pelargoniums pink red H H 520Peonidin peonin peonies red O.CH3 H 532Delphinidin delphin delphiniums blue OH OH 546Petunidin petunin petunias red O.CH3 OH 543Malvidin malvin mallows pink O.CH3 O.CH3 542
a For the meaning of R1 and R2, see Figure 8.13.bHorticulture has produced a vast range of colour types in all of these flower groups. Only the native colour is given in the table.c In methanol solution.
O
O
O
OH
OH
O-Glc
R2
H
H
HO
HO
HO
OH
OH
OH
OH
OH
O-Glc
R1
OH
OH
5
5
5
3
3
3
7
7
7
4′
4′
4′
generic anthocyanidin
(a)
(b)
(c)
–Cl
–Cl cyanin chloride
cyanidin chloride
Figure 8.13 (a) The general structure of an anthocyanidin, where R1 and R2 represent groups such as those listedin Table 8.2. (b) Cyanidin chloride, with R1¼OHandR2¼H. (c) Cyanin chloride, the 3,5-diglucoside of cyanidin,where Glc represents the glucoside residue
329 Colour from Molecules
outer layers of the skins of black grapes and are incorporated into the wine by allowing the skins to remain in
contact with the pressed grape juice. The anthocyanin colorants are in equilibrium and the various forms show
different colours (Figure 8.14, for example), including red, violet and blue forms. In newly fermented red
wines,which are relatively acidic, the flavyliumcations provide themajority of the bright red colour associated
O O
O-Glc O-Glc
OH OHHO O
OH O
O-Glc O-Glc
OH OH
Strongly acidic: flavylium ion (red) Neutral: anhydrobase: purple
+–H
++H
H
(b) (c)
aqueous solution
(a)
O
OH
HHO
OH
OH
OH
5
3
7
4′–Cl cyanidin chloride
O O
O-Glc O-Glc
OH OHO HO
O O
O-Glc O-Glc
O O
A kaline: anhydrobase anion: blue Metal complex: blue
–H+ –Mn+
++H n++M
H
n+M(d) (e)
Figure 8.14 Some of the forms taken by the cyanin molecule in aqueous solution: (a) cyanin ion; (b) flavyliumion (red); (c) anhydrobase (purple); (d) anhydrobase anion (blue); (e) metal complex (blue). These molecularspecies are in dynamic equilibrium which shifts under a change of pH. Other forms, not shown, can also exist inaqueous solution
Colour and the Optical Properties of Materials 330
O
O
O-Glc
O-Glc
OCH3
OCH3
HO
HO
OH
OH
OH
OH
OCH3
OCH3
5
5
3
3
7
7
4′
4′
(a)
–Cl malvin chloride
(b)
flavylium cation (red)
O
O
O
O-Glc
O-Glc
O-Glc
OCH3
OCH3
OCH3
HO
HO
HO
OH
OH
OH
OH
OH
OH
OCH3
OCH3
OCH3
5
5
5
3
3
3
7
7
7
4′
4′
4′
(c)
Figure 8.15 The colour of red wine: (a) the structure of the anthocyanin salt, malvin chloride (malvidin3-glucoside); (b) the flavylium cation derived frommalvin (malvidin 3-glucoside), found in the skins of red grapesand which contributes significantly to red wine colour; (c) possible structure of a fragment of polymericanthocyanin monomers which leads to the change in colour of red wine from red to tawny as it ages
331 Colour from Molecules
with the young wine (Figure 8.15b). Because of the complex equilibria holding, only about 30% of the
anthocyanins present actually contribute to the initial red colour.
It is well known that the colour of redwine changes over time from an initial bright ruby red via purple red,
plum and brick red to a pale tawny colour. While the chemistry of the changes is not fully understood, it is
believed that the overall cause is polymerization of the flavylium cations.Within about 1 year of beingmade,
about 50% of all of the anthocyaninmaterial is in the form of short polymer chains known as oligomers. The
polymeric forms are complex and are difficult to analyse structurally. Figure 8.15c shows one of many
possible forms that may occur. Initially, these polymeric molecules enhance the red colour of the wine
because the conjugated bonding is more extensive in the oligomers than in the monomers. As the
polymerization increases, the polymer tends to precipitate and the colour starts to change, leading to
the colour sequence mentioned.
8.8 Autumn Leaves
Deciduous trees have green leaves throughout summer. This colour is due to the presence of chlorophyll,
which is found in regions of the cell called chloroplasts, although the overall visual ‘greenness’ of a leaf is a
result not only of the chlorophyll but also of size, surface texture and surface coating. The production of
leaves is controlled by photoperiodic behaviour, and as the day length shortens in the Northern Hemisphere,
leaf senescence commences and leaves start to die. This produces a blaze of colours in favourable years
(Figure 8.16).
The colour change is due to the fact that the dominant colour generator, chlorophyll, is no longer
synthesized and green no longer swamps the other pigments that may be present. These include
carotenoids, which are present within the chloroplasts and aid photosynthesis. Chlorophyll absorbs
mainly in the red and blue (Section 8.6) and much of the incoming sunlight is wasted. Carotenoids have
absorption maxima nearer to the green, and so can harvest a portion of the spectrum unavailable to
chlorophyll. The carotenoids pass this energy to chlorophyll molecules to use in photosynthesis, hence
improving the photosynthetic efficiency of the chloroplasts. When the chlorophyll production ceases in
autumn, the carotenoid pigments become visible and leaves turn yellow. This is the normal autumn colour
for many trees (Table 8.3).
Nevertheless, many of themost spectacular of trees show brilliant orange, red and scarlet colours. These are
the result of anthocyanin production as the leaf approaches death. At the same time, a layer of semipermeable
cells, called the abscission layer, forms at the leaf base. The abscission layer acts as a barrier to themovement of
sugars from the leaf to the branch, and these sugars are converted into anthocyanin pigments in some species.
Theproductionof anthocyaninsvaries greatlywithin a species, fromspecies to species andas a functionof local
weather. In some groups, such as the Japanese maples, breeders have produced autumn leaf colour variation
from gold via reds and scarlets to deep purples.
There is still controversy over why leaves produce anthocyanins at this stage. A number of theories exist,
includingtheideathatanthocyaninsprotectfromdamageduetoharmfulultravioletlight,orthattheyscavengefree
radicalsandotherreactivedamagingmolecules, thattheyreduceosmoticpressureintheleavespriortoleafdrop,or
that they act as signals to pests such as aphids. At present no consensus exists, and maybe all of these ideas and
perhapsothersaswell,contributetoanthocyaninproduction.Eitherway,autumnredsandscarletsremainadelight.
These colours are often brief, coming as a prelude to the final colour changewhen the leaves turn brown. The
brown colours are due to tannins (Section 8.9.2) that may be naturally present in the leaves, of oaks, for
example, or they may be produced by breakdown of other cell components.
Colour and the Optical Properties of Materials 332
8.9 Some Dyes and Pigments
Technically, dyestuffs are soluble in the medium in which they are applied, whereas pigments are insoluble.
Pigments are thus used most frequently in a finely ground solid state, mixed with a carrier medium. They are
incorporated in this form into paints, inks and mixed with plastics to obtain opaque coloured products.
Nevertheless, there is no fundamental difference between dyes and pigments.Many compounds can be used as
dyes in one liquid and as pigments in another. The phthalocyanines and themetal organic complexes described
earlier in this chapter are, for example, both important pigments and dyes.
Figure 8.16 Autumn leaf colours: (a) Virginia creeper (Parthenocissus sp.); (b) peony (Paeonia sp.) showingyellow and red leaves; (c) Japanese maple (Acer sp.) showing orange leaf colours; (d) maple (Acer sp.), showingpurple-red leaf colours
333 Colour from Molecules
Many of the naturally occurring molecules discussed above can be considered to be dyes, but most modern
dyestuffs are synthetic chemicals. This is because commercial dyes must have certain properties apart from
colour before they are useful. The dyemust be fixed to thematerial to be coloured in someway, and it must not
be fugitive. These terms mean that the dye must be firmly attached to the material and be stable with respect to
light and the normal conditions of washing. The actual mechanism by which a material becomes dyed is
complex and depends upon both dye and fabric. All aspects of dyes and dying are the subject of extensive and
continuing study.
There are over 7000 commercial dyes and pigments available, which go under more than five times asmany
trade names. They are ubiquitous in daily life, in paints, inks, hair dyes, cosmetics, coloured plastics and so on
(Figure 8.17). Here, we will only mention one or two of particular interest.
Figure 8.16 (Continued)
Colour and the Optical Properties of Materials 334
8.9.1 Indigo, Tyrian purple and mauve
Indigo is one of the oldest dyes known to man and has been in use since Neolithic times. It is the colouring
derived from the plant woad (Isatis tinctoria), and preparation of this dyewas an important industry in Europe
until the seventeenth century. This industrywas displaced by imported indigo obtained from plants indigenous
Table 8.3 Typical autumn leaf colours
Tree Typical colour
Ash yellow, later purpleBeach yellowBirch yellowHazel yellowHorse chestnut yellow, orangePoplar yellowSycamore yellowWillow yellowWitch hazel yellowHawthorn yellow, redMaples gold, red, scarlet, purpleOaks orange, redVirginia creeper red, scarlet
Figure 8.17 Plastic polyhedra brightly coloured by organic pigments. (The model shows the crystal structureof the compound spinel, MgAl2O4)
335 Colour from Molecules
to Bengal, Java and other parts of Asia. The invention of synthetic indigo in the 1890s had a severe economic
effect upon the Asiatic indigo industry and led to the demise of production from plant sources.
The trans form of the indigo molecule (Figure 8.18a) is only one of several forms of the molecule, but is the
onemainly responsible for the characteristic colour of this dye.Although the structureof indigomaynot bewell
known, its colour, that of ‘blue’ jeans, will be familiar to everyone.
The important dyestuff of the classical ancient world ofRomewasTyrian purple (fromTyre, inAsiaMinor).
This was manufactured in a messy process that involved the mashing of molluscs, especially Murex and
Purpura species. The amount of dye produced from a large number of animals was minuscule, and hence the
cost was prohibitive except for those with unlimited wealth in Roman days, mainly the Emperor. The major
colorant is 6,60-dibromo-indigo, structurally very similar to indigo. The trans form (Figure 8.18b), so called
because the nitrogen and bromine groups lie on opposite sides of the C¼C double bond found in the middle of
the molecule, is the most stable isomer and the main contributor to the colour of the dyestuff. This latter
structure also exhibits a form in which the H�N groups hydrogen-bond to the nearby oxygen atom
(Figure 8.18c). The cis form, in which the nitrogen and bromine groups are on the same side of the central
N
N
N N
N
N
NBr
Br
BrBr
Br
Br
N
H
H
H H
H
H
H
H
O
O
O
O
O
O
O
O
(a)
(b)
(c)
(d)
Figure 8.18 Indigo and Tyrian purple: (a) the trans structure of the dye indigo, which occurs in both crystals andsolutions, imparts the colour to ‘blue’ jeans; (b) the trans structure of themajor colourmolecule in the dye Tyrianpurple; (c) the hydrogen-bonded form of (b); (d) the cis structure of the molecule, which may have had a minorrole in the perceived colour of the dye produced from molluscs
Colour and the Optical Properties of Materials 336
double bond (Figure 8.18d), is not considered to play a significant role in the dye colour. The method of
production of the dyewould probably lead to amixture of components, and it is possible that all of these forms
contributed in subtle ways to the colour prized by the Romans.
Mauveinewas the first commercial synthetic dyemade, in 1856, and its productionmarked the birth of the
synthetic dyestuffs industry. The discoverer,William Perkin, found that the material, which he extracted via
the oxidation of aniline sulfate, could be used as a purple dye. Initially it was successfully used on silk under
the name of aniline purple or Tyrian purple. (This latter name was incorrect, as Tyrian Purple, described
above, has a different structure to mauveine.) In 1857, Perkin discovered how to apply this dye to cotton
using tannin as a mordant (a compound used to attach the dye molecules to the fabric), leading to its very
widespread use. In France, the dyewas extensively used under the name ofmauve (the French for themallow
flower) and chemically the compound is now known as mauveine. In recent times, reanalysis of Perkin’s
original dye samples have shown that he actually produced a complex mixture of molecules, mauveine A,
mauveine B, mauveine B2 and mauveine C, all of which have a single absorption maximum in the visible
close to 550 nm. Only two, mauveine A and mauveine B, contribute significantly to the colour mauve
(Figure 8.19).
8.9.2 Tannins
Asun tan, golden-browncolouringof pale skin through exposure to sunlight, is highlyprizedby some.The term
derives from the word used to describe the transformation of raw hides into leather. In past times this process
employed natural products, including oak tree galls, bark and wood that were rich in the appropriate
compounds, now called tannins. Tannins are astringent polyphenol molecules found in many plants, where
they are suspected of being a deterrent against predators. In daily life they are notably present in tea and red
wine. Like many natural materials, tannins are complex polymeric molecules which are not easily defined
chemically or physically. They are generally yellow brown in colour, and the adjective tan applies to many
objects with a similar colour, such as shoes and the associated shoe polish. Tannins find application in brown
wood stains.
Tannin polymers are divided into two groups: the hydrolysable tannins, which are derived from gallic acid
and similarmolecules, and the condensed tannins, nowcalled proanthocyanidins, derived fromflavone. Tannic
acid, a commercial product, is also ill-defined, with a chemical formula dependent upon the source of the
material.
8.9.3 Melanins
Sun tan, as the previous section suggests, does not involve tannins at all, but the generation of pigment
molecules called melanins. These form in specialized organelles called melanosomes, in specialized cells
called melanocytes, which lie near the surface of the skin. Melanins are responsible for not just skin tone, but
also for most of the black and brown colours found in nature, including the brown colour of hair and the brown
colour which appears on damaged or cut fruit. They are a group of colorants whose structure, and the
relationship between structure and colour, is still poorly understood. In fact, the melanins are heterogeneous
materials that may not have a unique structure in the crystallographic sense.
Eumelanin, mainly responsible for blacks, is produced by the oxidative polymerisation of the amino acid
tyrosine (Figure 8.20a). Initial reaction gives rise to two indole derivatives (Figure 8.20b and c). Further
polymerisation produces many complex polymer species (Figure 8.20d and e). Many browns, red browns
and tans are attributed to the presence of another melanin variant, phaeomelenin. The structure of this
material is less well understood than that of eumelanin, and further studies in this area are needed before the
nature of the various colour forms is clarified. Figure 8.21 shows a yellow water-lily (Nymphaea hybrid,
337 Colour from Molecules
N
N
N
+N
+N
+N
H C3
H C3
H C3
CH3
CH3
CH3
CH3
CH3
H N2
H N2
H N2
NH
NH
NH
(a)
(b)
(c)
N
+N
H C3
CH3
CH3
CH3
H N2 NH
(d)
Figure 8.19 The dye mauveine. The first synthetic dyestuff prepared was a complex mixture consisting of thecomponents (a) mauveine A, (b) mauveine B, (c) mauveine B2, (d) mauveine C, of which themolecules (a) and (b)are the principal colorants
Colour and the Optical Properties of Materials 338
NH2
tyrosine
COOH
HO
N
HO
HOH
5,6 dihydroxyindole
5,6 dihydroxyindole-2-carboxylic acidNH
COOH
HO
HO
(a)
(b)
(c)
N
N
N
N
N
H
H
H
H
H
COOH
COOH
COOH
COOH
COOH
HO
HO
HO
HO
HO
HO
HO
HO
HO
HO
(d)
N
N
HO
HO
HO
HO
OH
OH
HO
HO
H
H
NH
NH
(e)
Figure 8.20 Melanins. (a) The structure of the melanin precursor molecule, the amino acid tyrosine; twoinitial reaction products. (b) 5,6-Dihydroxyindole. (c) 5,6-Dihydroxyindole-2-carboxylic acid. (d) Possiblestructure of a fragment of a polymer derived from (c). (e) Possible structure of a fragment of a polymerderived from (b)
339 Colour from Molecules
Chromatella) containing a flavone colorant and a Meadow Brown butterfly (Maniola jurtina) mainly
coloured by melanins.
Eumelanin absorbs light across thevisible and behaves as an organic semiconductor (Chapter 10).At present
it is being studied for possible device use.
8.10 Charge-Transfer Colours
8.10.1 Charge-transfer processes
Acharge-transfer transition is one inwhich a relatively large redistribution of electron density occurs across the
molecule. The electron involved in the transfer is excited from amolecular orbital localizedmainly in one part
of themolecule into amolecular orbital mainly localised in another part. This can occur in several ways.When
two ormoremetal cations are involved the electron redistribution can involve electron transfer fromone cation
to another, in a cation-to-cation or intervalence charge transfer. Cations can also give or receive electrons from
surrounding nonmetal atoms in cation-to-ligand or ligand-to-cation charge-transfer processes. Finally, the
electron redistribution might simply involve charge transfer between orbitals that are largely localized on
different ligands togivea ligand-to-ligand charge transfer.Generally, charge-transfer colours are intense; those
involving transition metal cations, for example, are much more intense than the crystal-field transitions
described in Chapter 7. Although it is often possible to be sure that change transfer is taking place, it is not
always easy to decide which of the transfer options listed is responsible for the colour of a compound.
Figure 8.21 A yellow water lily flower (Nymphaea hybrid Chromatella) coloured by flavonoid pigments and abutterfly (Maniola jurtina) coloured by melanin-related pigments
Colour and the Optical Properties of Materials 340
8.10.2 Cation-to-cation (intervalence) charge transfer
For intervalence charge-transfer transitions to occur the cations must be able to adopt two different valence
states; for example:
M2þ ½site 1� þM3þ ½site 2�!M3þ ½site 1� þM2þ ½site 2�
Many cation-to-cation charge-transfer bands lie in the infrared and overlap into the red end of the spectrum,
giving rise to visually perceiveddark blue black colours. There aremanyexamples of this among the transition
metals. Hydrated oxides of tungsten (called tungsten blue) and molybdenum (called molybdenum blue) are
poorly characterized dark blue black colloidal precipitates formed by reducing aqueous solutions of tungstate
or molybdate ions. Slight reduction of niobium pentoxide (Nb2O5) gives a series of blue black oxides with
complex ‘block’ structures and slight reduction of titanium dioxide (rutile) gives a series of blue black
crystallographic shear oxides (Figure 8.22).
If the ions arewidely separated or if the site geometry of one cation is quite different from that of the other the
transition will not occur. As an illustration, spinels contain cations in two different site geometries: octahedral
and tetrahedral. Charge transfer is possible between two cations situated in neighbouring octahedral sites, but
not, in general, between two cations one of which is situated in an octahedral site and the other in a tetrahedral
site. Similarly, the compound BaBiO3 contains equal numbers of Bi5þ and Bi3þ ions (i.e. it is better written as
Ba2Bi3þBi5þO6). The two Bi ions occupy quite different anion coordination polyhedra, as the Bi3þ ions
possess lone pair electrons. The differences in site geometry make charge transfer impossible and the
compound is colourless.
8.10.2.1 Prussian blue
One of the best known examples of cation-to-cation charge-transfer coloration is provided by the dark
blue compound known asPrussian blue or Turnbull’s blue. Prussian blue, long used as a pigment in inks, is
Figure 8.22 Plastic coatings on wire containing titanium dioxide (TiO2) coloured by charge transfer, induced inthe oxide by laser irradiation. [Reproduced by permission of Spectrum Technologies PLC]
341 Colour from Molecules
a precipitate prepared by adding an aqueous solution of pale yellow K4[Fe2þ (CN)6] to a pale yellow
green aqueous solution of any Fe3þ salt. Turnbull’s blue, which seems to be chemically the same as
Prussian blue, is made by mixing an equally pale coloured aqueous solution of K3[Fe3þ (CN)6] with a pale
green aqueous solution of an Fe2þ salt. The reaction in each case is quite spectacular. The mixing of two
virtually colourless solutions instantly produces a dark blue black-coloured material containing iron in
both the Fe2þ and Fe3þ forms.
Having said that, the composition of Prussian blue and even the naming of this compound are both subject to
some uncertainty. Apart from Turnbull’s blue, the blue-coloured pigment may be called (among other names)
Berlin blue, Chinese blue, Hamburg blue or Paris blue, and there is also Prussian green and Prussian white to
contend with. All of these names probably refer to slightly different materials. During preparation, a variable
amount of water and alkali are incorporated into the precipitated pigment. Prussian blue is generally given
the formula Fe3þ4 [Fe2þ (CN)6]3�xH2O) (14 < x< 16). One well-investigated form, sometimes called ‘soluble’
Prussianblue, has the formulaKFe3þFe2þ (CN)6 andcontains equal quantities ofFe2þ andFe3þ . TheFe3þ and
Fe2þ ions form a face-centred cubic array and the large Kþ cations occupy alternate cube centres. Prussian
green, the all-Fe3þ -containing phase has the formula Fe3þFe3þ (CN)6 and Prussian white is the all-Fe2þ -containing phase K2Fe
2þFe2þ (CN)6 (Figure 8.23).The charge-transfer transition involves the displacement of an electron from an Fe2þ to an Fe3þ ion. The
electron moves from a (t2g)6 configuration on Fe2þ to a (t2g)
5 configuration on Fe3þ , reversing the oxidationstates in the process:
Fe2þ ½site 1� þ Fe3þ ½site 2�! Fe3þ ½site 1� þ Fe2þ ½site 2�
This produces aband in the absorption spectrumcentred at approximately700 nm(14 200 cm 1), effectively
removing the red end of the visible spectrum, leaving dark blue. Clearly, this transition is not available to either
the Prussian green or Prussian white pigments. The green colour is due to crystal field transitions (Chapter 7),
while the colourless phase has no crystal-field transitions in the visible.
8.10.2.2 Blueprints
The first half of the nineteenth century was a time when many scientists were exploring the idea of capturing
images using light as the writing medium and light-sensitive chemicals as the record producer. One of these
scientists, Sir John Herschel, tried many materials, including anthocyanins, but these mainly proved to be
unsatisfactory. One process, however, was successful, the cyanotype. Details were first published in 1842,
although the process was not truly exploited until 1872, in the form of the architectural, and later, engineering
blueprint.
Herschel used a number of compounds in arriving at his cyanotype process, but found best results with the
water-soluble salts ‘ferrocyanate of potash’ (now potassium iron(III) cyanide) and ‘ammonio’ (ammonium
iron(III) citrate, an ill-defined material containing 7.5 9 % ammonia, 14.5 18.5 % Fe and 65 75 % hydrated
citric acid). A solution of the reactants was spread upon paper and then exposed to an image, formed by a lens,
for example. The exposure resulted in a blue imagewhich was preserved by washing away surplus chemicals.
After drying, the imagewas permanent and stable to light. However, the problemwith the imagewas that it was
a negative bright areas of the original became dark in the image and viceversa.Marion, in Paris in 1872, found
this not to be a problem and used the process (renamed as Ferroprussiate Paper) for the creation of copies of
architectural drawings. A drawing, made upon tracing paper, was placed upon a sheet of Ferroprussiate Paper
and exposed to light, after which the paperwaswashed inwater. A negative copy of the drawingwas obtained
a blueprint. This copy was, of course, completely adequate for the purposes of the architect and shortly
afterwards was also adopted for copying engineering drawings (Figure 8.24).
Colour and the Optical Properties of Materials 342
The chemistry of the process is reasonablywell understood. In principle, twowater-soluble iron compounds
are used to prime the paper, potassium iron(III) cyanide and ammonium iron(III) citrate. The action of light on
the solution of the citrate causes a redox reaction to occur in which the Fe3þ ions are reduced to Fe2þ and the
carboxylic acid (COO ) groups on the citrate are oxidized to CO2. In outline:
ultraviolet lightþ Fe3þ þCOO ! Fe2þ þCO2
Fe(III)
Fe(II)
K
(a)
(b)
(c)
Figure 8.23 Prussian-blue-related structures: (a) Fe3þFe3þ (CN)6, Prussian green; (b) KFe3þFe2þ (CN)6, soluble
Prussian blue; (c) K2Fe2þFe2þ (CN)6, Prussianwhite. The linearCN� ions (not shown) sitmidwaybetweeneach of
the Fe cations. Crystals also contain a variable amount of water in the structure
343 Colour from Molecules
Under ordinary circumstances the Fe2þ ions are slowly oxidized back to Fe3þ by the oxygen in air. Tomake an
image it is necessary to prevent reoxidation. This is the role of potassium iron(III) cyanide,which reacts rapidly
with Fe2þ ions to yield an ill-defined compound which can be approximated to Prussian blue,
KFe3þFe2þ (CN)6. This compound forms in greatest amounts where the light irradiance was strongest,
thus producing darkest coloration where the image is lightest; a negative image. Note that too much light is
detrimental to image formation because oxidation of the Fe3þ in Prussian blue in the presence of excess citrate
can occur, following the chemical equation above, to produce a similarly ill-defined material, Prussian white,
potassium iron(II) cyanide, approximately K2Fe2þFe2þ (CN)6. As the Fe ions are in a single oxidation state,
Fe2þ , intervalence charge transfer cannot occur and the material is no longer coloured. This will cause
subsequent fading of the blueprint and is prevented by thewashing stage, which removes the unreacted citrate.
Blueprints have now been superseded by photocopies of various types. Nevertheless, the use of the
cyanotype process for copying plans was so widespread that the term ‘blueprint’ has now come to mean
‘plan’. Thus, one can talk about a ‘blueprint for success’, meaning a ‘plan for success’.
8.10.2.3 Aquamarine and some other minerals and gemstones
The colour of a charge-transfer material depends upon the concentration of ions present. When the
concentration of the ions involved is low the charge-transfer bands give rise to less-intense colours. For
example, Fe2þ Fe3þ charge-transfer transitions are responsible for the blue colour of aquamarine, which is
a form of the mineral beryl (Be3Al2Si6O18) containing small amounts of iron as an impurity. The structure of
beryl is hexagonal and, when pure, is a colourless mineral. The structural framework is composed of Si6O18
Figure 8.24 An engineering blueprint, circa 1937. [Reproducedwith permission ofMrAndrewDulley, AssistantCounty Archivist, West Glamorgan Archive Service]
Colour and the Optical Properties of Materials 344
rings forming tunnelsparallel to the c-axis linkedbyBe-centredoxygen tetrahedra andAl-centredoctahedra. In
aquamarine, a trace of iron in two valence states substitutes for some Al3þ . The blue colour of the aquamarine
becomes deeper and darker as the concentration of iron increases. When the impurity concentration becomes
very high the mineral appears black rather than blue.
A similar cation-to-cation charge transfer is responsible for the colour of the black mineral magnetite or
lodestone. This material has the spinel structure with a formula (Fe3þ )t[Fe2þ Fe3þ ]oO4. Half of the Fe3þ
cations in this structure are found in tetrahedral sites, written as (Fe3þ )t and the remainder, together with the
Fe2þ cations, are in octahedral sites, written as [Fe2þ Fe3þ ]o. Charge transfer does not occur between the ionson octahedral and tetrahedral sites because the change in geometry between the two sites is too large.However,
it does occur between the ions which reside only on octahedral sites. Interactions between the iron ions at such
high concentrations broadens the absorption band so much that all visible wavelengths are absorbed and the
material looks black (see also Section 8.10.4).
Intervalence transitions need not involve only one type of cation. The gemstone sapphire is formed from
colourless corundum(Al2O3) containing less than1%ofbothTi4þ andFe2þ . Theseoccupyneighbouring face-sharing octahedra in the structure that run in chains along the c-axis of the crystals. The charge transfer takingplace is:
Fe2þ ½site 1� þTi4þ ½site 2�! Fe3þ ½site 1� þTi3þ ½site 2�
Aswith Fe2þ Fe3þ transitionsmentioned above, when the concentration of the cations becomes very high the
beautiful blue colour is lost and the material becomes black. This occurs, for example, in the mineral ilmenite
(FeTiO3),which has a similar structure toAl2O3but theAl3þ ions are replaced by anordered arrangement of Fe
and Ti. It is jet black in colour and occurs as black sands on beaches in several parts of the world.
Intervalence charge transfer can alsooccurwhen twodifferent cations occupyoctahedral sites in spinels.The
spinel Li0.5Fe2 xCrxO4provides an example inwhich one of the cations, Cr, adopts the unusual valence state of
Cr4þ . In these spinels the charge-transfer colour arises from:
Fe3þ ½site 1� þCr3þ ½site 2�! Fe2þ ½site 1� þCr4þ ½site 2�
The absorption band, centred at 690 nm, overlaps into the red end of the spectrum, colouring the spinel blue.
As the concentration of the two charge-transfer cations becomes more equal, the colour deepens.
8.10.3 Anion-to-cation charge transfer
Anions tend to be electron rich, while cations tend to be electron poor, so that anion-to-cation charge transfer is
not unexpected and is responsible for many of the brightest colours shown by inorganic compounds. These
transitions are usually of higher energy than cation-to-cation charge-transfer transitions and lie in the
ultraviolet. Colour arises when the ultraviolet peak tails into the blue end of the visible spectrum, giving
red, orange and purple hues to the compounds. For example, potassium permanganate (KMnO4) forms dark
purple, almost black, crystals. The crystals are only slightly soluble in water, but produce an intense purple-
coloured solution. The colour is associated with the (MnO4) ion, as Kþ ions never show colours in solution.
Although it might be thought that the manganese alone could be responsible for the colour, owing to crystal-
field transitions (Chapter 7) this is not so. Themanganese ion has a formal charge ofMn7þ , which indicates thatit has lost all the d-electrons and sowill not show crystal-field colours. In addition, the absorption spectrum of
the solution is quite unlike crystal-field-induced absorption. In fact, the colour is attributed to a charge transfer
between an oxygen ion in the (MnO4) unit and the central Mn7þ ion. This is an anion-to-cation or ligand-to-
metal charge-transfer process.
345 Colour from Molecules
Anumber of other transitionmetal anions also show intense anion-to-cation charge-transfer colours.Among
the most familiar is the dichromate ion (Cr2O7)2 , which gives crystals of potassium dichromate (K2Cr2O7)
a red colour and yields intense orange yellow colour in aqueous solutions. The bright colours of PbCrO4
(artists’ chrome yellow) and BaCrO4 (artists’ lemon yellow) also arise from similar ligand-to-metal
charge transfer.
Because the absorption is usually in the ultraviolet, there is interest in using the anion-to-cation charge-
transfer process in sunscreens. The need is to effectively screen out ultraviolet A (�320 400 nm) and
ultraviolet B (�290 320 nm). Currently, fine particles of zinc oxide (ZnO) and titanium dioxide (TiO2)
are used (Sections 5.7 and 10.1). However, these are not totally perfect from the point of view of the cosmetics
industry and other materials are being sought. The important factors are that the particles should not absorb in
the visible and that their refractive index shouldmatch that of the spreadingmedium, so that they are, in effect,
invisible. Compounds of cerium, including borates (CeBO4 and CeB3O6) and cerates (SrCeO7 and Sr2CeO4),
seem tobe suitable alternatives.The important anion-to-cation charge-transfer step is fromoxygen to the empty
5d levels on the Ce4þ ion, leading to a strong absorption at �189 nm (borates) and �300 350 nm (cerates).
8.10.4 Iron-containing minerals
The orange yellow brown colours of iron-containing minerals are derived form a combination of anion-to-
cation charge-transfer and crystal-field effects. Ferric oxide (haematite, Fe2O3) and various Fe(III)-containing
iron oxide hydroxides give many soils and rocks a ruddy colour (Figure 8.25a). The common red brown
colour of bricks, flower pots andmanybaked-clay artefacts arises from the same source, as do the familiarwarm
tones of limestone containing Fe3þ ions, much prized in buildings.
The discoloration of streams and rivers in old coal-mining areas is also frequently due to the presence of
ferric oxy-hydroxides. Deep underground, fairly large amounts of iron sulphide FeS2 exist within coal
deposits. When mining operations cease, water builds up in the workings and dissolves the sulfide to give
Fe2þ ions in solution. These are eventually transported to the surface where they emerge as Fe2þ in streams.
At this stage the water still looks clear. However, it rapidly becomes a bright yellow brown colour because of
the oxidation of Fe2þ to Fe3þ and the subsequent appearance of the colour of the hydrated Fe3þ species. To
make matters worse, the rather insoluble complex iron oxy-hydroxides formed are deposited as a glutinous
mass on weeds and rocks. These not only look unattractive, but prevent the plants from continuing
photosynthesis and clog the gills of many aquatic animals. In severe cases the result is a discoloured stream
devoid of plant and animal life (Figure 8.25b).
At the simplest level the colour derives from charge transfer between O2 or OH and Fe3þ . Fe3þ is a d5 ion
and can readily accept an extra electron in this half-filled shell to become Fe2þ (d6):
OH ½site 1� þ Fe3þ ½site 2�!OH½site 1� þ Fe2þ ½site 2�
This results in a strong absorption band in the ultraviolet at about 250 nmwhich extends into the blue region
of the visible spectrum and tends to shift towards red as the concentration of iron increases, so that colours
change from pale yellow in, for instance, limestones containing traces of Fe3þ , to intense yellows and orangesin rocks with higher concentrations.
However, the root cause of the intense colours displayed by these minerals is more complex than just charge
transfer, and twoothermechanisms play an important role in generating the rich tones of iron-containing rocks.
The first of these is crystal field related (Chapter 7). Normally, crystal-field transitions are forbidden and so of
low intensity. This is typified by the colour of many ordinary ferric salts, such as ferric nitrate, where colour
arises in the Fe3þ (H2O)6 unit and gives rise to a pale purplish colour. However, the crystal-field intensities are
greatly enhancedwhenFe3þ ions occupy a pair of face- or edge-sharing octahedra; a very commonoccurrence.
Colour and the Optical Properties of Materials 346
Figure 8.25 Iron oxy-hydroxide charge-transfer colours. (a) A section of limestone deeply coloured by iron–oxygen charge transfer. The greyish area in the centre of the view indicates where the rock face has been newlyexposed, revealing that Fe2þ ions are present here and do not contribute to the yellow–orange coloration.Subsequent oxidation will change these to Fe3þ ions andmake this area indistinguishable from the surroundings.(b) A stream discoloured by deposits of iron oxy-hydroxides due to the transport of Fe2þ to the surface fromdisused mine workings. [Reproduced with kind permission of Dr A. Eddington]
347 Colour from Molecules
These Fe3þ ions can interact magnetically and this gives rise to a new set of selection rules that bypass the
limitations normally found for isolated atoms. It results in transitions by simultaneous excitation of both
cations, called electron pair transitions. These give a strong band at about 475 nm, which considerably
enhances the orange-yellow of the material.
As well as this magnetic interaction, materials containing both Fe2þ and Fe3þ ions in suitably situated sites
can also show intense intervalence charge-transfer bands, as described earlier, and is responsible, among other
things, for the black colour of Fe3O4 (magnetite) described above.
8.10.5 Intra-anion charge transfer
Although the blue colours derived from litmus, indigo and woad, mentioned above, were suitable for some
coloration of fabrics, theywere not found to be satisfactory for art work. This is because they are sensitive to pH
changes and are also prone to lose colour. Paintings from the Middle Ages until close to 1830 used very little
blue at all, and the blues thatwere adopted tended tobe produced fromcopper or cobalt compounds.Thesewere
also regarded as unsatisfactory by artists and only employed reluctantly. There was, however, one exceptional
blue pigment available, made from themineral lapis lazuli (Figure 8.26). This is a rare dark blue stone found in
isolated depositsmainly inAsia. Lengthy treatment of themineral produced thefine blue pigment ultramarine.
However, it was expensive (of the order of FF10 000 per kilogram in 1830) and onlymanuscripts and paintings
commissioned by thewealthiest of patrons, who alsowished to advertise their wealth, used any large quantities
of ultramarine.
The purple blue colour in lapis lazuli is due to lazurite, an aluminosilicatewith an approximate composition
given by (Ca,Na)8(Al,Si)12O24(S,SO4,Cl)xwith x taking a value of 1 4. The colour arises from the presence of
a polysulfide anion with an approximate formula S3 . The unit consists of a triangle of three sulfur atoms
together with one additional electron. The molecular orbitals of this cluster are not fully occupied and
a transition between the filled and empty levels produces a strong absorption band at about 600 nm in theyellow
region of the spectrum. (Note that the charge transfer occurs within this group of three sulfur atoms. It involves
a redistributionof the chargeswithin theS3 unit itself, not fromoneS3 group to another.)The colour reflected
by ultramarine is thus blue with purple overtones. In natural lazurite and ultramarine the colour depends upon
the exact amounts of calcium, sulfur, chlorine and sulfate present and in particular is deepened by increased
calcium and sulfur content, which encourages S3 formation.
Figure 8.26 Lapis lazuli beads. The dark blue mineral was once used to make the pigment ultramarine
Colour and the Optical Properties of Materials 348
The cost of ultramarinewas so high that the French Soci�et�e d’Encouragement pour l’IndustrieNationale and
the British Royal Society of Arts both set up prizes for the discovery of an artificial method of ultramarine
fabrication. A process using the easily obtainable clay kaolin was discovered by Guimet in 1828 and from that
time ultramarine has not been excessively expensive. The approximate equation of formation is:
Al2Si2O7 � 8H2O ðkaolinÞþNa2CO3 þNa2SO4 þ SþC!Na7CaAl6Si6O24S3SO4
However, the process is not straightforward; it involves a reduction step and a reoxidation step, all of which
produce coloured intermediates, which are approximated as:
S8ðyellowÞ! ðreduceÞ! S2 ðgreenÞ! ðreoxidizeÞ! S3 ðblueÞ
In fact, the details of the process are still in dispute. For example, if there is insufficient sulfur present a green
colour, arising from S2 , will appear.
The production of a relatively cheap blue pigment was an important factor in the blossoming of the
Impressionistmovement of painters, andmanyof the classic paintings of this style contain copious quantities of
synthetic ultramarine.
Many other polysulfides are coloured. The formation of extended groups of sulfur atoms gives rise to
molecular orbitals which can participate in intra-anion charge-transfer redistribution of electrons, and in so
doing generates intense absorption colours.
8.11 Colour-Change Sensors
Because the eye is so sensitive to subtle colours, the use of a colour change to give information about the
physical or chemical stateof a systemhasbeen longexploited.For this, a compound, the sensingchemical,must
change colour significantly in the presence of the analyte (thematerial being tested for). This can be qualitative,
when just the colour change itself is significant, or quantitative, when the depth of colour change is measured
with a spectrometer or similar instrument and comparedwith the colour change induced by standard solutions.
In both cases, the sensing chemical must react with the analyte.
The strength of the interaction and its specificity are important. Weak interactions such as van der Waals
forces or dipole dipole interactions that are involved in physical absorption or adsorption are reversible and
may prove of use only for qualitative studies. An example was given above. Flavones react readily with
ammonia (NH3) to producemuchdeeper yellowcolours, a reaction that provides an easy test for the presence of
flavones innature.However, the interaction isweak and thedeepyellowcolour returns to theoriginalwhite tone
when the ammonia fumes are removed.
The interactions that give rise to p p� transitions and charge transfer are stronger and are open to
modification so are able to provide more selective data. For example, the presence of Fe(III) in solution is
easily confirmed by adding iron(II) cyanate, to yield a Prussian blue precipitate. Chemical bonding at specific
sites, involving acid hydrogen atoms (pH changes), hydrogen bonding or cation binding can be highly specific,
and is the means chosen in biological systems to control many important life-supporting reactions in cells.
These are frequently the best suited for analysis, due to the specific nature of the interactions taking place. The
challenge is to adapt them for analytical purposes via colour change.
8.11.1 The detection of metal ions
There are many reasons for needing to detect small quantities of metal ions in solution, and an appealing way
has been, formanyyears, to use colour changes to indicate the presence of particular cations. This objective can
349 Colour from Molecules
be achieved in a number of ways; two of the best known are to react the metal ion with an organic molecule to
form a coloured derivative, such as a porphyrin, or to link ametal cation to an organicmolecule so as to shift its
ultraviolet absorption into the visible, similar to the mechanism that produces the change in colour of lobster
shell on boiling.
The reactions of metallic cations with organic molecules to form brightly coloured complex molecules or
complexes are well tried and have been used to detect small quantities of metals in a solution for many years.
The reactions utilised are those in which the cation in question reacts with a component in solution, often an
organic molecule, to form a complex, the colour of which is indicative of the cation present. This technique
is simple to apply and can readily give qualitative information on impurities at parts per million level of
discrimination. The procedure is known as a spot test. A drop of the solution to be tested is placed on a filter
paper or into a well on a white ceramic plate. To this is added a drop of the necessary reagent, and the colour
produced, if any, is observed. Difficulties lie in ensuring that the solutions are free from contamination and that
the pH is correct, as the colours seen are often pH dependent. If the amount of product is evaluated, themethod
becomes quantitative.
An illustration of this technique is provided by the detection of nickel and palladium using the organic
compound dimethyl glyoxime. In a basic solution this moleculewill produce an intensely scarlet precipitate in
the presence of Ni2þ cations. The structure of the complex (Figure 8.27a) shows the central Ni cation to be
surrounded by a square of four nitrogen atoms fromcoordination to two glyoximemolecules. The compound is
called bis(dimethylglyoximate)nickel(II). The colour is due toHOMO LUMOtransitionswithin the extended
organic framework, not due to d d transitions within the Ni2þ ion as such, which only acts to bring the organic
parts into conjunction. If the solution is acidified, then the scarlet colour will disappear. Should any palladium
be present, then a yellow compound with a similar structure is formed in place of the red material.
The numbers of organicmolecules that can bind tometal ions to produce coloured products is enormous, and
so the majority of cations can be conclusively identified using spot tests (see this chapter’s Further Reading).
Many organic molecules show peaks in the ultraviolet absorption spectrum due to HOMO LUMO
transitions. Binding of a metal ion to the surface of some of these molecules moves the absorption peaks
slightly. If this change is sufficient to move the absorption into the visible, then the presence of the cations will
be revealed by a change in colour. An example of this technique is provided by a method of detection of Cu2þ
and Fe3þ ions. The organicmolecule is bound to the surface of a quantum dot consisting of a ZnS-coated CdSe
nanoparticle of about 15 nm diameter (Figure 8.27b). In this configuration the absorption spectrum is
characterized by two bands in the ultraviolet, at 275 and 355 nm. The organic molecules are constrained
by the quantum dot surface so that they are able to react only with Cu2þ and Fe3þ in solution. A reaction of the
boundmolecules with Fe3þ ions increases the strength of the absorption enough tomove the tail of the band at
355 nm into the violet end of the visible (Figure 8.27c). The solution takes on an orange colour, which is
indicative of the cation. Reaction of the boundmoleculeswithCu2þ ions in solution shifts the absorption peaks
towards the visible, to 295 and 410 nm (Figure 8.27c). This changes the appearance of the solution from
colourless to green. Other cations do not change the visible colour of the preparation, which thus becomes
a sensitive test for the presence of the two reactive species.
8.11.2 Indicators
There are many molecules that are sensitive to the acidity of the surroundings, including the anthocyanins
described earlier. The change of colour of the cyaninmolecule from red in acid solution, through pale violet in
neutral solution to blue in alkaline solution was the basis of the pH theory of flower colours. Indicators, which
are molecules of weak organic acids that change colour as a function of the acidity (pH) of the surrounding
aqueous solution, are further examples of this widespread feature. They are widely used in titrations to
determine the progress of reaction between acidic and alkaline solutions. The best known indicator, litmus, is
Colour and the Optical Properties of Materials 350
a blue colouring matter derived from various lichens. It is chiefly composed of two compounds, azolitmin and
erythrolitmin, combined with alkalis. It becomes red in acid solution and blue in alkaline solution.
Besides litmus, there are a large number of other indicators which operate over varying pH ranges and which
display a variety of colour changes (Table 8.4).
The reason for the colour change in an indicator is that some hydrogen atoms (acidic hydrogens) are lost or
gained by the indicator molecule depending upon the pH of the solution. This hydrogen exchange causes
Ni
CH3H C3
H C3 CH3
O
O
O
O
H
H
NN
N N
(a)
R
R
RR
RR
RR
RR
RRCH N
OH S
R =
(b)
CdSe
ZnS
1.0
2.0
300 400250 350 450
(c)
Wavelength / nm
Abs
orba
nce
Cu2+
Fe3+
Figure 8.27 Cation sensors. (a) The structure of the intensely scarlet complex of nickel (Ni) with dimethylglox-ime to form bis(dimethylglyoximate)nickel(II); dotted lines represent hydrogen bonds. (b) The structure ofa quantum dot sensor for Cu2þ and Fe3þ (schematic). (c) The absorption spectra of the sensor solution in thepresence of Fe3þ ions (red), Cu2þ ions (green) and other cations (black). [Data for (b) and (c) adapted from N.Singh et al., Chem. Commun. 4900–4902 (2008)]
351 Colour from Molecules
a change inmolecular structure,which, in the indicators, producesmolecules of different colours. For example,
the colourless form of the phenolphthalein molecule is the acidic form, which includes acidic hydrogen
(Figure 8.28a). Removal of this hydrogen atom, which occurs in alkaline solutions, generates a number of
possible structures (Figure 8.28b) which produce a subsequent shift of the p to p� absorption band into the blueregion of the visible. The indicator then takes on a pink red colour.
The general reactions taking place for an indicator in solution are:
HIn ðaqÞþH2O ðlÞ!H3Oþ ðaqÞþ In ðaqÞ
Table 8.4 Colours of some indicators
Indicator Colour: acid Colour: alkali pKa
Methyl orange red yellow 3.4Bromophenol blue yellow blue 3.9Bromocresol green yellow blue 4.7Methyl red red yellow 5.0Litmusa red blue �7a
Bromothymol blue yellow blue 7.1Thymol blue yellow blue 8.9Phenolphthalein colourless pink 9.4Alizarin red purple 11.7
a Litmus is a complex mixture of molecules, the principal indicator components of which are polymeric. For this reason
litmus does not have awell-defined value forKa. It is useful for qualitative study, especially as litmus paper, but is not often
used for quantitative work.
HO OH
O
O
colourless (acid)
(a)
O O–O –O
–COO –COO
pink (alkaline)
(b)
Figure 8.28 The indicator phenolphthalein: (a) the principal colourless (acid) formof the indicatormolecule; (b)two of a number of possible structures (resonance hybrids) occurring in alkaline solution are pink–red in colour
Colour and the Optical Properties of Materials 352
where HIn represents the un-ionised form of the indicator and In the ionised form. The colour change is
brought about because the ionised form is different in colour to the un-ionised form. The reaction can be treated
by means of normal chemical equilibrium theory, which allows us to write the expression:
Ka ¼ ½H3Oþ �½In �
½HIn�
whereKa is the acid dissociation constant of the indicator and square brackets indicate concentrations. The ‘end
point’ of a titration is arrived at when [In ] is equal to [HIn]. At this point the relationship:
½H3Oþ � ¼ Ka
holds.Now the pHof the solution is givenby�log10[H3Oþ ] and thevalue of the analogous acid constant pKa is
�log10[Ka] (Table 8.4) so that the end point of the titration is given by:
pH ¼ pKa
Thus, the colourof an indicator changeswhen thepHof the solutionpasses thepKa value listed in the table.With
many indicators the colour change is sharp enough for the end point to be gauged by eye to within one drop of
added solution.
8.11.3 Colorimetric sensor films and arrays
Themethods described in the previous two sections are essentially single tests, giving a yes/no answer, that are
rather old-fashioned. However, the same, or similar, methods can be accommodated into modern devices that
can record the presence of analytes automatically. The simplest idea, conceptually, is to incorporate a colour-
changing entity into amembrane. Themembrane can be gas permeable,mounted on a glass plate or enclose the
end of an optical fibre. The membrane is illuminated by a suitable light source, frequently an LED or a diode
laser (see Chapter 10) and the output, reflected or transmitted light, is analysed to give the desired information.
For example, the detection of acid or alkali gases can bemeasured by the incorporation of a pH indicator into
a polymer membrane. For example, writing the acid form of an indicator as HIn and the dissociated (alkaline)
form as In , the ideal reaction with an alkali gas is:
alkali gasþHIn ðcolour 1Þ! ðalkali gas-Hþ Þþ In ðcolour 2Þ
For ammonia and bromothymol blue:
NH3 þHIn ðyellowÞ!NHþ4 þ In ðblueÞ
Theamount of ammoniapresent canbe related to the colour change,which in this casewouldbea rise in theblue
appearance of the membrane. The colour change can be varied by using indicators with different pH values,
which then allows flexibility in the amounts of the ammonia which can be detected.
In the case of acid gases, such asSO2,SO3,CO2and soon, reactionwithwater is needed to form the acid, as in
thereactionofCO2toproducecarbonicacid(H2CO3),whichthendecomposestobicarbonate(HCO3 )andHþ :
H2OþCO2 !H2CO3 !Hþ þHCO3
353 Colour from Molecules
It is thus imperative to provide a water-containing membrane in order to measure such acid gases in dry
conditions. The incorporation of liquid water into a membrane or film can be difficult. One solution to this
limitation is to use indicator molecules that incorporatewater into the structure in a similar way to thewater of
crystallisation that is found in many inorganic materials, such as NiSO4�7H2O. One successful application of
this idea utilises Qþ In �xH2O salts of indicators instead of the normal acid Hþ In form. In these molecules,
the cation, Qþ , must be soluble in the polymer from which the membrane is fabricated. It has been found that
quaternary alkyl ammonium ions NR4þ (R is an alkyl group (CnH2nþ 1)
þ , n > 6) fulfils this requirement and
can react with dry acid gases thus:
NRþ4 In � xH2O ðbasic colourÞþCO2 !NR þ
4 HCO3 � ðx�1ÞH2OþHInðacid colourÞ
Changing the indicator molecule allows for different gases and concentration ranges to be sampled.
The idea of using thin films containing a single colour-change indicator can be extended to envisage a film
containing a two-dimensional array of sensors, so that the overall effect of colour changes in several cells canbe
used to assess a range of substances present in the atmosphere. One such application is the detection of volatile
organic compounds or inorganicmolecules fromvehicles, such as SO2,NO,NO2 andO3,which can contribute
to smog and haze as well as being injurious to health.
The designof such an arraywill depend upon the range of analytes to be detected.However, overall, the array
must contain at least somemolecules thatwill reactwith every potential analyte. Second, the reactionmust lead
to a discernable colour change. Thus, pH indicators,metal-ion reactants such as porphyrins and charge-transfer
reactants are all potential components to make up an array. Additionally, specialized molecules which react
onlywith specified shapes of analyte, as in the lock-and-keymolecular pairs encountered in biology, canmake
the array very selective (see haemoglobin, Section 8.6). At present, there is considerable research into these
devices (see this chapter’s Further Reading).
8.11.4 Markers
Many substances are taxed and there is a considerable interest in distinguishing those on which revenues have
been paid from those which may be illicit. In Europe, for instance, fuels such as petrol, (gasoline), diesel fuel
and paraffin (kerosene) are charged at two rates of duty. The normal rate applies to the everyday consumer, but
a preferentially low duty rate applies to certain industrial sectors such as agriculture. It is clearly a very
profitable enterprise to take low-duty fuels and resell them at normal prices, pocketing the difference. A simple
(in principle) way to separate the two sorts is to add a dye to one group of fuels, usually the low-tariff segment.
A simple visual inspection will then allow an officer to tell if the fuel falls into the low- or high-duty category.
However, these dyes can be difficult to seewhen present in small quantities (such as when legal and illegal fuel
has been blended) or when viewed in dark conditions.
The method chosen by many countries is to use a test analogous to the spot tests already described. A
chemical is added to, say, the low-duty group of fuels. This must be invisible and blend in with the fuel, adding
little colour and being hard to remove. A simple test is then applied and a colour is produced, proclaiming that
the fuel is in the low-dutygroup. Ideally, the test solution shouldbewater based, and thewater, being immiscible
with the fuel, should initially form a colourless layer. Shaking this fuel with the test solution should produce
a coloured dye that was soluble in water. Allowing the fuel to settle then gives two layers, one of fuel and one
coloured if the marker is present. This is readily seen irrespective of any other colorants added to the fuel itself
(Figure 8.29a). Formany years theUKGovernment added small amounts of the compound quinizarin to diesel
fuel to produce ‘red diesel’. Shaking an alkali solutionwithmarked fuel changed the colour of the ‘water’ layer
to purple (Figure 8.29b). Other substances used for fuel marking include diphenylamine and 2-ethylanthra-
quinone (Figure 8.29c and d). More recently, Europe has adopted a standard additive, C.I. solvent Yellow 124.
Colour and the Optical Properties of Materials 354
This is a pale yellow dye that is fuel soluble. Treatment with aqueous acid (Hþ -containing) solution gives
a water-soluble red dye that is easily seen in the test solution (Figure 8.29e).
8.12 Dye Lasers
Dye lasers use a solution of organic dye molecules of the type described earlier in this chapter as the laser
medium. Dye lasers differ from the solid-state and gas lasers described above (Chapter 7) in a significant way.
