scattering of guided waves in thick gratings at … · gratings at extreme angles submitted by...

QUEENSLAND UNIVERSITY OF TECHNOLOGY SCHOOL OF PHYSICAL AND CHEMICAL SCIENCES

SCATTERING OF GUIDED WAVES IN THICK GRATINGS AT EXTREME ANGLES

Submitted by Martin KURTH to the School of Physical and Chemical Sciences,

Queensland University of Technology, in partial fulfilment of the requirements of

the degree of Master of Applied Science.

August 2006

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Abstract The aim of this project was to develop a passive optical compensating arrangement that

would allow the formation and continued stability of interference patterns over a long

timescale and also to investigate optical wave scattering in thick gratings at extreme

angles of scattering.

A novel passive arrangement based on a Sagnac interferometer is described that

produces interference patterns more stable than those produced by a conventional

arrangement. An analysis of the arrangement is presented that shows it to be an order of

magnitude more stable than an equivalent conventional approach. The excellent fringe

stability allowed holographic gratings with small periods (~ 0.5 μm) to be written in

photorefractive lithium niobate with low intensity writing fields (~mW/cm2) produced

by a He:Ne laser, despite long grating fabrication times (~ 1000 s). This was possible

because the optical arrangement compensated for phase shifts introduced by

translational and rotational mirror motion caused by environmental perturbations. It

was shown that the rapid introduction of a phase shift in one of the writing fields can

change the direction of energy flow in the two-wave mixing process.

It was found that the improvement in stability of the modified Sagnac arrangement over

a conventional interferometer decreased when the crossing angle was increased and that

the point about which the mirrors are rotated greatly affects the stability of the

arrangement. For a crossing angle of 12 degrees, the modified Sagnac arrangement is

more than twice as stable when the mirrors are rotated about their midpoints, rather than

their endpoints.

Investigations into scattering in the extremely asymmetrical scattering (EAS) geometry

were undertaken by scattering light from a 532nm Nd:YAG laser off gratings written in

photorefractive barium titanate and lithium niobate. Despite the difficulties posed by

background noise, there was very good agreement between the observed scattered field

and that predicted by a previously established theoretical model. Thus, this work

represents the first experimental observation of EAS in the optical part of the spectrum.

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Keywords: holographic grating fabrication, Sagnac interferometer, photorefractive effect,

extremely asymmetrical scattering, passive stabilisation, stable interference patterns

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List of publications Jaatinen, E. and Kurth, M. Fabrication of holographic gratings in photosensitive

media with a passively stable Sagnac optical arrangement. Journal of Optics A:

Pure and Applied Optics, 2006. 8(6): p. 594-600

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Table of contents

Abstract ii Keywords iii List of publications iv Table of contents v List of diagrams vii Statement of original authorship xii Acknowledgements xiii1 Literature review 1.1 Introduction 11.2 Types of periodic gratings 41.3 Types of scattering in periodic gratings 51.4 Extremely Asymmetrical Scattering 61.5 The photorefractive effect 101.6 Two-wave mixing in photorefractive materials 161.7 Beam fanning 191.8 The laser beam 201.9 Scope of this project 232 Fabrication of holographic gratings in photosensitive media with a passively stable Sagnac optical arrangement 2.1 Introduction 252.2 Evaluation of stability of the Sagnac interferometer 272.3 Stability of holographic grating 332.4 Formation of holographic gratings in a photorefractive crystal 352.5 Conclusion 403 Modelling of the Modified Sagnac Interferometer and Comparison of Its Stability to that of a Conventional Interferometer 3.1 Introduction 413.2 Overview of the model 41 3.2.1 Determination of crossing angle 43 3.2.2 Determination of path length difference 433.3 Effect of Mirror Rotation on PLD for the Modified Sagnac Arrangement 463.4 Effect of Mirror Translation on PLD for the Modified Sagnac Arrangement 503.5 Comparison of interferometers 53 3.5.1 Stability against rotational motion with mirror rotation axes 55 symmetrically placed 3.5.2 Stability against translational motion 56 3.5.3 Stability against rotational motion with mirror rotation axes 58 asymmetrically placed 3.6 Conclusions 59

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4 The Experimental Observation of Scattering in the Extremely Asymmetric Scattering Geometry 4.1 Introduction 60 4.2 EAS Theory 61 4.3 The photorefractive materials 4.3.1 Lithium niobate 62 4.3.2 Barium titanate 63 4.4 Experimental rationale 64 4.5 Image analysis 67 4.6 Results in lithium niobate 70 4.7 Results in barium titanate 4.7.1 Grating written with two thin beams 73 4.7.2 Two thin beams, with a half-wave plate inserted prior to grating 75 formation 4.7.3 Two thin beams, with a half-wave plate inserted prior to grating 79 formation, but removed when the beam was blocked 4.7.4 A wide beam and a thin beam in the EAS geometry 81 4.8 Conclusions 88 5 Conclusions 90 6 Bibliography 93

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List of diagrams Figure 1.1 - The phase relationship between rays diffracted from a grating with a

blazed profile

Figure 1.2 - Two waveguides coupled via a grating layer [14]

Figure 1.3 - Examples of different types of gratings (a) represents a phase grating,

(b) represents an amplitude grating and (c) represents a surface relief grating

Figure 1.4 - (a) Scattering from a reflection grating, (b) Scattering from a

transmission grating.

Figure 1.5 - Extremely asymmetrical scattering in a periodic grating

Figure 1.6 - The methods of photoconduction – (a) the band transport model and (b)

the hopping model

Figure 1.7 - The intersection of the normal surface with the xz plane for a negative

uniaxial crystal.

Figure 1.8 - The construction for finding the indices of refraction and polarisation of

a negative uniaxial crystal for a direction of propagation given as s [1].

Figure 1.9 - (a) Contour of the Gaussian beam and (b) Amplitude distribution of a

Gaussian beam

Figure 1.10 - The interaction geometry of finite optical beams, where A1 and A4 are

the wave amplitudes. In this case, A1 is a function of x and A4 is a function of y.

Figure 2.1 - The displaced Sagnac interferometer, where BS1 and BS2 are beam

splitters, PD is a silicon photodetector, and M2-M4 are mirrors. The solid line

indicates the beam that travels around the loop in a clockwise direction, while the

dotted line indicates that travels around the loop in a counter clockwise direction.

Figure 2.2 – The Michelson interferometer

Figure 2.3 - Three typical noise spectra of the output of (a) a displaced Sagnac

interferometer and (b) a Michelson interferometer under similar ambient conditions.

Figure 2.4 - Three typical noise spectra of the output of (a) a displaced Sagnac

interferometer and (b) a Michelson interferometer when both systems suffered

similar vibrational perturbations.

Figure 2.5 - The output of (a) a displaced Sagnac interferometer and (b) a

Michelson interferometer as observed over 15 minutes.

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Figure 2.6 - The Allan Variance of (a) the displaced Sagnac interferometer output

shown in figure 4(a) and (b) the Michelson interferometer output shown in figure

4(b).

Figure 2.7 - The modified Sagnac arrangement that allows the formation of

interference patterns produced by two beams that cross at large angles.

Figure 2.8 - The evolution of the intensities of (a) beam 1 and (b) beam 2 when a

grating is formed using a ‘traditional’ arrangement.

Figure 2.9 - The evolution of the intensities of (a) beam 1 and (b) beam 2 when a

grating is formed using the modified Sagnac arrangement shown in figure 2.7. A 180

degree phase shift was introduced into beam 2 after 1500 seconds and removed after

another 260 seconds.

Figure 2.10 - The theoretically predicted evolution of the intensities of (a) beam 1

and (b) beam 2 assuming a square wave variation in the phase of beam 2 with an

amplitude of 22 degrees.

Figure 3.1 (a) Shows symmetric location of rotation axis in comparison to beam

intersection points on mirror. (b) Shows asymmetric location of rotation axis in

comparison to beam intersection points on mirror.

Figure 3.2 An illustration of how the intersection point shifts when a mirror is

rotated. Point O is the original crossing point of ray A and ray B, while N is the

crossing point of the two rays after the system is perturbed in some way

Figure 3.3 – The effect of setting the angle of M4 to (a) 133 degrees, (b) 133.5

degrees, (c) 134 degrees, and (d) 134.9 degrees before rotating M5

Figure 3.4 – The effect of setting M5 and M6 at different sets of supplementary

angles while M4 remains unchanged for the angle of M5 equal to (a) 34 degrees, (b)

39 degrees, (c) 41 degrees, and (d) 42 degrees

Figure 3.5 – The derivatives of the curves shown in figure 3.4

Figure 3.6 The effect of the relationship between mirror angles on system stability

for individual translations of (a) M4, (b) M5, (c) M6 when the modified Sagnac

interferometer was configured with the parameters described in case 1

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Figure 3.9 - Ratio of phase jitter in the conventional arrangement to phase jitter in

the modified Sagnac arrangement as a result of mirror rotations of 10’ arc sec around

symmetric axes


the modified Sagnac arrangement as a result of mirror translations of 1 μm


the modified Sagnac arrangement as a result of mirror rotations of 10’ arc sec around

asymmetric axes

Figure 4.1 – Scattering in the geometry of EAS. k0 is the wavevector of the incident

wave, k1 is wavevector of the scattered wave, L is the grating width, θ0 is the angle

of incidence. Λ is the grating vector, and E00 is the amplitude of the incident field.

Figure 4.2 – wavevectors of the three waves from the grating

Figure 4.3 – the geometry used to write a grating in LiNbO3

Figure 4.4 – the geometry used to write a grating in BaTiO3

Figure 4.5 – the experimental setup, where L are lenses, M are mirrors, BS is a

beam splitter, BE is a beam expander, and CCD is a CCD camera

Figure 4.6 – The interface of the bitmap analyser with an image already loaded

Figure 4.7 – the line indicates the cross-section that was selected for analysis, and

the circles indicate artefacts

Figure 4.8 – the intensity profile of the image in figure 4.6

Figure 4.9 – a non-ideal profile, due to the flat peaks caused by saturation of the

CCD

Figure 4.10 – (a) modelled interaction of two overlapping Gaussian beams that are

slowly diverging from each other (b) modelled profile of a Gaussian beam

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Figure 4.11 – (a) experimentally obtained profile of the interaction between two

overlapping Gaussian beams that are slowly diverging from each other (b)

experimentally obtained profile of a Gaussian beam

Figure 4.12 – CCD image of the field produced at and near the Bragg angle when a

horizontally polarised thin beam was scattered off a grating written with two

horizontally polarised thin beams in the EAS geometry

Figure 4.13 – Intensity profile of the field produced at and near the Bragg angle

when a horizontally polarised thin beam was scattered off a grating written with two


Figure 4.14 – CCD image of scattering near the Bragg angle for a vertically

polarised thin beam scattered off a grating written with two vertically polarised thin

beams

Figure 4.15 – Profile of scattering near the Bragg angle for a vertically polarised

thin beam scattered off a grating written with two vertically polarised thin beams

Figure 4.16 – Scattering at the wide angle of divergence (thinhwp2)

Figure 4.17 – Profile of scattering away from the Bragg angle for a vertically

polarised thin beam scattered off a grating written with two vertically polarised thin

beams

Figure 4.18 – A beam is scattered off a grating written with two beams of a different

polarisation at an angle of θi, which results in a divergence from the Bragg angle of

θr in air.

Figure 4.19 – Image of scattering away from the Bragg angle when a horizontally

polarised thin beam was scattered off a grating written with two vertically polarised

thin beams in the EAS geometry

Figure 4.20 – Profile of scattering away from the Bragg angle when a horizontally



Figure 4.21 - Image of scattering away from the Bragg angle when a horizontally



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Figure 4.22 – Profile of scattering away from the Bragg angle when a horizontally



Figure 4.23 – Image of the field produced when a horizontally polarised thin beam

was scattered off a long, thin grating produced by two horizontally polarised beams

after a short evolution time

Figure 4.24 – (a) profile of scattering near the Bragg angle when a horizontally

polarised thin beam was scattered off a long, thin grating produced by two

horizontally polarised beams after a short evolution time (b) modelled scattered

wave profile for the conditions described in (a) [2]

Figure 4.25 - – Image of the field produced when a horizontally polarised thin beam

was scattered off a long, thin grating produced by two horizontally polarised beams

after a moderate evolution time

Figure 4.26 - (a) profile of scattering near the Bragg angle when a horizontally


horizontally polarised beams after a moderate evolution time (b) modelled scattered


Figure 4.27 – Scattering off a long, thin grating after the grating was allowed to

evolve fully

Figure 4.28 – Profile of scattering near the Bragg angle when a horizontally


horizontally polarised beams that was allowed to evolve fully

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STATEMENT OF ORIGINAL AUTHORSHIP

The work contained in this thesis has not been previously submitted for a degree or diploma at any other tertiary educational institution. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made.

Signed_____________________________

Date_______________________________

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Acknowledgements First and foremost, I’d like to thank my supervisor Dr Esa Jaatinen for all of his

assistance with my research and, most importantly, for his patience with me. I’d

also like to thank my associate supervisor Dr Dmitri Gramotnev for not giving up on

me and for always having time for me. I’d like to thank my collaborator Steven

Goodman for modelling the results for my thesis and for his general assistance with

my project. To Dan, DJ, Jye, Ben, Gillian and Kristy – I’m sorry if any of my bad

habits have rubbed off on you. Thank you all for keeping me mildly sane. Finally,

I’d like to thank my family for putting up with me, with particular thanks going to

my mother for the immense patience and support that she has shown towards me.

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1.1 Introduction The invention of the laser in 1960 heralded a new wave of interest in the areas of

pure and applied optics and photonics, which resulted in discoveries that have

had a tremendous impact on both industry and everyday life [1]. Photonics deals

with the control, manipulation and transfer of information using photons. This

area of optics seeks to replace existing electronic devices with devices that are

based on light signals rather than electric signals. Photonic devices are able to

generate optical waves and/or manipulate the properties of optical waves. The

benefit of using photonic devices over electronic devices is that optical waves

propagate at high speed, allowing for the fast processing of information. In

addition optical signals are able to carry a higher density of data then electrical

signals. Wave multiplexing and demultiplexing in fibre optic cables enables this

high-density transfer of data [2, 3]. The density of data transfer increases with

increasing frequency [4].

The driving force behind photonics research is the need for greater information

carrying capacity, faster rates of transfer and more compact devices. This has led

to the development of new devices, as well as attempts to expand the data

transfer capacities of existing integrated optical devices. For instance, it has been

suggested that nonlinear photorefractive materials may be utilised in integrated

processors for optical computing. It is predicted that such devices will process

data at a much faster rate than conventional computers [5]. Photonics has

allowed the successful miniaturisation of optical devices based on bulk

electromagnetic waves by using localised electromagnetic waves. This is

achieved through the use of waveguides that use total internal reflection to

confine and guide the light waves [6, 7].

Waveguides are the backbone of photonics because guided waves are able to

propagate over long distances with minimal losses from scattering. In addition,

optical devices employing waveguides can be miniaturised [3]. Waveguides are

also used as switching elements and as connections between switching elements

[8]. However, a major obstacle is encountered when trying to use guided optical

waves in a variety of applications. When dealing with bulk optical waves, it is

possible, for instance, to use a mirror to reflect the waves in a particular

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direction. However, the complex field structure of guided waves means that the

use of an isolated element, such as a mirror or a single groove on the surface of a

waveguide, is highly inefficient [9]. Using a single element would lead to the

uncontrollable loss of energy from the guided mode. The solution to this

problem is to assemble a large number of periodically spaced elements. Such an

array is known as a diffraction grating.

When light is incident onto a grating, it is diffracted by the fringes of the grating.

Due to the periodicity of the grating, virtually all waves scattered by one fringe

of the grating will be eliminated by waves scattered by the other fringes because

of destructive interference [10]. Only waves that are of a certain wavelength (or

wavelengths) and that propagate in a particular direction(s) will be in phase, and

therefore result in a large scattered wave amplitude. The criteria governing

whether or not scattered waves will be in phase is determined by the grating

period and the angle of incidence of the wave [3, 11].

