Propagation of Low Power Low Divergence Gaussian Fields …eprints.qut.edu.au/45618/1/Michael_Jones_Thesis.pdf · QUEENSLAND UNIVERSITY OF TECHNOLOGY FACULTY OF SCIENCE AND TECHNOLOGY

QUEENSLAND UNIVERSITY OF TECHNOLOGY

FACULTY OF SCIENCE AND TECHNOLOGY

APPLIED OPTICS AND NANOTECHNOLOGY PROGRAM

Propagation of Low Power Low Divergence Gaussian

Fields in Unbiased Self-defocusing Photorefractive

Media and their Interactions

Submitted by Michael William Jones, to the Faculty of Science and Technology,

Queensland University of Technology, in partial fulfilment of the requirements

of the degree of Doctor of Philosophy.

2011

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Key Words

Photorefractive media, nonlinear optics, solitons, soliton interactions, self-focusing,

self-defocusing, unbiased photorefractive media, Gaussian beam optics, GRIN lens,

intensity dependent absorption, intensity dependent transparency, Ce:BaTiO3.

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Abstract

Many optical networks are limited in speed and processing capability due to the

necessity for the optical signal to be converted to an electrical signal and back again.

In addition, electronically manipulated interconnects in an otherwise optical network

lead to overly complicated systems. Optical spatial solitons are optical beams that

propagate without spatial divergence. They are capable of phase dependent

interactions, and have therefore been extensively researched as suitable all optical

interconnects for over 20 years. However, they require additional external

components, initially high voltage power sources were required, several years later,

high power background illumination had replaced the high voltage. However, these

additional components have always remained as the greatest hurdle in realising the

applications of the interactions of spatial optical solitons as all optical interconnects.

Recently however, self-focusing was observed in an otherwise self-defocusing

photorefractive crystal. This observation raises the possibility of the formation of

soliton-like fields in unbiased self-defocusing media, without the need for an applied

electrical field or background illumination.

This thesis will present an examination of the possibility of the formation of

soliton-like low divergence fields in unbiased self-defocusing photorefractive media.

The optimal incident beam and photorefractive media parameters for the formation

of these fields will be presented, together with an analytical and numerical study of

the effect of these parameters. In addition, preliminary examination of the

interactions of two of these fields will be presented.

In order to complete an analytical examination of the field propagating

through the photorefractive medium, the spatial profile of the beam after propagation

through the medium was determined. For a low power solution, it was found that an

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incident Gaussian field maintains its Gaussian profile as it propagates. This allowed

the beam at all times to be described by an individual complex beam parameter,

while also allowing simple analytical solutions to the appropriate wave equation.

An analytical model was developed to describe the effect of the

photorefractive medium on the Gaussian beam. Using this model, expressions for the

required intensity dependent change in both the real and imaginary components of

the refractive index were found. Numerical investigation showed that under certain

conditions, a low powered Gaussian field could propagate in self-defocusing

photorefractive media with divergence of approximately 0.1 % per metre.

An investigation into the parameters of a Ce:BaTiO3 crystal showed that the

intensity dependent absorption is wavelength dependent, and can in fact transition to

intensity dependent transparency. Thus, with careful wavelength selection, the

required intensity dependent change in both the real and imaginary components of

the refractive index for the formation of a low divergence Gaussian field are

physically realisable.

A theoretical model incorporating the dependence of the change in real and

imaginary components of the refractive index on propagation distance was

developed. Analytical and numerical results from this model are congruent with the

results from the previous model, showing low divergence fields with divergence less

than 0.003 % over the propagation length of the photorefractive medium. In addition,

this approach also confirmed the previously mentioned self-focusing effect of the

self-defocusing media, and provided an analogy to a negative index GRIN lens with

an intensity dependent focal length. Experimental results supported the findings of

the numerical analysis.

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Two low divergence fields were found to possess the ability to interact in a

Ce:BaTiO3 crystal in a soliton-like fashion. The strength of these interactions was

found to be dependent on the degree of divergence of the individual beams.

This research found that low-divergence fields are possible in unbiased self-

defocusing photorefractive media, and that soliton-like interactions between two of

these fields are possible. However, in order for these types of fields to be used in

future all optical interconnects, the manipulation of these interactions, together with

the ability for these fields to guide a second beam at a different wavelength, must be

investigated.

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Publication List

Journal Articles:

Jaatinen, E. & Jones, M.W. Theoretical description of low divergence Gaussian

fields in self-defocusing photorefractive media. Opt Commun. 281 (11) 2008,

3201-3207

Jones, M. W. & Jaatinen, E. Transition from intensity dependent absorption to

transparency in Ce:BaTiO3. Opt. Matter. 31 (2) 2008, 122-125

Jones, M. W., Jaatinen, E., & Michael, G. Propagation of low-intensity Gaussian

fields in photorefractive media with real and imaginary intensity-dependent

refractive index components. Appl. Phys. B. doi: 10.1007/s00340-010-4309-y

Jones, M. W., Jaatinen, E., & Michael, G. Soliton-like interactions of bright low-

power low-divergence Gaussian fields in an unbiased self-defocusing

photorefractive BaTiO3 crystal. Opt Eng. 50 (1) 2011, 019701-1-019701-3

Conferences:

Jones, M. W. & Jaatinen, E. Field propagation with nearly constant Gaussian beam

parameters in unbiased self-defocusing photorefractive media. Proc. SPIE.

6801, 2008, 68011P-1-68011P-9

Jones M. W., Jaatinen, E., & Michael, G. (Peer Reviewed) Experimental

investigation of bright photovoltaic spatial soliton-like fields in unbiased self-

defocusing photorefractive BaTiO3. AIP Congress, ISBN 1-876346-57-4,

2008, 184-187.

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Jones M. W., Jaatinen, E., & Michael, G. Analysis of the interactions of bright

photovoltaic low-divergence soliton-like fields in unbiased self-defocusing

photorefractive BaTiO3. Proc. SPIE. 7197, 2009, 719711P-1-719711P-8

Jaatinen, E., Jones, M. W. Numerical modeling fields with forced Gaussian spatial

distributions when they propagate through nonlinear media. ACOLS, 2009, 97-

98.

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Statement of Original Authorship

The research contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education 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

Signature

Date

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Acknowledgement

This thesis would not have eventuated without the assistance of many people. First

and foremost, I would like to thank my supervisory team, Esa Jaatinen and Greg

Michael. The fountain of knowledge of all things optical that is Esa was drawn upon

many times, and his insight and support was greatly appreciated. With thousands of

images to analyse, Greg provided the means to a streamlined and automated process,

which not only allowed me to concentrate on interpreting the results, but also cut

months, possibly years, off the analysis.

I am indebted to the entire Applied Optics and Nanotechnology Program at

QUT. Each member of the group provided untold amounts of assistance and support.

The community spirit within the group goes a long way in such a large project.

A PhD thesis does not stay out of the home, and I would like to thank my

wonderful wife, Rachel, for her support throughout the process. I know she’ll miss

hearing about the adventures of BaTiO3. I would also like to thank my family, who

were always there to support me, and my friends, who were always there for

distraction.

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Table of Contents

Key Words................................................................................................................... ii  Abstract....................................................................................................................... iii  Publication List............................................................................................................vi  Statement of Original Authorship ............................................................................ viii  Acknowledgement.......................................................................................................ix  Table of Contents .........................................................................................................x  Table of Figures......................................................................................................... xii  1. Introduction ............................................................................................................1  

1.1 Research problem ..................................................................................1  1.2 Overall aims of the study.......................................................................3  1.3 Specific objectives of the study and linking the research papers ..........3  

2. Theory and background information .................................................................11  

2.1 Introduction .........................................................................................11  2.2 Interaction of light with matter............................................................12  

2.2.1 Linear response..........................................................................13  2.2.2 Nonlinear response ....................................................................14  

2.3 The photorefractive effect ...................................................................15  2.3.1 Charge migration .......................................................................17  

2.3.1.2 Bulk photovoltaic current ......................................18  2.3.1.3 Diffusion current ...................................................19  

2.3.2 Charge transportation ................................................................20  2.3.2.1 Single centre model ...............................................20  2.3.2.2 Hole-electron competition .....................................23  2.3.2.3 Two centre model ..................................................24  

2.3.3 Effective charge carrier density.................................................27  2.3.4 Light induced absorption...........................................................28  2.3.5 Complex refractive index ..........................................................30  

2.4 Photorefractive materials.....................................................................31  2.4.1 Self-focusing photorefractive media .........................................32  2.4.2 Self-defocusing photorefractive media......................................33  2.4.3 Biased photorefractive media ....................................................34  

2.6 Gaussian Beam Optics.........................................................................35  2.6.1 Lenses ........................................................................................38  

2.6.1.1 Thick Lenses..........................................................41  2.6.1.2 GRIN Lenses .........................................................42  

2.7 Gaussian beam propagation in inhomogeneous media .......................44  2.8 Spatial optical solitons.........................................................................47  

2.8.1 PR screening solitons ................................................................50  2.8.2 Photovoltaic solitons .................................................................51  2.8.3 Soliton Waveguides...................................................................52  2.8.4 Soliton Interactions....................................................................53  

2.9 Methods of analysis .............................................................................56  

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2.9.1 Far field image analysis.............................................................57  2.9.2 Z-scan ........................................................................................59  2.9.3 Numerical wavefront study .......................................................60  

3. Theoretical description of low divergence Gaussian fields in self-defocusing photorefractive media ..............................................................................................62  

4. Transition from intensity dependent absorption to transparency in Ce:BaTiO3 .................................................................................................................71  

5. Propagation of low-intensity Gaussian fields in photorefractive media with real and imaginary intensity-dependent refractive index components ...............77  

6. Soliton-like interactions of bright, low power, low-divergence Gaussian fields in unbiased self-defocusing photorefractive BaTiO3 crystal ................................86  

Soliton-like interactions of bright, low power, low-divergence Gaussian fields in unbiased self-defocusing photorefractive BaTiO3 crystal ....................................87  

7. Conclusion.............................................................................................................91  

References ..................................................................................................................95  

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Table of Figures

Figure 2.1: Schematic of charge migration due to the drift current (a) and the

diffusion current (b), with arrows representing the migration of charge carriers from

areas of illumination………………………………..…………………….…………19

Figure 2.2: Band diagram for the single centre model, with arrows indicating

excitation and recombination of electrons and the photorefractive centre C− /C 0

……………………………………………………….……………………………...21

Figure 2.3: Band diagram for the two centre model, with arrows indicating

excitation and recombination of electrons at the photorefractive centres and

……………………………………………………………………….…......25

Figure 2.4: Phase fronts of a normally diffracting Gaussian beam (dashed lines), and

a Gaussian beam propagating in self-focusing media (solid line). ……………........32

Figure 2.5: Phase fronts of a normally diffracting Gaussian beam (dashed lines), and

a Gaussian beam propagating in self-defocusing media (solid line)………………..33

Figure 2.6: Profile and phase fronts of a Gaussian beam as functions of the

propagation distance, z, from the waist position, w(0). The beam divergence in the

far field is given by θ. …………………………………………………………...…38

Figure 2.7: Transformation of a Gaussian beam due to a thin lens, where q10 and q20

are the complex beam parameters before and after propagating through the thin lens.

