optical radiation and matter

7/30/2019 Optical Radiation and Matter

1/385

Optical Radiation and Matter

J. M. OHare and R. J. BrechaPhysics Department

Revision, August 2004


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Contents

1 Review of Electromagnetic Radiation 7

1.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Maxwells Equations in Free Space . . . . . . . . . . . . . . . 9

1.3 The Free-Space Wave Equation . . . . . . . . . . . . . . . . . 10

1.3.1 Plane Wave Solution to the Wave Equation . . . . . . 10

1.3.2 Spherical Wave Solution to the Wave Equation . . . . 13

1.3.3 Gaussian Beam Solution to the Wave Equation . . . . 14

1.4 Phase and Group Velocity . . . . . . . . . . . . . . . . . . . . 20

1.5 Energy Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.5.1 The Free-Space Poynting Vector . . . . . . . . . . . . . 27

1.5.2 Time Averaging of Sinusoidal Products: Irradiance . . 30

1.6 Resonator Electric Field . . . . . . . . . . . . . . . . . . . . . 32

1.7 Problems - Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . 42

2 Polarization of Light 47

2.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . 47

2.2 Polarization of Light Waves . . . . . . . . . . . . . . . . . . . 48

2.2.1 Elliptical Polarization . . . . . . . . . . . . . . . . . . 49

2.2.2 Linear or Plane Polarization . . . . . . . . . . . . . . 54

2.2.3 Circular Polarization . . . . . . . . . . . . . . . . . . . 54

2.2.4 Polarization in the Complex Plane . . . . . . . . . . . 55

2.3 Jones Vector Representation of Polarization States . . . . . . . 57

2.3.1 Superposition of Waves using Jones Vectors . . . . . . 60

2.4 Optical Elements and Jones Matrices . . . . . . . . . . . . . . 61

2.5 Longitudinal Field Components . . . . . . . . . . . . . . . . . 70


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4 CONTENTS

3 Radiation and Scattering 77

3.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . 773.2 Summary of Maxwells Equations . . . . . . . . . . . . . . . . 783.3 Potential Theory and the Radiating EM Field . . . . . . . . . 793.4 Radiation From a Dipole . . . . . . . . . . . . . . . . . . . . . 833.5 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.5.1 Scattering by a Dipole . . . . . . . . . . . . . . . . . . 933.6 Polarization of Rayleigh Scattered Light . . . . . . . . . . . . 97

3.6.1 Polarization of scattered light . . . . . . . . . . . . . . 993.7 Radiation in the Coulomb Gauge . . . . . . . . . . . . . . . . 1013.8 Problems - Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . 106

4 Absorption and Line Broadening 1094.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . 1094.2 Extinction By a Dipole . . . . . . . . . . . . . . . . . . . . . . 1114.3 Field from a Sheet of Dipoles . . . . . . . . . . . . . . . . . . 1164.4 Propagation in a Dilute Medium . . . . . . . . . . . . . . . . . 1184.5 Beers Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.6 Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.6.1 Natural Line Broadening . . . . . . . . . . . . . . . . . 1294.6.2 Doppler Broadening . . . . . . . . . . . . . . . . . . . 1314.6.3 Collision Broadening . . . . . . . . . . . . . . . . . . . 1354.6.4 Voigt Profile . . . . . . . . . . . . . . . . . . . . . . . . 138

4.7 Absorption Spectroscopy Experiment . . . . . . . . . . . . . . 1384.8 Problems - Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . 144

5 Macroscopic Electrodynamics 1475.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . 1475.2 The Local Field . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.3 The Macroscopic Maxwell Equations . . . . . . . . . . . . . . 1505.4 The Polarization Density and Constitutive Relation . . . . . . 152

5.4.1 Monochromatic Waves . . . . . . . . . . . . . . . . . . 1545.5 Dielectric and Impermeability Tensors . . . . . . . . . . . . . 156

5.6 The Electromagnetic Wave Equation . . . . . . . . . . . . . . 1565.7 Plane Waves in Dense Matter . . . . . . . . . . . . . . . . . . 1585.8 Classification of Wave Types . . . . . . . . . . . . . . . . . . . 159

5.8.1 Homogeneous and Inhomogeneous Waves . . . . . . . . 1605.8.2 Transverse Electric and Magnetic Waves . . . . . . . . 160


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CONTENTS 5

5.8.3 Additional Properties of Plane Waves . . . . . . . . . . 161

5.9 Reflection and Transmission at an Interface . . . . . . . . . . 1645.9.1 Nature of Reflected and Transmitted Waves . . . . . . 1645.9.2 Amplitude of Transmitted and Reflected Waves (Fres-

nels Equations) . . . . . . . . . . . . . . . . . . . . . . 1665.9.3 Jones Matrices for Reflection . . . . . . . . . . . . . . . 1735.9.4 Stokes Relations . . . . . . . . . . . . . . . . . . . . . 1755.9.5 Specific Angles of Reflection . . . . . . . . . . . . . . . 176

5.10 Thin film anti-reflection (AR) coating . . . . . . . . . . . . . . 1815.11 Waves at a Conducting Interface . . . . . . . . . . . . . . . . 188

5.11.1 Reflection and transmission for a conducting medium . 1915.11.2 Characteristic Angles . . . . . . . . . . . . . . . . . . . 194


6 Optical Properties of Simple Systems 2036.1 Normal Modes of Motion . . . . . . . . . . . . . . . . . . . . . 2046.2 Local and Collective Modes . . . . . . . . . . . . . . . . . . . 209

6.2.1 Local Modes . . . . . . . . . . . . . . . . . . . . . . . . 2096.2.2 The Linear Monatomic Lattice . . . . . . . . . . . . . 2106.2.3 One Dimensional Diatomic Lattice . . . . . . . . . . . 212

6.3 Optical Properties of Simple Classical Systems . . . . . . . . . 2156.3.1 Frequency Dependence of the Dielectric Constant of

Bound Charges . . . . . . . . . . . . . . . . . . . . . . 2166.3.2 Electronic Contributions to the Dielectric Constant . . 2186.3.3 Reststrahlen Bands . . . . . . . . . . . . . . . . . . . . 221

6.4 Drude Theory of Metals . . . . . . . . . . . . . . . . . . . . . 2276.4.1 Low Frequency Region ( 1): The Hagen-Rubens

Relation . . . . . . . . . . . . . . . . . . . . . . . . . . 2296.4.2 The High Frequency Region ( 1) . . . . . . . . . 2316.4.3 Plasma Reflection . . . . . . . . . . . . . . . . . . . . . 232

6.5 Semiconductors - Example of InSb . . . . . . . . . . . . . . . 2336.6 Kramers-Kronig Relations . . . . . . . . . . . . . . . . . . . . 2366.7 Problems - Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . 241

7 Crystal Optics 2477.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . 2477.2 Polarizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2497.3 Birefringence (Double Refraction) . . . . . . . . . . . . . . . . 250


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6 CONTENTS

7.3.1 Uniaxial Crystals . . . . . . . . . . . . . . . . . . . . . 256

7.3.2 Ray Direction and the Poynting Vector . . . . . . . . . 2597.3.3 Double Refraction at the Boundary of an AnisotropicMedium . . . . . . . . . . . . . . . . . . . . . . . . . . 259

7.3.4 The Optical Indicatrix . . . . . . . . . . . . . . . . . . 2617.3.5 Wave Velocity Surfaces . . . . . . . . . . . . . . . . . . 267

7.4 Retarders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2677.5 Optical Activity . . . . . . . . . . . . . . . . . . . . . . . . . . 270

7.5.1 Susceptibility tensor of optically active medium . . . . 2707.6 Faraday Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 2737.7 The k-Vector Surface of Quartz . . . . . . . . . . . . . . . . . 2777.8 Off-axis waveplates . . . . . . . . . . . . . . . . . . . . . . . . 279


8 Electro-optic Effects 2958.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . 2958.2 Optical Indicatrix Revisited . . . . . . . . . . . . . . . . . . . 2968.3 Electro-optic Effects . . . . . . . . . . . . . . . . . . . . . . . 298

8.3.1 Crystal symmetry effects on the tensor rijk : . . . . . . 2998.3.2 Deformation of the Optical Indicatrix . . . . . . . . . . 304

8.4 Electro-optic Retardation . . . . . . . . . . . . . . . . . . . . 3168.4.1 The Longitudinal Electro-optic Effect . . . . . . . . . . 3168.4.2 The Transverse Electro-optic Effect. . . . . . . . . . . . 319

8.5 Electro-optic Amplitude Modulation . . . . . . . . . . . . . . 3218.6 Electro-optic Phase Modulation . . . . . . . . . . . . . . . . . 3248.7 The Quadratic E-O Effect . . . . . . . . . . . . . . . . . . . . 3278.8 A Microscopic Model for Electro-optic Effects . . . . . . . . . 331

8.8.1 Pockels Effect . . . . . . . . . . . . . . . . . . . . . . . 3318.9 High-Frequency Modulation . . . . . . . . . . . . . . . . . . . 3358.10 FM Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 3388.11 Problems - Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . 342