For some gratings, the fringes may take the form of periodically spaced grooves

on the surface of the material. Essentially, each of the grooves then becomes a

very small source of reflected and/or transmitted light. The principle of

interference states that the diffracted light will be in phase if the condition

di ddm θθλ sinsin −= is satisfied.

Figure 1.1 - The phase relationship between rays diffracted from a grating with a

blazed profile [3]

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This figure is not available online. Please consult the hardcopy thesis available from the QUT Library

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Figure 1.1 shows this phase relation. In this case, d is the groove spacing, θi is

the angle between the incident ray and the normal to the grating, θd is the angle

between the diffracted ray and the normal to the grating, and m is a real integer.

In other words, light will be in phase when the path difference between the light

diffracted by successive grooves is equal to m wavelengths of the light [3].

Therefore, reflection should only occur in certain directions that satisfy the

condition. Waves that do not fulfil this condition are cancelled by destructive

interference.

Diffraction gratings play an invaluable role in many integrated optical devices.

These include analog-to-digital converting, antennas, beam coding, beam

coupling, beam shaping, beam splitting, data storage, diagnostic measurements,

holographic optical elements, image amplification, image processing, phase

conjugation, spectral analysis, etc. [3, 8, 10-13].

One example of the numerous, widely used applications of gratings is a

waveguide coupler for frequency selective coupling between two waveguides

[14, 15]. This application is illustrated in figure 1.2. In integrated optical

devices, it is often necessary to transfer energy from one planar waveguide to

another for processing or other purposes. Consequently, grating couplers are

employed to couple this energy from one planar waveguide to another [14].

Grating couplers are the method of choice when connecting two thin-film

waveguides because the waveguides do not have to share a common substrate

[14].

Figure 1.2 - Two waveguides coupled via a grating layer [14]

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1.2 Types of periodic gratings A wide variety of gratings exist and can be characterised by, for example,

geometry, method of manufacture, efficiency and usage [3]. There are

essentially three methods of fabricating gratings. These are mechanical,

lithographic and holographic methods, as shown in figure 1.3. For example,

mechanical ruling is capable of producing gratings with either a blazed,

triangular or trapezoidal profile [3]. Most modern ruled gratings are ruled in a

layer of metal, usually aluminium or gold, that has been vacuum-deposited on an

optically flat glass substrate [11]. However, this form of grating is difficult to

manufacture and can sometimes take one to two weeks to complete [11].

Holographic gratings are formed by the interference of two beams that form

bright and dark fringes in a material. If the material has photosensitive

properties, the interference pattern can be recorded as a grating [10], since the

regions of brightness and darkness will affect the material differently. If this

interference pattern causes a change in the permittivity or refractive index, the

resultant grating will be called a phase grating [3, 10, 11], because the refractive

index of the material mimics the interference pattern. If, however, the

interference pattern causes a change in the surface of the material, it will produce

a surface relief grating that separates two media with different optical properties

[3, 10].

Figure 1.3 - Examples of different types of gratings (a) represents a phase

grating, (b) represents an amplitude grating and (c) represents a surface relief

grating [3]

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Lithographic gratings are manufactured by covering a substrate with a layer of

photoresist. A periodically spaced mask is then placed on top of the layer of

photoresist. This mask prevents light from reaching portions of the photoresist

layer, thereby preventing light from affecting these regions. The photoresist

layer is first exposed to light and then chemically etched, as is the underlying

material [3]. This process produces a relief grating. Diffraction caused by the

mask usually prevents the production of gratings with a period less than 1-

1.25μm [3, 16].

1.3 Types of scattering in periodic gratings

Two main regimes of scattering exist in gratings. They are the Raman-Nath

regime and the Bragg regime [3, 10]. A major characteristic of the Raman-Nath

regime of scattering is the presence of multiple diffracted orders. A grating that

produces Raman-Nath scattering is categorised as a “thin” grating [3, 10]. The

width of such a grating is less than or equal to the wavelength of the incident

wave. A major characteristic of the Bragg regime of scattering is the presence of

only one diffracted order with a significant amplitude. A grating that produces

Bragg scattering is categorised as a “thick” grating [3, 10]. The width of such a

grating is larger than the wavelength of the incident wave.

There are three types of scattering in thick grating structures. They are

reflection, transmission and extremely asymmetrical scattering (EAS) [17-22].

Reflection occurs when the scattered wave leaves the grating through the same

boundary as the incident wave entered through, as shown in figure 1.4(a). In this

case, the amplitudes of the incident and scattered waves decrease exponentially

with distance into the grating [17, 22]. Transmission occurs when the scattered

wave passes through the front boundary and leaves through the opposite

boundary [17], as shown in figure 1.4(b). In this case, the amplitudes of the

incident and scattered waves vary as sine or cosine functions with coordinates

inside the grating [22].

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x

y

k0

k1 (a)

x

y

k0 k1

(b)

Figure 1.4 - (a) Scattering from a reflection grating, (b) Scattering from a

transmission grating.

Extremely asymmetrical scattering occurs when the scattered wave propagates

parallel to the boundary of the grating [23-28], as shown in figure 1.5.

Figure 1.5 - Extremely asymmetrical scattering in a periodic grating [9]

1.4 Extremely Asymmetrical Scattering

Extremely asymmetrical scattering (EAS) is radically different from regular

Bragg scattering. The main distinctive feature of EAS is the strong resonant

increase in the scattered wave amplitude compared to the amplitude of the

incident wave at the front grating boundary [24, 27].

grating fringe

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Research into EAS was conducted in the 1970s and 1980s [18-21], but was

confined to the analysis of scattering of x-rays and neutrons in crystal plates, and

mainly for the case when the incident wave propagated almost parallel to the

boundary of the crystal [29]. Both theoretical and experimental investigations

were carried out, with calculations based on the approximate and rigorous

methods [24, 30]. The main attention was paid to this geometry of scattering

because EAS with grazing incidence allowed highly precise diagnostics of the

near surface regions of crystals and ultra-thin films. In addition, highly efficient

collimators of x-rays and neutrons resulted from the investigation of this type of

scattering [29]. However, these applications were of little interest to researchers

working in the optical range of frequencies. Instead, there was interest in the

strongly resonant effects observed in EAS when the scattered wave propagates

parallel to the grating boundary.

The analysis of EAS using the dynamic theory of scattering produces extremely

awkward calculations and is therefore unsuitable for the analysis of EAS of bulk

waves in anisotropic and nonlinear media and it is also inappropriate for the

analysis of guided and surface waves [27]. In addition, there was previously no

method that was applicable to the treatment all types of waves. That is, a new

rigorous method needed to be developed for each of the different types of waves

to be investigated. As a result, a new, universal approximate approach was

developed [24]. This method can be applied to the EAS of all types of waves,

including surface and guided optical and is applicable to different gratings,

including periodic groove arrays [22, 23, 25, 27, 31]. This new method of

analysis is based on the consideration of the diffractional divergence of the

scattered wave, which has been found to be one of the main physical reasons for

EAS [24]. This universal approximate approach has also been shown to provide

excellent agreement with the rigorous analysis [30] for the most interesting cases

that feature strong resonant increases of the scattered wave amplitude.

The necessity for taking diffractional divergence into account can easily be seen

from the following consideration. As can be seen in figure 1.5, the incident wave

enters the grating through the front boundary at x = 0. Due to the translational

symmetry of the structure along the y-axis (the structure is assumed to be infinite

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along the y-axis), the amplitudes of both the incident and scattered waves are

independent of the y coordinate. The amplitude of the scattered wave must

increase because of the scattering of the incident wave inside the structure.

Simultaneously, the scattered wave interacts with the periodic grating and re-

scatters. The re-scattered wave then propagates in the direction of the incident

wave and is in antiphase with this wave. Consequently, the incident wave’s

amplitude decreases with distance into the grating. The larger the scattered wave

amplitude, the more rapid the decrease in the amplitude of the incident wave.

If only this effect is considered, then the scattered wave would contract to the

front boundary of the grating and its amplitude would be infinitely large, yet the

incident wave would barely penetrate into the grating. However, as the scattered

wave contracts to the front boundary, its diffractional divergence becomes more

significant. The divergence causes the scattered wave to decrease in amplitude

due to the spreading of its energy in space. At the same time, the amplitude of

the scattered wave must increase due to scattering of the incident wave in the

grating. The steady-state amplitudes of all of the waves involved in EAS are the

result of the competition between the two opposing mechanisms described above

[24].

EAS has been analysed under a wide range of conditions. These conditions

include uniform [25, 32] and non-uniform gratings [26, 27, 31], wide [27, 33]

and narrow [24, 27] gratings and gratings with varying structural parameters

[34]. A narrow grating is classified as a grating for which the width of the

grating is L < LC, while L > LC is for a wide grating. Physically, LC/2 is the

distance within which the scattered wave can be spread across the grating by

means of diffractional divergence, before it is rescattered by the grating [28, 31].

These investigations have highlighted the following major and distinctive

characteristics of EAS:

• The amplitude of the scattered wave is usually much larger than that of the

incident wave at the front boundary. The smaller the amplitude of the

grating, the larger the amplitude of the scattered wave [19, 20, 23-26].

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• Step-like phase variations in the grating were shown to have a stronger effect

on EAS than the variation of the magnitude [31].

• Since it is a strongly resonant effect, EAS is extremely sensitive to small

dissipation (absorption) of waves inside and outside the grating [35].

• The pattern of EAS in narrow gratings with a gradually varying magnitude of

the grating amplitude is almost exactly the same as for a uniform grating with

the same width and grating amplitude equal to the average amplitude of the

grating in the non-uniform grating [27].

• EAS in wide gratings results in a scattered wave amplitude that is mainly

determined by the local values of the grating amplitude [27].

• There is a highly unusual sensitivity of the incident and scattered wave

amplitudes in the EAS geometry to small step-wise variations of mean

structural parameters at the grating boundaries. For example, these variations

can be represented by the mean dielectric permittivity or the mean waveguide

thickness for EAS of slab modes [34]

• Drastically different patterns of scattering have been observed in narrow and

wide gratings. In narrow gratings, where L < LC, varying the mean structural

parameters at either of the boundaries has a strong effect on the scattered

wave amplitude throughout the structure [34]. In wide gratings, where L >

LC, then the effect of varying mean parameters on the scattered wave

amplitude is usually significant only within a half of the critical width from

the boundaries [34].

• An additional unique resonance with respect to the angle of scattering has

been observed if the scattered wave propagates almost parallel to the grating

boundaries [36].

• An additional exceptionally strong resonance in the frequency response of

EAS and grazing angle scattering (GAS), when the diffracted order satisfying

the Bragg condition propagates at a grazing angle with respect to the

boundaries, has been predicted in the side-lobe structure at a frequency larger

than the Bragg frequency [37].

• For EAS in non-uniform gratings that are formed by joining two uniform

gratings with different phases, Double-Resonant Extremely Asymmetrical

Scattering (DEAS) occurs. DEAS is characterised by a unique combination

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of two simultaneous resonances in the grating. One resonance is with respect

to frequency and other is with respect to phase variation between the joined

uniform gratings [28, 31].

However, these characteristic effects of EAS have only been theoretically

predicted. As of this point in time, there has been no experimental verification of

these predictions. Consequently, the observation and characterisation of

scattering in this geometry is an extremely important step towards the

development of applications utilising EAS. Such applications may include EAS-

based resonators, high sensitivity sensors and measurement techniques, narrow-

band optical filters, couplers, switches and lasers [27].

1.5 The photorefractive effect

The photorefractive effect is a phenomenon through which the local index of

refraction of a material is changed by the spatial variation of light intensity [38].

This effect was first observed in lithium niobate (LiNbO3) in 1966 [39] and has

since been observed in barium titanate (BaTiO3) and other inorganic crystals

such as SBN, BSO, BGO, GaS, InP [38]. The photorefractive effect has also

been observed in organic crystals [40], as well as in polymers [41, 42]. The

effect was initially considered to be an impediment because it limited the

usefulness of lithium niobate in certain applications [39]. It was observed that

the light induced refractive index change scattered the beam propagating through

the crystal. However, additional research revealed that the “optical damage”

reported by Ashkin et al. [39] could be utilised as a method of holographic

storage [43]. Since then, photorefractive materials have been used in

applications such as resonators, self-pumped phase conjugators, real-time

holography and nonlinear optical information processing [38, 44-46]. All of

these applications rely on the formation of volume gratings in the material.

Photorefractive materials exhibit a nonlinear response when exposed to light.

Light stimulates charges in the material, causing them to migrate [47]. Charge

transfer within the crystal occurs through diffusion due to non-uniform free

carrier distribution [45, 48], drift (when an electric field is applied externally [45,

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49]) and/or the photovoltaic effect [45, 50]. The properties of the photorefractive

material govern the method of charge transfer, with diffusion being the dominant

method of charge transfer in BaTiO3 [38, 45].

Diffusion is a result of the gradient of the electron density, with electrons moving

from regions of high concentration to regions of low concentration. Free

electrons migrate from the bright regions of the crystal to the dark regions, where

they are subsequently trapped. The regions vacated by the electrons are then

positively charged, while the regions where electrons are trapped become

negatively charged [44]. This results in the formation of an electric field

between the light and dark regions. The electric field then distorts the crystal

lattice, causing a change in refractive index in certain regions [51]. Since

diffusion is a thermal process, it is relatively slow [45]. For example, BaTiO3

has a response time of the order of seconds [52]. It is possible to hasten the rate

of response of photorefractive materials to light by increasing the light intensity.

The use of intense laser beams has increased the speed of the photorefractive

effect from the order of seconds to the order of picoseconds for some materials

[53].

There are currently a number of models that describe the photorefractive effect.

The most widely accepted models are the band transport model (figure 1.6(a))

and the hopping model (figure 1.6(b))[54].

Figure 1.6 - The methods of photoconduction – (a) the band transport model and

(b) the hopping model [54]

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12

The band transport model was developed by Kukhtarev and Vinetskii [55, 56],

and is a three-step process for electron transport. This model assumes that

photorefractive materials contain donor and acceptor traps that arise from

imperfections in the crystal. These traps create intermediate electronic energy

states in the bandgap of the insulators [38]. The first step in this process is the

photoionisation of a carrier at a particular defect site, followed by the drift and

diffusion of the carrier through the conduction band and finally, recombination at

a new defect site [57].

The following set of equations can be used to determine the space-charge field

produced by the migration of charge carriers [38, 58].

)4.1()().(

)3.1()log(

)2.1())((

)1.1(.1

+

+++

+

−+=∇

+∇−=

−−+=∂∂

∇−∂∂

=∂∂

DA

sc

DRDDD

D

NNNeE

cpINe

kTENeJ

NNNNsINt

Je

Nt

Nt

ε

μ

γβ

The rate of carrier generation is (sI+β)(ND-ND

+), while the rate of ionised donors

catching carriers is γRNND+. Here, s is the cross section of photoionisation, β is

the rate of thermal generation, γR is the carrier-ionised trap recombination rate, N

is the concentration of carriers and ND and ND+ stand for the concentration of

donors and ionised donors respectively. c is the unit vector along the c axis of

the crystal, I is the light intensity, NA is the acceptor concentration, μ is the

mobility, T is the temperature, k is the Boltzmann constant, ε is the dielectric

tensor, pI is the photovoltaic current, sc

E is the space-charge field, and p is the

photovoltaic constant.

This “deep-trap” band transport model has been used to describe a wide array of

steady state behaviour in photorefractive materials. However, it has been found

that this model cannot explain some phenomena. Deviations from the deep-trap

13

model include predictions that photorefractive gratings decay exponentially in

the dark [59, 60] and the nonlinear relationship between photoconductivity and

light intensity [61-64]. Consequently, modifications were made to the band-

transport model to incorporate shallow traps [61, 65]. Modifications were also

made to account for both electrons and holes being able to act as charge carriers

[54, 59]. These additions to the original model were necessary to explain

transient photorefractive effects [47].