…………………………………………………………………………………...…40

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Figure 2.8: Transformation of a Gaussian beam due to a thick lens, where q10 and

q20 are the complex beam parameters before and after propagating through the thick

lens. ………………………..…………………….………………………….………41

Figure 2.9: Transformation of a Gaussian beam due to a GRIN lens, where q10 and

q20 are the complex beam parameters before and after propagating through the thick

lens. ………………………………………………………………..…………..……42

Figure 2.10: Comparison between a spatial optical soliton (a) and a normally

diffracting beam (b) ……………………………………………………....………...48

Figure 2.11: Images and beam profiles illustrating one-dimensional PR screening

soliton interactions in a 6 mm long SBN crystal; a, parallel input beams with 18 µm

separation; b, output under normal diffraction; c, output when beams launched

separately; d, output for in-phase soliton collision showing soliton fusion; e, output

for soliton collision with phase difference showing energy transfer and slight

repulsion, with separation equal to 35 µm; f, output for out of phase soliton collision

showing repulsion, with separation equal to 46 µm ………………………..…..…..54

Figure 2.12: Sample far field image (left) with the cross section (dots) taken at the

horizontal line, and fitted Gaussian profile (line, right). …….………………..........59

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1. Introduction

1.1 Research problem

Optical devices, which use light to control light, have long been declared as one

possible solution to improve information processing capacity and speed, as well as

offering the possibility to be completely reconfigurable when created in

photosensitive media[1-3]. These devices would utilise the ability of low divergence

beams to create a waveguide that could then be used to guide a second wavelength

where the material is insensitive. In addition, these devices would display optically

manipulated interactions controlled by the light itself, removing the need for

additional electronic circuitry.

As with electronic circuitry, the functionality and operating speed of such

devices would increase if the size of the device was reduced[4-6]. Optically, this

requires minimising the divergence of each beam over the distance it travels to avoid

crosstalk and to maintain uniform waveguides. One example of a suitable low

divergence field that can be used for devices where light controls light is a

photorefractive (PR) spatial optical soliton[7]. PR spatial solitons display phase

dependent interactions, and can be configured into ‘X’ and ‘Y’ junctions by

manipulating the relative phases of the two soliton beams[8].

To integrate with existing communications technology, these low-divergence

fields will be required to operate at low powers, and be capable of controlling light at

optical wavelengths in the infrared[9]. Running at low optical power would not only

lower overall electrical power consumption resulting in low heat generation, but

additionally reduce safety concerns. Current optical communications use low power

compact semiconductor lasers, with Gaussian intensity profile outputs[9]. As such, to

2

create low divergence waveguides capable of light controlled interactions, the effect

that nonlinear media has on low power beams with a Gaussian spatial profile is of

importance.

PR media is one type of medium where the refractive index has a component

that is intensity dependent, allowing low divergence fields such as spatial solitons to

be created. Photorefractive spatial optical solitons are available in many varieties

such as bright, grey, and dark, all of which have been investigated over many years

and have many common fundamental similarities[10, 11]. In a PR medium where

the refractive index is increased in regions where light is present, it is possible for the

converging effects of the medium to exactly counteract natural beam divergence,

resulting in a beam that propagates with a constant planar wavefront and without

divergence as a single bright spatial soliton[11]. These types of solitons are

theoretically described by exact solutions to the wave equation which incorporates

the self-focusing nonlinearity of the medium[11, 12]. However, if the beam intensity

is too high, the beam can overly self-focus, causing irreversible damage to the PR

medium.

In self-defocusing PR media however, a planar field will propagate with

increased divergence, with the defocusing effects of the medium exacerbating the

natural divergence of the beam. As a consequence of this, single bright spatial

solitons with planar wavefronts do not exist in unbiased self-defocusing media.

However, in 2006, self-focusing was observed in both doped and undoped unbiased

self-defocusing BaTiO3 where the PR media was placed at the beam waist position

of a tightly focused TEM00 Gaussian field[13]. This apparent self-focusing of

Gaussian beams in unbiased self-defocusing PR media has not been theoretically

explained, and could offer the potential for single bright spatial solitons to be realised

3

in self-defocusing PR media. This is advantageous, as unlike self-focusing, self-

defocusing cannot result in catastrophic over-focusing of the optical radiation

causing irreparable damage to the medium.

1.2 Overall aims of the study

As discussed in the section 1.1, low divergence fields in PR media created with low

power Gaussian beams could provide a solution for increasing the speed and capacity

of current information processing. A simple solution that avoided the possibility for

destructive self-focusing would be ideal, as it would be more likely to survive spikes

in laser power. The recent observation of self-focusing effects in unbiased self-

defocusing PR media[13] has lead to the possibility of stable bright low divergence

fields being created in such media. This study therefore aims specifically to

investigate the possibility of generating such fields in unbiased self-defocusing PR

media, and more generally, to gain a greater understanding of the propagation of low

power Gaussian beams in PR media in order to theoretically explain, for example,

the previously observed apparent self-focusing.

1.3 Specific objectives of the study and linking the

research papers

Investigation of the formation of stable bright low power, low divergence fields in

unbiased self-defocusing PR media comprises of a series of integrated analytical,

numerical, and experimental studies. Initially, a general analytical approximation of

low Gaussian beams propagating in nonlinear media was developed, and applied to

several scenarios. One such scenario was the main objective of the study - the

4

generation of low divergence fields in self-defocusing media. Additional scenarios

were investigated to test the analytical approximation against known outcomes.

Following this, several conditions arose from the development of the analytical

approximation that required experimental investigation. To fully explain the

observations, comprehensive numerical and analytical models were developed. The

predictions of this model were compared to the experimental observations of the

defined scenarios. This is described in the chapters of this thesis:

• Determination of the effect of self-defocusing PR media on the spatial

intensity profile of an incident Gaussian beam (Chapter 3).

Methods and outcomes:

After propagating through the self-defocusing PR medium, the far field

intensity profile of a low power Gaussian beam was measured. Far field

profiles were obtained for the medium was placed at various distances from

the beam waist position. In all cases under low power illumination, the

incident Gaussian beam was found to maintain its Gaussian profile after

propagating through the medium. This result allows the beam within the

medium to be described by a complex beam parameter that uniquely

describes the diameter and wavefront curvature of the Gaussian field. An

important consequence of this is that it allows the refractive index variation

inside the medium to be treated as a quadratic function of distance from the

beam axis - greatly simplifying the wave equation and its solution.

5

• General analytical approximation of low power Gaussian beams propagating

in nonlinear media with a quadratic refractive index profile with low

divergence. The case for both self-focusing PR media and self-defocusing

media are examined (Chapter 3).

Methods and outcomes:

The real and imaginary parts of the intensity dependent refractive index

required to form a low divergence field in unbiased self-defocusing and self-

focusing PR media are obtained by substituting the Gaussian field into the

quadratic form of the nonlinear wave equation. In the absence of an

imaginary component of the intensity dependent refractive index, a stable

non-diverging Gaussian field with a plane wavefront is obtained in self-

focusing PR media, as predicted elsewhere[14]. Following this analysis, the

intensity dependent part of the refractive index was expanded to contain both

real and imaginary parts. In this case, the conditions for low divergence fields

from converging incident Gaussian beams in self-defocusing PR media were

obtained for both one and two transverse dimensions. Although the fields

predicted were not true spatial optical solitons, numerical simulations

revealed that they could propagate over relatively long distances with

divergences several orders of magnitude smaller than that of a comparable

Gaussian field propagating in a homogeneous medium.

Stable low divergence fields are predicted to occur for a converging

Gaussian beam in self-defocusing PR media where the imaginary part of the

change in refractive index is negative. The feasibility of realising these fields

in a typical self-defocusing PR medium was investigated in Chapter 4.

6

• Determination of whether the PR Ce:BaTiO3 crystal possesses the required

real and imaginary intensity dependent refractive index for the formation of

bright low divergence fields. In particular, this part of the research project

aims to characterise the wavelength dependence of intensity dependent

absorption, which is related to the imaginary component of the change in

refractive index, of the PR crystal (Chapter 4).

Methods and outcomes

Unbiased Ce:BaTiO3 is a self-defocusing PR crystal. When irradiated with

light at a wavelength of 532 nm, it exhibits an absorption that increases as the

intensity of the incident light increases. An increasing intensity dependent

absorption corresponds to a positive change in the imaginary part of the

refractive index. By comparing the intensity of the beam before and after

propagating through the PR crystal, the absorption was calculated using the

Beer-Lambert law[15, 16]. Unwanted PR effects, such as beam fanning, were

minimised by careful selection of the incident angle relative to the optical c-

axis of the PR crystal, allowing the intensity dependent change in absorption

to be calculated. Results from this study showed that at an incident

wavelength of 532 nm, Ce:BaTiO3 displays an absorption that increases with

incident field intensity, while at an incident wavelength of 790 nm, the same

crystal displays an increase in transparency as the intensity increases. This

result shows that the conditions required for the formation of bright low

divergence fields in self-defocusing PR media are physically realisable

through choice of wavelength.

7

• Development of a theoretical description of the complex beam parameter of a

Gaussian beam propagating in nonlinear media with a quadratic refractive

index profile with real and imaginary parts of the intensity dependent change

in refractive index (Chapter 5).

Methods and outcomes

While previous sections of the research project (Chapter 3), dealt with an

approximation of the problem, this section aimed to develop a more complete

theoretical description of the propagation of a Gaussian beam in nonlinear

media with a quadratic refractive index profile. In this approach, the Gaussian

complex beam parameter, q, is obtained as a function of the propagation

distance through the PR media. Using boundary conditions obtained from the

incident beam parameters, and appropriate values for the real and imaginary

parts of the intensity dependent refractive index, the evolution of the

Gaussian beam through the medium can be completely described as a

function of the propagation distance, z. This theoretical description was then

applied analytically and numerically to the same scenarios described in

Chapter 3. In addition it was also shown that a PR medium under low power

illumination could be described as an intensity dependent gradient-index

(GRIN) lens. This model was also able to explain the previously observed

apparent self-focusing of Gaussian beams in unbiased self-defocusing PR

media[13], as being a direct result of the imaginary component of the

intensity dependent refractive index.

8

In implementing the analytical model, it was assumed that the change

in intensity over the medium length was small, this allows the intensity

dependent change in the refractive index to be treated as constant throughout

the medium. Under this approximation, the model accurately described the

well-known scenario of the formation of bright spatial solitons in self-

focusing PR media, and confirmed the conditions for the formation of bright

low divergence Gaussian fields in self-defocusing PR media found in Chapter

3. In addition, when an imaginary intensity dependent refractive index

component was included, the analytical solution also explains the apparent

self-focusing in unbiased self-defocusing PR media previously

experimentally observed[13]. The self-defocusing PR media was also found

to behave as a GRIN lens with an intensity dependent negative focal length

within a range of incident intensities.