9 Acousto-optic Effects 347

9.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . 3479.2 Interaction of Light with Acoustic Waves . . . . . . . . . . . . 3489.3 Elastic Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 3519.4 The Photoelastic Effect . . . . . . . . . . . . . . . . . . . . . . 3559.5 Diffraction of Light by Acoustic Waves . . . . . . . . . . . . . 360


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CONTENTS 7

9.5.1 Raman-Nath Diffraction . . . . . . . . . . . . . . . . . 361

9.5.2 Bragg Scattering . . . . . . . . . . . . . . . . . . . . . 3649.6 Problems - Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . 373

A Vector Theorems and Identities 375A.1 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375A.2 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

B Operators in Different Coordinate Systems 377B.1 Rectangular Coordinates . . . . . . . . . . . . . . . . . . . . . 377B.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . 377B.3 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . 378

C Coordinate Transformations 379C.1 Rectangular (x, y, z) - Spherical (r, , ) . . . . . . . . . . . . 379C.2 Rectangular (x, y, z) - Cylindrical(, , z) . . . . . . . . . . . 379


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8 CONTENTS


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Chapter 1

Review of ElectromagneticRadiation

The main part of this text deals with the propagation of electromagneticwaves in matter. In this first Chapter, however, after a brief historical in-troduction, we begin by considering the properties of electromagnetic wavespropagating in a vacuum, ignoring for the moment the possible presence ofany material medium. Equivalently, we can imagine that our observationpoint for the waves under consideration is very far from any possible sourcesof radiation, such as oscillating charges (more on this in the next chapter)or currents. First we will take a look at how Maxwells equations relate theelectric and magnetic fields of radiation. We will see how Maxwells equa-tions, in turn, can lead to the free space wave equation, which connects thespatial and time dependance of the electric field. This equation may be usedto find solutions describing several different types of waves. We will learnto describe the phase and group velocities of waves, which tell us about the

information transmission speed possible by a wave. Finally, we discuss theenergy flux and the time averaged power carried by a wave. The last sectionof the Chapter provides an example of how to tie some of these bits of infor-mation together, through an example studying the electric field of an opticalcavity or resonator.

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10 CHAPTER 1. REVIEW OF ELECTROMAGNETIC RADIATION

1.1 Historical Background

Although it is impossible to pick out any one given point in time to begin thestudy of the development of electromagnetic theory, a fairly good argumentcan be made that the beginning of the 19th century was a pivotal epoch. Wecan start with the Danish physicist Hans Christian Oersted (1777-1851), whohypothesized that there might be a relation between electric and magneticphenomena. In fact, Oersted performed a series of experiments in 1820 inwhich the effects of current flow on the deflection of a compass needle wereobserved. A few months later, in France, Andre-Marie Ampere (1775-1836)was able to show that the current in an electrical circuit was equivalent to amagnet oriented in a plane perpendicular to that of the circuit. He even came

up with a first attempt at a molecular theory of magnetism based on theseearly observations. Another Frenchman, Dominique Francois Jean Arago(1786 - 1853), and independently, Humphry Davy (1778 - 1829)in England,soon showed that a helically-wound wire would be able to magnetize a pieceof iron placed inside the helix. The result of all these investigations was thebasis of the first qualitative understanding of the connection between whathad previously been seen as two distinct physical phenomena.

The concept of fields was still relatively new, but Michael Faraday (1791- 1867) was able to use this idea to construct an experiment to demonstratethat nature of the lines of magnetic force surrounding a straight wire. Itshould be noted that Faradays experiments were performed in late 1821;all of the above-mentioned progress had taken place in the remarkably shortperiod of about one year! Faraday was also key in making one of the nextimportant discoveries linking electricity and magnetism. He found (in 1831)that a changing magnetic flux can create an electric current in a circuit. Fromstart to finish Faraday needed about ten days to perform his experiments onthis phenomenon and to arrive at what are essentially the correct conclusions,results which in turn form the basis for all electric motors.

James Clerk Maxwell (1831 - 1879) wished to find a mathematical frame-work capable of containing all of the aforementioned experimental results,and in particular the concept of electric and magnetic fields. Maxwell pos-

tulated an electric displacement, the time rate of change of which makes acontribution to the total current and thus to the magnetic field. As with allof the previous workers, one fundamental conviction of Maxwells was that allof space is pervaded by the ether, undetected but crucial conceptually forall theoretical models. The experimental and theoretical accomplishments of


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1.2. MAXWELLS EQUATIONS IN FREE SPACE 11

this period still stand unchanged, although based on the false premise of the

ether.One of the key results of Maxwells work was the realization that changesin magnetic and electric fields propagate through the surrounding medium ata finite speed. His theory showed further that this speed of propagation wasexactly the same as the known speed of light propagation. The predictionsarising from Maxwells theory led to a great deal of experimental activityattempting to verify the speed of wave propagation in various materials.

1.2 Maxwells Equations in Free Space

We begin our study of the interaction between optical radiation and mat-ter with a review of the basic elements of electromagnetic theory, startingwith the Maxwell equations for fields in the absence of currents and charges.These equations, formulated in terms of E, the electric field vector and B,the magnetic induction vector, are all we need to understand some of the fun-damental properties of electromagnetic radiation. In MKS units (the unitswhich will be used throughout this text), the free-space Maxwell equationsare:

E = 0 (1.1)

B = 0 (1.2)

E = B

t(1.3)

B = 1c2

E

t(1.4)

In writing the above we have assumed that neither charges nor currentsare present. The constant c is the speed of light in vacuum, considered oneof the fundamental physical constants and defined to be exactly 299,792,458m/s. As a side note, the meter itself, which used to be defined by the length ofa standard bar, is now the distance light travels in vacuum in 1/299,792,458

s. To complete this somewhat circular set of definitions, the second is definedin terms of a specific transition in atomic cesium (an atomic oscillator, if youlike) as follows: The second is the duration of 9 192 631 770 periods of theradiation corresponding to the transition between the two hyperfine levels ofthe ground state of the cesium 133 atom. [1]


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1.3 The Free-Space Wave Equation

The free-space electromagnetic wave equation is readily obtained from Maxwellsequations by taking the curl of Eqn. 1.3 and substituting Eqn. 1.4:

E

=

t[ B]

= t

1

c2E

t

(1.5)

=

1

c2

2 E

t2

Using a standard vector identity (see Appendix A, Vector Theorems andIdentities) on the left hand side of the above equation, we obtain

( E) 2 E = 1c2

2 E

t2(1.6)

Next, using Eqn. 1.1 we obtain the following form for the free-spaceelectromagnetic wave equation,

2 E =1

c22 E

t2 (1.7)

This result is one of the key starting points for what follows in the rest of thischapter. To summarize the assumptions we have thus far made in arrivingat Eqn. 1.7, we recall that the fields in which we are interested are far fromany sources of radiation and thus the Maxwell equation source terms weretaken to be zero. In the following sections we will investigate three differentsolutions to the wave equation, starting with the one that will be used mostfrequently throughout this text.

1.3.1 Plane Wave Solution to the Wave Equation

The most straight-forward and algebraically simple solutions to Eqn. 1.7are the so-called plane wave solutions. A monochromatic plane wave in freespace can be written as,


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1.3. THE FREE-SPACE WAVE EQUATION 13

E(r, t) = 12 E(k, ) ei( k0rt) + c.c. (1.8)

=

E( k0, ) ei( k0rt)

(1.9)

where c.c. stands for complex conjugate and indicates the real part of acomplex number. E(k0, ) is a complex vector amplitude, k0 is the free spacewavevector, with (|k0| = 2/0) being the spatial frequency of the wave ,and is the angular frequency of the wave (k0 = /c). The complex vectoramplitude is a notational device to designate the polarization of the wave;this will be discussed in much greater detail in Chapter 2. The wave in Eqn.1.9 is a harmonic wave and could just as easily have been represented by

a sine or cosine rather than the complex expression that we have chosen.Let us write this out once for the sake of completeness and to get a betterfeeling for the notation. We consider a monochromatic (single frequency)

wave travelling in the z-direction, which means that k0 r becomes kzz andthat we can drop the arguments for E. From Maxwells equations we find thatthe electric field vector must lie in a plane perpendicular to the propagationdirection, i.e. in the x y plane. Thus we write the vector amplitude as

E= Ax eix x + Ay eiy yIn the above expression, x and y represent a relative phase of the wave with

respect to a chosen starting point. Substituting this into Eqn. 1.9 givesE(r, t) =

Ax e

ix x + Ay eiy y

ei(

k0rt)

= Ax cos(kz z t + x) x + Ay cos(kz z t + y) y ,which is the final result we are after.

Alternatively we could have used Eqn. 1.8 to find

E(r, t) =1

2

Ax e

ix x + Ay eiy y

ei(

k0rt)

+1

2 Ax e

ix x + Ay eiy y

ei(

k0rt)

= 12

Ax {exp[i (kz z t + x)] + exp [i (kz z t + x)]} x

+1

2Ay {exp[i (kz z t + y)] + exp[i (kz z t + y)]} y

= Ax cos(kz z t + x) x + Ay cos(kz z t + y) y .