The output can be tuned over a range ofwavelengths. In the other lasersmentioned the energy levels utilized for
laser transitions were fairly narrow and the output consists of several sharp lines. In order to alter the output
significantly one has to use frequency doubling or tripling, linear parametric oscillators (Section 4.9) or up-
conversion (Section 9.9). Molecules have rather broad energy bands, due to the addition of vibrational and
rotational energy levels to each electronic level. The output from a dye laser thus has a significant width
(Figure 8.30).
fuel
water-based test reagent
initial mix final
(a)
OH OH
OH OH
O O
O O
+Na
+Na
colourlessfuel soluble
purplewater soluble
NaOH solution
quinizarin(b)
colourlessfuel soluble
violetwater soluble
acid solution+NH Ph+PhNH
diphenylamine
Ph PhNH
Ph = C H = 6 5
(c)
Figure 8.29 Marker reagents: (a) schematic use of marker reagent; (b) quinizarin, colourless in fuel to purple inaqueous solution; (c) diphenylamine, colourless in fuel to violet in aqueous solution; (d) 2-ethylanthraquinone,colourless in fuel to deep red in aqueous solution; (e) C. I. Solvent Yellow124, pale yellow in fuel to red in aqueoussolution
355 Colour from Molecules
O O Na- +
O-Na+Ocolourlessfuel soluble
deep redfuel soluble
2-Ethylanthraquinone
Na2S2O4 solution
(d)
pale yellowfuel solubleR is an organic group
redwater soluble
N
N H+
N
N
O
OH
OR
N
N
C. I. Solvent Yellow 124
acid solution
(e)
Figure 8.29 (Continued)
400 500 600 700
Wavelength / nm
Abs
orpt
ion/
emis
sion
arbi
trar
y un
its
absorption
emission
laserrange
Figure 8.30 Absorption and emission spectra of the laser dye rhodamine 6G. The useful range of laser output fora dye laser using this molecule is relatively broad
Colour and the Optical Properties of Materials 356
When a dye molecule is excited, an electron moves from the lower HOMO (ground-state term symbol S0) to the
upper LUMO (excited-state term symbol S1) (Figure 8.31a). Because both of these states have associated vibrational
levels, the absorption spectrum is broad as the excitation can take themolecule from the ground state intomany of the
vibrational levels associatedwith the excited energy level (Figure 8.31b). Energy is rapidly lost, by collisions, and the
molecule rapidly ends in the lowest vibrational level of the excited state. Laser action can now occur when the
molecule drops to any of the empty vibrational energy levels of the ground state. Like the absorption spectrum, the
emission spectrum is broad because of the number of vibrational levels, and it is also displaced slightlywith respect to
the absorption spectrum due to the loss of energy as the excited state decays to its lowest level.
In practice, many dyemolecules can be used, but thosewith efficient fluorescence are naturally preferred. In
use thedye isdissolved ina suitable solvent, oftenmethanol or ethanol.Theenergy loss as the excitedmolecules
decay is transmitted to the solution as heat, which can seriously impair the performance of the laser. To avoid
this, the dye is circulated continuously from a temperature-controlled reservoir, so as to keep the solution at the
optimum temperature. Laser action takes place in a glass cell or across an air gap (Figure 8.32). In order to
ground state S0
excited state S1
LUMO
HOMO
(a)
electronic level
v brational levels
laser transition
electron
(b)
(c)
energy lost by collisions
excitation
Figure 8.31 Dye laser molecular transitions: (a) the ground state S0 and (b) the excited state S1 of a typical dyemolecule. Excitation promotes an electron from the HOMO to the LUMO. The molecule then loses energy viacollisions to reach the lowest level. A laser transition (c) returns the molecule to the ground state
357 Colour from Molecules
achieve a population inversion an intense pump illumination is needed, from flash lamps or other lasers. If
power levels and dye flow rates are adjusted, dye lasers can operate in a continuous mode as well as a pulsed
mode. The strong absorption of dye molecules allows the laser cell to be small, and a path length of several
centimetres will suffice in the majority of cases. As the emission spectrum is broad, the output wavelength can
be selectedusingadiffractiongrating, prismorother standardoptical components as a tuner.Themultiplicityof
dyes available means that the whole of the visible spectrum is easily accessible. Some of the commoner dyes
used are listed in Table 8.5.
8.13 Photochromic Organic Molecules
A photochromic organic compound is one that undergoes a major reversible colour change, usually from
colourless to deeply coloured, on irradiation with light. The reaction can be represented by the equation:
A ðcolourlessÞþ hn1 !B ðcolouredÞAsA is colourless it does not absorb in thevisible, and the ideal frequency for the activating photon is in the near
ultraviolet. The reverse reaction takes place when the coloured form of the molecule absorbs light with
a frequency near to the absorption maximum to yield the colourless product again:
B ðcolouredÞþ hn2 !A ðcolourlessÞThis second step is known as bleaching.
pump
mirror
dye out
dye in
partialmirror
laseroutput
tuner
Figure 8.32 Dye laser (schematic). The dye solution flows through a cell from a temperature-controlledreservoir. The tuning element, which could be a diffraction grating, selects the output wavelength from thebroad emission band
Table 8.5 Dye molecules used in lasers
Dye Output range/nm
Coumarin 9 430 530Rhodamine 6G 540 605Rhodamine B 580 655Oxazine 9 644 709
Colour and the Optical Properties of Materials 358
The first photochromic reaction of an organicmolecule to be reported, by terMeer in 1876, was that of the
potassium salt of dinitroethane, which changes from colourless to red in sunlight and back to colourless in
the dark. Since the mid 1950s there have been a vast number of studies of photochromic molecules, and at
present many hundreds of photochromic organic compounds are known. They have found uses in
applications such as photochromic sunglasses and ski goggles and are actively explored for displays
and information storage.
As with all ‘chromic’ reactions, there is no single mechanism of colour change, and every system has to be
treated separately. To illustrate organic photochromic systems, two widely explored and closely related
groups will be described: the naphthopyrans and the spiro-naphthoxazines. These have found application in
photochromic plastic lenses. (Note that the silver-based photochromic system used in glass lenses (Section
10.18) cannot be used with plastic lenses, necessitating the need for compatible organic photochromic
compounds.)
The strategy for the formation of photochromic molecules in the naphthopyrans is based upon inclusion of
a relativelyweakpyran ring in the structure (Figure8.33a).Typical of thesegroupsofmolecules arebenzopyran
and naphthopyran (Figure 8.33b and c). The pyran ring is opened to form a newmolecule under the influence of
light. In general, the ring-opened form exists in a number of conformationswhich exist in equilibrium. The ring
reformswhen the light source is removed (Figure 8.33d). For example, the colourless molecule (Figure 8.33e)
changes to a purple form under irradiation with ultraviolet light (Figure 8.33f). A similar strategy is employed
O O
Opyran benzopyran
naphthopyran
(a) (b)
(c)
O O O
cis-form trans-form
↔hν
(d)
O O
hν
NEt2 NEt2
Me Me
colourless purple(e) (f)
Figure 8.33 Photochromic molecules: (a) pyran; (b) benzopyran; (c) one form of naphthopyran; (d) ringopening in benzopyran; (e) colourless and (f) purple forms of a naphthopyran derivative. In (d) the two forms ofthe product molecule exist in equilibrium, possibly with other forms. In (f) only one of a number of possiblecoexisting structures is drawn. Me: methyl; Et: ethyl
359 Colour from Molecules
with the spiro-naphthoxazines, but in this case, although a ring is again broken at an oxygen, the ring also
contains a nitrogen atom (Figure 8.34). As before, the resultant molecules can exist in a number of isomeric
forms, two of which are shown.
The photochromic colours generated are tuned by changing the groups attached to the molecular
naphthopyran or spiro-naphthoxazine skeleton. In both cases, ring opening enables the molecule to adopt
a more planar configuration when the bond is broken, allowing for a greater degree of electron delocalisation,
whichmoves the absorptionmaximum into the visible. Naturally, for this to happen in a plastic the host matrix
must be relatively open or flexible.
The inherent problem of organic photochromic materials is one of fatigue, in which the active molecules
degrade with every colouring and bleaching cycle. Naphthopyran and spiro-naphthoxazine derivatives are
fairly resistant to fatigue and are used commercially in a number of applications.
With respect to ring opening, if this can be triggered by a rise of temperature, thematerial changes colour and
is said to be thermochromic. Many molecular species related to the photochromic molecules described above
show this feature and find application from novelty articles to security inks.
O O
O
N
(a) (b)
(c)
N
N
hνN N
N
R R
R
X X
X
Figure 8.34 Photochromic spiro-naphthoxazines. (a) General formof themolecules; R andX represent possiblesubstituent groups. Ringopening in the spiro-naphthoxazines to give a cis isomer. (b) and a trans isomer. (c)Otherisomeric forms are also possible
Colour and the Optical Properties of Materials 360
Further Reading
Background reading on the molecular orbital theory of molecular energy levels of molecules, together with
a description of the associated spectroscopies, is given by
P. W. Atkins, Physical Chemistry, 5th edition, Oxford University Press, 1994 (especially Chapters 16
and 17).
P.W.Atkins, J. de Paula, R. Friedman,Quanta,Matter andChange, OxfordUniversity Press, 2008 (especially
Chapters 10 and 11).
Fireworks are described in
M. S. Russell, The Chemistry of Fireworks, Royal Society of Chemistry, Cambridge, 2000.
The colour of water is discussed by
C. F. Bohren, Clouds in a Glass of Beer, Dover, New York, 2001, Chapter 20 (originally published by John
Wiley and Sons, Inc., New York, 1987.
The colours of water and ice and of liquid and solid D2O are discussed by
T.Quickenden andA.Hanlon,Chem.Br.36 (12), 37 39 (2000), the references cited therein and the subsequent
correspondence: Chem. Br. 37 (2), 19 (2001); 37 (3), 18 (2001).
Sonoluminescene can be tracked starting from
H. Xu, N. G. Glumac, K. S. Suslick, Angew. Chem. Int. Ed. 49, 1079 1082 (2010).
An introduction to organic chemistry is found in
J. McMurry, Organic Chemistry, 6th edition, Brooks/Cole, Belmont, CA, 2004.
Information on the structures and colours of organic molecules will be found in
P. F. Gordon, P. Gregory, Organic Chemistry in Colour, Springer-Verlag, Berlin, 1983.
J. Griffiths, Colour and Constitution of Organic Molecules, Academic Press, London, 1976.
P. Rys, H. Zollinger, Fundamentals of the Chemistry and Applications of Dyes, John Wiley and Sons, Inc.,
New York, 1972.
Information on many aspects of the materials in this chapter is given by
R. M. Christie, Colour Chemistry, Royal Society of Chemistry, Cambridge, 2001.
P. Bamfield, Chromic Phenomena, Royal Society of Chemistry, Cambridge, 2001.
A vast amount of information on porphyrins is contained in
K. M. Kadish, K. M. Smith, R. Guilard (eds), The Porphyrin Handbook, Vols 11 19, Academic Press, San
Diego, 2003. Of relevance to this chapter, see Vol. 19, Applications of Phthalocyanines.
An extensive description of he colours found in plants is given by
D. Lee, Nature’s Palette, The Science of Plant Colour, University of Chicago Press, Chicago, IL, 2007.
T. Bechtold, R. Mussak (eds), Handbook of Natural Colorants, John Wiley and Sons, Ltd, Chichester,
2009.
The following review, and the references cited therein, gives much information on flower colours:
K. Yoshida, M. Mori, T. Condo, Blue flower colour development by anthocyanins: from chemical structure to
cell physiology. Nat. Prod. Rep. 26, 884 915 (2009).
The history of the discovery of mauve is given by
S. Garfield, Mauve, Faber and Faber, London, 2000.
361 Colour from Molecules
For information on blueprints, see
M. Ware, Cyanotype: The History, Science and Art of Photographic Printing in Prussian Blue, Science
Museum, London, 1999.
Pigments, from the point of view of an artist, are the subject of
V. Findlay, Colour; Travels through the Paintbox, Folio, London, 2009.
Details of analysis of metal ions using colour tests is given in
F. Feigl, V. Anger, Spot Tests in Inorganic Analysis, Elsevier, Amsterdam, 1971.
Earlier editions of this volume, author F. Feigl alone, are equally useful
Colorimetric sensor arrays are described by
K. S. Suslick, Mater. Res. Soc. Bull. 29, 720 725 (2004).
An excellent introduction to photochromic materials is given by
H. G. Heller, Photochromics for the future, in Electronic Materials, from Silicon to Organics, L. S. Miller,
J. B. Mullin (eds), Plenum, New York, 1991, p. 471.
For further information, see
J. Crano, R. J. Guglielmetti (eds), Organic Photochromic and Thermochromic Compounds, New York,
Plenum, 1999.
Colour and the Optical Properties of Materials 362
9
Luminescence
. How do fluorescent tubes produce light?
. How do plasma displays produce colours?
. How do glow sticks produce colours?
This chapter is concerned with a number of aspects of colour that are of commercial and social importance.
In day-to-day life, the most important of these may well be fluorescent lighting, which is widespread in homes
and offices, providing a relatively low-energymethod of illumination. Fluorescencemicroscopy and the use of
green fluorescent protein have had an impact upon medicine and health which cannot be overstated. Both of
these are described, together with some other aspects of luminescence which are of related interest. Here, it is
useful to note that the term luminescence essentially means ‘light production’. It does not imply that a single
process operates in all cases. Thus, the mechanism of light emission by a firefly is quite different from light
emission by a fluorescent bulb, although both may be termed luminescence.
9.1 Luminescence
The emission of light by bodies at relatively low temperatures, ‘cold light’, is generally called luminescence,
which can be contrasted with light emission by a hot body, called incandescence (Section 1.6). Solids that give
rise to luminescence are called phosphors or, latterly, luminescentmaterials. Investigations into luminescence
have a long history. The term phosphor is derived from the element phosphorus. This element was first isolated
in about 1674or 1675by the alchemistBrandt,whodiscovered that thematerial shonewith a palegreenish light
in the dark and gave it the namephosphorus, which is from theGreek phos (light) and phero (I carry). The name
Colour and the Optical Properties of Materials Richard J. D. Tilley
� 2011 John Wiley & Sons, Ltd
for the glow from phosphorus, phosphorescence, was taken over and quitewidely applied tomany other forms
of ‘cold light’, including that from decaying organisms.
Themodern story of luminescence can be thought to start with Ritter, in 1801, who, investigating solids that
would glow after being illuminated with daylight, discovered that the effect was greatest when the samplewas
placed in the dark region beyond the violet end of the spectrum.He postulated, therefore, that an invisible form
of ‘light’, termed ultraviolet, existed (Figure 9.1).
Figure 9.1 (a) A pale yellow phosphor based on zinc sulfide (ZnS) in normal daylight. (b) The same materialirradiated with ultraviolet light with a wavelength range of 350–380 nm, showing a bright yellow–greenfluorescence
Colour and the Optical Properties of Materials 364
In 1852 Stokes published the results of an extensive investigation inwhich he invented the term fluorescence
for the light that he observed emerging from crystals of the mineral fluorspar after illumination. Stokes’ law,
proposed at this time, stated that the radiation emitted has a longer wavelength (lower energy) than the exciting
radiation. Thewavelength difference is known as the Stokes shift. A.-E.Becquerel, in France, also studying the
emission of light by solids after illumination, was of the opinion that fluorescencewas simply a short-duration
form of phosphorescence in which hewas essentially correct. Stokes’ fluorescence is characterised by the by
the immediate release of the exciting energy as light, while A.-E. Becquerel’s phosphorescence is typified
by the slow conversion of the exciting energy into light, so that light emission is delayed by a length of time
that can vary from milliseconds to hours or days. Now it is appreciated that the two expressions represent
extremes on a continuum that can be defined in terms of a quantum mechanical probability of the emission of
visible radiation.
Roughly coincident with the fluorescence and phosphorescence studies of Stokes andBecquerel, researches
by Faraday, Giessler, Crookes and others showed that gases and some solids produced light when bombarded
with ‘cathode rays’ (Section 7.5). To distinguish this light from fluorescence and phosphorescence the effect
was called cathodoluminescence.
Yet another formof luminescencehadanenormous impact upon twentiethcentury science.H.Becquerel, the
son ofA.-E.Becquerel,was studying phosphorescencewhen he discovered, in 1896, that uraniumsalts emitted
a radiation, ‘uranium rays’, thatwere new. Pierre andMarieCurie followed this up anddiscovered radioactivity
and the highly radioactive element radium. The radioactivity is intense and this causes manymaterials to emit
light, now called radioluminescence.1 Radium was widely used in luminous paints for watch and other
instrument dials that could be seen in the dark. This has now ceased due to the harmful effects of the radiation
emitted.
These preceding studies pointed the way towards a general understanding of luminescence. It is now clear
that luminescent materials are able to gain energy from an energetic, often ‘invisible’ source (ultraviolet light,
electric fields,X-rays, energetic particles from radioactive decay, and so on2) and re-emit someof this energy in
the form of light. For this reason, luminescence is now subdivided into a number of categories depending upon
the nature of the exciting source (Table 9.1).
Recall that there are no general mechanisms for luminescence, and apart from fluorescence and phospho-
rescence, which are two aspects of the same process, each type needs to be treated independently.
Phosphors are widely used in, for example, fluorescent lamps (ultraviolet to visible), old-fashioned cathode
ray (CR) tube TV (electron impact to visible) and scintillators (X-rays, g-rays and energetic subatomic particle
impact to visible). Molecular fluorescence is of increasing importance in the study of living organisms and
medical sciences via fluorescence microscopy and related techniques.
9.2 Activators, Sensitisers and Fluorophores
The first commercial phosphor, ‘Balmain’s paint’, calcium sulfide (CaS), was produced in 1880. Partly as a
result of the desire to make better commercial materials, it was discovered that in many instances pure
compounds would not show luminescence, although the samematerial when contaminated with minute traces
of impurities was luminescent. Moreover, the colour of the luminescence was dependent upon the chemical
nature of the impurity.
1 Marie Curie described the phenomenon thus: ‘One of our joys was to go into our workroom at night; we then perceived on all sides the
feebly luminous silhouettes of the bottles or capsules containing our products. It was really a lovely sight and one always new to us. The
glowing tubes looked like faint, fairy lights.’2 Materials that emit light in response to the impact of high energy particles or X rays are often termed scintillators rather than phosphors.
365 Luminescence
This can be illustrated with respect to the fluorescence of minerals when irradiated with ultraviolet light.
(No distinction will be made between fluorescence and phosphorescence here.) A few examples are:
white or
bluish white:
agate (SiO2), aragonite (CaCO3), calcite (CaCO3), gypsum (CaSO4�2H2O), fluorite
(CaF2), halite (NaCl), wollastonite (CaSiO3);
red: barite (BaSO4), calcite (CaCO3), corundum (Al2O3), halite (NaCl), sphalerite (ZnS);
orange: barite (BaSO4), calcite (CaCO3), scheelite (CaWO4), sphalerite (ZnS), wurtzite (ZnS),
zircon (ZrSiO4);
yellow: agate (SiO2), calcite (CaCO3), diopside (CaMgSi2O6), scheelite (CaWO4),
talc (Mg3Si4O10(OH)2), wollastonite (CaSiO3), zincite (ZnO), zircon (ZrSiO4);
green: agate (SiO2), aragonite (CaCO3), calcite (CaCO3), opal (SiO2), willemite (Zn2SiO4);
blue: albite (NaAlSi3O8), calcite (CaCO3), fluorite (CaF2), gypsum (CaSO4�2H2O), sphalerite
(ZnS), wollastonite (CaSiO3).
The first point of note is that many minerals appear frequently and show different fluorescent colours. This
indicates that the crystal matrix is simply acting as a (nominally inactive) host that has a small quantity of
impurity or activator (A), incorporated within it. The role of the host structure or of the host activator
combination is to absorb an excitation in the form of a photon of energy hn1. The activator re-emits the
excitation as a photon of energy hn2. The colour emitted is dependent upon the nature of the activator.
Sometimes it is found that the activator-containing material cannot absorb the exciting radiation directly, in
which case a helper species, a sensitiser, is needed as well. In this case the sensitiser absorbs the exciting
photons, of energy hn3, and passes the energy to the activator. The sensitiser can be the crystal matrix itself or
a specially introduced centre such as another cation (Figure 9.2a).
Irrespective of whether the luminescence is derived from an activator sensitiser pair, or just from the
activator alone, a rapid decay of light is characteristic of fluorescence. On the other hand, a slow decay is
characteristic of phosphorescence (Figure 9.2b). In this case, the energy is often regarded as being stored in
Table 9.1 Types of luminescence
Type Definition Source of energy
Fluorescence Electronic decay between allowed states Ultraviolet and visible photonsPhosphorescence Electronic decay between forbidden states Ultraviolet and visible photonsBioluminescence Luminescence in a living organism Gibbs energy of chemical reactionsCathodoluminescence Luminescence due to electron
bombardment (cathode ‘rays’)Electron kinetic energy
Chemiluminescence Luminescence during a chemical reaction Gibbs energy of chemical reactionElectroluminescence Luminescence resulting from the
application of an electric fieldElectrical potential energy
Photoluminescence Luminescence after irradiation by visible orultraviolet light
Ultraviolet and visible photons
Radioluminescence Luminescence as a result of radioactivity Energetic particles and g raysTenebrescence Reversible darkening under irradiation Photon or particle energyThermoluminescence Luminescence following an increase of
temperatureThermal energy
Triboluminescence Luminescence following fractureor friction
Chemical bond energy
Colour and the Optical Properties of Materials 366
a reservoir fromwhich it slowly leaks. This feature is more commonly associated with heavy atoms, and is one
of the reasons why H. Becquerel was interested in uranium compounds.
Although a large amount of study and research has focused on inorganic phosphors, because of applications
in lighting, TV tubes anddisplays, the fluorescence of organicmolecules is equally important. In these systems,
the activator is usually the molecule itself, and such molecules are said to show autofluorescence. In larger
molecules, only a specific group of atomsmight be involved in the fluorescence, and, by analogy with the term
chromophore, the group is labelled a fluorophore. In this sense, a fluorophore in an organic molecule is the
equivalent of an activator in an inorganic phosphor.However, this term is used rather imprecisely, and often it is
applied to any small fluorescent molecule. Thus, the molecule fluorescein (see below) is often called a
fluorophore.
As with inorganicmaterials, organic fluorescent organic compounds may need a sensitiser. This may just be
a different part of the same large molecule or be the surrounding matrix or solvent.
h in3h in1
h out2
AS
energy transfer
heat
heat
(a)
Time
Lum
inou
s in
tens
ity
(b)
phosphorescence
fluorescence
Figure 9.2 Schematic representation of energy absorption and emission processes taking place in a luminescentmaterial. (a) Absorption of radiation. A represents an activator centre and S a sensitizer centre. The photonsabsorbed and emitted, hn1, hn2 and hn3, need not necessarily all be different. Some energy is also lost to the hoststructure as vibrational energy (heat). (b) Emission of radiation. Fluorescence is characterised by a rapid decayof intensity, while phosphorescence is characterized by a slow decay
367 Luminescence
Fluorescence from organic molecules has long been used by the manufacturers of detergents used for
washing white clothes. A colourless dye is incorporated into the detergent which fluoresces blue when
irradiated with ultraviolet light. The small amount of ultraviolet in natural daylight is sufficient to create this
effect, but illumination by ultraviolet light in, for example, a club can make these garments almost shine.
The structures of these fluorescent brighteners generally consist of linked benzene rings, together with
groups to aid in the solubility in water and incorporation into the cloth (Figure 9.3a). Similar additives can be
found in cosmetics which are designed to glow when illuminated by the ultraviolet light in discos and
nightclubs.
Fluorescein (C20H12O5, Figure 9.3b), a fluorescent compound, is one of a remarkable family of coloured
materials closely related to phenolphthalein (Figure 8.28). Fluorescein is a yellow red powder with an intense
green fluorescence. Fluorescein itself is rather insoluble and is more often met with as ‘soluble fluorescein’,
which is the disodiumsalt,Na2C10H10O5,which is freely soluble inwater.The excitation radiationmaximum is
close to 495 nm (blue green) and the fluorescence wavelength is 519 nm (green). The effect of the absorption
and fluorescence is to impart an unmistakable intense yellow green fluorescence to solutions. It is widely used
to colour safety garments and is the familiar yellow greenmarker colour used to highlight passages of text. It is
also the bright yellow green colour that is used in eye examinations and contact lens fitting.
9.3 Atomic Processes in Photoluminescence
There are two basic atomic processes that must take place during photoluminescence: (i) photon absorption;
(ii) photon emission. In addition, energy transfer between excited and nonexcited states is often important and,
indeed, vital when sensitisers are a necessary component of the luminescent system. Some of these processes
are listed in Table 9.2 and are discussed at various points throughout this chapter.
9.3.1 Energy absorption and emission
The initial process that takes place in fluorescence is the absorption of a photon of the exciting radiation,
E1¼ hn1. (No distinction will be made between fluorescence and phosphorescence here.) For simplicity,
NN NNHH HH
CC
OO
CCHH
SO Na2NaO S2
+ -Na O - +O Na
O
O
(b)O
(a)
Figure 9.3 The structures of fluorescent molecules: (a) C. I. fluorescent brightening agent 30, used indetergents; (b) soluble fluorescein, used in protective clothing and ophthalmic medicine
Colour and the Optical Properties of Materials 368
assume that the absorption takes place at the activator, which is excited from the normal low-energy ground
state A to an excited state A�. The activator subsequently emits the fluorescent photon and returns to the
ground state. Because the emitted radiation is at a greater wavelength (lower energy) than that absorbed,
some energy DE1 is redistributed from light energy into another form. Schematically, the activator drops
down through a closely spaced set of energy levels to a new lower state (Figure 9.4a). The transitions
responsible for this degradation are generally called nonradiative transitions, decay or relaxation. The
energy deficitDE1 generally ends upwithin the phosphormatrix, in the form of lattice vibrations or phonons,
i.e. heat. The activator then returns to the initial state, emitting a photon as it does so, in a radiative transition
conforming to E2¼ hn2. The lower energy level reached may not be the ground state, but one of several
higher states associated with the lower level. Radiationless transitions will again disperse this extra energy
DE2 as vibrational energy in the host matrix until the centre reaches the final state of lowest energy
(Figure 9.4b). Note that these nonradiative transitions are often drawn as if they occur between vibrational
energy levels. This is not mandatory, and transitions can be between electronic energy levels as long as the
energy can be carried away by phonons successfully. The 4T2g! 2E transition in ruby is an example
(Sections 7.10 and 7.11).
The difference in energy between the exciting radiation and the emitted radiation is:
DE ¼ DE1þDE2
Table 9.2 Photoluminescence processes
Process Examplea Lifetime/s
Absorption of photonsGround state absorption (GSA) A þ hn ! A� 10�16 10�15
Excited state absorption (ESA) A� þ hn ! A��
Multiphoton absorption A þ nhn ! A�
Emission of photonsFluorescence (spontaneous emission) A� ! A þ hn þ phonons (allowed transition) 10�12 10�6
Phosphorescence (spontaneous emission) A� ! A þ hn þ phonons (forbidden transition) 10�6 1Stimulated emission A� þ hn ! A þ 2hn
Photon conversionUp conversion (UC) A þ nhn1 ! A� ! A þ hn2 (n1 < n2)Quantum cutting (QC) A þ hn1 ! A� ! A þ nhn2 (n1 > n2)
Energy distribution and quenchingMolecular collision A� þ Q ! A þ Q þ phononsDefect A� þ De ! A þ De þ phononsInternal conversion (IC) A� ! A þ phononsIntersystem crossing (ISC) 1A*! 3A* 10�12 10�4
Energy transfer (ET) S� þ A ! A� þ S; A� þ A ! A þ A�;A� þ Q ! A þ Q�
Cross relaxation (CR) A� þ A� ! A þ A��
aA: activator, luminescent centre ground state; S: sensitiser ground state; Q:molecule; De: surface or bulk defect; �: excited state; ��: doubly excitedstate; phonons are equivalent to heat energy.Note that intersystemcrossing can also involve othermultiplicities and is not confined to singlet–triplet
pairs. The lifetime gives an approximation to the length of time that the process takes.
369 Luminescence
The wavelength equivalent of this energy difference:
Dl ¼ hc
E1
� hc
E2
ð9:1Þ
constitutes the Stokes shift (Figure 9.4c).
Under intense irradiation, an excited state A� can absorb a second photon to reach a higher energy state A��,a process called excited-state absorption. The excited state can then undergo similar vibrational losses before
return to the ground state (Section 9.9). Similarly, a ground state A can absorb several low-energy photons
simultaneously to jump to the excited state A� (Section 9.11.4).
9.3.2 Kinetic factors
The difference between fluorescence and phosphorescence can usefully be discussed in terms of the kinetics of
allowed and disallowed transitions. For light atoms, organic molecules, proteins and so on the instantaneous
production of light (fluorescence) can be regarded as due to a spin-allowed transition (DS¼ 0). The delayed
production of light (phosphorescence) is attributed to a spin-forbidden transition (DS¼ 1, 2, etc.). Frequently,
ground state A
excited state A*
E = hν1 E = hν2
(a) ΔE1
ΔE2
(b)
excitation
emission
Wavelength λ
Stokes shift
Inte
nsity
(c)
λ1 = hc /E1
λ2 = hc /E2
Figure 9.4 Energy transfer in phosphors (schematic): (a) absorption by activator; (b) luminescent photonemission; (c) the Stokes shift between the wavelength of the excitation and the emission pulses
Colour and the Optical Properties of Materials 370
molecular phosphorescence is associated with the transformation of a singlet excited state 1A* into a triplet
excited state 3A* that takesplacemore rapidly than thedownwardfluorescence transition.These changes canbe
displayed on energy-level diagrams called Jablonski diagrams, which set out the electronic and vibrational
energy levels of a molecule in a schematic way, with singlet and triplet states shown in separate columns.
Phosphorescence ariseswhen amolecule in a singlet excited state is transformed into a triplet state (Figure 9.5).
In this process, called intersystem crossing, the vibrational energy levels of both states coincide and
the molecule can transform from one multiplicity to the other without requiring energy input. In the triplet
state, radiationless transfer of energy continues until the molecule lies at the lowest energy level of the
triplet state. Further emission by a photon is slow because of the selection rules prohibition.
In the case of heavy atoms such as transition metals and lanthanoids, mixing of the wave functions on the
atoms leads to spin orbit coupling and in reality the spin states are not aswell defined as themultiplicity symbol
suggests. Thus, although the 4T2! 4A2 transition in ruby can be labelled fluorescence (DS¼ 0) and the2E! 4A2 transition as phosphorescence (DS¼ 2), in practice transitions are often allowed or disallowed to
a varying degree by virtue of both the spin and parity selection rules.
9.3.3 Quantum yield and reaction rates
The quantum yield, which measures the efficiency of the fluorescence, is given by:
FðlÞ ¼ Npe
Npa
ð9:2Þ
where Npe is number of photons emitted during fluorescence and Npa is the number of photons of the exciting
radiation of wavelength l absorbed. The quantum yield reflects the number of ways that the excited state can
lose energy. If every excited centre loses energy by only one reaction, rapid photon emission, then the quantum
yieldwill be unity.Quantumyields of 10%ormoremay be satisfactory for some applications, butmuch higher
quantum yields are always desirable and are mandatory for some specialist devices.
Because the numbers of photons emitted and absorbed are measured over a certain time span, the quantum
yield is a measure of the rate of fluorescence, which can be treated in terms of chemical kinetics. The rate of
decay of the excited state A�will simply be given by the sum of the rates of all the deactivation reactions that
contribute to the loss of energy of the excited state, including fluorescence.
1ground state A
1excited state A* 3excited state A*
E = hν3
fluorescence
phosphorescence
ISC
E = hν2E = hν1
Figure 9.5 Intersystem crossing (ISC) in which a fluorescent molecule changes from an excited state with spinmultiplicity 1 to an excited state with spin multiplicity 3, leading to phosphorescence
371 Luminescence
The simplest case to consider is when the rate of decay of the exciting centres is simply a function of the
number of excited centres that are formed; in chemical terms, the concentration ofA�, written [A�]. The rate ofthe reaction is given by the differential form:
d½A*�dt¼ �k A*½ � ð9:3Þ
where d[A�]/dt is the rate of decay of the excited state and k is the rate constant of the process. This dependenceis termed first order and in this case the rate of decay follows first-order kinetics:
½A*�t ¼ ½A*�0e kt ¼ ½A*�0e t=t ð9:4Þ
where [A�]t is the concentration of excited centres at time t after a pulse of excitation radiation has generated an
initial population of [A�]0, t is the elapsed time and t is the fluorescence lifetime. The fluorescence lifetime is,
then, the time taken for the number of excited centres to decay to a value of 1/e of the initial value at t¼ 0.
The luminescence lifetime and phosphorescence lifetime are defined in the same way. Note, though, that the
difference between fluorescence and phosphorescence is simply a matter of rate of reaction and there is no
value of t that arbitrarily separates one from the other.
The number of excited centres is assessed by measuring the radiant exitance, and this can be substituted for
[A�] in these equations.Note that in almost all literature this is termed ‘intensity’, given in arbitrary units, and is
plotted in the form:
It ¼ I0et=t
A plot of the (natural) logarithm of the radiant exitance (or It) emitted against time will give a straight line of
slope k¼ 1/t (Figure 9.6). Departure of the plotted curve from an exponential form is evidence that the
mechanism of light emission is more complex than that supposed.
If a luminescent material loses energy by first-order processes due to fluorescence and phosphorescence,
both will depend upon [A�] in the way described by Equations 9.3 and 9.4. The rate constant k of the overall
reaction is now given by the sum of the rate constants for fluorescence and phosphorescence, kF and kP:
k ¼ kFþ kP
The quantum yield for fluorescenceFF(l) is given by the number of fluorescent photons emitted in a certain
time comparedwith the total number absorbed and then used up in the two competingprocesses of fluorescence
and phosphorescence. It is then possible to write Equation 9.2 in terms of the rate constants as:
FFðlÞ ¼ kF
kFþ kP
Similarly, the quantum yield for phosphorescence FP(l) is given by the number of phosphorescent photons
emitted in a certain time compared with the total number absorbed and then used up in the two competing
processes of fluorescence and phosphorescence, leading to:
FPðlÞ ¼ kP
kFþ kP
Colour and the Optical Properties of Materials 372
It follows that if the excited centre loses energy by a number of other first-order processes that compete with
fluorescence, the overall reaction rate constant is given by:
k ¼ kFþ kPþ kXþ kYþ � � � ð9:5Þ
and the fluorescence quantum yield is given by:
FFðlÞ ¼ kF
kFþ kPþ kXþ kYþ � � � ð9:6Þ
A number of luminescent materials exhibit light production, often at a very low level, for much longer than
the lifetime indicates. This is called afterglow, and is distinct from phosphorescence. Afterglow is, in broad
1.0
0.5
00.5 1.0 1.5 2.0 2.5 3.0
Time / arbitrary units
Time / arbitrary units
Rad
iant
exi
tanc
e / a
rbitr
ary
units
Ln (
Rad
iant
exi
tanc
e)
slope = –1/τ
(a)
(b)
Figure 9.6 First-order kinetics of fluorescence: (a) exponential decay of radiant exitancewith time (schematic);(b) the slope of a plot of ln(radiant exitance) versus time gives the reciprocal fluorescence lifetime 1/t
373 Luminescence
terms, due to electrons becoming trapped at a sitewhere they are prohibited from losing energy. It is prevalent in
semiconductors (Chapter 10), where both trapped electrons and trapped holes can cause afterglow. Generally,
the electrons (or holes) are released from trapping sites by thermal energy, after which the normal luminescent
process can take place. Afterglow is a problem for some applications.
9.3.4 Structural interactions
The role of the surroundingmedium is often important in processes that lead to luminescence. For example, the
widthof the absorptionandemissionpeaks is verydependent upon the interactionof theorbitals on the activator
with the surroundingmatrix. In the case of transitions that take place betweenwell-shielded inner orbitals, such
as f f transitions, the importance of the external structure is masked, and fluorescence emission lines are
narrow. However, if the orbitals involved interact with the outer matrix, such as d d transitions (Cr3þ ) or d f
transitions (Eu2þ ) or p s transitions (Sb3þ ), the fluorescent emission bands are wide. These differences are
of considerable importancewhen the performance of fluorescent lamps and other fluorescence-based displays
is evaluated.
Solvatochromism, the change in colour of a material due to a change in solvent polarity, provides another
example of the way in which the surroundings influence fluorescence. The change in colour is described as
negative solvatochromism if the colour shift moves the emission to shorter wavelengths (a hypsochromic or
blue shift) as the polarity of the solvent increases. It is called positive solvatochromism if the colour shiftmoves
the emission to longer wavelengths (a bathochromic or red shift) as the polarity of the solvent increases. The
colour shift comes about because the ground-state energy, the excited-state energy, or both aremodified by the
surrounding solvent. Solvatochromism is then a manifestation of a change in the position of the electronic
absorption and emission bands from a fluorophore. It is often displayed by polarmolecules, i.e. moleculeswith
an observable dipolemoment. The energy of the ground statewill be influenced by the interaction of the solvent
with thedipoleon themolecule. If the solvent is nonpolar, typically ahydrocarbon solvent, little interactionwill
occur. If the solvent is polar, such as water or an alcohol, the interaction may be large. In such a molecule,
the excitation of an electron from the ground-state orbital to the excited-state orbital will significantly alter the
dipole on the molecule. The interaction with the solvent will then be different in the excited state to that in the
ground state. Thus, a change in the polarity of the medium containing the fluorophore will alter the relative
positions of the excited- and ground-state energy levels. Adding these effects together gives rise to the overall
change in colour.
Charge-transfer colours that are associated with cation-to-ligand or ligand-to-ligand electron transfer are
also susceptible to solvatochromic effects. Theorbitals on the ligands aregenerally exposed to the surroundings
and this has an effect upon theground-state energy.Transfer of an electron fromacation to a ligand, or fromone
ligand to another, creates an excited state, the energy of which is also influenced by the surroundings. The net
difference between the two will then vary if the surroundings change.
Although solvatochromism was originally described in terms of molecules in solution, the definition now
includes colour change due to the influence of any external surroundings, including a solid matrix.
9.3.5 Quenching
In many circumstances the ability of a normally luminescent centre to emit fluorescence is suppressed or
inhibited. This feature is called fluorescence quenching. Quenching is said to be dynamic if the inhibition
involves the excited state and static if it involves the ground state in such a way as to prevent the excited state
from forming. Quenching is not the result of just a single process, but can be caused by a multiplicity of
reactions that are able to compete with the fluorescence mechanism. Several are described below.
Colour and the Optical Properties of Materials 374
9.3.5.1 Thermal quenching
In solid phosphors, thermal quenching, the reduction or suppression of luminescence due to increase in
temperature, is of importance inmany applications (Figure 9.7a). Some lamps, for example, can become quite
hot during operation. Thermal quenchingwill then drastically reduce the amount of light given out. The reason
for thermal quenching lies in the vibrations of the surrounding matrix. At very low temperatures these are
minimal. Electronic excitation will promote a luminescent centre from a low vibrational level in the ground
state to a lowvibrational level in the excited state (Figure9.4a).Aconsiderablegapbetween theupper and lower
energy levels is present that is bridged by the emission of a photon. As the temperature increases, higher and
higher vibrational energies are occupied in the ground state and excited state. Ultimately, the ground-state and
ground state
E = hν1
(b)
Temperature / K
Rel
ativ
e ex
itanc
e
(a)
400 500 600 700 800 900
5
10
CaWO4 Eu3+:Gd2O3
excited state
Figure 9.7 Thermal quenching: (a) relative exitance emitted by CaWO4 (WO42� fluorophore) and Eu3þ doped
into Gd2O3 (Eu3þ activator); (b) schematic depiction of the vibrational energy levels of the ground and excited
states at high temperatures. The excited centre can move into the ground state entirely via nonradiative steps
375 Luminescence
excited-state vibrations appear as if merged and the potential energies of both the ground and excited states are
virtually the same. In this case the differentiation between the two configurations is blurred and it becomes
possible for the excited state to pass from an excited-state vibrational energy level to a ground-state vibrational
energy levelwithout photon emission (Figure 9.7b). Inmolecular terms, themultiplicity of the two states is the
same and the transfer of the system from the excited electronic state to upper echelons of the ground state
is called internal conversion (IC). Further transitions down the ground-state vibrational energy-level ladder
return the system to the ground state solely via nonradiative phonon transitions. IC is a first-order reaction
obeying the kinetics given by Equations 9.3 and 9.4. The rate constant kIC can be incorporated into
Equations 9.5 and 9.6.
9.3.5.2 Energy transfer
Energy transfer away from the excited centre to another centre or the surroundings will quench the
fluorescence. This is a process akin to IC, in that the energy transfer is radiationless. The best understood
mechanism of energy transfer is F€orster resonance energy transfer (FRET), sometimes called fluorescence
resonance energy transfer. The energy absorbed by the fluorescent centre in the reaction:
Aþ hn!A*
is given by:
DEA ¼ hnA
The frequencynA is called the resonant frequency for the transition. If the resonant frequencymatches a similar
frequency on a nearby quencher molecule Q (the resonance condition), i.e.:
DEA ¼ hnA ¼ DEQ ¼ hnQ
then energy can be transferred from A� to Q thus:
A*þQ!AþQ*
The centre which provides the energy, A�, is often called the donor, and the centre which receives the energy,Q, is often called the acceptor. In addition to the resonance condition, energy transfer can only take place if
a suitable interaction is present between the two centres. This can be the overlap of suitable wavefunctions,
electric or magnetic dipole interactions or, more rarely, other multipole interactions. This latter condition
implies that resonant energy transfer will only occur when the two centres are very close (Figure 9.8a). These
conditions canbe summarizedgraphically in termsof theoverlap of the absorption spectrumofQ(the acceptor)
and the emission spectrum of A� (the donor) for the transition in question. The rate of energy transfer is
proportional to the area of overlap between the two spectra (Figure 9.8b).
In general, energy transfer will be in competition with other processes, such as fluorescence. The relative
rates of all of these processeswill then contribute to the effectiveness of the energy transfermechanism. For the
sake of simplicity, assume that the only two processes that occur are either fluorescence from A� or energytransfer to Q. The critical separation R0 of the centres, the F€orster distance, can be taken as that at which therates of these two processes are equal; that is:
kF ¼ kET
Colour and the Optical Properties of Materials 376
If the distance R between the centres is greater than R0, then fluorescence from A� occurs, whereas energytransfer is preferred if R is less than R0. For many systems the efficiency of energy transfer ZFRET is
given by:
ZFRETR60
R60þR6
The value of R0 is of the order of 2 5 nm.
9.3.5.3 Concentration quenching
Concentration quenching occurs in many inorganic phosphors. In this phenomenon, the fluorescent centres
show good quantum yields when present at low concentrations. However, when the concentration of the
luminescent centres increases beyond a certainvalue, whether because of distortions of the surrounding crystal
structure, because of clustering or simply because the luminescent centres are close enough for their electron
orbitals to interact, excitation energy can be passed to an adjoining centre and so does not result in emission of
a photon.
There are a number of mechanisms for this energy transfer, which depend upon the closeness of the centres
and the way in which they interact. The two main routes are known as direct energy transfer (ET) and cross-
relaxation (CR). Energy transfermay allow energymigration through the structure by jumping fromone centre
Wavelength
Inte
nsity
(b)
A* Q*
A Q
ET
F
R
Q absorptionspectrum
A* emissionspectrum
ΔE = hνA hνQ=
(a)
overlap
Figure 9.8 F€orster resonance energy transfer (FRET). (a) Two competing processes, energy transfer (ET) andfluorescence (F), are possible when the energy-level separation of the A and A� and the S and S� centres are equal.(b) The efficiency of the energy exchange is proportional to the degree of overlap of the emission spectrum of A�
and the absorption spectrum of Q
377 Luminescence
to another. Energy transport in thisway iswell known in crystals containing a high concentration ofGd3þ ions,
where energy is efficiently transferred through theGd3þ sublattice jumping from one ion to another. Although
energy transferred in thisway does notmean that photon emission cannot eventually occur, quantumefficiency
is frequently impaired if significant energy transfer occurs.
Concentration effects are not confined to the luminous centres alone. Impurities or defects in the solid,
especially those near to surfaces, can accept energy. If these reach critical concentrations, then luminescence is
throttled. For this reason, much effort is directed towards improving the crystalline perfection of phosphor
powders.
Concentration quenching is often observed in solutions of fluorescent molecules. The concentration effect
can involve only the active molecule, in which case the effect is also called self-quenching, or it may involve
an added quenching molecule called a quencher. Fluorescein is a self-quenching molecule and anthracene
(C14H10) is quenched by indole 2,3-benzopyrrole (C8H7N) molecules.
In solution, concentration quenching is frequently modelled in terms of the collisions between the
fluorescent species and the quenching species. The simplest form that this can take is a bimolecular
reaction:
A*þQ!AþQ
where Q is a molecular quencher. The rate of such a reaction is given by:
Rate ¼ kM½A*�½Q�
wherekMis the rate constant of the reaction, [A�] is the concentration of thefluorescentmolecules and [Q] is the
concentration of the quencher molecules. In the case of dynamic quenching, the quantum yield is given by the
Stern Volmer equation:
F0ðlÞFðlÞ ¼ 1þKSV Q½ �
where F0(l) is the quantum yield in the absence of a quenching molecule, F(l) is the quantum yield when
the concentration of quenching molecules is [Q] and KSV is the Stern Volmer constant. The value of the
Stern Volmer constant is given by:
KSV ¼ t0kQ
where t0 is the fluorescence lifetime of the fluorescent species in the absence of the quenching molecules and
kQ is the rate constant of the quenching reaction due to molecular species Q. The Stern Volmer equation is
often written in terms of the radiant exitance, M0/M, or the lifetimes, t0/t:
M0
M¼ t0
t¼ 1þKSV Q½ � ¼ 1þ t0kQ Q½ �
whereM0 is the radiant exitance in the absence of the quencher,M is the radiant exitance in the presence of the
quencher, t0 is the fluorescence lifetime in the absence of the quenching molecules and t is the fluorescencelifetime in thepresenceof thequencher.Aplot ofF0/F,M0/Mor t0/twill yield a lineargraphwith a slopeofKSV
(Figure 9.9).
Colour and the Optical Properties of Materials 378
9.4 Fluorescent Lamps
Fluorescent lamps utilize photoluminescence for light generation. Fluorescent lighting for advertising was
first used in 1925, and development of phosphors during the 1930s led to the commercial introduction of
low-voltage fluorescent lamps in 1939. The intensity of the luminescence is roughly proportional to the
amount of phosphor that is exposed to exciting radiation. Early phosphors were not especially efficient, and
the first fluorescent lamps were in the form of tubes about 1m in length. Improvements in phosphor
specification have made the efficiency greater, and since the 1980s compact fluorescent lighting has become
commonplace.
These lamps, used for indoor lighting, contain an inert gas and a small quantity of mercury vapour at a low
pressure.Under electron bombardment from the current passing through the lamp theHg atoms are excited and
emit copious ultraviolet radiation. This consists mainly of line emissions with wavelengths 185, 254 and
365 nm, as well as some radiation in the visible (Section 7.7). Conversion of the ultraviolet radiation to visible
is by way of a phosphor coated onto the inside of the tube (Figure 9.10).
9.4.1 Halophosphate lamps
These lamps use modified calcium fluorophosphate (Ca5(PO4)3F) as the host matrix. When doped with Sb3þ
ions as activator (written asCa5(PO4)3F:Sb), ablue emission isproduced.TheSb3þ ions absorbvia an s2 to s1p1
transition centred at 254 nm,which closelymatches themercury vapour output. Aminor problemwith Sb3þ is
that the blue emission gives the lamps a rather cool colour. If Mn2þ is also incorporated into the system as a
coactivator then a warmer tone is produced, as this ion produces an orange red emission (Figure 9.11).
Variation in the proportions of Sb toMnvaries the tone of the light. Note that the emission bands are very broad
because the orbitals involved in the electron transitions producing the light output interact strongly with the
surrounding crystal matrix.
There are a number of other aspects of the phosphorwhich are of interest. First, although theMn2þ has good
emission characteristics, it is found to be unsatisfactory when used alone. Fortunately, the Sb3þ acts as a
sensitiser for the Mn2þ ion, thus avoiding another component in the phosphor. Second, the Mn2þ and Sb3þ
ions occupy the Ca2þ positions in the host matrix. Now, while Mn2þ incorporation will not pose an
electroneutrality problem, as the Mn2þ ions have the same charge as the Ca2þ ions that they replace, this
is not so with Sb3þ . The introduction of Sb3þ ions into the phosphate will thus cause an internal charge
slope = KSV
Φ / Φ0
Concentration [Q]
1
Figure 9.9 Idealised Stern–Volmer plot.F0 is the quantum yield in the absence of a quenchingmolecule,F is thequantumyieldwhen the concentration of quenchingmolecules is [Q] andKSV, the Stern–Volmer constant, is givenby the slope of the graph
379 Luminescence
imbalancewhichwill result in adegradationofperformance.Toovercome this, chargebalance ismaintainedby
adding one F or Cl ion to the phosphate for each Sb3þ ion. It has been found that an empirically derived
composition for the host matrix of Ca10P6F1.8Cl0.2O24 is most satisfactory. These lamps are still available and
work continues on improving their performance.
400 500 600 700
blue (Sb3+)
orange-red (Mn2+)
Wavelength / nm
Em
issi
on (
arbi
trar
y un
its)
Figure 9.11 Emission spectra from Sb3þ (blue emission) and Mn2þ (orange–red emission) in a typicalhalophosphate fluorescent tube phosphor
Hge–
ultraviolet
visible light
vis ble light
glass tube
phosphor
(a)
(b)
Figure 9.10 Fluorescent lamps: (a) schematic fluorescent tube lamp; (b) processes occurring in the lamp.Electrons (e�) from the cathode collide with mercury (Hg) atoms, which emit ultraviolet radiation which isconverted into visible light by a phosphor coating on the inside of the glass tube
Colour and the Optical Properties of Materials 380
9.4.2 Trichromatic lamps
Over the years, improvements have occurred in fluorescent lighting, especiallywith respect to the colour of the
light produced. Trichromatic (Colour 80) lamps produce a very good spectral balance by using a phosphor
mixture which emits equal amounts of the colours red, blue and green.
The commonestfluorescent centres used are lanthanoidor transitionmetal ions.Lanthanoid ionshavea set of
unfilled 4f orbitals. Electron interactions give rise to a large number of electronic energy levels. The 4f orbitals
are shielded from the surroundingmatrix by outer 5p, 5d and 6s orbitals which contain electrons, so that the 4f
energy levels are sharp and similar to those in isolated atoms or ions. The orbitals, 4d, 5p and 6s, all interact
stronglywith the environment and instead of presenting sharp energy levels they are broadened intowider band
of energy (see Chapter 10 for more information on this). Moreover, transitions from a 4f energy level to one of
these orbitals are quantum mechanically allowed. This combination of factors means that excitation can be
achievedby awide range of excitingwavelengths.Energy loss via photon emissionusually takes placebetween
f energy levels. These transitions are forbidden for the same reasons as transitions between the 3d orbitals
discussed earlier, but spin orbit coupling and the admixture of other orbital states means that f f transitions
are available (Section 7.15).
The favoured red emitter in trichromatic lamps is Eu3þ doped into a Y2O3 matrix, Y2O3:Eu, with the
Eu3þ ions occupying the Y3þ sites. The ground state of the Eu3þ 4f6 ion is 7F0. The broad band at higher
energy is due to a charge transfer transition in which an oxygen 2p electron is transferred to the Eu3þ to
make a 4f7 configuration (Figure 9.12a). This charge transfer band absorbs efficiently at 254 nm and
accordingly readily takes up the ultraviolet radiation given off by the excited mercury atoms. Subsequent
nonradiative decay allow the ion to end up in one of the 5D levels, from which a return to the ground state is
by photon emission. The main transition is between the energy levels 5D0! 7F2, leading to emission at a
wavelength near to 611 nm.
The green emission is from Tb3þ . This ion absorbs the mercury emission poorly and is coupled with a
sensitiser, usually Ce3þ , which is able to absorb the 254nm wavelength mercury radiation efficiently due to
a charge-transfer band between the oxygen 2p orbitals and the Ce3þ 5d orbitals. This absorbed energy is then
transferred to the Tb3þ ions. The green emission is at wavelength close to 540nm, mainly from a
0
10000
20000
30000
40000
1
2
3
4
5
Ene
rgy
cm–1
64f - 5d band
74f - 5d band
2+ 7Eu (4f )
5D0
5D3
5D2 5D1
5D3
5D4
Eu3+ (4f6) Tb3+ (4f8)
eV O 2p - 4f band
7F0
7F07F6
8S7/2
7F6
Figure 9.12 Schematic partial energy-level diagrams for the luminescent ions Eu3þ , Tb3þ and Eu2þ . The Tb3þ
ions do not absorb ultraviolet radiation but gain it by energy transfer from a Ce3þ sensitiser
381 Luminescence
5D4--7F5 transition (Figure 9.12b). Three other peaks of lesser intensity occur: 5D4--
7F6, 489nm; 5D4--7F4,
589nm; 5D4--7F3, 623 nm. Host matrices are La(Ce)PO4, LaMg(Ce)Al11O19 and La(Ce)MgB5O10. In each case
the Tb3þ and Ce3þ ions replace La3þ ions and no charge compensation is needed.