While the analytical treatment gives good qualitative agreement with

experimental observations a numerical model, that includes variations in

intensity as a function of the propagation distance, is necessary to provide an

accurate quantitative description. The results of this research were congruent

with the previous analytical results in Chapter 2 and Chapter 3. Of particular

interest, the numerical model differed from the theoretical model in the GRIN

lens analogy, with the numerical model providing a much better fit to

experimental results (discussed in the following research section) over a

wider range of incident intensities than the analytical model. In addition, the

numerical model predicted a greater degree of apparent self-focusing than the

analytical model.

9

• Experimental verification of the output of the numerical model (Chapter 5).

Methods and outcomes

To verify the results obtained from the numerical model, an experimental and

numerical z-scan was undertaken using an unbiased self-defocusing

Ce:BaTiO3 PR crystal and a 532 nm Nd:YAG laser. This combination was

selected due to the well documented PR response of the material to light at a

wavelength of 532 nm. Under similar incident beam parameters, and with

appropriate values for the intensity dependent refractive indices (real and

imaginary) of the PR medium, both the experimental and numerical results

showed good agreement. This agreement was demonstrated for an array of

experimental conditions, such as, changes in incident beam power and

incident beam waist size. This outcome showed that the numerical model, and

by extension, the theoretical description, accurately described the low power

light propagation through PR media.

• Investigation into the possible interactions of two bright low divergence

fields in unbiased self-defocusing PR media.

Methods and outcomes

Bright solitons in self-focusing PR media interact with each other, attracting

or repelling each other depending on the relative phase of the two beams[17].

Therefore, this section of the research project aimed to determine if soliton-

like interactions between two bright low divergence fields in unbiased self-

defocusing PR media were possible. A thin wedge was used to create two

10

beams, which crossed within the PR medium. The angle between the two

beams was less than the beam divergence, resulting in a interference pattern

forming in the far field intensity profile of the beam. Shifts in the band

spacing in the interference pattern allowed any change in the angle that the

two beams exited the PR medium to be calculated. These results showed a

small increase in the angle with which the beams exited the PR medium for a

small range of z positions, relative to the incident beam waist, where low

divergence fields could exist. The increase in angle indicates that the two

fields interacted within the PR medium, repelling each other. This result is

important as it suggests that bright low divergence Gaussian fields, although

not considered classical spatial solitons, are capable of soliton-like

interactions.

11

2. Theory and background information

2.1 Introduction

Controlling the propagation of light within matter requires an understanding of the

interactions that light has with matter. When the incident light has a relatively high

intensity, such as a laser beam, the response of the material can be a nonlinear

function of the incident intensity[18]. These nonlinear interactions can result in a

light induced change in the refractive index of the material, which in turn alters the

direction and divergence of the light as it propagates through the medium.

Photorefractive (PR) media is one class of material that has a light induced change in

refractive index, called the PR effect, which could be used in a variety of ways to

control the propagation of laser beams through the media.

The PR effect is a result of charges (electrons or holes) being excited into the

conduction or valance band, and then migrating to areas of lower intensity before de-

exciting and becoming trapped again[19, 20]. This causes an electric field to develop

within the medium, called a space-charge field, which modulates the refractive index

through the Pockels effect[21]. Charge carriers become excited from traps at a range

of energy levels, with their excitation dependent on the intensity and wavelength of

the incident light[22-24]. As a result, the change in refractive index is also dependent

on these parameters. The charge excitation and transportation process responsible for

the space-charge field is also responsible for the light induced absorption of a PR

medium[16, 25]. Like the change in refractive index, the light induced absorption

can be positive or negative, and is intensity and wavelength dependent[26-28].

By altering the incident beam parameters, the PR effect can be exploited for a

variety of purposes such as creating phase conjugate mirrors, holographic storage of

12

data, and creating PR spatial solitons to be used as waveguides[14]. Of particular

interest to this research are PR spatial solitons. PR spatial solitons are optical beams

that propagate in PR media without spatial divergence. They are formed when the

intensity dependent change in refractive index causes the beam to self-focus, exactly

counteracting the beams natural divergence[29].

Spatial solitons have many interesting properties. As they manipulate the

refractive index of the medium they propagate through, they can be used to fabricate

waveguides. In addition, the wavelength dependence of the PR effect leads to the

possibility of the spatial soliton to guide a second beam of a different wavelength at

much greater intensity[30, 31]. Multiple spatial solitons propagating through the

same medium have displayed phase dependent interactions, with any guided beams

taken along with their respective soliton waveguide in the interaction[32-34].

2.2 Interaction of light with matter

The interaction of light with matter has received both academic and public interest

for centuries. Various phenomena such as using a prism to split white light into a

spectrum of colours, magnification through microscopes, and data transmission

through fibre optics all utilise a light-matter interaction. When the incident light has a

relatively low intensity, the material responds in a way that is linearly dependent of

the intensity of the incident light. However, when the incident light power is

increased to relatively high levels, the response of the material can become

nonlinearly dependent on the intensity of the incident light. The study of the

nonlinear response of materials to light, called nonlinear optics, is of particular

interest to researchers at present, with its application to optical computing coming

close to being physically realised.

13

2.2.1 Linear response

Visible light is an electromagnetic wave that can interact with individual atoms in a

material. The response of a material to light can thus be explained in the same way as

the response of a material to an applied oscillatory electric field. When an electric

field is applied to an atom, an induced dipole moment (p) is produced as a result of

the separation of the positive and negative charge within the atom. Put simply, at

visible wavelengths, the applied field drives the electrons bound to nuclei as simple

harmonic oscillators. When an electric field is applied to a large number (N) of

identical atoms, their individual induced dipole moments combine, resulting in a

macroscopic polarisation (P) of the material[9]:

P = Np (2.1)

If the electric field is relatively weak, the polarisation of the material is proportional

to the strength of the local electric field multiplied by the electric susceptibility (χ) of

the material[35];

P = ε0χE (2.2)

where ε0 is the permittivity of free space. When an oscillating electric field such as

light is applied to the material, the individual dipole moments of the atom will

oscillate. These oscillating dipoles then act as secondary radiators with the same

frequency as the incident radiation. The electric field inside the medium will

therefore be a combination of the applied electric field and the electric field resulting

from the polarisation[35].

14

The interaction between the incident field and the dipoles is not instantaneous

but occurs over a finite time interval. As a result the subsequent dipole radiation can

be shifted in phase to the applied field, which alters the phase speed of the local field

to some fraction of the speed in a vacuum. This retardation of the plane wave is

given by the refractive index, n, of the medium[18, 35];

n2 = c2

u2= 1+ χ( )µr (2.3)

where c is the speed of light in a vacuum, u is the speed of light in the medium, and

µr is the relative permeability. If the medium contains no magnetic response; the

relative permeability is one, and the refractive index reduces to:

n2 = 1+ χ (2.4)

Therefore, the refractive index of the material is a parameter that completely captures

the materials linear response to the dipole inducing applied field.

2.2.2 Nonlinear response

As the applied electric field strength increases, electrons behave much more

energetically and their oscillations become anharmonic. As a result the polarisation

of the material becomes a nonlinear function of the field strength[18, 35];

P = ε0 χE + χ2EE + χ3EEE + ...( )= PL + P2

NL + P3NL + ...

(2.5)

15

where the nonlinear susceptibility coefficients χ2 ,χ3...are constants that describe the

materials’ response to the field, and the superscripts L and NL refer to linear and

nonlinear terms. In most materials, one nonlinear polarisation term will dominate all

others, and the polarisation can therefore be reduced to[35];

P = PL + PjNL (2.6)

where PjNL is the dominate jth order term of the nonlinear polarisation.

An inspection of equation (2.5) reveals that each nonlinear polarisation term

is dependent on the intensity of the incident light field, with the refractive index

dependence on the polarisation given by the following relationship:

(2.7)

It is therefore evident that the effective refractive index is intensity dependent when

the nonlinear effects are included. The photorefractive effect is one example of an

intensity dependent refractive index.

2.3 The photorefractive effect

The photorefractive effect is a light induced change in the refractive index of electro-

optic materials. The effect was first observed in LiNbO3 and BaTiO3 in 1966 with

focused green and blue laser beams[36]. Ashkin et al observed that this effect

occurred at different rates in different materials and was wavelength dependent, in

that at some wavelengths, light induced refractive index inhomogeneities failed to

16

materialise[36]. The effect was initially called ‘optical damage’, as it was thought of

as a hindrance to common nonlinear applications of electro-optic materials[21, 37,

38]. In the late 1960’s, the inhomogeneous change in refractive index was attributed

to the excitation and redistribution of charges due to inhomogeneous illumination,

resulting in the formation of space-charge fields[19, 20]. It was observed that the

change in refractive index (Δn) was proportional to the space charge field (ESC) that

was in turn proportional to the density of excited free charge carriers (Neff)[20]:

Δn ∝ ESC ∝ Neff (2.8)

At relatively low intensities, the space charge fields modulate the refractive index of

the medium through the linear electro-optic Pockels effect[18, 21, 39-41];

(2.9)

where n is the background or dark refractive index and rijk is the electrooptic tensor.

The Pockels effect had been used for many years to phase modulate fields in electro-

optic materials by creating a time varying refractive index by applying an oscillatory

electric field. The materials used for these modulators, (eg. LiNbO3) were chosen for

their large electro-optic properties. It is therefore, little wonder that 'optical damage'

was first observed in electro-optic phase modulators.

The space charge field and therefore the change in refractive index are a

result of a three step charge migration process of charges excited from donor ions.

Free charge carriers are excited by inhomogeneous illumination and transported from

17

donor ions to the conduction band. Once in the conduction band, the free charge

carriers migrate to other areas of the electro-optic material, where they are trapped by

acceptor ions[42]. There are several processes by which charges can migrate within

the conduction band, and several models that explain various aspects of the charge

transportation processes, which will be dealt with individually.

2.3.1 Charge migration

Free charge carriers redistribute due to either one or a combination of charge

migration currents within the electro-optic material. The charge migration process

that occurs within the electro-optic crystal is dependent on the characteristics of the

individual crystal, and any applied external electric field or illumination[21, 39]. The

three charge migration currents are drift current, bulk photovoltaic current, and

diffusion current.

2.3.1.1 Drift current

When a homogeneous electric field is applied to the electro-optic crystal, charge

carriers that have been excited by the incident light are forced to move in the

direction dictated by the field, as shown in Figure 2.1 (a). The applied field interacts

with the free charge carriers through Coulomb interactions[21], and the resulting

current obeys Ohms’ law and is given by[21, 39];

jdrift = σ̂E (2.10)

18

where σ̂ = eµ̂e,hNe,h , and E is the electric field, consisting of the applied electric

field, the space charge field, and the pyroelectric field; e is the electron charge,

is the free charge carrier mobility, and Ne,h is the density of free charge carriers. The

drift current is therefore directly proportional to the strength and direction of the

applied external electric field.