The hopping model [66], while different in its approach, predicts the same results

as the band transport model, as long as the electron (or hole) recombination time

is short compared to any optical transients and there is only a small relative

charge modulation [57]. This model assumes that there are a certain number of

charges that can occupy a larger number of sites in any of a large number of

permutations [66]. Each charge stays fixed at a site when the material is in

darkness, but “hops” to an adjacent site when the material is exposed to an

optical beam. The probability of hopping is proportional to the intensity of the

beam [66]. The rearrangement of charges ultimately produces a space-charge

field in the material [57].

The presence of a space-charge field in a photorefractive material produces a

change in the refractive index which is induced via the linear electrooptic effect

[44]. This change in refractive index is given by the following equation:

where rijk is the electro-optic coefficient and Ek is the “kth” component of the

space charge field E1 [38, 47].

Barium titanate has become a popular photorefractive material for many

applications because of its large electrooptic coefficient [67, 68], and therefore

the large refractive index change that is possible. Barium titanate is categorised

as a negative uniaxial crystal. It is uniaxial because the index ellipsoid can be

simplified to [69]:

)5.1(1 302

sckijk

sckijk

i

ErnnErn

rr−=Δ⇒=⎟

⎠⎞

⎜⎝⎛Δ

14

)6.1(12

2

20

2

20

2

=++en

zny

nx

In other words, two of the principal indices are the same. This means that the z-

axis is touched at two points by the two sheets of the normal axis, thereby

making the z-axis the only optic axis [69]. This is illustrated in Figure 1.7.

Figure 1.7 - The intersection of the normal surface with the xz plane for a

negative uniaxial crystal [69].

BaTiO3 is negative because the ordinary index, n0, is larger than the

extraordinary index, ne. The refractive indices for BaTiO3 for λ = 633nm are n1 =

n2 = no = 2.404 and n3 = ne = 2.316 [45].

In uniaxial crystals, the electric field vector E (and displacement vector D) for

ordinary waves is always normal to both the c axis (optic axis) and the

propagation vector. However, the displacement vector D of extraordinary waves

is normal to the propagation vector, while the electric field vector E is generally

not normal to the propagation vector. Instead, it lies in the plane formed by the

propagation vector and the displacement vector. The electric field vectors of

these two waves are mutually orthogonal [69].

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15

Figure 1.8 - The construction for finding the indices of refraction and

polarisation of a negative uniaxial crystal for a direction of propagation given as

s [69].

From figure 1.8, it is evident that if the angle θ is changed, the polarisation of the

ordinary ray remains fixed and its index of refraction is constant (no), and is

equivalent to the length OB. However, the direction of De is always dependent

on the angle θ. Consequently, the index of refraction can vary between ne(θ) =

no (at θ = 0ο) and ne(θ) = ne (at θ = 90ο). The index of refraction is equivalent to

the length OA in figure 1.8 [69].

Barium titanate is a ferroelectric crystal that has a phase transition temperature of

TC = 120°C. Above this temperature, the linear electrooptic effect disappears and

the crystal symmetry becomes m3m (cubic). Below this temperature, the linear

electrooptic effect is dominant and the crystal is acentric with point group 4mm

[69].

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16

The nonvanishing electro-optic coefficients for barium titanate are given as [44]

The electro-optic coefficients for BaTiO3 are r51 = r42 = 1640 × 10-12 m/V and r33

= -r13 = 108 × 10-12 m/V [69].

1.6 Two-wave mixing in photorefractive materials

Two-wave mixing (TWM), also known as two-beam coupling, is the process by

which phase and energy are exchanged between two beams that intersect within a

photorefractive material [47, 56]. If two coherent laser beams of the same

frequency intersect within a photorefractive material, they will interfere and

produce a pattern of bright and dark regions whose intensity varies sinusoidally

with position in the crystal [5]. Electric charges will move from the light regions

to dark regions and create an electric field that will vary in strength sinusoidally.

This electric field then distorts the crystal lattice via the linear electro-optic

effect, resulting in the formation of a volume holographic grating. The index of

refraction is given as [38]:

where n0 is the index of refraction when no light is present, φ is the phase shift

between refractive index pattern and the interference pattern, n1 is real and

positive, 12 kkKrrr

−= is the grating vector, *1A is the complex conjugate of the

wave amplitude A1 , A2 is a wave amplitude and 22

21210 AAIII +=+= .

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

0000000

000000

42

42

33

13

13

rr

rrr

)7.1(..).exp(2 0

2*11

0 ccrKiIAAennn i +−+=

rrφ

17

Once the grating has been formed, a fraction of the light from one beam will be

deflected or diffracted in the direction of the other beam (and vice versa). As a

consequence, the two deflected beams will interact constructively with one of the

original beams, and destructively with the other [5]. Both of the diffracted

beams will acquire a phase shift of π/2 via the normal diffraction process [47].

In photorefractive materials that operate by diffusion, such as BaTiO3, there will

be an additional phase shift of π/2 due to the phase shift between the diffraction

grating and the interference pattern [38, 47]. The sign depends on the direction

of the c axis [38, 66], since the charges migrate in one direction along that axis.

This results in an additional +π/2 phase shift for one beam and a shift of -π/2

shift for the other. The beam formed by constructive interference will emerge

from the crystal with more energy than when it entered, while the beam formed

by destructive interference will emerge weaker. The intensities of the two beams

are given by the following equations [44]:

where m is the input intensity ratio )0(/)0( 21 IIm = , γ is the gain coefficient, α

is the absorption coefficient and z is distance along the z axis.

However, two-wave mixing is rarely found in most nonlinear materials because

they respond “locally” to optical beams. As a consequence, the optical pattern

and the grating overlap exactly. The light that is deflected by the grating

interferes with each of the undeflected beams in exactly the same way, resulting

in the two beams exchanging an equal amount of energy. Therefore, neither of

the beams grows in energy [5].

One of the most interesting properties of holographic gratings formed in

photorefractive materials is that they can be easily erased without damaging the

crystal [66]. Gratings can be erased from the crystal by flooding the crystal with

)9.1(1

1)0(2)(2

)8.1(11

11)0(1)(1

zezme

mIzI

zezem

mIzI

αγ

αγ

−−+

+=

−−+

−+=

18

light incident from directions that do not satisfy the Bragg condition. An erasing

beam with a wavelength of 633nm will erase a grating formed using a 514nm

laser, and vice versa [66]. The beam that is used to read the holographic grating

in the material also has the effect of slowly erasing the grating. This decay rate

increases with intensity [66]. However, this phenomenon is undesirable in

applications that require a constant, uniform grating. It is possible to thermally

fix the grating in the material, thereby preventing the reading beam from erasing

the grating [70]. However, this treatment is impractical when the intention is to

use the crystal for multiple recordings and erasures. The grating can last for

years if the crystal is stored in the dark, which makes it ideal for the storage of

information [5].

It has been shown that the two-beam coupling gain coefficients, the parameters

that describe the level of amplification that the two-wave mixing in a

photorefractive material can provide, decrease with wavelength. This was

explained by showing that the effective carrier number density is a linearly

decreasing function of wavelength [71]. The wavelength dependence on the

effective charge number density has been determined by taking into account the

dispersion of the effective electro-optic coefficient reff [72] and can be explained

by using the deep- and shallow-level model [65, 72]. It has also been shown that

index gratings in photorefractive materials are also dependent on intensity. The

strength or modulation of index gratings is influenced by the intensity of the

writing beams, by their absolute intensity, as well as by their intensity ratio [73].

It was shown that the modulation of the refractive index, Δn, as a function of the

modulation, m, has a linear dependence, where )/(2 2121 IIIIm += . The

dependencies are not only valid for small modulation, but for all the range of m

between zero and unity [73].

Many photorefractive materials are doped with other substances to enhance their

photorefractive properties. Substances such as rhodium, cerium and cobalt have

been used as dopants [74-76] in barium titanate. It has been reported that doping

the crystal with cerium (Ce) [75] or with cobalt (Co) [77] increases the beam-

coupling gain by increasing the effective trap density and intensity dependent

19

factor. By increasing the Ce concentration from 30 ppm to 50 ppm, the effective

trap density is doubled and the intensity dependent factor is increased by more

than 20 percent [75]. These increases, in turn, cause the electro-optical

coefficient to increase. However, increasing the doping concentration also

increases the photorefractive response time [75, 78], meaning that grating

formation can take longer than in undoped crystals.

It is also possible to alter the properties of photorefractive materials by reduction

and oxidation at high temperatures [57]. It was shown that the relative

contributions of electron and hole photoconduction were altered when treated in

different oxygen atmospheres to form or fill oxygen vacancies. It was found that

hole photoconduction dominated at high oxygen partial pressures, while at low

oxygen partial pressures, electron photoconduction dominated [57]. When

oxidised, BaTiO3 exhibited a sublinear photoresponse and delivered a linear

photoresponse when reduced. A linear response means a quicker response at

lower optical intensity [57]. However, the heating of photorefractive crystals is

extremely risky because the crystals are susceptible to stress fractures when

heated or cooled quickly [79].

1.7 Beam fanning

The most common form of noise found when fabricating diffraction gratings in

photorefractive BaTiO3 during two-beam coupling is beam fanning [80]. Beam

fanning is a common phenomenon in photorefractive crystals and many

photorefractive devices, such as various self-pumped phase conjugators, rely on

it [81-83]. Beam fanning is caused by two beam coupling between the incident

beam and its own scatter. Scattering can be caused by surface roughness, index

inhomogeneity and impurities in the crystal [44]. Defects and impurities in the

crystal scatter small amounts of light propagating through it.

Beam fanning is characterised by the bending of the incident beam towards the c-

axis, as well as the self-defocusing of the beam in its spatial distribution [84].

The first two-dimensional analysis of beam fanning [84] reported that the

formation and bending of a Gaussian beam is due to the competition of the beam

20

coupling between the incident and scattered beam with the beam coupling

between different directions of the scattered beam. This fanning has the

deleterious consequence of amplifying the scattered beam within the crystal [46].

Beam fanning may be highly undesirable in some experiments because it is a

source of noise that can destroy the grating formed in the crystal by two well-

defined Gaussian beams.

Since some experiments require the minimisation or elimination of beam

fanning, investigations have been made into noise reduction techniques. For

example, noise can be suppressed in photorefractive image amplifiers by

performing two-wave mixing in a slowly rotating crystal [80]. However, this

technique is rather difficult to implement [85]. Another method uses the large

time constant for beam fanning in BaTiO3 crystals to its advantage by reading the

grating with a short pump pulse that is shorter than the time constant for beam

fanning. A large amount of energy can then be transferred from the strong pump

beam to the signal beam without being exposed to beam fanning [85]. It is also

possible to rotate the crystal to find an optimum balance between the signal-to-

noise ratio and high gain in the signal beam in two-beam coupling [68]. This

compromise is necessary because in photorefractive crystals, the noise is at a

maximum in a direction that is close to the direction of signal beam propagation

[46]. In high gain media, regardless of its homogeneity, there will be beam

fanning. This is a consequence of natural phenomena such as Brillouin

scattering [44].

1.8 The laser beam Unlike the idealised plane waves considered in many publications, the intensity

profile of laser beams is almost Gaussian in nature. This intensity profile

corresponds to the theoretical TEM00 mode, which is closely approximated by

helium neon lasers [86]. This means that the phase fronts are slightly curved and

the intensity distribution is not uniform across the beam, but instead,

concentrated near the axis of propagation [87]. In some cases, it may be possible

to ignore this Gaussian profile, while in other cases, neglecting to take into

account the Gaussian profile of the beam may result in a difference between

21

experimental observations and theoretical predictions [88]. Figure 1.9 below

highlights some of the important characteristics of the Gaussian beam.

Figure 1.9 - (a) Contour of the Gaussian beam and (b) Amplitude distribution of

a Gaussian beam [87]

As indicated in figures 1.9(a) and (b), the radius of curvature of the wavefront,

given by R(z), intersects the axis at z. The beam contour w(z) is a measure of the

decrease of the field amplitude E with the distance from the beam axis. The

distance at which the field amplitude is 1/e times that on the axis is given by w.

This parameter is often called the beam radius. The beam has a minimum beam

radius at the beam waist. The beam waist is the position where the radius of

curvature of the wavefront is planar [89]. At the beam waist, the diameter of the

beam is 2w0 [90].

Most models of wave interactions in photorefractive crystals have been one-

dimensional, assuming the interacting beams to be infinite plane waves [91].

However, laser beams are finite and generally have a Gaussian profile [88].

Even if the Gaussian wavefront were made perfectly flat at some plane, the beam

would eventually begin to diverge as a result of diffraction. The radius of

curvature of the wavefronts of the beam, R(z), and the beam radius, w(z), vary

with the distance the beam has propagated, z, as given in the following equations

[87]:

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22

)10.1(1)(22

0

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

zw

zzRλπ

)11.1(1)(

2/12

20

0⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

wzwzw

πλ

where λ is the wavelength of light.

The profiles of both of the interacting beams in the photorefractive material

change as they propagate through the interaction region as a result of the

exchange of energy between the beams. Solymar et al. [92] developed a two-

dimensional theory that described the interaction of finite beams via refractive

index gratings. This model was then modified by Królikowski and Cronin-

Golomb [88] to consider the treatment of coupling via the dynamic gratings of

photorefractive crystals. The geometry of the interaction of the beams is shown

in Figure 1.10.

Figure 1.10 - The interaction geometry of finite optical beams, where A1 and A4

are the wave amplitudes [88]. In this case, A1 is a function of x and A4 is a

function of y.

This research showed that the energy transfer is greatest in the region of small x

because the beam A1 is amplified at the expense of pump beam A4. The profile

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23

degradation was shown to be dramatically different when Gaussian beams were

considered instead of uniform beams. The deformation of the amplified beam

increased when the coupling strength γd was increased because of the stronger

energy transfer.

In some applications, especially those involving photorefractive materials, it may

be desirable to have a beam with a plane profile. This is because the

photorefractive effect is sensitive to the intensity profile of the beams [78]. Since

the intensity of a Gaussian beam is not uniform across the profile, the refractive

index change of the material will not be uniform. As a result, the grating will not

be uniform, which may be undesirable in some applications.

A number of different methods have been proposed to produce a more uniform

intensity profile from a laser beam [93-98]. These include a technique for

generating focal-plane flat-top laser beam profiles [97], an aspheric lens that

converts a Gaussian laser beam into a flat-top beam [98] and an optical device

that produces a square flat-top intensity irradiation area from a pulsed laser beam

with a Gaussian profile [95]. It may also be possible to use relay imaging [93] to

image the flat section of a Gaussian beam’s wavefront.

1.9 Scope of this project

As has been previously outlined, extensive research has been conducted into

Bragg scattering in the extremely asymmetrical geometry. However, this

research has been solely theoretical in its approach in the optical range of

frequencies. Prior to this work, no experimental observations and/or

investigations of the predicted phenomena have been reported in the visible part

of the spectrum. Consequently, the aim of this project is to experimentally

observe EAS- related resonances for guided optical electromagnetic waves in

holographic gratings in a planar waveguide.

This experiment will utilise the nonlinear response of photorefractive materials to

light. By employing two-wave mixing, temporary holographic gratings will be

formed in photorefractive barium titanate and in lithium niobate. BaTiO3 has an

24

electro-optic coefficient two orders of magnitude larger than LiNbO3, allowing

the investigation of EAS with grating amplitudes two orders of magnitude

greater. These gratings will ideally have a width between 10μm and 100μm and a

length of 3mm.

Since grating fabrication times in excess of 1000 seconds can be required in

photorefractive materials, the stability of various optical arrangements will be

investigated. The aim here is to find the most stable interferometric arrangement

for fabrication of holographic gratings in photorefractive materials.

Once stable photorefractive gratings have been fabricated, a variety of wave

scattering will be observed in the EAS geometry. Once the scattering has been

investigated using a certain set of parameters, the grating will be erased and a

grating with different parameters will be formed. Thus, the same crystal will be

reused many times for the investigation of different grating parameters and

angles of incidence and scattering.