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The reason for using the complex notation is that it offers algebraic sim-

plicity for use with Maxwells equations. From now on we will write Eqn.1.9 without the term in the expression but with the convention that itis implied.

Eqn. 1.9 is referred to as a plane wave solution because the surfaces ofconstant phase, i.e. for which k r is a constant, used to characterize thewave geometrically are planes. This is illustrated in Fig. 1.1 where we havedrawn the spatial variables of the phase of the wave of Eqn. 1.9. As shownin the figure, for any radius vector r drawn from the origin to a point on aplane which is perpendicular to the wavevector, the phase of the wave, k r,will have the same value.

x

y k

k

r

Figure 1.1: Surfaces of constant phase for the plane-wave solution to the wave

equation. To be a truly accurate representation, the planes should extend toinfinity.

We will return often to the plane wave solution to the wave equation.


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Although idealized, in the sense that no real electromagnetic wave is truly

a plane wave, the mathematical simplicity of starting out by treating planewaves is a great bonus. For most problems we will treat in this text, theresults obtained by using plane waves is sufficient for our purposes. How-ever, it is also useful to look at two other wave equation solutions commonlyencountered in optical and laser physics.

1.3.2 Spherical Wave Solution to the Wave Equation

In this section we will briefly present the so-called spherical wave solution tothe wave equation. An application of spherical waves will be encountered inChapter 3.

In complex notation the spherical wave is represented by

E(r, t) =1

rexp i(kr t) (1.10)

where r is the spherical coordinate (r =

x2 + y2 + z2). This solution rep-resents a wave emanating from a point and propagating uniformly in alldirections in space. The amplitude of the wave falls off as 1/r, and as weshall see shortly, this leads to a total propagating energy which is constantwhen integrated over a sphere of radius R, independent of the magnitude ofR. We can show that the spherical wave is a solution to the wave equationby using the Laplacian in spherical coordinates (see Appendix B):

2E = 1r2

r

r2

E

r

+ [derivatives with respect to and ]

=1

c22E

t2

We ignore the derivatives with respect to the angular variables and sinceour guessed solution has no dependence on those variables. Substitutingthe spherical wave solution into the above,

1

r2

rr2 ikr

1

r2 ei(krt) =1

c2

2E

t2

k2

rei(krt) =

1

r

2c2

ei(krt)

which confirms that Eqn. 1.10 satisfies the wave equation, since k2 = 2/c2.


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1.3.3 Gaussian Beam Solution to the Wave Equation

There is another set of solutions to the wave equation that is very important,and that must be treated extensively when considering optical resonators andlasers. These fields are known as Gaussian beams, the derivation of whichwill be the subject of this section. Further information on Gaussian beamsand more generalized solutions to the free-space wave equations can be foundin many textbooks, for example, Ref. [2, 3, 4]. An early and valuable reviewof this material is found in a paper by Kogelnik and Li [5]. In Chapter 2 wewill look at a way to think of a Gaussian beam as being a superposition ofplane waves; that result will serve to connect some of the concepts coveredin these first two chapters.

We begin by showing in Fig. 1.2 some features of Gaussian beams. Whenseen from the side, a Gaussian beam with waist w0 propagates to the right,increasing in radius as a function of position z. The surfaces of constantphase are shown as curved lines. The intensity of the beam has a maximumat the center, and decreases radially with a Gaussian functional dependence,as shown in Fig. 1.2b, which might represent a burn pattern formed by anintense laser incident on an absorbing medium. Fig. 1.2c represents a plotof the intensity of a Gaussian beam that would be measured by scanning apinhole or slit across the beam profile and recording the intensity transmittedthrough the aperture.

With the preceding qualitative discussion, we can now start the mathe-matical formalism needed to arrive at the specific result we are seeking. Webegin by assuming a harmonic wave proportional to eit. The wave equationfor the electric field, Eqn. 1.7, may be written as

2 E+ k2 E = 0

where k2 =

2

c2

. Before proceeding towards a solution to the wave equa-

tion, we consider a concrete physical situation to which the mathematical

formalism will commonly apply. A laser, such as a common red helium neonlaser, has a necessary component two round mirrors, through one of whichthe laser light is emitted. We will make use of the fact that for such an opticalcavity, or resonator, a natural coordinate system to choose might be one withcylindrical symmetry. Thus we will rewrite the wave equation in cylindrical


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w0

z

a)

b) c)I0

r

r

Figure 1.2: A qualitative illustration of different Gaussian beam character-istics. a) Viewed from the side, with spot size as a function of position z.The curved lines represent surfaces of constant phase, each with radius ofcurvature R(z). b) Viewed head-on, the beam intensity is highest in thecenter and decreases radially. For the only profile considered here, there isno dependence of the intensity on the azimuthal angle . c) A sketch of

the intensity as a function of radial distance for the beam shown in part b),showing the Gaussian functional dependence.

coordinates yielding (see Appendix C for the coordinate transformation)

2 E(r,,z) + k2 E = 1r

r

r

E

r

+

1

r22 E

2+

2 E

z2+ k2 E = 0 (1.11)

We postulate that E depends only on r in the transverse direction, i.e. that

there is cylindrical symmetry. Thus2 E

2= 0

This assumption is not a general one, and will restrict us in the type of


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solutions we will find to the very simplest ones. We can now write

2 E

r2+

1

r

E

r+

2 E

z2+ k2 E = 0 (1.12)

If we were to consider, for example, a laser beam propagating in free spaceafter leaving a laser cavity, the energy flow will be predominantly in onedirection, the z-direction as we have the problem set up here. Thus we canuse the intuition we have built up in the previous sections and assume asolution to the wave equation which is a modified plane wave:

E(r, z) = (r, z)eikz (1.13)

Here we have dropped the vector notation for simplicity; we can assume thatthe field is polarized in, for example, the y direction, but this will play nofurther role in what follows. The amplitude (r, z) varies slowly in space,i.e. it represents an envelope in the z-direction along with variations inthe direction transverse to the propagation direction with a spatial scalelarge compared to the wavelength, while the exponential factor takes intoaccount rapid spatial variations in the z-direction. As we shall see later, thepossibility of absorption of energy from the beam may be included in thisterm as well. In the context of laser physics, the amplification, or gain, ofbeam energy propagating through the medium may also be part of this latterterm.

Now we can substitute this ansatz into the wave equation, yielding

2

r2+

1

r

r+ 2ik

z+

2

z2= 0. (1.14)

Our assumption of the slowly varying field allows us to make further simpli-fications in the above. If is a slowly varying function of z, we can drop thesecond derivative term (with respect to z) as being small compared to thefirst derivative or itself. This leaves us with

2

r2

+1

r

r

+ 2ik

z

= 0. (1.15)

To solve the above we try a solution of the form

(r, z) = exp

i

P(z) k2q(z)

r2

(1.16)


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where P(z) and q(z) may both be complex numbers. Although not necessar-

ily the most obvious choice for a trial solution, we are here most interestedin arriving at a result, and not in the mathematical details behind solvinga partial differential equation. Substituting this into the above form of thewave equation givesk2r2

q2 (z)+

2ik

q(z)+ 2k

P (z)

z+

k2r2

q2 (z)

q(z)

z

(r, z) = 0. (1.17)

For this relation to be satisfied, the expression in brackets must vanish, andfor that to happen, the individual powers of r must vanish separately. Thisleads to two conditions,

k2q2 (z)

+k2

q2 (z)

q(z)

z= 0

2ik

q(z)+ 2k

P (z)

z= 0.