2.3.1.2 Bulk photovoltaic current

The bulk photovoltaic current is due to free charge carriers moving in a preferred

direction due to the anisotropic nature of the crystal. Therefore, it is the dominant

effect in highly non-centrosymmetric photorefractive crystals like LiNbO3 [21]. The

magnitude and direction of the photovoltaic current is given by[39];

jphv( )i =12

βijkE j*Ek + c.c.( )

(2.11)

where βijk is the third rank bulk photovoltaic tensor, and Ej and Ek are components of

the incident light. The magnitude and direction of bulk photovoltaic current is

therefore proportional to the bulk photovoltaic tensor and the intensity of the incident

light.

19

2.3.1.3 Diffusion current

The diffusion current is due to the thermal movement of free charge carriers from

areas of high intensity to areas of low intensity due to inhomogeneous illumination,

as shown in Figure 2.1 (b). In this case, the excited charge carriers move as a result

of thermal energy. Charge carriers that move away from areas of high intensity

become re-trapped as their energy is lost. This results in a spatial variation of free

charge carriers, with a current given by[39];

jdiff = −QD̂∇Ne,h (2.12)

where is the diffusion tensor, related to the mobility of free charges and the

temperature, T, through D̂ = µ̂e,hkBTe−1 . Q is the charge of the carrier, equal to either

–e for electron migration, or +e for hole migration. The diffusion current is therefore

proportional to the thermal energy transferred to the crystal from the incident light.

Figure 2.1: Schematic of charge migration due to the drift current (a) and the

diffusion current (b), with arrows representing the migration of charge carriers from

areas of illumination.

20

2.3.2 Charge transportation

Several charge transportation models have been developed to explain the origin of

the charge carriers, and also their dynamics under various experimental scenarios

such as the electro-optic medium under inhomogeneous illumination. Examples of

these models that are relevant to the work discussed here are: the single centre

model, hole electron competition, and two centre model. The energy levels from

where the free charge carriers are excited from within the band gap influence bulk

material properties such as light induced change in absorption, conductivity, and

holographic sensitivity[39]. Free charge carriers can be electrons, holes, or both. In

this thesis, the charge carrier will be assumed to be electrons when there is only one

charge carrier present.

2.3.2.1 Single centre model

The single centre model of the photorefractive effect was the first charge

transportation model used to describe the excitation and recombination of free charge

carriers in electro-optic materials[41]. In this model, electrons are excited into the

conduction band from a single filled centre, C- in the band gap, as can be seen in

Figure 2.2.

21

Figure 2.2: Band diagram for the single centre model, with arrows indicating

excitation and recombination of electrons ad the photorefractive centre C− /C 0 [39].

Initially, these centres were thought to result from Iron (Fe2+/Fe3+) impurities

in LiNbO3, with free charges resulting from the reaction[21]:

Fe2+ + hυ = Fe3+ + free carrier (2.13)

These centres were later attributed to a host of other impurities including copper

(Cu1+/Cu2+) and manganese (Mn2+/Mn3+)[21]. The free electrons are then trapped out

of the conduction band by unfilled centres, C0[39]. The energy level C-/C0

corresponds to the thermal excitation energy required to excite an electron from C-

into the conduction band, creating C0. Depending on the impurity, either electrons or

holes can be charge carriers in the single centre model, with each being excited into

the conduction or valance band respectively.

22

Using terms for the recombination of charges first used in 1971[43], rate

equations for the charge migration process in a single level model were formalised in

1975[44];

dN −

dt= − SI + β( )N − + γ RN

0Ne

(2.14)

N = N − + N 0 (2.15)

where N- and N0 are the concentrations of the donor centres C- and C0, β is the

thermal generation rate, S is the photon absorption cross section, I is the light

intensity, γR is the recombination coefficient, Ne is the concentration of electrons in

the conduction band, and N is the total concentration of electrons in the

photorefractive centre.

The single centre model for the excitation of charge carriers in the

photorefractive effect agrees well with experiments in both copper and iron doped

LiNbO3[39], pure BSO and BaTiO3[41], and Rhodium doped BaTiO3[45]. However,

anomalies were observed in some photorefractive crystals which could not be

explained by the single centre model. For example, multiple dark decay rates of

photorefractive gratings[46], signal gratings changing sign as a function of the

grating period[47], and intensity dependent factors[39, 48].

Extending the single centre model to include two different photorefractive

centres, as shown in Figure 2.3, related to two different impurities, allows for the two

dark decay rates observed in Ce:BaTiO3[46, 49]. Since the energy levels

corresponding to these impurities may differ, the relative effect of each dopant is

wavelength dependent, leading to wavelength dependent dark decay rates. Therefore,

23

to explain this and other anomalies, additional terms need to be included in the single

centre model.

2.3.2.2 Hole-electron competition

To explain some of the anomalies observed in some electrooptic crystals, the single

centre model was expanded to include the simultaneous excitation of both electrons

and holes. In this model, the wavelength of the incident light determines the relative

contribution of each electron and hole excitation to the space charge field[50].

Including two species into the charge transportation process also helps explain how

photorefractive grating erasure rates could occur over two different time scales[46,

49].

In 1986, two models for the inclusion of simultaneous hole-electron

excitation in the single level model were proposed[51, 52]. In the first model, holes

and electrons are produced from a single set of recombination centres. For example,

electrons are photoionised from donors (Fe2+) and recombine with traps (Fe3+), while

holes follow the opposite path, photoionised from hole donors (Fe3+) and

recombining with hole traps (Fe2+) in LiNbO3[52]. In this model, the strength and

direction of the space charge field are dependent on the grating wave vector,

allowing the signal grating to change sign as a function of the grating period[51].

However, it fails to explain the two time scales observed in grating erasure rates,

since the model leads to an erasure rate that is proportional to intensity[52].

In the second model of simultaneous hole-electron competition, electrons and

holes are photoionised from separate recombination centres. In this model, the

parameters such as the light excitation cross section, recombination cross section,

and charge density of one species favours electrons, while those for the other favours

24

holes. The competition between two separate recombination centres, with each

favouring a different charge carrier, allows for two time scales observed in grating

erasure rates[52].

While both simultaneous hole-electron competition models correctly explain

photorefractive phenomena under certain conditions[53], neither model allows for

observations where the grating erasure rate is proportional to a fractional power of

the optical intensity (ie. has a sublinear dependence on intensity) [52]. Simultaneous

hole-electron competition models also fail to predict light induced absorption or a

nonlinear dependence of the space charge field on incident intensity[39]. In 1987,

Motes and Kim[27, 54] suggested that the light induced change in absorption plays a

role in the photorefractive effect, while others suggest that a second set of traps exist

between the donor (acceptor) level and conduction (valence) band[52, 55-57].

2.3.2.3 Two centre model

A secondary trap at an energy level between the donor level and conduction band

was formally developed in 1988[16] as shown in Figure 2.3. In this model, the

shallow traps at the intermediate energy level are partially unfilled at room

temperature. Electrons can therefore be excited into the conduction band from either

the shallow trap level or the deep donor site level. Electrons can also be excited from

the deep donor sites to the shallow trap energy level and then re-excited into the

conduction band. Therefore, charge carriers can be both excited or trapped at two

separate energy levels, where the shallow traps compete with the deep sites for

excited charge carriers[16, 25].

25

The rate equations for the transfer of the concentration of free carriers in the

two centre model is given by[16];

dN1dt

= − S1I + β1( )N1 + γ 1 N1T − N1( )Ne (2.16)

dN2

dt= − S2I + β2( )N2 + γ 2 N2T − N2( )Ne

(2.17)

where Ne is the concentration of free carriers, which will be assumed in this case to

be electrons, and N1, S1 and N2, S2 are the charge concentrations and light excitation

cross-sections of the deep and shallow sites respectively. The addition of the shallow

level traps correctly allows for two dark decay rates observed in a range of

photorefractive materials, such as BSO, BaTiO3[25], reduced KNbO3[58], and

Bi12SiO20[57].

Figure 2.3: Band diagram for the two centre model, with arrows indicating

excitation and recombination of electrons at the photorefractive centres and

[39].

26

The light excitation cross section of the shallow traps and therefore the

shallow trap charge concentration is intensity dependent[25, 59], leading to the

intensity dependence of the photorefractive effect[16, 25]. If the charge density of

the shallow traps is comparable to that of the dark density of the deep sites, then the

photorefractive effect will be a function of intensity, however, if the shallow trap

charge density is relatively low, then the photorefractive effect will not be a function

of intensity[25].

The light excitation cross sections of the deep sites and shallow traps are

wavelength dependent, with their dependence given by[23];

S1 λ( ) = λN1T hc

αwi λ( ) (2.18)

S2 λ( ) = λN1hc

α li λ( ) + S1 λ( ) (2.19)

where αwi and αli are the weak-intensity and light-induced photo absorption

coefficients, respectively. The wavelength dependence of the two levels has been

attributed to different shallow energy levels becoming more active at different

wavelengths[26]. As will be seen in Chapters 3 and 4 the two centre model plays a

key role in ensuring that the imaginary component of the intensity dependent

refractive index has the appropriate sign and magnitude to form low divergence

fields.

27

2.3.3 Effective charge carrier density

The magnitude of the photo-induced refractive index change is determined by the

effective charge carrier density, Neff, which has been shown to be strongly

wavelength dependent in Ce:BaTiO3[24] and other photorefractive crystals[60].

This is not a surprising result, since the energy of the photon that is absorbed must

have sufficient energy to excite the electron (hole) into the conduction (valence)

band. Therefore, it is expected that a cut-off photon frequency will exist, below

which no photorefractive effect is observed. Between wavelengths of 500nm and

750nm, the effective charge carrier density rapidly decreases with increasing

wavelength in Ce:BaTiO3[23]. The effective charge carrier density increases with

intensity[22], to a point where the shallow traps become saturated, after this point,

the effective charge carrier density is independent of intensity as the charge carriers

no longer redistribute[23].

Doping has been shown to increase the total effective trap density[61, 62],

due to the additional charges that are available to be excited. It has been suggested

that this could be due to the increased shallow trap densities causing the deep centres

to become more active[23].

28

2.3.4 Light induced absorption

While the single centre model predicts no light induced absorption, in doped BaTiO3

and LINbO3, light-induced absorption has been attributed to transition metal

impurities at deep donor sites[63-65]. The absorption in this case is related to both

the incident intensity, impurity concentration, and directly related to the absorption

band of the impurity[64].