The blue emission is produced by Eu2þ ions, which have a 4f7 electron configuration. This leads to a
particularly simple energy-level diagramwhere theground state is 8S7=2 and theupper energyband corresponds
to the transfer of an electron into the outer 5d orbital to give a configuration 4f6 5d and ensuring that ultraviolet
radiation is absorbed efficiently (Figure 9.12c). The d orbitals interact with the surrounding anions and the
exact position of the band depends upon the host crystal. Thus, the luminescent colour of the Eu2þ centre will
bemodified by changing the site in the host lattice and the type of host structure. The emission spectrum of the
usual tricolour lamp phosphor, BaMgAl10O17:Eu, has a maximum at 450 nm.
An emission spectrum from a trichromatic fluorescent tube is illustrated in Figure 9.13. The emission lines
are narrow compared with those in Figure 9.11 because the f orbitals involved in the process are shielded from
the surrounding matrix. As with the fluorophosphates lamps, the overall emission colour can be modified by
changing the relativeamountsof the threephosphorspresent so as to emphasise the red, greenorblue endsof the
spectrum.
9.4.3 Other fluorescent lamps
Thecolour spectrumsof thefluorescent lampsdescribed above, although satisfactory formanypurposes, donot
give an accurate impression of the colour of an object compared with that perceived when the same object is
viewed indaylight. Toovercome this, deluxe (Colour 90) lamps canbeused. These employmodifiedphosphors
so that the emissions are shifted slightly and a fourth phosphor is added to the blend. This latter phosphor,
Y3Al5O12dopedwithCe3þ , absorbs someof the blue violet light emitted byEu2þ and emits yellow light in its
green
(Ce3+, Tb3+)
400 500 600 700
Wavelength / nm
Em
issi
on (
arbi
trar
y un
its)
blue (Eu2+)
orange-red (Eu3+)
Figure 9.13 The emission spectrum of a trichromatic fluorescent lamp (schematic)
Colour and the Optical Properties of Materials 382
place. The Ce3þ absorption is from the 4f1 ground state into the 5d orbitals (Figure 9.14). These higher energy
orbitals interact strongly with the environment surrounding the ion and are split due to the crystal-field
interaction. The two lowest absorption bands are at 342 and 460 nm, and it is this latter transition that is
important for absorption of the 450 nm blue emission fromEu2þ . Following absorption, luminescence is from
the lower edge of the d band to the ground state, 2F5=2 and the close2F7=2 level (Figure 9.14). These transitions
give an output luminescence with a wavelength maximum close to 565 nm.
Mercury lamps for street lighting use a high pressure ofmercury vapour andproduce an emission that ismore
or less continuous between the limits of 250 and 550 nm (Section 7.7). This output is unbalanced from a visual
viewpoint and it is desirable to introduce an ultraviolet-absorbing phosphor that will emit in the red, so as to
balance the output. A favoured phosphor for this purpose is a mixed strontiummagnesium phosphate using tin
as an activator, (Sr,Mg)3(PO4)2:Sn2þ , which emits at 630 nm. However, this phosphor is not ideal, and many
other materials are currently being explored.
Suntanning beds also make use of phosphors, but in this case the main output is required to be in the
ultraviolet. UVA, wavelength range 320 400 nm, and UVB, wavelength range 280 320 nm, are both used for
this purpose. However, as health concerns over the relationship between ultraviolet irradiation and skin cancer
have surfaced, phosphors have been modified to alter the ratios of these components. Initially, sun bed tubes
used SrMgSi2O7:Pb, in which Pb2þ is the activator. These gave a broad emission centred on 350 nm and
spanningbothUVAandUVB.Unfortunately, thismaterial has a lowstability andwas later replacedbyBaSi2O5
with Pb2þ activator, which gave a narrower band centred at 350 nm, limited toUVA. Today, sun beds often use
tubes containing a mixture of BaSi2O5:Pb and SrAl12O19 containing Ce3þ activator, which has an emission
peak centred at approximately 310 nm, thus providing some UVB output.
9.5 Plasma Displays
Plasma displays are, in essence, gas-discharge lamps and theworking principle of these displaysmirrors that of
fluorescent lamps. Monochromatic plasma displays were first used in some portable computers in about 1988
(seeFigure 7.8). These used an ionised gas to produce anorange red colour in a similar fashion to that exploited
0
10000
20000
30000
40000
1
2
3
4
5
Ene
rgy
/ eV
Ene
rgy
/ cm
–1
2F7/22F5/2
Ce3+ (4f1)
5d band
Figure 9.14 Schematic partial energy-level diagram for Ce3þ . The transitions shown correspond to theabsorption of blue light of wavelength approximately 466 nm and emission at approximately 537 nm. A chargetransfer band from O 2p to Ce3þ 5f that absorbs strongly at 256 nm is not shown
383 Luminescence
by neon signs (Section 7.5). The gas was confined in a series of wells, and two grids of transparent electrodes,
one running horizontally and one vertically, provided the necessary current and voltage to ionise the gas.
These monochromatic displays rapidly gave way to full-colour displays, which are now commonplace
(Figure 9.15). Currently (2010), full-colour plasma display screens dominate the large-screen market. They
have an advantage over other display technologies in that the light emitted does not vary greatly with viewing
angle. In addition, plasmadisplay television sets arewidely available and at themoment appear to have the edge
in providing high-definition television over competing technologies.
A display panel consists of a pair of glass plates containing a series of cells each of which acts as a small
fluorescent lamp. To form the lamp array, the region between the glass sheets is divided up into sub-pixels by
a series of ribs or separators controlled by two sets of electrodes arranged at right angles to each other
(Figure 9.16a). Each lamp is several hundred micrometres in size, and there are several million such lamps in
a display. Each pixel consists of three lamps, giving off red, blue and green light, making the luminance
and resolution uniform across the display that does not vary with viewing angle.
Theworking gas in the cells is amixture of helium and xenon.When a high voltage is applied across the two
electrodes above and below a well the gas is excited into a state resembling a plasma; that is, an electrically
neutralmixture of electrons, positive and negative ions. The excited gases emit ultraviolet radiation.Aswith all
the inert gases, the energy-level diagrams are complex and numerous wavelengths are emitted. The principal
wavelengths, though, are at 147 nm from excited Xe and at 172 nm from an Xe excimer.3 Each well is coated
internally with a red, green or blue phosphor (Figure 9.16b). The layer of magnesium oxide (MgO) serves as
a dielectric to enhance the electric field present in the well. The ultraviolet light excites the phosphors to emit
in the red, green or blue, in the sameway that the ultraviolet light from amercury vapour excites the phosphors
in a fluorescent tube.
Figure 9.15 Plasma screen, circa 2009: (a) side view showing the slim profile compared with CRT displays;(b) detail of display image showing colour rendition
3 An excimer is an electronically excited pair of atoms that are not bonded in the ground state.
Colour and the Optical Properties of Materials 384
Themain phosphors used at present are yttriumgadoliniumborate dopedwith europium ((Y,Gd)BO3:Eu3þ )
for red emission, bariummagnesium aluminate dopedwith europium (BaMgAl14O23:Eu2þ ) for blue emission
and zinc silicate dopedwithmanganese (ZnSiO4:Mn2þ ) forgreen light.Noneof these is ideal in everyway.Forexample, the red emitter is rather too orange in hue, the blue emitter degrades rather too rapidly under the
intense ultraviolet irradiation and the green emitter has a long decay time, which can lead to image blurring.
Researchon improving these phosphors is intense. In addition, the use of phosphors that utilise quantumcutting
(Section 9.10), so that more than one visible photon is emitted per ultraviolet photon absorbed, is highly
attractive from the point of view of improving luminosity whilst lowering damage and the other drawbacks
mentioned.
9.6 Cathodoluminescence and Cathode Ray Tubes
9.6.1 Cathode rays
Cathodoluminescence is light emission due to irradiation with electrons. This effect was discovered during
early researches on the effect of electric fields on gases at low pressures (Section 7.5). A pair of electrodeswere
sealed into an evacuated glass tube and subjected to a high voltage. Electrons are expelled from the cathode and
are subsequently accelerated towards the anode by the applied voltage. A hole in the anode allowed these
‘cathode rays’ to pass through and hit the glass (or later a phosphor coating), which then gave out light. The
glass
insulating layer
separator
phosphor(r, g, b)
electrode
electrode
glass
glass
glass
phosphor
magnesium oxide
neon +xenon
(a)
(b)
Figure 9.16 (a) The layer structure of a full-colour plasma display. Each well in the structure contains a red,green or blue phosphor. (b) Detail of one subpixel. A mixture of neon and xenon emits ultraviolet light whichinteracts with the phosphor to give out either red, green or blue light
385 Luminescence
process was called cathodoluminescence, and the evacuated tube assembly as a whole was, and still is, called
a cathode ray tube, often abbreviated to CRT (Figure 9.17).
The first CRT device, invented by Braun in 1897, was the oscilloscope. These instruments display voltage
variation with time. The voltage to be displayed is applied to a set of deflection plates or magnetic coils within
the tube. These displace the electron beamby an amount proportional to thevoltage.Abeam scanned across the
screen of the CRTwill then not follow a straight line, but display a wave-form, mimicking the signal voltage.
Radar (an acronym for radio detection and ranging)was a development ofCRT technology that took place in
the 1930s and 1940s. In a radar detector, the screen is circular and the linear display of an oscilloscope is now
rotated in the plane of the screen every few seconds or so. An antenna sends out a pulsed radio-frequency signal
which is reflected by various objects and hence returned to the antenna, which also acts as a receiver, or to a
specialized receiving antenna. A disturbance on the rotating signal line, bright spots or breaks, indicates the
reception of a radar signal. The position on the line gives the distance to the disturbance and the orientation
of the line at that moment gives the direction. Initially used for military applications, such as the detection of
enemyaircraft or shipping, problemsarosebecause the signal can alsobe reflected by storms, rain, snowand ice
crystals in the atmosphere.These early ‘problems’ havebeen exploitedandare now thebasis ofweather radar in
common use for meteorological purposes.
9.6.2 Television tubes
Themassmarket forCRTswas the development of television at the end of the 1940s and on. Towards the end of
the century, the market expanded when television-like CRTs became used as computer terminal displays.
The electrons are produced by heating a metal filament, which forms the cathode of the device, in an
evacuated glass envelope. They are accelerated towards a perforated cylindrical anode by the application of a
high voltage. The anode and cathode assembly is often referred to as an electron gun. The far end of the tube
assembly is flattened and phosphor coated to form the screen upon which the light-emitting image is formed.
The electrons emerge from the anode as a narrow collimated beam which is scanned horizontally and
vertically across and down the screen in a predetermined raster pattern by electrostatic or electromagnetic
means. The beam ismade to cover the screen in a fraction of a second.As the focused beam sweeps past a dot of
phosphor, light is emitted. More than a million phosphor dots of approximately 300mm diameter for each of
three primary colours are deposited on the curved screen in routine manufacturing operations. The screen,
therefore, is lit up by small spots of phosphor whose emission is refreshed at each pass of the electron beam.
Light emission is controlled by variation of the electron flux as the beam scans across the screen.
cathode -
anode +deflection
plates
cathode rays
phosphor coatedscreen
evacuated glass tube
Figure 9.17 CRT (schematic). The cathode is a pointed hairpin filament and the anode consists of a cylinder,allowing the ‘cathode rays’ to pass through. The anode and cathode are usually contained in an assembly called anelectron gun. There can be several sets of magnetic or electrostatic deflection plates after the gun, allowing thecathode ray spot to be displaced so as to display information
Colour and the Optical Properties of Materials 386
In both monochrome and colour televisions the light emitted is perceived by the eye in the same way that
impressionist pointillist paintings are, by way of additive coloration. For black-and-white displays, two
phosphors are used: a blue emitter and a yellow emitter. Colour television utilises three primary colours,
arranged in an array (Figure 9.18). Just as in pointillist paintings, it is important that the phosphor spots do not
overlap in colour displays, otherwise the picture quality is degraded.
Apart from this, there are a number of important parameters that have to be closely controlled to give a good
picture. The efficiency of a fluorescent material depends upon many factors, including whether an activator
alone is needed or whether a sensitiser and an activator are involved. The concentrations of the activator
and sensitiser play an important part in controlling efficiency, and optimal concentrations have to be found
experimentally. Impurities generally have a negative effect, and so high purity is a necessity.
In addition, thedecay timeof the phosphormust be suitable. The exitance (invariably termed the intensity) of
the light given out by a phosphor after excitation is removed is frequently given by an equation of the type:
It ¼ I0 expð�t=tÞ
where It is the exitance after time t has elapsed, I0 is the initial exitance the moment excitation ceases and t isthe decay time (Section 9.3). The decay time is the time taken for the luminescent radiation to decay to 1/e of
its initial value.Clearly, for a continuouspicture to beobserved, the decay timemust be longer than the time that
it takes the electron spot to complete its round trip and refresh the emission again. However, this must not be so
long that shadow images persist after the action has moved on. In this respect, afterglow can be troublesome.
Afterglow is said to occur when the luminescence decays at the rate expected until a certain value is reached
and then decays much more slowly. This is frequently due to the presence of impurities in the material which
contribute differing mechanisms to the emission process.
In summary, a phosphor must have (a) a high efficiency, (b) a suitable decay time, (c) a suitable emission
spectrumand (d) a lowafterglow level.Thesewere achieved somedecades ago and the technologyof television
tube manufacture is mature from this point of view.
Black-and-white TVuses silver-activated zinc sulfide (ZnS:Agþ ) to give a blue colour plus silver-activatedzinc cadmium sulfide ((Zn,Cd)S:Agþ ) to give a yellow. The ratio of these two phosphors controls the overallhue of the screen.
The blue colour produced by ZnS:Agþ is due to an electron transition between defects in the ZnS structure
that are deliberately introduced by silver doping. The nominal reaction to make the phosphor is between
silver sulfide (Ag2S) and ZnS, to produce silver (Agþ ) substituted for zinc on Zn2þ sites4 (AgZn0) and
Figure 9.18 The arrangement of red (r), green (g) and blue (b) phosphor dots on a colour cathode-ray televisiontube.
4 Defect notation follows that of Kroger and Vink (see this chapter’s Further Reading).
387 Luminescence
sulfur vacancies (V 2.
S ):
Ag2S ðZnSÞ! 2AgZn0 þ SSþV 2.
S
The transition giving rise to light emission is due to the transfer of an electron trapped on a sulfur vacancy V 2.
S
to a silver ion AgZn0 (Figure 9.19a). (In semiconductor terminology (see Chapter 10), the sulfur vacancy forms
a donor level just below the valence band and the silver ion forms an acceptor level just above the conduction
band in the ZnS band gap, and the transition is from a donor level to an acceptor level.) The colour-producing
electrons arise in the following way. To a first approximation, the valence band is fully occupied by electrons
and the conduction band is completely empty and the defects introduce energy levels into the energy gap
between the valence and conduction bands. Irradiation by cathode rays promotes electrons from the valence
band into the conduction band. This is an extremely energetic process. Whereas one ultraviolet photon will
promote one electron froma lower energy level into the conduction band, a cathode ray electronmight promote
3000. A proportion of these, about one-third, end at V 2.
S defects, from whence they return to AgZn0 defects
valence band
conduction band
2•VS2•VS2•VS
AgZn′ AgZn′AgZn′
blue
blue blue green
blueyellow
(a)
(d) (e)
(c)(b)
(f)
CuZn′AgZn′AgZn′
Cls•
Cls•AlZn
•
Figure 9.19 Schematic diagram of the defect energy levels in ZnS and (Zn,Cd)S phosphors used in televisionCRTs: (a) B&W, blue; (b) B&W yellow; (c) colour, blue; (d) colour, blue; (e) colour, blue; (f) colour, green
Colour and the Optical Properties of Materials 388
and emit photons. The separation of the defect energy levels is approximately 3 eV, giving an emission at
approximately 410 nm.
The yellow emission is obtained by dopingZnSwith CdS and employing the same silver activator. AsCdS is
added to the ZnS, the energy-level separation of the V 2.
S and AgZn0 defects decreases, and as the composition
becomes richer in CdS the emission moves towards the red end of the spectrum, so that a continuum of
colours can form between ZnS:Agþ (blue), Zn0.68Cd0.32S:Agþ (green), Zn0.5Cd0.5S:Ag
þ (yellow) and
Zn0.13Cd0.87S:Agþ (red). Awidely used composition for yellow emission is Zn0.5Cd0.5S:Ag
þ (Figure 9.19b).
Colour TVuses similar phosphors for the blue and green emission, relying upon the same donor acceptor
recombination, as above. For blue, silver-activated zinc sulfide (ZnS:Agþ ) or zinc sulfide doped with AgCl
to form ZnS:Agþ ,Cl is used (Figure 9.19c and d). In this latter material, the important defects for colour
emission are AgZn0 as before and chlorine ions (Cl ) substituted for sulfur (S2 ) to form ClS
.centres:
AgCl ðZnSÞ!AgZn0 þClS
.
The colour of the emission can be tuned somewhat by altering the defect energy, and for this the ZnS is
sometimes dopedwithAlCl3, to introduce additional AlZn.defects (Figure 9.19e). The transition fromAlZn
.to
AgZn0 is at a slightly different wavelength and changes the hue of the display.
Thegreenphosphor is zinc sulfidedopedwithCuCl to introduceCuþ ions ontoZn sites to formCuZn0 defects
and acceptor levels and ClS.defects as before:
CuCl ðZnSÞ!CuZn0 þClS
.
The CuZn0 defects give rise to higher energy levels than those introduced by AgZn
0, resulting in an emission
centred at 530 nm, comparedwith approximately 410 nm for the blue emitter (Figure 9.19f). The colour can be
tuned by incorporation of CdS, or other defects as described above.
Early red phosphors for colour television also used ZnS-based materials, especially silver-doped zinc
cadmium sulfide ((Zn,Cs)S:Agþ ) described above. This colour was not satisfactory, and emission from Eu3þ ,similar to that used in fluorescent lamps (Section 9.4) was an early replacement. The first host matrix used was
YVO4. In thismaterial, cathode ray energywas absorbedby theVO43 group and transferred to theEu3þ ions in
a charge-transfer process. This has been replaced by Y2O2S:Eu3þ , which gives a brighter emission. Light
emission is due to transitions between excited 5D levels and lower energy 7F levels that lie in the band gap of
the host. There are five of the former of importance, 5D0;5D1;
5D2;5D3;
5D4, and seven of the latter,7F0;
7F1;7F2;
7F3;7F4;
7F5;7F6. To obtain the desired red colour output it is necessary to ensure that the
dominant transition is 5D0! 7F2. (Although the terms suggest that this transition should give only one emission
line, a small crystal-field splitting (Section 7.9) gives rise to a close doublet.) The result is a pair of lines at
approximately at 612 and 628 nm (Figure 9.20). The limitation imposed upon the emission is achieved by a
careful selection of the host material (Y2O2S is preferred for this reason) and by adjusting the concentration of
Eu3þ ions, which quench emissions from the 5D1 and higher D levels by cross-relaxation (Section 9.9).
In working TV tubes the phosphor layers are backed by an aluminium film to increase the brightness of the
image.
9.6.3 Other applications of cathodoluminescence
There are a number of other devices which use CRT technology widely. The most familiar are computer
monitors, which are essentially television monitors, flying spot scanners, oscilloscopes and radar screens.
These need different decay characteristics than TV screens, and the phosphor technology in each is tailored
to the exact requirements of the product.
389 Luminescence
Cathodoluminescence has long been used in electron microscopes. In effect, a transmission electron
microscope is rather like a long CRT. Electrons are emitted from an electron gun and traverse the specimen
before hitting a fluorescent screen coated with a cathodoluminescent phosphor which displays the image.
The brightness of the image is a simple measure of the intensity of the electron beam. To maintain a good
performance, the phosphor coating has to be replacedperiodically, as the intense irradiation causes degradation
of performance.
Cathodoluminescence is also used as an analytical tool, particularly in scanning electron microscopes. In
these latter instruments, an electron beam is scanned over the sample in a raster fashion. The sample can emit
light by direct cathodoluminescence, or else cathodoluminescence can be generated from secondary electrons
emitted under the primary beam. In both cases the cathodoluminescent spectra can be recorded and used for
analytical purposes. In this way, a wide variety of opaque objects, ranging from archaeological or fine art
artefacts to semiconductor devices can be analysed using a nondestructive method.
9.7 Field-Emission Displays
Field-emission displays (FEDs; also called field-effect displays) use electron-excited phosphors as the light-
emitting mode. They are similar to plasma displays, in that the display screen is composed of many tiny cells,
but colour production in each cell is comparable to that in a CRT. In an FED, each cell contains a microscopic
cathode. A voltage is applied to a cathode, which then emits electrons the process of field emission. The
electrons are accelerated by the voltage differential and strike the phosphor as in a conventional CRT and,
hence, produce light emission.
The difficulty lies in the ejection of electrons from the cathode. The energy needed to force electrons out of
a metal is called the work function. Electron emission in a normal CRT electron gun is usually accomplished
by heating: the thermionic effect. This is not possible in FED. An alternative is to pull electrons out of the
cathode using an electrostatic field.Under ordinary conditions, an extremely high electrostatic field is required.
The problem is overcome in FEDs bymaking the cathode in the form of a sharp spherical tip. The static electric
0
10000
20000
30000
40000
1
2
3
4
5
Ene
rgy
cm–1
5D3
7F6
7F0
5D0
5D1
5D2
Eu3+ (4f6)
eV O 2p - 4f band
Figure 9.20 Schematic energy-level diagram of Eu3þ used as the red emitter in colour television CRTs. Theimportant transition is 5D0! 7F2 with emission at approximately 612 and 628 nm
Colour and the Optical Properties of Materials 390
field F generated by an applied voltage V to a tip of radius r is:
F ¼ V
r
Thus, the field is considerable if the tip has a radius of a fewnanometres, evenunder the imposition of only a few
volts. To capitalize on this fact, the cathode in each cell is made of a microscopically pointed spike, using a
refractory metal such as molybdenum, carbon nanotubes or low work-function materials such as synthetic
diamonds. Field emission then occurs even with a relatively low external voltage.
The light emission is similar to that described in a CRT. Electrons, leaving the cathode, are accelerated by
the electric field and strike a phosphor coated on a glass substratewith energy of about 100 eV. This principle is
now actively explored for flat-panel displays, but none are yet in commercial production.
9.8 Phosphor Electroluminescent Displays
Electroluminescent displays containing a thin film of a phosphor, called thin-film electroluminescent (TFEL)
displays, are, like FEDs, also akin to CRTs, in that the colour is generated by stimulation of a phosphor by
energetic electrons. These devices find use as display panels, backlighting in products such as instrument
panels, in flat-panel colour displays and in some aspects of lighting.
The principle of operation involves the excitation of a chosen phosphor by high-energy electrons created
within the phosphor film itself. Electrons enter the phosphor at the junction with a surface insulating coating
and are accelerated under the influence of a high electric field. These electrons collide with the luminescent
centres in the phosphor, transferring energy in the process. The excited luminescent centres then fall back to the
ground state and release energy by light emission (Figure 9.21a).
Themost promising devices useAC supplies in a TFEL (ACTFEL) display. The device is built up in thin layers
on a glass substrate (Figure 9.21b). A layer of a transparent electrical conductor, most often indium tin oxide, of
about 400 nm thickness, is laid down first as one electrode. This is covered with about 400 nm of a transparent
insulator. The active layer, about 700 nm of phosphor, and then another 400 nm layer of transparent insulator are
then added. Finally, a 200nm thick layer of aluminium is deposited on the stack. This serves as an electrode and
also reflector. The display is viewed through the glass substrate, which acts as a protective surface.
The accelerating field in these devices is of the order of 1 2MVcm 1. This is generated by applying a lower
voltage across a thin insulating layer, which acts as a capacitor. The whole arrangement is, in fact, a series of
capacitors. This design is chosenbecause the highvoltageswhich are needed in the phosphor layer are generated
from low electrodevoltages byway of the capacitance of the thin insulating layers. To optimise the high fields in
the phosphor, the dielectric needs to have a high relative permittivity (dielectric constant) and high breakdown
strength, as well as being transparent to light. The ferroelectric barium titanate (BaTiO3) is often chosen.
The most efficient electroluminescent thin-film phosphors consist of zinc sulfide containing manganese
(ZnS:Mn2þ ) as the luminescent centre. The Zn2þ ions in the host are in tetrahedral coordination in both the
cubic zinc blende form and the hexagonal wurtzite form.Mn2þ ions readily replace Zn2þ in the crystals. The
ground state of the free Mn2þ (d5) ion is 6S, which transforms to a single 6A1 energy level in the tetrahedral
crystal field of the surrounding 4S2 ions in ZnS. The first free-ion higher energy term is 4G, which splits into
four in the tetrahedral crystal field, 4T1,4T2,
4A1 and4E (Figure 9.22; also see Section 7.9). The lowest of these
is 4T1 and the electroluminescent emission is due to the transition from this level to the ground state. The strong
influence of the surrounding structure means that the spectrum, centred upon yellow, wavelength 585 nm, is
broad.Optimal doping levels are close to 1%Mn2þ , as higher concentrations lead to concentration quenchingof the emission.
391 Luminescence
A number of other colours can be generated using doped ZnS containing Cuþ , Al3þ and Cl as described
above. ZnS:Cuþ ,Cl gives out blue light with a wavelength near to 460 nm or green light with a wavelength
near to 510 nm, depending upon the concentration of Cl present. The defects formed are identical to those
described earlier. The presence of Al3þ donors is used to adjust the exact emission colour.
Colours can also be generated by the incorporation of lanthanoids into the host phosphor. For example, red
emission is produced by calcium sulfide dopedwith europium (CaS:Eu2þ ). The Eu2þ ions have a 4f7 electron
configurationwith a ground state 8S7=2, and an upper energy band corresponds to the transfer of an electron into
4T2
4T1
6S 6A1
4A1
4E14G
free ion ion in ZnS
ground state
~585 nm
Figure 9.22 The energy levels of Mn2þ free ions and in the tetrahedral crystal field of ZnS (schematic)
insulator insulatorphosphor
–eA
photon
(a)
glass substrate
indium tin oxideinsulator
phosphor
insulatoraluminium
light emission(b)
Figure 9.21 TFEL displays. (a) Schematic representation of the process taking place in an electroluminescentmaterial. Electrons enter the phosphor from an insulator–phosphor interface and accelerate under a high voltage.These transfer energy to luminescent centres A via collisions and these, in turn, lose energy by emitting light.(b) Idealised electroluminescent thin-film unit
Colour and the Optical Properties of Materials 392
the outer 5d orbital to give a configuration 4f6 5d. The colour-generating transition is from this upper band back
to the ground state (Figure 9.23a). At first sight this is surprising, as the usual blue tricolour lamp phosphor,
BaMgAl10O17:Eu2þ , has amaximumat 450 nm (Section 9.4). However, the exact position of the upper energy
band depends upon the interaction of the d orbitals with the surrounding crystal, and in ZnS the softer bonding
gives a broad emission centred close to 640 nm.
A strong green line emission is produced by zinc sulfide doped with terbium (ZnS:Tb2þ ). The green
emission is at wavelength close to 545 nm,mainly from a 5D4--7F5 transition (Figure 9.23b). Three other peaks
of lesser intensity occur: 5D4--7F6, 489 nm; 5D4--
7F4, 589 nm; 5D4--7F3, 623 nm. Note that these emission lines
are very similar to those given out from oxide host crystals, because the 4f energy levels are well shielded and
the f f transitions are thus insensitive to the surroundings.
Blue emission still poses a problem for these displays, but onematerial that has been used is zinc sulfide doped
with both thulium and fluorine (ZnS:Tm3þ ,F ). In this latter case the thulium ions (Tm3þ ) occupy positions
normally filled by zinc (Zn2þ ) ions. The fluoride (F ) ions are needed to compensate for the excess charge on the
thulium ions, (Tm3þ þ F ) being equivalent to (Zn2þ ). Theemission, at 450 nm, is sharp, due to an f f transition1G4! 3H6 (ground state). However, the emission is also weak and much energy is given out in the far red due to1G4! 3F4 and in the infrared due to 1G4! 3H5 transitions (Figure 9.23c). Superior blue emission has been
obtained from the thiogallates CaSr2S4, SrGa2S4 and BaGa2S4 doped with the 4f1 ion Ce3þ . The transitions
64f - 5d band
5d band
74f - 5d band
8S7/27F6
2F5/2
2F7/2
7F0
5D3
5D4
Tb3+ (4f8)Eu2+ (4f7)
Ce3+ (4f1)
Tm3+ (4f12)
~640 nm
(a) (b) (c)
489 nm589 nm
623 nm
450 nm
650 nm
790 nm
3H6
3F4
3F3
3F2
3H5
3H4
1G4
1D2
~459nm~445 nm
(d)
Figure 9.23 The energy levels of (a) Eu2þ , (b) Tb3þ , (c) Tm3þ and (d) Ce3þ in TFEL host structures(schematic)
393 Luminescence
between the excited 5d state and the 4f1 ground-state doublet 2F5=2 and2F7=2 are both in the blue region of the
spectrum (Figure 9.23d). Because of the involvement of themore exposed 5d electron energy levels, the emission
is quite broad, centred at 459 nm for the Ca compound and 445nm for the Sr and Ba phases.
White-light emission can also be achievedusing combinations of luminescent centres.Oneway inwhich this
has been achieved is to use stacked layers of ZnS:Mn2þ and SrS:Ce3þ . These broad-emission band phosphors
combine to produce a light with peaks in the green and yellow that appear white to the eye (Figure 9.24).
Full-colour displays can be constructed in a number of ways. The simplest, conceptually, is to use a white-
emitting phosphor such as that just described and to incorporate a coloured filter into the device (Figure 9.25a).
Analternative is to build up subarrays of pixelswith red, blue andgreen phosphor subpixel units (Figure 9.25b).
By varying the voltage distribution between the aluminium and indium tin oxide electrodes, any of the red,
blue or green phosphors can be excited to luminesce. Other device geometries have been used, including stacks
of single emitting devices.
9.9 Up-Conversion
The detection of infrared radiation and subsequent conversion to visible has many possible applications,
encompassing the generation ofwhite light fromLEDs (Chapter 10) and the study of nocturnalmammals. One
method of achieving this objective is variously known as frequency up-conversion, anti-Stokes fluorescence or
cooperative luminescence. In this effect, low-energy radiation, typically in the infrared, is ‘up-converted’ into
visible radiation. Up-conversion is thus the opposite of photoluminescence, in which high-energy ultraviolet
radiation is ‘down-converted’ into visible light. Note that up-conversion is distinct from frequency doubling,
which uses nonlinear polarisation rather than luminescence (Section 4.9).
The up-conversion efficiency of a system can be defined as the ratio:
Efficiency ¼ power emitted ðvisibleÞpower absorbed ðinfraredÞ
In general, the up-conversion efficiency is low and varies with the concentration of the activator and sensitiser
ions. In general, low concentrations of the active ion, of the order of 1%, are used. At these concentrations, the
20
40
60
80
100
500400 600 700Wavelength / nm
Rel
ativ
e in
tens
ity
Figure 9.24 The emission spectrum of the stacked TFEL phosphors ZnS:Mn2þ and SrS:Ce3þ , which give whiteoutput to the eye
Colour and the Optical Properties of Materials 394
ions form point defects well isolated from each other. At higher concentrations, dopant ions tend to cluster and
other energy loss mechanisms interfere with up-conversion.
The exitance of the up-converted output Iup is related to the irradiance of the exciting radiation Iex by the
formula:Iup / Inex
where n is the number of photons absorbed per up-converted photon emitted. A graph of log Iup versus log Iexwill give information on the mechanism of the process, as detailed below. (The data given in the literature
often uses ‘intensity’ in arbitrary units for the quantities Iup and Iex, or ‘intensity’ in arbitrary units for Iup and
‘pump power’ for Iex.)
The majority of studies of up-conversion have dealt with the behaviour of lanthanoid ions, especially Er3þ ,Tm3þ , Ho3þ and Yb3þ . A number of host structures have been used for these ions, including binary oxides
Y2O3, Gd2O3 and ZrO2, perovskite structure BaTiO3, SrTiO3 and PbTiO3, fluorides such as NaYF4 and oxide
and oxyfluoride glasses.
9.9.1 Ground-state absorption and excited-state absorption
The energy for up-conversion can be gained by the active ion in several ways, and there are often many
competing energy transfer processes taking place during up-conversion (Table 9.2). In principle, the simplest
(a)
aluminiumelectrodes
insulator
white phosphor
insulator
indium tin oxideelectrodes
glass substrate
light emission
Figure 9.25 ACTFEL device structures (schematic): (a) colour display element using a white-emitting phosphorand colour filters; (b) colour display element using phosphor subpixels emitting red, blue and green
395 Luminescence
is for the activator ion to pick up photons in two distinct steps. The first photon excites the ion from the ground
state to an excited energy level, a process referred to as ground-state absorption (GSA). A subsequent photon
is then absorbed to further promote the excited ion to a higher energy level again excited-state absorption
(ESA).
The oxide CeO2 doped with approximately 1 % Er3þ exhibits up-conversion in this way. The Er3þ ions
substitute for Ce4þ to form a low concentration of Er3þ ions randomly distributed within the oxide matrix.
Irradiation with near-infrared photons with a wavelength close to 785 nm, the pump wavelength, excites the
Er3þ ions from the 4I15=2 ground state to the 4I9=2 level, that is; a GSA mechanism:
4I15=2þ hn ð785 nmÞ! 4I9=2
These ions lose energy nonradiatively to phonons (lattice vibrations) to reach the 4I11=2 and4I13=2 energy levels
(Figure 9.26a):
4I9=2 ðErÞ! 4I11=2 ðErÞþ 4I13=2 ðErÞþ phonons
Ions in both these levels are further excited by an ESA mechanism via the same pump wavelength to gain
energy from the irradiating 785 nm radiation. For example, those in the 4I11=2 energy level are excited to the4F3=2; 5=2 doublet:
4I11=2þ hn ð785 nmÞ! 4F3=2;5=2
aluminiumelectrodes
insulator
red, blue, greenphosphors
insulator
indium tin oxideelectrodes
glass substrate
light emission
(b)
Figure 9.25 (Continued )
Colour and the Optical Properties of Materials 396
These states subsequently relax via internal energy loss to the 2H11=2;4S3=2 and
4F9=2 energy levels
(Figure 9.26b):
4F3=2;5=2! 2H11=2þ 4S3=2þ 4F9=2þ phonons
The ions in the 4I13=2 energy level follow a similar path, being excited to the 2H11=2 energy level:
4I13=2þ hn ð785 nmÞ! 2H11=2
then subsequently relax to the 4S3=2 and4F9=2 energy levels (Figure 9.26c):
2H11=2! 4S3=2þ 4F9=2þ phonons
Ene
rgy
/ 100
0 cm
–1
4I15/2
4I15/2
4I13/2
4I13/2
4I11/2
4I11/2
4I9/2
4I9/2
4F9/2
4F9/2
4S3/2
4S3/2
4F7/2
4F7/2
4F3/2;5/2
4F3/2;5/2
2H9/2
2H9/2
2H11/2
2H11/2
785 nm
785 nm
(a)
(b)
Er3+ GSA + relaxation
ESA + relaxation
25
5
10
15
20
0
hν ESA
hν GSA
Figure 9.26 Up-conversion in CeO2 doped with Er3þ (CeO2:Er): (a) GSA plus decay; (b) ESA plus decay;(c) ESA plus decay; (d) emission; (e) emission spectrum following up-conversion
397 Luminescence
The result of this is that the energy levels 2H11=2;4S3=2 and
4F9=2 arepopulated tovaryingdegrees, depending
upon the precise details of the excitation and relaxation steps. Subsequent loss of energy from these levels gives
rise to green and red emission (Figure 9.26d):
2 H11=2! 4I15=2þ hn ð526 nm; greenÞ4S3=2! 4I15=2þ hn ð547 nm; greenÞ4F9=2! 4I15=2þ hn ð658 nm; redÞ
The up-conversion spectrum consists of three major peaks (Figure 9.26e).
All up-conversion spectra from Er3þ (including those using different mechanisms, below), are similar, but
the relative intensities and positions of the three peaks vary with concentration of activator ions, sensitiser ions
and the nature of the host matrix. Some of these aspects are outlined in the following sections.
4I15/24I15/2
4I13/24I13/2
4I11/24I11/2
4I9/24I9/2
4F9/24F9/2
4S3/24S3/2
4F7/24F7/2
2H9/22H9/2
2H11/22H11/2
526
nm (
gree
n)
547
nm (
gree
n)
658
nm (
red)
785 nm
(c) (d)
ESA+relaxation emission
500 600 700
Wavelength / nm
Inte
nsity
(ar
bitr
ary
units
)
24
H →
I11
/215
/2
44
S →
I3/
215
/2
44
F →
I9/
215
/2
(e)
4F3/2;5/2
hν ESA
Figure 9.26 (Continued )
Colour and the Optical Properties of Materials 398
9.9.2 Energy transfer
When the concentration of the dopant Er3þ rises, other processes become important. In this section the two
simplest of these are described. Pump energy can be picked up by a sensitiser and transferred directly to the
emitter; a process called energy transfer (ET). (This is alsowhat happens in a phosphor containing a sensitiser.)
Energy transfer canbe illustrated byup-conversion in a host containing the co-dopantsYb3þ /Er3þ ,which givea strong green emission. The ion that absorbs the incoming infrared radiation is the Yb3þ ion, which then
transfers energy to the Er3þ active ion. Generally, the concentration of the absorbing Yb3þ centres is about
20 %, while the concentration of the activator Er3þ ions is about 1 %.
The energy levels of the infrared radiation suited to the energy transfer process match the Yb3þ ion energy
transition from the ground state 2F7=2 level to the2F5=2 level and lasers with an output of 975 nm are usually
employed. This pump energy also matches the 4I15=2 to4I11=2 GSA transition of Er3þ centres, but energy
transfer from the Yb3þ centres dominates the process:
GSAðYbÞ 2 F7=2 ðYbÞþ hn ð975 nmÞ! 2F5=2 ðYbÞET 2 F5=2 ðYbÞþ 4I15=2 ðErÞ! 2F7=2 ðYbÞþ 4I11=2ðErÞ
This is followed by nonradiative relaxation of some ions to the 4I13=2 level (Figure 9.27a):
4I11=2 ðErÞ! 4I13=2 ðErÞþ phonons
The second stage in the excitation process can use any of three mechanisms. Two are similar to those just
described; that is, further gain of energy from Yb3þ (ET) or absorption of a photon (ESA):
ET 2 F5=2 ðYbÞþ 4I11=2 ðErÞ! 2F7=2 ðYbÞþ 4F7=2 ðErÞESA 4 I11=2 ðErÞþ hn ð975 nmÞ! 4F7=2 ðErÞ
Some ions relax to the 2H11=2 (Er) and4S3=2 (Er) levels (Figure 9.27b and c):
4F7=2 ðErÞ! 2H11=2 ðErÞþ 4S3=2 ðErÞþ 4F9=2 ðErÞþ phonons
The third mechanism that operates involves energy transfer between two excited Er3þ ions in a process
called cross-relaxation (CR), which results in further excitation of one ion and loss of energy of the other
(Figure 9.27d):
CR 4 I11=2 ðErÞþ 4I11=2 ðErÞ! 4F7=2 ðErÞþ 4I15=2 ðErÞ
The populated 4F7=2 (Er) level is able to lose energy nonradiatively as above. The end result is that the2H11=2
(Er), 4S3=2 (Er) and4F9=2 (Er) energy levels are populated.
The 4F9=2 (Er) level can also be populated by transitions from the 4I13=2 (Er) level (which itselfwas populated
by nonradiative relaxation from the 4I11=2 (Er) level) in these three ways (Figure 9.27e):
ESA 4 I13=2 ðErÞþ hn ð975 nmÞ! 4F9=2 ðErÞET 2 F5=2 ðYbÞþ 4I13=2 ðErÞ! 2F7=2 ðYbÞþ 4F9=2 ðErÞCR 4 I13=2 ðErÞþ 4I11=2 ðErÞ! 4F9=2 ðErÞþ 4I15=2 ðErÞ
399 Luminescence
The end result is that the energy levels 2H11=2;4S3=2 and
4F9=2 are populated to varying degrees, giving rise to
the same photon emissions as detailed above (Figure 9.26e):
2 H11=2 ðErÞ! 4I15=2 ðErÞþ hn ð525 nm; greenÞ4S3=2 ðErÞ! 4I15=2 ðErÞþ hn ð550 nm; greenÞ4F9=2 ðErÞ! 4I15=2 ðErÞþ hn ð655 nm; redÞ
A maximum efficiency is observed with concentrations of about 1 3 % of the active centre.
Note that these are only an outline of the many processes that can occur. It is easy to imagine back transfer
from Er3þ to Yb3þ and Yb3þ Yb3þ energy transfer, both of which will lower the efficiency of the process.
In addition, increasing interactions between both lanthanoid ions can lead to cluster formation. In effect, this
changes the site symmetry and surrounding matrix experienced by the ions, again limiting the efficiency.
4I15/2
4I15/2
4I13/2
4I13/2
4I 11/2
4I11/2
4I 9/2
4I9/2
4F9/2
4F9/2
2F7/2
2F7/2
2F5/2
2F5/2
4S3/2
4S3/2
4F7/2
4F7/2
4F3/2;5/2
4F3/2;5/2
2H9/2
2H9/2
2H11/2
2H11/2
975 nm
975 nm
(a)
(b)
3+Yb
3+Yb
3+Er
3+Er
ET
ET
hν GSA
Ene
rgy
/ 100
0 cm
–1) 25
5
10
15
20
0
Figure 9.27 Up-conversion in a matrix containing the co-dopants Yb3þ /Er3þ : (a), (b) ET from Yb3þ to Er3þ ;(c) ESA in Er3þ ; (d) CR in Er3þ ; (e) ET, CR and ESA
Colour and the Optical Properties of Materials 400
9.9.3 Other up-conversion processes
Other up-conversion processes are known. The blue emission from a Yb3þ /Tm3þ couple in which the active
emitters are Tm3þ centres is mainly due to the efficient triple excitation ET process from Yb3þ centres
(Figure 9.28), althoughCR and other complexities cannot be ignored in a detailed interpretation of the process.
The interest in this process is partly because, if combinedwith aYb3þ /Er3þ couple in the same host lattice, the
red, green and blue emissions produce a white light output.
Two-frequency up-conversion has been investigated using Pr3þ defects in a fluoride glass matrix.
Illumination with one pump wavelength, 1014 nm, results in GSA to the metastable 1G4 energy level. No
further excitation is possible with this pump, but simultaneous irradiation with a second appropriate pump
wavelength, 850 nm, further excites the GSA centres via ESA to the 3P3 level. The doubly excited ions lose
energy by nonradiative decay to the 3P0 level. These then drop to the3H6 level and emit red light (Figure 9.29).
Up-conversion and visible output only takes place at the intersection of the two beams.
Note that these are only an outline of the many processes that can occur in a phosphor. In systems that rely
upon a sensitiser, energy transfer must take place between the two centres. Energy transfer in the reverse
4I15/2
4I13/2
4I11/2
4I9/2
4F9/2
4S3/2
4F7/2
4F3/2;5/2
2H9/2
2H11/2
3+Er
hν ESA
(c)
4I15/2
4I13/2
4I11/2
4I9/2
4F9/2
4S3/2
4F7/2
4F3/2;5/2
2H9/2
2H11/2
(d)
3+Er
CR
Ene
rgy
4I15/2
4I13/2
4I11/2
4I9/2
4F9/2
2F7/2
2F5/2
4S3/2
4F7/2
4F3/2;5/2
2H9/2
2H11/2
975 nm
3+Yb 3+Er
CREThν ESA
(e)
Figure 9.27 (Continued )
401 Luminescence
direction may happen as the concentrations change, which leads to concentration quenching (Section 9.3).
Back energy transfer from Er3þ to Yb3þ and Yb3þ to Yb3þ energy transfer both lower the efficiency of the
process, as does the presence of defects in the phosphormatrix. In addition, the degree of phosphor crystallinity
and particle size are important. For this reason, many compositions and dopant levels are explored
systematically before an optimum composition and preparation route are achieved.
9.10 Quantum Cutting
In up-conversion, several low-energy photons are processedwithin a luminescentmatrix to give out one higher
energy photon, typically infrared to visible. Quantum cutting is the reverse of this, as one high-energy photon is
processed (i.e. cut) to give out several lower energy photons, typically ultraviolet to visible. One aim of this
work is to improve the efficiency of phosphors in fluorescent lamps. Here, the driving force is to eliminate the
mercury vapour component of the lamp and replace it with less toxic gases, such as xenon. However, the main
Xe emissions are at 147 and 172 nm, comparedwith 254 nmofmercury. Thus, newphosphors need to be found
that are stable under these intense ultraviolet rays and can compete with mercury vapour lamps in terms
1D2
0.5
1.0
1.5
2.0
2.5
3.0
3.5
eV
5000
10000
15000
20000
25000
30000
cm–1
Ene
rgy
3+ 12Tm (4f )
2F7/2
2F5/2
975 nm
3+Yb
ET
ET
ET
1G4
3F2
3H4
3F4
3H6
3H5
3F3
Figure 9.28 Schematic energy-level diagram for Tm3þ . Red arrows indicate excitation, via energy transferfrom Yb3þ sensitiser ions. Dotted arrows show nonradiative losses and the blue arrow indicates the main blueoutput
Colour and the Optical Properties of Materials 402
of luminosity. Quantum cutting is valuable in this context, as one input ultraviolet photon can be cut to yield
several visible photons. Further applications in colour plasma display panels are also being explored for this
technique.
There are twomainmechanisms for quantum cutting. The first is photon cascade emission, typified by Pr3þ
(4f3) ions. Initial absorption of high-energy 185 nm ultraviolet photons causes excitation to the 4f2 5d band
ofPr3þ (Figure 9.30). Subsequent relaxation takes the ion to the 1S0 level.Thereafter, the transitions giving rise
to visible output are:
1S0! 3P3 at � 400 nm; then 3P0! 3H4 ground state at � 480 nm
1S0! 1D2 at � 330 nm; then 1D2! 3H4 ground state at � 605 nm
The second mechanism explored involves cross-relaxation involving Tb3þ ions, in, for example, GdPO4:
Tb3þ . As with Pr3þ , initial absorption of the ultraviolet photons causes excitation of the active Tb3þ (4f8)
ions to the 4f7 5d band (Figure 9.31). Absorption of 210 nm ultraviolet photons leads to the following steps:
1. Tb3þð1Þ ð4f7 5dÞ! 5D3 and via cross-relaxation simultaneously gives Tb3þð2Þ 5D4 7F6 ground state,
then 5D4! 7F5 plus green emission at approximately 550 nm (Figure 9.31a). Transitions 5D4! 7FJ , with
J¼ 3, 4 and 5, also occur with lesser intensity.
850 nmpump red
emission
0.5
1.0
1.5
2.0
2.5
3.0
3.5
5000
10000
15000
20000
25000
30000
–1cm
Ene
rgy
1014 nmpump
eV
Pr3+ 4f33H4
3H5
3H6
3F2
3F3
1G4
1D2
3P0
3P3
Figure 9.29 Schematic two-frequency up-conversion process in Pr3þ resulting in the production of redlight
403 Luminescence
2. Tb3þð1Þ 5D3! 5D4 nonradiative transition ! 7F5 plus green emission at approximately 550 nm
(Figure 9.31b). Transitions 5D4! 7FJ , with J¼ 3, 4 and 5, also occur with lesser intensity.
Other energy transfer processes also occur involving the host Gd3þ ions, but these do not give rise to strong
emissions in the visible.
185 nm
400 nm
330 nm
605 nm
480 nm
3P0
3P3
1I6
1S0
1D2
1G4
3F3
3H6
3H5
3H4
3F20.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
5000
10000
15000
20000
25000
30000
35000
40000
45000
50000
–1cm eV
Pr3+ 4f3
24f - 5d bandE
nerg
y
Figure 9.30 Quantum cutting of one 185 nm ultraviolet photon to give two photons at 400 and 408nm or at330 and 605nm. Both processes occur
Colour and the Optical Properties of Materials 404
9.11 Fluorescent Molecules
9.11.1 Molecular fluorescence
Theenergy levels giving rise tofluorescence inmolecules aremost oftenof associatedwith theHOMO LUMO
pair of molecular orbitals that are formed by delocalised electrons. This implies that many of the aromatic
compounds, conjugated molecules and dyes described in Chapter 8 are also fluorescent, and many fluorescent
molecules are referred to as fluorescent dyes. Typical examples include the anthraquinones, xanthenes,
cyanins, phthalocyanines andporphyrins. Fluorescein (Figure9.3) is typical of thesegroups,with anabsorption
0
10000
20000
30000
40000
1
2
3
4
5
Ene
rgy
–1cm
74f - 5d band
74f - 5d band
7F6
7F6
7F0
7F0
5D3
5D3
5D4
5D4
3+ 8Tb (1) (4f )
3+ 8Tb (1) (4f )
3+Tb (2)
eV
CR
550 nm
550 nm
(a)
(b)
Figure 9.31 Quantum cutting of one 210 nm ultraviolet photon into two 550nm green photons by Tb3þ ions(schematic)
405 Luminescence
peak at 495 nm and an emission peak at 519 nm, giving rise to the characteristic yellow green hue of materials
coloured with this substance. Another group of fluorescent molecules that is being actively explored consists
of an electron-accepting group (aLewis acid, such as phenol) connected to an electron-donating group (aLewis
base, such as methylamine) by a series of aromatic (benzene) rings. These types of compound are known
generally as donor p-bridge acceptor molecules. They display intense fluorescence; and because of the
exposed donor and acceptor groups, they often show pronounced solvatochromism.
There is also considerable work in progress on the incorporation of luminescent molecules into polymers,
thin films and liquid crystals, for potential optoelectronic applications. In addition, fluorescent tags attached to
molecules can be exploited to follow the course of chemical reactions, including catalysis, inwhich the amount
of catalyst is small and the sensitivity of the fluorescence technique is vital.
The schematic energy-level diagram of a typical molecular fluorophore (Figure 9.32a) shows that the
absorption of energy is from the lowest vibrational level (J¼ 0) of the ground state to the various vibrational
energy levels of the LUMO,whilst the emission is from the lowest vibrational level (J¼ 0) of the LUMO to the
various vibrational levels of the ground state. The absorption transition from J¼ 0 ground state to J¼ 0 excited
state is at the sameenergyas the emission from J¼ 0excited state to J¼ 0ground state. From this it follows that,
ideally, the emission and absorption curves from a molecule are approximately mirror images around this
energy (or wavelength). In ideal cases, the fine structure peaks on the absorption spectrum indicate the
vibrational energy-level spacing of the excited state and the fine structure peaks on the fluorescence spectrum
indicate the vibrational energy-level spacing of the ground state (Figure 9.32b).
1ground state A
1excited state A*
fluorescence
J=0
J=0
absorption
(a)
fluorescenceabsorption
ground state vibrationallevels
excited statevibrational levels
Wavelength
Stokes shift(b)
J = 0 to J = 0 transition
Figure 9.32 Idealised absorption and fluorescence from a molecule: (a) energy level (Jablonski) diagram;(b) absorption and emission spectra
Colour and the Optical Properties of Materials 406
9.11.2 Fluorescent proteins
Althoughmanyfluorescentmolecules are important, there is some justification to the argument that fluorescent
proteins are the most important, particularly in the light of current biological research. Indeed, ‘life may be
defined as the ordered interaction of proteins’ (Further reading, Section 9.17, D.Whitford, p. 2).Moreover, the
‘central dogma’ ofmolecular genetics, viz.DNAmakesRNAmakes protein, puts the importance of fluorescent
proteins into context. Using standard laboratory techniques, fluorescent proteins can be incorporated into
cellular pathways via DNA modification. The course of action of the subsequent fluorescent engineered
molecules can then be observed using fluorescence microscopy to study gene expression, protein protein
interactions and cell reaction pathways in a multiplicity of organisms, from the simplest to the most complex.