Finally, there will be a theoretical determination of EAS in the structures that

have been analysed experimentally. This analysis will utilise the universal

approximate method that takes diffractional divergence into account. The

experimental and theoretical results will then be compared.

25

Chapter 2

Fabrication of holographic gratings in photosensitive media with a passively stable Sagnac optical arrangement 2.1 Introduction Bragg gratings have numerous applications in photonics and applied optics,

including wave multiplexing and demultiplexing, beam splitting, image

processing and spectral analysis [3, 10]. One of the most common forms of

Bragg grating is the holographic grating. Holographic gratings are formed by the

interference of two coherent beams in a photosensitive medium. These

photosensitive materials include porous glass, dichromated gelatin and dynamic

media, such as photorefractive crystals [99].

Holographic gratings formed in photorefractive materials are of particular

interest because they enable the materials to be used in applications such as real-

time holography and nonlinear optical information processing [38]. Here,

gratings are formed when the interference pattern produced by the interacting

beams creates bright and dark regions within the photorefractive material. In

response to an electric field, the nature of which depends on the material,

electrons migrate from bright regions to dark regions, thus producing a static

electric field inside the material. This field produces a periodic refractive index

change via the Pockels’ effect, which is essentially a volume holographic grating.

The photorefractive effect is intensity dependent, as the time constant for grating

formation is inversely proportional to intensity [78]. With low power cw lasers

(~ mW), a grating can take of the order of 103 seconds to form. Focusing the

beams to increase the light intensity can reduce the grating writing time but this

reduces the volume over which the two beams overlap and therefore the size of

the grating. Therefore, if gratings that are greater than a few millimetres in

length and width are desired, beams with large diameters are necessary. Larger

gratings are of interest, since the spectral resolution depends on the physical size

of the grating.

26

When fabricating a holographic grating, it is necessary to create and maintain a

stable interference pattern so that the grating period and amplitude remains

uniform over the extent of the grating. Perturbations such as vibrations of optical

components and temperature changes can cause the relative phase of the

interfering fields to fluctuate, thereby degrading the grating that is written.

Existing stabilising techniques include active techniques like the use of

piezoelectrically driven mirrors to apply an optical phase offset [100] and passive

strategies such as using a Sagnac interferometer [101]. Active systems monitor

mirror motion and then correct for this movement, while passive systems are

designed to minimize fluctuations in optical path difference. Reducing the

fluctuations in the optical path difference improves the stability of the

interference pattern, resulting in a more homogeneous grating.

Described in this chapter is a novel optical system that can be used to generate

stable interference patterns for writing gratings in photosensitive media. This

passive arrangement is based on a rectangular Sagnac interferometer that

compensates for translational motion of the mirrors. A unique feature of the

system described here is the novel technique used to interfere the two writing

fields at large crossing angles (>20°) as is required for producing gratings with

small periods (~ 0.5 μm).

Sagnac-style interferometers in both triangular [102] and rectangular [101, 103]

configurations have been previously investigated and have been shown to

provide greater stability than a conventional Michelson interferometer. A

modified rectangular Sagnac has been shown to produce stable interference

fringes from a broadband nanosecond laser [103]. The triangular Sagnac has

been used to write photolithographic gratings with a setup that enables precise

adjustment of the fringe spacing of an interference pattern while providing

excellent stability against external perturbations [102].

This chapter provides the first analysis of stable holographic grating formation in

a photorefractive crystal through the utilisation of a unique modified rectangular

Sagnac interferometer. As will be discussed, the improvement in fringe stability

27

that the arrangement provides allows a better understanding of the dynamics of

photorefractive grating growth. In particular, environmental perturbations that

typically affect the grating writing process are investigated and analysed. This

analysis is then expanded to explain the reversal of the direction of energy flow

during photorefractive two-wave mixing when a phase shift is introduced into

one of the beams. Some of the findings discussed in this chapter were published

in [104].

2.2 Evaluation of stability of the Sagnac interferometer The optical arrangement is based on the rectangular Sagnac interferometer [101]

shown in figure 2.1.

Figure 2.1 The displaced Sagnac interferometer, where BS1 and BS2 are beam

splitters, PD is a silicon photodetector, and M2-M4 are mirrors. The solid line

indicates the beam that travels around the loop in a clockwise direction, while the

dotted line indicates that travels around the loop in a counter clockwise direction.

Laser

M4

M2

M3

BS2 BS1

PD

28

Figure 2.2 Michelson interferometer

The polarisation of the output from a 633nm He:Ne laser is rotated by a half-

wave plate to ensure that light enters the photorefractive Fe:LiNbO3 (0.05%)

crystal as ordinary rays. An acousto-optic modulator (AOM) acts as an isolator

to prevent optical feedback from causing instabilities in the laser output. The

output from the AOM is filtered by an aperture before striking a plate

beamsplitter (50/50) which splits the amplitude of the incident light into two

beams that counter propagate through the interferometer. This results in each

beam striking each reflective surface once as it travels around the loop. The

three mirrors in the interferometer are arranged at an angle of approximately 90

degrees with respect to the perpendicular direction of the optical axis of each

arm.

The translational stage attached to the base of mirror M2 is used to separate the

paths of the two beams. Translating the mirror lengthens or shortens the distance

travelled by both beams by the same amount, keeping the optical path difference

constant. The two beams are recombined by the beam splitter. The output from

the interferometer is detected via a reflection off a second beamsplitter. An

aperture ensures that only one interference fringe falls onto a silicon

photodetector.

29

The stability of the output from the Sagnac was compared to that of a Michelson

interferometer set up in the same laboratory environment. Mirror M3 of the

Sagnac was used to adjust the fringe pattern until a bright fringe was visible at

the output of the interferometer. No shift in the fringe pattern occurred when the

mirror M2 was translated over a range of 5 millimetres. A simple analysis of the

displaced Sagnac interferometer reveals that the optical path length difference is

insensitive to translational motion of BS2, M2 or M3. A frequency analyser was

used to measure the level of fringe jitter that resulted from vibrations and thermal

changes that affect the mirrors at frequencies between 1 Hz and 20 kHz. A

typical frequency spectrum of the resultant noise for this configuration is shown

as figure 2.3. A comparison of the outputs from the Michelson and Sagnac

reveals that the Michelson is much more sensitive to mirror motion under

ambient conditions, particularly at frequencies below 10 Hz.

Figure 2.3 – Three typical noise spectra of the output of (a) a displaced Sagnac interferometer and (b) a Michelson interferometer under similar ambient conditions.

The frequency analyser was used to measure the effect of perturbations such as

sudden, abrupt vibrations caused by striking the supporting bench/ breadboard.

0

0.05

0.1

0.15

0.2

0.1 1 10 100 1000 10000 100000

Frequency (Hz)

Noi

se a

mpl

itude

(hal

f-wav

elen

gths

)

b

a

30

The data in figure 2.3 indicates that when the systems were not deliberately

perturbed, the instability of the Michelson fringe pattern was approximately four

times larger than that produced by the Sagnac across the frequency range from

1Hz to 10000Hz under ambient conditions. However, when deliberately

perturbed by striking the optical bench, the fringe pattern of the Michelson was

noticeably more sensitive to noise than the Sagnac, particularly at frequencies

below 10Hz, as is shown in figure 2.4.

Figure 2.4 - Three typical noise spectra of the output of (a) a displaced Sagnac interferometer and (b) a Michelson interferometer when both systems suffered similar vibrational perturbations.

The Sagnac’s frequency spectrum in figure 2.4 shows a peak at a frequency of

about 600Hz, which corresponds to a resonant frequency of the optical

arrangement used. The specific amplitude and frequencies of mechanical

resonances of the interferometer will depend on the particular components used

in the arrangement and the manner in which they are supported.

A comparison of figure 2.3 and figure 2.4 shows that the Michelson was

extremely sensitive to vibrational frequencies between 60Hz and 500Hz, as well

as frequencies below 2Hz. At resonant frequencies below 2Hz, and those

between 60 and 500Hz, the instability of the Michelson more than doubled. In

the presence of these deliberate perturbations some degradation in the stability of

0

0.1

0.2

0.3

0.1 10 1000 100000Frequency (Hz)

Noi

se a

mpl

itude

(Hal

f-wav

elen

gths

)

b

a

31

the output of the Sagnac was also observed, indicating that these perturbations

caused rotation of the mirrors as well as translation. Analysis of the Sagnac

shows that the optical path length difference is sensitive to rotation of the mirrors

of the interferometer, and therefore, rotational perturbations are not compensated

by the arrangement. From the comparison of the fringe stability of both

interferometers shown in figure 2.3, it can be concluded that the dominant

motion of the optics caused by ambient perturbations is translational.

The intensity of the interfering beams was approximately 10 mW/cm2. Writing

photorefractive gratings in the Fe:LiNbO3 crystal with such low intensity light

was observed to take in excess of 30 minutes. The effect of perturbations such as

thermal expansion of components and optical paths that typically occur over such

long timescales was investigated. The low frequency fringe stability of both the

Sagnac and the Michelson were analysed simultaneously over a period of 15

minutes. Data was recorded immediately after the mirrors were adjusted to

optimise interferometer output. A phase shift of 180 degrees corresponds to a

change in the optical path length difference by a distance of λ/2. The data in

figure 2.5 shows that the output of the Michelson cycles through multiple

maxima during the sampling period, indicating that the path length difference

had changed by many wavelengths over that time.

32

Figure 2.5 - The output of (a) a displaced Sagnac interferometer and (b) a Michelson interferometer as observed over 15 minutes. As expected, the variation in the output from the Sagnac is considerably smaller

over the same time interval. The slow drift that was observed in the Sagnac

output was determined to be due to fluctuations in laser power, and not caused by

movement of the mirrors. One trend seen in figure 2.5 (b) is that the rate of

change of the optical path length difference in the Michelson decreased as the

time elapsed. This is most likely due to relaxation of mirror springs and the

dissipation of thermal energy imparted to the mirrors during their final

adjustment prior to data collection. It was observed that even minimal

adjustments to the mirrors of the Michelson resulted in fluctuations in the optical

path length difference of the order of wavelengths that persisted for many

minutes. Thus, in applications requiring good fringe stability, the Michelson

requires a significantly longer settling time than the Sagnac.

The Allan variance [105] was used to analyse the fringe stability of both the

Michelson and Sagnac at very low frequencies (f ≤ 1Hz). The fringe stability

was observed by recording the outputs from both interferometers for up to 3000

0

0.4

0.8

1.2

0 200 400 600 800 1000

Time (s)

Shift

(Hal

f-wav

elen

gths

)

b

a

33

seconds. The data in figure 2.6 indicates that the fluctuations in the fringes

produced by the Sagnac are at a lower level that those of the Michelson at low

frequencies.

Figure 2.6 - The Allan Variance of (a) the displaced Sagnac interferometer output shown in figure 4(a) and (b) the Michelson interferometer output shown in figure 4(b).

This significant increase in stability over long time intervals is extremely

desirable when using low intensity beams to create stable interference patterns

for grating fabrication as is required when working with photorefractive

materials like lithium niobate.

2.3 Stability of holographic grating The period of a holographic grating is of the order of the wavelength of the light

that is diffracted. Gratings with such small periods can only be written by two

coherent fields that intersect at large angles (>10°). Therefore, the output

directly from an interferometer is unsuitable since the two interfering fields very

nearly co-propagate. A modification to the displaced Sagnac is necessary to

allow the beams to cross at large angles as is shown in figure 2.7.

0.01

0.1

1

0.001 0.01 0.1 1

Frequency (Hz)

Shift

(Hal

f-wav

elen

gths

)

b

a

34

Figure 2.7 - The modified Sagnac arrangement that allows the formation of interference patterns produced by two beams that cross at large angles.

The two beams are separated by adjusting the angle of mirror M4 causing the

beams to diverge after returning to the beamsplitter. However, both beams still

strike the same reflective surfaces, ensuring that all translational motion of the

mirrors is compensated, keeping the optical path length difference constant.

The maximum separation of the two beams is determined by the diameter of the

recombining beamsplitter (M1) and is limited to approximately half the width of

the beamsplitter by its mounting. A single mirror (M7) picks off both beams

after the recombining beamsplitter. A 1 cm diameter circular aperture was cut

through the mirror to allow the input beam from the laser to reach the

interferometer. Two mirrors (M5 and M6) direct the two beams onto the

photorefractive medium. The mirror angles are arranged so that each beam was

reflected from both mirrors, thus ensuring that the optical path difference remains

constant in the presence of translational motion of the mirrors. The Fe:LiNbO3

crystal was placed along the bisector between the two diverging beams and M5

and M6 were adjusted until the beams crossed inside the crystal. With the

arrangement shown in figure 2.7 beam crossing angles greater than 90º were

possible.

Laser

Fe:LiNbO3

M4

M2

M3

M6 M5

M7 M1

35

2.4 Formation of a holographic gratings in a photorefractive crystal The interference pattern formed inside the photorefractive crystal produces a

refractive index grating that diffracts the two writing fields, since both satisfy the

Bragg condition. A portion of each field is diffracted in exactly the direction of

the other field. The intensity of one of the fields increases because the

diffraction from the grating adds constructively, while the intensity of the second

field decreases, since the diffraction from the grating is out of phase. Thus, via

this two wave mixing process, energy flows from one field to the other. The

direction of this energy flow is dependent on the orientation of the optical axis of

the crystal and the directions and polarisations of the two fields.

The change in intensity of the two writing fields depends on the amplitude of the

grating. Therefore, by monitoring the intensity of the two beams after they

emerge from the crystal, it was possible to view the rate of grating formation and

hence the stability of the interference pattern inside the crystal. If the

interference pattern shifts due to changes in phase of the two writing fields,

regions that were previously dark may become bright and vice versa. Therefore,

in general, changing the phase relationship between the interference pattern and

the partial grating already present acts to erase the grating and diminish its

amplitude. This weakening of the grating will manifest itself as a change in the

amount of energy diffracted between the two beams that write the grating.

The data in figure 2.8 shows the typically observed evolution of the intensities of

the two fields when a simple ‘traditional’ arrangement consisting of two mirrors

and a beam splitter was employed to form the interference pattern inside the

photorefractive crystal.

36

Figure 2.8 - The evolution of the intensities of (a) beam 1 and (b) beam 2 when a grating is formed using a ‘traditional’ arrangement.

In this arrangement, no compensation for translational motion of the mirrors

occurs. The optical axis of the crystal was orientated so that the energy is

transferred from beam 1 to beam 2. As expected from the conservation of

energy, the change in energy of one beam is equal and opposite to the change

experienced by the other beam. However, the evolution of the intensities of the

beams was irregular and appears to meander with no recognisable pattern. Also,

the direction of energy flow was observed to fluctuate randomly over the course

of many minutes and, in some instances, reverse so that beam 1 is amplified at

the expense of beam 2.

This change in the direction of the energy flow is due to phase variations in one

or both of the fields that occur too quickly for the grating to respond to. A

qualitative understanding can be gained using a simple analysis of the diffraction

of beams 1 and 2 from a ‘permanent’ grating. Assuming that the optical axis of

the crystal is orientated so that energy flows from beam 1 to beam 2, the

amplitudes of the two fields that exit the medium are given by:

3.0

3.2

3.4

3.6

3.8

4.0

0 500 1000 1500 2000

Time (s)

Inte

nsity

(mW

/cm

2 )

a

b

37

)2.2(

)1.2(

122

211ININOUT

ININOUT

EEE

EEE

α

α

+=

−=

Here α represents the diffraction efficiency of the grating that has been written.

If α is positive, beam 2 gains the energy supplied by beam 1. If a phase shift of

180 degrees is introduced to beam 2 and the geometry of the interaction remains

fixed, the amplitude of the beam 2 is given by:

)3.2(22*2

ININiIN EEeE −== π

Consequently the amplitude of the two output fields are given by:

)5.2(

)4.2(

122

211ININOUT

ININOUT

EEE

EEE

α

α

+−=

+=

Under these conditions, it is evident that the energy flow direction has reversed

since beam 2 loses energy, while beam 1 gains energy.