In the two centre model, the secondary shallow traps are responsible for the

intensity dependent absorption[16, 66], with the light induced absorption given

by[16];

α li = S1 N1 − N1 0( )⎡⎣ ⎤⎦ + S2 N2 − N2 0( )⎡⎣ ⎤⎦ (2.30)

where N1(0) and N2(0) are the dark concentrations of the deep and shallow traps

respectively. The light induced absorption generalised for many different energy

levels, which can exist due to a range of impurities, as[28]:

α li = Sk Nk − Nk 0( )⎡⎣ ⎤⎦k=0,1,2∑ (2.31)

The absorption was further investigated by Tayebati and Mahgerefteh[25],

who gave the total absorption of the photorefractive media as;

α = S1N1 0( ) + S2 − S1( )N2⎡⎣ ⎤⎦hν (2.32)

29

where the first term is the absorption in thermal equilibrium, or the dark absorption,

and the second term accounts for the light induced change in absorption. The relative

magnitudes and intensity dependence of the light excitation cross sections of the

donors (S1) and shallow traps (S2) influence whether light induced absorption ( S2 >

S1 ) or transparency ( S2 < S1 ) will occur. From this equation, it is evident that the

shallow traps have no effect on the absorption in thermal equilibrium, or the dark

absorption.

The light induced absorption in the two centre model is related to the relative

concentrations of the charge carrier densities at the deep and shallow sites, or a

redistribution of free charges between energy levels in the band gap[67]. However, if

the charges are redistributed to different energy levels at roughly the same location,

they will have no effect on the space charge field and therefore on the change in

refractive index. Under these circumstances, the redistribution of free charge carriers

will only influence the intensity dependent absorption[59].

The light induced absorption of photorefractive materials is a function of both

the incident wavelength, and the impurity levels in the material. In Ce:BaTiO3, an

increase in Cerium levels leads to an increase in the light induced absorption at a

wavelength of 514.5nm[68]. Increasing the Cerium concentration from 30 ppm to 50

ppm leads to an increase in the light induced absorption of approximately two

times[61]. In the same material, the light induced absorption decreases as a function

of increasing wavelength in the visible spectrum when pumped at 514.5nm[68]. The

same trend is also observed in undoped BaTiO3 when pumped at the probe

wavelength between 450nm and 500nm[27]. In Rh:BaTiO3, the light induced

absorption spectrum peaks at approximately 800nm, with a local minimum about

600nm[26, 28]. Interestingly, the local minimum observed in the light induced

30

absorption in Rh:BaTiO3 is negative around 600nm[26, 28], indicating light induced

transparency. Similar trends are observed in Cerium doped Lead Barium Niobate

crystals[66]. These results suggest that various PR materials can display both a light

induced increase in absorption or transparency, depending on the incident

wavelength[26, 28].

2.3.5 Complex refractive index

The refractive index of a dispersive medium can be written in terms of real and

imaginary parts[35];

n = nR + iκ (2.33)

where κ is the attenuation coefficient and is related to the absorption coefficient, α

through the relationship[35];

(2.34)

where λ0 is the wavelength in a vacuum. When a dispersive medium exhibits both a

light induced change in refractive index and absorption, the change in refractive

index can be expanded to include both real and imaginary parts[69]

Δn = n2R + in2 I (2.35)

31

where n2R and n2I are the change in the real and imaginary parts of the refractive

index respectively. In this case, the light induced change in absorption is related to

the imaginary part of the refractive index, n2I[69];

Δα =4πn2 Iλ0

(2.36)

Photorefractive materials are an example of dispersive materials that exhibit a

light induced change in refractive index and absorption. Therefore, photorefractive

materials have a refractive index, n, that can be described by;

n = n0 + n2R + in2 I (2.37)

where n0 is the dark component of the refractive index and n2R and n2I are the change

in the real and imaginary parts of the refractive index respectively.

2.4 Photorefractive materials

Photorefractive materials are electrooptic materials that display a change in refractive

index that is dependent on both the intensity and wavelength of the incident

light[21]. When a beam with an inhomogeneous spatial intensity profile is incident

on a photorefractive medium, the resulting change in the refractive index also has an

inhomogeneous spatial profile. For a beam with a Gaussian, or similar, spatial

intensity profile, the greatest change in refractive index is on the beam axis,

decreasing as the distance from the beam axis increases.

32

2.4.1 Self-focusing photorefractive media

In a material where the change in refractive index is positive, the areas of the

Gaussian beam close to the beam axis will encounter a larger refractive index than

the areas further from the beam axis. Under these circumstances, the areas close to

the beam axis will slow relative to the areas further from the beam axis, according

to[35];

(2.39)

where v is the velocity of the light in the medium, c is the speed of light in a vacuum,

and n is the refractive index of the medium. According to Snell’s Law, the areas of

the beam not on the beams axis will bend towards the beam axis, with the net result

being an overall focusing of the beam, as illustrated in Figure 2.4.

Figure 2.4: Phase fronts of a normally diffracting Gaussian beam (dashed lines), and

a Gaussian beam propagating in self-focusing media (solid line).

33

As the beam becomes more focused, the intensity increases and the beam focuses

further. This process can continue to a point where the intensity is so great that

permanent damage can be caused to the photorefractive material.

2.4.2 Self-defocusing photorefractive media

In photorefractive materials where the change in refractive index is negative, the

spatial refractive index profile created in the medium has a minimum on the beam

axis, with the refractive index increasing as a function of the distance from the beam

axis. Therefore, the centre of the beam speeds up relative to the edges, and according

to Snell’s Law, the beam will bend away from the beam axis. As a result, the beam

expands or defocuses as it propagates, as seen in Figure 2.5.

Figure 2.5: Phase fronts of a normally diffracting Gaussian beam (dashed lines), and

a Gaussian beam propagating in self-defocusing media (solid line).

34

Unlike self-focusing media, a Gaussian beam propagating in a self-defocusing

photorefractive medium will decrease in intensity as the beam expands, and the

resulting change in refractive index will also decrease. Therefore, the rate at which

the beam expands as it propagates decreases as it propagates. Examples of nominally

self-defocusing photorefractive materials are LiNbO3 and BaTiO3. Recently, a self-

focusing effect was observed in unbiased self-defocusing photorefractive media[13].

The mechanism behind this effect is currently unexplained, and is examined in

Chapter 5.

2.4.3 Biased photorefractive media

Photorefractive materials where the photorefractive effect is due to the bulk

photovoltaic current or the diffusion current can be biased with an external field.

This external field gives rise to the drift current described in Section 2.3.11. If the

applied field is large enough, then the drift current can become the dominant current

in the photorefractive material. In this case, the change in refractive index becomes a

function of the applied external field, and both the strength and direction of the space

charge field can be externally controlled and can become independent on the

intensity of the incident beam. Therefore, by applying a sufficient external field, self-

defocusing media can behave as self-focusing and vice-versa.

The external field required for biasing photorefractive materials can be found

in a variety of forms. By applying a DC bias voltage of the order of 1 kV, BaTiO3

has been observed to display a stationary self-focusing photorefractive effect at an

incident wavelength of 633nm[70]. Similar results have been reported when the

photorefractive material is illuminated with a homogeneous background beam at a

different wavelength in LiNbO3[71, 72].

35

2.6 Gaussian Beam Optics

Laser beams are examples of electromagnetic fields that can be described as having

an intensity distribution concentrated on the beam axis, decreasing as the distance

from the beam axis increases. As the beam propagates from the minimum spot size

where the wavefront is planar, the beam energy spreads out, and the wavefront

curvature becomes finite. The output from many laser systems have been engineered

so that the electromagnetic field distribution has a TEM00 Gaussian spatial profile

that is given by[14];

E r, z( ) = E0w0w z( ) exp i kr2

2q z( ) + kz − tan−1 z

zR

⎧⎨⎩

⎫⎬⎭

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪ (2.40)

where E0 is the maximum field amplitude, w(z) is the radius of the beam, defined at

the radial (transverse to the beam propagation direction) distance where the beam

intensity is of the maximum intensity. The minimum beam radius, w0, is the

radius of the minimum spot size, called the beam waist and occurs at the position z =

0. The variable k is the wavenumber in the medium, defined as;

k = 2πnλ

(2.41)

36

where n is the refractive index of the medium, and λ is the wavelength of the

electromagnetic field, and q(z) is the complex beam parameter, defined as[14, 18, 73,

74]:

1q z( ) =

1R z( ) − i

λnπw2 z( ) (2.42)

The complex beam parameter describes the profile of the Gaussian beam as a

function of z position in terms of the wavefront curvature, R(z), and the beam radius,

w(z). At the beam waist, the beam has a planar wavefront, and the complex beam

parameter reduces to[73]:

q0 = inπw0

2

λ (2.43)

After propagating a distance z from the beam waist, the complex beam parameter is

given by[73];

q = q0 + z = inπw0

2

λ+ z (2.44)

37

Combining equations (2.42) and (2.44), and equating real and imaginary parts, gives

the evolution of the beam as a function distance from the beam waist in terms of the

beam radius and the wavefront curvature[14, 73];

w z( ) = w0 1+zzR

⎛⎝⎜

⎞⎠⎟

2⎡

⎣⎢⎢

⎤

⎦⎥⎥

1 2

(2.45)

R z( ) = z 1+ zRz

⎛⎝⎜

⎞⎠⎟2⎡

⎣⎢⎢

⎤

⎦⎥⎥

(2.46)

where zR is the Rayleigh range and is defined by[14]:

zR =nπw0

2

λ (2.47)

For large distances from the beam waist, z » zR, both the beam radius and the

wavefront curvature can be approximated as linear functions of the z position[75],

with a divergence angle in the far field equal to:

(2.48)

However, within the Rayleigh range, geometrical optics is no longer valid, and the

beam radius is approximately constant and the wavefront curvature becomes a

strongly dependent on z, having a minimum value at z = zR, before rapidly

approaching infinity at the beam waist[75]. The evolution of a Gaussian beam can

be seen in Figure 2.6.

38

Figure 2.6: Profile and phase fronts of a Gaussian beam as functions of the

propagation distance, z, from the waist position, w(0). The beam divergence in the

far field is given by θ[73].

2.6.1 Lenses

When a Gaussian beam is incident on an ideal thin lens, the wavefront curvature is

transformed, producing a new beam waist location and position. An ideal thin lens is

one where the transverse field distribution is not altered by the lens element.

However, the transformation of a Gaussian beam by a thin lens also introduces a

focal shift[75-79] which is different to what is predicted by geometrical optics. For

thin lenses, the beam size on either side of the lens are equal[18, 76];

w1 = w2 (2.49)

39

and the wavefront curvature of the beam at the incident (R1) and exit (R2) faces of the

lens are related to the focal length of the lens (f) through the equation[18, 73, 76];

(2.50)

In terms of the complex beam parameters, the Gaussian beam at each face of the thin

lens is defined as[18, 73];

(2.51)

If the beam waist of the incident beam is a distance d1 from the thin lens, then the

beam waist after propagating through the lens will be at a position d2. The complex

beam parameters in this case can be found by combining equations (2.51) and

(2.44)[73];

q20 =1− d2 f( )q10 + d1 + d2 − d1d2 f( )

− q10 f( ) + 1− d1 f( ) (2.52)

where q10 and q20 define the waist of the incident and exit Gaussian beams

respectively. Equation (2.52) therefore describes the transformation of a Gaussian

beam by a thin lens, which is illustrated in Figure 2.7.