The first fluorescent protein to be discovered, green fluorescent protein (GFP), was isolated from a
coelenterate medusa (jellyfish) Aequorea victoria. It has a barrel-like structure composed of 11 antiparallel
b-sheets approximately 3 nm in diameter and 4 nm in length (Figure 9.33a). The fluorophore is part of a single
a-helix positioned in the centre of the barrel. It is formed from three adjacent amino acids in the helix, serine
(Ser 65), tyrosine (Tyr 66) and glycine (Gly 67), where the numbers refer to the position of the amino acid in
the chain. This triplet of amino acids occurs commonly in proteins; the important difference between the
sequence inGFPandother proteins is in the location of thegroup. This is such as to allow the amino acids to link
up in a specific cyclic way to form the fluorophorewith a notable sequence of conjugated bonds (Figure 9.33b
and c).Naturally occurring or ‘wild-type’GFP (wtGFP) absorbsmainly at 395 nm (via a protonated formof the
fluorophore), and to a lesser extent at 475 nm (via a deprotonated form of the fluorophore). These two forms are
present in varying amounts depending upon the pH and temperature of the surroundings. The emission is at
509 nm, irrespective of the absorption, and the quantum efficiency is about 0.75.
Proteins are folded and coiled in very specific ways, and a mistake in this will prevent the protein from
carrying out its normal cellular function. This specific folding is usually achievedwith the help of other cellular
molecules produced in the cell. Remarkably, the complicated folding required to produce functioning GFP
occurs without the necessity of co-reactants only found in the cells of the living animal, so that the fluorescent
protein can bemade in the laboratory and fused to a variety of enzyme and other protein targets so as tomonitor
cell processes usingfluorescencemicroscopy.Despite this enormous advantage, there are drawbacks towtGFP.
First, the complex folding needed occurs efficiently at 28 �C, the normal ambient temperature encountered
by A. victoria, but not very well at 37 �C, the typical mammalian cell temperature. Second, the two absorption
peaks are inconvenient.
To offset these disadvantages, a variety of mutations have been made to thewtGFP. These changes are often
brought about by just one or two modifications in the amino acid sequence making up the protein. At present,
fluorescent proteins that form at 37 �C and fluoresce with a variety of colours from blue (BFP), cyan (CFP),
green (GFP) andyellow (YFP) are available.However, theGFPbasic structure cannot bemodified toyield a red
fluorescing form. This gap has been filled by the isolation of a red fluorescent protein from corals the protein
that gives many corals a pink tone. This material, initially isolated from the coral Discosoma striata, is
known as DsRed, and has an emission peak at 583 nm. As with the GFP family, the DsRed family have also
been chemically modified, so that a considerable number of red and orange fluorescent proteins are now
commercially available.
Despite the enormous usefulness of these families of fluorescent proteins, they suffer from a significant
drawback. They need to be excited with radiation in the ultraviolet blue green region of the spectrum. These
wavelengths do not penetrate tissues, and so most studies using them are confined to thin reaction volumes
in vitro (literally, inglass, i.e. onglassmicroscope slides or shallowdishes).Anewgroupoffluorescent proteins
has now been produced that absorb infrared radiation and emit in the near infrared or deep red. The initial
fluorescent protein was obtained from the bacterium Deinococcus radiodurans, which, as its name suggests,
is able to survive in extreme environments. The great advantage of these fluorescent molecules is that the
407 Luminescence
absorption and emission wavelengths readily penetrate mammalian tissue and bone. This means that studies
can be carried out on living organisms buried deep inside the body. For example, malignant tumours can be
tagged with fluorescent proteins and the processes taking place at a cellular level can be imaged using infrared
detectors.
The increasing importance of fluorescent proteins has become apparent in the last few years, and progress
on the use of these molecules will undoubtedly continue unabated.
CH2 CH
(b)
CH2 C
C
C
CH
CH2
O
O
O
OH
OHHOtyrosine (Tyr)
OH
HOserine (Ser)
NH2
NH2
NH2
glycine (Gly)
Figure 9.33 GFP: (a) barrel-like structure of the protein; (b) molecular building blocks of the fluorophore ofGFP; (c) the structure of the fluorophore of GFP
Colour and the Optical Properties of Materials 408
9.11.3 Fluorescence microscopy
A principal application of molecular fluorescence is in fluorescence microscopy. The first fluorescence
microscope was invented in 1911 and the first practical epi-fluorescence microscope in 1929, and thereafter
fluorophores (also referred to asfluorochromesorfluorescentdyes)wereused to stain tissues andbacteriabefore
observation. Since then the technique has become amajor instrument in the life sciences. It is powerful because,
in a normal optical microscope, incident light is scattered by the object and then collected and passed to the
observer via the optical train of the instrument. However, the fluorophore emits light, rather than just scattering
incident light, and for this reason it is possible to investigate submicroscopic cellular components. Moreover,
the realisation that a combination of fluorescent dyes and sophisticated imaging techniques would make it
possible to surpass the conventional diffraction limit of an optical microscope has now led to the production
of images of subcellular structures with detail resolved far below that thought possible even a few years ago.
At its most basic, there are several distinct steps to obtaining fluorescence microscope images. First, the
component of interest must be linked to a fluorophore if it is not self-fluorescent. There are a large number of
commercial products available for this, comprising (i) small molecules such as fluorescein derivatives, made
up of 30 or so atoms, (ii) fluorescent proteins, which are large molecules, made up of thousands of atoms, and
latterly (iii) quantum dots, which are 10 100 atoms in size. Many of these are highly specific, and this variety
makes it possible (in theory) to study a wide range of cellular processes simultaneously.
Second, thesemarkersmust be inserted into the host tissue. Thismay involve the temporarymodification of the
fluorescentmolecule to enable it to penetrate living tissue, afterwhich the change is reversedwithin the cell so that
the fluorescent form is regenerated. Of course, it is important that the fluorophore is neutral with respect to the
biology that is being investigated.Somenanoparticles are toxic, for instance,which rules out use in live cells unless
they are treated to avoid this difficulty. Note that the fluorescence observed may also reflect the nature of the cell
fluids. For example, if a solvatochromic fluorophore is exposed to a watery cell fluid it may have a different
fluorescent wavelength than if is in encapsulated in the interior of a hydrophobic region. However, these problems
are often bypassed, and in such cases several different molecules can be used simultaneously to observe separate
cellular functions simultaneously by recording the different fluorescent wavelengths emitted.
CH2
C
CH2
C
C
CH
O
HO
tyrosine (Tyr 66)
OHserine (Ser 65)
HN
N
CCH2
ON
glycine (Gly 67)
(c)
Figure 9.33 (Continued )
409 Luminescence
Third, the treated material must be illuminated with a powerful source of exciting radiation, usually in the
ultraviolet or blue region of the spectrum. The relatively weak fluorescence signal must be separated from the
powerful beamused to excite themolecules and then be observed and recorded. This task ismademore difficult
by the fact that the contrast of thefluorescence is diminished by bothRayleigh andMei scattering,which occurs
from many of the organelles in living matter and can seriously degrade the image. (The technicalities of
operation of a fluorescence microscope can be best appreciated by reference to manufacturer’s literature; see
this chapter’s Further Reading).
The likelihood of absorption of a photon of the exciting radiation by a fluorophore is quantified by the
attenuation coefficient (formerly extinction coefficient) of the dye. Clearly, a high attenuation coefficient is a
necessity. Similarly, the fluorescence quantum yield must be as close to unity as possible, to ensure that a light
signal can be detected.
One advantage of fluorescencemicroscopy is that it offers the opportunity of imaging singlemolecules. It is
achieved because of the advent of strongly fluorescentmolecules, particularly of the donor p-bridge acceptor
type, that are able to emit 106 ormore photons over a period ofminutes. A remarkable example of this ability is
demonstrated in experiments to determine how DNAmolecules respond to stretching. This is of considerable
relevance, as cellular life processes are centred upon the coiling, folding and unzipping of double-stranded
DNA (see this chapter’s Further Reading).
F€orster resonant energy transfer (FRET) (Section 9.3), first explored in biological microscopy in the mid
1970s, is also a widely used technique. In this application, both the absorbing molecule A and the receiving
moleculeQ are designed tofluoresce efficiently at differentwavelengths. Thismeans that, in the absence of any
energy transfer, fluorescencewill be characteristic ofA. If, however,A andQapproach close enough for energy
transfer to occur, thefluorescence fromAwill be partly or completely quenched,whilefluorescence fromQwill
appear. In this way, cellular processes such as protein folding can be observed. A protein containing two
fluorescent centres, A and Q, will show fluorescence from A if the folding does not bring the centres into
coincidence, while fluorescence from Q will appear if the centres are juxtaposed after folding. Similarly,
processes taking place across cell walls can be investigated. If a fluorescentmoleculeA is bound to the external
surface of a cell wall and a fluorescentmolecule Q is introduced or formedwithin the cell, fluorescence fromQ
will be seen as a result of FRET if Q is attached or very close to the internal cell wall. As can be imagined, there
are many variations on this technique that are now in use.
Another property of fluorescentmolecules that is used in the study of cellular dynamics, includingmolecular
diffusion, is photobleaching. Under the intense irradiation needed to image fluorescentmolecules successfully
in a microscope, many of the molecules decompose. The irradiated area (volume) then ceases to fluoresce and
photobleaching is said to have occurred. In a dynamic situation, as in diffusion, for example, new fluorescent
molecules can penetrate into the bleached region. A measurement of the rate of this recovery will give
information about the mechanism of the molecular mobility taking place.
9.11.4 Multiphoton excitation microscopy
Multiphoton excitation microscopy, which in practice relies mainly upon two-photon excitation, is a
complementary technique to fluorescence microscopy. Although the theory of two-photon absorption was
worked out in 1931, the application to microscopy had towait until the 1990s and the availability of lasers that
could deliver the required light intensities. The recorded signal is againfluorescence, involvingafluorophore in
the sample. However, in multiphoton excitation microscopy the fluorophore is excited by the simultaneous
multiple absorption of low-energy photons. In this respect it resembles up-conversion, but is quite distinct
and should not be confused. The fluorophore is able to pick up two or more photons to bridge the energy gap
between the ground state and the fluorescent excited state without the necessity of populating or involving
intermediate energy levels (Figure 9.34). For example, the simultaneous absorption of two infrared photons of
Colour and the Optical Properties of Materials 410
1050 nm can be used to excite a 525 nm absorbing fluorophore. A considerable advantage of the technique is
that it allows access to ultraviolet absorbing fluorophores. In practice, it is difficult to build satisfactory optical
systems for use at many ultraviolet wavelengths. Thus, two-photon absorption of 480 nm light can be used to
excite a 240 nm absorbing fluorophore without the necessity of using ultraviolet sources.
For two or more photons to be absorbed at the same time requires an extremely high photon density in the
neighbourhood of the absorbing centre. This can only be achieved if an intense laser beam is focused into
a small region using, in two-photon microscopy, the objective lens of the microscope (Figure 9.35). Thus, the
image obtained is only from fluorescent centres that are in focus, but only a very small field is available. To
offset this, the laser beam is scanned over the sample and the image recorded digitally. Aswith all fluorescence
microscopy, scattering is a problem and can reduce image quality.
9.12 Fluorescent Nanoparticles
The subject field covered by the term nanoparticles is enormous. In this section, attention is focused on the
organic and inorganic materials described earlier in this chapter. These materials are often nontoxic compared
ground state A
fluorescence
ground state A
up-conversion
ground state A
excited state A* excited state A*
excited state A*
E = hν1
E = hν1
E = hν1
E = hν1
E = hν1
E = hν2
E = 2hν1
E = 2 hν1
two-photonfluorescence
(a) (b)
(c)
Figure 9.34 A comparison of the absorption mechanisms for (a) normal fluorescence, (b) up-conversion and(c) two-photon fluorescence
411 Luminescence
with semiconductor quantumdots prepared fromCdS and related chalcogenides (see Section 10.10) and so are
preferred for many tagging purposes.
Broadly speaking, nanoparticles of inorganic phosphors have fluorescence spectra similar to those of bulk
materials.However, the large relative surface area and frequently lower crystallinity of these samplesmean that
the quantumefficiencyof a nanoparticle cluster is usually lower than that of a similar bulk sample. This is due to
energy transfer to the surface, surface defects and surface quenching of ions. This shortcoming can be
ameliorated by enveloping the nanoparticles in a suitable shell of a similar material; for example, fluorescent
lanthanoid-doped CePO4 nanoparticles can be given a shell of LaPO4. Many of the surface problems are now
suppressed and the quantum efficiency of such core shell composites can be high. An alternative approach is
to precipitate nanoparticles within a solid matrix, such as a glass, by, for example, heating or laser irradiation.
The fluorescence from these inclusions is only rarely influenced by the surrounding solid. In the case of
fluorescent molecules, the active phase can be coated onto the exterior of an inert nanoparticle such as silica
(SiO2). In this case, although the surface effects are not removed completely, the fluorescent efficiency of the
molecules can be adequate for many purposes, such as in the study of living cells.
The small dimensions of nanoparticles mean that they do not scatter light. If such particles are embedded
in transparent materials, the result is a clear composite. This opens the possibility of making transparent
and fluorescent thin films or coatings on a wide range of substrates. Such films have found applications as
sensors.
9.13 Fluorescent Markers and Sensors
Fluorescent markers are commonplace. Many banknotes, passports and other security documents are treated
with fluorescent dyes that are incorporated in printing inks. Thefluorescentmarkings need to be invisible under
normal circumstances,whichmeans that themolecules should not be strongly colouredor at least bemasked by
another dye molecule so as to be rendered undetectable. These invisible markings become bright when
illuminated with ultraviolet light.
Sensors to detect molecules need to be more sophisticated, and there are a number of ways in which
these operate. Broadly speaking, the sensor can change in one of two ways. (i) The intensity of
microscope objective lens
cover glass
sample
glass slide
laser beam
fluorescent centres
focal plane
Figure 9.35 Two-photon fluorescence microscopy: the fluorescence only occurs where the intensity of theincident beam is high, generally in the focal plane of themicroscope objective. This means that all the fluorescingcentres are in focus
Colour and the Optical Properties of Materials 412
fluorescent emission can vary as a function of the concentration of the analyte. This may simply be an effect
related to fluorescence quenching or be caused bymore complex interactions. (ii) The position of the emission
band can change as a function of the concentration of the analyte. This is related to the solvatochromic effect
described above and can occur if a degree of bonding occurs between the analyte molecules and the
fluorophore.
An ideal response is one in which a change in fluorescence is sharp, in which case the sensor may display a
digital (on/off) output. An example of this digital-type response is provided by a pH sensor that switches from
nonfluorescent to fluorescent as the pH passes a required value. The sensor consists of a pH-sensitive polymer
containing a water-sensitive fluorophore. Typically, the polymer adopts an open hydrated form in low pH
(i.e. Hþ -rich) environments. The solvent has access to the fluorophore, interacts with it and successfully
quenches emission. At high pH (i.e. low Hþ ) environments, the polymer dehydrates and contracts upon itself
into a globular structure. The solvent does not have access to the fluorophore, which is able to fluoresce under
the appropriate excitation. A careful tailoring of the polymer structure and the fluorophore chosen allows this
on/off effect to be sharp and to be tuned to operate over a range of specific pH values. That is, the fluorescent
molecules light up at a specific pH.
A similarmethod has been used tomeasure temperatures inside living cells. In this instance, the polymer that
carries the water-sensitive fluorophore is heat sensitive rather than pH sensitive. At lower temperatures (with
respect to normal metabolism) the polymer adopts an expanded and open state. The fluorophore is able to
interact with thewatery cell contents, allowing a degree of quenching that results, at best, inweakfluorescence.
As the temperature increases, the polymer contracts and the fluorophore molecules become increasingly
protected, resulting in an increase in fluorescent emission output. The intensity of the emission then acts as a
temperature indicator. As before, the temperature range and colour of the emission can be tuned by varying the
polymer and fluorophore.
The detection of explosives is an important area of research where fluorescence can be helpful. An example
is given by a sensor for the volatile components of explosives, especially TNT (2,4,6-trinitrotoluene). Here,
the sensor is laid onto a glass surface as a thin film. The fluorescence is excited at 370 nm and the emission
is at 408 nm. Once again, detection of the explosive relies upon emission quenching in the presence of the
TNT vapour. Clearly, for such a sensor to be effective, it is important that other common molecules, such as
perfume or fruit odours, do not interfere with the operation of the fluorophore. In addition, it must be
reversible. That is, once the explosive is removed, the sensor must return to its initial state and show
fluorescence again.
Details of these and other fluorescent sensors are given in Further Reading.
9.14 Chemiluminescence and Bioluminescence
Chemiluminescence is light emitted as a result of a chemical reaction that leaves product molecules in a high-
energy state from whence they return to the ground state by the emission of photons. The best known
chemiluminescent reactions are associated with glow sticks. A glow stick consists of a transparent plastic tube
containing one of the active chemicals and a fragile glass ampoule or inner tube containing the other reactant
(Figure 9.36). To activate the glow stick, the outer tube is twisted or bent in order to fracture the inner glass tube,
thus allowing the chemicals tomix. The resulting chemical reaction excites incorporated dyemoleculeswhich,
in turn, give out light.
The energy-providing chemical is hydrogen peroxide (H2O2), contained in the outer part of the stick. The
glass tube contains diphenyl oxalate and the chosen dye. Hydrogen peroxide oxidizes the diphenyl oxalate
to phenol and the very unstable intermediate 1,2-dioxetanedione, which decomposes immediately to carbon
413 Luminescence
dioxide, exciting the dye molecules in the process (Scheme 9.1). The concentrations of the chemicals and the
temperature influence the length of time over which the glow stick is luminous.
A large number of dyes have been used in glow sticks. Two of the commonest are rubrene (5,6,11,12-
tetraphenylnaphthacene), which gives a yellow orange fluorescence, and 9,10-diphenylanthracene, which
gives blue (Figure 9.37).
Bioluminescence is a form of chemiluminescence in which light is emitted by living organisms.
Bioluminescence occurs widely, from bacteria, single-celled algae, many marine organisms, cnidarians
(jellyfish and relatives) to insects. The energy emission from the excited molecules can be rapid, in
biofluorescence, or delayed, in which case it is correctly referred to as biophosphorescence.
Bioluminescence is typified by insects such as the European glow-worm Lampyris noctiluca, which emits
green light, and the North American firefly,Photinus pyralis, which has a yellow emission. The light-emitting
organs, often called lanterns, differ in size and disposition from species to species and often from one sex to the
other. For example, the wingless female of L. noctiluca has two large lanterns and four smaller ones along the
sides of the abdomen, while the winged males have just two small lanterns at the tip of the abdomen.
O O
O
O
O
O
O
O
H O +2 2
diphenyl oxalate
OH
2
O
O
O
O
phenol 1,2-dioxetanedione
1,2-dioxetanedione
+
+ dye 2 CO + dye*2
dye* dye + light
(a)
(b)
(c)
Scheme 9.1 The reactions taking place in glow sticks: (a) hydrogen peroxide and diphenyl oxalate form phenoland 1,2-dioxetenedione; (b) 1,2-dioxetanedione decomposes to carbon dioxide and the energy released is passedto the dyemolecule to produce an excited state dye�; (c) the excited dyemolecule releases the energy in the formof light
transparent plastic outer case
H2O2 solutiondiphenyl oxalate + dye solutionin thin-walled glass tube
Figure 9.36 Glow stick (schematic). The inner thin-walled glass tube is broken when the flexible outer plastictube is bent, allowing the reactive chemicals to mix and generate light by chemiluminescence
Colour and the Optical Properties of Materials 414
Because of the large numbers of organisms that display bioluminescence, there are awidevariety of adaptive
purposes that light production is put to, including mate attraction and prey attraction. There are similarly
a number of different light-producing mechanisms involved. It is possible, with a certain loss of precision, to
generalize the light-producing mechanisms into two main pathways. In beetles such as glow-worms, the
reaction usually involves a molecule of a light-producing chemical generally called a luciferin which, in the
presence of oxygen, the energy providing molecule ATP (adenosine triphosphate) and a catalytic enzyme,
a luciferase, produces an unstable dioxetanone intermediate (similar to those produced in glow sticks), which
decomposes spontaneously to an excited oxidized oxyluciferin, which loses energy by photon emission
(Scheme9.2a). The processes involved in this last step frequently generate light by transitions from anupperp�
state to a lower p energy level.
Luciferin is not a single compound, and luciferins and luciferases differ from one species to another, thus
accounting (at least in part) for the different colours produced. One of the most widely studied is luciferin
extracted from the North American firefly P. pyralis. It is optically active and only one enantiomer is
biochemically active in light production. It has the (daunting) name 2,4-dihydro-2-(6-hydroxy-2-benzothia-
zolyl)-4-thiazolecarboxylic acid (Scheme 9.2b). In this insect it appears that the flashing is controlled by the
production of nitric oxide (NO) in the lantern. This reactive gas is an oxygen scavenger. It is believed that NO
formation inhibits the consumption of oxygen for respiration, allowing it to be used in the light-producing
reaction via the oxidation of luciferin.
A different strategy is used in the coelenteratemedusa (jellyfish)A. victoria, famous as the original source of
GFP (Section 9.11). In this and similar species, the luciferin, coelenterazine, is combined with the luciferase
(a) (b)
Figure 9.37 The idealized structures of (a) rubrene (5,6,11,12-tetraphenylnaphthacene) and (b) 9,10-diphenylanthracene
N
NS
S
HO COH
O(b)
luciferin + O + luciferase → oxyluciferin* → oxyluciferin + light2(a)(excited) (ground state)
Scheme 9.2 (a) The generalized reaction sequence of light production in the North American firefly Photinuspyralis; (b) the structure of the North American firefly luciferin
415 Luminescence
catalyst and oxygen into a single photoprotein called aequorin. In the presence of a trigger, in this case Ca2þ
ions, the luciferin, if isolated, emits blue light of wavelength approximately 470 nm. However, in the living
animal, the luciferin is in close proximity to a molecule of GFP. Energy transfer occurs and the complex emits
green fluorescence at 509 nm instead of blue light.
Because of the vast range of bioluminescent species, it is certain that other variations on these themes will
be discovered in the future.
9.15 Triboluminescence
This is the property of amaterial to emit light on crushing or scratching. It is easily seenwhen sugar is ground in
a darkened room or when a sugar-rich sweet is bitten in two. Besides sugar, many minerals, including calcite,
various feldspars, fluorite and sphalerite, show the phenomenon. It is also seen if a strip of tape is rapidly pulled
from a roll of adhesive tape in the dark.
The source of the light emission in these cases is often not clear and a number of mechanisms have been
proposed to account for it. One set of explanations centres upon the fact that the release of mechanical energy
at a crack tip over small time scales can create very high temperatures, sufficient to raise the surface to
incandescence. In the case of metals, oxidation of the newly formed surface also contributes greatly to the
temperature and can enhance light emission. For example, metallic glasses with compositions near to
Zr41.2Ti13.8Cu12.5Ni10Be22.5 emit a broad band of intense light equivalent to a black body temperature
of about 3175K when fractured in air. It is surmised that the fracture exposes fresh metallic surfaces and
that oxidation of these, a strongly exothermic process, causes the increase in temperature and the emission
of light.
The rupture of chemical bonds at the growing tip gives rise to unpaired electrons. These can react with
gaseous molecules, providing a further source of light emission. In insulating materials, the separation of the
surfaces can also create considerable electric fields. The recombinationof separated chargesmaygiveout light.
Alternatively, the field created might be of sufficient strength to excite gases such as nitrogen to such an extent
that they can emit ultraviolet radiation. In fact, it is believed that when sugar is fractured the blue glow is due to
the visible tail of the ultraviolet emission from excited nitrogen molecules. Ultraviolet light generated in this
way can be absorbed by a fluorophore, should any be present, and also emit visible light.
There is some interest in using triboluminescence in sensors to detect fracture damage that may occur; for
instance, when a bird strikes an aircraft. Strongly triboluminescent crystals are embedded in resin and applied
to the structure to bemonitored. Upon fracture, the triboluminescent crystals emit a brief flash of light that can
be collected by an optical-fibre cable and transferred to a recording instrument. The intensity of the light gives
an indication of the severity of the fracture damage. In addition, a variety of materials that give out different
colours can be distributed over the component to be monitored, so that the location of the damage becomes
readily apparent. The threshold at which damage can be detected is a function of the type of triboluminescent
crystals used, and the density and particle size of crystals contained in the resin. This refinement allows trivial
impacts to be discounted.
9.16 Scintillators
Scintillators are luminescent materials that emit light in response to the impact of heavy particles or highly
energetic radiation. They are used for the detection of electrons, neutrons, a-particles, X-rays, g-rays and so on.Applications range from security scanners, industrial inspection units, medical diagnostic imaging and
high-energy physics.
Colour and the Optical Properties of Materials 416
Considering that these applications spanmany orders of magnitude of energy and particle characteristics,
it is not surprising that a large number of different scintillator materials have been tested. These
include inorganic crystals, polycrystalline ceramics and glasses, organic plastics and liquids, and inert
gases. Despite this variety, there are a number of properties that thesematerials must have in common. These
are:
1. Transparency. Themethod of detection of the high-energy radiation is by visible light emission, so that the
scintillator must be transparent at the appropriate wavelengths.
2. High attenuation. Clearly, the stopping power of the scintillators in important. If this is low, then many of
the particles to be detected will pass straight through the detector without giving a signal.
3. High light output. The output power must be sufficiently high that photodetectors record every event.
The neutron scintillator Cs2LiYCl6 doped with Ce3þ is able to emit 70 000 photons per neutron
absorbed.
4. Decay time. A high intrinsic decay time is important for many particle physics applications, as each event
needs to be recorded separately. However, in imaging applications, longer decay times are often necessary
to give a brighter image.
5. Low afterglow. As with the decay time, significant afterglow can interfere with the recording of
single events. However, it may be useful in some imaging equipment in contributing to a brighter
image.
6. High threshold for radiation damage. It is obvious that, as the radiation to be detected is highly energetic,
the detector must be able to withstand considerable exposure.
Liquid scintillation counters are widely used to record radioactivity from b-emitters. These materials are
generally unstable radioactive isotopes used for medical diagnostics, the b-radiation consisting of energetic
electrons expelled from the radioactive nuclei. The liquid in the counter is frequently benzene or toluene. These
organic molecules absorb the energy of the b-rays and are excited to higher energy levels. Usually, they do notemit light directly, but transfer energy to a fluorescent dye molecule dissolved in the liquid medium and this
in turn emits photons.
X-ray tomography uses ceramic specimens of (Y,Gd)2O3 doped with Eu, or Gd2O2S dopedwith a mixture
of PrF3 and CeF2, single crystal caesium iodide (CsI) dopedwith Tlþ (CsI:Tl), as well as some of the single-
crystal detectors listed below. X-ray detectors for other purposes include amixed barium halide BaFBr1 xIxdoped with Eu2þ , (CsI:Tl) and calcium tungstate CaWO4. In the systems containing a lanthanoid the
detectors fluoresce in the same way as the lanthanoid-containing luminescent materials described above.
That is, the energy from the X-rays excites an electron into a high energy band. Thereafter, radiationless
decay allows the energy to degrade until an f energy level on the dopant activator is reached. A photon is
subsequently emitted to lower the energy to either the ground state or close to it by an f f transition. In the
case of CaWO4, the energy input results in charge transfer within theWO42 units. The return to the ground
state is by photon emission.
Positron emission tomography uses radioactive nuclei that emit positrons as the radiation source. Positrons
are positive electrons, and in ordinarymatter they are short lived. Each positron soon collides with an electron.
Both particles are eliminated and give rise to two energetic photons. The detectors employed are mainly
single-crystal sodium iodide (NaI) doped with thallium iodide (NaI:Tl), CsI:Tl or single crystals of the oxide
Bi4Ge3O12. The Tl-doped crystals emit a green light of wavelength close to 550 nm from transitions between
the Tl energy levels.
High-energy physics uses a wide variety of mainly single-crystal detectors. These include NaI:Tl, CsI:Tl
and lead tungstate (PbWO4). This latter material is used in high-energy particle detectors in the recently
commissioned Large Hadron Collider at CERN. A charge transfer process, similar to that in CaWO4, is
responsible for light emission.
417 Luminescence
Further Reading
The Kroger Vink notation for defects is explained in
R. J. D. Tilley, Defects in Solids, John Wiley and Sons, Inc., Hoboken, NJ, 2008.
Material relevant to this chapter is contained in several articles in
J. P. Hornak (ed.), Encyclopedia of Imaging Science and Technology, John Wiley and Sons, Inc., New York,
2002, including: Optical microscopy, p. 1106; Laser fluorescence imaging, p. 861; Cathode ray tubes, p. 44;
X-ray fluorescence microscopy, p. 1475.
The luminescent phosphors used in fluorescent lamps, CRTs and other devices are described by
G. Blasse, B. C. Grabmaier, Luminescent Materials, Springer-Verlag, Berlin, 1994.
T. J€ustel, H. Nikol, C. Ronda, Angew. Chem. Int. Ed. 37, 3084 3103 (1998).
H. A. H€oppe, Angew. Chem. Int. Ed. 48, 3572 3582 (2009).
A survey of competing flat-panel display types with emphasis on plasma displays is given in
A. Sobel, Sci. Am. 278 (May), 48 55 (1998).
More technical information on displays is given inMater. Res. Soc. Bull., 23 (March), (1996), which includes:
Y. Yang, Polymer electroluminescent devices, p. 31.
J. Hanna, I. Shimizu, Active matrix liquid crystal displays, p. 35.
T. Tsutsui, Molecular thin films, p. 39.
P. D. Rack, A. Naman, P. H. Holloway, S-S. Sun, R. T. Tuenge, Electroluminescent displays, p. 49.
L. F. Weber, J. D. Birk, Colour plasma displays, p. 65.
For an introduction to protein chemistry, see
D. Whitford, Proteins, Structure and Function, John Wiley and Sons, Ltd, Chichester, 2005.
GFP and related matters are reviewed by
O. Shimomura, Angew. Chem. Int. Ed. 48, 5590 5602 (2009).
M. Chalfie, Angew. Chem. Int. Ed. 48, 5603 5611 (2009).
R. Y. Tsien, Angew. Chem. Int. Ed. 48, 5612 5626 (2009).
Much relevant up-to-date information on fluorescence-related microscope techniques is to be found on the
websites of microscope manufacturers, including Olympus, Nikon and Zeiss.
The experiments on DNA uncoiling are in
J. vanMarneren, P.Gross,G. Farge, P.Hooijman et al., Proc. Nat. Acad. Sci. U. S. A.106, 18231 18236 (2009).
Triboluminescence is discussed by
C. J. Gilbert, J.W.Ager, V. Schroeder, R. O.Ritchie, J. P. Lloyd, J. R. Graham,Appl. Phys. Lett. 74, 3809 3811
(1999).
I. Sage, Chem.Br. (February), 24 27 (2001).
Information on scintillators is given by
C. Greskovich, S. Duclos, Annu. Rev. Mater. Sci. 27, 69 88 (1997).
http://scintillator,lbl.gov/.
Colour and the Optical Properties of Materials 418
10
Colour in Metals,Semiconductors and Insulators
. How can colourless boron impurities tint diamond blue?
. What produces light in a light-emitting diode (LED)?
. Why are copper and gold coloured, whereas most
metals resemble silver?
So far, the broad scheme of bonding in solids has been ignored. This chapter addresses this lack, and the colours
and optical properties that arise as a consequence of this bonding are described. In order to achieve this, it is
necessary toknowsomethingof theway that theouter electronson the component atomsof thematerial are held
inmetals, semiconductors and insulators. This is described by band theory. In this approach, the outer electron
energy levels are shown to be broadened into energy bands as the atoms coalesce into a molecule and then a
solid. Themain energy landscape in a solid is then the energy band structure. This process can be viewed as an
extension of the ideas that delocalize atomic orbitals into molecular orbitals, but now these orbitals extend
throughout the solid rather than just over the molecule. Thus, transitions in an atom, between sharp energy
levels, change into transitions between a HOMO and a LUMO in a molecule and then into transitions between
a lower energy valence band and a higher energy conduction band in a solid. In the simplest depictions, the
upper energy band (the conduction band) is separated from the lower energy band (the valence band) by a
constant band gap. This is called the flat band model. In real structures, the band architecture is complex.
Note that these concepts are reversible. The properties of a bulk solidwill change as the degree of division of
the solid increases until, at the smallest dimensions, the properties become less and less referable to energy
bands identical to thoseof thebulk, andmust be considered in termsofmolecular and then atomic energy levels.
These changes come to the fore in nanoparticles, described in more detail below.
Colour and the Optical Properties of Materials Richard J. D. Tilley
� 2011 John Wiley & Sons, Ltd
It is most convenient to start with a consideration of insulators, as these provide links with the previous
chapters.
10.1 The Colours of Insulators
Insulators have the upper energy band completely empty and the lower energy band completely filled by
electrons (Figure 10.1a). The filled energy band is called the valence band and the empty energy band is called
the conduction band. The energy difference between the top of the valence band and the bottom of the
conduction band is the band gap, magnitude Eg. A typical insulator is characterized by a large band gap
(Table 10.1). If light falls onto an insulator, it will not be absorbed unless the energy of the incident photons is
high enough to promote an electron from the valence band to the conduction band. The photon energy at this
point is a measure of the optical band gap. In a flat band model (i.e. Figure 10.1a), this is a single energy and
a sharp step in the absorption spectrum would be expected, called the band edge or the absorption edge
(Figure 10.1b). In real solids the band gap is of more complex geometry, and the transition is not so sharp in
practice. This means that there is an uncertainty involved in the estimation of the band gap from spectra, and
a range of values are found in the literature. (Other techniques are also used to measure the band gap of solids,
and these also give values slightly different from those obtained from spectra, adding to the spread of recorded
values.)
Inmost insulators, oxides for example, the optical band gap lies in the ultraviolet part of the spectrum. Thus,
although there is very strong absorption at these wavelengths, the visible spectrum is not affected. Crystals of
insulators are then transparent and powders are white, due to scattering and reflection, as detailed in earlier
chapters.However, the absorptiondue to thevalenceband conductionband transitionhas a certainwidth. If the
optical band gap energy falls just into the ultraviolet, roughly speaking below 3.1 eV, the band gap absorption
spectrum impinges into the violet end of the spectrum. This tends to give oxides a yellow tint, typified by lemon
yellow tungsten trioxide (WO3; Figure 10.1c) and pale yellow ceria (CeO2; Section 7.15).
The fact thatmany insulators strongly absorb in the ultraviolet but are transparent to visible radiation ismade
use of in sunscreens (Section 5.7). In particular, ZnO and TiO2 are very widely applied. These absorb the
incident harmful ultraviolet radiation and, provided that the particle size is sufficiently small, are invisible to
the eye.
The optical band gap of a solid varieswith particle size.Although this has little consequence for ordinaryfine
powders and polycrystalline thin films, such as those used in paints and sunscreens, a change is observed at the
smallest particle sizes. Nanoparticles show a considerable shift in band gap energies, with the band gap
increasing as the particle size drops below approximately 10 nm diameter (Figure 10.2).
The optical band gap ofmanymaterials has also been found to decrease slightlywith temperature. Although
this effect is small, it can lead to interesting colour changes. White zinc oxide (ZnO) absorbs in the near
ultraviolet. At high temperatures the decrease in the band gap means that some violet light is absorbed. The
material will then become yellow to the eye. This effect is also noticeablewith the yellowoxide In2O3.At room
temperature this absorbs in the green blue. As the temperature increases, the absorption shifts towards the
lower energy red, causing the oxide to take on a much deeper yellow brown colour. The effect of colour
variation with temperature is known as thermochromism. This term was encountered in Section 6.9. with
respect to liquid-crystal thermometers.1
1 Note that the name thermochromism applies to the change of colour with temperature. That is, it does not describe the mechanism, only
the observed effect. The two examplesmentionedhavequite differentmechanisms.Othermechanisms for various thermochromic changes
are also found, especially among organic thermochromic materials.
Colour and the Optical Properties of Materials 420
10.2 Excitons
One of the most important aspects of the band theory of solids is that, when an electron is promoted from the
valence band to the conduction band, two free charged ‘particles’ form: an electron now in the conduction
bandandan electronhole,more commonly just calledahole, in the valenceband.Holes contribute significantly
to the electronic properties of the solid and can be somewhat loosely considered to behave as if they were
valence band(full)
conduction band(empty)
(a) (b)
(c)
band gap, energy Eg
Abs
orpt
ion
EnergyEg
0
100%
400 500 600 700 800
50
100
Ref
lect
ance
%
WO3
ZnO
Wavelength / nm
Figure 10.1 (a) Simple ‘flat band’ approximation of the band structure of an insulator. (b) The absorption ofenergy by an insulator (schematic). (c) The reflectance spectra ofwhite zinc oxide (ZnO) and pale yellow tungstentrioxide (WO3)
421 Colour in Metals, Semiconductors and Insulators
‘positive electrons’.2 Because these two species have opposite charges, they attract via Coulomb forces. The
bound pair is called an exciton. The energy required to create an exciton is equal to the energy required to
promote the electron into the conduction band, the bandgap energyEg,minus the binding energyof the exciton.
This is depicted as a new energy level just below the conduction band (Figure 10.3a). The binding energy
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
1 2 3 4 5 6 7
bulk
Particle size / nm
Ban
d ga
p / e
V
Figure 10.2 Variation of the band gap of ZnO nanoparticles as a function of particle size. [Data fromR. Viswanathan et al., J. Mater. Chem., 14, 661–668 (2004); H.-M. Xiong et al., Angew. Chem. Int. Ed., 48,2727–2731 (2009)]
Table 10.1 Optical band gap Eg of bulk oxidesa
Oxide Band gap/eV Oxide Band gap/eV
MgO 8.7 MgAl2O4 7.5Al2O3 6.3 SrZrO3 5.4Y2O3 5.8 La2Ti2O7 4.0Hf2O3 5.2 LiNbO3 3.8Ga2O3 4.8 LiTaO3 3.8ZrO2 4.6 NaTaO3 3.8Ta2O5 3.9 MgTiO3 3.7SnO2 3.5 Al2TiO5 3.6Nb2O5 3.3 KNbO3 3.3TiO2 3.2 BaTiO3 3.2ZnO 3.3 SrTiO3 3.1Sb2O3 3.0In2O3 2.8Bi2O3 2.8WO3 2.7
aData from F. Di Quarto, C. Sunseri, S. Piazza, M. C. Romano, J. Phys. Chem. B101, 2519–2525 (1997). For
alternative data, see J. Portier et al., Prog. Solid State Chem., 32, 207–217 (2004).
2 These are not genuine positive electrons (positrons), because such a particlewould be eliminated instantaneously by combinationwith a
normal electron. They are virtual particles equivalent to the absence of an electron.
Colour and the Optical Properties of Materials 422
depends upon a number of factors, and there may be several closely spaced levels present rather than one. The
presence of excitons in a crystalwill then be revealed by absorption peaks corresponding to transitions between
the valence band and the exciton energy levels which lie on the low energy side of the absorption edge
(Figure 10.3b).
Generally, this interaction energy is weak in insulating (and semiconductor; Section 10.6) crystals. The
electron and hole are not close and the exciton is considered to extend over several unit cells of the structure.
Excitons representing this situation are called Mott Wannier excitons or free excitons.
A free exciton can be thought of as analogous to an extended hydrogen atom with a hole replacing a proton
and the energy levels that lie below the absorption edge analogous to hydrogen atom energy levels. The idea of
considering an exciton as an ‘atom’ is, in fact, close to the original conception of an exciton. In the 1930s,
Frenkel suggested that some aspects of the ultraviolet absorption spectra of insulators could be explained if
some atomswere in a state of excitation; that is, were excitons. In this case, the excited electron of the electron
hole pair is in an upper atomic orbital rather than the conduction band of a solid and the hole is in a lower atomic
orbital rather than the valence band of a solid. The hole and electron remain in close proximity, within the
atomic orbital structure of the excited atom, and the hole electron interaction energy is high. Such excitons
are called Frenkel excitons or tightly bound excitons. Frenkel excitons may occur in insulating solids such as
valence band
conduction band
(a)
(b)
Eg
Abs
orpt
ion
EnergyEg
exciton energy levels
exciton peaks
Figure 10.3 Exciton energy levels: (a) energy levels just below the conduction band in a semiconductor;(b) exciton absorption peaks close to the absorption edge in a semiconductor
423 Colour in Metals, Semiconductors and Insulators
the ionic oxides andhalides. For example, in the alkali-metal fluorides,where excitons are located on anions, an
electron is promoted from one of the ground-state (1s2 2s2 2p6) orbitals to one of the higher (3s, 3p, . . .) empty
ones. The exciton energy levels will now take values close to those of the ionic energy levels of the F ion.
Excitons can also form inmolecular crystals, such as those of the dyemolecules and conjugated or aromatic
hydrocarbons. In these cases the bondingwithin themolecule is strong and represented by a series ofmolecular
orbitals. Bonding between the molecules is rather weak. Excitons will then be associated with a hole in a
low-energy molecular orbital, typically a HOMO, and an electron in a higher energy molecular orbital,
typically a LUMO. Exciton energy levels will then be similar to the molecular orbital energy ladder.
The concept of an exciton, therefore, spans the range from a strongly bound electron hole pair on an atom to
a weakly bound pair of virtually free particles moving through the band structure of the solid. In all cases, the
excitons are revealed by extra absorption peaks in the spectrum of the material. However, these are mostly
observed when the sample is at low temperatures, as thermal vibrations smear out the absorption peaks at
normal temperatures.
10.3 Impurity Colours in Insulators
Impurities are the commonest way to introduce colour into colourless insulators. These impurities are usually
regarded as defects; the commonest of these are point defects, which are located at a particular atom sitewithin
the solid matrix.
‘Coloured’ transition metal or lanthanoid ion impurities in glasses, gemstones and phosphors, which have
already been described, fall into this category (Chapters 7 and 9). In these compounds, the impurity atomor ion
occupies a position normally filled by one of the component atoms of the structure, such as impurity Cr3þ in an
Al3þ site in ruby. The energies of the colour-producing optical transitions, d d or f f transitions, are much
lower than the band gap (�2 eV compared with 6.3 eV for ruby) and the energy levels giving rise to colour lie
in the band gap between the valence and conduction band of the solid. In the case of lanthanoid phosphors,
exciting radiation is often energetic enough to promote an electron from the impurity atom into the conduction
band originating in the outer electron orbitals of the surrounding matrix atoms. Energy is lost by nonradiative
transitions until an energy level located on the lanthanoid ion is reached, afterwhich avisible photon is released
(Figure 10.4).
Newenergy levels can alsobe introduced into thebandgapby the additionof ‘colourless’ impurity atomsand
other point defects, including the absence of an atom froma normally occupied site a vacancy. The impurities
are classed as donors if they normally contribute electrons to the conduction band or as acceptors if they
normally take up electrons from the valence band. Donor dopants may give energy levels close to the
conduction band or far from it. Similarly, acceptor dopants may give energy levels close to the valence band or
far from it. Those close to the band edges are called shallow levels, while those towards the centre of the band
gap are called deep levels. The excitation of electrons to and from these levelswill give rise to colourswhen the
energy difference falls in the visible range (Figure 10.5).
10.4 Impurity Colours in Diamond
The processes leading to colour when ‘colourless’ dopants are introduced into a transparent insulator are well
illustrated by the coloration induced in diamonds, which varies from brownish (low value) coloration through
orange, pink and purple to the highly prized and rare blue and yellow gems.
The diamond structure is built up of carbon atoms each surrounded by four carbon atom neighbours in a
tetrahedron, the linking being via sp3 hybrid bonds (Figure 10.6a). Diamond has a band gap (of about 5.5 eV)
Colour and the Optical Properties of Materials 424
which is too large to absorb visible light; therefore, perfect diamonds are clear. The commonest impurity in
natural diamonds is nitrogen (N).Most of these nitrogen atoms substitute for carbon onnormal tetrahedral sites
in the crystal. Diamonds are often subjected to temperatures of 1000 1200 �C over geological timescales,
allowing these nitrogen atoms to diffuse through the structure, and most end up in clusters, some of which
produce a straw-yellow colour. Such stones are known as Cape yellows and are of considerable value in
jewellery.More rarely, some diamonds contain nitrogen as isolatedN atoms located on carbon sites; this group
also includes yellow diamonds (canaries) that are similarly highly prized.
valence band
conduction band
acceptor levels
donorlevels
Figure 10.5 Schematic band structure of an insulator containing defects that introduce additional energy levelsin the band gap. Any transitions that are of a suitable energy can cause the solid to become coloured
valence band
conduction band
lanthanoidimpurity
transition metalimpurity
Figure 10.4 Absorption and emission of radiation by transition and lanthanoid metal ions in an insulator. Theenergy levels important for colour are situated in the band gap of the insulator. Efficient fluorescence often usesexcitation to the conduction band as a preliminary to the emission of visible light. [Nonradiative transitions areshown as dotted lines]
425 Colour in Metals, Semiconductors and Insulators
The colour of these latter diamonds is caused by the isolated nitrogen impurities in the following way.
Nitrogen, with an electron configuration 1s2 2s2 2p3, has five bonding electrons, one more than carbon, with
configuration 1s2 2s2 2p2. Substitution of nitrogen for carbon on a normal carbon atom site in the crystal creates
an N 0C defect3 with an effective negative charge (Figure 10.6b). Four of the electrons around each impurity
nitrogen atom are used to fulfil the local sp3 bonding requirements of the crystal structure and one electron
remains unused.The extra electron, oneper nitrogenatom impurity, canbe excitedby suitable radiation into the
conduction band and so the defect is a donor impurity. On an energy-level diagram this is often represented as
a deep donor level in the energy gap centred at approximately 2.2 eV below the conduction band, but because
of lattice vibrations and other interactions it is better regarded as a narrow band of energies centred at 2.2 eV
and extending to 1.7 eV, the quoted ionization energy of the nitrogen atom in diamond (Figure 10.6c). This
centre is able to absorball visible light ofwavelengths longer thanabout 564 nm,giving the stones a faint yellow
aspect. As the nitrogen concentration increases, the colour intensifies.
N
C
a
b
c
electron
(a)
(c)
(b)
conduction band
valence band
~5.5 eV
~2.2 eV
NC′
Figure 10.6 (a) The structure of diamond. Each carbon atom is surrounded by four others at the vertices of atetrahedron. (b) Idealized representation (ignoring atomic relaxation andmolecular orbital formation) of a singlenitrogen substitutional impurity N0
C. (c) Energy levels of a substitutional N impurity N0C
3 The nomenclature is that of the standard Kroger Vink notation (see this chapter’s Further Reading).
Colour and the Optical Properties of Materials 426
The most common impurity in diamond appears to consist of a pair of nitrogen atoms on adjacent carbon-
atom sites (NC NC)20. These form deep levels at approximately 4 eVbelow the conduction band. These centres
absorb 310 nm radiation and do not contribute to the colour of stones.
From among the many other nitrogen-containing clusters known, the N3 centre, which consists of three
nitrogen atoms on adjacent carbon sites in a planar fashion around a carbon vacancy (i.e. a missing carbon
atom), ðNC--NC--NC--VCÞ., seems to be responsible (at least in part) for the pale straw colour of Cape yellow
diamonds (Figure 10.7a). The N3 centre absorbs just in the blue end of the visible, at 415 nm, giving yellow
stones. The N3 centres are often accompanied by N2 centres consisting of two nitrogen atoms on normal
carbon sites adjacent to a carbon vacancy, i.e. (NC VC NC) clusters. These can be electronically neutral, in
which case they absorb at approximately 475 nm, giving a yellow colour to the stones and adding to that
contributed by theN3 clusters. They emit a strong green light at 531 nmafter excitationwith laser light. TheN2
3E
1.945eV
3A ~ 6 meV
~ 3 meV
b
VCVC
N
C
electron
(a)
(c)
(b)
Figure 10.7 Idealised representations (ignoring atomic relaxation andmolecular orbital formation) of nitrogen-containing defect centres in diamond: (a) an N3 centre, ðNC--NC--NC--VCÞ.; (b) an N–V centre, (NC–VC ).(c) Energy levels of an N–V centre. (These energy levels fall within the band gap of diamond)
427 Colour in Metals, Semiconductors and Insulators
cluster can also be negatively charged, (N V N)0, in which case the absorption is at approximately 989 nm in
the infrared. This absorption band can spill over into the red part of the visible spectrum, leading to stones with
a blue tone. When all these clusters are present in roughly equal quantities, a green colour is perceived.
The cluster studied in most detail is that consisting of a single nitrogen impurity located next to a carbon
vacancy, (NC VC), mostly written as (N V), as these endow diamond thin films with interesting electronic
properties. These centres are readily created by the irradiation of artificial diamonds or diamond thin films,
which normally incorporate nitrogen impurities during synthesis, with high-energy protons. The proton
irradiation results in the formation of carbon vacancies, and if the crystals are then annealed at above 600 �C,the temperature at which the vacancies becomemobile, they diffuse through the structure until they encounter
a nitrogen impurity. The strain around the nitrogen atom effectively traps the vacancy, preventing further
migration. In the resultant (N V) centres, the tetrahedron surrounding the carbon vacancy is composed of
three carbon atoms and one nitrogen atom (Figure 10.7b). These centres can be electronically neutral, but
the most studied cluster is the negatively charged (N V) centre [(NC VC)0]. These absorb strongly at
approximately 575 nm, giving stones a pink hue. They are strongly photoluminescent, and excitation with
laser light in the range from 490 to 560 nm results in a strong red emission peak with a maximum at
approximately 670 nm.
The electron trapped at the centre has an orbital that extends over the cavity, so that it encompasses not only
the nitrogen impurity, but also the three carbon atoms that also surround the vacancy. The energy levels of this
cluster can be calculated bymolecular orbital theory. To a first approximation, the ground-state term is 3A and
the first excited state is 3E (Figure 10.7c). However, spin orbit coupling (Section 7.2) splits the ground state
and the excited state into two. These have different energies, and because of the spin-multiplicity rule, when
an (N V) centre emits a photon, the transition is allowed from one of these and forbidden from the other.
Moreover, the electron can be flipped from one state to another by using low-energy radio-frequency
irradiation. Irradiation with an appropriate laser wavelength will excite the electron and as it returns to the
ground statewill emit fluorescent radiation. The intensity of the emitted photon beamwill depend upon the spin
state, which can be changed at will by radio-frequency input.
In addition, the application of a magnetic or static electric field splits the levels again. The extent of this
splitting depends upon themagnitude and relative orientation of the appliedfield and the defect. The transitions
between the ground-state and excited-state levels are subject to different transition rules than those in the
absence of these fields. Thus, the intensity of the output fluorescence may be modulated by the imposition of
radio-frequency radiation, by magnetic fields and electrostatic fields. Unsurprisingly, these centres are under
active exploration for use as components for the realization of quantum computers.
Although nitrogen impurities give rise to the highly valued yellow-hued diamonds, other colourless
impurities are also important. For example, prized blue diamonds are the result of boron impurities. In
this case, each boron impurity atom occupies a carbon position, again forming a substitutional defect, BC.
(Figure 10.8a). Boron,with an electron configuration 1s2 2s2 2p1, has only three outer bonding electrons instead
of the four found on carbon. These three are used in fulfilling the bonding requirements of the structure, but
one bond of the four is incomplete and lacks an electron, giving the defect an effective positive charge. In
semiconductor physics terms, each boron atom dopant has an accompanying hole in proximity to the occupied
site and is an acceptor impurity. This is represented by the creation of a set of new acceptor energy levels
approximately 0.4 eV above the valence band (Figure 10.8b). The transition of an electron from the valence
band to this acceptor level has an absorption peak in the infrared, but atomic vibrations and other imperfections
broaden this into anarrowbandof energies allowing thehigh-energy tail of the absorptionband to encroach into
the red at 700 nm. The boron-doped diamonds, therefore, absorb some red light and leave the gemstone with
an overall blue colour.
Other colourless ions, such as of hydrogen, sulfur and phosphorus, have also been introduced into diamonds,
especially with a view to altering the electronic properties rather than colour.
Colour and the Optical Properties of Materials 428
10.5 Colour Centres
In the 1920s and 1930s there was considerable interest in the fact that synthetic alkali halide crystals could be
made intensely coloured in a number ofways, including irradiation byX-rays, electrolysis (with colourmoving
into the crystal from the cathode), or heating the crystals at high temperatures in the vapour of an alkali metal.
The principal investigator, Pohl, in Germany, attributed the colour to the presence of Farbzentren (lit. colour
centres). It is now well known that exposure of transparent solids, both glasses and crystals, to high-energy
radiation frequently makes them coloured and the colour arises because the treatment has introduced defects
into the material. The defects responsible for this are known as colour centres. Many different colour centres
have now been characterised. (Note that there is a certain degree of imprecision in the literature, and colours
caused by impurities, described above, are also sometimes said to be due to colour centres.)
10.5.1 The F centre
The first colour centre to be characterised was the F centre, a term derived fromFarbezentrum (colour centre),
before it was clear thatmany different colour centres can form. F centres were first produced by exposing alkali
halide crystals to high-energy radiation such as X rays. This causes the crystals to become brightly coloured
with fairly simple bell-shaped absorption spectra. The peak of the absorption curve lmax moves to higher
wavelengths as both the alkali metal ion size and halide ion size increase (Table 10.2).