Simplifying these gives two partial differential equations which can be solvedeasily. From the first of the above equations we have

q(z)

z= 1 = q(z) = z+ q0 (1.18)

This solution can in turn be used in the second equation:

P (z)

z=

iq(z)

=i

z+ q0(1.19)

or

P(z) = i ln

1 +z

q0

. (1.20)

Now we need to start working backward through all of the various sub-stitutions we have made, so that we find out the form of the electric field forthis solution. Recalling how we defined (r, z) initially, we can substitute

the expressions for q(z) (Eqn. 1.18) and P(z) (Eqn. 1.20) into Eqn. 1.16which gives, to within an overall amplitude and phase factor,

(r, z) = exp

ln

1 +

z

q0

+

ikr2

2q(z)

. (1.21)


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As r we require that (r, z) 0 since physically the fields must vanish

at very large distances from the axis. Examining the expression above, wesee that this requires that q0 be imaginary. Setting q0 = iz0, where z0 is areal variable, we can write Eqn. 1.21

(r, z) =1

1 +

zz0

2 ei tan1(z/z0)e+ ikr22q(z) (1.22)To arrive at this result, we have used the relation

ln1 + ziz0 = ln1 iz

z0 = ln 1 + z

z02

i tan1 z

z0Finally, we may define q(z) in terms of two real parameters R(z) and

w(z) such that1

q(z)=

1

R(z) i

w2(z), (1.23)

we can rewrite Eqn. 1.22 as

(r, z) =1

1 + zz02

ei tan1(z/z0)e+ikr

2/2R(z)er2/w2(z). (1.24)

It will be left to the problems to show that

R(z) = z

1 +

w20z

2= z

1 +

z0z

2(1.25)

w2(z) = w20

1 +

z

w20

2= w20

1 +

z

z0

2(1.26)

In the above we have set z0 = w20/. The real quantities R(z) and w(z)

represent respectively the radius of curvature of the wavefront of the beamand the spot size of the beam. Of particular interest are the values of theseparameters at the position z = 0, namely R(z) = , thus representing planewave, and w(z) = w0, the minimum value of the beam spot, called the beamwaist. Also from the above we can gain a feeling for the parameter z0, called


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the Rayleigh parameter or Rayleigh range. For z = z0 we have a beam

radius equal to

2 of the waist size, or an area of twice the minimum spotsize. Thus, the Rayleigh range gives an indication of how quickly the beam

is diverging.Returning to the definition of E(r, z) in terms of (r, z), we can write

finally

E(r, z) = E01

1 +

zz0

2 ei tan1(z/z0)e+ik z+ r22R(z) er2/2(z) (1.27)Here the factor E0 represents the maximum amplitude of the field, i.e. theamplitude at z = 0 and r = 0. This result represents a wave which varies

slowly in amplitude along the propagation direction, z. In addition, the fieldamplitude decreases as a function of the distance away in a radial directionfrom the z-axis. The Gaussian functional dependence gives the name to thissolution to the wave equation.

The phase of the electric field, given by the imaginary arguments in theexponential of Eqn. 1.27, is now a function of not only the position alongthe z-axis, but also of the distance from the axis, as well as of the beamradius, R(z). Two limiting cases are of some interest to us. We have alreadyseen that the radius of curvature of the wavefront is infinite at the beamwaist position z = 0. However, the phase of the wave, due to the factor

exp(i tan1

(z/z0)), shifts as the beam passes through the focus. In anycase, we see that the waist of a focused beam provides one example of agood approximation to a plane wave . Equivalently, for a beam that is well-collimated, we have the same result.

In the opposite extreme, for z z0 the Gaussian solution reduces on axis( r = 0 ) to

E 1z

e+ikz (1.28)

which is just the result for a spherical wave. In Fig. 1.2a the tangent linesdrawn to the profile for large values of z appear to converge at a point more

or less near to the beam waist, depending on the Rayleigh parameter. Look-ing at the beam at a distance z z0 one could approximate the wavefrontsby spherical waves with an effective origin shown schematically by this con-vergence point. Off-axis there is an additional rdependent phase shift, butthe main characteristic of the 1/z amplitude dependence remains.


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If we had not thrown away the -dependence early on in our derivation,

the situation would have been more complicated. The interested reader maywish to consult a textbook on laser theory to see examples of some othermodes which appear when the total solution including is considered.[2]

1.4 Phase and Group Velocity

We return now to consider the relationship between the frequency of an elec-tromagnetic wave and its wavevector. The goal of this section is to investigatethe speed of travel of a plane wave, or of a group of plane waves. Two keyconcepts are needed at the outset. First, the fact that the electromagnetic

wave equation is a linear function of the electric field implies mathematicallythat, for any two solutions to the wave equation E1 and E2, a linear super-position of the two, E = a E1 + b E2 is also a valid solution. This principleof superposition is a powerful tool to which we will refer repeatedly. Forthe purposes of the present Chapter we wish to consider the superpositionof two fields that can be taken as a description of a pulse of electromagneticradiation. The second concept needed for this section is that of the index ofrefraction. We will discuss the index of refraction in far more detail later;for now it suffices to note that the index of refraction describes the ratio ofthe speed of propagation of light in vacuum to the speed of propagation in amedium. The index of refraction is usually a (weak) function of the frequency

of radiation propagating in a medium, a phenomenon known as dispersion.For simplicity we will only look at a wave propagating in the z-direction,

E = E0 cos(kzt). In Fig. 1.3 a sinusoidal wave is sketched at two differenttimes, t and t + t, a time interval during which the wave moves to the right(+z-direction) a distance z. If we consider any given point on the waveand ask how fast it is moving, we will thereby define the phase velocity ofthe wave. Since we are looking at a point on the wave at which the phase isconstant, we must have, for the two different times,

[kz t] = [k(z+ z) (t + t)]

Cancelling common terms, we find for the displacement z

z =

kt = vpt,

which gives the definition of the phase velocity we seek, vp = /k. We have


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1.4. PHASE AND GROUP VELOCITY 23

z

t

t + t

z

Figure 1.3: Two snapshots of the motion of a plane wave along the z-axis,taken at times separated by an interval t.

already seen that in free space, /k0 = c, and will find later that for wavesin matter, vp = c/n, where n is the index of refraction of the medium.

Since a purely monochromatic wave is an idealization, and because suchan unchanging, unmodulated wave does not carry useful information in any

case, we can extend our considerations to a more complicated situation inwhich two waves of slightly differing frequencies are considered. In the processwe will define the group, or information velocity of the wave. For two waves,of wavenumbers k k and frequencies we can write (assuming hereunit amplitude)

cos [(k + k) z ( + ) t] + cos [(k k) z ( ) t] (1.29)

which can be rewritten using trigonometric relations in the form

2cos(kz

t)cos(kz

t) . (1.30)

To arrive at the above result, we assumed that we could write our wave asthe sum of two different waves with differing frequencies and wavenumbers.Mathematically speaking, we took advantage of the linear nature of the waveequation which gives rise to the principle of superposition, as mentioned in


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the introduction to this section. Again, we can state this principle as follows:

If A is a solution to the wave equation and B is a solution to the waveequation, than any linear superposition aA + bB is also a valid solution tothe wave equation.

The result given in Eqn. 1.30 shows a wave propagating with frequency and wavenumber k, but with a modulated amplitude. Looking at the resultas shown in Fig. 1.4, we see that two different time scales are present. For thepurposes of communication of information, it is most useful to determine therate of arrival of pulses, or in the case shown here, of peaks in the modulatedamplitude. In analogy with the approach taken to determining the phasevelocity, we can define the group velocity of the wave as being the rate ofadvance of the pulse peaks,

vg k

, (1.31)

which becomes, in the limit k 0, vg = d/dk.

z

z

Figure 1.4: Two snapshots of the motion of a wavepacket along the z-axis,

taken at times separated by an interval t. It is the motion of these packetsthat is equivalent to the transmission of information.

There are various forms in which we can express the group velocity, and


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more specifically, to relate the phase and group velocities. For example, using

the fact that = vpk we can write immediately

vg =d

dk= vp + k

dvpdk

= vp

1 k

n

dn

dk

(1.32)

which shows us that the group and phase velocities are in general different.For example, in the presence of material dispersion, or dependence of theindex of refraction on wavelength, the phase velocity will also depend onwavelength, and the two velocities will not be equal. Expressing the resultof Eqn. 1.32 directly in terms of the index of refraction and wavelength, wefind

vg = vp 1 + n dnd (1.33)Likewise, we may express the group velocity in terms of the frequency,

vg =c

n + dnd

(1.34)

We will leave to a later chapter the detailed discussion of index of refraction,stating here only that usually (so-called normal dispersion) dn/d is posi-tive and that both dn/d and dvp/dk are negative, implying that the groupvelocity is less than the phase velocity.

To close out this section we note one additional important point. Theprinciple of superposition may be extended to any number of solutions; oneexample is that of a pulse of light (we referred to something like a pulsein the group velocity development), which may be considered as a sum ofmany different frequencies. The mathematical machinery used to rigorouslydescribe functions of many frequencies is that of Fourier analysis, the detailsof which we leave for an Appendix.

Example

A specific example may help with the visualization of the idea of group ve-

locity and phase velocity. We will have to introduce the concept of index ofrefraction, which will be discussed in more detail in later chapters and willbe taken as a characterization of a medium in which a wave propagates. Fur-thermore, we will see in later Chapters that the index of refraction may varydepending on the frequency of the wave travelling through a given medium,


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a concept known as dispersion. A wave propagating in the x-direction has a

two-frequency electric field given by

E =

sin(k1x 1t)

k1x 1t +sin(k2x 2t)

k2x 2t

z (1.35)

Note that the expression we are using as a description of the field is notsimply a plane wave. However, substitution into the wave equation showsthat this field is a valid solution to the equation; instead of a plane wave ofinfinite extent, it represents a localized electric field pulse. The frequencies,chosen simply to correspond to two visible wavelengths, are given by 1 =2c/600nm and 2 = 2c/550nm. The wavevector magnitude is kn/c,where n is the index of refraction and is equal to one for propagation in avacuum. First we can plot the electric field as a function of the distancepropagated along the x-axis at two different times. Assume a propagationtime of 20 fs (1 fs = 1015 s). The plot of E(t) is shown in Fig. 1.5. Wecan calculate the velocity of propagation of the pulse from the change inposition and the elapsed time and find that the peak is traveling at vg =3 108 m/s. Thus, in a vacuum the group velocity is identical to the phasevelocity.