The underlying assumption is that the grating remains unchanged when the phase

shift is introduced. Due to the low beam intensities used to write the gratings, this

assumption is valid, since the rate of change of the phase occurs over a time scale

much shorter than the writing time for the grating (approximately 3000s), and the

grating cannot respond quickly enough. In this event, the grating acts like a

quasi-permanent structure and the intensity of the two fields depends on the

phase difference between the interference pattern and the grating. If the phase

difference between the interference pattern and grating changes by 180 degrees,

the direction of the energy flow reverses. As shown in figure 2.5, without

compensation for the translational motion of mirrors, thermal changes and

fluctuations can result in phase shifts that are in excess of 180 degrees. Therefore

it is likely that the apparent reversal of energy flow direction seen in figure 2.8 is

due to phase fluctuations that occur too rapidly for the grating to respond.

38

To reduce the effect of phase shifts introduced by mirror motion, the Sagnac

arrangement in figure 2.7 was used to write the grating. Over a number of trials,

the intensities of the two fields were found to evolve in a uniform manner, with

the direction of energy flow always in the expected direction. A typical example

is shown in figure 2.9.

Figure 2.9 - The evolution of the intensities of (a) beam 1 and (b) beam 2 when a grating is formed using the modified Sagnac arrangement shown in figure 2.7. A 180 degree phase shift was introduced into beam 2 after 1500 seconds and removed after another 260 seconds.

Such uniform evolution of the intensities of the two fields implies that the

fluctuations in the phase difference between the interference pattern and the

grating is considerably less than 180 degrees during the entire 2000 second

grating fabrication time.

As indicated in figure 2.9, the amplitude of the noise of both beams grows

linearly as the grating evolves. This is thought to arise from residual phase jitter

between the interference pattern and the grating. Some residual phase fluctuation

is expected since the Sagnac optical arrangement cannot compensate for

rotational motion of the mirrors. Also air turbulence, changes in air pressure and

temperature can also produce phase fluctuations since the two beams are

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 500 1000 1500 2000

Time (s)

Inte

nsity

(mW

/cm

2 )

a

b

39

spatially separated. Figure 2.10 shows the theoretically predicted evolution of the

two beams in the presence of a ‘randomly occurring’ phase change of 22º

between the interference pattern and grating.

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 500 1000 1500 2000

Time (s)

Inte

nsity

(mW

/cm

2 )

a

b

Figure 2.10 - The theoretically predicted evolution of the intensities of (a) beam

1 and (b) beam 2 assuming a triangular wave variation in the phase of beam 2

with an amplitude of 22 degrees.

For simplicity this noise was modelled by assuming that the phase difference

between the interference pattern and the grating varied as a triangular wave

function of time. Despite this over simplification, there is some similarity with

the evolution of the noise actually observed in figure 2.9.

Since the use of the Sagnac arrangement allowed the reproducible production of

gratings written with low intensity light, it can be used to test the response of the

system to phase changes that occurred at rates much faster than the grating

writing time. For this investigation, an instantaneous phase shift of 180 degrees

40

was introduced to one of the beams by inserting a thin piece of glass into the

beam path after the grating had formed sufficiently.

As the data in figure 2.9 shows, a grating was allowed to form with a flow of

energy from beam 1 to beam 2. After 1500 seconds the 180 degree phase shift

was introduced causing the energy flow direction to reverse with beam 1

receiving energy from beam 2. After another 260 seconds, the phase shift was

removed and the grating was observed to continue growing from where it had

previously left off.

It is interesting to note that the residual noise of the intensity of both beams was

reduced when the phase shift was introduced, but then increased to its previous

level when the phase shift was removed. These observations will be the subject

of further investigations. Future work will also focus on applying this phase

dependence of the energy flow direction for optical switching.

2.5 Conclusion This chapter described a modified Sagnac interferometer that produces stable low

intensity interference patterns for grating fabrication in photosensitive media like

Fe:LiNbO3. The modified Sagnac interferometer described here has been shown

to produce interference patterns with a fringe drift less than 45 degrees over

timescales in excess of 1000 seconds without requiring active stabilisation. The

excellent stability of this arrangement makes it suitable for fabrication of

holographic gratings that require a long exposure time as is the case when

gratings have large dimensions or the intensity of the writing fields is low.

Grating formation in photorefractive Fe: LiNbO3 was affected by phase

variations in the two writing fields. If the phase shift occurred too rapidly for the

grating to respond, the amount of energy diffracted into the two beam directions

changes. It was shown that a rapidly introduced phase shift of 180 degrees can

reverse the direction of energy flow in the two wave mixing process. This was

shown to be particularly problematic with low intensity writing fields since phase

shifts of this magnitude caused by thermal and vibrational motion of the mirrors

can occur too rapidly for the grating to respond.

41

Chapter 3

Modelling of the modified Sagnac interferometer

and comparison of its stability to that of a

conventional interferometer 3.1 Introduction Chapter 2 of this thesis analysed the modified Sagnac interferometer from an

experimental point of view, and came to the conclusion that it provided a

significant improvement in stability over a conventional interferometer in the

scenarios tested. The purpose of this chapter is to model the modified Sagnac

interferometer in an attempt to gain a better understanding of the way in which

this interferometer acts to compensate against the different types of mirror

perturbations expected to be encountered in a laboratory and to directly compare

the stability of this arrangement with that of a conventional interferometer.

The modified Sagnac arrangement described in Chapter 2 and shown in figure

2.7 is comprised of 6 mirrors, which means that there are 12 independent

variables that described any given configuration. There are 6 sets of coordinates

specifying where the mirrors are located and 6 angles that determine the direction

that light is reflected by each mirror. As such, the model to be developed needed

to be able to analyse clockwise and anti-clockwise rotations about the midpoints

of the mirrors, forward and backward translation of the mirrors in the plane

parallel to the face of the mirror, and combinations of rotation and translation.

With these parameters to be investigated, a model of the modified Sagnac was

developed, thus enabling the effect of rotating and translating various mirrors to

be analysed. Research contained in this chapter was conducted in collaboration

with Esa Jaatinen and some of the findings were published in [104]

3.2 Overview of the Model The modelled interferometer is as shown in figure 2.7. The location of the

interference pattern is found by ray tracing both of the beams around the optical

arrangement. It is possible to produce a single equation to take into account the

42

twelve degrees of mirror freedom, but such an equation is unmanageable in

practice. Therefore, for the purpose of this analysis, only mirrors M5 and M6

(the crossing mirrors) and M4 (the mirror diagonally opposite the beam splitter)

were subject to rotation and translation in an effort to simplify the calculations.

Such a situation could be realised in practice if the square part of the arrangement

is fixed inside a block of transparent material to further increase stability,

meaning that only M5 and M6 would be subject to movement. However, the

angle of M4 would need to first be adjusted to separate the beams, and then M5

and M6 could be used to vary the crossing angle of the final two beams.

While the model allowed the mirrors to be rotated around any axis, rotation about

the midpoint was deemed to be better than around a fixed end since this is the

way that most real mirrors are mounted and used in practice. Figure 3.1 (a)

shows rotation about an axis placed symmetrically and figure 3.1 (b) shows

rotation about an asymmetrically placed axis.

Figure 3.1 (a) Shows symmetric location of rotation axis in comparison to beam

intersection points on mirror. (b) Shows asymmetric location of rotation axis in

comparison to beam intersection points on mirror.

In addition, when a new set of mirror positions and angles to be modelled were

set, the model was reset so that the two counter propagating beams incident onto

each mirror struck equal distances either side of the midpoint of the mirrors in an

attempt to made the arrangement symmetric.

Ray A

Ray B

d

d Axis of rotation into page

Ray A

Ray B

2d

(a) (b)

Axis of rotation into page

43

3.2.1 Determination of crossing angle

Since the impetus for developing this system was to cross two beams to produce

stable interference patterns, it is essential to know how the angles of the

individual mirrors affect the crossing angle because the crossing angle

determines the periodicity of the interference pattern. Simple trigonometry

shows that the expression for the crossing angle between ray 1 and ray 2 is given

by:

α = )180654321(4 −+−+−−× MMMMMM (3.1)

(where all angles are expressed in degrees, and are measured with respect to the

positive x axis)

To remain consistent with the experimental conditions described in Chapter 2,

this model assumes mirrors M2 and M3, and the beam splitter M1 to be fixed at

45 degrees.

Under such conditions, the above equation can be simplified to:

α = )225654(4 −+−× MMM (3.2)

Once this relationship had been established, the objective was to find the most

stable arrangement for a given crossing angle that is experimentally practical.

However, it was first necessary to determine the path length difference (PLD),

and thus the phase difference, between the two rays when they intersected.

3.2.2 Determination of path length difference

The modified Sagnac arrangement was analysed by determining the shift in the

interference pattern that occurs when the mirrors (M1 – M6) are perturbed from

their initial values. This involves ray tracing the paths of the two counter-

propagating beams (rays A and B) around the assembly from the beam splitter to

the crossing point and comparing the position of the interference pattern to that

found before the perturbations were introduced.

44

The interference pattern can be characterised as:

)4.3()2cos(0 yIIΛ

=π

Where I0 is the intensity of the interference pattern at point O in figure 3.2, I is

the intensity of the interference pattern at point N, y is the distance in the y-

direction from point O to point N, and Λ is the periodicity of the interference

pattern and is given by:

⎥⎦⎤

⎢⎣⎡×

=Λ

2Sin2 αλ (3.5)

whereλ is the wavelength of the interfering beams and α is the crossing angle of

the two interfering beams that form the grating..

When a mirror (M4, M5, or M6) is rotated or translated, the point at which ray A

(the ray ultimately reflected by M6) in figure 2.7 and ray B (the ray ultimately

reflected by M5) in figure 2.7 intersect shifts. In order to produce a stable

interference pattern, it is vital to establish whether or not the phase of the

interference pattern at point O has changed.

Figure 3.2 illustrates how the intersection point of ray A and ray B shifts when

the arrangement is perturbed and shows the link between the phase of the

wavefronts of the two rays and the stability of the interference pattern.

Prior to any mirror perturbation, the two beams are in phase at point O, since the

lengths of ray A and ray B are equal. However, after a mirror has been

perturbed, the rays will be not necessarily be in phase at the new crossing point N

since the overall path length of ray A will not be equal to the path length of ray

B. But, while the crossing point has shifted, the resulting interference pattern

may still be in phase with the original pre-perturbed interference pattern if point

IA on the wavefront of ray A and point IB on the wavefront of ray B and in

phase with each other.

45

Figure 3.2 An illustration of how the intersection point shifts when a mirror is

rotated. Point O is the original crossing point of ray A and ray B, while N is the

crossing point of the two rays after the system is perturbed in some way.

In this event, the phase of the interference at N is given by:

)6.3()()(21 IBNDISTIANDIST −−−=

λπϕ

This phase value is then equal to the phase of the original interference pattern

when:

⎟⎠⎞

⎜⎝⎛=

Λ=

2sin22.2.2

αλππφ yy (3.7)

Therefore, combining (3.6) and (3.7), we get:

)9.3()()(2

sin.2y

and)8.3(21

IBNDISTIANDIST −−−=⎟⎠⎞

⎜⎝⎛

=

α

φφ

The aforementioned method was then employed to observe the effect that

rotation and translation of various mirrors had on the path length difference

between two rays crossed using the modified Sagnac interferometer, with larger

Y

O

αN

IA

IB

Ray A

Ray B

46

PLDs indicative of a large shift of the interference pattern in response to the

mirror perturbation.

3.3 Effect of Mirror Rotation on PLD for the Modified Sagnac

Arrangement First, the effect of rotating the mirrors independently without translation was

analysed. M5 and M6 were set at 42 and 138 degrees respectively, and M4 was

set at 133 degrees. This resulted in a crossing angle of 16 degrees. The angle of

M5 was then increased, with the PLD being calculated for angles of M5 ranging

from 42 to 45. Additional trials were then conducted with the same initial values

(42 degrees for M5 and 138 degrees for M6), except that the value of M4 was

changed to 133.5, 134, and 134.9 degrees, before the angle of M5 was changed.

The PLD was calculated as a function of the angle of M5 for the various values

of M4 to observe the effect of setting M4 at different angles and is shown in

figure 3.3. Understanding the effect of the angle of M4 on the stability of the

arrangement is important, especially if the square section of the arrangement is to

be fixed inside a block of material for additional stability.

It was found that the path length difference reduces dramatically as the angle of

M4 approaches 135°. The sensitivity of the system was monitored to observe the

effect of rotational perturbations to M5 or M6 when M4 was varied. A function

was fitted to the results and the sensitivity of the systemwas found to be

proportional to )45tan(

1−γ

, where γ is the angle of M4 in degrees. This is

displayed in figure 3.3, as the gradient of the plots of PLD vs angle of M6

decrease at a non-linear rate as the angle of M4 is increased.

47

-0.015

-0.005

0.005

0.015

41.75 42.25 42.75 43.25 43.75 44.25 44.75 45.25

Angle of M5 (Degrees)

PLD

(cm

)

(a)

(b)

(c)

(d)

Figure 3.3 – The effect of setting the angle of M4 to (a) 133 degrees, (b) 133.5

degrees, (c) 134 degrees, and (d) 134.9 degrees before rotating M5

However, setting M4 at an angle close to 135 degrees is impractical in

experimental situations because the angle of M4 determines the angles at which

the two beams propagate as they leave the Sagnac. If the angle of propagation

with respect to the x-axis is too small, the distance in the x-direction between M4

and the mirror pair M5/M6 needs to be large to ensure sufficient separation

between the beams to allow the beams to be crossed. This may be difficult to

achieve if there is limited space. For the previously mentioned dimensions of the

loop, when M4 = 134.9 degrees, the beam separation at the beam splitter is

approximately 0.2cm, compared with approximately 2.7cm separation if M4 =

133 degrees.

Due to the symmetry of the arrangement, if M5 and M6 are set at supplementary

angles (i.e. the angles add up to 180 degrees), the crossing point will always fall

on the x-axis, and the PLD will be zero, regardless of the angle of M4. This

initially was believed to be the ideal condition for optimum stability against

mirror perturbation.

48

However, even if M5 and M6 are set at supplementary angles, small rotational

movements will change the PLD. As a result, it is important to set M4, M5, and

M6 at angles that will minimise the effects of any rotational motion.

To investigate this, M4 was fixed, and M5 and M6 were set at supplementary

angles to each other. M6 was then fixed, while the angle of M5 was increased.

As can be seen from figure 3.4, the relationship between the PLD and the angle

of M5 is polynomial, rather than linear. The angle of M5 corresponding to the

single maxima of the curve is the most stable angle for the particular values

selected for M4 and M5. By keeping M4 fixed while changing the initial angles

of M5 and M6 (they remain at supplementary angles), a number of curves can be

plotted. What is interesting is that no matter what the initial angles of M5 and

M6, as long as M4 is kept constant, the curves plateau at the same angle for M6.

Taking the derivatives of the dependence of the PLD on the angle of M5

confirmed that the lines cut the x-axis at M5 = M4-90.

0

0.1

0.2

0.3

34 36 38 40 42 44 46


PLD

(cm

)

(a)

(b)

(c)(d)

Figure 3.4 – The effect of setting M5 and M6 at different sets of supplementary

angles while M4 remains unchanged for the angle of M5 equal to (a) 34 degrees,

(b) 39 degrees, (c) 41 degrees, and (d) 42 degrees

49

The sensitivity of the optical arrangement to the rotation of M5 is given by the

following equation:

θΔΔ

∝PLDySensitivit (3.5)

where PLDΔ is the change in the path length difference due to the rotation of a

mirror and θΔ is the angle through which the mirror is rotated.

If 0=Δ

Δθ

PLD , then the arrangement is in its most stable configuration

As can be seen from figure 3.5, the arrangement is most stable from the effects of

the rotation of mirror M5 when M5=M4-90, while the angle of M6 is virtually

independent of the angle of M5. Figure 3.5 illustrates the case when M4 = 133.5

degrees and M5 is varied while M5+M6 = 180 degrees meaning, as can be seen

from the point at which the lines cut the x-axis, that the greatest stability is

achieved when M5 = 43.5 degrees.