C
B
hole
conduction band(empty)
valence band(full)
~ 0.4 eV
(a)
(b)
Figure 10.8 (a) The idealised structure of a substitutional B atom in diamond BC.. (b) Acceptor energy level of
the defect
429 Colour in Metals, Semiconductors and Insulators
F centres can be introduced in several ways, apart from using ionising radiation. One of these involves
heating the crystals at high temperatures in the vapour of an alkali metal. It is notable that the exact metal does
not matter as long as it is an alkali metal. That is, if a crystal of potassium chloride (KCl) is heated in an
atmosphere of sodium vapour, typical violet KCl F centres are formed, not the orange brown NaCl colour
centres. Another way of introducing F centres into alkali halide crystals is to pass an electric current through
heated samples and electrolyse them. In this case, the typical F centre colour is seen to move into the crystal
from the cathode region. Once again, the colour depends upon the crystal being electrolysed, not on the exact
nature of the cathode. Thus, F centres in sodium chloride (NaCl) always give the crystal an orange brown
colour irrespective of the method of generation.
Theseobservations suggest that the centres are defects in the crystal structure that donot involve the chemical
nature of the components of the material in a direct fashion. This is so, and it has long been known that the
F centre is an anionvacancyplus a trapped electron (Figure 10.9). The trapping is due to the fact that themissing
anion creates a vacancy that has an effective positive charge and it is this charge that attracts the electron to form
a ðVX.e0Þ centre where X represents the missing anion. The F centre in its ground state forms a deep level in the
band gap of the alkali halide solid. The electron in this location behaves rather like the electron surrounding a
hydrogen atom, and is able to absorb electromagnetic radiation, causing it to bepromoted fromone energy level
to another. These transitions give rise to the colour of the solid. If enough energy is supplied then the electron is
promoted into the conduction band, where it is no longer trapped.
10.5.2 Electron and hole centres
Since the original studies of F centres, many other colour centres have been characterised which may be
associated with either trapped electrons or trapped holes. These are called electron-excess centres when
electrons are trapped and hole-excess centres when holes are trapped.
The F centre is an electron-excess centre and arises because the crystal contains a small excess of metal.
Similar metal-excess F centres exist in compounds other than the alkali halides. An example is provided by the
mineral Blue John.4 This is a rare, naturally occurring form of fluorite (CaF2). The coloration is caused by
Table 10.2 Alkali metal halide F centres
Compound Absorption wavelength lmax/nm Coloura Lattice parameter/nm
LiF 235, UVb colourless 0.4073NaF 345, UV colourless 0.4620KF 460, blue yellow brown 0.5347RbF 510, green magenta 0.5640LiCl 390, UV (just) yellow green 0.5130NaCl 460, blue yellow brown 0.5641KCl 565, green violet 0.6293RbCl 620, orange blue green 0.6581LiBr 460, blue yellow brown 0.5501NaBr 540, green purple 0.5973KBr 620, orange blue green 0.6600RbBr 690, red blue green 0.6854
aThe appearance of the colour centre-containing crystal is the complementary colour to that removed by the absorption band.bUV¼ultraviolet.
4 The name ‘Blue John’ is a corruption of the French term ‘bleu jeune’whichwas used to describe the blue form of the normally yellowish
fluorite crystals found in nature.
Colour and the Optical Properties of Materials 430
electron-excess F centres, each consisting of an anion vacancy plus a trapped electron. It is believed that the
colour centres in Blue John were formed when the fluorite crystals were fortuitously located near to uranium
compounds in the rock strata. Radioactive decay of the uranium produced the energetic radiation necessary to
form colour centres.
One of the best understood hole-excess centres gives rise to the colour in smoky quartz and some forms of
amethyst. These minerals are essentially crystals of silica (SiO2) which contain small amounts of either Al3þ
or Fe3þ as substitutional impurities, Al0Si or Fe0Si. Charge neutrality is preserved by way of incorporated
hydrogen as Hþ . The colour centre giving rise to the smoky colour in quartz is formed when an electron is
liberated from an [AlO4]5 group by ionising radiation and is trapped on one of the Hþ ions present. The
reaction can be written as:
½AlO4�5 þHþ ! ½AlO4�4 þH
The colour centre is the [AlO4]4 group, which can be thought of as [AlO4]
5 togetherwith a trapped hole. The
colour arises when the trapped hole absorbs radiation.
The situation in amethyst, containing Fe3þ impurities, is similar. These crystals are a pale yellow colour due
to the crystal-field splitting of the d-electron levels on the Fe3þ ions. In this form, natural crystals are known as
citrine, a semiprecious gemstone. On irradiation, [FeO4]4 groups form by interaction with Hþ ions, as
described for [AlO4]4 above.The colour centre, an [FeO4]
5 group containing a trapped hole, is able to absorb
light, giving the crystals the purple amethyst coloration (Figure 10.10). If these crystals are heated to high
temperatures the purple coloration is lost is and replaced by pale yellow crystal-field colours due to Fe3þ . Thistechnique is sometimes used to convert relatively inexpensive amethyst into an artificial form of the rarer
and more costly semiprecious stone citrine.
There is a great deal of interest in the formation of colour centres in minerals by irradiation. In part this is
because of the possibility of creating an impressive gemstone from an inexpensive precursor. Themost widely
e′
Figure 10.9 Idealised representation of an F centre, an anion vacancy plus a trapped electron, in an alkali metalhalide crystal
431 Colour in Metals, Semiconductors and Insulators
available irradiated stone is topaz, Al2SiO4(F,OH)2. Normally, good-quality topaz is clear and of little value.
The structure contains [AlO4F2]7 octahedra which, like the [AlO4]
4 above, are able to form stable colour
centres under irradiation. These endow the stones with a beautiful blue colour (Figure 10.11).
Although the exact cause of the coloration is not completely clarified, the rutile form of the white pigment
titanium dioxide (TiO2) seems to be coloured by hole centres formed as a consequence of the incorporation
Figure 10.11 Blue topaz stones. The blue colour is induced in the colourless topaz crystals by irradiation
Figure 10.10 Crystals of amethyst from Brazil. The purple coloration is due to hole centres, the intensity of thepurple hue being proportional to the number of centres present in a crystal
Colour and the Optical Properties of Materials 432
of colourless Ga3þ ions. When crystals of rutile are heated with gallium oxide (Ga2O3), small quantities of
Ga3þ impurity are readily incorporated into the structure and initially clear single crystals of rutile take on
ayellow orange colour (Figure10.12).The impurityGa3þ ions enter the rutile structure and substitute forTi4þ
ions in octahedral sites to formGa0Ti defects. These impurities have an effective negative charge, allowing them
to trap positively charged holes. The liberation of the holes absorbs energy towards the violet end of the
spectrum and colours the crystals yellow orange.
Colour centres can give rise to a variety of useful colour effects. The oxide SrAl2O4 is a long-life phosphor
giving a green output colour when doped with B, Eu2þ andDy3þ . The origin of the colour lies in two complex
colour centres formedby the impurity cations. The structure of this phase is a distorted formof tridymite,which
is composed of corner-linked AlO4 tetrahedra that enclose Sr2þ ions in the cavities so formed. The B3þ
substitutes for Al3þ to create BO4 tetrahedra and BO3 triangular groups. The Dy3þ substitutes for Sr2þ to
form DySr.defects. Charge is balanced by the creation of Sr2þ vacancies, V20
Sr:
Dy2O3 ð3SrAl2O4Þ! 2DySr. þV20
Sr þ 6AlAl þ 12OO
Twocomplex centres form: ðDy--BO4--V0Sr--h
.Þ, which are hole centres formed thermally from ðDy--BO4--V20SrÞ,
and ðBO3--VO.--e0Þ, which are electron centres formed from ðBO3--V
2O
.Þ under violet light. Under normal
conditions, the electron and hole centres are metastable and the holes and electrons gradually recombine. The
energy liberated is transferred to the Eu2þ ions, to give a green fluorescence. As there is no radioactivity
Figure 10.12 Crystals of rutile, one form of titanium dioxide (TiO2), coloured yellow–orange by the inclusionof small quantities of gallium trioxide (Ga2O3)
433 Colour in Metals, Semiconductors and Insulators
involved, these materials can be used for luminous dials on clocks and watches, replacing a historic use
involving radioactive materials, or as cold-light displays.
10.5.3 Surface colour centres
The concept of colour centres has been extended to surfaces to explain a number of puzzling aspects of surface
reactivity. For example, in oxides such as MgO an anion vacancy carries two effective charges, V2O
.. These
vacancies can trap two electrons to form an F centre or one electron to form an Fþ centre.When the vacancy is
located at a surface, the centres are given a subscript s, i.e. Fþs represents a single electron trapped at an anion
vacancy on anMgO surface. As the trapping energy for the electrons in such centres is weak, they are available
to enhance surface reactions.
The concentration of Fþs centres can be increased by irradiation with energetic radiation such as X-rays or
ultraviolet light, as well as by reaction with hydrogen. This latter reaction has led to the suggestion that several
new colour centres could form involving hydroxyl. The Fþs (OH ) centre is imagined to form in the following
way. A hydrogen atom reacts on the surface to form a hydroxyl group, OH . This leaves the surface to link to a
nearbymetal cation in the exposed surface, at the same time creating an oxygen vacancy and leaving a trapped
electron to create an Fþs (OH ) defect (Figure 10.13).
The properties of defects of this type are difficult to determine experimentally, although absorption spectra
do give information about electron or hole binding energies. Much information is obtained by calculation,
using density functional or other quantumcomputationalmethods. In thisway, the relative stabilities of defects
on plane faces, steps, terraces and corners is being explored.
10.5.4 Complex colour centres: laser action
The fabrication of lasers based upon colour centres adds a further dimension to the laserwavelengths available.
Ordinary F centres do not exhibit laser action, but F centres that have a dopant cation next to the anion vacancy
are satisfactory. These are typified by FLi centres, which consist of an F centre with a lithium ion neighbour
(Figure 10.14a). Crystals of KCl or RbCl doped with LiCl, containing FLi centres, have been found to be good
lasermaterials yielding emission lineswithwavelengths between 2.45 and 3.45 mm.Aunique property of these
crystals is that in the excited state an anion adjacent to the FLi centre moves into an interstitial position
(Figure 10.14b). This is type II laser behaviour, and the active centres are called FLi (II) centres.
These complex defects are introduced in a series of steps. Take KCl doped with Li as an example. Initially,
KCl crystals are grown from a solution containing LiCl as an impurity. The Liþ cations form substitutional
Mg2+
O2-
OH-
VO2• + e′
Figure 10.13 An F þs (OH�) centre on an MgO (100) surface (schematic)
Colour and the Optical Properties of Materials 434
LiK impurity defects distributed at random throughout the crystal. F centres are introduced by irradiation using
Xrays.These are not usually locatednext to adopantLiþ cation.Toconvert theFcentres intoFLi (II) centres the
crystal is subjected to aprocess calledaggregation. In this step, the crystals are cooled to about�10 �Cand then
exposed to white light. This releases the electrons trapped at the F centres, leaving ordinary anion vacancies,
which are then able to diffuse through the crystal before recombiningwith the electrons oncemore to reform the
F centre. Ultimately, each vacancy ends up next to an Liþ ion. At this position it is strongly trapped and further
diffusion is not possible. Recombination with an electron forms the FLi centre required. This process of
aggregation is permanent if the crystal is kept at �10 �C and in this state the crystal is laser active.
10.5.5 Photostimulable phosphors
Photostimulable phosphors are widely used in X-ray imaging, particularly by dentists, where it has largely
replacedX-ray film recording. In dentalX-ray imaging, a plate coveredwith a thin layer of a phosphor is placed
into themouth and exposed to X rays. The X rays generate electrons and holes that are trapped at defects in the
phosphor and do not recombine. This process is said to generate a latent image in the phosphor. Subsequent
irradiation of the plate with a light source of the correct wavelength gives the electrons or holes sufficient
e′
e′
Cl
K
Li
(a)
(b)
Figure 10.14 Schematic diagram of FLi colour centres in KCl: (a) ground-state FLi centre; (b) excited-state type IIFLi centre responsible for laser output
435 Colour in Metals, Semiconductors and Insulators
energy to escape the trapping defects, allowing them to recombine. This leads to light emission, usually via
energy transfer to an activator. The subsequent fluorescence is recorded as a digital image. The number of
trapped electron and holes and, therefore, the amount of fluorescent emission is proportional to the X-ray
intensity. The optical image thus records accurately the degree to which the X rays have penetrated
the subject.
A suitable phosphor must be very efficient at absorbing X rays, to lower any danger to the patient, and must
have a high luminous output when irradiated after the X-ray exposure. There should be no afterglow, which
would seriously degrade image resolution. In addition, the phosphor must be reusable. The first commercial
material to fulfil these requirements, introduced in 1983, was BaFBr doped with Eu2þ . Since that time many
other systems have been explored for use in X-ray imaging, especially other binary and ternary alkali halides
doped with Eu2þ as activator. At present (2010) the detailed mechanism by which these phosphors work is
not altogether clear. However, it is well established that an important component of the process is the formation
of F centres. These are produced as a result of X-ray irradiation and are similar to those in alkali halides
described above (Section 10.5.1), consisting of an anionvacancy together with a trapped electron. Thesemake
up the electron trapping centres. For the commercial phosphor BaFBr:Eu2þ , the radiation used to liberate
the electrons trapped at the F centres is usually from a helium neon laser at 633 nm. The electrons, promoted to
the conduction band, can then recombinewith holes in thevalence band. The energy is transferred to Eu2þ ions
which give out visible light at 420 nm (see Section 9.4).
This is not the only proposed mechanism of light emission. In some phosphors it has been suggested that
X-ray irradiation forms Eu3þ ions, which are equivalent to Eu2þ together with a trapped hole. Electrons
liberated by irradiation then recombinewith holes at an Eu3þ ion without involving energy transfer. The result
is blue emission from Eu2þ as before.
Considerable research is ongoing to unravel the mechanisms by which photostimulable phosphors produce
light and to produce new phosphors with greater resolution.
10.6 The Colours of Inorganic Semiconductors
10.6.1 Coloured semiconductors
In an (inorganic) insulator, the upper conduction energyband is completely emptyand the lower energyvalence
band is completely filled. As the band gap shrinks, a profound change comes over the colour (and electronic
properties) of the insulator,which gradually becomes an (inorganic) semiconductor. Intrinsic semiconductors5
have a similar band picture to insulators except that the separation of the empty and filled energy bands is small.
How small is small? The original definition of a semiconductor as a poor electrical conductor suggests that the
bandgapmust be such that someelectronshaveenoughenergy tobe transferred from the topof thevalenceband
to the bottom of the conduction band at room temperature. The band gap of silicon, one of the most important
intrinsic semiconductors, is approximately 1.1 eV, and this may be taken as representative for semiconductor
bandgaps.A remarkable propertyof intrinsic semiconductors is that each electron transferredwill leavebehind
a hole in the valence band. In an intrinsic semiconductor, both holes and electrons contribute equally to the
electrical conductivity. In the idealised band picture, both of these particles are able to move through the solid
in unhindered fashion, and so are often called free electrons or free holes.
5 Intrinsic semiconductors are pure materials with, ideally, no impurities. The majority of semiconductors used in devices are extrinsic
semiconductors, in which impurities (dopants) are deliberately added to confer specific electronic properties on the material. p type
semiconductors are doped so as to electrically conduct mainly by way of holes. n type semiconductors are doped so as to electrically
conduct mainly by way of electrons.
Colour and the Optical Properties of Materials 436
The colour of a pure semiconductor is, to a first approximation, governed by the band separation energy.
When the energy gap is relatively large, light photons are not energetic enough to excite an electron from the
valence band to the conduction band and so are not absorbed. The material will appear transparent to the eye.
(This is so for diamond, with a band gap of approximately 5.5 eV, and titanium oxynitride (TiON), with a band
gap of approximately 4.12 eV, although these compounds are generally regarded as insulators rather than
semiconductors.) On the other hand, if the energy gap is quite small, corresponding to the infrared region, the
semiconductor will absorb the entire visible spectrum and take on a black ormetallic appearance. Silicon, with
a band gap of approximately 1.1 eV, is typical of this group. In powder form it is black. Single crystals appear to
look metallic.
If the band gap energy falls in the visible, between approximately 1.77 and 3.10 eV, the semiconductor will
absorb all photons with energy greater than the band gap energy and not those with a smaller energy. This will
cause the material to be strongly coloured. For example, the pigment vermilion, which is produced from the
mineral cinnabar, mercuric sulfide (HgS), has a band gap of approximately 2.0 eV. This energy corresponds to
the red orange region of the spectrum. All shorter wavelengths than this are associated with more energetic
photons, and these will be absorbed. These are the yellows, greens and blues. The colour perceived will be
due to the photons with energy less than 2.0 eV, which are not absorbed. These are the reds and oranges
(Figure 10.15a). The pigment cadmium yellow, cadmium sulfide (CdS), has a band gap of 2.42 eV, which
corresponds to the green blue part of the visible. Photons of lower energy, red, orange, yellow and green, will
not be absorbed, while the higher energy blue, violet and indigo will be. The net result is that the pigment
appears yellow to the eye.
Almost all coloured sulfides have figured as artist’s pigments in one context or another in earlier centuries.
For example, a lesswidely used pigment these days is orpiment, arsenic trisulfide (As2S3). Themineral name is
a corruption of theLatin auri pigmentum, golden paint, and it is also known as the artist’s colourKing’s yellow.
It is readily prepared as a canary yellow precipitate when hydrogen sulfide gas is passed into solutions
containing As3þ ions. The pigment has fallen into disfavour because of its toxicity and tendency to give off
poisonous vapour when exposed to damp air.
Apart from sulfides, many other materials are brightly coloured in the same way. These include the
decorative hard coating materials titanium nitride (TiN), which is golden (often seen as gold-coloured hard
tips on drill bits), zirconium nitride (ZrN), which is yellow green, tantalum nitride (TaN), which is blue grey,
and titanium carbide (TiC) and tungsten carbide (WC), both of which are dark grey. Thesematerials, as well as
some metal sulfides show a similarity to metals, both visually and electronically. This can happen if a large
number of electrons are present in the conduction band. In this case the electronsmay take on properties similar
to those ofmetals (Section 10.15). Themostwidely known example of this similarity is found in the compound
pyrite, FeS2, also knownas fool’s gold (Figure 10.15b). The physical properties are not at allmetallic, however;
pyrite is brittle rather than malleable, like gold is. Conductivity is still by way of both electrons and holes,
whereas in a metal only electrons are important. Admixture of copper sulfide (CuS) with pyrite produces the
mineral chalcopyrite, with a nominal formula Cu2Fe2S4. This material also has a metallic appearance and
takes on a variety of golden or purplish hues, depending upon the exact composition, for the same reason
(Figure 10.15c).
As with the insulators described above, the band gap of semiconductors tends to decrease with temperature,
leading to thermochromism.
10.6.2 Transparent conducting oxides
The electrical conductivity of a semiconductor depends upon the number of holes and electrons present.
Doping is widely used to modify these populations and so alter the measurable conductivity. If this can be
achieved in a semiconductorwith a fairly largebandgap the conductivitymaybe appreciablewhile thematerial
437 Colour in Metals, Semiconductors and Insulators
Figure 10.15 The colours of semiconductors: (a) cinnabar, mercuric sulfide (HgS); (b) pyrite (FeS2), fool’s gold;(c) chalcopyrite, nominally Cu2Fe2S4
Colour and the Optical Properties of Materials 438
remains transparent. This is the situation in transparent conducting oxides (TCOs), sometimes referred to as
transparent metals, which are widely used as transparent conducting electrodes. The best known of these
materials is indium oxide (In2O3) doped with between 5 and 15mol% tin oxide (SnO2), known as indium tin
oxide or ITO.
Surprisingly, in view of the importance of this material, there are (2010) ongoing attempts to explain its
electronic properties. (Indeed, there is still disagreement about the true band gap of pure In2O3, which is
reported to vary from about 2.8 to 3.75 eV.) Irrespective of the true value, In2O3 is a lemon yellow colour and
has an absorption spectrumvery similar to that ofWO3 (Figure 10.1c). The absorption spectrum just creeps into
the visible at the blue end of the spectrum, giving a resultant yellow tone to the bulk solid. The transparent
electrode material ITO is also a pale yellow colour in bulk, but when prepared as a thin film it appears
transparent to the eye.
Incorporation of SnO2 into In2O3 leads to the formation of defects which in turn leads to the increase
in conductivity whilst retaining the large band gap that makes the oxide transparent. Whilst there is
still considerable uncertainty about the nature of these defects and how the doping influences the band
structure of the host In2O3, the following broad-brush picture describes the state of affairs that is believed
to occur.
Figure 10.15 (Continued)
439 Colour in Metals, Semiconductors and Insulators
The Sn4þ ions are considered to mainly occupy In3þ sites forming SnIn.point defects.6 The additional
oxygen is accommodated as interstitial O2 ions, O20i :
2SnO2 ðIn2O3Þ! 2SnIn. þO20
i $ ð2SnIn OiÞx
There is some evidence to suppose that the tin and oxygen interstitial defects may aggregate into a neutral
defect complex (2SnIn Oi)x rather than remain isolated. In either case, the number of interstitial oxygen
defects will vary with the ambient oxygen pressure during thin-film preparation according to the reversible
equation:
O20i $ 1
2O2 þ 2e0
This means that interstitial oxygen defects are preferred at higher ambient oxygen pressures, while electrons
are produced at lower pressures. This accounts for the fact that highly conducting oxide films are prepared
under reducing conditions; that is, at relatively low oxygen partial pressures, incorporation of SnO2 into In2O3
leads to the production of electrons. These electrons enter the conduction band to enhance the n-type
conductivity of the oxide. As the dopant concentration rises, the number of electrons in the oxide increases. At
a dopant concentration of about 2� 1019 cm 3 the electrons behave as free electrons, rather similar to those in
a metal. Semiconductors that are so heavily doped that the conductivity approaches that of a metal are called
degenerate semiconductors. The band gap, although varyingwith dopant concentration, remains wide enough
for the material to appear transparent in thin-film form.
A number of other n-type transparent oxide conductors have been found, including tin oxide (SnO2) doped
with F, zinc oxide (ZnO) doped with Al2O3, and a number of oxides with structures related to that of fluorite
(CaF2).
Unfortunately, a matching transparent p-type oxide conductor has not yet been found, although delafossite-
structure oxides CuM3þO2, including CuGaO2, CuInO2 and CuScO2, have potential in this respect. Such
a material is considered to be important because it would allow for highly desirable transparent electrodes at
each face of a light-emitting device (Sections 10.8 and 10.11).
10.7 The Colours of Semiconductor Alloys
Band gaps of semiconductors can be finely tuned by making solid solutions spanning the composition range
between two isostructural parent phases. This can be illustrated with respect to cadmium sulfide (CdS) and
the very similar cadmium selenide (CdSe). Both of these compounds adopt thewurtzite structure, one of the
forms of zinc sulfide (ZnS). CdS, with a band gap of 2.42 eV, absorbs high-energy photons from violet to
blue. CdSe has a smaller band gap of 1.74 eVand absorbs all the visible wavelengths. It appears black to the
6 The Kroger Vink point defect notation is used; see this chapter’s Further Reading, for details.
Colour and the Optical Properties of Materials 440
eye. The sulfur and selenium atoms in these two compounds are of a similar size, which allows one to replace
the other readily. If a solid solution is made with a general formula CdS1 xSex the band gap gradually
changes from that appropriate to CdS at x¼ 0 to that appropriate to CdSe at x¼ 1.0. At x¼ 0 the photons
absorbed are only those in the violet to blue region of the visible, but, as x increases, the range of absorbed
photons moves towards red and infrared. The colour perceived gradually changes from yellow at x¼ 0
to orange to red and ultimately to black as x increases. The material CdS0.25Se0.75 is the pigment
cadmium orange.
Most isostructural pairs of semiconductors can form solid solutions in the sameway. In these instances, the
band gap can be manipulated at will. Note, though, that the dependence of band gap upon composition is not
linear, but tends to follow a shallow curve. For example, the band gap of the important semiconductor system
gallium nitride indium nitride (GaN InN) is given by:
Eg ðalloyÞ ¼ xEg ðGaNÞþ ð1�xÞEg ðInNÞ�xð1�xÞb
whereEg represents the relevant band gap and b is called the bowing coefficient or bowing parameter. Inserting
experimental values for the bandgapofGaN (3.30 eV) and InN (0.61 eV) and abowing coefficient of 1.43 gives
the quadratic function (Figure 10.16):
Eg ðalloyÞ ¼ 0:61þ 1:26xþ 1:43x2
It is seen that this system spans the visible. In this context, the isostructural insulator aluminium nitride
(AlN) has a band gap of 6.1 eV. Alloys can be fabricated with GaN that take the emission into the ultraviolet.
The alloy range AlN GaN InN can, therefore, give an output anywhere between the deep ultraviolet and
the infrared.
Many of the semiconductors mentioned above also form alloys with varying colours. The titanium
carbonitride TiCxNy, with (x þ y)� 1, varies from gold to red. The closely related zirconium
carbonitride ZrCxNy takes on hues between silver, gold and violet, depending upon composition. A
change of band gap with temperature can lead to a change in the perceived colour of the phase, giving
rise to thermochromism.
10.8 Light Emitting Diodes
10.8.1 Direct and indirect band gaps
Electrons in the conduction band can gain energy by dropping back to thevalence band and recombiningwith a
hole. This energy is frequently released as a photon, and so semiconductors can act as lamps given a continuous
power input to maintain the supply of charged particles. Suchmaterials display electroluminescence, which is
light emission following an input of electrical energy. The colour of light emitted in this way is naturally
influenced by the band structure of thematerial. However, the band structuremust be considered inmore detail
than before to understand emission from semiconductors.
The real band structure of a solid can be envisaged as a series of undulating surfaces, resembling stacked
sheets, which define the allowed energy states accessible to electrons and holes, usually plotted as a graph of
441 Colour in Metals, Semiconductors and Insulators
energy versus the wave vector k of the electron defined as 2p/l, where l is the electron wavelength.7 Each
surface consists of a series of hills and valleys, not the flat bands used in the simple descriptions given above. If
the highest ‘peak’ in thevalence band corresponds to the lowest ‘valley’ in the conduction band, an electron can
absorb a photon and be promoted directly across the band gap, leaving a hole in the valence band. This
characterizes a direct band gapmaterial (Figure 10.17a). The reverse process is also possible: an electron in the
conduction band can emit a photon and directly recombinewith the hole in the valence band (Figure 10.17b). If
the peak in the valence band does not correspond to the lowest point of the conduction band that is, if the two
energy features are displacedwith respect to eachother an electron canonly bepromoted from the lower to the
upper band if it is given an increment of momentum k. This ‘sideways kick’ is equivalent to the addition of a
phonon (a quantum of lattice vibration) to the process. The electron must interact with both a photon and a
phonon simultaneously to jump the band gap (Figure 10.17c). This situation characterizes an indirect band gap
material. The reverse process, inwhich an electron gives up both a photon and a phonon so as to recombinewith
a hole in the valence band, is of low probability, and generally indirect band gapmaterials do not emit radiation
efficiently (Figure 10.17d). Energy is lost instead by internal energy conversion, i.e. nonradiative transitions.
The nature of the band gap, direct or indirect, is of vital importance when luminous efficiency is concerned.
Indirect band gap materials are generally very poor light emitters.
0 0.2 0.4 0.6 0.8 1.00
2
1
3
Ban
d ga
p / e
V
Fraction x in Gax In1-xN
Figure 10.16 The band gap of the alloy series GaN–InN as a function of the composition
7 As the momentum of the electron is equal to kh/2p, the k axis is often labelled as momentum.
Colour and the Optical Properties of Materials 442
10.8.2 Idealized diode structure
Tomake an electroluminescent light-emitting device it is necessary to pumpelectrons into the conduction band
and holes into the valence band so that recombination can occur continuously. The arrangement suited to this
is that of a semiconductor LED. These are formed by the juxtaposition of a region of n-type and p-type
semiconductor, grown into a single crystal. In the p-type region of the material the semiconductor has been
dopedwith acceptors so that the top of thevalence band contains a high population ofmobile holes. It is, in fact,
a hole conductor. In the n-type region the semiconductor has been doped with donors, so that the material
contains a population of mobile electrons at the bottom of the conduction band. This region is an electron
conductor. When a p-type region abuts an n-type region, electrons move into the p-type region from the n-
type side and holes move into the n-type region from the p-type region, by diffusion. Most of the displaced
electrons and holes recombine and so are eliminated. However, as electrons leave the n-type region,
positively charged donor atoms are left behind, while negatively charged acceptor atoms are left in the
+k
+k
+k
+k
-k
-k
-k
-k
0
0
0
0
valenceband
valenceband
conductionband
conductionband
directtransition
indirecttransition
hν
photon
hνEg
Eg
phonon
(a) (b)
(c) (d)
Figure 10.17 Direct and indirect band gap materials. (a), (b) A direct band gap material can absorb and emitphotons equally efficiently. (c) An indirect band gap material requires that a photon and a phonon combine topromote an electron. (d) The reverse process is of low probability and indirect materials do not make satisfactorylight emitters
443 Colour in Metals, Semiconductors and Insulators
p-type region as holes leave. This net imbalance in the charges present is called the space charge. The result
of the changes is to create an electric potential, the contact potential, or built-in potential of about 0.3 V. At
equilibrium, the energy bands have been shifted give a distorted band structure in the junction region
(Figure 10.18a).
The transition region, which is also called the depletion region or active region, has awidth of about 1 mmin an ordinary diode. At equilibrium (thermal and electrical) there will still be an exchange of carriers at the
junction, but the current in each directionwill be the same.Dynamic equilibrium holds. This changeswhen a
voltage is applied across the junction. An applied voltage, which will drop across the transition region,
because of the absence ofmobile charge carriers, can be appliedwith the positive side connected either to the
p-type region or to the n-type region. The arrangement in which the positive voltage is connected to the
p-type region is called forward bias. This causes the potential barrier to be reduced. Under a forward bias
there is a rapid increase in the current flowing across the junction. Electrons and holes now enter the junction
conductionband
conductionband
valence band
valence band
p-type p-type
n-type n-type
junction region
++++++
(a) (b)
(c)
electrons electrons
holes holes
photon
photon
+
current
current
_
photon
p -type
n-type
junctionregion
Figure 10.18 A semiconductor LED (schematic): (a) equilibrium situation; (b) under a forward bias, light isemitted in the junction region; (c) schematic device construction of a homojunction LED
Colour and the Optical Properties of Materials 444
and recombine. The energy released, which is approximately equal to the band gap, can appear as light
(Figure 10.18b). The number of electrons and holes in the device and entering the active region is continually
replenished by the external power supply.
The size of LEDs is remarkably small and often compared with a pencil point or a grain of salt, although
dimensions much less than this can easily be realized. The simplest (conceptual) device configuration is just
a planar slab, made of the same material, one part doped to be n-type and one part to be p-type, to form a
homojunction LED (Figure 10.18c).
10.8.3 High-brightness LEDs
Of course, this simplified account does not do justice to the immense amount of hardwork needed to produce
the efficient LEDs now available at little cost. The initial studies were made on gallium arsenide in the early
1960s. This semiconductor is a direct band gap material with a band gap of 1.42 eV, giving out infrared
wavelengths. The first challenge was to convert this to visible. The material gallium phosphide (GaP) was a
suitable contender, with a band gap of 2.26 eV, but this compound is unfortunately an indirect band gap
material. However, combining GaAs with small amounts of GaP resulted in a direct band gap alloy, and
compositions close to GaAs0.6P0.4 were used to obtain successful light emission in the red region of
the spectrum.
The first red LEDswere commercially available in 1962, although theywere not very efficient and certainly
did not have a good brightness. They made use of a single semiconductor doped either p- or n-type, and are
homojunctiondevices.Unfortunately, the bandgapof theGaAs GaPalloys changes fromdirect to indirect part
way across the composition range. This means that not all of the composition range can be utilized, so that the
only colours available were towards the yellow orange red end of the spectrum.
The brightness of LEDs has been improved dramatically since then. This has been due to a number of
major advances. First, crystal defects, in particular dislocations running through the active layer, have been
greatly reduced. These defects provide sites at which electrons and holes can combine nonradiatively, hence
lowering device brightness. Second, more complex alloy systems have been developed to give a broader
spectral range. Thus, red, yellow and orange emitters rely on quaternary alloys of gallium arsenide (GaAs),
aluminium phosphide (AlP), gallium phosphide (GaP) and indium phosphide (InP) with typical composition
(AlxGa1 x)0.5In0.5P and a direct band gap between 1.9 and 2.26 eV, as a function of the Al content. The light-
emitting region, has been confined between semiconducting slabs with different overall composition and of
lower refractive index in heterojunction devices, which channel the light more effectively using total internal
reflection (Figure 10.19a). The active layer has also been fabricated into single or multiple quantum-well
configurations (see Section 10.9) approximately 2 mm thick. This confines the holes and electrons to a narrow
spatial region, increasing the likelihood of recombination and again increasing brightness. Reflecting
layers, either multiple thin-filmmirrors or simple reflecting cups in which the chip is mounted, also improve
apparent brightness. In 1993 the reds and yellows of the GaAs GaP system were supplemented with alloys
in the GaN InN system, which are direct band gap alloys across the whole of the composition range and are
able to provide blue and green light. (Note that, by including direct band gapAlN in the system, alloys can be
fabricated that are able to give an output anywhere between the deep ultraviolet and the infrared.)
Mechanical design changes are also important. In a planar material, light can only escape if it meets a
face at less than the critical angle given by Snel’s law (Section 2.2). In 1998, the design of the light-emitting
chip was changed to a truncated pyramid, with sides at 35� to the vertical, so as to optimize the escape of
photons (Figure 10.19b).
445 Colour in Metals, Semiconductors and Insulators
10.8.4 Impurity doping in LEDs
The colours produced in diamond by impurities suggest that it might be possible to place impurities into the
band gap of LEDs andmanipulate the system so that new colours are emitted. It has been found that this can be
done successfully with lanthanoid elements introduced into silicon, such as silicon doped with erbium (Si:Er).
However, the most successful colour-producing devices that have been fabricated contain lanthanoids
introduced into gallium nitride (GaN).
The advantage of using lanthanoids is that the important 4f energy levels of the atoms are shielded from
the influence of the surrounding host structure by filled 5s2 and 5p6 orbitals. The energy levels remain narrow
and the colours produced only spread over a very narrow range of wavelengths. Although light emission
from these dopants can be obtained via photoluminescence (Chapter 9), for LED use it is more desirable to
populate the upper levels by applying an electrical potential to the semiconductor. Electrons can then
continuously fill the upper energy levels of the lanthanoid dopant and light will be emitted as these excited
atoms return to the ground state. Electroluminescent devices GaN:Pr and GaN:Eu yield red output, GaN:Er
produces green and GaN:Tm violet (Figure 10.20). It is clear that a combination of these three devices
(sometimes referred to as light-emitting devices or, confusingly, LEDs) could be employed for flat-
screen displays.
The energy levels populated can be controlled by variation of the electrical input to the galliumnitride diode.
Thus, it has been demonstrated that GaN:Er can also emit in the infrared at 1538 nm. This is a very convenient
output, because it matches both theminimum attenuation of silica optical fibres (Section 2.9) and an important
energy region of Er-doped optical-fibre amplifiers (Section 7.17).
10.8.5 LED displays and white light generation
LEDs generate light of specific colours. Impressive full-colour displays using LEDs have been built using
millions of small red, green and blue units. From this point of view, using three LED colours to give the
electrode
electrode
n-Al0.8Ga0.2As layer, ~30 μm
active layer, p-Al0.35Ga0 65As, ~2 μm
p-Al0.75Ga0 25As layer, ~100 μm
reflecting substrate(a)
(b)
n-GaP ~ 200 μm
active layer, AlGaInP ~2 μmp-GaP ~ 55 μm
35°
Figure 10.19 LED structure: (a) typical planar heterojunction LED; (b) truncated prism design LED, cut at anangle of 35� to optimize brightness
Colour and the Optical Properties of Materials 446
impression of white light, by virtue of additive colour mixing (Section 1.10) does not give a particularly good
colour rendition, and four colours, red, yellow, green and blue are more satisfactory. The displays have no size
limitations and the intensity produced by each LED is also sufficient to make them easily visible in daylight.
There are, though, a number of limitations. The efficiency of the LEDs varies; in particular, green emitters are
less efficient than red or blue. Thismeans that the numbers ofLEDs selected for the displaymust take efficiency
into account.Moreover, efficiencies varywith the age of the LED, so this againmust be corrected if the display
is to retain high brightness and good colour rendition over time.
Formany purposeswhite light is essential. The simplest way tomake awhite light is to combine appropriate
LEDs, which, whenviewed from a distance, appear white byway of additive coloration. However, while this is
satisfactory for displays, it is not satisfactory for most lighting purposes. For this, the commonest way tomake
white light is to coat amonochromatic LEDwith a phosphor. It is these devices that are normally termed ‘white
LEDs’. The commonestwhite LEDconfiguration is to use a high-intensity (Ga,In)Nblue-emittingLEDcoated
with a yellow-emitting phosphor. At present, this is most often yttrium aluminium garnet (Y3Al5O12) doped
with Ce3þ , which gives a rather broad yellow emission. The coating is sufficiently thin to allow a certain
amount of blue light to be transmitted. This, together with the yellow luminescence, creates a cool bluishwhite
colour. These white LEDs are found in many applications, including cycle lamps, flash lamps, traffic lights,
headlamps, tunnel illumination and so on. The rather cold light is not entirely suitable for indoor lighting,
which ideally needs to be warmer. For this, two phosphors are used, a red and a green emitter in tandem with
a (Ga,In)N blue-emitting LED. The orange red phosphor is the nitrosilicide Sr2Si5N8 doped with Eu2þ .The red orange tone can be adjusted by replacing some of the Sr by Ca. The green phosphor used is often
650 nm
3P0
1D2
1G4
3F3
7F3
5D1
5D0
2H11/2
4S3/2
4I11/2
1G4
3H4
3H5
3H4
3H6
4I13/2
4I15/2
7F27F1
3F2
3H4
3H63H5
543 nm
600 nm
621 nm
663 nm
GaN band edge
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5E
nerg
y / e
V
Pr3+ Eu3+ Er3+
537 nm558 nm
Tm3+
477 nm647 nm
1538 nm
Figure 10.20 Energy levels of some lanthanoids in GaN: (a) Pr3þ , (b) Eu3þ ; (c) Er3þ , (d) Tm3þ
447 Colour in Metals, Semiconductors and Insulators
SrSi2O2N2 dopedwith Eu2þ . Once again, colour tuning is possible by substitution of the Sr byCa. Blue light is
contributed by the LED itself.
There is considerable work in this area, and new phosphors are continually being tested. Moreover,
phosphors for usewith ultraviolet-generating diodes are also available. The improvements inwhite LED lamps
are certain to be considerable in the near future.
10.9 Semiconductor Diode Lasers
If a forward bias voltage is applied to a suitable p n junction it will function as an LED. In principle, it is very
easy to turn anLED into a semiconductor laser, a diode laser. This is because (at least in principle), a population
inversion is achieved by using a heavily doped n-type region with respect to the p-type region. The population
inversion is generated in the following way. When a low forward current is applied to the diode, a small and
roughly equal number of holes and electrons enter the junction region, recombine and emit light. This is normal
LEDbehaviour and produces a radiancemore or less in proportion to themagnitude of the current. However, at
some critical threshold voltage, far more electrons start to enter the junction region than holes. This is because
the hole transport into the junction region is at a plateau, the height of which is controlled by the relatively low
doping level. However, the electron transport into the junction region can continue to rise because the n-type
segment is heavily doped and has a far higher plateau. At the point when the electron numbers outweigh the
number of holes, a population inversion occurs and the LED becomes a laser (Figure 10.21a). The light is
emitted from the active region as in an LED. Stimulated emission is achieved by using carefully polished
crystals so that any photon emitted will be reflected to and fro in the junction to promote the laser avalanche.
The change from LED to laser operation is marked by a large increase in both output and efficiency as the
current passes a point termed the threshold current (Figure 10.21b). Semiconductor lasers are small, being of
comparable size to ordinary LEDs.
Diode lasers were developed in tandemwith LEDs, and it is not possible to disentangle the evolution of one
device from the other. Initial diode lasers were made from materials related to gallium arsenide (GaAs) or
indium phosphide (InP). The output wavelengths fall into the ranges of approximately 630 980 nm for
GaAs-derived systems and 1300 1550 nm for InP-derived systems. An early problem was the construction of
a resonant cavity to ensure laser output rather than LED output. This was solved by polishing the ends of the
semiconductor crystals making up the LED/laser. These first homojunction devices were not very efficient and
operated best at liquid-nitrogen temperatures. Since then a large number of device structures have been
explored, including the first device architecture that led to successful room-temperature operation, in which a
double heterojunction constructionwas employed.This consists of a series of layers of differentmaterials, such
as n- and p-GaAlAs alloys of wide band gap surrounding an active layer of narrow band gap alloy. The buried
heterostructure design, in which the active region is confined to a narrow strip, has been successfully used
formany common laser devices. Photons emitted in the active layer are confined to this region due to refractive
index differences between the surrounding alloys. This confinement enhances stimulated emission and gives a
more collimated beam.
Semiconductor laser diodes are ubiquitous. Perhaps the widest distribution is in barcode readers, found in
every store. In the home, lasers are used for the recording and playing of CDs (785 nm GaAlAs red lasers),
DVDs (650 nm GaAlAs lasers), and Blu-ray and HD-DVD discs (405 nm InGaN lasers) (Section 3.1). Laser
measuring equipment is available for ordinary tasks such as room dimensionmeasurement. Laser pointers are
commonplace (670 nmGaAlAs or 650 nmAlGaInP) and are a good source of laser light for home experiments
(e.g. see Section 6.7). Fibre-optic communications (Section 2.9) use semiconductor laser light to carry
information. The list could be continued!
Colour and the Optical Properties of Materials 448
10.10 Semiconductor Nanostructures
10.10.1 Nanostructures
Nanostructures are structures of a dimension that endows a solid with properties that are noticeably different
from those of bulkmaterial. The dimension atwhich this transformation becomes apparent depends upon the
phenomenon investigated. In the case of thermal effects, the boundary occurs at approximately the value of
thermal energy kBT, which is about 4� 10 21 J. In the case of optical effects, nonclassical (i.e. diffraction)
behaviour is noted when the scale of the object illuminated is of the same size as a light wave, say
about 5� 10 7m. For particles such as electrons, the scale is determined by the Heisenberg uncertainty
principle, about 3� 10 9m. This is illustrated by the band gap of ZnO crystals, which starts to change from
that of bulk material as the particle size approaches about 6 nm (Figure 10.2). In this section, the optical
consequences of semiconductor nanostructures will be outlined. In these structures it is the electrons
and holes that are under consideration and the length scales of importance are accordingly of the order
of nanometres.
The overall consequences of limiting the dimensions of a material can be understood in terms of outer
electron interactions. The electrons on isolated atoms are associated with sharp energy levels located at the
atom in question. The outer (valence) electrons on atoms in a molecule are delocalised over the molecule in
LED region
laser region
thresholdcurrent
Current
Ligh
t out
put
+
current
current
_
photonsphotons
p-type
n-type
polished facepolished face
(a)
(b)
Figure 10.21 Semiconductor diode lasers. (a) Schematic device construction of a homojunction laser.(b) A current above a threshold value (ideally) converts an LED into a laser
449 Colour in Metals, Semiconductors and Insulators
molecular orbitals, but the energy of these remainsmostly sharp. The outer electrons associatedwith an atom in
a solid are spread over all of the atoms in thematerial and are associatedwith a transformation of narrowenergy
levels into energybands.Asnoted earlier, in the reverse situation, as a solid is imagined to fragment into smaller
and smaller units, the energy levels must change from typically bulk-like bands to more molecular and then
atom-like sharp levels.
Because of this, the manner in which the dimensions of a solid are constrained will have a major effect upon
the resultant properties of the body.A thin layer of amaterial will have bulk properties modified towards atom-
like properties in adirection normal to the layers.A thin layer of a semiconductor sandwiched between layers of
a different semiconductor, called a quantum well, will show this behaviour (Figure 10.22a). Semiconductor
electronic devices can increase the effect by stacking up several alternating thin layers to form multiple
quantumwell (MQW) structures (Figure 10.22b).Carbonnanotubes andnanorods of other compounds that are
small on an atomic scale in two directions are known as quantumwires or nanowires (Figure 10.22c). A cluster
of atoms, called a quantum dot, has properties approaching that of the isolated atoms. Electrons are distributed
between energy levels that resemble atomic or molecular orbitals (Figure 10.22d).
These structures have unique electronic and optical properties because of the way in which electrons are
localised, or confined. An electron or hole bonding energy much greater than thermal energy characterises
strongly confined charge carriers.
(a) (b)
(c) (d)
~10 nm
Figure 10.22 Semiconductor nanostructures: (a) a single quantumwell (SQW); (b) a series of quantumwells – amultiple quantum well (MQW) structure; (c) a quantum wire; (d) a quantum dot
Colour and the Optical Properties of Materials 450
10.10.2 Quantum wells
A single quantum well (SQW) is constructed by laying down a thin layer of a semiconductor with a
smaller band gap within a semiconductor with a larger band gap. In this structure, the electrons and holes
are essentially confined to the two-dimensional plane of the thin layers by the difference in the band
structures of the two materials. The electrons are trapped at the bottom of the conduction band and the
holes at the top of the valence band in the well. In an MQW structure, the electrons and holes are similarly
trapped in the low band gap layers. For example, quantum well structures formed from a layer of gallium
arsenide (GaAs) sandwiched in gallium aluminium arsenide (GaAlAs) trap electrons in the conduction
band ‘valleys’ and trap holes in the valence band ‘hills’ located in the GaAs layers (Eg� 1.42 eV) between
the GaAlAs layers, with a band gap of approximately 1.75 eV, between �1.42 (GaAs) and �2.16 eV
(AlAs) (Figure 10.23).
The energy of an electron in a quantumwell will bemore or less the same as the energy in the bulk for the two
directions in the plane of the confining sheet. In a direction normal to the sheet, the narrow dimension of the
layer, a very rough estimate of the allowed energy levels available to an electron can be calculated by assuming
that it is free and trapped by an infinite boundary potential. The electron, regarded as a wave, can only fit into
the volume if thewave has a node at each boundary (Figure 10.24). In this case, the energy E of a free electron
in a rectangular parallelepiped with edges a, b and c is given by
Eðnx; ny; nzÞ ¼ h2
8me
n2xa2
þ n2y
b2þ n2z
c2
!ð10:1Þ
where h is Planck’s constant,me is the mass of the electron and nx, ny and nz are the quantum numbers along
the three axes. Exactly the same equation will apply to a free electron confined to a slab ofmaterial, although it
is better to replace the electron mass with the effective mass me*. In the case of a quantum well, the electron is
confined in one dimension, say x, and unconfined in two directions, which can be taken as y and z, so it is
convenient to rewrite Equation 10.1 as
Eðnx; ny; nzÞ ¼ h2
8me*
n2xa2
� �þ h2
8me*
n2y
b2þ n2z
c2
!" #ð10:2Þ
The values of b and c can be taken as about 1 cm, while the value of a is about 10 8m. The energy,
therefore, is dominated by the first term in Equation 10.2. This introduces a new set of energy levels,
associated with electron waves trapped in the well (Figure 10.25a). The electron energy level in the
lowest, n¼ 1, state is raised by h2=ð8me*a2Þ compared with the base of the well. These energy levels are
called electron subbands, and when the energy levels trap the electron strongly the electrons are said to
be strongly confined.
Exactly the same equations apply to holes, when the effective massmh* replacesme
*. The energy levels that
arise from trapped holes are called hole subbands.
In a quantum well, the electrons and holes occupy these energy levels. The electrons in the upper energy
levels can drop to the lower hole levels and emit photons (Figure 10.25b). The energy separation of these levels
is greater than that of the bulk conduction band valence band energy gap Eg; hence, the photons will be of
higher energy, or shorter wavelength, than the bulk. The emission is said to be blue shifted compared with the
bulk, and the transitions are called interband transitions.
451 Colour in Metals, Semiconductors and Insulators
The photon energy derived from an interband transition is:
EðphotonÞ ¼ hn ¼ Eg þEelectron þEhole
¼ Eg þ h2
8a2n2
me*þ n2
mh*
� � ð10:3Þ
CB
VB
GaAlAs GaAs GaAlAs
holes
electrons
Eg GaAlAs Eg GaAs
(a)
(b)
(c)
(d)
CB
VBGaAlAs GaAlAs GaAlAs GaAlAs
GaAlAs
GaAs GaAs GaAs GaAs
GaAs GaAs GaAs GaAs
GaAlAs
GaAlAs GaAlAs GaAlAs GaAlAs
Eg GaAlAs Eg GaAs
Figure 10.23 Quantum wells (schematic). (a) A single quantum well (SQW) of gallium arsenide (GaAs) in agallium aluminium arsenide (GaAlAs) alloy. (b) Schematic energy band sequence. (c) A multiple quantum well(MQW) structure formed from the same materials. (d) Schematic energy band structure of (c). CB, conductionband; VB, valence band; Eg band gap
Colour and the Optical Properties of Materials 452
where h is Planck’s constant, n the frequency of the radiation emitted, Eg is the band gap of the bulk well
material, a is the dimension of the quantum well, me* is the electron effective mass and mh
* the hole effective
mass. In the approximation that the effective mass of the electron and the hole are identical and equal to m�:
EðphotonÞ ¼ hn ¼ Eg þ n2h2
4a2m� ð10:4Þ
CB
CB
VB
VB
SQW
n = 2
electron energy levels
hole energy levels
n = 1
n = 1
n =1 n = 2n =1
(a)
(b)
Figure 10.25 Energy levels in a single quantumwell (schematic): (a) electron (upper) and hole (lower) subbands;(b) interband transitions
n = 3
n = 2
n = 1
Ene
rgy
a/2-a/2 0
Figure 10.24 The first three energy levels of an electron (or hole) trapped in a quantum well correspond to thethree longest wavelength waves with nodes at the well boundaries
453 Colour in Metals, Semiconductors and Insulators
The selection rule for the transition isDn¼ 0; that is, transitions can only take place between levels with the
same quantum number. (As with all selection rules, these are never perfectly obeyed, and transitions between
levels with differing n values do occur infrequently, giving rise to weak lines in the emission spectrum.)
Electrons can also be excited from one electron level, say n¼ 1, to another electron level, say n¼ 2, both
levels lying in the electron subband. Holes can make similar transitions between levels in the hole subband.
These transitions, which give rise to extra peaks in the emission spectrum, are known as intersubband
transitions.
Because the dimensions of the quantum well can be varied, the emission spectrum can be varied or tuned.
This feature, in both quantum wells and in quantum wires and dots, discussed below, is called bandgap
engineering.
Quantumwell structures arewidely used in LEDs and laser diodes to improve device performance. They do
this in a number of ways: by confining electrons and holes into a limited space, so that recombination is more
likely, and by guiding the photons emitted by virtue of the differing refractive indices of the materials. Typical
of these device structures is the SQW structure used in the first green-emitting LEDs (Figure 10.26). A change
in the composition of the SQW active layer allows the colour emission to vary between 450 nm (blue) and
600 nm (yellow).
10.10.3 Quantum wires and quantum dots
The above considerations can be applied equally well to confinement in two or three dimensions, to give
quantumwires andquantumdots.For a quantumwirewith restricted dimensions alonga andb, the free electron
confined in an infinite potential well will have energy levels given by:
Eðnx; ny; nzÞ ¼ h2
8me*
n2xa2
� �þ h2
8me*
n2y
b2
!þ h2
8me*
n2zc2
� �ð10:5Þ
where a and b are small and c is large. An analogous equation for holes, with effective mass mh* can also be
written. The case of a quantum dot, Equation 10.5, is retained, but the third dimension, c, is also small. For
a roughly spherical quantum dot of radius r, Equation 10.3 then becomes:
EðphotonÞ ¼ hn ¼ Eg þ h2
8r2n2
me*þ n2
mh*
� �ð10:6Þ
p-electrode
n-electrode
0.5 μm p-GaN:Mg
100 nm p-Al0.2Ga0.8N:Mg barrier layer
3 nm undoped In0.45Ga0 55N SQW active layer
4 μm n-GaN:Si
Figure 10.26 Schematic diagram of the construction of a green-emitting LED containing a single quantum well(SQW) active layer
Colour and the Optical Properties of Materials 454
This indicates that the energy change is proportional to 1/r2. (Note, however, that these equations give only a
rough estimate of the energy levels of the confined particles. The rigorous calculation of the energy levels
available to electrons and holes in quantum dots can be assessed using quantummechanical routines that take
into account not only the wave functions of the constituent atoms, but also the strain in the structure and the
considerable surface effects that are present.)