40

Figure 2.7: Transformation of a Gaussian beam due to a thin lens, where q10 and q20

are the complex beam parameters before and after propagating through the thin lens.

Equation (2.52) can be applied multiple times when a Gaussian beam encounters

multiple thin lenses. More complicated lens elements can be thought of as multiple

thin lenses. In these cases, Equation (2.52) can be generalised to[18, 73];

q2 =Aq1 + BCq1 + D

(2.53)

where A, B, C, and D are elements of the ray transfer matrix. Therefore, providing

the elements of the ray transfer matrix are known, the form of the Gaussian beam

after propagating through the medium can be calculated. The ray transfer matrices

for several optical structures can be found in [73]. As well as thin lenses and

combinations of thin lenses, some other common lens elements are optically thick, or

transform the field by having a Graded Refractive Index (GRIN). These will be dealt

with separately.

41

2.6.1.1 Thick Lenses

A thick lens is one where the thickness of the lens is sufficient that equation (8) no

longer holds, and these are widely used for optical fibre coupling[80]. The focal

length of a thick lens is given as[81];

1f= n −1( ) 1

r1−1r2+n −1( )dnr1r2

⎡

⎣⎢

⎤

⎦⎥ (2.54)

where r1 and r2 are the radii of the incident and exit faces, d is the lens thickness, and

n is the refractive index of the lens material, as depicted in Figure 2.8.

Figure 2.8: Transformation of a Gaussian beam due to a thick lens, where q10 and

q20 are the complex beam parameters before and after propagating through the thick

lens.

The properties of a Gaussian beam after transmission through a thick lens are

therefore dependent on the curvature of each surface and thickness of the lens[80-

83]. When both surfaces of the lens have the same curvature, the dependence on the

thickness of the lens is significant when the focal length of an equivalent thin lens is

larger than the Rayleigh length[83]. When one surface of the thick lens is planar,

then the transmitted beam parameters are highly dependent on the incident

surface[82, 83]. If the incident surface is concave or convex, then the position of the

42

transmitted beam waist is particularly dependent on the lens thickness, while the

beam waist magnitude is only a function of the incident waist magnitude and

effective focal length[83]. If the incident surface is flat, then both the position and

magnitude of the transmitted beam waist are functions of the lens thickness[83].

Detailed expressions for the transformation of Gaussian beams by thick lenses are

presented in reference [83].

2.6.1.2 GRIN Lenses

A GRIN lens is a thick lens with a refractive index profile given by[73, 84-86];

n r( ) = n0 1−A0r

2

2⎛⎝⎜

⎞⎠⎟

(2.55)

where r is the transverse coordinate, and A0 is a positive constant defining the

refractive index gradient.

Figure 2.9: Transformation of a Gaussian beam due to a GRIN lens, where q10 and

q20 are the complex beam parameters before and after propagating through the thick

lens.

43

Due to their flat input and output surfaces, GRIN lenses are widely used to couple

radiation to and from optical fibres and other optical components[84, 85]. The

transformation of the complex beam parameter as a function of propagation distance,

z, of such a system is[86];

q z( ) =cos z A0( )q0 + 1

n0 A0sin z A0( )

−n0 A0 sin z A0( )q0 + cos z A0( ) (2.56)

Giving the beam radius w(z) and wavefront curvature R(z) using equation (2.42)

as[86];

1R z( ) =

AC + BD zR2

A2 + B zR( )2 (2.57)

λnπw2 z( ) = −

1 zR2

A2 + B zR( )2 (2.58)

where A, B, C, and D are found by comparing equations (2.56) and (2.42);

(2.59)

Therefore, the transmitted beam properties of the Gaussian beam after propagating

through a GRIN lens is determined by the GRIN lens parameter, A0, and the

44

propagation length z[86]. It is evident from equations (2.56-2.59) that as the beam

propagates within the medium, the radius of the Gaussian beam fluctuates

sinusoidally between maximum and minimum values determined by the GRIN lens

parameter, A0. The period of these fluctuations, P, is[86];

(2.60)

When the beam radius is at its maximum and minimum values, the wavefront of the

field is planar, analogous to a converging Gaussian beam passing through its beam

waist, and a diverging Gaussian beam being focused by a positive lens[86].

2.7 Gaussian beam propagation in inhomogeneous

media

In some cases, inhomogeneous media can act in a way that is similar to a lens[87]. In

such an inhomogeneous medium, where the variation in the wavevector, k, is equal

to k2, the complex beam parameter after a propagation length z, is equal to[18, 73]:

q z( ) =cos k2

kz

⎛⎝⎜

⎞⎠⎟q0 +

kk2sin k2

kz

⎛⎝⎜

⎞⎠⎟

− sin k2kz

⎛⎝⎜

⎞⎠⎟

k2kq0 + cos

k2kz

⎛⎝⎜

⎞⎠⎟

(2.61)

By comparing this equation to the ray transfer matrix given in equation (2.53), the

lens-like properties of an inhomogeneous medium can be seen. In addition, it can be

45

seen that the propagation of the Gaussian beam in inhomogeneous media is

dependent on the profile of the variation of the wavevector within the medium, k2.

In PR and other nonlinear optical media, the inhomogeneities, k2, become

functions of the incident light, and thus so are the lens-like properties of the medium.

In nonlinear self-focusing media, the medium can be modelled as a combination of a

positive thin lens, and a negative propagation distance[88]. The analysis was

however only valid for a narrow range of low incident beam powers, where self-

focusing effects are minimal[88]. More recently, a wavefront study of converging

Gaussian beams propagating in a self-focusing medium found that the radius of

curvature of the wavefronts increased after propagating through the medium[89].

The same study also found that the radius of curvature of the wavefronts decreased

for diverging incident beams, and that the opposite effect is observed for self-

defocusing media[89]. Recently, it has also been found that nominally self-

defocusing BaTiO3 can exhibit a self-focusing effect under intense illumination[13],

acting analogously to a positive focal length lens.

As discussed previously, most nonlinear media display an intensity dependent

absorption or gain. In 2006, the intensity dependent absorption or gain, related to the

imaginary part of the change in refractive index, in the propagation of optical beams

in nonlinear media was shown to be of great importance[90]. Including these

absorption effects, an initially converging Gaussian beam in nonlinear Kerr media

will continue to converge, and then either diverge or self-focus, depending on the

initial beam parameters and the maximum intensity[91]. The beam divergence occurs

as either the propagation distance or absorption cause the beam to lose sufficient

intensity, reducing the self-focusing effect that stops the beam from diverging. On

46

the other hand, if the incident beam is sufficiently intense, the self-focusing effect

continues to balance diffraction in the presence of further losses [91].

With no transverse refractive index variation, the general case of the

propagation of a Gaussian beam in a lens-like medium with either gain or absorption

variation reveals that the beam radius and wavefront curvature will fluctuate

according to[92];

1q2

=1q1

sinh 2 zR1

− δ⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ − i sin 2

zR1

−ϕ⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

cosh 2 zR1

− δ⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ + cos 2

zR1

−ϕ⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

(2.62)

where R1 is the incident wavefront curvature and δ and ϕ are determined by the

initial conditions of the beam. Here we can see that the addition of absorption or gain

to the nonlinear medium has resulted in the inclusion of hyperbolic trigonometric

functions which cause the beam to either expand or collapse, due to gain or

absorption, depending on the material and incident beam parameters[92]. In a

material with a quadratic gain profile, it is possible to theoretically realise a beam

which propagates with a finite wavefront curvature and beam radius given by[92];

(2.63)

(2.64)

47

where α2 is equal to the variation in the gain of the medium. Inspection of these

parameters reveals that this occurs at the Rayleigh range, where the wavefront

curvature is at a minimum. In this case, the gain profile of the medium counteracts

the expected divergence from this point, and the beam will propagate with no spatial

divergence as a spatial soliton[92]. Thus, spatial optical solitons are predicted in both

self-focusing nonlinear media, and in lens-like media with a quadratic gain profile.

2.8 Spatial optical solitons

Spatial optical solitons are optical beams that propagate without spatial divergence[7,

11]. In addition, they are stable to small perturbations, and capable of phase

dependent particle-like interactions[93]. In photorefractive media, they are formed

when a plane wave, such as the beam waist of a Gaussian beam, is incident on a self-

focusing (or biased self-defocusing) PR medium. If the self-focusing nature of the

PR medium exactly matches the beams natural divergence, then the beam will

propagate without divergence as a spatial soliton[10]. The comparison between an

optical spatial soliton and a normally diffracting Gaussian beam is illustrated in

Figure 2.10. An optical spatial soliton is a solution to the appropriate nonlinear

Schrodinger equation[14];

∂A∂z

−i2k

∂2A∂x2

= i 2n2kn0

A 2 A (2.65)

where A is the amplitude of the wave. The solution for a soliton, which propagates

without spatial variation, takes the form;

48

(2.66)

where the transverse field profile of the beam is described by a sech function, with a

peak magnitude equal to;

A02 =

n02n2

1kw

⎛⎝⎜

⎞⎠⎟2

(2.67)

where k is equal to the wavevector in the medium.

Figure 2.10: Comparison between a spatial optical soliton (a) and a normally

diffracting beam (b)[29]

Spatial solitons can also be formed in optical Kerr media with the same field

as PR spatial solitons[8, 29], although in this case, beam powers in excess of 10

orders of magnitude larger than for PR spatial solitons are required[94]. In addition,

49

two dimensional Kerr spatial solitons can not exist in bulk media, requiring a slab

waveguide to contain the beam in one transverse direction[8, 29, 94].

Photorefractive spatial solitons can be either bright, dark, or gray[95-98].

While bright spatial solitons possess a maximum intensity at the centre of the

propagating beam, dark spatial solitons are dark areas in a constant background

intensity[96]. Dark PR spatial solitons are the opposite of bright spatial solitons, and

are therefore formed in material that is either nominally self-defocusing, or in a self-

focusing medium with a reversed external biasing field[96, 98, 99]. Gray solitons are

related to dark solitons, however, they do not have a constant phase across the soliton

width[96].

The required space charge field and resultant change in refractive index for

the formation of PR spatial solitons can be a result of an external biasing field (PR

screening solitons), or due to background illumination (photovoltaic solitons)[11]. In

the absence of either of these influences, PR spatial solitons can be formed when the

natural self-focusing effect of the PR crystal exactly balances the beam divergence.

The observation of apparent self-focusing in unbiased self-defocusing PR media

could lead to the possibility of PR spatial solitons forming in unbiased self-

defocusing media. These types of spatial solitons, and the conditions required for

their formation are discussed in Chapters 3 and 5. PR screening solitons and

photovoltaic solitons will be discussed separately in the following sections.