We may repeat this calculation making the additional assumption thatn = 1, and that n depends on frequency, for example as n = 1+2/2r , wherer = 2c/500 nm. This frequency dependence of the index of refraction, andthus of the phase velocity of a wave, corresponds approximately to what onefinds for many materials such as glasses in the visible part of the spectrum.Now the result is as shown in Fig. 1.6. The pulse as been deformed, butwe can still make an estimate of the group velocity: vg = 3.4 106m/20 1015s = 1.7 108m/s. The different frequency components of the pulsebegin to separate, since each propagates at a slightly different speed. Thethird snapshot in Fig. 1.6 shows the two frequencies nearly separated. In a

more realistic situation in which a pulse consists of many frequencies that areclosely spaced, the net result of dispersion, or index of refraction dependenceon frequency, is that any pulse will begin to spread as it propagates througha medium. This is of particular importance in the world of fiber optics andtelecommunications.


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0

0.5

1

1.5

2

4e06 2e06 2e06 4e06 6e06 8e06 1e05

x

Figure 1.5: Two snapshots of the electric field E(t) as a function of positionalong the x-axis, taken at times separated by a time interval t = 20fs. Thefield is propagating in a vacuum (n = 1)

0.5

1

1.5

2

4e06 2e06 2e06 4e06 6e06 8e06 1e05

x

Figure 1.6: Three snapshots of the electric field E(t), now propagating in amedium. The time intervals are t1 = 20f s and t2 = 40f s


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1.5 Energy Flux

Up to this point we have considered only the electric field as an entity byitself. Now that we have expressions for the E-field vector as a functionof time and spatial variables, we would like to know how energy is actuallycarried by these fields. This means that we will need an expression for energytransport by the electromagnetic wave or an expression for power flux ofthe electromagnetic wave. This is readily dealt with through the PoyntingTheorem and a quantity called the Poynting vector. Our interest at this pointis with the Poynting vector, which will give us the appropriate expression forthe transport of electromagnetic radiation. In what follows we will consideronly plane waves.

From elementary electricity and magnetism texts [8] we can find an ex-pression for the energy density (Joules/m3) stored in electric and magneticfields, which for vacuum fields we express as

u =1

2

0E

2 +1

0B2

. (1.36)

In the end we will see that this result holds not only for static fields, but fortime-varying fields as well. To connect the expression for the energy densityin the above form to Maxwells equations we form the dot products of E withEqn. 1.4 and of B with 1.3, yielding

E B = 1c2

E Et

(1.37)

B E

= B

B

t(1.38)

Subtracting the second equation from the first and dividing by 0 we arriveat

0 E E

t+

1

0B

B

t= 1

0

E B

= S (1.39)

where we have used a vector identity (see Appendix A) to arrive at theright-hand side, and then defined the Poynting vector to be

S =1

0

E B

(1.40)


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1.5. ENERGY FLUX 29

The left-hand side of Eqn. 1.39 can be seen to be the time derivative

of the energy density defined by Eqn. 1.36. Integrating both sides over thevolume and using the divergence theorem we can write

1

2

t

V

dV

0E

2 +1

0B2

=

V

dV S

=

S

S n dA (1.41)

Eqn. 1.41 is referred to as Poyntings Theorem, and essentially describesconservation of energy. The left-hand side represents the time rate-of-changeof energy stored in a given volume. If that energy changes, it must be dueto a flux of energy through the surface bounding the same volume, as givenby the expression on the right-hand side of Eqn. 1.41.

1.5.1 The Free-Space Poynting Vector

The Poynting vector defined in Eqn. 1.40 is defined as the power flux of anelectromagnetic field; that is, it is the energy carried by the field per unit areaper unit time (energy/time/area) [7, 8]: Since the Poynting vector involvesthe vector product of two waves, the use of complex notation involves a littlepitfall of which we need to be aware. We must keep in mind the fact thatS is the product of the real part of each function rather that the real partof the product. An example of how the complex fields notation can lead oneinto an error is given below. Consider a harmonic function and the use of

complex notation to represent it:

E(t) = |E| cos(t + )= Eeit

where E= |E|ei so that the quantity E(t) can be written

E(t) = Eeit (where Re is implied) (1.42)

Examine the product of two functions (like those encountered with thePoynting vector) of this form using the sin/cos notation rather than complex

notation:E(t) = |E| cos(t + 1)B(t) = |B| cos(t + 2)

E(t)B(t) = |E||B| cos(t + 1)cos(t + 2)


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Using the identity

cos cos = 12

(cos( ) + cos ( + ))we obtain

E(t)B(t) =|E||B|

2[cos(1 2) + cos (2t + 1 + 2)] (1.43)

Now using complex notation,

E(t)B(t) = EeitBeit = E B e2it= |E||B| ei(2t+1+2)

Taking the real part of this last expression yields

[E(t)B(t)] = |E||B| cos (2t + 1 + 2) (1.44)Thus when comparing Eqns. 1.43 and 1.44 we see that the constant term12|E||B| cos(1 2) is missing in Eqn. 1.44. Again, the reason that we

obtained this result is because S is the product of the real part of eachharmonic function, not the real part of the product. In obtaining Eqn. 1.44,we incorrectly calculated the result by taking the real part of the product ofthe complex functions. In Fig. 1.7 the relationship between E, B, and S isshown for a given point in time.

Example:Given the electromagnetic wave,

E = xE0 cos(kz t) + yE0 cos(kz t)where E0 is a constant, find the corresponding magnetic induction and thefree-space Poynting vector.

Solution: We can write B = (1/) k E (the proof is left for the Prob-lems) using Maxwells equations (Eqn. 1.3). For this example, k has onlyone component, in the z-direction. Thus

B =

1

k z [x cos(kz t) + y cos(kz t)] E0=

E0k

[y cos(kz t) x cos(kz t)]

=E0c

[y cos(kz t) x cos(kz t)] .


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1.5. ENERGY FLUX 31

Figure 1.7: Illustration of the relative orientation of the vectors E, B and Sfor a plane wave.

Note first of all that the magnitude of the Bfield is smaller than the mag-nitude of the Efield by a factor of c. The fields E and B are seen fromthe above to be mutually orthogonal, and thus k, E and B form a triad of

mutually orthogonal vectors, a general result for plane waves in free space.

For the second part of the problem,

S =1

0E B

=1

0[xE0 cos(kz t) + yE0 cos(kz t)]

E0c

[y cos(kz t) x cos(kz t)]

=

E200c

2cos2(kz t) z .The key result here is that the direction of energy flow ( S) corresponds to

the direction of propagation since in this case k = k z.


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1.5. ENERGY FLUX 33

Solution: For

E = xE0 cos(kz t) + yE0 cos(kz t)

andB =

E0c

[y cos(kz t) x cos(kz t)]we can write the complex fields, respectively, as

E = xE0ei(kzwt) + yE0ei(kzwt)

andB =

x

E0

cei(kzwt) + y

E0

cei(kzwt)

The magnitude of the electric field is given by E2 = E E = E20 + E20 =

2E20 . Likewise, for the magnitude of the magnetic field we have B2 = E20 /c2.

Thus, from Eqn. 1.46 we can write E(t)B(t) = 2E20 /2c and from Eqn. 1.48S = 2E20 /20c.

Example:

A diode laser has a beam radius of 2.0 mm and an output power of 5.0 mW.

What is its irradiance (in W/m2

)? Calculate the amplitudes of the electricand magnetic fields.

Solution: The irradiance is the power per unit area. Thus

I S = power

area=

5 103 W (2 103 m)2 = 398 W/m

2 .

To find the amplitudes of the E and B fields, we use results from this section,along with the fact that the field amplitudes are related by

E = c Bwhere c is the speed of light. Using Eqn. 1.47 we findS = 1

20

E B = c20

B2


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and therefore

B2 = 2 (4 107 Wb/A m) 398 W/m23 108 m/s B = 1.83 106 T

Finally, the magnitude of the electric field is given by E = 3 108 m/s

1.83 106 T = 549 V/m.We can compare the results calculated above to some common values so

as to gain a physical feeling for these magnitudes. For example, the solarconstant, the amount of sun reaching the surface of the earth (an averagevalue, of course) is about 1370 W/m2, several times the irradiance of thediode laser in the example. As a comparison for the magnetic field magnitude,we note that the Earths magnetic field is on the order of 105 T.

One last point to notice about this example is that we have presentedthe diode laser on the one hand as a plane wave, which by definition hasan infinite extent, and on the other hand we have given the beam a definiteradius. Referring back to our discussion of Gaussian beams, we saw thatnear the focus the Gaussian beam electric field is very nearly equivalent to aplane wave, and at very large distances from the beam waist the approximatespherical wave character of the beam for which the radius of curvature issmall, can also be treated to first approximation as a plane wave. Thus we

take this as our justification for the approximations used in this example.