Quantum wires are difficult to construct and generally require sophisticated equipment. Quantum dots,
however, are fairly readily prepared.Conventional semiconductor techniques can be used to grow small islands
the dot on the surface of a semiconductor crystal. Isolated quantumdots are synonymouswith nanoparticles
and, as such, can be precipitated in glasses or solutions, or prepared as colloids.
The quantumdots that havebeen the subject ofmost studyare of the compounds cadmiumsulfide (CdS), zinc
sulfide (ZnS), cadmium selenide (CdSe) and zinc selenide (ZnSe). These emit fluorescent light that is a precise
function of the dimensions of the quantum dot. For example, CdSe quantum dots of radius 2.9 nm emit at
approximately 555 nm, those of radius 3.4 nm emit at approximately 580 nm and those of 4.7 nm radius emit at
approximately 625 nm.
The colour variation comes about because the band-like properties of the bulk semiconductor are
transformed into a closely lying set of discrete energy levels as the dimensions of the particle approach
the atomic scale, as described above. The higher group of states, derived from the (nominally empty)
conduction band, are, in principle, antibonding or nonbonding orbitals. The lower group of states derived from
the (nominally filled) valence band and are, in principle, bonding orbitals. Moreover, the energy gap between
the highest orbital in thevalence band group, equivalent to aHOMO, drops in energywhile the lowest orbital of
the conduction band group, equivalent to aLUMO, increases in energy, so that the effective bandgap appears to
increase steadily as the dot size falls (Figure 10.27a). The photoluminescence,which is relatively pure in colour
as the emission spectra are narrow, comes about in the followingway.Electrons are excited from the lower set of
orbitals to the upper set with ultraviolet radiation, as in the case of ordinary inorganic phosphors (Chapter 9).
These excited states subsequently lose energy by nonradiative transitions to end in the lowest orbital of the
upper set. Energy is then released as a photon as the electron drops to the topmost orbital of the lower set
(Figure 10.27b).
Semiconductor nanoparticles (Figure 10.28a) can now be produced with a definite size and narrow size
distribution. The relationship between size and band gap allows the photoluminescent colour to be controlled
precisely (Figure 10.28b and c). The colour of the photoluminescencewill varywith the chemical nature of the
nanoparticles, as well as the size of course, so that a wide range of colour tuning is possible, even between the
four semiconductors listed above.
There are many potential applications for photoluminescent quantum dots, because they constitute minute
verybright lamps that canbe activated atwill by anultraviolet or blue light probe.Moreover, the colour output is
pure in thesensethat theemissionspectrumisnarrow.Theyaremuchbrighter thanthefluorescentdyesdescribed
earlier (Section 9.7) and are less easily degraded under normal conditions than dyemolecules are. Applications
include the biological imaging of processes in living cells, production of quantum dot lasers and white LEDs.
These latter devices are made in a similar fashion to that described above, in which a phosphor is coated onto
the surface of a blue LED, typically a GaInN device with an output of approximately 460 nm (Section 10.8.4).
A layer of CdSe nanoparticles that emit at green, yellow or red wavelengths replaces the phosphor coating.
The simple quantum dots described above have a number of shortcomings. The relatively large surface area
of the dots reduces the light-generating efficiency considerably. This is in part due to the fact that many of the
bonds on the surface atoms are not complete. These dangling bonds serve to trap electrons and holes so that the
excited dot loses energy other than by emission of photons. The surface can thus be considered as a defect-rich
region that interferes with themechanism of luminescence. For biological imaging of processes in living cells,
not only is the luminous efficiency important, but also the quantum dots must be treated so that they are water
soluble; those above are not soluble.
455 Colour in Metals, Semiconductors and Insulators
The commonest approach to overcoming both of these difficulties is to coat the quantum dot with a thin
covering of another material to make core shell structures (Figure 10.29a). For example, a CdSe nanodot core
surrounded by a thin shell of ZnS has modified light-emitting properties, while a CdS dot coated with silica
(SiO2) or organic surfactants make quantum dots water soluble and less toxic.
A curious feature of core shell quantum dots is that they sometimes cease to emit light for a period before
returning to normal. This behaviour, termed blinking, is a feature of other fluorescent systems, including
fluorescent dyes andGFP. In quantumdots, the cause is believed to be due to the shell acting as an electron trap.
When the core of the dot is excited by ultraviolet radiation, an electron hole pair is generated. It appears that if
the electron is able tomigrate into the shell and become trapped at a surface site, photon emission is suppressed.
In effect, the core becomes positively charged; and although the core can still absorb photons, internal energy
loss takes over and the excess is lost as heat (Figure 10.29b and c). Eventually, the trapped electron reunites
valence band
conduction band
(a)
(b)
band gap Eg
antibonding orbitals
bonding orbitals
ultravioletexcitation
redemission green
emissionblue
emission
~6 nm ~4 nm ~2 nm
Figure 10.27 Quantum dot colours (schematic). (a) The change in the band gap and band structure of quantumdots as the size falls; (b) Fluorescence colours of different-sized CdSe dots (schematic). Nonradiative transitionsshown dotted
Colour and the Optical Properties of Materials 456
with the core, after which normal photoluminescence is restored. This on off sequence is blinking. Further
studies of this phenomenon are currently in progress.
Besides roughly spherical dots, many other dot geometries are being created, including rods, dipods,
tetrapods and so-called flowers. All are being tested for applications in medicine and biology, photovoltaics,
optical computing and other areas.
10.11 Organic Semiconductors and Electroluminescence
10.11.1 Molecular electroluminescence
Organicmolecules are generally insulators, with electrons occupyingmolecular orbitals confined to the spatial
region of the molecule itself. The unoccupied, higher energy orbitals are generally antibonding orbitals, and
the filled orbitals are generally bonding orbitals, although, as pointed out earlier, some orbitals are regarded as
neutral with respect to bonding and are called nonbonding orbitals. The lowest unoccupiedmolecular orbital is
given the acronym LUMO and the highest energy occupied orbital the acronym HOMO.
Organic electroluminescence the generation of light by passing an electric current through an organic
solid therefore seems unlikely. However, this is the basis of operation of organic light emitting diodes
300 400 500 600 700 800 900
Pho
tolu
min
esce
nt e
mis
sion
Wavelength / nm(a) (b)
(c)
r ~2 nm ~4 nm ~6 nm
Figure 10.28 Cadmium sulfide quantum dots: (a) a CdS dot approximately 8 nm diameter; (b) the photo-luminescent colours emitted by CdS quantum dots, schematic; (c) Photoluminescence of two different sizedquantum dots (left and centre) and a mixture of the two, giving an approximately white output. [(a) and (c)reproduced with permission from Dr. R. D. Tilley, Victoria University of Wellington, New Zealand]
457 Colour in Metals, Semiconductors and Insulators
(OLEDs) that use organic molecules as the active medium. In principle, electroluminescence comes about in
the followingway.Under the influence of an applied voltage, electrons are forced into the antibonding orbitals,
notionally the LUMO, and electrons are pulled out of the bonding orbitals, notionally the HOMO,which is the
same as saying that holes are forced into the HOMO. The electrons and holes recombine and release energy in
the form of visible light (Figure 10.30a).
The molecules that can be used for this are exactly the same dye molecules that are of use in photo-
luminescence (Section 9.11). That is, they contain conjugated double bonds that overlap to give delocalized pand p� orbitals (Chapter 8). The electrons do not flow through these molecular orbitals in the same way that
electrons flow through the band structure of an inorganic semiconductor, but can be imagined to jump or ‘hop’
fromone atom to another via the bond structure of themolecule. Eachmobile electron causes a small distortion
of the surrounding atoms and associated bonds, and the distortion has to be dragged along as the electron hops.
This combination of electron plus distortion is called a polaron. Polaronmovement requires more energy than
the equivalent electron transport in an inorganic semiconductor, and a fairly high voltage must generally be
used to obtain electroluminescence in such a system.
It has been found that the electrons and holes injected into the organic light-emitting medium do not simply
annihilate each other directly, but initially interact to form excitons (Section 10.2). In an exciton the spin of the
electron in the HOMO may be antiparallel to the spin of the electron in the LUMO, resulting in singlet state.
However, many more of the molecules end up in a triplet state, in which the spin of the electron in the HOMO
is parallel to the spin of the electron in the LUMO (Figure 10.30b). Both states may lose energy internally or
photonin
(a)
(b)
(c)
photonin
photonin
photonout
no emission
no emission
+
+
+
-
-
-
Figure 10.29 Blinking: (a) under normal conditions core–shell nanodots emit photons efficiently; (b) sometimesan electron is trapped on the surface of the shell, preventing emission of a photon; (c) as long as the electron istrapped, photons are absorbed but no light is emitted
Colour and the Optical Properties of Materials 458
emit photons. The singlet excited state ismore likely to fluoresce and give out electroluminescence. The triplet
state has amuch longer lifetime, and ismore likely to degrade by losing energy internally as heat, butmay give
out light via phosphorescence (long-lifetime fluorescence) (Figure 10.31). Thus, unless steps are taken to alter
this situation, the light output of the system will be fairly low.
10.11.2 Organic light emitting diodes
The transformation of the preceding principles intoworking devices took place in the period from about 1990.
A typical early construction used a thin polymer film of the conducting material poly(2-methoxy-5,20-ethyl-hexloxy)-1,4-phenylene vinylene, abbreviated toMEH-PPV (Figure 10.32a). This has an emission maximum
close to 625 nm in the orange red portion of the spectrum. In order to introduce electrons into the film,
a conducting anode and cathode are needed.The energy levels of these two electrodesmustmatch theLUMOat
the cathode side and theHOMOat the anode side. The cathodematerial is calcium.Although this is not an ideal
cathodebecausecalcium ishighly reactive, the energy levelof the electrons in thismetal, theFermi level, is very
(electrons into LUMO)
(electrons out = holes into HOMO)
-
+
photonout
(a)
(b)
ground state
excited state
singlet, S = 0
triplet, S = 1
LUMO
LUMO
LUMO
HOMO
HOMO
HOMO
voltage
organic molecular solid
Figure 10.30 Organic electroluminescence. (a) Principle of an organic electroluminescent device; an appliedvoltage introduces electrons and holes into the material which recombine and emit light. (b) The applied voltagecan lead to either a singlet or a triplet excited state of the molecule
459 Colour in Metals, Semiconductors and Insulators
close to that of theLUMO in the polymer and allows easy flowof electrons across the interface (Figure 10.32b).
In order for light to leave the device, one electrodemust be transparent, and for this the transparent conducting
material indium tin oxide (ITO) is used. The Fermi level of thismaterial is similar to theHOMOof the polymer
(Figure 10.32b). The completed device (Figure 10.32c) gave out an orange red colour and had an external
quantum electroluminescent efficiency Z defined by:
Z ¼ number of photons escaping the device
number of charges entering
of about 2 %.
The early single-layer devices of the sort just described have been changed considerably in the ensuing years
and are now replaced by multilayer configurations (Figure 10.33a). Hole or electron injection layers may be
introduced next to the anode or cathode to improve the number ofmobile charge carriers entering the transport
layers. Thesemay adjoin a hole-transporting layer and an electron-transporting layer respectively, which serve
tomove the charge carriers into the emitting layer. These additional components are not often used at the anode
boundary, because the standard anode material, ITO, has an energy band structure that matches the HOMO of
many hole-transporting compounds, thus ensuring efficient direct injection of holes into the hole-transporting
layer.However, there ismoreof a problemat the cathode.Calcium is ideal fromanenergy level viewpoint, but it
has a high reactivity in air. A common replacement is magnesium or a calcium-silver or magnesium silver
alloy. The Fermi energy of these conductors does not match the LUMO energy of most electron-transporting
materials, and it is found that a thin (0.5 nm) layer of lithiumfluoride (LiF) as an electron injection layer greatly
increases the number of electrons entering the electron-transporting layer. The mechanism by which this
enhancement is achieved is not understood.
h e
from anode from cathode
electron - hole pair
singlet exciton
emission emission
external photons
deactivationdeactivation
triplet exciton
fast slow
Figure 10.31 Schematic processes taking place within an organic electroluminescent solid. Note that each stephas its own efficiency and that surface processes, important in real devices, are ignored
Colour and the Optical Properties of Materials 460
Although the transporting layers can be directly coupled to the emitting layer, they are often connected to
exciton blocking layers. These latter layers ‘reflect’ diffusing excitons and effectively confine them to the
emitter layer (Figure 10.33a). The electroluminescent layer that emits photons may just rely on the energy
levels of theHOMOandLUMO,ormaycontaindopantswith adifferent energygap. In actual devices, not all of
these layers are present, and those that are have been chosen carefully so that the energies of the HOMO and
LUMO pairs are matched up (Figure 10.33b).
Much effort hasbeen expended indiscoveringnewmaterials for thesedevices.An important advancewas the
incorporation of heavymetals such as platinum or iridium into the emitting layer. These heavymetals interact
strongly with triplet states, so that they emit photons rather than lose energy by nonradiative transitions, thus
immediately improving the external quantum efficiency by a factor of three or four.
MEH - PPV
MEH - PPVemitter
(a)
O
O
n
2.8 eV
4.9 eV4.7 eV
2.7 eV
π*
π
calcium cathode
ITOanode
(b)
-
+
glasssubstrate
light output
(c)
Figure 10.32 (a) The skeletal structure of poly(2-methoxy-5,20-ethyl-hexloxy)-1,4-phenylene vinylene(MEH-PPV). The segment in brackets is repeated many times in the polymer. (b) Energy-level diagram for theanode, emitting polymer and cathode. (c) Schematic of OLED
461 Colour in Metals, Semiconductors and Insulators
Apart fromfilmsmade fromsmallmolecules or polymers,moleculardendrites are of increasing importance.
Dendrites are ‘bushy’ organic molecules that consist of a core surrounded by branching linear chains of atoms
to give a tree-like structure. The core of the dendrite is the luminescent centre. The branches need to transport
charge and are often conjugated units. The outer ends of the branches can bemodified by attaching a variety of
cathode (-)electron injection layer (EIL)
electron transport layer (ETL)
hole transport layer (HTL)
hole injection layer (HIL)
exciton blocking layer (EBL)
exciton blocking layer (EBL)
emitting layers (EL)
anode (+)
light(a)
(b)-2
-3
-4
-5
-6
-7
electron generationand transport
hole generationand transport
light production
LUMO energy
HOMO energy
ITO (+)
Ca:Ag (-)EL
ETL
EBLHTL
40 n
m
15 n
m
25 n
m
40 n
m
dopant energy levels in EL
Figure 10.33 (a) Generalized schematic diagram of amultilayerOLED. (b) Schematic energy-level diagram of ablue-emittingOLED; the top of each coloured band represents the LUMOenergy of that material and the bottomof each coloured band represents the HOMO energy. The Fermi energy of the ITO anode and Ca:Ag cathode arealso included. [Adapted with permission from S. Ye et al., ‘Wide-Energy-Gap Host Materials for BluePhosphorescent Organic Light-Emitting Diodes’, Chem. Mater. 21, 1333–1342. Copyright 2009 AmericanChemical Society]. HTL¼N,N0-di(naphthalene-1-yl)-N,N0-diphenylbenzidene; ETL¼ 2,9-dimethyl-4,7-diphe-nyl-1,10-phenanthrolene; EBL¼N,N0-dicarbazolyl-3,5-benzene; EL ¼ 1,3-bis(9-phenyl-9H-fluoren-9-yl) ben-zene host plus Ir(III) bis(40,60-difluorophenylpyridinato)tetrakis(1-pyrazolyl) borate
Colour and the Optical Properties of Materials 462
surface groups to modify the ease of processing of these materials. Many dendrimers that have been explored
for use in OLEDs have iridium at the core, thus making use of the ability of this metal to induce luminescence
from triplet energy states.
White-light-emitting OLEDs can be produced in the same way as described with inorganic LEDs; that is,
a blue emitter can be combined with a yellow phosphor. In displays, additive colour mixing can give the
impression of white if three separate OLEDs, emitting red, green and blue, combine. With OLEDs it is also
possible to combine two ormore different dendrimers in the emitting layer to give two separate colour outputs,
such as yellow and blue, so as to achieve white light output.
In all OLEDs, light extraction is hampered because of the differing refractive indices of thematerials, which
results in total internal reflection at the many interfaces. This remains a problem. At present the internal
quantum efficiency of an OLED is about five times the external quantum efficiency, i.e. only about one in five
of the photons generated in the emitting layer leaves the device.
10.12 Electrochromic Films
Electrochromic materials are compounds which change colour reversibly when subjected to an electric field.
Colour change in electrochromic films ismediated by oxidation or reduction, usually accompanied by counter-
ion transport to maintain charge neutrality. An electrochromic device is thus a form of electrochemical cell.
Reduction processes, which are equivalent to a gain in electrons, take place at the cathode. This is also referred
to as n-doping. Materials in which the significant colour change is induced by reduction are said to be
cathodically coloured.Oxidation processes, which are equivalent to a loss of electrons, take place at the anode.
This is also called p-doping.Materials inwhich the significant colour change is induced by oxidation are said to
be anodically coloured. The transparent or colourless form of the electrochromic compound is often called the
bleached state. There are two basic types of electrochromic reaction:
coloured> bleached ðiÞcoloured A> coloured B ðiiÞ
From a practical point of view, electrochromic materials are mostly employed in thin-film form as elements
in electrochromic displays. The simplest, asymmetric arrangement, is one in which there is a single
electrochromic film next to either the anode or the cathode, linked to the counter-electrode by an electrolyte
which is also the source and sink for the ions involved in the colour changes. In these devices, an applied voltage
can be set up so as to drive ions and electrons into the electrochromic material, changing the colour of the film.
Reversal of the voltage drives the ions and electrons in the opposite direction, causing the film to revert to its
original state (Figure 10.34a and b). In a dual film arrangement, different electrochromic films are in contact
with both electrodes (Figure 10.34c). In this type of device the electrochromic films are coloured in tandem,
so as to increase the contrast developed. The electrodes are generally made from the transparent conductor
indium tin oxide (ITO). The efficiency of an electrochromic device Z is given by:
Z ¼ DAQ
¼ logðTox=TredÞQ
whereDA is the change in absorbance (optical density) produced by an injectionor removal of chargeQper unit
area of film and Tox and Tred are the transmittance of the film in the oxidized and reduced states respectively
(Section 1.13).
463 Colour in Metals, Semiconductors and Insulators
The speed of darkening of a film, or its opposite, bleaching, depends mainly on ionic diffusion in a weak
electric field. At ordinary temperatures this is too slow for fast displays such as television, but it is satisfactory
for electronic notice boards or shop signs and similar displays that are relatively permanent. Recently,
electrochromic films have been explored for use as dynamic camouflage, involving switching between green
andbrown tones.However, the best-knownapplication is in ‘smart’windowsormirrors that control the amount
of light reflected or transmitted.
glass support
glass support
glass support
glass support
ITO cathode
ITO cathode
ITO anode
ITO anode
electrochromic layer
electrochromic layer
electrolyte and source of ions
electrolyte and source of ions
V
V
e–
e–
e–
e–
(a)
(b)
(c)
glass support
glass support
ITO cathode
ITO anode
electrochromic layer 1
electrochromic layer 2
electrolyte and source of ionsV
e–
e–
Figure 10.34 Electrochromic devices (schematic). (a), (b) Asymmetric cell design in which the electrochromicfilm is located next to the cathode or the anode. (c) Dual-film device in which electrochromic films are locatedadjacent to both electrodes. ITO represents the transparent conductor indium tin oxide
Colour and the Optical Properties of Materials 464
10.12.1 Tungsten trioxide electrochromic films
Thematerial most widely explored for electrochromic devices is tungsten trioxide (WO3), which in the bulk is
pale yellow and an insulator (see Figure 10.1). Thin films are transparent. The crystal structure of this oxide is
rather openandbuilt of corner-linkedWO6octahedra (Figure10.35a).Electrochromicfilmsare colouredby the
formation of tungsten bronzes,MxWO3.There are a number of tungsten bronze structures, but the ones utilized
for electrochromic purposes are the perovskite bronzes, in which cations such as Li, Na or Koccupy cage sites
between the corner-linked WO6 octahedra of the parent phase (Figure 10.35b). The colour of these is dark
blue black. The phase range over which the perovskite bronze structure is stable is greatest for the Li phases
and smallest for the K phases. The hydrogen bronzes, HxWO3 are also deeply coloured blue black, although
these are rather different from the alkali metal phases and are probably best regarded as a nonstoichiometric
hydroxide, WO3 x(OH)y.
Although the colour of the tungsten bronzes has not been explained fully over all of the composition range,
at the low concentrations employed in electrochromic films the blue black colour induced is so similar to that
of reduced tungsten trioxide that it is presumed that charge transfer between two valence states of tungsten is
occurring. If so, colour may then be attributed to W5þ W6þ or W4þ W6þ couples.
The principle of an electrochromic device using tungsten trioxide films is not too difficult to envisage. It is
necessary to drive some appropriate metal, such as Li, into the WO3 film using an applied voltage. This will
make the tungsten trioxide turn into a blue black tungsten bronze. Reversal of the voltage must remove the
interpolated metal and regenerate the colourless state. The reaction can be schematically written as:
glass support
(a)
(c)
(b)
glass support
ITO cathode
ITO anode
WO3 electrochromic layer
M+ source / sink
electrolyteV
e–
e–
e–
e–
M+
M
Figure 10.35 Inorganic electrochromic films. (a) The idealized structure of tungsten trioxide (WO3) composedof corner-sharingWO6 octahedra. (b) The idealized perovskite tungsten bronze structure, AxWO3, in which largeA cations are interpolated into the cages in the parent WO3 phase. (c) An electrochromic device schematic usinga thin film of WO3 as the electrochromic phase
465 Colour in Metals, Semiconductors and Insulators
WO3 þ xMþ þ xe > MxWO3
transparent ðbleachedÞ filmþ electrons > dark film
In this material, the transparent (oxidized) film is transformed into the coloured (reduced) film by the gain
of electrons (that is, reduction or n-doping) and WO3 is a cathodically coloured substance. Devices are
constructedas a series of thinfilmsonglass.Transparent conductingelectrodes, usually ITO, sandwichafilmof
WO3, an Mþ ion conducting electrolyte and a separate reservoir (source/sink) of metal Mþ ions if needed
(Figure 10.35c).
Dark electrochromic films making use of sodium tungsten bronzes, NaxWO3, have been obtained using the
nonstoichiometric fast-ion conductor b-alumina (Na1þ xAl11O17þ x/2). This compound has a broad composi-
tion range, with x taking values between 0.15 and 0.3, and is an excellent conductor for Naþ ions so that it can
act as both the source and sink for these ions. The power supply can force Naþ ions to migrate into theWO3 to
form a dark NaxWO3 bronze, or remove them back into the b-alumina reservoir to turn the bronze back into
colourless WO3 (Figure 10.36a).
ITO cathode
ITO cathode
ITO cathode
ITO cathode
ITO anode
ITO anode
+ –ITO anode: x/2 H2O → xH + x/2 O + xe2
ITO anode
+ –WO + xNa + x e → Na WO3 x 3
+ –WO + xLi + x e → Li WO3 x 3
+ –WO + xH + x e → H WO3 x 3
+ –WO + xH + x e → H WO3 x 3
+Na Al O → NaAl O + xNa + x/2 O2 1+x 11 17+x/2 11 17
+ –gel electrolyte containing Li salt → xLi + xe
HUP proton conductor
HUP proton conductor
+
+
+
+
_
_
_
_
–e
–e
–e
–e
–e
–e
–e
–e
–e
–e
–e
–e
–e
–e
–e
–e
+Na
+Li
(a)
(b)
(c)
(d)
moist air
+H
+H
H2O
+ –NiO H → NiO H + xH + xey z y z–x
Figure 10.36 Electrochromic devices using tungsten trioxide (schematic). (a) Colour due to the formation ofsodium tungsten bronze,NaxWO3. (b) Colour due to the formationof lithium tungsten bronze, LixWO3. (c)Colourdue to the formation of hydrogen tungsten bronze, WO3�x(OH)y. (d) Colour due to the formation of hydrogentungsten bronze and oxidized nickel oxy-hydroxide
Colour and the Optical Properties of Materials 466
Lithium can be inserted into the WO3 thin film if a lithium reservoir is substituted for a sodium source.
A material that can be used for this purpose is an electrolyte consisting of a gel containing a readily ionized
lithium salt. Application of an electric field can drive Liþ out of or back into the gel reservoir at will
(Figure 10.36b).
In the case of hydrogen tungsten bronzes, HxWO3, it is possible to use the decomposition of water vapour in
the atmosphere as a source of Hþ . Decomposition takes place on the outer ITO electrode:
2H2O!O2 ðgÞþ 4Hþ þ 4e
This electrochemical decomposition requires about 1V at the electrode surface. To drive the protons into
theWO3film, a proton-conducting electrolyte, typically hydrogen uranyl phosphate, HUO2PO44H2O (HUP),
is utilized. The Hþ produced can pass through the proton-conducting electrolyte to form the bronze, using
electrons from the other electrode (Figure 10.36c).On reversal of the applied voltage theHþ ions are pulled out
from the bronze and the film becomes colourless once more.
10.12.2 Inorganic electrochromic materials
Apart from tungsten trioxide, a number of other inorganic materials show electrochromic behaviour. Among
the most important of these are hydrated nickel oxide, niobium pentoxide and Prussian blue.
Hydrated nickel oxide (Ni(II)OxHy,), a poorly characterizedmaterial, is pale green in bulk and transparent in
thin-film form. It is readily converted to a metastable brown oxyhydroxide containing Ni(III). The complex
chemistry of the reversible reaction can be approximated as:
NiOxHy > NiOxHy z þ zHþ þ ze
transparent ðbleachedÞ film > dark filmþ electrons
In this material, the transparent (reduced) film is transformed into the coloured (oxidised) film by the loss of
electrons (that is, oxidation or p-doping) and hydrated nickel oxide is an anodically coloured substance.
Hydrated nickel oxide can be used in conjunction with tungsten trioxide films, enhancing the darkening
effect of the tungstenbronze layer and so improving the darkeningcharacteristics of thedevice (Figure 10.36d).
Niobiumpentoxide (Nb2O5),which is colourless and forms transparent films, has a structure related to that of
tungsten trioxide and, like this latter material, can take in Liþ or Hþ to form a dark blue black phase on
reduction. The reaction can be written:
Nb2O5 þ xMþ þ xe > MxNb2O5
transparent ðbleachedÞ filmþ electrons > dark film
In this material, the transparent (oxidized) film is transformed into the coloured (reduced) film by the gain of
electrons; that is, reduction or n-doping, and Nb2O5 is a cathodically coloured substance analogous to WO3.
Dark blue Prussian blue, KFe3þFe2þ (CN)6, is also readily oxidized to colourless K2Fe2þFe2þ (CN)6,
Prussianwhite (see alsoSection8.10).Thus thinfilmsofPrussianblue canbemade transparent by the reduction
reaction and the transparent films darkened by oxidation. The essence of the reaction is the interconversion of
Fe2þ and Fe3þ :
Fe2þ Fe2þ > Fe3þ Fe2þ þ e
transparent ðbleachedÞ film > dark filmþ electrons
467 Colour in Metals, Semiconductors and Insulators
In this material, the transparent (reduced) film is transformed into the coloured (oxidized) film by the loss
of electrons (that is, oxidation or p-doping) and Prussian white is an anodically coloured substance. An
electrochromic cell using both WO3 and Prussian white to enhance contrast can be fabricated, operating
similarly to that with WO3 and hydrated nickel oxide described above.
10.12.3 Electrochromic molecules
Many organic molecules can be coloured via oxidation or reduction, making them ideal targets for electro-
chromic devices. In this section, organic conjugated polymers, a widely explored group, will be considered as
illustrative. The most important of these polymers are those that are also electrically conducting polymers,
notably poly(pyrrole) (PPy), poly(thiophene) (PTh) and poly(aniline) (PANI) and derivatives of these parent
phases. The colours of these polymers derives from a p p� (HOMO LUMO) transition. In the normal
insulating (or neutral) state these materials are semiconductors, with a band gap Eg defined by the separation
between the HOMO and the LUMO which also controls the colour of the neutral state. Narrow band gap
polymers will be coloured, whereas broad band gap polymers will be transparent. Oxidation or reduction
(p-doping or n-doping, depending upon the polymer) accompanied by ion insertion or removal turns these
materials into electrical conductors. The conducting phases showabsorption bands due to transitions involving
the newly created charge carriers, generally thought to be polarons (Section 10.11).As thematerial is gradually
oxidised or reduced, colour due to the original p p� transitions diminishes and new transitions to the modified
band structure and transitions involving the newly formed polarons and bipolarons change the colour of the
polymer.
As with inorganic materials, some electrochromic polymers may be more readily subject to reduction, in
which case they are cathodically coloured. These materials generally have a relatively small band gap, of
the order of 1.7 eV, and so tend to be coloured in the neutral (insulating) state. Other electrochromic
polymers are more readily subject to oxidation, in which case they are anodically coloured. These
materials tend to have a large band gap, of the order of 2.5 eV, and so are usually transparent in the neutral
(insulating) state.
An advantage of organic molecules over inorganicmaterials is that the colours available can bemodified by
inserting substituents into the structure usingwell-known organic chemistrymethodology. This leads to a third
type of electrochromic reaction:
coloured A> coloured B> coloured C ðiiiÞ
This last group, in which several different coloured forms can be cycled, can be found in some polymers as the
degree of doping changes, or may be manufactured from copolymers of monomers showing just two colour
states.
10.12.4 Electrochromic polymers
From the large number of polymeric electrochromic materials so far investigated, two are illustrated here
as representative, PANI and the alkoxy-substituted polythiophene poly(3,4-ethylenedioxythiophene)
(PEDOT).
PANI, derived from the small molecule aniline (Figure 10.37a), exists in three basic forms, each of which
shows a different colour; the polymer, therefore, is polychromic. The polymers can be regarded as built up
from two end-species: aromatic, reduced, leucoemeraldine, yellow, and clear in thin -film form; and quinoid,
oxidized, blue violet, pernigraniline. The 1:1 intergrowth of these two structures is the green blue emeraldine
(Figure 10.37b).
Colour and the Optical Properties of Materials 468
This simple description masks a complex polymer chemistry and physics. Generally, PANI is prepared in
a doped conducting emeraldine salt form ES-I. The dopant is often a simple acid such as HCl, and one or both
parts of the polymer can bemodified in this way (Figure 10.37c). The acid can be removed to yield an insulting
emeraldine base form, EB-I. Thematerial can also be prepared in a different insulating emeraldine base form,
EB-II, which can be made conducting by doping with, for example, HCl to form the conducting emeraldine
salt ES-II. All of these products show variable degrees of crystallinity.
For electrochromic device use, these materials are often dissolved in a solvent and cast as films, which are
partly crystalline and show conducting or insulating behaviour depending uponwhether they are doped or not.
In addition, the colour changes slightly on doping, so that ES forms are green and EB forms blue. In order to
improve the device characteristics and performance, the PANI films are frequently reacted with other dopants,
such as D,L-camphor-10-sulfonic acid (CSA) or poly(styrene sulfonic acid) (PSS) to give PANI CSA or
PANI PSS for instance. In this group of materials, the colour of the film can be cycled between pale yellow
(leucoemeraldine) and dark blue (pernigraniline). PANI in these devices is anodically coloured, as the reduced
transparent form is oxidized to the coloured form.
The widely used electrochromic polymer PEDOT is derived from polythiophene (Figure 10.38) and has
two coloured states: red and blue. Like PANI and its derivatives, polythiophene and derivatives also have
an aromatic-type and aquinoid-type structure, but in this case the quinoid type is of higher energy andnot found
in normal preparations. PEDOT itself shows two colour varieties: a transparent oxidized form and a reduced
blue form.The band gap of thematerial is about 1.78 eVand, likeWO3, becomes coloured on reduction, i.e. it is
cathodically coloured.
Devices usingpolymers are constructed in a similarway to those using inorganicmaterials. They can contain
one active electrochromic polymer layer, one electrochromic polymer layer coupled with an inorganic
N
N
N
N
N
N
N+ N+
N
H
H
H
H
+H +H
H
H
(a)
(b)
(c)
a
a
b
b
n
n
aniline
polyaniline (PANI)
polyaniline (PANI) salt
-An -An-An -An
leucoemeraldine yellow, a = 1, b = 0, reduced
emeraldine, green / blue, a = b = 1/2, neutral
pernigraniline, blue / violet, a = 0, b = 1, oxidized
H H
Figure 10.37 The idealised structures of polyaniline (PANI): (a) aniline; (b) leucoemeraldine, a¼ 1, b¼ 0;emeraldine, a¼ b¼ 1/2; pernigraniline, a¼ 0, b¼ 1; (c) polyaniline salts, in which the anions, An� and cations orHþ are combined with the polymer
469 Colour in Metals, Semiconductors and Insulators
material, or two organic polymer layers. As before, the electrochromic layers are separated by an electrolyte
layer, often in gel form, containing the ions needed to maintain charge balance. A cell made from a cathodic
film of PEDOTand an anodic film of PANI PSS is an example (Figure 10.39). The anode of the cell, where the
reduced form of the electrochromic polymer is to be reduced to a coloured form, is in contact with PANI PSS.
The cathode of the cell, where the oxidized form of the electrochromic polymer is reduced to a coloured form,
is in contact with PEDOT PSS. The electrolyte contains lithium chlorate, giving Liþ and ClO4 ions, in a gel
matrix. The schematic reactions taking place are as follows:
at the anode
PANI--PSS > ðPANI--PSSþClO4 Þþ e
pale yellow reduced form > blue oxidized form
at the cathode
PEDOT--PSSþ e þLiþ > ðLiþ PEDOT--PSSÞpale blue oxidized form > dark blue reduced form
In these reactions, both electrochromic films darken simultaneously, to give an overall transparent to blue
electrochromic change.
Many polythiophenes also show thermochromism. The reasons are related to the occurrence of photochro-
mic behaviour. The polythiophene molecules are usually planar when cold, have considerable electron
delocalisation and a smaller band gap. The ‘cold’ colour tends to be red. As the temperature increases, the
backbone of the polymer can buckle or twist. This inhibits the amount of electron delocalisation and has the
effect of increasing the band gap of themolecule. Thus, the colour change is from red towards the green blue as
the temperature increases.
ITO cathode
ITO anode
PEDOT–PSS: pale blue oxidised → dark blue reduced
LiClO4 containing gel electrolyte
–e –e
–e–e
+Li –ClO4
PANI–PSS: yellow reduced → blue oxidised
Figure 10.39 Schematic electrochromic device utilising PEDOT cathodic and PANI anodic electrochromic films
SS Sn n
O O
(a) (b) (c)
Figure 10.38 The structures of (a) thiophene, (b) polythiophene and (c) poly(3,4-ethylenedioxythiophene(PEDOT)
Colour and the Optical Properties of Materials 470
10.13 Photovoltaics
10.13.1 Photoconductivity and photovoltaic solar cells
If radiation of a suitable wavelength falls on a semiconductor, it will excite electrons across the band gap. One
result is that a voltage develops across the semiconductor and the conductivity of the material increases.
Materials that show a voltage on illumination are called photovoltaic materials. Themagnitude of the effect is
roughly proportional to the light intensity. These properties, called the photovoltaic effect or photoconductive
effect, have been used in light meters, exposure meters and automatic shutters in cameras, and many other
devices. The first exposure meters for the measurement of light amounts available for photography used
selenium (Se), cadmium sulfide (CdS) or silicon (Si). In the case of selenium, the photovoltage is large enough
to be measured directly and converted to an exposure value. Cadmium sulfide and silicon need voltage
amplification, and these materials need a power source, usually a battery, to give a reading. A DC voltage
applied to the ends of a semiconductor will also allow the photoeffect to be measured. The increase in
conductivity on illumination provides a means of measurement of the amount of incident light falling on
the device.
A p n junction can act in a similar way to a single piece of semiconductor. However, the control afforded by
the junction makes the device, called a photodiode, far more flexible; as a result, photodiodes arewidely used,
especially in solar cells. A solar cell is specialist large-area p n junctionwith a depletion region approximately
500 nm thick. (Solar cells must have a large area, to collect as much sunlight as possible.) In addition, the
normal built-in potential that exists across the junction, due to the space charge, is engineered to be high
(Figure 10.40a). The junction is not connected to any external power source. Holes and electrons produced in
photon
sunlight incident on anti-reflective coating
n-type material
p-type material
junction region
negativefront contact
n-type semiconductor–6~ 10 m
p-type semiconductor
positive back contact
space charge region
(a)
(b)
Ip
+
_
holeelectron
reflecting layer
Figure 10.40 Solar cell schematics. (a) Sunlight incident upon a p–n diode junction creates electron–hole pairsthat are swept into the external circuit by the built-in field in the junction region. (b) An operating cell needsan antireflection front coating, a junction region near to the illuminated surface and a back reflecting layer tooptimise cell efficiency
471 Colour in Metals, Semiconductors and Insulators
the junction region by sunlight are swept across the depletion region by the high built-in space charge present,
the electrons going from the p to n region and the holes from the n to p region. This process, called drift, charges
the p region more positive and the n region more negative, and produces a photocurrent current Ip across the
junction which generates a photovoltage. The photovoltage corresponds to a forward bias, and so will cause a
current I toflow.At equilibrium I¼ Ip. Should anexternal loadRbeconnected, somecurrent canflow through it,
and so do useful work.
Not all electron hole pairs are created in the junction region. Those generated outside itmust diffuse through
the solid until they approach the junction, where they are subjected to the internal field and can contribute to the
photocurrent.
This simple description obscures the amount of effort required to produce an efficient solar cell. The simplest
amendments are the requirement of an antireflection surface to maximize the number of photons reaching
the semiconductor, a thin initial semiconductor layer, so that the optimum number of photons reach the
depletion region, and an underlying reflective layer that redirects any photons that pass straight through the
lower semiconductor layer back towards the junction region (Figure 10.40b).
A number of basic considerations influence the design of solar cells. For example, because impurities and
defects trap mobile electron and holes, which greatly reduces the efficiency of the cell, high-purity materials
are mandatory, although this increases cell costs considerably. It is also clear that the band gap of the solar
cell materials chosen must utilize as much of the wavelength spread available in sunlight (approximately
350 2500 nm, 3.5 0.5 eV,with a peak in the yellow green at 550 nm) as possible.Moreover, indirect band gap
materials have a lower efficiency than direct band gap materials.
The cells that currently (2010) show the highest efficiency are based upon silicon. A drawback is that silicon
is an indirect band gap solid and does not absorb across all of the desired energy range very efficiently. In
an effort to overcome this problem, amorphous silicon, which behaves as a direct band gap material, is used
in many devices. In order to increase efficiency over that currently available, other cell materials investigated
include the semiconductors cadmium telluride (CdTe), cadmium sulfide (CdS), copper indium selenide
(CuInSe2) and mixed copper selenides (Cu(In,Ga)Se2) and quantum dots (Section 10.10).
Recently, much effort has been put into the construction of solar cells using polymers. These have the
great advantages of low weight and flexibility. However, efficiencies are not yet adequate for commercial
purposes.
Solar concentrators, mirrors or lenses that focus the sunlight onto the photoactive layers, arewidely used to
increase efficiency, as are mobile systems that are able to follow the motion of the sun throughout the day. To
absorb as much as possible of the high-irradiance part of the solar spectrum, cells have been stacked in series;
for example, GaInP2, GaAs and Ge, which is able to utilise photons from 590 to 1200 nm, 2.1 to 1.0 eV.
10.13.2 Dye-sensitised solar cells
In a conventional solar cell, the conversion of the light to free charge carriers is carried out by the solid
semiconductor, which then has to move these away from the junction in order to obtain energy. To achieve
good efficiency the photons need to be absorbed close to the p n junction. Electron hole pairs created
elsewhere have to diffuse to the junction region and, unless the materials are of high purity, recombination is
likely. The method of conversion of sunlight to energy of most importance on the Earth, photosynthesis, uses
slightly different methods of achieving the same objective. The central reactions are oxidation and reduction.
Photoelectrochemical cells, of which dye-sensitised solar cells (also called Gr€atzel cells) are an important
example, aim to mimic this process. The task of harvesting the light is left to a sensitiser, which is a
dye molecule, and the carrier transport task is allocated to a semiconductor. Because the charge separation
takes place in the dye, the purity and defect structure of the semiconductor are not crucial to satisfactory
operation.
Colour and the Optical Properties of Materials 472
The reactions in the cell are (Figure 10.41a):
1. Excitation of the sensitiser dye S by a photon:
Sþ hn! S*
2. The excited sensitiser S� loses an electron, which moves into the conduction band of the semiconductor:
S* ! Sþ þ e ðsemiconductorÞ
3. The electronmoves through the conduction band of the semiconductor to the transparent conducting anode,
also called the working electrode, which acts as the electron collector. Thereafter, electrons traverse the
external circuit to arrive at the cathode, also called the counter electrode.
4. Electrons arriving at the counter electrode reduce a redox couple, R/R , usually in a liquid electrolyte:
RðaqÞþ e !R ðaqÞ
where (aq) represents an aqueous solvent.
5. The sensitiser is regenerated by reaction with the reduced half of the redox couple:
Sþ þR ! SþR
A large number of different dyes have been tried in the role of sensitiser in conjunction with a variety of
inorganic oxides, including ZnO and Nb2O5 as the semiconductor. Currently (2010), the best efficiency is
hole
electron
R oxidized species
R– reduced species
TCO transparent conducting oxide
CB conduction band
VB valence band
FL Fermi level
RO redox potential
current
dye electrolyte
–R–R
R
S*
S
RO
R –ephoton
load
TCOanode
TCOcathode
CBFL
VB
V
(a)
(b)
semiconductor
Figure 10.41 Dye-sensitised solar cell schematics. (a) Sunlight absorbed by the dye liberates an electron into thesemiconductor. The dye is regenerated by interaction with an internal redox couple. (b) Schematic energy-leveldiagram. The cell voltage V is the difference between the Fermi level of the semiconductor and the redox potentialof the couple
473 Colour in Metals, Semiconductors and Insulators
obtained with cells using dye molecules containing Ru(II), such as cis-dithiocyanatobis-2,20-bipyridine-4-COOH, 40-COO Ru(II), in combination with nanocrystalline anatase (titanium dioxide, TiO2) as the
semiconductor. The charge states on the dye correspond formally to the conversion of Ru(II) to Ru(III)
via photon interaction:
RuðIIÞþ hn!RuðIIIÞþ e
The dye is absorbed onto the surfaces of the anatase crystallites to give a large surface area whilst maintaining
compact electrode geometry. The transparent conducting oxide electrodes are usually tin oxide doped with
fluorine (SnO2:F). The redox couple usually chosen is iodide triiodide in solution:
I3 ðaqÞþ 2e ! 3I ðaqÞ
In order to catalyse the oxidation-reduction equilibria in the electrolyte, the cathode electrode is coated with
a thin layer of platinum.
The cell output depends upon the relative positions of the energy levels in the adjoining components, which
must be matched for optimum efficiency. In the present cell design, this is given by the difference between the
redox potential of the oxidation reduction couple chosen and the Fermi level of the semiconductor
(Figure 10.41b). For the triiodide iodide couple the redox potential is þ 0.54V. The Fermi level of the
nanostructured anatase is about �0.4V, so that the cell voltage is approximately 0.54V þ 0.4V, i.e. 0.94V.
There is much current research directed towards improving the dyes used in these cells. The stability of the
dye above is limited by the thiocyanate (SCN) ligands within it, and dyes which do not contain these groups
are under active consideration as alternatives. The desirability of having a stable organic dye that does not
contain a heavy metal is also an important research goal. Similarly, there is interest in replacing the expensive
platinum cathode with an organic conductor such as PEDOT. Active research in this area means that new cell
specifications are continually appearing in the literature.
10.14 Digital Photography
10.14.1 Charge coupled devices
In the space of a few years, digital photography has become the standard recording technique for amateur
photographers, almost totally replacing film photography. Some time before this, digital imaging replaced
photographic film as an image storage medium in many areas of scientific and medical research, initially
starting with astronomy. The difference between digital and conventional photography is simply in theway in
which the informationcontent of light is capturedand stored, photographicfilm(Section10.16)versus (mainly)
charge coupled devices (CCDs).
The concept of the CCD was proposed by Boyle and Smith in 1969, as a contender for computer memory.8
Charges were localised in small volumes (bubbles) in a silicon chip. Each bubble could represent a 0 or 1,
depending upon the presence or absence of charge, and bubbles could be annihilated or moved around by
changing the voltage applied to an array of electrodes covering the silicon slice. The name ‘charge coupled
device’ springs from the mechanism by which the bubbles were moved in concert. Although the memory
storage aspect did not result in commercial applications, it was apparent at the time that CCDs had other
8 This was an attempt to mimic another related data storage technology, the magnetic bubble memory. Magnetic bubble memories never
became a commercial success as they were overtaken by other means of data storage, especially optical data storage.
Colour and the Optical Properties of Materials 474
applications. Among the first was as a light-collecting alternative to photographic film. The first commercial
image recording sensorwas in place at theKitt PeakObservatory in 1979, just 10 years after the initial concept.
CCDs employ similar basics to photodiodes (Section 10.13). Photons falling upon a silicon slice generate
electron hole pairs. The number of electron hole pairs that are produced during an exposure is measured and
form the datum that ends up as a pixel of the image.
The structure of a pixel consists of a photoactive layer of p-type silicon. This is covered with a thin layer of
silicon dioxide (SiO2) and this is topped with a conductor, originally aluminium, but now heavily doped
polycrystalline silicon, to form a metal oxide semiconductor (MOS) device (Figure 10.42a). The polycrys-
talline silicon (M) layer is called the gate. A positive voltage applied to the gate will create an electric field
across the thin layer of insulator, theO layer,mimicking the action of a parallel-plate capacitor. This field repels
holes in the adjacent volume of the p-type semiconductor, the S layer, which creates a depletion region similar
to that in a p n junction diode. This region acts as a potential well to trap charges, and so can be thought of as
a pocket or bucket in which charge is stored. As the gate voltage increases, the field increases and extends
the region where the holes are repelled, increasing the size of the pocket (Figure 10.42b).
A number of simultaneous processes occur in the pixel.
1. Electron hole pairs are continually created throughout the p-type silicon slab by thermal energy and by
incident light photons. The electrons that form in the bulk of the p-type region have a short lifetime before
they recombinewith the excess of holes present. However, those that are formed in the potential well under
the gate have a much longer lifetime because the hole population there is low. The field present sweeps the
electrons towards the oxide layer. However, this build up of charge gradually diminishes the field present.
2. Thermal diffusion causes some holes to enter the depletion volume despite the electric field. These will be
annihilated by the electrons present.
3. As more electrons accumulate at the oxide interface, the field that maintains the potential well weakens,
allowing more and more holes to diffuse in. Ultimately, a dynamic equilibrium is reached.
heavily doped poly-silicon gatesilicon dioxide
p-type silicon
(a)
(b)
(c)
V + V ++ V +++
+
- - - - -
+ + + +
Figure 10.42 CCDs: (a) schematicMOS pixel; (b) potential well formation under a gate voltage; (c) steady-statecapacitor-like charge distribution in an MOS pixel
475 Colour in Metals, Semiconductors and Insulators
4. The amount of charge accumulated in the potential well then depends upon the rate of creation of thermal
electron hole pairs, the rate of creation of photogenerated electron hole pairs and the rate of destruction of
electrons by hole diffusion. Provided that the rate of photogeneration is much greater than that of the
competing processes, charge will build up in the pocket in direct proportion to the amount of light incident
upon the pixel.
5. At this stage, there is a charge separation across the oxide layer that resembles that in a charged capacitor
(Figure 10.42c). As with a capacitor, if the initial gate voltage is removed, then the charges will remain in
place, or, more correctly, will slowly leak away. The amount of charge is measured before this happens.
The potential well is emptied and the next exposure can be recorded. (The sophisticated mechanisms by
which these records are read out and stored can be explored via the sources in this chapter’s Further
Reading.)
10.14.2 CCD photography
There are a number of features of CCDs that are of importance in photography. The first to note is that the CCD
recording device is linear. That is, the charge that accumulates is directly proportional to the light irradiance,
so that the data recorded in a pixel is proportional to the light irradiance. This is not true of photographic film,
and represents an improvement over the older technique, especially in scientific recording.A related advantage
is that the spectral range over which a silicon-based CCD is sensitive is far greater than that of photographic
film. Silicon can detect well into the infrared, although these wavelengths are not visible to the eye.
Naturally, there is a limit as tohowmuchchargecanbeaccumulated inanypixel.This is the full-well capacity
of the sensor,which is a function of the temperature of the device, the doping levels and the physical dimensions
of the various parts. After the full-well capacity has been reached during an exposure, no further information is
registered. The dynamic range of a sensor, therefore, is directly associated with the full-well capacity.
The rate of thermal generation of electron hole pairs is an important parameter. In low light level use, such as
astronomy,where long exposure times are needed toobtain an imageof a faint object, the thermal ‘dark current’
adds noise to the information. In these cases, cooling the detector with liquid nitrogen is employed. For day-to
day photography this background noise is not usually of importance.
Although CCDs are linear in their response, they do not respond evenly to all wavelengths unless some
modifications aremade to the basic structure described above. This is because incident photons have to traverse
the gate electrode before reaching the photoactive silicon layer. Electron hole pairs and other energy-
absorbing processes take place in the gate layer, and higher energy photons lose energy faster than lower
energyphotons.This ismeasuredby the linear absorption coefficient of silicon,which shows that photons at the
violet end of the spectrum frequently cannot penetrate the gate electrode. This means that the device is blind to
the blue end of the spectrum and only records at the red end. (This is the exact opposite of native silver halide
emulsion films; Section 10.16.) There are a number of ways in which this is corrected. The gate layer must be
made as thin as possible, allowing theviolet end of the spectrum to penetrate. Carefully positioned perforations
can be made in the gate to allow violet photons to reach the photoactive silicon layer. The device can also
be inverted, so that photons enter the photoactive silicon directly, not through the ‘front’ gate electrode layer.
All these are made use of in current cameras.
The sensitivity of each pixel, evenwhen the greater absorption of the violet end of the spectrum is balanced,
does not match the sensitivity of the eye. To this extent the pixels measure in radiometric units, while the eye
responds to photometric units. This sensitivity can be adjusted by using filters. The simplest method is to use
a Bayer filter. Each pixel in the array is covered by a filter that only allows one light frequency range through.
In order to match the sensitivity of the eye (Section 1.10) these are in a ratio of one red, one blue and two green
filters. The disadvantage of filters of any kind is that they significantly reduce the amount of light reaching the
photoactive region of the pixel, thus reducing sensitivity considerably. A better (and more costly) technique is
Colour and the Optical Properties of Materials 476
to use three separate CCDs. The image is passed through a prism or dichroic crystal so that the red, green and
blue parts of the image are received by a separate sensor array.
There are many other techniques for image acquisition that are used in specialist areas, ranging from deep-
space astronomy to light microscopy. These can be explored via this chapter’s Further Reading.
10.15 The Colours of Metals
Metals are definedasmaterials inwhich theuppermost energyband is onlypartlyfilled.This canbe imagined to
be the logical outcome of shrinking the band gap of a semiconductor to zero. The highest energy attained by
electrons in the resulting single band is called the Fermi energy or the Fermi level in a one-dimensional
situation. More correctly, this is known as the Fermi surface in the three-dimensional crystal.
The key point about a metal is that the higher empty electronic energy levels of a metal are so close to the
uppermost filled levels that they form an essentially continuous band of allowed energies. Above the Fermi
energy almost all the levels are empty (at absolute zero they are all empty) and so can accept electrons excited
from lower energy levels. To a first approximation this means that all incident radiation can be absorbed,
irrespective of its wavelength.
Intuitively, this would lead one to expect that a metal should appear black. However, each excited electron
can immediately fall back to the state that it came from at once, emitting exactly the same energy, causing a flat
piece of metal to appear reflective. Ordinary mirrors are metal films deposited onto glass. In a good mirror the
absorptionand reflection shouldbe identical over thewhole of the spectrumandall colours accurately reflected.
Exactly the same absorption and emission processes lead to finely powderedmetals having a black appearance.
This is because the re-emitted (i.e. ‘reflected’) photons are reabsorbed again in nearby grains and ultimately do
not emerge at the ‘angle of reflection’ and so do not enter the eye.