50

2.8.1 PR screening solitons

PR screening solitons were first predicted in the early 1990’s[10, 100] and observed

soon after[94, 95, 101]. They are formed when a self-defocusing PR media is biased

with an external electric field, allowing careful manipulation of the change in

refractive index, according to equations 2.9 and 2.10, to exactly counteract beam

divergence, with a negative refractive index change that is a function of the incident

intensity, applied voltage, and dark irradiance[94-96, 99]. In this case, the dark

irradiance is defined as the ratio of the dark thermal regeneration rate to the photon

absorption cross section[99]. The space-charge field created is due to charge

migration as a result of the drift current, and is therefore a function of the applied

voltage rather than the incident light intensity[94, 102]. As a result, the profile of a

PR screening soliton is maintained in the presence of either gain or loss in the

medium[10, 101].

In 1993, PR screening solitons were first observed in Strontium Barium

Niobate (SBN), with an external voltage in the order of 100 V/cm. These screening

solitons were found to be temporally unstable, or transient[101]. The following year,

steady-state PR screening solitons were observed in Barium Titanium Oxide

(BTO)[102]. Due to the weaker electro optic nature of BTO compared to SBN, a

biasing voltage in the order of kilovolts per centimetre was required for the

formation of PR screening solitons[102]. At high biasing voltages (kV/cm) and with

homogenous background illumination in SBN, two-dimensional PR screening spatial

solitons were first observed in 1995[94]. In this case, the homogenous background

illumination is used to create an artificial dark irradiance, thus allowing the relative

difference between this and the peak soliton intensity to be controlled, and reducing

the required applied biasing voltage[94]. This process allowed solitons with widths

51

less than 10 µm in both dimensions to propagate in the steady-state, with a change in

refractive index ofΔn = −13.5 ×10−4 [94]. In all these cases, the incident power of the

beam was low enough to ensure that beam fanning, and other PR effects detrimental

to the formation of PR screening solitons were not observed[101, 102]

PR screening solitons are stable against small perturbations[93], and can be

formed at incident intensities ranging from microwatts to hundreds of Megawatts per

square centimetre[99]. However, regardless of the intensity of the incident light, the

ratio of the peak soliton intensity to the background and dark irradiance is relatively

constant from two times at low intensities, and up to five times at high intensities[94,

99]. The shape of a PR screening soliton is unique, and dependent on the material,

irradiance, and applied voltage[93, 95, 96, 99].

2.8.2 Photovoltaic solitons

PR photovoltaic spatial solitons are formed in PR media where the charge migration

process is a result of the bulk photovoltaic current[98]. Due to the nature of the bulk

photovoltaic current, which according to equation 2.11 is proportional to the

intensity of the incident light, these type of solitons are dependent on the incident

intensity[98], and were first predicted in 1994[103]. Similarly to PR screening

solitons, the ratio between the intensity of the incident beam and the dark irradiance

is an important factor, with the dark irradiance manipulated with incoherent

illumination[103].

In 1995, photovoltaic spatial solitons were first observed in LiNbO3[104]. As

unbiased LiNbO3 is self-defocusing, displaying a negative refractive index

perturbation, these photovoltaic solitons were dark in nature, and the relatively weak

electo-optic coefficients led to steady state being reached in approximately 15

52

minutes[104]. The first two-dimensional bright photovoltaic spatial solitons were

observed in self-focusing Cu:KNSBN[105]. As the elecro-optic coefficient is

significantly larger than LiNbO3, steady state was reached in less than 0.5 s, with

incident intensities as low as 3 W/cm2[105].

In a way analogous to the applied electric field in PR screening solitons, the

background illumination can be used to influence the crystal parameters to change

from self-defocusing, to self-focusing[72]. As a result, both dark and bright

photovoltaic spatial solitons have been found in PR materials with a positive, and a

negative refractive index perturbation[71, 97]. In these cases, the wavelength of the

background illumination was used to manipulate the refractive index

perturbation[71].

2.8.3 Soliton Waveguides

A bright PR spatial soliton is formed in a refractive index profile that is greatest on

the beam axis, and decreasing towards the edges. The result is that the incident beam

is trapped in a self-induced waveguide[11]. Since the PR effect is wavelength

dependent, the waveguide formed by the soliton can be used to guide light at a

wavelength to which the material is less sensitive[30, 31, 106, 107]. As the material

is less sensitive to the guided beam, it can have much higher intensities than the

soliton beam[102, 108]. In addition, the beam will continue to be guided once the

soliton beam and background beam (or applied field) are turned off, until the

waveguide is erased due to dark decay or incoherent illumination[94, 108, 109].

Dark PR spatial solitons can also be used as waveguides to guide beams at

less sensitive wavelengths in one transverse dimension[104, 110]. In addition to

guiding cw beams, soliton waveguides can be used to guide femtosecond pulsed

53

light[111, 112]. In this case, the soliton was written with cw light, and the guided

pulses propagated with low temporal dispersion[109], making the waveguides

suitable for efficient optical interconnects[111].

Analogous to optical fibres, soliton waveguides can be used to guide a

number of different modes. Bright one dimensional soliton waveguides have been

reported to guide TEM00, TEM10, and TEM20 Gaussian modes, with each mode

requiring a different ratio between the peak soliton intensity and the sum of the dark

irradiance and background illumination[31]. For the case of the TEM20 mode, only

the central lobe is guided, while the outer lobes were diffracted[31]. On the other

hand, dark one dimensional soliton waveguides were only capable of guiding TEM00

beams, regardless of the intensity ratio[31]. In two dimensional bright soliton

waveguides, similar trends are observed, with TEM00, TEM10, TEM01, and TEM20

modes being guided with a dependence on the intensity ratio of the incident soliton.

2.8.4 Soliton Interactions

Of great interest to photonic applications of spatial solitons is their particle-like

interactions[17]. Soliton interactions can be attractive and repulsive, depending on

the relative phase of the interacting solitons[17, 32, 113-117]. The strength of the

interaction is a function of the relative amplitude of the solitons[93], their

separation[117], diameter[115, 118], and interaction angle[32, 113, 114, 116].

Bright spatial solitons display particle-like collisions[116], with each beam

maintaining its properties following the collision[7, 119]. In the case of bright

solitons, two in-phase solitons will attract each other, while out of phase bright

solitons will repel[7, 93, 116]. As the two spatial solitons interact, the beams begin to

overlap, causing the intensity in the region between the two beams to change[7].

54

When the two solitons are in-phase, the intensity increases, causing the refractive

index to increase, drawing the two beams together. Conversely, for out of phase

solitons, the opposite occurs, and the two beams are repelled[7].When the phase of

the two solitons is not completely in or out of phase, the solitons interact somewhat

repelling each other, with some energy transfer between the solitons[17, 115].

Experimental observations of these interactions are illustrated in Figure 2.11.

Figure 2.11: Images and beam profiles illustrating one-dimensional PR screening

soliton interactions in a 6 mm long SBN crystal; a, parallel input beams with 18 µm

separation; b, output under normal diffraction; c, output when beams launched

separately; d, output for in-phase soliton collision showing soliton fusion; e, output

for soliton collision with phase difference showing energy transfer and slight

repulsion, with separation equal to 35 µm; f, output for out of phase soliton collision

showing repulsion, with separation equal to 46 µm[115].

In addition, the interaction angle between the two solitons greatly effects the

outcome of the soliton collision[11]. When the interaction is small (crossing angle

inside the crystal less than 0.5°), the attraction between in-phase solitons causes them

to fuse together, as seen for solitons launched parallel to each other in Figure 2[32,

55

115, 116]. On the other hand, at large interaction angles (crossing angle inside the

crystal greater than 1.5°), no interaction is seen, with the solitons passing through

each other unaffected[116]. The interaction strength within the PR media is

dependent on the relative diameter and separation of the two spatial solitons[93]. It

has been found that the strength of the interaction increases exponentially as the

soliton separation decreases[17].

Spatial solitons propagating in PR media can also be bent due to the factors

such as PR diffusion, drift and applied electric fields[93, 120, 121]. Increases in the

applied electric field result in monotonic increases in the soliton deflection, while

increasing the ratio of the peak soliton intensity to the dark irradiance influenced

bending up to a maximum value of approximately six times the soliton

diameter[121]. Limits to the self-bending of solitons in this way are found when the

soliton becomes unstable and filamentation occurs[121]. PR drift and diffusion

mechanisms cause the soliton to bend in a way analogous to PR beam fanning

discussed previously, with no apparent filamentation occurring[93, 120].

Combining the waveguide and interaction properties of spatial solitons allows

for all rewritable optical switches to be created[114]. Spatial solitons will continue to

guide the probe beam through an interaction, creating an ‘X’ junction, or splitting or

combining the probe beam into two parts, creating a ‘Y’ junction[11, 32-34]. Thus,

configurable optical switches can be created by varying the relative phase of the two

soliton writing beams[11].

56

2.9 Methods of analysis

In this section, the experimental, analytical, and numerical methods will be

introduced. These methods are employed throughout the project to analyse the

propagation of Gaussian beams through self-defocusing PR media. Far field

measurements were taken to determine the profile of the Gaussian beam after

propagation through the PR medium under various circumstances. The intensity

distribution of the far field images was analysed, comparing the measured intensities

to theoretical derived profiles. Least squares data fitting was used extensively in

analysing the far field images. Measurements of the intensity dependent absorption

of the PR medium at a variety of wavelengths were conducted by comparing the

beam intensity before and after passing through the PR medium. Analytically and

numerically, the problem of a low power Gaussian beam propagating in PR medium

can be simplified in two ways. Firstly, at low beam powers, the intensity dependent

refractive index profile can be treated as quadratic, approximated by;

n r( ) ≈ n0 − 2n2I0w2 r

2 (2.68)

Secondly, an incident low power Gaussian beam maintains its Gaussian profile at all

times during propagation. These two simplifications allowed both analytical results

to be possible, and a detailed numerical analysis to be conducted.

57

2.9.1 Far field image analysis

Far field images depict the beam a distance significantly larger than the Rayleigh

length after it has propagated through the PR material. In the far field, the evolution

of the beam radius is a linear function of the distance from the beam waist, and the

wavefront curvature is approximately equal to the propagation distance. Far field

images were captured with a CCD array as colour bitmap images, and analysed using

Scilab software. The CCD arrays used were commercially available, Thorlabs®

DCU223C, with a resolution of 1024 × 768 pixels, and a pixel size of 4.65 µm × 4.65

µm.

Initial far field images were taken with the PR medium at various distances

relative to the incident beam waist to determine if a low power Gaussian beam

maintained its Gaussian spatial profile after propagating through the medium. In this

case, the intensity profile of the image was analysed, and a Gaussian spatial profile

was least squares fitted to the data. Results of this analysis are presented in Chapter

3, and confirm that the Gaussian spatial profile is maintained for the field intensities

considered.

Subsequent far field images were taken at two different propagation

distances. A 50 / 50 beam splitter was used to capture the two images simultaneously

with two CCD arrays placed at different distances after a beam splitter. By obtaining

the spatial profile of the far field at two different propagation distances, the profile of

the Gaussian beam can be completely described using equations (2.45 - 2.48) and

elementary trigonometry. Far field images were used in a z-scan experimental setup

(discussed in Section 2.9.2). This enabled the effect of the PR medium on a Gaussian

beam with different incident parameters to be measured.