1.6 Resonator Electric Field

In our earlier discussion about phase and group velocities it was pointedout that the description of the electric field of a temporal pulse of light isbest thought of in terms of a superposition of many electric fields of differingfrequencies. An example at the end of Chapter 2 will demonstrate how we candescribe a Gaussian beam in terms of a spatial superposition of plane waves.In this section we present an example of superposition in the time domain

of electric fields to describe the transient behavior of an optical resonator,such as a laser cavity. The type of resonator we will be investigating in thisexample is often known as a Fabry-Perot cavity or resonator; if one considersa solid piece of material such as quartz, the treatment below remains muchthe same, but one refers in that case to a Fabry-Perot etalon.


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1.6. RESONATOR ELECTRIC FIELD 35

First, we need to give a small amount of background. In its simplest form

a laser consists of a collection of atoms residing in a resonant optical cavity.The purpose of the resonator is to provide feedback for the laser atomic gainmedium. Here we will concentrate on the properties of the resonator in theabsence of any atomic medium. The resonator consists of two mirrors facingeach other, each of which reflects a fraction of the incident field, given byr1 and r2, and transmits a fraction t1 or t2. As we will see in Chapter 5,the intensity reflectance is given by R = r2 for each mirror. Throughout, weassume identical mirrors. Our aim is to evaluate the time dependance of theoutput intensity of the resonator, with a given input intensity.

We proceed as follows. The characteristics of a resonator can be foundin somewhat simplified form if one assumes a confocal configuration for the

resonator, with the mirrors radii of curvature equal to the cavity length das shown in Fig. 1.8. The resulting expressions may easily be generalizedto a cavity of arbitrary configuration. For an incident field of amplitudeEin(t) the transmitted field amplitude Eout(t) is simply the superposition offields transmitted by the second mirror after different numbers of round tripswithin the resonator[6].

Eout(t) = t1t2

n=0

Ein(t 0 n)(r1r2)nei(0+n) (1.49)

In Eqn. 1.49, Ein(t

0

n) is the input field evaluated earlier in timeby an amount corresponding to one single pass and n round trips. 0 is thetime taken for light to make a single pass from the entrance mirror to theexit mirror and is the round-trip time, 0 is the single-pass phase shift forthe electric field and is the round-trip phase shift (relative to that of aparticular cavity resonance), t1 and t2 are the mirror field transmissivities,and r1 and r2 are the mirror field reflectivities.

In the steady state, when Ein and Eout are independent of time, the outputintensity can be found analytically by realizing that the sum given in Eqn.1.49 is a geometric series. This is the key step for all that follows. We realizethat the output field (again, considering that the input has been on for an

infinitely long time) is the superposition of many fields, with contributionsdue to i) the input field multiplied by the transmission of each mirror, ii)the field that did not escape after traversing the resonator once, but afterbeing reflected once by each mirror, is transmitted on the second pass, iii)two full round trips, etc. The result is given by (the derivation is left for the


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d

Ein E

out

r1

r2t

1t2

Figure 1.8: A Fabry-Perot cavity of resonator is made up of two mirrors offield reflectivity r1 and r2 and field transmissivities t1 and t2, separated by adistance d.

problems)

T() = EoutEin 2

=t21t

22

(1 (r1r2)2)2

+ 4(r1r2)2 sin2

(/2)

(1.50)

where , introduced as a round-trip phase shift of the field, can also beinterpreted as a detuning of the resonator from its transmission maximum,which occurs for = 2m. The cavity intensity transmission function T()is not to be confused with the transmittance of a single mirror, e.g. T1 = t

21

or T2 = t22. The resonator transmission as a function of the phase argument

is shown in Fig. 1.9.The phase dependence is equivalent to a dependence of the transmission

of the resonator on pathlength, since the accumulated phase delay of the waveis proportional to the distance travelled, i.e. = k(2d) = (2 0 )(2d). The

transmission reaches a maximum for sin /2 = 0, which implies d = m0/2,meaning that the resonance condition is one for which there is a standingwave in the cavity. Converting to frequencies using = c/, we can write thecondition for the discrete frequencies at which the cavity is resonant are givenby m = m

c

2d

. The separation between successive transmission maxima is


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Iout/I

in

0 24 6

1.0

Figure 1.9: The steady-state transmission of an ideal Fabry-Perot resonatoras a function of detuning () from the cavity resonance frequency.

given by the free spectral range of the resonator,

F SR = m+1 m = c2d

(1.51)

We will now assume that the two mirrors are identical, r1 = r2 r andt1 = t2 t and use the fact that the intensity reflectance and transmittanceare given by R r2 and T t2, respectively. In the limit of high reflectivityand small detuning (1 R 1, 1) each of the the cavity transmissionpeaks shown in Fig. 1.9 show approximately what is known as a Lorentzianlineshape as a function of phase.

T() =

EoutEin

2

T2

(1 R)2 + R2 (1.52)

Using the above relation we may define several quantities. The full-widthat half-maximum (FWHM) of the transmission function is defined from Eqn.1.52 as (twice) the value of for which the function T() = 1/2, and isfound to be 1/2 = 2

1RR

. The finesse F is the ratio of the phase shift

between transmission peaks (2) (the free spectral range), to the full width


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frequency, oscillation during the decay takes place at the oscillators natural

frequency.If the cavitys input field is suddenly turned on at t = 0, the transmissionwill turn on in a stepwise manner, and the field will begin to build up in thecavity with a characteristic time we refer to as the filling time. We mayapproximate the filling behavior for short round-trip time and high finesseby an expression that depends on [6]:

Iout (t = + n)

Iout() =1 Rnein2 1 exp[ (t n0)] ein2 (1.56)

On resonance ( = 0) this becomes

Iout (t)

Iout() = (1 exp[ (t 0)])2 (1.57)

Thus, can also be found from the cavitys transient behavior, according toEqns. 1.55 (decay) and 1.57 (filling).

We wish to finish out this example and the Chapter with concrete the-oretical and experimental examples of the concepts discussed above. If theresonators round-trip time is not short compared to the time over whichthe input field switches on or off, the cavity transmission will show distinctsteps, as is illustrated in Figs. 1.10 and 1.11 for the case of a cavity excitedon resonance. Fig. 1.10 shows experimental results for dye laser excitationof a cavity, with mirrors of 88% reflectivity and a round-trip time of 12 ns,by an electro-optically-produced, nearly square pulse of length 50 ns withrise and fall times (10% - 90%) of 8 ns and 13 ns, respectively. In Fig. 1.11is shown the result of a numerical calculation of Eqn. 1.49 for the same pa-rameter values as used in the experiment of Fig. 1.10, with only the fillingof the cavity shown.

In the non-resonant ( = 0) filling of a resonator after switch-on of thedriving radiation, we see the presence of beating, or interference, betweenthe input and cavity fields quite dramatically, as shown theoretically in Fig.

1.12a. From Eqn. 1.56 we see that deep modulation at the frequency dif-ference between the incident light and the cavity resonance may be presentwhile the cavity fills if that offset frequency is a submultiple of the round-tripfrequency, that is, if the detuning is equal to divided by a small integer.This modulation is illustrated in the calculations of Fig. 1.12a, for a detuning


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Figure 1.10: The time-dependant transmission of a Fabry-Perot resonatorfor resonant excitation. In A) the input pulse is shown, while B) shows thestep-wise filling of the cavity, as well as the decay after the input pulse isswitched off.


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(ns)

T

Figure 1.11: The calculated time-dependant transmission of an ideal Fabry-Perot resonator for mirror reflectance R = 0.88 and resonator round-triptime = 12 ns. The inset shows a magnified view of the initial 100 ns of thefield building up inside the resonator.


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T

(ns)

T

(ns)

a) b)

Figure 1.12: For an input field detuned from the cavity resonance, the char-acter of the time-dependant transmission of an ideal Fabry-Perot resonatoris more complicated than for zero detuning. For mirror reflectance R = 0.88and resonator round-trip time = 12ns and detunings a) = /2 and b) = /4 the transmitted intensity shows oscillations occurring at a period oftwice and four times the cavity round-trip time, respectively.


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of, where the cavity output oscillates with a period of twice the round-trip

time (period-2 oscillations), and of Fig. 1.12b, for a detuning of /2, wherethe oscillations have a period of four times the round-trip time (period-4oscillations). These oscillations are nothing more than a manifestation ofinterference between fields, either constructive or destructive, depending onthe observation time.


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1.7 Problems - Chapter 1

1. The current density and charge density at a point in space are relatedby the continuity equation given below:

t+ j = 0

This equation is just a statement of conservation of charge. Derive thisequation from Maxwells equations.