To take this absorption into account, the refractive index N of a metal is written as:
N ¼ nþ ik
where n is the ‘normal’ refractive index defined in Chapter 2, k is the extinction coefficient, coefficient of
absorptionorattenuation coefficient, and i is the square root of�1 (Section 2.1). Thevalues ofn andk are often
called the optical constants of a material, although they vary considerably with the wavelength of the
irradiation used as a probe and are not constant at all. For a metal the extinction coefficient k and the refractive
index n are both strongly wavelength dependent (Table 10.3). The reflectivity of ametal depends upon n, k and
Table 10.3 Reflectivity of copper, silver and golda
Wavelength/nmCopper Silver Gold
n k R n k R n k R
705 0.21 4.205 0.956 0.04 4.838 0.993 0.13 4.103 0.971660 0.22 3.747 0.943 0.05 4.483 0.991 0.14 3.697 0.963620 0.30 3.206 0.900 0.06 4.152 0.987 0.21 3.272 0.931549 1.02 2.577 0.619 0.06 3.586 0.983 0.43 2.455 0.787496 1.22 2.564 0.576 0.05 3.093 0.981 1.04 1.833 0.447451 1.24 2.397 0.539 0.04 2.657 0.980 1.38 1.914 0.408397 1.32 2.116 0.464 0.05 2.070 0.963 1.47 1.952 0.407
aData from: P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370–4379 (1972).
477 Colour in Metals, Semiconductors and Insulators
the polarisation of the light. For light falling perpendicularly on a metal surface, polarisation can be ignored
and the reflectivity is given by:
R ¼ ðn�1Þ2 þ k2
ðnþ 1Þ2 þ k2
If k is omitted, the formula reduces to that for a normal insulator such as glass.
The colours of copper and gold are due to the fact that the absorption and emission of photons are noticeably
dependent on wavelength across the visible (Table 10.3). These data indicate that both gold and copper have
rather low reflectivity at the short wavelength end of the spectrum and so yellow and red will consequently be
reflected to a greater degree. This leads to the colours observed. Silver, on the other hand, has a high
reflectivity which does not vary significantly with wavelength, making it suitable for use in mirrors for
astronomical telescopes. It has now largely been replaced in this use by aluminium, which has a similar high
and uniform reflectivity over the visible spectrum and which forms a protective transparent oxide film over
the metal on exposure to air. Silver films, on the other hand, gradually degrade, especially in polluted
atmospheres.
Thin flakes of a ductile metal such as aluminium, produced by ball milling, are added to paints to obtain
a ‘metallic’ effect. Aluminium is especially suitable from this point of view and is the commonest metal used,
but bronzes and copper alloys are also employed for this purpose. Metal flakes are rarely employed alone,
usually being used in conjunction with other pigments to produce a shining and attractive finish.
A number of compounds, notably metal oxides, change from metallic to insulating behaviour at a definite
transition temperature. This is the case, for example, with the oxide VO2. At room temperature this oxide
behaves like a poor semiconductor. Above 68 �C it becomes a metal, with characteristic reflectivity. This
example of thermochromism is brought about by a change of bonding and, hence, of symmetry of the structure,
frommonoclinic at low temperatures to tetragonal at high temperatures. This type of transition obviously has a
value for the fabrication of ‘smart’ windows and similar devices, which can reflect sunlight when the day is hot
yet allow it in when the day is cool. The transition temperature is too high for this to be effective in normal
climatic conditions, but doping VO2, particularly with WO2 to form V1 xWxO2, with x equal to 1 or 2 at.%,
reduces the transition temperature to nearer normal room temperatures.
10.16 The Colours of Metal Nanoparticles
10.16.1 Plasmons
Asdescribed inChapter 5, the optical properties of smallmetal particles, often referred to asmetal sols, colloids
or nanoparticles, is dominated by absorption. Mie scattering theory is well able to describe the production of
colour of spherical particles by a combination of scattering and absorption, but it does not address the origin of
the absorption itself. The earliest theories to attempt thiswere developed in theyears between1899 and1903by
Drude and thenLorentz. The collective theory that emerged is knownas theDrude Lorentz free electron theory
of metals. It is remarkably successful and, besides acting as the nucleation site for many quantum-mechanical
theories of the solid state, is stillwidely applied today.The essence of the ideawas that ametalwas a solidwhich
contained free electrons that behaved rather like a gas and were confined to a ‘box’ that was a representation
of the shape of the bulk metal.
In terms of this classical theory, the absorption of the light (electromagnetic) wave falling upon a metal
induces an oscillation of the free electrons present. These, in turn, reradiate an electromagnetic wave which is
recorded as scattering, as exemplified by theRayleigh andMie theories described above.Aswith all oscillating
Colour and the Optical Properties of Materials 478
systems, most light frequencies will only interact slightly with the electrons but, when the properties of the
metal are appropriate, a particular frequency will be strongly absorbed, a phenomenon known as resonance.
Pushing a swing illustrates this effect for a mechanical system. The frequency of the pushes will generally be
out of synchronisation with the swing oscillations and notmuch energywill be transferred. However, when the
frequency of the pushes just matches the normal oscillation frequency of the swing, a large amount of energy is
transferred and the swing goes higher and higher. The frequencyof the oscillation of the electron gas for ametal
in air or a vacuum, which was found to be independent of thewave vector of the electromagnetic wave and the
shape of the metal, was called the plasma frequency and is given by:
o2p ¼
ne2
mee0
or, as n¼ 2po, by
n2p ¼ne2
4p2mee0
where n is the density of free electrons in themetal, e is the charge on the electron,me is the electronmass and e0is the permittivity of free space. Note that the electron mass in a nanoparticle, the effective mass, is usually
different from the mass of a free electron in a vacuum.
This equation suggests that a metal should be transparent for radiation with a frequency greater than the
plasma frequency. In simple terms, the electrons cannot oscillate fast enough to interact with the electromag-
netic field. For radiation at frequencies less than the plasma frequency, this interaction is total, irrespective of
wavelength. The electrons will absorb all of the incident radiation and then immediately reradiate it. Inserting
values for the constants in the equation shows that the plasma frequency falls in the ultraviolet. Hence, metals
should change from being transparent to being reflective in this radiation region, a prediction that is well
obeyed.
In reality, these classical collective electron oscillations are limited by quantummechanical considerations.
In this case, the oscillation is described as a plasmon, which is the quantum-mechanical particle corresponding
to the collective oscillation of the bulk electrons in the metal. The absorption part, k, of the optical constants of
the metal is now seen to arise from the plasmon oscillations in the metal. (Note, though, that the bulk optical
constants of a metal may not apply to small particles.)
10.16.2 Surface plasmons and polaritons
From the point of view ofmetal nanoparticles, the most important plasmons are those at the surface of ametal,
called surface plasmons. These can be imagined as electron densitywaves confined, like ripples inwaterwaves
on the surface of a pond, to the metal surface. The collective oscillation of the electrons is called surface
plasmon resonance. These may interact with incoming photons to form a hybridized quasiparticle called
a surface plasmon polariton.9 Note that surface plasmon polaritons are often just called surface plasmons,
although strictly speaking this is incorrect.
The colour of metal nanoparticles, especially those of silver and gold, the most studied metal nanoparticles,
is dominated by surface plasmon polariton formation. The peaks that appear in the extinction spectra of these
9 A polariton is hybridised state that forms when a photon couples with another excitation such as a phonon or a plasmon.
479 Colour in Metals, Semiconductors and Insulators
small particles (Section 5.9) are due to the interaction of photons with surface plasmons. The recording of the
peaks due to surface plasmons or surface plasmon polaritons is referred to as surface plasmon resonance
spectroscopy.
The detail of these surfacewaves is a function of the geometry of the surface and also of the optical constants
of the surroundingmedium. For example, a planar metal film in a vacuumwill support a surface plasmonwave
of frequencyop= 2p
, whereop is the bulk plasma frequency. The plasmon so formed canmoveover the surface
and is known as a propagating surface plasmon or propagating surface plasmon polariton. This is intriguing.
It means that a photon, with a wavelength much greater than that of the foil, can be moved along the surface in
this coupled way, opening the door to bypassing diffraction-limited optics (see below).
When the oscillations are confined on the surface of a nanoparticle they are no longer free to travel, and so
form localised surface plasmons or localised surface plasmonpolaritons. For a spherical particle, the frequency
of these confined oscillations is given by op= 3p
, where op is the bulk plasma frequency.
Experimentally, gold spheres give strong absorption peaks in thewavelength range between 510 and550 nm,
and silver spheres in the range from400 to440 nm.Although not apparent from the simple theory just given, the
absorption peak moves towards the red end of the visible as the sphere radius increases.
In reality, the number and frequency of the surface plasmons onmetal nanoparticles are dependent upon the
shape and size of the particle. For gold and silver, these lie in the visible, and so contribute to the well-known
colours displayed (Section5.5).Mucheffort is nowbeing focusedupon the synthesis of nanoparticles of precise
shapes, thus allowing tuning of the absorption. Cylindrical rods generally show two absorption peaks, one
corresponding to the long dimension of the rod, the longitudinal surface plasmon polariton, and the other to the
short dimension, the transverse surface plasmon polariton (Figure 10.43a). These produce two energy levels
(Figure 10.43b) that give rise to two absorption maxima which, when added together, produce the observed
colour of the particles. Because the energies are shape sensitive, the observed colour changes as the rod
dimensions change.
Shape tuning of the absorption characteristics of nanorods is limited, both by the geometry of the
material and by the considerable synthetic skills needed to prepare the desired shape in sufficiently large
quantities and purity. A second method of tuning the colour of the particles is to deposit shells of precious
metal on an insulating core. In essence, two surface plasmon polaritons are generated, one on the outside
of the metal sheath, corresponding to that for a solid spherical nanoparticle, and one on the inside of the
metal sheath, corresponding to a surface plasmon polariton on the surface of a cavity in the metal
(Figure 10.43b g). Because the metal shell is thin, these interact to give rise to two new energy levels
similar to the formation of two molecular orbitals by the interaction of two atomic orbitals (Chapter 8). The
process is called plasmon hybridisation. The energy levels of the new surface plasmon polaritons are given
approximately by:
hoþ ¼ hos þ hoc
ho ¼ hos�hoc
where os represents the plasmon contributed by the solid sphere component and oc represents the plasmon
contributed by the cavity component (Figure 10.43h). The thicker the shell is, theweaker is this interaction and
the closer the energies of the sphere and cavity plasmons become, until the two energy levels are the same and
equal to that of a solid sphere. Only the lower energy level ho interacts strongly with the electric field of the
light wave, but this is sufficient to provide awide range of tunable frequencies, because the shell thickness, the
core diameter and the total nanoparticle radius can be varied. Multiple shells can also be fabricated, giving yet
more flexibility to the colour-varying abilities of these materials. As would be expected, the colours shown by
these particles will change if they are embedded in materials other than air.
Colour and the Optical Properties of Materials 480
10.16.3 Polychromic glass
The relative contribution of scattering and absorption of small spherical metal particles is given byMie theory
(Sections 5.9 and 5.10). However, an explanation of the colours found required the introduction of the surface
plasmon concept outlined above. An example of how the colours of small needle-shaped particles depend
critically on particle dimensions is provided by a mid-twentieth century example of the fabrication of
‘polychromic’ coloured glass and highlights the careful processing that is necessary to achieve desired results.
The colour-forming centres are minute silver needles. These form in glass after a complex set of heating
cycles that initially results in the formation of sodium fluoride cubes that form around tiny silver grains as
nuclei. These cubes then act as nucleation sites for pyramids of amixed sodium silver bromide phase, (Na,Ag)
Br, that grow on the cubes of NaF, followed, finally, by the photochemical initiated growth of needle-like
crystals of silver on the tips of the (Na,Ag)Br pyramids (Figure 10.44). The glass remains colourless when the
crystals are below about 200 nm in size as they are too small to scatter light appreciably. If the crystallites
become much larger than this then they scatter light and the glass becomes hazy or opalescent and has to be
rejected. However, when the needles have dimensions of between 3 and 6 nm wide and between 3 and 36 nm
in length then they are too small to cause much light scattering, but they do absorb strongly and generate
brilliant colours when the glass is viewed in transmission. The precise absorption characteristics depend
critically on the needle shape, especially the ratio of the width to the length (Table 10.4). In order to achieve
longitudinalplasmon ω l
ω l
ωs
ωs
ω–
ω+
ωc
ωc
transverseplasmon ω t
ωt
plasmonon spheresurface ωs
plasmonon cavitysurface ωc
plasmonson shellω ω+–
Ene
rgy
Ene
rgy
Ene
rgy
Ene
rgy
0
0
0
0
(a) (b)
(c) (d)
(e) (f)
(g) (h)
�
�
�
�
�
�
�
�
Figure 10.43 Surface plasmon polaritons and associated energy levels for: (a), (b) metal nanorods; (c), (d) solidmetal spheres; (e), (f) a cavity in a solidmetal; (g), (h) a thinmetallic shell on an insulating core. The energy scale isnotional and the zero level is only to show relative positions
481 Colour in Metals, Semiconductors and Insulators
a uniform bright colour the silver needles must all be of a similar size, a task needing considerable processing
skill for success.
10.16.4 Photochromic glass
Photochromic glass is a material which is sensitive to light that owes its properties to silver nanoparticles.
Although many types of photochromic glass have been fabricated, the best known are those which darken on
exposure to high-intensity visible or ultraviolet light and regain their transparencywhen the light intensity falls.
Suchglasses arewidelyused in sunglasses, sunroofs and for architectural purposes. (Forphotochromicplastics.
see Section 8.13.)
The mechanism of the darkening transformation is similar to that involved in the photosensitive glass
described in the previous section. Photochromic glasses are complex materials which usually contain silver
(a)
(b)
(c)
NaF crystallite
(Na, Ag)Br pyramidal crystal
silver nanorod or elongated pyramid
Figure 10.44 The formation of silver nanoparticles in polychromatic glass: (a) initial heat treatment forms cubiccrystals of sodium fluoride (NaF); (b) further processing causes silver halide crystals, mainly consisting of sodiumbromide (NaBr), to grow on the cubic faces of the NaF crystallites; (c) needle tips become photoreduced to silvernanorods and pyramids
Table 10.4 Colour and needle dimensions in polychromic glass
Needle length/nm Needle width/nm Length/width Colour transmitted
3.0 3.0 1.0 yellow4.0 3.0 1.3 deep yellow5.0 3.0 1.7 orange6.0 3.0 2.0 red orange7.5 3.0 2.5 red10.0 3.5 2.9 magenta12.0 3.5 3.4 purple16.0 4.0 4.0 blue21.0 4.5 4.7 turquoise36.0 6.0 6.0 green
Colour and the Optical Properties of Materials 482
halides as the light-sensitive medium. The glass for this use would typically be an aluminoborosilicate (Pyrex
type)material containing about 0.2wt% of silver bromide or chloride. In addition, a small amount of a cuprous
chloride (CuCl) is also added.When the glass is first fabricated it is cooled rapidly. Under these conditions the
silver and copper halides remaindissolved in thematrix and theglass produced is transparent anddoes not show
any photochromic behaviour at all (Figure 10.45a and b). This material is transformed into the photochromic
state by heating under carefully controlled conditions of temperature and time, which might be, for example,
550 �C for 30min followed by 650 �C for 30min. The heat treatment is chosen so that the halides crystallise
in the glass matrix (Figure 10.45c). Care must be taken to ensure that the crystals do not become too large and
that they do not aggregate. A desirable size is about 10 nm diameter and the individual crystallites should be
about 100 nm apart. Contrary to the polychromic glass described above, in this case it is important that the
processing conditions give a wide range of particle sizes, so that in effect the whole of the visible is uniformly
absorbed.
It is important that the copper is in the monovalent state and incorporated into the silver halide crystals as
an impurity. Because theCuþ has the samevalence as theAgþ , someCuþ will replaceAgþ in theAgXcrystal
to form a dilute solid solution CuxAg1 xX (Figure 10.45d). The defects in thismaterial are substitutional CuAgpoint defects. These crystallites are precipitated in the complete absence of light, after which a finished glass
blank will look clear because the silver halide grains are so small that they do not scatter light appreciably.
Light photons incident on the clear glass will liberate electrons from the Cuþ ions which are converted to
Cu2þ ions (CuAg.) in the process. These electrons are trapped by interstitial silver ions, which exist as Frenkel
defects, to form neutral silver atoms:
hnþCuAg !CuAg. þ e0
e0 þAgi. !Agxi
Agxi þAgxi ! 2Agxi
This process continues until a small speck of silver is created. It is these clusters of silverwhich absorb the light
falling on the glass. The absorption characteristics of the silver specks depend quite critically upon their size
and shape.Asmentioned, photochromic glass production is carefully controlled so as to produce awidevariety
of shapes and sizes of the silver specks. For example, if the silver specks are rod shaped, each will have two
Melt Homogeneousglass blank
cast heat treatment
Photochromicglass
(a) (b) (c)
(d)
Cl
Ag
Cu
Figure 10.45 Photochromic glass: (a) glass melt containing CuCl and AgCl; (b) the melt is cast into ahomogeneous glass blank; (c) heat treatment precipitates nanocrystals; (d) sodium chloride structure of AgClcontaining copper impurities and Frenkel defects
483 Colour in Metals, Semiconductors and Insulators
absorption peaks, depending upon the ratio of length towidth. Awide variety of shapesmeans that thewhole of
the visible spectrum is covered, ensuring that the glass darkens uniformly.
In order for the glass to become clear again after irradiation it is essential that the silver particles release
electrons to the Cu2þ ions when the light is turned off, reforming Cuþ ions and making the whole process
reversible. This bleaching process is the reverse of the darkening process. In fact, the darkening and bleaching
reactions are taking place simultaneously under normal circumstances, in dynamic equilibrium. When the
amount of incident light is high, a large number of silver specks are present in the glass, hence leading to a high
degree of darkening. At low light intensity the number of silver particles present decreases and the glass
becomes clear again.
Commercially useful materials require that the rate of the combined reaction is rapid. If the darkening takes
place too slowly, or if the subsequent fading of the colour is too slow, the materials will not be useful. The
presence of the copper halide is essential in ensuring that the kinetics of the reaction are appropriate and that the
process is reversible.
10.16.5 Photographic film
Photographic film was the most widely used storage method for images throughout the twentieth century, and
still has an important part to play in image capture and storage. Both black and-white and colour photography
rely on nanoparticles of silver to capture images. The light-sensitive materials employed that give rise to the
nanoparticles are silver halides, notably AgBr, which are dispersed in gelatine to form the photographic
emulsion. Inorder toensure that the crystals are freeofmacroscopic defects suchasdislocations,whichdegrade
the perfection of the photographic images produced, the silver halide crystals are carefully grown within the
gelatine matrix itself. The crystals so formed are usually thin triangular or hexagonal plates, varying between
0.01 and 10 mm in size, and in photographic parlance are known as grains.
After illumination, some grains will have interacted with the light photons while some remain unchanged.
Despite the fact that not all details of the photographic process are completely understood, the overall
mechanism for the production of the silver particles is known and follows a path similar to that originally
suggested by Gurney and Mott in 1938:
1. Interaction of a light photonwith a halogen ion in theAgBr crystal. The energy from the photon hn liberates
an electron from this ion:
hnþBr ! e0 þBr.
2. The liberated electron is free to move in the structure and migrates to an interstitial silver ion Agþ (Agi.),
which is part of a Frenkel defect in the crystal, to form a neutral silver atom Ag (Agxi ):
Agi. þ e0 !Agxi
3. In many instances, the above reaction will then take place in the reverse direction, and the silver atom will
revert to thenormal stable state as aFrenkel defect.However, themetal atomseems tobe stabilised if another
photon activates a nearby region of the crystal before the reverse reaction can take place, to produce a cluster
of two neutral silver atoms:
Agxi þAgxi ! 2Agxi
4. Further aggregation of Ag atoms occurs by a similar mechanism.
Colour and the Optical Properties of Materials 484
In this state the emulsion is said to contain a latent image. The film is then put into a developer, which is a
reducing agent. A grain that has interacted with light is totally reduced to metallic silver. The reactions taking
place can be written down schematically as:
AgBr ðcrystalÞþ light photons!½AgBr crystalþ latent image�½AgBr crystalþ latent image� þ developer!Ag crystal
All other crystallites remain unchanged. The final step in the photographic process, fixing, removes the
unreacted silver bromide crystals from the emulsion, thus preventing further reaction (Figure 10.46).
light photons
(a)
(b)
(c)
(d)
(e)
crystal with latent image
silver crystal
silver halide crystal
Figure 10.46 Production of a negative image in a photographic emulsion: (a) film emulsion; (b) interaction ofsome crystallites with light; (c) crystallites containing latent images; (d) development transforms crystallitescontaining latent images into silver grains; (e) fixing removes all unreacted silver halide crystallites
485 Colour in Metals, Semiconductors and Insulators
The image is stored in the emulsion by the silver crystallites. These are densely packed where the irradiance
was high and are sparse where the irradiance was low. Bright areas on the image appear dark on the emulsion,
and the result is a negative (image). A photograph (that is, a positive image) is created by exposing another
emulsion layer, usually coated onto a sheet of paper, to light that has passed through the negative.One negative
can produce as many positives (or prints) as needed.
This simple picture ignores the fact that silver halides are not sensitive to the whole visible spectrum but
respond mainly to short-wavelength (violet) light. This causes severe tonal problems in black-and-white
photography,which relies upon agrey scale to indicate dark and bright parts of the image. Thus, on an untreated
silver-halide-derived negative, blues, indigos and violets appear black and the other colours are only poorly
registered. In a print (i.e. positive), the blues, violets and indigos appear to be far too pale. To broaden the
sensitivity, the silverhalide crystals are treatedwith sensitizingdyes so that they respond to longerwavelengths.
These dyes are adsorbed onto the surface of the silver halide crystals and absorb light energy, which is then
transferred to the halide crystal, initiating the sequence of steps described above. Themostwidely used of these
are derivatives of the cyanines (Section 8.6). The first dyes used extended the sensitivity into the yellow and
green region; the result being the orthochromatic films. The overreaction to blue, violet and indigo could be
corrected by using yellow filters, but because the film did not respond to red, the negative for red objects was
clear and the resultant prints show all red areas far too dark. Later black-and-white films, so called
panchromatic films, contained sensitizers that allowed all of the visible spectrum to be absorbed to an extent
that the finished print showed colours in the expected tonal range, with blues indigos and violets appearing
darkest and red appearing lightest.
Colour films also relied upon the same silver halide processes. In this case the emulsion consisted of three
layers, sensitive to blue, green and red. On processing, the exposed silver halides were replaced by dyes. Most
colour films used the substractive primaries cyan, magenta and yellow. Processing led to either a positive
(colour slides, for projection) or a negative (to be printed on paper) end result (see this Chapter’s Further
Reading).
10.16.6 Metal nanoparticle sensors and SERS
Nanoparticle sensors, using colour change for the detection of chemicals, are readily available. Studies of this
effect have centred upon the chemically inert precious metals gold and silver. Oneway in which colour change
can be initiated is by formation of clusters. Clearly, if gold or silver nanoparticles cluster significantly then they
are no longer quite as ‘nano’ and the observed colourwill change. If the nanoparticles are treatedwith a surface
layer that is sensitive towards an additive that can promote clustering, then the technique becomes an analytical
one. For example, nanoparticles treated with DNA fragments can combine with complementary DNA
fragments leading to a colour change that is detectable. Similarly, nanoparticles treated with surfactants so
as to bind to a specificmetal in solution canbe used as an analytical test for themetal using colour changes of the
nanoparticle suspension as the indicator. In this way, tests for toxic metals, such as arsenic in drinking water,
have been developed.
Apart from colour,metal nanoparticles have the unusual property of strongly enhancing the electromagnetic
field close to their surface. When a photon interacts with a metal nanoparticle to create a surface plasmon
polariton, the electromagnetic field is concentrated at the particle surface. In fact, the electromagnetic field can
be enhanced by a factor of 104 or more. This enhanced field has a spatial range and contour that is dependent
upon the shape of the nanoparticles. The effect has been used in a variety ofways, the best known being surface
enhanced Raman spectroscopy (SERS).
Raman spectroscopy is a well-established chemical analysis tool. It is based on inelastic scattering of
photons from molecules. The majority of photons scattered by a molecule are scattered elastically and the
behaviour is described byRayleigh scattering theory (Sections 5.2 5.4). In essence, themolecule is treated like
Colour and the Optical Properties of Materials 486
an antenna that simply reradiates the incident disturbance. However,molecules have rotational and vibrational
energy levels, and in some circumstances a scattered photon can give up some of its energy to a molecule,
exciting it to newvibrational or rotational levels. The inelastically scatteredphoton is thendepleted in this small
increment of energy, and has a lower frequency than the original. The output is called Stokes radiation.
Similarly, if molecules are in excited vibrational and rotational levels then a scattered photon can pick up an
increment of energy, allowing the molecule to drop down to lower energy levels. The departing inelastically
scattered photon then has more energy and a higher frequency than the incident photons. This output is called
anti-Stokes radiation. (The elastically scattered radiation, which suffers no energy change, is called Rayleigh
radiation.) The change in energy of the inelastically scattered radiation is called the Raman effect. It occurs
in about 1 in every 107 or so incident photons, and so the effect is very weak; nevertheless, monochromatic
lasers with a high-intensity beam have allowed Raman spectroscopy to become an important tool in the study
of molecular energy levels. This is because each molecule has a unique Raman spectrum that can be used as a
fingerprint.
Theweakness of theRaman signal can be greatly increased by the surface electromagnetic field of ametallic
nanoparticle. In effect, nanoparticles are coated with the molecules to be studied. The electric field of the
incident photon is greatly enhanced at themetal surface, thus greatly increasing theRamaneffect. The scattered
photons are also similarly enhanced, so that theRaman signal is amplifiedby a factor of 106 108. This lies at the
heart of SERS.
To use the technique, nanoparticles of mainly gold (but silver and copper are also used) in a colloidal
suspension or on a thin film are brought into contact with the material to be analysed. Molecules attach to the
nanoparticleswhich are then examinedbyRaman spectroscopy.The amplificationof the signal nowmeans that
even single molecule attachment can be detected. However, the surface enhancement of the electromagnetic
field is sensitive to nanoparticlegeometry, and carefullyprepared colloidswith auniformshape andnarrowsize
distribution is essential for the work.
10.17 Extraordinary Light Transmission and Plasmonic Crystals
In 1989 Ebbesen discovered that a thin gold film perforated by small holes, of diameter much less than the
wavelength of visible light, deposited onto a glass slide, was able to transmit light very well, although,
simplistically, no light should be transmitted at all. The phenomenon, called extraordinary light transmission,
was fairly complicated, in that although some wavelengths of light were transmitted with an unusually high
intensity, otherwavelengthswere not transmitted aswell, so that objects viewed through the foilwere coloured.
The explanation of the effect, which took 10 years to unravel, was that the incoming photons interacted with
surface plasmons at themetal dielectric interface, were transported through the holes in the foil and were then
reradiated. The colours transmitted are those near to the natural oscillation frequency of the plasmons; that is,
the transmission spectrumof the film shows peaks at frequencies corresponding to the excited surface plasmon
modes. These, however, depend upon the geometry of the array of holes and their sizes. The consequence is
that the optical transmission of the perforated foil can be changed by adjusting the geometry and disposition of
the holes. In the simplest cases, holes, circular or square, are arranged on a crystal-like lattice, with a repetition
defined by a ‘unit cell’. Other surface geometries have also been explored, including regular arrays of
nanopyramids.
As well as the hole geometry, the surrounding medium is also important. If this is different on each side of
the film, as when a glass slide is used as a substrate, and the whole is viewed in air, the surface plasmons
formed on each side of the film have different frequencies. This means that the transmission spectra consist of
two sets of peaks, offset by the difference in the refractive indices of the insulating medium in contact with the
metal film.
487 Colour in Metals, Semiconductors and Insulators
Metal films perforated or patterned in a regular array, by dimples, holes slits or grooves and so on, separated
on a nanometre scale are called plasmonic crystals. The optical behaviour of these objects is dependent upon
the nature of the metal, the nature of the patterns and the interface between the metallic and insulating
surrounding medium. This latter property allows the device to be used as a sensor for molecules or molecular
layers deposited on the surface. In thisway it has been possible to detectminute quantities of absorbedmaterial
and to differentiate between closely related molecular species.
Further Reading
An introduction to band theory adequate for this book is
R. J. D. Tilley, Understanding Solids, John Wiley and Sons, Ltd, Chichester, 2004.
Much information on colour centres and the colours of irradiated minerals is given by
K. Nassau, Gems Gemol. XIV, 343 355 (1980).
K. Nassau, The Physics and Chemistry of Color, 2nd edition, Wiley-Interscience, New York, 2001.
Dental phosphors and related materials are described by
J.-M. Spaeth, Radiat. Meas. 33, 527 532 (2001).
For a complete discussion of solid-state lighting, especially with reference to LEDs, see
C. J. Humphreys, Mater. Res. Soc. Bull. 33, 459 470 (2008).
LEDs and diode lasers:
S. Nakamura, Mater. Res. Soc. Bull. 22 (February), 29 35 (1997).
S. Nakamura, Mater. Res. Soc. Bull. 23 (May), 37 43 (1998).
N. Holonyak Jr, Mater. Res. Soc. Bull. 30, 509 517 (2005).
M. Fox, Optical Properties of Solids, Oxford University Press, 2001, Chapter 5, esp. 5.4.
B. E. A. Saleh and M. C. Teich, Fundamental of Photonics, John Wiley and Sons, Inc., New York, 1991,
Chapters 15 and 16.
OLEDs:
M. Thompson, Mater. Res. Soc. Bull. 32, 694 701 (2007).
S. So, J. Kido, P. Burrows, Mater. Res. Soc. Bull. 33, 663 669 (2008).
S. Ye, Y. Liu, C.-A. Di, H. Xi et al., Chem. Mater. 21, 1333 1342 (2009).
Dendrimers in OLEDs:
J. Li, D. Liu, J. Mater. Chem. 19, 7584 7591 (2009).
Electrochromic displays, including ‘smart’ windows are described by
P. Monk, R. Mortimer, D. Roseinsky, Electrochromism and Electrochromic Devices, Cambridge
University Press, 2007.
A. A. Argun, P.-H. Aubert, B. C. Thompson, I. Schwendeman et al., Chem. Mater. 16, 4401 4412 (2004).
R. Baetens, B. P. Jelle, A. Gustavsen, Sol. Energ. Mater. Sol. Cells 94, 87 105 (2010).
The PANI PSS/PEDOT PSS electrochromic device that served as the basis for text discussion is detailed in
L.-M. Huang, C.-H. Cheng, T.-C. Wen, Electrochim. Acta 51, 5858 5863 (2006).
The historical evolution of solar cells can be followed via the following sources:
Y. Hamakawa, Sci. Am. 256 (April), 77 82 (1987).
Colour and the Optical Properties of Materials 488
Various authors, Mater. Res. Soc. Bull. 18, (October), 18 66 (1993).
Various authors, Mater. Res. Soc. Bull. 30, 10 52 (2005).
Various authors, Mater. Res. Soc. Bull. 32, 211 247 (2007).
D. Ginley, M. R. Green, R. Collins, Mater. Res. Soc. Bull. 33, 355 364 (2008).
The historical evolution of dye-sensitized solar cells can be followed via the following sources:
B. O. Regan, Nature 353, 737 740 (1991).
M. Gr€atzel, Mater. Res. Soc. Bull. 18, (October), 61 66 (1993).
M. Gr€atzel, J. Photochem. Photobiol. C Photochem. Rev. 4, 145 153 (2003).
M. Gr€atzel, Mater. Res. Soc. Bull. 30, 23 27 (2005).
S. Ahmad, J.-H. Yum, Z. Xianxi, M. Gr€atzel, H.-J. Butt, K. Nazeerruddin, J. Mater. Chem. 20, 1654 1658
(2010) and references cited therein.
Photodiodes and charge coupled devices are described in:
B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, John Wiley and Sons, Inc., New York, 1991,
Chapter 17.
Much technical information concerning digital photography can be obtained from thewebsites of microscope
and camera manufacturers, including Nikon, Olympus, Canon and so on.
Polychromic glass is described in
D. M. Trotter, Sci. Am. 264 (April), 56 61 (1991).
S. D. Stookey, G. H. Beall, J. E. Pierson, J. Appl. Phys. 49, 5114 5123 (1978).
For a concise introduction to electromagnetic waves in solids and the derivation of the plasma frequency, see
L. Solymar, D. Walsh, Electrical Properties of Materials, 7th edition, Oxford University Press, Oxford,
2004, Chapter 1.
More detail, clearly presented, is in
N. Braithwaite (ed.), Electromagnetism, Book 3, Electromagnetic Waves, The Open University, Milton
Keynes, 2006.
A survey of the surface plasmonic properties ofmetallic nanoparticles is given in a series of articles by various
authors in Mater. Res. Soc. Bull. 30, 338 389 (2005).
The photographic process is described, together with many references, in the following reviews and in much
technical literature produced by the manufacturers of film:
F. C. Brown, The photographic process, in Treatise on Solid State Chemistry, Vol. 4, Reactivity of Solids,
N. B. Hannay (ed.), Plenum, New York, 1976, Chapter 4.
J. A.Kapecki, J. Rodgers, Colour photography, inEncyclopedia of Imaging Science and Technology, Vol. 1,
J. P. Hornak (ed.), John Wiley and Sons, Inc., New York, 2002.
S. H. Mervis, V. K. Walworth, Instant photography, in Encyclopedia of Imaging Science and Technology,
Vol. 1, J. P. Hornak (ed.), John Wiley and Sons, Inc., New York, 2002.
Plasmonics and plasmonic crystals are described by
H. A. Atwater, Sci. Am. 296 (April), 38 45 (2007).
Various authors, Mater. Res. Soc. Bull. 30, 338 380 (2005).
J. Heber, Nature 461, 720 722 (2009).
T. W. Odom, Mater. Res. Soc. Bull. 35, 66 73 (2010).
The first report of extraordinary transmission was
T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, Nature 391, 667 669 (1998).
489 Colour in Metals, Semiconductors and Insulators
Index
Note: Figures and Tables are indicated (in this index) by italic page numbers, footnote by suffix ‘n’
Abb�e number, 67Abb�e numerical aperture value, 204Abb�e V-value, 67abscission layer, 332absorbance, 36absorption, 17, 18, 33 4
double, 297Einstein coefficient, 20rate of, 18 21
absorption band, 65, 66absorption coefficient, linear (Napierian), 35absorption edge, 290, 420absorption efficiency (factor), 186absorption index/coefficient, 51, 477absorption spectrum
Nd3þ -doped glass, 289and polarized light, 151, 279 80rhodamine 6G dye, 356solar radiation, 255, 256transition metal ions, 264, 265, 275, 278water and ice, 315 16
acceptor dopants, 424acceptor (in quenching), 376achromats, 68ACTFEL (AC thin-film electroluminescent) display, 391, 395–6actinide, 301nactinoid compounds, colours, 295actinoids, 249activators, 366, 379active matrix display, 39, 170additive colour mixing, 29 31
on displays, 170, 447, 463ADP (ammonium dihydrogen phosphate), 153aequorin, 416aether drift, 2afterglow, 373 4, 387agate, 366aggregation, 435aglycons, 324, 329AGS (silver gallium sulfide), 153AGSe (silver gallium selenide), 153air lens, 64airlight, 180, 181Airy disc, 202, 203
Airy rings, 202, 203, 226Airy wavefront, 205alanine, 165albite, 366Alexander’s dark band, 69, 71, 75alexandrite, 283 4, 286algae, chlorophylls in, 320alizarin, 352alkali metal halides, F centres, 430, 431alkali metals, (energy) term for, 252alkaline earth compounds, colours, 255, 315alumina, 43, 189aluminium, 478aluminium nitride, 441aluminium oxide: see alumina; corundumaluminium oxynitride, 43amethyst, 431, 432amino acids, enantiomers, 165, 166ammonites, 140ammonium dihydrogen phosphate (ADP), 153ammonium iron(III) citrate, 342, 343amplification, 18, 19, 22amplitude diffraction grating, 198, 206, 207amplitude grating, 198amplitude hologram, 241amplitude object, 198amplitude of wave, 7, 44analyser, polarization, 137anatase, 121, 474angular frequency, 44anhydrobase, 330aniline purple, 337anisotropic materials, 138anodically coloured materials, 463, 467, 468, 469anodized films, 103 4anthocyanidins, 324 6, 329anthocyanins, 323, 324, 325, 329, 332anthracene, 378antibonding molecular orbital, 310, 457antimony ions, in phosphors, 379 80antireflection (AR) coatings, 62, 64, 105 10
graded-index, 108 10moth-eye, 64, 108, 109, 231 2in solar cell, 471, 472
Colour and the Optical Properties of Materials Richard J. D. Tilley
� 2011 John Wiley & Sons, Ltd
antireflection (AR) layer, 106 7, 109tuneable, 107
anti-Stokes fluorescence, 394anti-Stokes radiation, 487appearance of objects, 40 3apple, colours, 328aquamarine, 344 5aragonite, 366Archer fish, 49argon-ion laser, 115arsenic compounds, colours, 297arsenic poisoning, 297arsenic triselenide glass, 80arsenic trisulfide, 437astaxanthin, 319astronomical telescopes, 204, 478atomic absorption analysis, 255atomic orbitals, 264, 266, 267, 300attenuation, 34, 176extrinsic, 79factors affecting, 176in optical fibres, 79, 80
attenuation coefficient, 34, 36, 186, 477attenuation cross-section, 36aurora australis, 313aurora borealis, 313autofluorescence, 367autumn leaf colours, 332, 333–4, 335auxochrome, 317auxochromic shift, 324Avogadro’s number, estimation of, 178axisfast, 138optic, 139, 140slow, 138
azolitmin, 351azurite, 284, 285
b-radiation, measurement of, 417Babinet’s principle, 200, 201, 205backscattering efficiency, 194, 195bacteriorhodopsin, 28Balmain’s paint, 365Balmer series, 250, 251bandconduction, 4, 5, 388, 419, 420energy, 419valence, 4, 5, 388, 419, 420
band edge, 420band gap, 4, 419, 420, 421direct, 442, 443indirect, 442, 443listed for various oxides, 422optical, 420, 422
band structure, 4, 5, 420, 421band theory, 419bandgap engineering, 454bandpass filters, 114barite, 366barium chromate, 346barium compounds, colours, 255, 315barium fluorobromide, 436barium magnesium aluminate, 385barium titanate, 391bathochromes, 26bathochromic shift, 26, 317, 374Bayer filter, 476BBO (beta barium borate), 153, 157beam splitter, 56, 57
Beer Lambert Bouguer law, 35Beer Lambert law, 35, 36, 176Beer’s law, 34, 176beetles, colours, 223, 230bending mode, water molecules, 315Benton holographs, 239benzopyran, 359Berlin blue, 342beryl structure, 283, 286, 344 5biaxial crystaldouble refraction in, 144 7trichroism in, 149
bimolecular reaction, 378biofluorescence, 414bioluminescence, 366, 414 16biophosphorescence, 414birefringence, 138, 141, 147circular, 167colour produced by, 147phase matching and, 160stress, 148
bis(dimethylglyoximate)nickel(II), 350, 351bismuth borate, 155black vacuum, 257black-and-white photographic film, 486black-and-white television, 387 9black-body radiation, 11 13law, 11 13, 15, 20, 44, 45, 119
bleachingelectrochromic film and materials, 463, 467, 468eye pigments, 24, 26photochromic materials, 360, 484
blinking, of core shell quantum dots, 456, 458Blu-Ray discs, 94, 448blue butterflies, 122, 123–4, 188, 324, 326blue diamonds, 428blue eyes, 188blue feathers, 188, 234 5Blue John, 430 1blue moon, 180, 188‘blue remembered hills’, 179 80, 181blue shift, 374, 451blue sky, 23, 40, 178 9, 180blue sun, 188blueprints, 342 4Boltzmann constant, 17, 43Boltzmann’s equation (for specific rotation dispersion), 167Boltzmann’s law, 17 18, 20bonding molecular orbital, 310, 457boron-doped diamonds, 428, 429Bouguer’s law, 34, 176see also Beer Lambert Bouguer law
bowing coefficient/parameter, 441Brackett series, 251Bragg angle, 213Bragg equation, 211 13modified (for opal), 215 17
Bragg fibre grating, 115 19Bragg reflector, 114 15, 117Bragg’s law, 211 13applications, 223, 229, 230dynamical form, 224 5see also Bragg equation
Brewster angle, 134, 135Brewster’s law, 133 4Brewster window, 134, 135brightening agents, 368brightness, 28see also illuminance
Index 492
bromocresol blue, 352bromophenol blue, 352bromothymol blue, 352, 353butterfly
colours, 122, 123–4, 324, 326, 340eye, 108, 109wing markings, 114, 115wing scales, 29 30, 122, 123–4, 188
cadmium orange, 441cadmium selenide, 441, 455
ZnS-coated (quantum dot), 350, 351, 456cadmium sulfide, 437, 441
photovoltaic cell, 471, 472quantum dot, 455, 457
cadmium telluride, 472cadmium yellow, 437calcite
double refraction of, 139 40, 140, 142fluorescence, 366structure, 139, 141, 142, 143
calcium carbonate, 139minerals, 188, 366
calcium chromium silicate, 295calcium fluorophosphate, 379calcium sulfide, 365calcium tungstate, 292candle flame, 11, 14, 314carbon bonds
conjugated double, 317double, 317
carbon dioxide (CO2) laser, 156carbon nanotubes and nanorods, 136, 450carborundum, 103a-carotene, 317, 318b-carotene, 317, 318carotenoids, 317 19, 332carrot, colour, 317cathode rays, 385 6cathode-ray tube (CRT), 365, 386
television tube, 386 9cathodically coloured materials, 464, 467, 469cathodoluminescence, 365, 366, 385 90cationic configurations, 301Cauchy’s equation, 65 6CCD: see charge couple deviceCD: see compact discceramics, 188, 189 90
pigments for, 295 7transparent, 43, 189 90
ceria (CeO2), 51, 420, 421cerium compounds
absorbance spectrum, 291in sunscreens, 346
cerium ionscolours, 289, 290energy levels, 291, 383, 393in phosphors, 393 4
cerium oxide, europium-doped, 397–8chalcone, 323, 325chalcopyrite, 437, 439charge carriers, strongly confined, 450charge couple device (CCD), 474 6
‘dark’ current, 476dynamic range, 476full-well capacity, 476photography using, 476 7spectral response, 476 7
charge-transfer colours, 340, 341 2, 344 9, 346, 374
charge-transfer processes, 340charge-transfer transition
anion-to-cation, 345 6cation-to-cation, 340, 341cation-to-ligand, 340, 374intervalence, 340, 341intra-anion, 348 9ligand-to-cation, 340ligand-to-ligand, 340, 374
chemical analysis, 254 5chemiluminescence, 366, 413 14china clay, 188Chinese blue, 342chiral carbon atom, 164, 165, 166chiral centre, 164, 165, 166chiral molecule, 164chiral nematic (liquid crystal) phase, 228, 229chlorophyll, 320, 321–2, 332
cessation of production, 332chloroplasts, 332cholesteric blue phases, 230cholesteric liquid-crystal mesophase, 168, 228 30cholesterol-based compounds, 168chroma, 28chromatic aberration, 68chromaticity diagrams, 30 1, 32–3chrome alum, 284, 286chrome green, 277, 295chrome yellow, 346chromic oxide, 277, 295chromium compounds, colours, 286, 295chromogen, 317chromophore, 316 17chrysoberyl, 283, 284, 286C.I. fluorescent brightening agent 30, 368C.I. Solvent Yellow 124, 354 5, 356CIE 1931 chromaticity diagrams, 31, 32–3cinnabar, 437, 438circular birefringence, 167circular dichroism, 167circularly polarized light, 130, 131citrine, 431Clebsch Gordon rule, 304, 305close-packing of spheres and atoms, 218, 219CMK (cyan/magenta/yellow) colour model, 37CMYK colour model, 39co-activator, 379cobalt aluminate, 295 6cobalt chromite, 296 7cobalt compounds
colours, 285, 286, 295 7in glass, 285, 287, 288
cobalt silicate, 296coelenterazine, 415coherence length, 160coherent light, 7, 8, 17, 235coherent scattering, 197cold light, 363collagen, 162, 163, 233, 234
of inorganic molecules, 311 15colloidal crystals, 218, 220, 230colloids, 478coloration
additive, 29 31, 170, 447, 463 4subtractive, 37 9, 194
colorimetric sensor films and arrays, 353 4colour
and birefringence, 147of butterflies, 122, 123–4, 324, 326, 340
493 Index
colour (Continued)charge-transfer: see main entry: charge-transfer colourscomplementary, 31, 38of copper compounds, 264, 265, 275, 284 5, 286crystal-field, 264 70, 284 6and diffraction, 198 203, 205 8of electrochromic polymers, 468, 469, 470and energy, 3 4of eye, 188of flowers, 323 8, 329of fluorescent proteins, 407gamut of, 31of gemstones, 150 1, 214 15, 277, 283 4, 285, 286,
344 5of incandescent objects, 11, 13 14of insulators, 420iridescent, 91, 94, 122, 230of lanthanoid ions, 288 90of leaves, 321–2, 332, 333–4meaning of term, 1, 3of metallic nanoparticles, 478 87of metals, 477 8of minerals, 122, 124, 125, 284 5, 286mixing of, 29 31, 37 9, 170, 447, 463 4of nickel compounds, 264, 265, 274, 277, 278, 286perception of, 10, 23, 28, 29, 45of pigments, 295 7, 322, 333 40, 437of polychromic glass, 482primary, 30, 37quantum dot, 455, 456of red wine, 328 32and reflection, 91 128and refraction, 67 75of ruby, 150, 277 81, 286saturated, 31of semiconductor alloys, 441of semiconductors, 436 9of shells, 122as structural probe, 287suppression of, 120of thin films, 99 104, 126–7of transition metal ions and compounds, 264 70,
284 5, 286of water, 315 16
colour blindness, 24, 31colour centre, 429 36complex, 433, 434 5electron-excess, 430 1hole-excess, 430, 432 3surface, 434see also F centre
colour centre laser, 434 5colour changeauxochromic, 324bathochromic, 26, 317, 374hyperchromic, 317hypochromic, 317hypsochromic, 317, 374
colour-change sensors, 349 55colour confusion, 31loci of, 31, 33
colour filters, 37, 38colour models, 28CMY, 37, 486CMYK, 39HCL, 28HIS, 28HSB, 28, 29HSL, 28
HVC, 28RGB, 30
colour photographic film, 486colour printers, 37, 39colour rendition, sodium-vapour lamps, 263colour spaces, 28colour television, 170, 384 5, 388, 389colour temperature, 13correlated, 13 14of incandescent objects, 14
colour triangle, 30 1combinatorial tones, in water molecules, 315, 316Commission Internationale de l’Eclairage, 31see also CIE
Common Blue butterfly, 122, 123–4, 324, 326compact discs (CDs), 57, 94, 209 10, 448laser used, 448recordable (CD-R), 94reflection grating colours from, 209 10rewritable (CD-RW), 94
complementary colours, 31, 38in thin films, 100, 103, 126–7
complex numbers, 51ncomplex refractive index, 51, 191, 477complexes, 350computer displays, 170, 173computer memory, 474nconcentration quenching, 377 8conduction band, 4, 5, 388, 419, 420cone opsins, 24configuration interaction energy, 276, 277conjugated double bonds, 317, 458in cyclic compounds, 319 23
constructive interference, 8, 96, 97, 98, 228, 235continuous spectrum, 11conversion factors, 43cooperative luminescence, 394coppercolour, 478reflectivity, 477
copper acetoarsenate, 297copper compoundsabsorption spectrum, 264, 266, 275colours, 264, 265, 275, 284 5, 286flame colour, 255Orgel diagram, 275
copper indium selenide, 472copper ions, detection of, 350, 351copper nanoparticles, 487copper phthalocyanine, 322, 323copper selenides, mixed, 472copper sulfide, 437corals, colours, 407cordierite, 184core shell composites/nanodots, 412, 456, 458cornea (of eye), 190, 198, 233 4cornflower, colour, 324corona, 226corpuscular theory of light, 1, 2, 3correlated colour temperature, 13 14corundum (Al2O3)Cr3þ in, 277, 286doping of, 284, 345fluorescence, 366refractive index, 61, 66Ti3þ in, 282, 283Ti4þ and Fe2þ in, 345
cosmic microwave background radiation, 13counter electrode (in solar cell), 473
Index 494
coupling, 252see also j j coupling; Russell Saunders coupling; spin orbit
couplingcrests and troughs, 6, 7critical angle, 54, 55, 134, 135, 445crocetin, 318, 319crocin, 318, 319cross-relaxation, 369, 377, 399, 401, 403crustacean
blood, 322 3colour, 319
a-crustacyanin, 319crystal
anorthic (triclinic), 137biaxial, 144 7, 149birefringence of, 138centrosymmetric, 155colloidal, 218, 220cubic (isometric), 138, 143dichroic, 149, 150hexagonal, 138, 143, 165liquid: see main entry: liquid crystalmetamaterial, 85, 86monoclinic, 138, 144non-centrosymmetric, 137, 157nonlinear, 151 3, 154 5optically negative, 143, 147optically positive, 143, 146orthorhombic, 138, 144, 283photonic, 85, 218, 220, 223pleochroic, 149, 151tetragonal, 138, 143trichroic, 149triclinic, 138, 144trigonal (rhombohedral), 138, 140, 143uniaxial, 143 4, 149unit cell of, 138
crystal defects, 445crystal field
intermediate, 273 7octahedral, 271, 272, 274 6splitting, 266, 268, 269 70strong, 270 1tetrahedral, 271, 272 3weak, 271 3
crystal-field colours, 264 70, 284 6, 345, 346crystal symmetry
and refractive index, 137 9in ruby crystal, 279 80
crystal systems, 137cyanidin, 325, 329cyanin, 326, 327, 329cyanotype process, 342 4cyclamen, 321
d-orbitals, 264, 266, 300crystal-field splitting, 266, 268degeneracy of, 267, 269shapes, 266, 267
Daltonism, 24ndangling bonds, in quantum dots, 455dark adaptation, 28data storage, 94, 297, 474ndaylight white, 31decorative coatings, 210defects: see crystal defects; Frenkel defects;
point defectsdefoliation, mechanical, 104degeneracy of orbitals, 267, 269, 271
degenerate semiconductors, 4403-dehydro-retinal (retinal2), 26delafossite-structure oxides, 440delphinium, colours, 329dendrimers, 462dendrites, 462density, and refractive index, 60 2, 138destructive interference, 8, 9, 96, 99, 235detergents, fluorescent brightener in, 368dextrorotatory molecules, 164, 165, 166DFG (difference frequency generation), 155DFM (difference frequency mixing), 155diamond
band gap in, 424 5, 437blue, 428boron impurities in, 428, 429‘canaries’, 425Cape yellow, 425impurity colours in, 425 8, 446N V centre, 428N2 centre, 427 8N3 centre, 427nitrogen impurities, 425 8spectral colours (‘fire’), 68structure, 424, 426thin films, 428yellow, 425
dichroic glass, 193dichroic sheet polarizer, 136dichroism, 149, 150
circular, 167in gemstones, 150 1, 279 81, 283
dichromated gelatine, 242dichromats, 24dielectric constant, 58dielectric mirrors, 111 12dielectric susceptibility, 152diffraction, 33
by amorphous material, 225 6, 227Bragg’s law, 211 13by a circular aperture, 202 3colour production by, 198 203, 205 8by crystals, 211 25by disordered gratings, 225 31by droplets, 226 7by dust, 226 7dynamical theory, 213, 224 5of electrons: see main entry: electron
diffractionFraunhofer, 198Fresnel, 198from disordered gratings, 225 31images limited by, 87kinematical theory, 213by moth-eye structures, 231 3by opal, 213 18by a rectangular aperture, 200 1by a slit, 198 200by specks, 226 7by sub-wavelength structures, 231 5wavelength estimation by, 210 11of X-rays: see main entry: X-ray diffraction
diffraction grating, 198colour production by, 205 8disordered, 225 31linear, 205 8moth-eye surface as, 109, 232 3see also grating(s)
diffraction grating equation, 205
495 Index
diffraction limit, 203 5Abb�e criterion, 204Rayleigh criterion, 203, 204
diffraction patternfrom amorphous material, 225 6Fraunhofer, 198, 199orders in, 198, 205from random droplets or specks, 226 7spectra, 200Whewell Qu�etalet, 227
digital camera display screen, 170, 172digital photography, 170, 172, 474 7dimethyl glyoxime, 350diode: see light-emitting diodediode laser, 448diopside, 3661,2-dioxetanedione, 413, 414diphenylamine, 354, 3559,10-diphenylanthracene, 414, 415directionallowed, 135fast, 138slow, 138, 141vibration, 135, 140, 141
dispersion, 65anomalous, 65, 66intermodal, 82modal, 82normal, 65, 66in optical fibres, 81 2, 83production of colour by, 67 75specific rotation, 167and wavelength, 82
dispersive power, 67display(s)electroluminescent, 391 4field-emission/field-effect, 390 1interference-modulated, 110 11liquid-crystal, 169 73plasma, 259, 383 5thin-film electroluminescent (TFEL), 391 4
distributed Bragg reflectors, 114 15DNA molecules, response to stretching, 410donor dopants, 424donor (in quenching), 376donor p-bridge acceptor molecules, 406Doppler effect, 248double absorption, 297double refraction, 139in biaxial crystal, 144 7of calcite, 139 40, 142in uniaxial crystal, 143 4
doublet states, 303drift, 472Drude Lorentz free electron theory, 478drying agents, 285DsRed fluorescent protein, 407DVDs (digital versatile/video discs), 57, 94, 210, 448DWDM (dense wavelength division multiplexing), 119dye lasers, 355 8dye-sensitized solar cells, 472 4dyes, 322, 333 40in glow sticks, 414in solar cells, 473 4
dynamic quenching, 374, 378
e-books, 39 40e-ink process, 39, 40e-ray (extraordinary ray), 139, 141, 142
effective refractive index, inverse opals, 221 3effective temperatures (of stars), 14egg yolk, colour, 319Egyptian blue, 297Einstein coefficientfor absorption of radiation, 20for spontaneous emission, 19
elastic scattering, 33, 175elbaite, 149electric dipole transition, 254electric field vector, 5, 6, 129electrochromic device, 463, 464asymmetric arrangement, 463, 464dual arrangement, 463, 464with tungsten trioxide film, 465 6
electrochromic film, 463, 464 70bleaching of, 464
electrochromic materialsanodically coloured, 463, 467, 468, 469bleached, 464cathodically coloured, 464, 467, 469inorganic, 465 8organic, 468polymeric, 468 70
electrochromic reactions, 464, 466, 467, 468electroluminescence, 366molecular, 457 9organic, 457 9, 460
electroluminescent displays, 391 4, 446electromagnetic spectrum, 2electronconfined, 450effective mass, 451, 479energy in quantum structures, 451subbands (energy levels), 451, 453, 454
electron configurationslanthanoids, 301, 303lighter atoms, 300 1listed for various atoms, 249, 300 1, 302mercury, 263neon, 261sodium, 263transition metals, 301, 302
electron diffraction, 213dynamical theory, 213
electron diffraction pattern, 214electron electron repulsion, 252, 253, 254, 266,
271, 302electron-excess centres, 430 1electron gun, 386electron hole, 421electron hole pair(s), 423, 456, 471, 472, 475, 476electron microscopy, 87, 390electronic energy levels, 310, 311electronic ‘paper’, 39 40electronic transitions, 258, 262, 263 4absorption due to, 79
electrophoresis, 39, 40elliptically polarized light, 130, 131embossed holograms, 242 3emerald, 283, 286emeraldine, 468, 469base form, 469salt forms, 469
emissionEinstein coefficient, 19rate of, 18 21spontaneous, 17, 18, 19stimulated, 17, 18, 19, 21, 22, 259, 260, 282, 292, 369
Index 496
emission spectrumphosphors, 380, 382, 394rhodamine 6G dye, 356sodium vapour lamp, 263tuning in quantum structures, 454
enantiomers, 164, 165 6energy
absorption and emission, 368 70units, 43
energy bands, 419energy exchange efficiency, 376, 377energy exchange equation, 309, 310energy level(s), 4, 253
deep, 424in intermediate crystal field, 273 7of many-electron atom, 306 7molecular, 309 11shallow, 424in strong crystal field, 270 1in weak crystal field, 271 3
energy-level diagramsinert gases, 258lanthanoid ions, 291, 292, 293, 381, 383, 390, 393, 447lasers, 261, 293mercury atoms, 264molecular fluorophore, 406OLED, 462sodium atoms, 262
energy-level populations, 17 18energy transfer (in quenching), 369, 377 8, 399 401erbium-doped optical-fibre amplifiers, 294 5, 446erbium ions
colours, 289energy levels, 447
erythrolitmin, 3512-ethylanthraquinone, 354, 356eumelanin, 337, 340europium ions
cerium oxide doped with, 397–8colours, 289, 290energy levels, 292, 381, 390, 393, 447in phosphors, 381, 382, 385, 389, 392 3, 436
evanescent waves, 54, 56, 57, 87, 89excited state, 17excited-state absorption (of photons), 369, 370, 396 7, 399, 401exciton, 421 4, 458
free (Mott Wannier), 423in molecular crystals, 424singlet, 458, 459, 460tightly bound (Frenkel), 423 4triplet, 458, 459, 460
exciton blocking layer, 461, 462exciton energy levels, 422 3exitance
radiant, 46, 372, 378spectral, 12
explosives, detection of, 413exposure meter, 471extinction, 34, 175 6
see also attenuationextinction coefficient, 34, 51, 477extinction cross-section, 186extinction efficiency (factor), 186, 187extinction position, 147extraordinary light transmission, 487extraordinary ray (e-ray, E-ray), 138, 141, 142eye
colour of, 188colour sensitivity, 10, 24, 25, 476
compound eye, 108, 109, 231dark adaptation, 28diseases, 162insect eye, 108, 231 2mirror eye, 125 6moth eye, 64, 108, 109, 232photoreceptors: see main entry: photoreceptorsscallop eye, 125sensitivity, 24, 25structure, 24, 190
F centre(s), 429 30listed for various alkali metal halides, 430
Fabry P�erot etalon, 110face-centred cubic structure, 218Fairy primrose, 323feathers, 188, 234 5feldspars, 122, 124Fermi energy, 460Fermi level, 459, 460, 474ferric oxyhydroxides, 346, 347ferroelectric crystals, 242ferroprussiate paper, 342fibre Bragg gratings (FBGs), 115 19fibre optics, 75, 77 84
see also optical fibresfield-emission display (FED), 390 1filling factor, 108film
anodized, 103 4birefringent, 151electrochromic: see main entry: electrochromic filmphotographic: see main entry: photographic filmpolymer, 148, 149, 151see also thin film
filterbandpass, 114, 116interference, 114longpass, 114, 116optical, 38, 114, 224polarization, 136, 137shortpass, 114, 116
firefly, 415fireworks, colours, 255, 315first-order kinetics, fluorescence and phosphorescence, 372, 373fishnet structures, 85, 86flame colours, 255, 255, 315flame test, 255flashlamps, 257flashtubes, 257flat band model, 419, 420, 421flavone(s), 323, 325
reaction with ammonia, 324, 326, 349flavonoid pigments, 323 32flavonol, 323, 325flavylium cation(s), 325, 330, 330, 331
polymerization of, 331, 332flint glass, 60, 61flower colours, 323 8, 329fluorescein and derivatives, 367, 368, 378, 405 6, 409fluorescence, 34, 365, 366
absorption mechanism for, 410, 411anti-Stokes, 394compared with phosphorescence, 367, 369, 370 1molecular, 405
fluorescence lifetime, 372, 378fluorescence microscopy, 363, 409 10fluorescence quenching, 374fluorescence resonance energy transfer, 376 7
497 Index
fluorescent brighteners, 368fluorescent centres, 381fluorescent coatings, in vapour lamps, 263 4, 383fluorescent dyes, 405 6, 409, 456fluorescent lamps, 379 83Colour 80, 381 2Colour 90, 382 3colour temperature, 14halophosphate, 379 80phosphors in, 365, 379, 381, 382 3trichromatic, 381 2
fluorescent markers, 409, 412fluorescent molecules, 405 11fluorescent nanoparticles, 411 12fluorescent proteins, 407 8, 409colours, 407
fluorescent sensors, 412 13fluorescent tags, 406fluorite, 366fluorochrome, 409fluorophore, 367, 409foam, 62as antireflection coating, 108
food wrap film, 148, 149fool’s gold, 437, 438F€orster distance, 376F€orster resonance energy transfer (FRET), 376 7, 410forward bias, 444 5, 472fovea (in eye), 24fracture damage, detection of, 416Fraunhofer diffraction, 198, 199Fraunhofer lines, 255 6Frenkel defects, 483, 484Frenkel excitons, 423 4frequency, 7, 44angular, 44relationship to velocity, 7temporal, 7, 44
frequency doubling, 115, 151, 153, 355see also second-harmonic generation
frequency matching, 157frequency mixing, 155 6frequency trebling, 153see also third-harmonic generation
frequency up-conversion, 394Fresnel diffraction, 198Fresnel’s laws, 132Fritillary butterfly, 115frontier molecular orbitals, 310fuchsia, colours, 327fuels, markers in, 354 5fused-fibre coupler, 56, 57
gallium aluminium arsenide, 155, 448, 451gallium arsenide, 448gallium arsenide gallium phosphide alloys, 445gallium nitride, with lanthanoids, 446, 447gallium nitride indium nitride system, 441, 442, 445, 455gallium phosphide, 445gallium trioxide, 433gamut of colours, 31garnet, 286Garnet Star, 14garnet structure, 286, 295gas analysis, 353 4gas discharge lamps, 256 9gas plasma display, 259, 260, 383 5Geissler tube, 257gelatine, 242, 484
gemstones, 149, 150 1, 213 18, 277 81, 283 4, 285, 286, 344 5irradiation of, 431 2see also amethyst; aquamarine; diamond; emerald; opal; ruby;
sapphire; topazgeranium, colours, 328GFP (green fluorescent protein), 407, 408–9, 456Gladstone Dale formula, 61 2refractive coefficient, 61 2
listed for various oxides, 63glassaluminoborosilicate, 483chalcogenide, 80, 242chemical impurities in, 80 1Co2þ in, 285, 287, 288devitrified, 188dichroic, 193flint, 60, 61fluoride, 80, 287lanthanide-containing, 190, 191lead ‘crystal’, 60metallic, 416Nd3þ in, 289opal, 42, 188photochromic, 482 4photosensitive, 118polychromic, 481 2Pyrex, 483ruby, 191 3second-harmonic generation in, 160 1selenide, 80silicate, 226stained, 37, 194structure, 287window, 79, 119ZBLAN, 80
glass ceramics, 188glass fibres, 77attenuation in, 79, 80
glaucoma, 162glow stick, 413 14reactions in, 413 14, 414
glow-worm, 414 15glycosides, 324, 329goldcolour, 478reflectivity, 477
gold nanoparticles, 192, 479 80colours, 191 2in Raman spectroscopy, 487
gold sols, 191graded-index materials, 64, 65disordered, 225 31see also GRIN...