-0.03

0

0.03

0.06

0.09

34 36 38 40 42 44 46


Rat

e of

cha

nge

of P

LD (a)

(b)(c)

(d)

Figure 3.5 – The derivatives of the curves M5 equal to (a) 34 degrees, (b) 39

degrees, (c) 41 degrees, and (d) 42 degrees as shown in figure 3.4

50

Only a slight increase in stability is observed when the angle of M6 increased

towards 180 degrees. The fact that the angle at which M6 should be set is not

dependent on the angle of M5 gives a tremendous degree of flexibility when

choosing the crossing angle, and suggests that M6 should be the mirror that could

be used to control the crossing angle. It is also evident that the stability increases

as M5 approaches 45 degrees (which means that M4 is approaching 135

degrees), but as has been previously stated, an angle of M4 that is close to 135

degrees is highly impractical because it dramatically limits the range of crossing

angles. It should also be highlighted that the roles of M5 and M6 are

interchangeable because, as indicated by figure 3.5, the sensitivity of the system

to rotational mirror movement is only dependent on the angle of M4. Thus the

interferometer is insensitive to rotational perturbation of M6 when M6 is set to

M6=270-M4 degrees.

3.4 Effect of mirror translation on PLD for the modified Sagnac

arrangement The effect of pure mirror translation without rotation on the PLD was analysed

under different sets of conditions. Here the results of three different scenarios

are discussed.

In the first case (case 1 shown in figure 3.6), the arrangement was symmetric, yet

not optimised against rotational motion. The angle of M4 was varied, M5 was

set at angles ≠ M4-90 degrees, and M6 was set at angles of 180-M5 degrees. In

this case, the angle of M5 was 42 degrees and the angle of M6 was 138 degrees.

In the second case (case 2 is shown in figure 3.7), the arrangement was

symmetric and optimised against rotational motion. The angle of M4 was varied,

M5 was set at angles of M4-90 degrees, and the angle of M6 was set at angles of

180-M5 degrees. In the third case (case 3 is shown in figure3.8), the

arrangement was asymmetric and not optimised against rotational motion. The

angle of M4 was varied, the angle of M5 was set at angles ≠ M4-90 degrees, and

the angle of M6 was set at angles ≠180-M5 degrees. In this case, the angle of

M6 remained constant at 138 degrees.

51

The motivation behind the selection of these particular sets of conditions was the

desire to analyse the system under what were thought to be semi-ideal (case 1),

ideal (case 2), and less-than-ideal (case 3) conditions, based on the

configurations that produced the maximum stability against rotational

perturbations.

0

0.1

0.2

132.9 133.3 133.7 134.1Angle of M4 (Degrees)

PLD

(per

uni

t len

gth)

(a)

(b) (c)

Figure 3.6 The effect of the relationship between mirror angles on system

stability for individual translations of (a) M4, (b) M5, (c) M6 when the modified

Sagnac interferometer was configured with the parameters described in case 1

52

-0.04

0

0.04

0.08

0.12


PLD

(per

uni

t len

gth) (a)

(b) (c)



Sagnac interferometer was configured with the parameters described in case 2

0

0.04

0.08

0.12


PLD

(per

uni

t len

gth)

(a)

(b)

(c)



Sagnac interferometer was configured with the parameters described in case 3.

53

An interesting observation is that when M4 was translated, the PLD was

dependent on the angle of M4 only. That is, for instance, if M4=133 degrees, the

PLD for a given translation is independent of the angles of M5 and M6, and also

independent of the angular relationship between those two mirrors. In all three

cases, the arrangement became less sensitive to the translation of M4 as M4

approached 135 degrees. This trend of increased stability was similar to that

observed when the mirrors were rotated while M4 was made closer to 135

degrees.

When M4 was varied between 133 and 134 degrees, it was observed that case 2

was always the most stable configuration, followed by case 1, and then case 3.

This supports the earlier finding that it is more advantageous to set M5 = M4-90,

rather than M5+M6=180 when the minimisation of the effects of translational

perturbations is desired.

In case 3, for different values of M4, it can be seen that translating M5 will result

in a different PLD than translating M6. This is because M5+M6≠180, and the

arrangement is asymmetric as a result. When the arrangement is symmetric, the

translation of either M5 or M6 will result in the same PLD.

3.5 Comparison of interferometers After analysing the intricacies of both interferometers, a comparison of the two

stabilising arrangements was made. In order for the comparison to be valid, the

two models had to be normalised. The model of the conventional interferometer

was similar in appearance to “beam-crossing section” in the model of the

modified Sagnac interferometer, but with one important difference. In order to

properly replicate the effects of a conventional interferometer, a ray was not

reflected onto M5 by M6, nor from M6 onto M5. This was achieved by

effectively placing the source of laser radiation between M5 and M6, and having

only a single reflection between the source and the point at which the

interference pattern was produced. The angles at which the beams propagated

from the source to the respective mirrors were able to controlled, as were the

angles of M5 and M6. Thus, it was possible to set the initial conditions of the

54

conventional interferometer to have the same angles for M5 and M6, the same

crossing angle and same crossing point as the modified Sagnac interferometer,

meaning that a direct comparison of the two systems could be made.

Previously in this chapter, the modified Sagnac arrangement was modelled for

cases of large perturbations in order to observe the general characteristics of the

interferometer. However, when making a direct comparison between the two

interferometers, the perturbations were limited in size to those typically expected

in a laboratory when using kinematic mirror mounts. For one such mirror, the

translational motion was estimated to be of the order of 1 μm and rotational

motion estimated to be approximately 10’ arc sec.

Equation (3.6a) gives the sensitivity of the interference pattern to a 1 μm

translation perturbation of mirror M1 while equation (3.6b) gives the sensitivity

to a 10’ arc sec rotation of M1.

)(-m) 1( 111 xxTM φμφδφ += (3.6a)

)(-arcsec) '10( 111 θφθφδφ +=RM (3.6b)

Here x1 represents the coordinates of mirror M1 and φ is the phase of the

interference pattern. In evaluating equations (3.6a) and (3.6b) all other mirror

angles and positions are kept constant. Similar expressions were used for

calculating the sensitivity of the interference pattern to rotational and

translational perturbations of the other 5 mirrors in the arrangement.

Assuming that the motion of any single mirror is uncorrelated with the motion of

the other mirrors in the arrangement, the overall phase fluctuation of the

interference pattern due to translational and rotational mirror motion can be

written in the form of equation (3.7a) and (3.7b) respectively:

26

25

24

23

22

21 TMTMTMTMTMTMT δφδφδφδφδφδφφ +++++=Δ (3.7a)

2

62

52

42

32

22

1 RMRMRMRMRMRMR δφδφδφδφδφδφφ +++++=Δ (3.7b)

55

3.5.1 Stability against rotational motion with mirror rotation axes

symmetrically placed

To test the stability against rotational mirror motion, M4 was set at 133 degrees,

M5 was set at 138 degrees, M6 was set at 42 degrees, and M1, M2 and M3 were

set at 45 degrees. The crossing angle produced by the arrangement was changed

by decreasing the angle of M6. For each crossing angle, the sensitivity of the

system was evaluated for 10’ arc sec perturbations to M4, M5, and M6. The

model for the conventional interferometer was then set with the same parameters

for M5 (42 degrees) and M6 (138 degrees) so as to produce the same crossing

angle and crossing point as the modified Sagnac arrangement. As above, the

crossing angle was varied by adjusting the angle of M6, and then M5 and M6

were subjected to a rotational perturbation of 10’ arc sec to evaluate the

sensitivity.

Given that ΔφRCon is the phase fluctuation in the interference pattern due to

rotation in the conventional arrangement, and ΔφR is the phase fluctuation in the

interference pattern due to rotation in the modified Sagnac arrangement, the ratio

of the stability of the modified Sagnac arrangement to the conventional

arrangement against rotational motion (R) can be expressed as:

R

RConRφφΔΔ

= (3.8)

R was then plotted against the crossing angle to show the angular dependence on

R, which is shown in figure 3.9

56

0

20

40

60

80

100

10 12 14 16 18 20

Crossing Angle (Degrees)

R

Figure 3.9 - Ratio of phase jitter in the conventional arrangement to phase jitter

in the modified Sagnac arrangement as a result of mirror rotations of 10’ arc sec

around symmetric axes

It can be seen that the modified Sagnac provides superior stabilisation against

rotational perturbations than did the equivalently configured conventional

arrangement at small crossing angles. At a crossing angle of 12 degrees, the

modified Sagnac arrangement is approximately 80 times more stable than the

conventional setup. Once the crossing angle is increased to 20 degrees by the

rotation of M6, the modified Sagnac arrangement is approximately 15 times

more stable than the conventional arrangement.

3.5.2 Stability against translational motion

Next, the stabilities of the two arrangements against translational motion of the

mirrors were compared. As was the case when comparing the stability against

rotational motion, M5 and M6 were initially set at 42 degrees and 138 degrees

respectively, M4 was set at 133 degrees, and M1, M2, and M3 were set at 45

degrees. In the modified Sagnac configuration, the crossing angle was once

again varied by adjusting M6, and the stability of the arrangement was evaluated

for 1 μm translations of M4, M5, and M6 at these different crossing angles. This

same procedure was applied to the model of the conventional model to test its

sensitivity.

57

Given that ΔφTCon is the phase fluctuation due to translation in the conventional

arrangement, and ΔφT is the phase fluctuation due to translation in the modified

Sagnac arrangement, the ratio of the stability of the modified Sagnac

arrangement to the conventional arrangement against translational motion (T) can

be expressed as:

T

TConTφφΔΔ

= (3.9)

T was then plotted against the crossing angle to show the angular dependence of

T, which is shown in figure 3.10

0

10

20

30

40

50

10 12 14 16 18 20


T


in the modified Sagnac arrangement as a result of mirror translations of 1 μm

As with the plot for R versus crossing angle, it can be seen that the modified

Sagnac arrangement provides superior compensation against translational motion

when the crossing angle is small. As the crossing angle is increased, T decreases

in a non-linear fashion. For a crossing angle of 12 degrees, the modified Sagnac

arrangement is approximately 40 times more stable than a conventional

interferometer, while this drops to an overall improvement of approximately 10

times when the crossing angle is increased to 20 degrees.

58

3.5.3 Stability against rotational motion with mirror rotation axes

asymmetrically placed

Finally, the stabilities of the two arrangements were compared when the

rotational axes of M4, M5, and M6 were asymmetrically placed, as shown in

figure 3.1 (b). As can be seen from the plot of R against the crossing angle

shown in figure 3.11, the modified Sagnac arrangement is approximately 30

times more stable than a conventional interferometer for a crossing angle of 12

degrees, while this drops to an improvement of approximately 10 times when the

crossing angle is increased to 20 degrees.

0

10

20

30

40

50

10 12 14 16 18 20


R


in the modified Sagnac arrangement as a result of mirror rotations of 10’ arc sec

around asymmetric axes

Rotation around such an axis does not result in the “pure” rotation that occurs

when mirrors are rotated about a symmetrically placed axis. As a result, the

stability ratio is lower than for an arrangement with a corresponding set of mirror

angles where the mirrors rotate about symmetrically placed axes. A possible

explanation is that rotating about an asymmetric axis results in simultaneous

rotational and translational perturbations, which combine to increase the

instability.

The PLD for this combination of rotational and translation motion is given by:

59

22TRANSROTTOTAL PLDPLDPLD += (3.10)

where PLDROT is the PLD due to rotational mirror motion and PLDTRANS is the

PLD due to translational mirror motion.

3.6 Conclusions The modelling of both the modified Sagnac interferometer and conventional

interferometer confirms the experimental observations reported in Chapter 2.

Both arrangements were subjected to rotational and translational perturbations of

the order of magnitude of those expected in a laboratory, and the modified

Sagnac arrangement proved to provide greater stability under all of the

circumstances tested.

This chapter analysed the parameters needed to achieve maximum stability and

discussed the way in which the stability degrades as the arrangement moves

away from these optimal conditions. It was shown that the modified Sagnac

arrangement is most stable against the effects of rotation when M6=M4-90.

Improvements in stability of 80 times were predicted for a crossing angle of 12

degrees when the mirrors were rotated about symmetrically placed rotation axes,

with the improvement decreasing with increasing crossing angle. When M5 and

M6 were adjusted so that the axes of rotation were asymmetrically placed, an

improvement in stability of approximately 30 times was able to be achieved for a

crossing angle of 12 degrees, with that improvement decreasing with increasing

crossing angle. Thus, it can be concluded that the rotation axes of the mirrors

should be symmetric with respect to the points at which the two rays strike the

mirrors in order to produce maximum stabilisation against rotational

perturbations. The modified Sagnac interferometer was also 40 times more

stable than the conventional interferometer against a translation of 1 μm when

the crossing angle was 12 degrees and, as in the other cases, the improvement

decreases with increasing crossing angle.

60

Chapter 4

The experimental observation of scattering in the

Extremely Asymmetric Scattering geometry 4.1 Introduction Although there has been extensive research conducted into scattering in the

extremely asymmetric scattering (EAS) geometries with grazing incident or

scattered wave angles [23, 24, 27, 28, 35], experimental investigations have been

confined to the scattering of x-rays at grazing angles of incidence to a crystal

face. Research has shown that a monochromatic X-ray beam can scatter from the

same Bragg plane in a form of extremely asymmetric diffraction [106].

However, the experimental observation of EAS in the optical range of

wavelengths is still yet to be observed. The optical wavelength range is of

interest because of the potential applications of this type of scattering. Possible

applications may include EAS-based resonators, high sensitivity sensors and

measurement techniques, narrow-band optical filters, couplers, switches, and

lasers.

While theoretical analysis of optical wave scattering in the EAS geometry has

been undertaken, the parameters for a grating such as its periodicity, length and

orientation that will produce the characteristic asymmetric scattering pattern are

not precisely known. Consequently, it would be unfeasible to produce a range of

permanent gratings with different parameters and chance upon a suitable set of

parameters to observe this characteristic scattering. For this reason, semi-

permanent volume holographic gratings written in a class of materials known as

photorefractives were used. The advantage of using photorefractives is that once

a grating is written and scattering off the grating has been observed, it can be

erased, and a grating with different parameters written. The two materials used

for grating fabrication were barium titanate (BaTiO3) and lithium niobate

(LiNbO3). Since the maximum possible refractive index change of barium

titanate differs by approximately two orders of magnitude from that of lithium

niobate, comparing the scattered wave profiles produced by gratings written in

the two materials can provide an insight into the effect of grating amplitude on

61

the angular divergence of the scattered waves in the EAS geometry. The

modelled profiles in section 4.7.4 of this chapter were produced by Steven

Goodman.

4.2 EAS theory

Figure 4.1 shows wave scattering in the EAS geometry in a holographic grating.

Here the diffracted wave travels in the direction along the boundary between the

grating and the non-grating region.

Figure 4.1 – Scattering in the geometry of EAS. k0 is the wavevector of the

incident wave, k1 is wavevector of the scattered wave, L is the grating width, θ0

is the angle of incidence. Λ is the grating vector, and E00 is the amplitude of the

incident field.

1. Approximate theory

Steady-state approximate theory of EAS gives the solution for the scattered wave

as [24]:

E(x,y,t) = E(x)exp(iky - iωt) (4.1)

where

E(x) = C1exp(ixλ) + C2exp[-xλ(31/2+i)/2]+ C3exp[-xλ(31/2-i)/2] (4.2)

where C1, C2, and C3 are complex constants and

λ = (|ε1|2ω4)1/3(2c4k0cosθ0)-1/3 (4.3)

62

ε1 is the grating amplitude, ω is the frequency, c is the speed of light, k0 is the

incident wavevector in the medium, and θ0 is the angle of incidence of the 0th

diffracted order.