2. A sphere of radius a is uniformly charged with a density . If the sphererotates with a constant angular velocity , show that the magneticinduction, B, at the center of the sphere is (0/3)a

2. Hint: Use theBiot-Savart law for the magnetic induction.

3. A particle with mass m and charge qmoves in a constant magnetic fieldB. Show that, if the initial velocity is perpendicular to B, the path iscircular and the angular velocity is

=

q

mv B

4. Show that the relation j = E is equivalent to the usual statement ofOhms law, V = IR, or (voltage) = (current) (resistance).

5. Starting with the Lorentz force and Maxwells equations for vacuum,derive Coulombs law.

6. Prove the following relationships for plane-wave fields,

E(r, t) = ik E(k, )ei(krt) E(r, t) =

ik E(k, )ei(krt)

2 E(r, t) =

(ik)2 E(k, )ei(krt)


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1.7. PROBLEMS - CHAPTER 1 45

7. Using Maxwells equations show that,

E = c2

k B

for a free-space plane wave solution to the wave equation.

8. Using Maxwells equations show that,

B =1

k E

for a free-space plane wave solution to the wave equation.

9. Change the expression below for a transverse sinusoidal travelling wave

moving in the positive x-direction into an expression involving k,x,,and t in the complex notation,

E = E0 sin

2

(x vt)

where v is the wave velocity.

10. Use Maxwells equations to show that free space plane waves are trans-verse, i.e. that k, E and B are all mutually perpendicular.

11. Show that for a Gaussian beam

R(z) = z

1 +w20

z

2and

w2(z) = w20

1 +

z

w20

2

(Hint: try taking the inverse of q(z) = q0 + z and relate the real andimaginary parts to R(z) and w(z) ).

12. Make plots of R(z) and w(z) for a Gaussian beam, for 0 < z < 5cm,

assuming a Helium-Neon laser with beam waist w0 = 50m and =632.8 nm. Be especially careful when plotting R(z). Your plot willlook different depending on the number of points you use. Explainphysically what is happening for z 0. If you extend the plot forR(z) to z = 50 cm, what happens? Explain physically.


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13. Show that for the electric field given by Eqn. 1.27 there is non-

transverse component.14. The peak power of a certain CO2 laser ( = 10.6m) is 100 watts. If the

beam is focused to a spot 10 microns in diameter, find the irradianceand the amplitude of the electric field of the light wave at the focalpoint.

15. The radiant energy from the sun is about 8 Joules/cm 2 per minute.Assuming this to be in the form of a plane wave travelling in the zdirection in vacuum, plane polarized with its electric field in the x-y plane, and having a wavelength of 6000 A, find expressions for the

electric and magnetic induction fields as functions of z and t and withall constants evaluated.

16. (a) Find the magnitude of the magnetic induction, B, of a free spaceelectromagnetic wave in terms of the amplitude of the electric field, E,of that wave. (b) Find an expression for the free space irradiance interms of the amplitude of the electric field.

17. (a) Given E = x25 cos(kz t)V/m in free space, find the averagepower passing through a circular area of diameter 5 m in the planewhere z = 5 m.

(b) Given that in free space

E =

200

sin

rcos(kr t)

Determine the average power passing through a hemispherical shell ofradius r = 102m and 0 /2.

18. Show that when complex notation is used, the time-averaged Poyntingvector can be written,

S = ( E B/20)19. The energy flow associated with sunlight, striking the surface of the

earth in a normal direction is 1.33 kW/m2.


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1.7. PROBLEMS - CHAPTER 1 47

(a) If the corresponding electromagnetic wave is taken to be a plane

polarized monochromatic wave, determine the maximum values of Eand B.

(b) Taking the distance from the earth to the sun as 1.5 1011 m, findthe total power radiated by the sun.

20. a) Consider a long straight conductor, of conductivity and radius r,carrying a current density j. Find the magnitude and direction of thePoynting vector in the conductor conductor in terms of r, , and j.

b) Suppose that the very long coaxial line is used as a transmissionline between a battery and resistor. Designate the emf of the batteryas V. The battery is connected to one end of the coaxial line while the

other end of the coaxial line is connected to a resistor of resistance R.Find the magnitude and direction of the Poynting vector in the regionbetween the conductors of the coaxial cable. Find the total powerpassing through a cross section of the line. Will the direction of energyflow change if the connections to the battery are interchanged?

21. Compute the irradiance of a plane wave with electric field vector givenby

E = (xExei + yEye

i)ei(kzt)

22. Consider a superposition of two independent orthogonal plane waves:

E = x E1ei(kzt+1) + y E2ei(kzt+2)

Find

S

and show that it is equal to the sum of the average Poynting

vectors for each component.

23. Consider the superposition of parallel independent plane waves

E = x E1ei(kzt+1) + x E2ei(kzt+2)

Find

S

and show that it is not equal to the sum of the average

Poynting vectors for each component. What is the difference physicallybetween this result and that of the previous problem?

24. Consider an electromagnetic wave,

E = xE0ei(kzt) + xE0ei(kzt)


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Find S and S.

25. Derive the result given by Eqn. 1.50

26. Show that the transmission function for a Fabry-Perot resonator canbe described approximately by Lorentzian lineshape as a function ofphase as in Eqn. 1.52

T() =

EoutEin2 T2(1 R)2 + R2 (1.58)

27. Plot the time dependence of the output intensity of a resonator with

identical mirrors of field reflectivity r = 0.9 and with a separation ofd = 100cm between the mirrors.


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Chapter 2

Polarization of Light

In Chapter 1 we reviewed and derived some of the basic properties of commonly-encounterd solutions to the electromagnetic wave equation. Our emphasis inChapter 1 was mainly on free-space propagation properties of these wavesand on the spatial profile of the wavefronts. In this second Chapter we willconcentrate on the vector nature of the (usually transverse) electromagneticwaves, as manifested in the phenomena of polarization.

We first introduce the definitions and algebraic formalism used to treatthe polarization of electromagnetic waves. Later in the Chapter we present aconvenient matrix formalism that can be used for treating problems involvingpolarized light. Specific applications of this formalism will be encounteredin Chapters 7 and 8. To close out Chapter 2 we will look in some detail atthe polarization of a Gaussian beam, with the specific aim of showing that,in contrast to plane waves, there must be a component of polarization in thedirection of wave propagation. Although this is a minor point, it does serve toremind us of the fundamental properties of waves and of the approximationswe will be using for most of our work.

2.1 Historical Background

Etienne Louis Malus (1775 - 1812) first gave the name polarization toproperties of light that had been discovered earlier, for example by ChristiaanHuygens (1629 - 1695) in the Netherlands. Malus was experimenting withlight transmitted by crystals known to split an incident beam into two parts.(This property is known as birefringence; we will return to this in Chapter

49


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52 CHAPTER 2. POLARIZATION OF LIGHT

E

Ey

x

b

a

Figure 2.1: Illustration of the polarization ellipse described by Eqn. 2.2.

The ellipticity is defined in terms of the lengths of the major and minoraxes:

=

b

aassuming that we choose such that b a. This convention implies that theellipticity satisfies

0 || 1.The complete description of the general polarization state of an electric

field requires, in addition to the ellipticity and the orientation of the ellipse,one more parameter to be specified. We must decide on a convention definingthe sense of rotation or handedness of the light. In the definition of thereis an arbitrary sign assignment that can be made. Here we choose to use theconvention that the + direction corresponds to right-elliptically-polarized

(REP) light, that is, for an observer facing the source of the wave, right-handed means that the electric field vector rotates in a counterclockwisedirection. Likewise, for an observer facing the source left-handed polarizationmeans that the electric-field vector appears to rotate in a clockwise direction.Put in another way, for REP light, an observer riding on the light wave would


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2.2. POLARIZATION OF LIGHT WAVES 53

observe a sense of electric (or magnetic) field rotation equivalent to that of

a normal screw. This convention is chosen because of a correspondenceto the quantum-mechanical definition of the angular momentum of a photonfor which the z-component, Lz = for right-circularly-polarized (RCP) light,and Lz = for left-circularly-polarized light (LCP).

The handedness is determined by the sign of sin . For sin > 0 thesense of rotation is counterclockwise (0 < < ) and we refer to the lightas right-elliptically polarized (REP); conversely for sin < 0 the rotationsense is clockwise and one refers to left-elliptically polarized (LEP) light.Several examples of phase angles and the corresponding polarization statesare illustrated in Figs. 2.2 and 2.3. For each of these figures, the propagationdirection of the field is to be taken as out of the page, i.e. we are looking

into the approaching beam of light.

= 0 = /4 = /2 = 3/4

= /4 = /2 = 3/4 =

Ax = Ay

Figure 2.2: Polarization ellipses for different phase angles , with field am-plitudes Ax = Ay. The orientation and sense of rotation correspond to ourconventions, as described in the text. We imagine ourselves to be lookingalong the -z-axis, i.e. into an oncoming elliptically polarized planewave.