graphene, 10grating(s)amplitude, 198, 206, 207blazed, 208chirped, 117diffraction: see main entry: diffraction gratingdisordered, 225 31fibre Bragg, 115 19Hill, 115one-dimensional, 208phase, 198reflection, 198, 205, 206, 207, 210 11, 232three-dimensional, 211 25, 231transmission, 198, 206, 207, 209two-dimensional, 208 10, 230 1ultrahigh spatial-frequency, 109, 232
Index 498
uniform, 117see also diffraction grating
grating equation, 205Gr€atzel cell, 472green fluorescent protein (GFP), 407, 408–9, 456GRIN antireflection coatings, 108 10, 121GRIN materials, 64, 65GRIN optical fibres, 82, 84Grotrian diagram, 258, 258, 262, 264ground state, 4, 17, 249, 253ground-state absorption (of photons), 369, 396 7, 399, 401ground-state term, 306gypsum, 366Gyricon process, 39, 40
haem, 320, 322, 323haematite, 322, 346haemocyanin, 323haemoglobin, 322halite, 61, 138, 366halo, 75, 76halogen vapours, colours, 311 12halophosphate fluorescent lamps, 379 80Hamburg blue, 342Han blue, 297Han purple, 297HD-DVDs (high-definition digital versatile/video discs), 94, 448heavy water, 316Heidinger’s brushes, 184nHeisenberg uncertainty principle, 449helium, 256, 257helium neon (He Ne) laser, 21, 211, 259 62, 436high-brightness LEDs, 445, 445high-reflectivity surfaces, 110highest occupied molecular orbital (HOMO), 310, 455, 457, 458, 459
see also HOMO LUMO separationHill gratings, 115hole, 421
effective mass, 451subbands (energy levels), 451, 453, 454
hole-excess centres, 430, 432 3hologram(s), 235 43
amplitude, 241Benton, 239embossed, 242 3and interference patterns, 235master, 239phase, 241, 242planar, 237polarization, 241, 242rainbow, 239 40, 242recording media for, 240 2reflection, 237 9thick, 237 9thin, 237, 242transfer, 239, 240transmission, 235 7, 239volume, 237 9
holographic image, reconstruction of, 235 6, 237, 238HOMO (highest occupied molecular orbital), 310, 455, 457, 458, 459HOMO LUMO transitions, 317, 350, 357, 419, 468homochiral molecules, 166‘horse and rider’ double star, 204HSB (hue/saturation/brightness) colour model, 28, 29hue, 28Hund’s rules, 306, 307hydrangea, colour, 327hydrogen bonding, 316hydrogen peroxide, 413, 414
hydrogen spectrum, 249 51hydrogen tungsten bronzes, 465, 466, 467hyperchromic shift, 317hyperlenses, 87 9hyperpolarizability, 161hypochromic shift, 317hypsochromic shift, 317, 374
IC (integrated circuit) manufacture, 106, 204Iceland spar, 139, 140, 147illuminance, 46ilmenite, 345image: see holographic image; latent imageimage reconstruction (of holograms), 235 6IMOD displays, 110 11impurity colours
in diamond, 424 8, 446in insulators, 424
impurity ion, and frequency doubling, 151incandescence, 11, 363
and colour, 11, 13 14spectrum, 11, 247
incidenceangle of, 51, 52, 92normal, 92plane of, 51, 92
incoherent light, 7, 11, 16, 17incoherent scattering, 197index of refraction: see refractive indexindicators, 350 3indicatrix, optical, 143, 144, 145 6indigo, 335 6, 336indium oxide, 420, 441indium phosphide, 448indium tin oxide (ITO), 439, 460, 464indole-2,3-benzopyrrole, 378inelastic scattering, 33 4, 175, 486 7‘inert’ gases
colours in gas discharge lamps, 257electron configurations, 261, 301see also argon; helium; krypton; neon; xenon
infrared radiation, 10insects
bioluminescence, 414 15detection of polarized light by, 184eyes, 10, 64, 108, 109, 231wing markings, 91, 115, 188see also butterfly; firefly; moth
insulators, 420colours, 420impurity colours in, 424
intensity, 372, 395interaction energy, 276, 277intercombination band, 279, 281interface
reflection at, 92 4refraction at, 54, 55second-harmonic generation at, 161
interference, 7 9constructive, 8, 96, 97, 98, 228, 235destructive, 8, 9, 96, 99, 235of polarized light, 131, 148at thin films, 94 9
interference filters, 114interference-modulated (IMOD) displays, 110 11internal energy conversion, 369, 376intersystem crossing (ISC), 369, 371invisibility, 41, 62, 87, 135ionizing radiation, F centres produced by, 429, 431 2
499 Index
iridescent colours, 91, 94, 122, 230iron compounds, colours, 284, 286iron-containing minerals, 149, 322, 345, 346 8, 437iron ions, detection of, 350, 351iron pyrites, 437, 438, 439irradiance, 46changes detected by eye, 92diffraction patterns, 199, 201factors affecting, 154
irradiance profile, diffraction by a slit, 199isomers: enantiomersisotropic substances, 54, 137, 138strain analysis, 148
ITO (indium tin oxide), 439, 460, 464
j j coupling, 253, 303Jablonski diagrams, 371, 406Japanese maple, leaf colours, 334jellyfish, 62, 135, 407, 414, 415jewellery, 110, 122, 425see also gemstones
kaempferol, 323, 325KDP (potassium dihydrogen phosphate), 153keratin, 234kinetics, luminescence, 370 1King’s yellow, 437Kitt Peak Observatory, 475Kroger Vink point defect notation, 426n, 440n
labradorescence, 122, 125labradorite, 122, 124, 125laevorotatory molecules, 164, 165, 166Lambert’s law, 34, 176lamp(s), 16 17fluorescent: see main entry: fluorescent lampsgas discharge, 257 9mercury-vapour, 247, 248, 263 4sodium-vapour, 189 90, 247, 262 3tungsten-filament, 14, 16
lanthanide, 190n, 301nlanthanoid compounds, colours, 295lanthanoid-doped crystals, 297 9lanthanoid elements, 249, 254electron configurations, 301, 303
lanthanoid ionscolours, 288 90electron configurations, 303, 382energy-level diagrams, 291, 292, 293, 381, 383, 390, 393, 447in gallium nitride, 446, 447in glass, 190, 191in insulators, 424, 425in phosphors, 381 2, 385, 389, 392 4
lapis lazuli, 348Laporte selection rule, 254 5, 270Large Hadron Collider, 417large particles, scattering by, 184 7laser, 17argon-ion, 115carbon dioxide (CO2), 156colour centre, 434 5continuous mode, 294, 358dye, 355 8first demonstrated, 17, 21, 259, 281four-level, 290, 292 4helium neon (He Ne), 21, 211, 259 62, 436neodymium (Nd:YAG, Nd:YLF), 156, 160, 290, 292 4pulsed mode, 282, 294, 358ruby, 17, 21, 259, 276, 281 2
semiconductor diode, 155, 448, 449three-level, 281 2titanium sapphire, 282 3type II behaviour, 434
laser cavity geometry, 21 2laser cavity modes, 21 3laser light, interference observed using, 8laser measuring equipment, 448laser pointer, 211, 448latent image, 435, 485lazurite, 348LBO (lithium triborate), 153, 156LCD: see liquid crystal displaylead chromate, 346lead crystal glass, 60lead oxide, in flint glass, 60, 61lead tungstate, 417leaf colours, 321–2, 332, 333–4, 335leaf senescence, 332LED: see light-emitting diodelemon yellow, 346lensair, 64eye, 64, 190photochromic, 358super, 87 9, 204
leucoemeraldine, 468, 469, 469level, energy: see energy levellever rule, 31LIDAR, 156lifetimeof excited states, 282fluorescence, 372, 378of spectral holes, 299
ligand-field splitting, 266, 268see also crystal field
lightabsorption and emission of, 17, 18absorption of, 4coherent, 7, 8, 17, 235diffraction of, 33generation of, 10 13incoherent, 7, 11, 16, 17interaction with materials, 33 6monochromatic, 7particle/corpuscular theory, 1, 2, 3polarized: see main entry: polarized lightreflection of, 33, 34scattering of, 33 4unpolarized, 7, 11, 16, 129, 228velocity in vacuum, 1 2, 7, 43wave theory, 1, 2, 3
light waves, 5 7and colour, 9 10interference of, 7 9polarization of, 129 35
light-emitting devices, 446light-emitting diodes (LEDs)active layer in, 445applications, 82, 173blue, 445, 447, 448depletion region in, 444direct band gap materials, 442, 443displays using, 446 7green, 445, 447, 454heterojunction, 445, 446high-brightness, 445, 445homojunction, 444, 445idealized structure, 443 5
Index 500
impurity doping in, 446indirect band gap materials, 442 3, 443organic: see main entry: organic light-emitting diodesphotometric characteristics, 45red, 445, 447transition region in, 444white light generation, 447 8yellow, 445, 447
lighting, 189 90, 247, 248, 262 4, 383, 447lightness, 28limestone, 188limonene, 166line(s)
Fraunhofer, 255 6‘persistent’, 257sodium D, 263, 264spectral, 248telluric, 256
line spectrum, 248, 249 51linewidth, natural, 248liquid crystal, 168 73liquid-crystal display (LCD), 169 73
active matrix display, 39, 170light source in, 170, 173passive display, 170
liquid-crystal mesophases, 168, 169calamitic, 168chiral nematic, 228, 229cholesteric, 168, 228 30columnar, 168director in, 168, 169, 228disclinations in, 168discotic, 168nematic, 136, 168, 169smectic, 168, 169twisted nematic, 168, 228
liquid-crystal thermometer, 230, 420liquid scintillation counters, 417lithium compounds, colours, 255lithium iodate, 153lithium niobate, 153lithium triborate, 153lithium tungsten bronzes, 466, 467litmus, 350 1, 352lodestone, 345longitudinal cavity modes, 22longpass filters, 114Lorentz Lorenz equation, 59, 61Lorenz Mie theory, 184nlow-emissivity windows, 119 21low-reflectivity films, 105 10lowest unoccupied molecular orbital (LUMO), 310, 455, 457,
458, 459LS coupling: see Russell Saunders couplingLucalox, 190luciferase, 415luciferins, 415luminance, 28, 46luminescence, 16, 363 418
cooperative, 394early studies, 363 5meaning of term, 363types, 366
luminescent materials, 363luminiferous aether, 2luminous efficiency, 25, 443, 455luminous exitance, 46luminous flux/power, 46luminous intensity, 46, 367
luminous paints, 365, 434lutein, 318, 319lycopene, 317, 318Lycurgus Cup, 193 4Lyman series, 251
magnesium aluminosilicate, 184magnesium fluoride, 105, 106magnesium oxide, 190, 384
surface colour centre on, 434magnetic bubble memory, 474nmagnetic dipole transition, 254magnetic field vector, 5, 6magnetite, 345malachite, 284 5, 286mallow, colours, 329Malus law, 137malvin, 328, 329, 331manganese compounds, colours, 286manganese ions, in phosphors, 379, 380maple, leaf colours, 334, 335Marbled White butterfly, 324, 326marine animals, 41, 62, 125, 135marker reagents (for fuels), 354 5markers, fluorescent, 409, 412masers, 21mask
fibre Bragg grating, 118integrated circuit, 106
mass absorption coefficient, 35mauveine, 337, 338Meadow Brown butterfly, 340melanins, 337, 339, 340melanocytes, 337melanosomes, 337mercuric sulfide, 437, 438mercury
energy-level diagram, 264ground-state configuration, 263
mercury-vapour lamps, 247, 248, 263 4, 383replacement of mercury with xenon, 402 3
meso- form, 165metal ions, detection of, 349 50metal nanoparticle sensors, 486metal oxide semiconductor (MOS) device, 475metallic glass, 416metallic mirrors, 111, 478metallic nanoparticles, colours, 478 87‘metallic’ paints, 478metalloanthocyanin, 327 8metals
colours, 477 8reflectivity, 477 8
metamaterials, 84 6metarhodopsins, 26, 27metastable trapping, 299methyl orange, 352methyl red, 352mica, 147microscopy
electron, 87fluorescence, 363, 409multiphoton excitation, 410 11optical, 162polarizing, 168resolution of, 57, 87, 203 5second-harmonic, 162, 163two-photon fluorescence, 411, 412
microwave absorption and emission, 310
501 Index
microwaves, 2, 16Mie Debye theory, 184nMie scattering theory, 184 7, 190, 194, 410,
478, 481Miller indices, 218nminerals, 122, 124, 125, 149, 184colour centres in, 430 1crystal-field colours, 284 5, 286fluorescent, 366iron-containing, 149, 322, 345, 346 8, 437
mirages, 64mirror eye, 125 6mirrorsdielectric, 111 12metallic, 111, 478‘smart’, 464
molar (decadic) attenuation coefficient, 36molar refraction, 61molecular crystals, excitons in, 424molecular fluorescence, 405molecular orbital, 309 10antibonding, 310, 457bonding, 310, 457frontier, 310highest occupied, 310, 455, 457, 458, 459lowest unoccupied, 310, 455, 457, 458, 459nonbonding, 310, 457p, 310p�, 310
molecular orbital theory, 310molecular polarizability, 161molecule(s)chiral, 164dextrorotatory, 164, 165, 166donor p-bridge acceptor, 406, 410energy levels of, 309 11homochiral, 166laevorotatory, 164, 165, 166organic: see main entry: organic moleculesphotochromic, 358 60as scattering centres, 178, 179
molluscs, dye derived from, 336molybdenum blue, 341molybdenum trioxide, 100monochromatic light, 7monodisperse suspension, 218, 230mooncolour, 180, 188eclipse, 180, 182
mordants, 337MOS (metal oxide semiconductor) device, 475motheyes, 64, 108, 109, 232wing scales, 29 30
mother-of-pearl, 122Mott Wannier excitons, 423Mount Palomar telescope, 204mullite, 42multilayer stacks, 111, 113 14disordered, 114, 115tunable, 114
multiphoton absorption (of photons), 369, 370multiphoton excitation microscopy, 410 11multiple quantum well (MQW) structure, 450, 451, 452multiple scattering, 190 1multiplexing, dense wavelength division, 119multiplicity selection rule, 270, 278Munsell colour cylinder, 28, 29Munsell colour solid, 28
n-doping, 464, 467N517 dye, 474N719 dye, 474nacre (colour of shell), 122nanoparticle sensors, 486nanoparticlesin antireflection coating, 109 10in coloured glass, 191, 193fluorescent, 411 12metallic, 478 87
nanorods, 109 10, 450tuning of absorption characteristics, 480
nanostructures, 449 50nanotubes, 136, 450nanowires, 450naphthopyrans, 359, 360National Ignition Facility (US fusion research), 295negative-index materials, 84 9metamaterials, 84 6superlenses, 87 9
neodymium (Nd:YAG or Nd:YLF) lasers, 156, 160, 290, 292 4neoncolour, 257energy levels, 260, 261Grotrian diagram, 258line spectrum, 258in plasma display, 384, 385
‘neon’ signs, 16, 257Nernst glowers, 16net curtains, 209Newton’s black film, 100nickel compoundsabsorption spectrum, 264, 266, 271, 277, 278colours, 264, 265, 274, 277, 278, 286detection of, 350Orgel diagram, 276
nickel oxide, hydrated, 467NIMs: see negative-index materialsniobium pentoxide, 467reduction of, 341, 467
nitric oxide, formation in firefly, 415nitrogen molecules, ionization of, 313noble gases: see ‘inert’ gasesnonbonding molecular orbitals, 310, 457non-crossing rule, 276nonequilibrium state, 18nonlinear crystals, 151 3, 154 5nonlinear effects, 151 7colour production by, 153
nonlinear optical materials, 153, 155nonlinear optics, 152nonradiative transition, 279, 281, 282, 293, 369, 455, 456
o-ray (ordinary ray), 139, 141, 142oak, leaf colours, 332, 335object beam (holograms), 235octahedral coordination, 266 7, 268, 269, 270 1octahedral crystal field, 271, 272, 274 6, 282oil film, 99, 104OLEDs (organic light-emitting diodes), 459 64oligomers, 332olivine structure, 284, 286, 296ommatidia, 108, 231opalartificial, 218, 220Bragg equation for, 215 17colloidal, 218colours, 214, 215, 216common (potch opal), 213
Index 502
diffraction by, 213 18fluorescence, 366inverse, 218, 220
effective refractive index, 221 3precious, 214 17total internal reflection in, 216, 217
opal glass, 42, 188opalescence, 214opsin proteins, 24, 26optic axis, 139, 140optical activity, 162, 164 8optical band gap, 420
listed for various oxides, 422optical communications, 75, 77optical constants, 51, 477optical density, 36optical fibre(s), 77 8
attenuation in, 79, 80chemical impurities in, 80 1cladding of, 77core, 77coupler, 57, 57dispersion in, 81 2, 83fibre-drawing process, 78graded-index, 82, 84monomode, 84, 115new materials, 80preform for, 78, 295refractive index modulation in, 115, 117repeaters, 294second-harmonic generation in, 160 1signal addition/removal from, 119signal amplification in, 294 5, 446stepped-index multimode, 82, 84
optical filtering, 38, 114, 224optical indicatrix
biaxial crystal, 145 6uniaxial crystal, 143, 144
optical masers, 21see also lasers
optical parametric amplifiers, 156 7optical parametric oscillators, 156 7, 355optical path, 53optical pumping, 156, 157, 282, 290, 292optical thickness, 53optically absent layer, 105optically anisotropic materials, 138optically isotropic substances, 54, 137optically negative crystal, 143, 147optically positive crystal, 143, 146orbital(s)
atomic, 264, 266, 267, 300d, 264, 266, 267, 300eg, 267, 269, 271molecular: see main entry: molecular orbitalp, 300s, 300t2g, 267, 269, 271
ordinary ray (o-ray, O-ray), 138, 141, 142organic electroluminescence, 457 9, 460organic light-emitting diodes (OLEDs), 459 64organic materials, second-harmonic and sum-frequency generation
by, 161 2organic molecules
as insulators, 457interaction with light, 136photochromic, 358 60
organic semiconductor, 340, 457 64Orgel diagram, 274, 275, 276
Orion type stars, 14orpiment, 437orthochromatic photographic film, 486oscilloscope, 386, 389overtones, in water molecules, 315, 316oxidation processes, 463, 467, 468oxygen atoms, 314
p-doping, 463, 467, 468p-orbitals, 300p-wave, 87, 131
reflection of, 131 3, 134see also ray, extraordinary
paint, 188 9‘metallic’, 478
palladium compounds, detection of, 350panchromatic photographic film, 486PANI: see polyanilineParis blue, 342Paris green, 297parity selection rule, 254, 278, 279particle detectors, 417particle theory of light, 1, 2, 3Paschen series, 251path difference, 96, 98, 126–7paua shell, 122, 125Pauli exclusion principle, 300, 305PEDOT: see poly(3,4-ethylenedioxythiophene)pelargonium, colours, 329peony, colours, 329period
spatial, 44temporal, 44
periodic table, 249permittivity, relative, 58pernigraniline, 468, 469, 469perovskite, 61perovskite bronzes, 465perovskite type structure, 395, 465petunia, colours, 329Pfund series, 251pH indicators, 350 3pH sensor, 413pH theory of flower colours, 326phaeomelenin, 337phase difference, 147phase grating, 198phase hologram, 241, 242phase matching, 158 60
birefringent crystals, 160phase object, 198phase speed/velocity, 7, 44phase of wave, 7phenolphthalein, 352phonon absorption, 79phonon-assisted transition, 279phosphor electroluminescent displays, 391 4phosphorescence, 364, 366, 367, 370
compared with fluorescence, 367, 369, 370 1phosphors
in cathode-ray television, 387 9in fluorescent tubes, 363, 364, 365, 379, 381, 382 3photostimulable, 435 6
photobleaching, 410photochromic bleaching, 358, 484photochromic glass, 482 4photochromic organic compounds, 358 60photochromic plastics, 359photochromic reactions, 24, 26, 28, 359
503 Index
photochromic sunglasses and ski goggles, 359photoconductive effect, 471photodiode, 471photoelectric effect, 3, 14 15photoelectrochemical cells, 472photoelectrons, 2photographic film, 484 6black-and-white, 486colour, 486orthochromatic, 486panchromatic, 486
photographydigital, 170, 172, 474 7film, 241 2, 484 6
photoionization, gated, 299photoluminescence, 366, 446atomic processes in, 368 78in quantum dots, 455
photometric units, 45, 46, 476photon(s), 3, 14 16absorption of, 369conversion, 369emission of, 369energy, 4, 15interaction with electron, 249
photon cascade emission, 403photonic band gap (PBG), 223 4photonic crystals, 85, 218, 220photonic engineering, 111in nature, 121 6
photonic stopband, 115, 223photoprotein, 416photoreceptor cellscones, 24
L (red) cone receptors, 24M (green) cone receptors, 24S (blue) cone receptors, 24
rods, 24, 26photorefractive materials, 60photoresist, 106, 205n, 242photosensitive materials, 118, 240, 241 2photostimulable phosphors, 435 6photovoltage, 472photovoltaic effect, 471photovoltaic materials, 471photovoltaic solar cells, 471 2phthalocyanines, 322 3Pigment Blue 15, 322pigments, 295 7, 322, 333 40, 437pixels, 39, 169, 170Planck constant, 11, 13, 15, 43Planck law of black-body radiation, 11 13, 15, 20, 44, 45plasma, 259plasma displays, 259, 383 5, 403plasma frequency, 479plasmon, 478 9surface: see main entry: surface plasmon
plasmon hybridization, 480plasmonic crystals, 488plastic films, 148, 149, 151plastics, appearance, 42platinum, in glass, 192pleochroism, 149, 151point defects, 395, 424, 440, 483pointillism/pointillist painting, 29, 122, 387polariton, 479npolarizabilityelectronic, 58molecular, 161
and refractive index, 58 60polarizationcolour produced by, 148linear, 6, 7, 130 1and optical activity, 166 8and phase matching, 159by reflection, 131 5and scattering, 181 4
polarization hologram, 241, 242polarized lightcircular, 130, 131colour affected by, 279 80detection of, 137, 184elliptical, 130, 131interference of, 131, 148plane (linear), 7, 130rotation of, 162, 164 6
polarizer, 136dichroic sheet, 136
Polaroid sheet/sunglasses, 136, 148, 168, 184polaron, 458polars, 135 7crossed, 137, 148sheet form, 136, 151in tandem, 136
poling, 161thermal, 161
polyaniline (PANI), 468, 469electrochromic device using, 470
polychromic glass, 481 2colours, 482
polychromic materials, 468polycrystalline materials, 160poly(3,4-ethylenedioxythiophene) (PEDOT), 469, 470electrochromic device using, 470
polymer films, 148, 149, 151polymers, electrochromic, 468 70poly(2-methoxy-5,20-ethylhexyloxy)-1,4-phenylenevinylene (MEH-
PPV), 459, 461poly(methyl methacrylate) (PMMA), 218, 220polypyrrole, 468polysulfides, 348 9polythiophene, 468, 470alkoxy-substituted, 468, 469, 470
poly(vinyl alcohol) (PVA), 136population inversion, 18, 255, 259, 281, 283, 292, 448porcelain, 42, 188porous coatings, 108porous materials, 62porphyrins, 319 22positron emission tomography, 417potassium dichromate, 346potassium dihydrogen phosphate (KDP), 153, 154 5potassium iron(III) cyanide, 342, 343, 344potassium permanganate, 345potential, built-in/contact, 444, 471potential well (in quantum structures), 451, 454praseodymium ionscolours, 289energy levels, 447in quantum cutting, 403, 404up-conversion and, 401, 403
primary coloursadditive, 30subtractive, 37, 486
principal refractive indices, 138, 140principle of superposition, 7 8prism, spectrum formed by, 67 8, 139proanthocyanidins, 337
Index 504
propagation number, 45propagation vector, 5proteins, folding and coiling of, 407Prussian blue, 341 2, 344, 349, 467 8Prussian green, 342, 343Prussian white, 342, 343, 344, 467, 468puddle, oil film on, 99, 104PVA (polyvinyl alcohol), 136pyramid, truncated, 445, 446pyran, 359pyrite, 437, 438
quality factor, 21quantum/quanta, 13quantum computers, 428quantum cutting, 402 4, 405quantum dots, 350, 351, 409, 450, 455 7
colours, 455, 456quantum optics/electrodynamics, 3quantum wells, 450, 451 4
energy levels in, 452, 453multiple, 450, 451, 452
quantum wires, 450, 454, 455quantum yield, 371, 372quarter-wave stack, 111 12quartz, 165
smoky, 431quenching, 374 8
concentration, 377 8by defects, 369dynamic, 374, 378by energy transfer, 376 7fluorescence, 374by molecular collisions, 369static, 374thermal, 375 6
quercetin, 323, 325quinizarin, 354, 355
racemic acid, 162, 164racemic mixtures, 164, 165radar, 386radar backscattering efficiency, 194, 195radiance, 46radiant exitance, 46, 372, 378radiant flux/power, 46radiant intensity, 46radiation
absorption of, 17, 18emission of, 17, 18
radiationless transition, 279, 281, 282, 293, 369radiative transition, 369radio waves, 2, 16radioactivity, 365
recording of, 417radioluminescence, 365, 366radiometric units, 12 13, 45, 46, 476radium, 365rainbow, 68 75
deviation of rays, 69, 70, 73, 74, 75impact parameter, 69, 70, 73, 74, 75polarized, 75primary, 68, 69 71, 69, 71, 72, 73secondary, 68, 69, 71 2, 71, 74, 75ternary and high-order, 75
rainbow holograms, 239 40, 242raindrops, reflection within, 69, 72, 73–4, 134, 135Raman effect, 487Raman spectroscopy, 486 7
rare earth elements, 301see also lanthanoids
rayextraordinary (e-ray, E-ray), 139, 141, 142ordinary (o-ray, O-ray), 139, 141, 142
Rayleigh criterion (for resolution), 203, 204Rayleigh Gans theory, 184Rayleigh radiation, 487Rayleigh scattering, 177 8, 184, 185, 486
in biological tissues, 190effects, 180, 188, 190, 410in optical fibres, 79, 81and wavelength, 179
Rayleigh scattering pattern (polar diagram), 178, 182rays of light, 1, 3reaction
bimolecular, 378electrochromic, 464, 466, 467, 468photochromic, 24, 26, 28, 358 9
reaction ratefluorescence and phosphorescence, 372 3photochromic glass, 484
red-hot object, radiation from, 11, 12red shift, 316, 374red sunset, 179red wine, 328 32reduction processes, 463, 467reference beam (holograms), 235reflectance, surface, 92, 132reflection, 33, 34
angle of, 92coefficient of, 92colour production by, 91 128data storage using, 94diffuse, 42from transparent plate, 92 4perpendicular to film, 96 7polarization by, 131 5total internal: see main entry: total internal reflection
reflection diffraction gratings, 198, 205, 206, 207, 210 11, 232reflection holograms, 237 9reflectivity
high, 110of metals, 111, 477 8surface, 92, 93 4, 132of thin film in air, 101 2
refraction, 33, 49, 50angle of, 51, 52colour production by, 67 75double: see main entry: double refractionat interface, 54, 55molar, 61specific, 61
refractive coefficient, 61 2listed for various oxides, 63
refractive index, 49, 51absolute, 52average, 62, 108complex, 51, 191, 477and crystal structure, 140 3and density, 60 2, 138effective, 221
of inverse opals, 221 3of foam, 62graded, 64listed for various substances, 61of metals, 477of mixtures, 62negative, 84
505 Index
refractive index (Continued)nonlinear, 54and polarizability, 58 60of porous materials, 62principal, 138, 140and symmetry, 137 9and wavelength, 52 4, 102, 158, 178
refractive index grating, 115relative permittivity, 58resolution limit, of optical instruments, 57, 87, 203 5resonance, 22, 479resonant condition, 376resonant frequency, 376retardation, 96and colours, 126–7relative, 147
retina, 24, 125, 162, 163retinal, 24, 2611-cis-retinal (retinal1), 26, 27all-trans-retinal, 26all-trans-retinal rhodopsin, 26, 27RGB colour model, 30rhinestones, 110rhodamine 6G dye, 356rhodopsin, 24, 26, 27, 28ring silicate pigments, 297rods and cones: see photoreceptorsrose, colours, 327rosemary, 321rotational energy levels, 310, 311roughness, surface, 40, 42rubrene, 414, 415ruby, 150, 277colour of, 150, 277 81, 286dichroism in, 150 1, 279 81
ruby glass, 191 3ruby laser, 17, 21, 259, 276, 281 2Russell Saunders coupling, 252, 253, 302rutile, 121, 138 9, 432 3Ryberg constant, 250
s-orbitals, 300s-wave, 131reflection of, 131 3, 134see also ray, ordinary
saffron, colour, 319sage, 322St Elmo’s fire, 259, 315sapphire, 345see also titanium sapphire laser
saturation, 28, 31scallop, eye, 125scanning electron microscopy, 390scatteringcoherent, 197colour production by, 175 96elastic, 33, 175incoherent, 197inelastic, 33 4, 175, 486 7meaning of term, 175multiple, 190 1and polarization, 181 4subsurface, 28and transparency, 42see also Mie scattering; Raman effect; Rayleigh scattering;
Tyndall scatteringscattering coefficient, linear (Napierian), 35, 176scattering efficiency (factor), 186, 187scattering length, 176
Scheele’s green, 297scheelite, 366schiller, 122, 124, 125schlera (in eye), 190schorl, 149scintillators, 365, 416 17properties required, 417
second-harmonic generation (SHG), 151, 153 4,155, 156
colours produced by, 154 5, 160at interfaces, 161microscopy using, 162, 163in organic materials, 161 2in polycrystalline materials, 160
selection rulesLaporte rule, 254 5, 270multiplicity rule, 270, 278parity rule, 254, 278, 279
selenium exposure meter, 471self-cleaning windows, 121self-quenching, 378Sellmeier constant, 66Sellmeier equation, 66semiconductorcolours, 436 9degenerate, 440extrinsic, 436ninorganic, 436 41intrinsic, 436isostructural pairs, 441organic, 340, 457 64see also transparent conducting oxide
semiconductor alloys, colours, 440semiconductor diode lasers, 155, 294, 448, 449semiconductor LED, 443semiconductor nanostructures, 449 50, 449 57sensitizer, 366, 367, 472, 473 4SERS (surface enhanced Raman spectroscopy),
486 7SFG (sum frequency generation), 155, 156SFM (sum frequency mixing), 155shells, colour of, 122SHG: see second-harmonic generationshims (for embossed holograms)child/stamper, 242 3mother/master, 242, 243
shortpass filters, 114SI units, 43, 46see also main entry: units
SiAlONs, 43signal beam, 235silica optical fibres, 77 8, 84chemical impurities in, 81
silica spheres, in opal, 214 15, 216silicon, band gap, 436, 437silicon carbide, 103silicon dioxide, 62, 103, 121see also quartz; silica
silicon oxynitride, 107silicon photovoltaic cell, 471, 472silvercolour, 478in glass, 192, 483 4reflectivity, 111, 477
silver gallium selenide (AGSe), 153silver gallium sulfide (AGS), 153silver halidesin photochromic glass, 483in photographic film, 484 5
Index 506
silver nanoparticles, 479 80in polychromic glass, 481 2in Raman spectroscopy, 487
single quantum well (SQW), 450, 451, 452singlet states, 303, 371, 458, 459ski goggles, 359sky
colour, 23, 40, 178 9, 180polarization of light from, 183 4
small particles, scattering by, 184, 185‘smart’ mirrors, 464‘smart’ windows, 119 21, 464smoky quartz, 431Snel’s law (Snell’s law), 51
applications, 67, 87, 216, 218, 445soap film, 91, 99, 100 1, 100sodium
Grotrian diagram, 262ground-state configuration, 263line spectrum, 262
sodium D lines, 263, 264sodium racemate, 162, 164sodium tartrate, 162, 164sodium tungsten bronzes, 466sodium vapour lamps, 189 90, 247, 262 3solar cells, 471 2
dye-sensitized, 472 4solar concentrators, 472sols, 191 3, 478solvatochromic fluorophore, 409solvatochromism, 374, 406Solvent Yellow, 354 5, 356sonoluminescence, 315space charge, 444, 471spatial period, 44
see also wavelengthspecific refraction, 61specific rotation, 164specific rotation dispersion, 167spectral exitance, 12spectral hole
homogenous linewidth, 298inhomogenous linewidth, 298lifetime, 299
spectral-hole burning, 297 9mechanisms, 299method, 298
spectral-hole formation, 297 300spectral irradiance, 12, 46spectral lines, 248spectral radiance, 11spectrometer, 247spectroscope, 247n, 255spectrum
absorption: see main entry: absorption spectrumof atoms, 247 51, 254 5band, 312, 313continuous, 11, 247electromagnetic, 2, 247emission: see main entry: emission spectrumformation of, 67 75of ions, 247 51line, 247 8solar, 255 6stellar, 256visible, 2, 9, 10, 10
sphalerite, 366spin-allowed transitions, 270, 276, 281spin-forbidden transitions, 278
spin orbit coupling, 253, 277, 302 3, 306, 371, 381, 428spin quantum numbers, 304spinels, 287, 295 6, 296 7, 345spiro-naphthoxazines, 359, 360spontaneous emission, 17, 18
Einstein coefficient for, 19spot test, 350SQW (single quantum well), 450, 451, 452stained glass, 37, 194stars
and diffraction limit, 204effective temperatures, 14spectra, 256
stepped-index multimode optical fibres, 82, 84Stern Volmer constant, 378, 379Stern Volmer equation, 378stimulated emission, 17, 18, 19, 21, 22, 259, 260, 282, 292, 369Stokes radiation, 487Stokes shift, 365stopband, photonic, 115strawberry tree, 321street lighting, 189 90, 247, 248, 262 4, 383stress, 148stress birefringence, 148stretching modes
antisymmetrical mode, 315symmetrical mode, 315
strontium aluminate phosphor, 433 4strontium compounds, colours, 255, 315strontium magnesium phosphate, 383strontium nitrosilicide phosphor, 447structural interactions, in luminescence, 374structural probe, colour as, 287subpixels, 170subtractive coloration, 37 9, 194, 486sum frequency generation (SFG), 155, 156, 161sum frequency mixing (SFM), 155sun
colours, 179, 180, 188effective temperature, 14radiation from, 11, 12, 179, 180, 313spectrum, 255 6
sun bed tubes, 383sun tan, 337sunglasses, 136, 148, 359sunscreens, 51, 190, 346, 420sunset, 179, 180sunstone, 184superlenses, 87 9, 204surface colour centres, 434surface enhanced Raman spectroscopy (SERS), 486 7surface plasmon, 479surface plasmon polaritons, 479
energy levels, 480, 481localized, 480longitudinal, 480, 481propagating, 480transverse, 480, 481
surface plasmon resonance, 479surface plasmon resonance spectroscopy, 480surface reflectivity, 92, 93 4, 132surface roughness, 40, 42symmetry operators, 137
talc, 366tannins, 332, 337tantalum nitride, 437tartaric acid, 162, 165, 166
meso-form, 165
507 Index
TE (transverse electric) wave, 131, 132, 133telescopes, 204, 478television sets, 170, 384, 386 9telluric lines, 256TEM (transverse electromagnetic) waves, 5 6temperature sensor, 413temporal frequency, 7, 44temporal period, 44tenebrescence, 366terbium ionsenergy levels, 381, 393in phosphors, 381 2, 393in quantum cutting, 403 4, 405
term (of atom or ion), 251, 252, 303multiplicity of, 252, 271
term splitting, 271 3term symbol, 252, 303tetrahedral coordination, 268, 269tetrahedral crystal field, 271, 272 3TFEL (thin-film electroluminescent) displays, 391 4thermal poling, 161thermal quenching, 375 6thermochromic materials, 270, 360thermochromism, 230, 270, 420, 437, 441, 470, 478thermoluminescence, 366THG (third-harmonic generation), 153, 154, 155, 156thin film(s)anodized, 103 4colour of
in air, 99 102on substrate, 102 4
interference at, 94 9reflected beams, 96 7transmitted beams, 98 9
multiple, 111 15, 121 2OLED, 459 60reflectivity of, 104 5on substrate
colour of, 102 4reflectivity of, 104 5
tapered/wedge-shaped, 96 7thin layer, properties, 450thin-film coatingsantireflective, 105 10high-reflectivity, 110
thin-film electroluminescent (TFEL) displays, 391 4thin-film engineering, 111thinning film, 100 1thiophene, 470third-harmonic generation (THG), 153, 154, 155, 156thulium ions, 393colours, 289energy levels, 420, 447
thymol blue, 352tin oxide, 119 20, 440titanium carbide, 437titanium carbonitride, 441titanium compounds, colours, 286titanium dioxide, 39, 62, 121, 138 9, 188, 190, 346reduction of, 341
titanium nitride, 437titanium oxynitride, 437titanium sapphire laser, 282 3TM (transverse magnetic) wave, 87, 131, 132, 133TNT (2,4,6-trinitrotoluene), detection of, 413tomato, colour, 317topaz, 286432total internal reflection, 54 7, 77frustrated, 56, 57
in LEDs, 445, 446in opals, 216, 217
tourmaline, 149 50, 286transitionallowed, 270charge-transfer, 340, 342, 345 6, 348 9electric dipole, 254electron pair, 348electronic, 258, 262, 263 4, 275, 280, 281forbidden, 270, 278, 281HOMO LUMO, 317, 350, 357, 419, 468interband, 451 4intersubband, 453, 454laser, 260 1magnetic dipole, 254n to p, 310nonradiative, 279, 281, 282, 293, 369, 455, 456p to p
�, 310, 322, 468
parity-forbidden, 281phonon-assisted, 279radiationless, 279, 281, 282, 293, 369radiative, 369rates, 281spectroscopic, 274spin-allowed, 270, 276, 281spin-forbidden, 278vibrational, 310
transition metal compoundscolours, 284 5, 286, 295 7pigments, 295 7
transition metal elements, 249, 301transition metal ioncolours, 264 70crystal-field splitting, 270 7electron configurations, 302in glass, 196in insulators, 424, 425
transition-metal-ion lasers, 281 3see also ruby laser; titanium sapphire laser
translucency, 42transmission diffraction gratings, 198, 206, 207, 209transmission electron microscopy, 390transmission holograms, 235 7transmissivity, 36transmittance, 36transparency, 34, 41transparent animals, 41, 62, 190transparent ceramics, 43, 189 90transparent conducting oxides (TCOs), 103, 437,
437 8transparent insulating solids, 91transparent materials, 34, 42, 43transparent plate, reflection from, 92 4transparent solids, 41 3transverse electric wave (TM wave), 131, 132, 133transverse electromagnetic (TEM) wave, 5 6transverse magnetic wave (TM wave), 87, 131, 132, 133trapping, metastable, 299triboluminescence, 366, 416trichroism, 149, 284trichromatic fluorescent lamps, 381 2trichromaticity, 24tridymite, 433triiodide iodide redox couple, 474triplet states, 304, 371, 458, 459tungsten blue, 341tungsten bronzes, 465 7tungsten carbide, 437tungsten-filament lamps, 11, 14, 16
Index 508
tungsten trioxideelectrochromic film, 465 7reflectance spectrum, 420, 421
tuning, optical parametric oscillator, 157Turnbull’s blue, 341, 342turquoise, 286two-frequency up-conversion, 401two-photon fluorescence, absorption mechanism for, 410 11, 411two-photon fluorescence microscopy, 411, 412Tyndall blue, 176, 187, 188Tyndall scattering, 176, 187Tyndall spectra, 185Tyrian purple, 336 7
ultramarine (pigment), 348 9artificial, 349
ultraviolet radiation, 10, 118, 364in IC manufacture, 106, 204in sun beds, 383
uniaxial crystaldichroism in, 149double refraction in, 143 4
unit cell of crystal, 138units
energy, 43photometric, 45, 46radiometric, 12 13, 45, 46spectral, 10
up-conversion, 151, 355, 369, 370, 394 402absorption mechanism for, 410, 411two-frequency, 401
up-conversion efficiency, 394uranium compounds
colours, 295, 296radioactivity, 365, 431
vacancy, 424vacuum, black, 257valence band, 4, 5, 388, 419, 420vanadium compounds, colours, 286vector, 5vector model of atom, 302 4velocity vector, 5, 6vermilion, 437vibration ellipse, 131vibrational energy levels, 310, 311vibrational transition, 310viewing angle
rainbow, 68, 70, 71, 71, 72thin film, 97 8
Virginia creeper, leaf colours, 333, 335visible spectrum, 2, 9vision, 23 8visual pigments, 24, 26 8visual purple, 24vitamin C, 166volatile organic compounds (VOCs), detection of, 354
walk off, 159water
colour of, 315 16heavy, 316refractive index, 70
water air surface, reflection at, 134 5water-lily, 340, 377wave
coherent, 7, 8, 17evanescent, 54, 56, 57, 87, 89idler, 156, 157
incoherent, 7, 11, 16, 17monochromatic, 235progressive, 5propagating, 5standing (non-travelling), 44transverse electric, 131, 132, 133transverse electromagnetic, 5 6transverse magnetic, 87, 131, 132, 133travelling, 5, 44
wave amplitude, 7wave equation, 6, 43 4wave particle duality, 15 16wave theory of light, 1, 2, 3wave vector, 5wave velocity, 5, 7
relationship to frequency, 7wavelength, 7, 44
and energy, 44 5estimation by diffraction, 210 11and Rayleigh scattering, 179and refractive index, 52 4, 102, 158, 178visible spectrum, 10
wavelength dispersion, 82wavenumber, 45Whewell Qu�etalet diffraction pattern, 227white-hot object, 11white light
generation by LEDs/OLEDs, 447 8, 463 4perception of, 14refraction by prism, 67 8standard daylight, 31
Wien displacement law, 13willemite, 366window glass, 79, 119windows
low-emissivity, 119 21self-cleaning, 121‘smart’, 119 21
windscreen, 148wine colour, 328 32woad, 335wollastonite, 366work function, 3working electrode (in solar cell), 473wurtzite, 366, 440
X-ray diffractionBragg’s law, 212 13dynamical theory, 213kinematical theory, 213‘powder’ method, 231
X-ray imaging, 435 6X-ray tomography, 417xanthophyll, 318, 319xenon, 257
line spectrum, 258in plasma display, 384, 385replacement in mercury-vapour lamps, 402 3
xenon arc lamp, 257xenon flashlamp, 257, 295xenon flashtube, 257, 281
ytterbium ions, energy transfer processes, 399 401yttrium aluminium garnet (YAG), 292, 447yttrium gadolinium borate, 385yttrium oxide, in fluorescent lamps, 381
ZBLAN glasses, 80zinc cadmium sulfide, silver-activated, 387, 389
509 Index
zinc oxide, 346, 420, 421reflectance spectrum, 420, 421
zinc oxide nanoparticles, band gap, 422, 449zinc selenide, 455zinc silicate, 385zinc sulfidecrystal structures, 441with cuprous chloride, 389
fluorescence, 364, 455with manganese ions, 391minerals, 366silver-activated, 387 9
zincite, 366zircon, 366zirconium carbonitride, 441zirconium nitride, 437
Index 510