Eq.(4.1) suggests that the scattered wave inside the grating is represented by

three waves with amplitudes C1, C2, and C3 and the wave vectors (λ,k1,0), (-

λ,k0,0), and (-λ/2,k0,0) respectively. Therefore, the wavevectors of the scattered

wave inside the grating are not equal to k1 (as predicted by the Bragg condition –

figure 4.1), but rather have small additions λ and -λ/2 as shown in figure 4.2.

Figure 4.2 – wavevectors of the three waves from the grating

4.3 The photorefractive materials

4.3.1 Lithium niobate

The LiNbO3 had dimensions of 10mm x 8mm x 5mm, with the optic axis

running the length of the longest side of the crystal. The front and rear faces of

the crystal were the only surfaces that were polished (but uncoated), and had a

reflectivity of 17%.

The geometry used to write volume holographic gratings in this particular

LiNbO3 crystal is shown in figure 4.3. Since only two faces of the crystal were

polished, it was necessary to have both beams enter the crystal through the same

face, and then exit through the opposite face. Therefore, in order to achieve co-

directional two-wave mixing, the crossing angle between the beams had to be

63

less than 90 degrees in air. Crossing angles of 60 degrees were achievable which

corresponds to a beam crossing angle of 24 degrees in the crystal.

Figure 4.3 – the geometry used to write a grating in LiNbO3

4.3.2 Barium titanate

The barium titanate crystal had dimensions of 5mm x 5mm x 5mm. Its refractive

indices were no = 2.416 and ne = 2.364, and the reflectivity of an end face was

17%. The crystal had six polished faces, with the optic axis running from one

polished face to the other along the length of the crystal, and normal to the other

two polished faces. As a result, it was possible to interfere the two beams by

passing one beam through the face parallel to the optic axis, and the other beam

through the face normal to the optic axis. The geometry used to write volume

holographic gratings in this particular BaTiO3 crystal is shown in figure 4.4.

Rotating the crystal in this geometry resulted in only slight changes in the

coupling coefficient, which in turn altered the grating amplitude. This is because

a change in the angle in air converges to a much smaller change in angle inside

the crystal due to the large refractive index.

64

Figure 4.4 – the geometry used to write a grating in BaTiO3

4.4 Experimental rationale As has been discussed previously, a lower power (2mW) 633nm He:Ne laser was

used in the initial experiments with the BaTiO3. With such low powers at

633nm, photorefractive gratings could take in excess of 15minutes to fabricate.

From these trials it was determined that a stabilisation system was needed at this

wavelength to maintain a stable interference pattern, so one was developed, as

described in chapters 2 and 3. However, it soon became apparent that the

frequency doubled Nd:YAG laser was a more suitable source due to the larger

photorefractive response of BaTiO3 at 532nm. This allowed large amplitude

gratings to be formed over much shorter timescales (in seconds) than could be

achieved at 633nm. Therefore, it was found that a conventional optical setup as

shown in figure 4.5 could be used to fabricate photorefractive gratings at 532nm.

The 532nm Nd:YAG laser, operating with an output of approximately 10mW,

was used to write the grating in the LiNbO3 crystal. A beam splitting cube was

inserted to allow the beams from the 633 nm and 532 nm lasers to co-propagate

and follow exactly the same path. It was intended that the arrangement would be

able to write gratings with either wavelength of light while keeping the geometry

identical in both cases. However, in practice, a number of deficiencies were

found in this setup.

65

Firstly, the He:Ne laser had a low power output, and putting the beamsplitting

cube in place reduced the power by an additional 50%. Since the photorefractive

effect is intensity-dependent, this translated into a longer time required for

grating formation. In addition, the nonlinear response of photorefractive

materials is wavelength-dependent, so light with a wavelength of 532nm

produces a stronger response than light with a wavelength of 632.8nm [71].

Figure 4.5 illustrates the experimental setup that was used to write volume

holographic gratings in photorefractive materials.

Figure 4.5 – the experimental setup, where L are lenses, M are mirrors, BS is a

beam splitter, BE is a beam expander, and CCD is a CCD camera

The theory suggests that for EAS to be observed, the grating must be thin, and

have sufficient length with respect to the width of the grating. As the LiNbO3

crystal has a length of 5mm, theoretical analysis of EAS shows that a grating

BLOCK

CCD

BS

M

M LL

532nm Nd:YAG

BE

LiNbO3

66

width of 100μm is required. Lenses with focal lengths of 5mm and 300mm were

used to focus the beam down to the desired diameter, and a cylindrical lens was

used to expand one of the beams to a width of approximately 3mm.

A pellicle beam splitter was initially used in an effort to prevent beam ghosting.

Ghosting occurs when a beam is reflected off both the front and rear face of a

beam splitter, and can degrade image quality. As the profiles of the scattered

waves were to be analysed, it was essential to get as clear an image as possible.

However, the use of the pellicle beam splitter turned out to be impractical

because the airflow in the laboratory made the ultra-thin membrane vibrate,

thereby dramatically degrading the stability of the interference pattern. As a

result, it was replaced with regular glass beam splitter. The use of the thicker

beam splitter appeared to have no detrimental effect on the beam quality.

After a profile of the scattered beam had been captured, the crystal was raised or

lowered to find a “clean” area in which to write the next grating. This allowed

many gratings to be written in the one crystal. Holographic gratings are semi-

permanent in LiNbO3, meaning that areas in which gratings have been written

cannot be reused immediately. If the crystal is rotated slightly, the beams will no

longer satisfy the Bragg condition for the previous grating, so Bragg scattering

will not occur off the old grating. However, a refractive index change will still

be present in that region of the material, and will act to scatter light in random

directions. As has previously been noted, such scatter can result in undesirable

effects such as beam fanning. It is for this reason that care was taken to write

new gratings in regions that were free from grating remnants. The crystal was

exposed to light from a pair of halogen lamps overnight to erase the gratings and

return it to its original state.

67

4.5 Image analysis The beam profiles of the scattered waves that emerged from the crystal were

captured with an Electrim EDC-1000 CCD camera. The array was 4.84mm x

3.67mm in size, with a non-interlaced spatial resolution of 652x494. The

relevant specifications for the camera are provided in table 4.1. The size of the

array placed a limitation on the distance that the crystal could be placed from the

camera because the scattered waves could diverge significantly from the Bragg

angle, and hence, not get captured. Therefore, the goal was to fill as much of the

array as possible with the scattered wave to obtain the best resolution, while at

the same time ensuring that all of the image data was captured. The camera was

mounted on a translation stage so that the orientation of the camera with the

respect to the crystal remained unchanged if the camera had to be translated.

Although the laser light could be considered fairly weak in intensity

(~15mW/mm2), it was more than enough to saturate the CCD. Exposure of the

CCD to intense light could have caused damage by burning pixels. For this

reason, attenuators were placed in front of the camera to prevent it from being

damaged. Once the intensity of the light was reduced to a suitable level, the

software on the computer was used to adjust the gain and bias of the camera

Camera Model Electrim EDC-1000E

CCD sensing area 4.84mm x 3.67mm

Non-interlaced pixel size 7.4μm x 7.4μm

Non-interlaced spatial resolution 652x494

Exposure time resolution 1ms

Table 4.1 – CCD camera specifications

The beam capture was loaded into specifically developed software to analyse the

image, and the image was displayed on the screen. Figure 4.6 illustrates the

bitmap analyser interface with a beam capture loaded.

68

Figure 4.6 – The interface of the bitmap analyser with an image already loaded

After capture, a cross-sectional profile of the scattered field in the horizontal

direction was taken. The height at which the profile was taken was chosen by

looking for the “clearest” profile that was devoid of artefacts caused by

diffraction or scattering caused by imperfections.

The criteria used to determine an appropriate cross-section was as follows:

1) the cross-section should include no artefacts, as these can give an

inaccurate representation of the intensity

2) the cross-section should be free of saturated pixels, as they distort the

profile

The horizontal line in Figure 4.7 indicates what would be an ideal cross-section,

and the circled regions indicate artefacts that distort the intensity profile.

69

Figure 4.7 – the line indicates the cross-section that was selected for analysis,

and the circles indicate artefacts

This profile was checked to ensure that cross-section selected was free from

saturated pixels. Figure 4.8 below is an example of a profile that was deemed to

be suitable.

Figure 4.8 – the intensity profile of the image in figure 4.6

70

Moving the cursor over the plotted profile showed the height of the peak at that

point, and also indicated the location of that pixel on the image.

Figure 4.9 is an example of a plot of the profile when the image contains

saturated pixels. Instead of containing sharp peaks, the saturated pixels resulted

in peaks with flat tops, which are obviously undesirable.

Figure 4.9 – a non-ideal profile, due to the flat peaks caused by saturation of the

CCD

Once a suitable profile had been selected, the profile was imported in the form of

a data file into Microsoft Excel. In Excel, a moving average over 30 points was

applied to the data to smoothen it.

4.6 Results from experiments conducted with LiNbO3 A beam was expanded to a width of approximately 3mm and interfered with a

thin beam that had a diameter of 100μm at a crossing angle of 60 degrees in

LiNbO3. An intensity profile of the scattered field was obtained and compared

with the profile produced by a model that simulated two overlapping Gaussian

components of the scattered field that had a small angular shift between them.

This angle was varied until a match between the two profiles was found. This

value for the angular separation was substituted into a re-arranged version of

equation (4.3):

71

)5.4(cos2 0

4

4213

θωε

λkc

=

where λ is the component of the scattered wavevector in the x direction (see

figures 4.1 and 4.2), ε1 is the grating amplitude, ω is the frequency, c is the speed

of light, k0 is the wavevector in the medium, and θ0 is the angle of incidence of

the 0th diffracted order.

This can be simplified to

)6.4(cos231 θβε n=

where 0kλβ = is approximately the angle between two of the Gaussian

components of the scattered wave and n is the refractive index of the lithium

niobate crystal.

Since β is smaller than the beam divergence, interference between the two

components occurs. By matching the modelled profile (figure 4.10) with the

profile obtained experimentally (figure 4.11), the angular separation, β, was

determined to be 5mrad. This corresponds to grating amplitude of the order of

2x10-5. As will be seen later, this grating amplitude is approximately two orders

of magnitude smaller than that found for EAS in BaTiO3. This is to be expected

since BaTiO3 has an electro-optic coefficient that is two orders of magnitude

larger than that of LiNbO3.

72

0

0.2

0.4

0.6

0.8

1

1.2

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02

(a) (b)

Figure 4.10 – (a) modelled interaction of two overlapping Gaussian beams that

are slowly diverging from each other (b) modelled profile of a Gaussian beam

0

0.2

0.4

0.6

0.8

1

1.2

-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

(a) (b)

Figure 4.11 – (a) experimentally obtained profile of the interaction between two

overlapping Gaussian beams that are slowly diverging from each other (b)

experimentally obtained profile of a Gaussian beam

73

4.7 Results from experiments conducted with BaTiO3

4.7.1 Grating written with two thin beams

Two beams with diameters of 100μm were interfered in the BaTiO3 crystal. A

grating was allowed to form, and the beam travelling parallel to the optic axis

was then blocked. Figure 4.12 shows a CCD image of the scattered field which

illustrates that when two thin beams were interfered in BaTiO3, the field was

Gaussian in nature at the Bragg angle, with noise spreading out to one side.

Much of the noise present is due to parasitic processes such as beam fanning.

Figure 4.12– CCD image of the field produced at and near the Bragg angle when

a horizontally polarised thin beam was scattered off a grating written with two


Taking a cross-section of the image in Figure 4.12 revealed that the scatter at the

Bragg angle (almost parallel to the optic axis of the crystal) had a deformed

Gaussian shape, as can be seen in Figure 4.13 A moving average of the data was

taken over 30 points to reduce the level of noise. This revealed that in addition

to a large peak at the Bragg angle, there was a weaker peak at an angle of 5 mrad

to the right of the Bragg angle.

74

0

0.2

0.4

0.6

0.8

1

-10 -5 0 5 10 15

Deviation from Bragg angle (mrad)

Rel

ativ

e in

tens

ity

Figure 4.13– Intensity profile of the field produced at and near the Bragg angle

when a horizontally polarised thin beam was scattered off a grating written with

two horizontally polarised thin beams in the EAS geometry

The additional scattering at 5 mrad to the right of the Bragg angle was not

expected, as general two-wave mixing theory suggests that all of the energy is

scattered at the Bragg angle [38]. Consequently, this additional scatter may be

indicative of the conditions for the observation of EAS being met with the

emergence of a scattered wave component with a wavevector not exactly at the

Bragg angle as predicted by the theory. However in this case, because the grating

was not sufficiently long with respect to its width, the amplitudes of the EAS

components were weak.

However, an alternative explanation is that the scatter at 5 mrad to the right of

the Bragg angle and beyond was due solely to beam fanning. Looking at the plot

to the right of 5 mrad, it can be seen that the tail of the peak with a maxima at 5

mrad doesn’t drop away suddenly, as would be expected if the peak was

Gaussian in nature. The tail dropping away relatively slowly indicates that the

scatter was spreading out from the Bragg angle in a manner consistent with beam

75

fanning. Thus it is possible that what was observed is a mixture of beam fanning

and weak EAS.

4.7.2 Two thin beams, with a half-wave plate inserted prior to grating

formation

A half wave plate was inserted after the laser, and two beams with diameters of

100μm were interfered in the BaTiO3 crystal. It was observed that, in addition to

the scatter propagating at the Bragg angle (almost parallel to the optic axis of the

crystal), additional scatter emerged from the crystal at approximately 65 mrad

from the Bragg angle during grating formation. This scatter appeared almost

immediately after the two beams were interfered, and persisted for a short time

after the beam parallel to the optic axis was blocked. In fact, blocking that beam

had no initial effect on the intensity of the second scattered beam. However, a

few seconds after blocking the incident beam, the scattered waves both near and

away from the Bragg angle faded away.

Rather than single spots being observed at the Bragg angle and an additional

large angle of divergence, it was noted that there were two spots at both of these

angles. Both pairs of spots contained one spot that was a number of times more

intense than the other spot, with these spots also being at different heights. It

was difficult to take an intensity profile of the image for two reasons. Firstly,

there was a disparity between the intensity of the bright dot and the dim dot.

Attenuation of the scatter to reduce the intensity of the bright peak made it

difficult to obtain any useful information about the scatter in the dim peak.

Conversely, if the beam was attenuated to a level where the dim scatter could be

registered properly on the CCD camera, the light in the bright peak saturated that

region of the image. As a result, the intensities of the peaks near the Bragg angle

could not be compared with the intensities of the peaks at the large angle of

divergence. However, what can be seen is that the spot at the Bragg angle, as

shown on the left in figure 4.14 is much more intense than the spot slightly to the

right of the Bragg angle. On the other hand, the peaks located between 5mrad

and 65mrad to the left of the Bragg angle had comparable intensities, as shown in

76

figure 4.16. The intensity profiles of the images in Figure 4.14 and 4.16 are

shown as Figure 4.15 and Figure 4.17 respectively.

Figure 4.14 – CCD image of scattering near the Bragg angle for a vertically

polarised thin beam scattered off a grating written with two vertically polarised

thin beams

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Figure 4.15 – Profile of scattering near the Bragg angle for a vertically polarised

thin beam scattered off a grating written with two vertically polarised thin beams

77

Figure 4.16 – Scattering at the wide angle of divergence

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Figure 4.17 – Profile of scattering away from the Bragg angle for a vertically

polarised thin beam scattered off a grating written with two vertically polarised

thin beams

After examination of the arrangement, two possible reasons for scattering at a

large angle of divergence observation were determined. One possible reason was

that the polarisation of one or both of the beams may have changed after being

reflected off the mirror in the vertical plane, resulting in a mixture of

polarisations. This was caused by an undetected rotation in the half-wave plate

because it was loose in its mount. Therefore the incident beam had a large

78

component polarised horizontally. Rotating the half-wave plate enabled the

amount of scatter at the large angle of divergence to be increased or decreased.