We can distinguish right and left circular (elliptical) polarization by not-ing that, if the vertical component lags in phase, the E-field vector rotatesin a counterclockwise direction, which we have defined to be right ellipticalpolarization. If the vertical component leads in phase the E-field vector ro-tates in a clockwise direction, giving left elliptical polarization. Clockwise


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= 0 = /4 = /2 = 3/4

Ax = Ay

= /4 = /2 = 3/4 =

Figure 2.3: Polarization ellipses for different phase angles , with field ampli-tudes Ax =

12 Ay. Note that in this case it is not possible to achieve circular

polarization.

and counterclockwise refer to the rotation direction as seen by an observerlooking into the approaching light wave, i.e. looking along the z direction.All of this, of course, is within the framework of the conventions adapted forthis course. Note that many optics texts use a convention which is exactly

opposite ours, while engineering texts tend to use a convention based on thesense of rotation as seen by an observer riding with the wave.

We can investigate this possible confusion a bit further, and give a methodto determine with certainty the polarization of a given field. For RCP lightwe can begin with Eqn. 1.9, dropping the subscript 0,

E =

E(k, ) ei(krt)

=

12

(x + iy)E ei(krt)

in which we have defined explicitly the vector nature of the electric field

amplitudeE

by writing the amplitude in terms of the magnitude E andthe unit vectors x and y. The factor of2 arises from the normalization ofthe unit vector terms. Consider an observer standing at z = 0 (r = 0 and

therefore k r = 0) and looking back into the beam of light. Starting at t = 0,the electric field points in the +x-direction, along x. A quarter period later,


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at t = (2/) /4 = /2, we have

E =

12

(x + iy)E ei(/2)

=

12

(ix + y)E =

E

2y .

Thus the electric field points along the +y-direction. The observer thereforesees an electric field rotating counterclockwise as a function of time; again,this is our definition of right-circularly-polarized light. Resorting to this type

of analysis is the key to resolving any confusion which crops up because ofnotational differences between texts. In Fig. 2.4 we show an example of thetime evolution of the electric field vector for right-circularly polarized light.

E = Exi

E = -Exi E = -Eyj

E = Eyj

t = 0 t = T/4

t = 3T/4t = T/2

Figure 2.4: The E-vector for RCP shown in the z = 0 plane for one full timecycle, in increments of one-quarter of a period. At t = 0, Ax is a maximumand Ay is zero. At t = T /4, Ax is zero and Ay is a maximum. As time

progresses, the E-vector rotates counterclockwise. This is a consequence ofthe /2 phase difference between the x- and y-components of E.


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2.2.2 Linear or Plane Polarization

One limiting case of elliptically polarized light occurs for an ellipticity pa-rameter || = 0. Equivalently, we can see that in terms of the phase shift, = m. We refer to linear polarized light in this case. In Fig. 2.5 the con-trast between a field of well-defined linear polarization and an unpolarizedfield is illustrated.

y y

x x

z z

EE

Figure 2.5: (a) Plane polarization; the wave is propagating along the z-axisand the electric field vector has a well-defined direction. (b) Unpolarized

wave; the E-vector for the wave propagating in the z-direction lies in the x-yplane but varies rapidly as a function of time. The E-vector appears to haverandom orientations.

2.2.3 Circular Polarization

In Eqn. 2.2, if Ax = Ay, then we must have = /4, unless = /2.In the latter case is indeterminate, but if we plot the time-variation of

the field vector for these parameters, we see that = /2 is simply thecondition that the field traces out a circle in the x y plane, and thereforethe coordinate-axis rotation is meaningless. We refer to this polarizationstate of the field as circularly polarized. In terms of the ellipticity parameter,circular polarization corresponds to || = 1. Pictorially, the time evolution


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of the electric field vector is shown in Fig. 2.6 for right-circularly polarized

light.

y y

x

E

E

(a) (b)

x

Figure 2.6: a) Left circular polarization (LCP). As time evolves, the E-vectorrotates clockwise on a circle in the z=0 plane. b) Right circular polarization(RCP); the E-vector rotates counterclockwise on a circle. In both cases weimagine looking into the source of the light, i.e. the light is propagatingalong the +z-direction.

2.2.4 Polarization in the Complex Plane

We stress again that the field amplitude E

k,

is a complex vector quantity

and contains all the polarization information for the light field. Concentratingon light propagating in the z-direction (polarization in the x-y plane), we canrelate the complex amplitude to the components Ax and Ay. If we define acomplex number as

AyAx

ei = tan ei

with = y x and tan AyAx for 0

2 , the parameters and

will completely define the state of polarization. Although it may seem anadditional complication to introduce yet another notation for the polariza-tion, the representation in terms of the azimuthal angle will be useful indiscussions of ellipsometry occurring in Chapter 5, and thus we include it at


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2.3. JONES VECTOR REPRESENTATION OF POLARIZATION STATES59

points far away from the origin in Fig. 2.7, the field approaches more and

more closely a state of linear polarization. Be careful with this notation:tan represents the length of the vector in the complex plane. Relatingback to the field amplitudes, a large value of tan implies that Ax 0 (orat least that Ax Ay) and thus a field polarized in the y-direction.

For circular polarization, as we have seen, the E-vector has orthogonalcomponents, equal in magnitude, Ax = Ay, with a phase difference =/2. Circularly polarized light can only be represented by a point alongthe imaginary axis in the complex plane, as shown in Fig. 2.7. Furthermore,only the point with tan = 1 (again, Ax = Ay) is possible as a representationof circularly polarized light.

Now that the general definitions for various polarization states have been

specified, we present in the next section a particularly easy method for car-rying out calculations involving polarized light.

2.3 Jones Vector Representation of Polariza-

tion States

The polarization of a light wave can also be conveniently represented by a2 1 matrix, called a Jones vector, which expresses the relative amplitudeand phase of the two orthogonal components of the E-vector. For example,

if the complex amplitude is written as

E= xEx + yEy (2.3)

with

Ex = Axeix and Ey = Ayeiy

a corresponding Jones vector can be expressed as

J

Axeix

Ayeiy

.

Returning briefly to the full form of the plane wave electric field, so that wedo not lose track of that, we would write

E (r, t) =

J ei(krt)

=

x Axe

ix + y Ayeiy

ei(krt)


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The important thing to realize about the Jones vectors is that the ma-

trix elements express the relative amplitude and phase of the orthogonalcomponents of the E-vector. By convention, the first element in the vectorcorresponds to the x-component of the E field, and the second element givesthe y-component, assuming propagation in the z-direction.

The Jones vectors are usually normalized, that is,

J J = JT J = 1where the stands for the hermitian conjugate of the vector, T meansthe transpose of the vector, and * is the complex conjugation operation.

Example: We can represent a linearly polarized electric field oriented atan angle by the vector

J1 =

cos sin

Given another linearly polarized field

J2 =

sin cos

we see first of all that both represent normalized states, e.g.

JT1 J1 = (cos sin ) cos sin

= cos2 + sin2 = 1 .

Furthermore, these two states are orthogonal, i.e.

JT1 J2 = (cos sin ) sin cos

= sin cos + sin cos

= 0 .

Whenever a pair of states satisfies the orthogonality condition

JT1J2 =

0, that pair may be used as a basis set: any polarization lying in that sameplane may be described as a linear combination of the two basis vectors.


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2.3. JONES VECTOR REPRESENTATION OF POLARIZATION STATES61

Example: For an angle = /4, the Jones vector for linearly polarized

light isJ/4 =

cos sin

=

1/21/

2

=

12

11

.

This can be described alternatively as a combination of x- and y-polarizedlight, for which the Jones vectors are given by

ex =

10

and ey =

01

,

respectively. Thus we can write

J/4 =

1

2 ex +1

2 ey .A few examples of Jones vectors which will illustrate this point are given

below: 10

= Wave linearly polarized in the x-direction.

01

= Wave linearly polarized in the y-direction.

12

11

= Wave linearly polarized at 45 to the x-axis.

12

1i

= (-) - left circularly polarized, LCP;

(+) - right circularly polarized, RCP

We can summarize by giving the Jones vector in slightly more generalterms by calling the y-axis the vertical component of the E-vector and the x-axis the horizontal component. Thus the vector can be written schematically

as HorizontalVertical

.

The most general form for the Jones vector is that for elliptically polarizedlight. As we have already seen, elliptically polarized light allows for arbitrary


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relative amplitude of the two field components as well as arbitrary relative

phase; the Jones vector can be written in several different forms:

J =

cos

sin ei

=

Ax

Ayei

=

Ax

b ic

(2.4)

where we have written the polar-form complex number Ayei in the alter-

native Cartesian form with Ay =

bc + c2, and with the relative phase anglebetween the Ax and Ay components given by

= y x = tan1c

b

.

The ellipse is oriented with respect to the x-axis such that the angle betweenthe major axis and the x-axis is as shown earlier.

Example: Given the Jones vector

3

2 + i

which is in the standard form

Axb + ic

, find the relative phase angle between the two field components,

along with the ellipse orientation.Solution: First we can identify the given field as being right-elliptically

polarized using our sign convention. The relative phase angle is given by

= tan1 cb = tan1 1

2 = 26.6

The amplitude Ay is given by Ay =

b2 + c2. The ellipse orientation is foundfrom

optical radiation and matter

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