When it was rotated further, the scattered wave at the large angle of divergence

returned, thus giving conclusive proof that this additional scattered wave was due

to the diffraction of extraordinarily polarised light from the grating formed by the

interference of two ordinarily polarised beams. Since the refractive index for the

extraordinary ray is different to the ordinary ray, the extraordinary ray effectively

has a different wavelength to the ordinary ray and is scattered at a different

angle.

Furthermore, a theoretical analysis of this scatter predicts that the extra ordinary

component of the incident beam should be scattered at 63 mrad from the Bragg

angle, which is in very good agreement with what was observed (see figure 4.17)

Take the idealised case of one beam travelling parallel to the optic axis and the

other normal to the optic axis, as can be seen in figure 4.18

Figure 4.18 – A beam is scattered off a grating written with two beams of a

different polarisation at an angle of θi, which results in a divergence from the

Bragg angle of θr in air.

C axis

θi

θr

grating

Incident beam

79

The main difference between the idealised case and what occurred

experimentally is that the incident beams were not parallel and normal to the

optic axis, since the crystal was rotated prior to grating formation in an attempt to

reduce the influences of undesirable nonlinear effects. As a result, there were

both no and ne components present, and this is the reason that two spots were

observed at the large angle of divergence.

4.7.3 Two thin beams, with a half-wave plate inserted prior to grating

formation, but removed when the beam was blocked

Once again, a half wave plate was inserted after the laser, and two beams with

diameters of 100μm were interfered in the BaTiO3 crystal. The grating was

allowed to evolve, but the half wave plate was removed when the beam parallel

to the optic axis was blocked. As was previously observed, some energy was

scattered at a large angle of divergence from the Bragg angle as the grating was

written. However, when the beam parallel to the optic axis was blocked,

virtually all of the scattered energy shifted to the large angle of divergence,

which increased the brightness of the spot at that angle. This occurred because

the new polarisation had a different wavelength in the material to the old

polarisation as discussed in section 4.7.2

Below are two examples (figure 4.19 and figure 4.21) of captures of scattered

waves at an angle of 64 mrad from the Bragg angle. The intensity profiles of

these captures are shown as figure 4.20 and figure 4.22 respectively. Note how

the peak on the right can have an intensity almost equal to that of the peak on the

left (figure 4.20), or it can have an amplitude that is significantly smaller (figure

4.22). A possible reason for this is that the grating may have faded in a non-

uniform manner, so some of the light that was incident onto the grating was

scattered more efficiently than other light.

80

Figure 4.19 – Image of scattering away from the Bragg angle when a


vertically polarised thin beams in the EAS geometry

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Figure 4.20 – Profile of scattering away from the Bragg angle when a



81

Figure 4.21 - Image of scattering away from the Bragg angle when a



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Figure 4.22 – Profile of scattering away from the Bragg angle when a



4.7.4 A wide beam and a thin beam in the EAS geometry

The beam propagating approximately normal to the long face of the crystal and

optic axis was expanded to a width of approximately 3mm. This beam was then

interfered with a thin beam that had a diameter of 100μm. The grating was

82

allowed to form until parasitic processes such as beam fanning became

noticeable. Once a grating had formed, the thin beam was blocked, and the wide

beam was scattered off the grating.

The first observation noted was that the scatter from the grating was a lot more

intense than the light scattered off the grating written by two thin beams. This

was to be expected, since there were more reflective elements in the grating

because of the additional length. In a qualitative sense, the observation of three

peaks, with the two outer peaks appearing asymmetrically at either side of the

middle peak, is in keeping with the EAS theory detailed in section 4.2 and

highlighted in figure 4.2.

When the grating formation was stopped early in its evolution, the energy was

concentrated mainly in the peak at the Bragg angle as can be seen in figure 4.23,

and in the intensity profile of this capture which is shown in figure 4.24.

Figure 4.23 – Image of the field produced when a horizontally polarised thin

beam was scattered off a long, thin grating produced by two horizontally

polarised beams after a short evolution time

83

Figure 4.24 – (a) profile of scattering near the Bragg angle when a horizontally


horizontally polarised beams after a short evolution time (b) modelled scattered


However, as the grating was allowed to evolve, it appears as though the energy

spread to the left and depleted the scatter in the direction of the Bragg angle as

can be seen in the captures in figure 4.25 and 4.27, and in its intensity profile

shown in figure 4.26 and 4.28. The rate at which the scatter to the left of the

Bragg angle appeared was related to the angles that the two beams made with the

optic axis. If the two beams travelled normal and parallel to the optic axis, the

parasitic process of self phase conjugation occurred much more quickly.

Therefore, the crystal was rotated slightly to reduce these undesirable non-linear

effects

halla


84

Figure 4.25 - – Image of the field produced when a horizontally polarised thin

beam was scattered off a long, thin grating produced by two horizontally

polarised beams after a moderate evolution time

Figure 4.26 - (a) profile of scattering near the Bragg angle when a horizontally


horizontally polarised beams after a moderate evolution time (b) modelled

scattered wave profile for the conditions described in (a) [107]

halla


85

Figure 4.27 – Scattering off a long, thin grating after the grating was allowed to

evolve fully

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Figure 4.28 – Profile of scattering near the Bragg angle when a horizontally


horizontally polarised beams that was allowed to evolve fully

From the outset, it must be said that there were a number of experimental

limitations that made it impossible to experimentally replicate the conditions

described in the theoretical investigations of EAS. For one, the theoretical

investigations assumed an incident beam of infinite width and a grating of

infinite length. Experimentally, the length of the grating was limited by the

86

length of the photorefractive crystal and the width of the beam was limited both

by the length of the crystal and by the extent to which the incident beam could be

expanded without its intensity decreasing so much as to make the photorefractive

process extremely slow.

As EAS is a resonant process, the time taken to reach steady state will be

influenced by the length over which the scattering occurs and the coupling

strength between the incident and scattered waves. The experimental parameters

used suggest that only non-steady state EAS can be achieved because the light

scattered by the grating appears to have been insufficient to achieve a steady

state resonance.

In conjunction with this course of research, a model was developed to enable the

theoretical investigation of EAS under the specific parameters used in the

experiment [107]. It was hoped that this model would allow the direct

comparison of experimental results with the predicted field profile for a given set

of parameters. The field inside the grating due to EAS was first calculated, and

then the field at the CCD camera as a result of the diffraction of the field as it

propagated from the crystal boundary to the CCD camera was determined.

The field inside the grating was calculated using the non-steady state theory of

EAS in a uniform grating and the field was then decomposed into a series of

plane waves via Fourier analysis. Another transform was used to recombine

these plane waves at the CCD camera in order to produce a profile of the electric

field, and therefore the intensity profile of the scattered wave could be

determined. The two incident beams were assumed to have rectangular intensity

profiles and the mean permittivity and amplitude of the written grating were

assumed to be uniform throughout the photorefractive material.

In these simulations, all grating parameters except for the grating amplitude

remained constant to reflect the variable parameter in the experiment.

The intensity profile shown in figure 4.24 features three peaks, which is in

agreement with the theory. Peak A is a result of the two waves that travel to the

87

left, peak B is a result of the field outside of the grating, and peak C is a result of

the wave that travels to the right. Peak C has the largest amplitude and is located

at the Bragg angle, and peaks A and C are located to the left of the Bragg angle.

The spread of the peaks is consistent with EAS theory, as peaks A and C fall

asymmetrically to either side of peak B. However, the fact that the largest peak

falls at the Bragg angle is consistent with two-wave mixing theory. It is believed

that the observed number and separation of peaks is due to the scatter being

produced by a long grating, since the three peaks weren’t produced when two

thin beams wrote the grating. It is also believed that a significant peak is

observed at the Bragg angle because EAS is still not the clearly dominant effect

at low grating amplitudes.

The modelled intensity profile in figure 4.24 (b) [107] bears a striking

resemblance to the intensity profile of the scattered field obtained when light was

scattered off a long, thin grating that wasn’t allowed to fully evolve. It must be

noted that the profile in figure 4.24 (b) was shifted horizontally so that the largest

peak angle was aligned with the peak at the Bragg angle in profile (a). This is

because the largest peak in the theoretical profile did not correspond with the

Bragg angle, which is clearly different to what was observed experimentally. So,

while the absolute angles at which the beams propagated were different, the

relative angles of the peaks with respect to each other match closely. This gives

a strong indication that EAS scattering has been experimentally observed in the

optical frequency range, although at this stage the reason for the discrepancy

cannot be ascertained.

The intensity profile shown in figure 4.28 was produced by a grating that was

allowed to evolve for a longer period of time than the grating that produced the

field in figure 4.24. It is evident that the profile now features four peaks, rather

than three. Although this isn’t in agreement with the theory described in section

4.2, it has been found that the number of peaks in the scattered field can increase

with increased grating amplitude [107]. It can be seen that the middle peak from

the previously analysed profile has become two peaks and this region has spread

out, resulting in the peaks on the left and right separating. Comparing the

experimentally obtained profile with the modelled profile [107] shows that the

88

relative angles of the peaks with respect to each other are once again very

similar, thus indicating that the observed pattern of scatter is consistent with

EAS. However, once again, the profile in figure 4.26 (b) had to be shifted

horizontally in order to match the largest peaks. Given that the largest peak was

once again observed experimentally to appear at the Bragg angle, it suggests that

the theory might not be taking into account some important aspect of the

scattering process.

The intensity profile shown in figure 4.28 was produced by a grating that was

allowed to evolve for a longer period of time than the grating that produced the

field in figure 4.28. The profile now contains five peaks and there has clearly

been an energy shift away from the Bragg angle. The largest peak is found at

10mrad to the left of the Bragg angle and there is a separation of approximately

15mrad between the peak on the left and the one on the right. This is

approximately twice the separation as is observed in the profile from figure 4.24,

which clearly demonstrates that the energy spreads as the grating amplitude

increases. The evidence here suggests that as the grating amplitude increases,

EAS overtakes the Bragg scattering associated with two-wave mixing to become

the dominant form of wave scatter, which is supported by the fact that energy at

the Bragg angle is depleted and a new peak formed to the left of the Bragg angle

is observed to have formed approximately 10mrad to the left of the Bragg angle.

Conclusions In this chapter, wave scattering in the EAS geometry was investigated in gratings

with a variety of different parameters

Two-wave mixing was initially performed using two horizontally polarised thin

beams in order to produce a baseline with which scattering in a long, thin grating

could be compared. The profile of the scattered field agreed with two-wave

mixing theory, with the peak to the right of the Bragg angle attributed to beam

fanning.

Next, a half-wave plate was used to alter the polarisation of the two thin beams

89

used to write the grating. Scattering was observed at a large angle of divergence,

in addition to regular scattering at the Bragg angle, regardless of whether or not

the half-wave plate was in place before light was scattered off the grating. It was

concluded that this large angular deviation is a result of a mixture of

polarisations being present in the incident wave, which means that some

components of the wave didn’t satisfy the Bragg condition of the grating and

were therefore scattered off the grating at some angle other than the Bragg angle.

Finally, the first experimental observations of extremely asymmetrical scattering

in volume holographic gratings written in the photorefractive materials lithium

niobate and barium titanate were made. A scattered field profile comprised of

three peaks was observed at and near the Bragg angle when the grating amplitude

was relatively small and this field expanded to four and then five peaks as the

grating amplitude increased.

These experimentally obtained profiles were compared with the profiles

generated by a theoretical model that was based on the experimental parameters.

Despite the model making assumptions such as no absorption, infinite plane

waves and uniform grating strength, there was remarkable agreement between

the relative distribution of the peaks for the experimental and modelled intensity

profiles.

90

Chapter 5

Conclusions This thesis has investigated both a new method of passive stabilisation that

allows the production of stable interference patterns on a timescale in excess of

1000 seconds and also optical wave scattering in thin volume holographic

gratings written in photorefractive materials.

A modified Sagnac interferometer has been shown to produce interference

patterns with a fringe drift less than 45 degrees over timescales in excess of 1000

seconds without requiring active stabilisation. Due to the excellent stability

provided by this arrangement, it is suitable for applications such as the

fabrication of holographic gratings that require a long exposure time, as is the

case when gratings have large dimensions or the intensity of the writing fields is

low.

This modified interferometer was also modelled and compared with a modelled

conventional interferometer in order to ascertain the increase in stability that it

provides. It was observed that the stability decreased when the crossing angle

was increased by adjusting the angles of M5 and M6, the mirrors responsible for

ultimately interfering the two beams. Improvements in stability of 80 times were

achieved for a crossing angle of 12 degrees when the mirrors were rotated about

symmetrically placed rotation axes and approximately 30 times when M5 and

M6 were adjusted so that the axes of rotation were asymmetrically placed. Thus,

it can be concluded that the axes of rotation of the mirrors should be symmetric

with respect to the points at which the two rays strike the mirrors in order to

produce maximum stabilisation against rotational perturbations. The modified

Sagnac interferometer was also 40 times more stable than the conventional

interferometer against a translation of 1 μm when the crossing angle was 12

degrees and the stability degraded with increased crossing angle.

The parameters needed to achieve maximum stability were also identified and

the way in which the stability degrades as the arrangement moves away from

these optimal conditions was discussed. It was shown that the modified Sagnac

91

arrangement is most stable against the effects of rotation when M6=M4-90

degrees

Grating formation in photorefractive Fe: LiNbO3 was affected by variations in

phase of the two writing fields. If this phase shift occurred too rapidly for the

grating to respond, the amount of energy diffracted into the two beam directions

changes. It was shown that a rapidly introduced phase shift of 180 degrees can

reverse the direction of energy flow in the two wave mixing process. This was

shown to be particularly problematic with low intensity writing fields since phase

shifts of this magnitude caused by thermal and vibrational motion of the mirrors

can occur too rapidly for the grating to respond. Further investigation of this

reversal in the direction of energy flow due to rapid phase change may be of

interest as it could lead to the development of optical switches.

In addition, investigations were undertaken into the effect of changing the

polarisation of two beams used to write a grating in barium titanate through the

insertion of a half-wave plate. Scattering was observed at a large angle of

divergence, in addition to regular scattering at the Bragg angle. This additional

scattering was observed regardless of whether or not the half-wave plate was in

place before light was scattered off the grating. The presence of the half-wave

plate did, however, lead to an increase in the amount of energy scattered at this

large angle of divergence. It was concluded that this large angular deviation is a

result of a mixture of polarisations being present in the incident wave, which

means that some components of the wave didn’t satisfy the Bragg condition of

the grating and were therefore scattered off the grating at some angle other than

the Bragg angle. By rotating the crystal and measuring the angle at which the

scattered wave appears, it may be possible to plot the index ellipsoid of the

crystal.

Finally, experiments were conducted using parameters that were believed to

result in the scattering described by EAS theory. When lithium niobate was

employed to produce gratings, there was a small angle between the scattered

components. The experimentally obtained profile was found to be an excellent

match with a modelled profile produced by simulating two overlapping Gaussian

92

components of the scattered field that had a small angular shift between them.

The angular separation was determined to be 5mrad, which corresponds to a

grating amplitude of 2x10-5.

When a wide beam was scattered off a long, thin grating in barium titanate that

had not been allowed to evolve fully, the scattered field featured three peaks as

was predicted by the EAS theory. When compared with a profile generated by a

model developed to analyse EAS in photorefractive gratings, the relative angular

separation between the peaks was in excellent agreement.

When a wide beam was scattered off a long, thin grating that had been allowed to

evolve more substantially, the scattered field featured four peaks, which wasn’t

predicted by the original EAS theory. However, collaborative research has

shown that scattered wave fields with more than three peaks are indeed possible,

since the number of peaks appears to increase with increased grating amplitude.

This relative angular separation between the peaks in the experimental and

modelled profiles was also in excellent agreement.

Any further research would have to include the implementation of a simple

method by which to determine the grating amplitude at any given time. It would

also be desirable to write a program to periodically capture images to allow

grating evolution to be observed. In addition, it would also be of interest to write

gratings in photorefractive materials of greater length in order to try to observe

EAS of a more steady-state variety.

93

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scattering of guided waves in thick gratings at … · gratings at extreme angles submitted by...

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