58

During the course of this PhD, over 50,000 far field images were analysed.

To ensure repeatability and consistency in the measured far field profiles, Scilab

code was written to automatically calculate the horizontal Gaussian beam radius,

w(z), together with an estimate of the uncertainty. A series of steps were carried out

to produce this result. The code loaded a bitmap file, and identified the area of

highest intensity. Next, five vertical cross sections of the image were taken around

the previously identified area, separated by 5 pixels. Gaussian spatial profiles were

least squares fitted to these five cross sections, with the peak value of the five plots

taken as the peak intensity value of the image. A total of 41 adjacent horizontal cross

sections were then taken about this point, comprising of the cross section at the peak

value, and the 20 cross sections above and below. Gaussian spatial profiles were

least squares fitted to these 41 cross sections. From these 41 sets of data, the

Gaussian beam radius, w(z), and the standard deviation were calculated.

Figure 2.12 shows a typical far field image, with a cross section and fitted

Gaussian beam profile. The horizontal line on the far field image represents where

the cross section was taken from in this case. From the cross section, the associated

noise in the image can be seen. In addition to sampling a large area of each image for

analysis, the z-scan (discussed in Section 2.9.2) was repeated numerous times. This

aided in ensuring the methods used were repeatable, and eliminating the effects of

noise as much as possible. In the case of a far field interference pattern, as is the case

in Chapter 6, a similar process was done, except that a Gaussian spatial profile was

least squares fitted to each peak, with the horizontal co-ordinate of each peak with a

corresponding error recorded.

59

Figure 2.12: Sample far field image (left) with the cross section (dots) taken at the

horizontal line, and fitted Gaussian profile (line, right).

2.9.2 Z-scan

A z-scan technique involves translating an optical medium along the direction of

propagation, z, of an incident Gaussian beam[89, 122-124]. Under these

circumstances the total beam energy is constant, however, by varying the distance

from the beam waist, the beam is incident on the medium with a different complex

beam parameter[124]. As a result, a z-scan enables the variation of both the

wavefront curvature and beam radius simultaneously. Different combinations of

wavefront curvature and beam radius are obtained by varying the incident beam

waist, w0, according to equations (2.45) and (2.46). The intensity of the incident

beam can be adjusted via the total beam power, or the incident beam waist.

60

2.9.3 Numerical wavefront study

A numerical wavefront study of the Gaussian beam propagating through the PR

medium allowed the complete properties of the beam to be determined during

propagation. With the incident beam parameters known, the initial wavefront can be

decomposed into numerous point sources, as in a geometrical optics approach. The

intensity, I0, and the beam radius, w(z), allows the quadratic refractive index profile

to be calculated through equation (2.68). The effect of this refractive index profile on

each individual point source can then be calculated using Snell’s Law;

n1 sinθ1 = n2 sinθ2 (2.69 )

where θ1 and θ2 are the angles of incidence and refraction, and n1 and n2 are the

relative refractive indices of the two effective media. Thus, after propagation through

the section of PR media with thickness, Δz, each point source will be refracted by the

quadratic refractive index profile. The point sources can then be reconstructed to

form a new wavefront, which is then incident on the next section of PR media with

thickness, Δz.

To eliminate the propagation and amplification of numerical noise, the

wavefronts after propagating through each section of the PR media were least

squares fitted to a Gaussian spatial beam profile. The fitting parameters for the least

squares fit were the maximum field amplitude, E0, and the beam radius, w(z).

Therefore, after each section of PR media, the transmitted field can be uniquely

described by its own complex beam parameter, q, and as a result, the Gaussian beam

can be completely described at all times within the PR medium and in the far field.

61

Comparison between far field measurements of both the numerical model and the

experimental z-scan allowed the validity of this numerical approach to be confirmed.

62

3. Theoretical description of low divergence Gaussian

fields in self-defocusing photorefractive media

Esa Jaatinen1, Michael W. Jones1

1School of Physical and Chemical Sciences, Queensland University of Technology, 2

George Street, Brisbane QLD 4000, Australia

Published in: Optics Communications 281, 3201 (2008)

Corresponding Author

Dr E. Jaatinen

School of Physical and Chemical Sciences

Queensland University of Technology

2 George Street

Brisbane QLD 4000, Australia

Fax: +61 7 3138 9079

Email: [email protected]

63

The authors listed below have certified* that; 1. they meet criteria for authorship in that they have participated in the

conception, execution, or interpretation, of at least that part of the publication in their field of expertise;

2. they take public responsibility for their part of the publication, except for the responsible author who accepts overall responsibility for the publication;

3. there are no other authors according to these criteria; 4. potential conflicts of interest have been disclosed to (a) granting bodies, (b)

the editor or publisher of journals and their publications, and (c) the head of the responsible academic unit, and

5. they agree to the use of the publication in the student’s thesis and its publication on the Australasian Digital Thesis database consistent with any limitations set by publisher requirements.

In the case of this chapter: Theoretical description of low divergence Gaussian fields in self-defocusing photorefractive media E. Jaatinen, M. W. Jones Optics Communications 281, 3201 (2008)

Contributor Statement of contribution*

Esa Jaatinen Original idea, conducted analysis (numerical and analytical),

paper writing, discussion

Michael W. Jones

Signature

Date

Undertook experiments, conducted experimental analysis,

discussion, paper writing

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I have sighted email or other correspondence from all Co-authors confirming their

certifying authorship

______________________ _____________________________ ____________

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64

Theoretical description of low divergence Gaussian fieldsin self-defocusing photorefractive media

Esa Jaatinen *, Michael W. Jones

Applied Optics Program, Queensland University of Technology, GPO Box 2434, Brisbane Qld 4001, Australia

Received 14 December 2007; received in revised form 28 January 2008; accepted 8 February 2008

Abstract

Experimental evidence is presented that shows that low intensity optical fields preserve their Gaussian transverse amplitude distribu-tion as they propagate through self-defocusing Ce:BaTiO3 photorefractive media. The Gaussian nature of the field is used in a theoreticaltreatment to derive conditions under which bright solitons are formed in photorefractive media that have a light induced refractive indexthat is approximately quadratic. This analysis shows that while it is not possible to produce a single bright soliton in self-defocusingmedia that it is possible to minimize the field’s divergence such that the change in beam radius is small (<1%) over large propagationdistances (!1 m). An imaginary light induced refractive index component is necessary to generate the low divergence fields in bothself-focusing and self-defocusing media when illuminated with Gaussian fields that have a non-planar wavefront.! 2008 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/optcom

Available online at www.sciencedirect.com

Optics Communications 281 (2008) 3201–3207

71

4. Transition from intensity dependent absorption to

transparency in Ce:BaTiO3

Michael W. Jones1, Esa Jaatinen1

1School of Physical and Chemical Sciences, Queensland University of Technology, 2

George Street, Brisbane QLD 4000, Australia

Published in: Optical Materials 31, 122 (2008)

Corresponding Author

Michael W. Jones

School of Physical and Chemical Sciences

Queensland University of Technology

2 George Street

Brisbane QLD 4000, Australia

Fax: +61 7 3138 9079

Email: [email protected]

Keywords: Ce:BaTiO3, Intensity dependent transparency, Intensity dependent

absorption

72

The authors listed below have certified* that; 1. they meet criteria for authorship in that they have participated in the

conception, execution, or interpretation, of at least that part of the publication in their field of expertise;

2. they take public responsibility for their part of the publication, except for the responsible author who accepts overall responsibility for the publication;

3. there are no other authors according to these criteria; 4. potential conflicts of interest have been disclosed to (a) granting bodies, (b)

the editor or publisher of journals and their publications, and (c) the head of the responsible academic unit, and

5. they agree to the use of the publication in the student’s thesis and its publication on the Australasian Digital Thesis database consistent with any limitations set by publisher requirements.

In the case of this chapter: Transition from intensity dependent absorption to transparency in Ce:BaTiO3 M. W. Jones, E. Jaatinen, Optics Materials 31, 122 (2008)

Contributor Statement of contribution*

Michael W. Jones

Signature

Date

Original idea, undertook experiments, conducted analysis

(experimental and numerical), paper writing, data analysis,

discussion

Esa Jaatinen Interpretation of results, discussion, paper writing

Principal Supervisor Confirmation

I have sighted email or other correspondence from all Co-authors confirming their

certifying authorship

______________________ _____________________________ ____________

Name Signature Date

73

Transition from intensity dependent absorptionto transparency in Ce:BaTiO3

Michael W. Jones *, Esa Jaatinen

Applied Optics Program, Queensland University of Technology, GPO Box 2434, Brisbane QLD 4001, Australia

Received 27 September 2007; received in revised form 2 February 2008; accepted 3 February 2008Available online 19 March 2008

Abstract

The intensity dependent absorption and transparency of Ce:BaTiO3 is examined in relation to the two trap model of the photorefrac-tive e!ect. At wavelengths of both 532 nm and 633 nm, the material exhibits intensity dependent absorption, while at a wavelength of790 nm, intensity dependent transparency is observed, with a transition between absorption and transparency at approximately650 nm. Experimental data is matched to the two trap theoretical model with good agreement. Analysis of this match shows that thelight excitation cross-section of the shallow traps are the main contributors to the observed intensity dependent transition.! 2008 Elsevier B.V. All rights reserved.

PACS: 78.20.C; 42.70.N

Keywords: Ce:BaTiO3; Intensity dependent transparency; Intensity dependent absorption

www.elsevier.com/locate/optmat

Available online at www.sciencedirect.com

Optical Materials 31 (2008) 122–125

77

5. Propagation of low-intensity Gaussian fields in

photorefractive media with real and imaginary

intensity-dependent refractive index components

Michael W. Jones1, Esa Jaatinen1, G. W. Michael

1School of Physical and Chemical Sciences, Queensland University of Technology, 2

George Street, Brisbane QLD 4000, Australia

Published in: Applied Physics B. doi: 10.1007/s00340-010-4309-7

Corresponding Author

Michael W. Jones

School of Physical and Chemical Sciences

Queensland University of Technology

2 George Street

Brisbane QLD 4000, Australia

Fax: +61 7 3138 9079

Email: [email protected]

Keywords: Photorefractive optics, spatial solitons

78

The authors listed below have certified* that; 1. they meet criteria for authorship in that they have participated in the

conception, execution, or interpretation, of at least that part of the publication in their field of expertise;

2. they take public responsibility for their part of the publication, except for the responsible author who accepts overall responsibility for the publication;

3. there are no other authors according to these criteria; 4. potential conflicts of interest have been disclosed to (a) granting bodies,

(b) the editor or publisher of journals and their publications, and (c) the head of the responsible academic unit, and

5. they agree to the use of the publication in the student’s thesis and its publication on the Australasian Digital Thesis database consistent with any limitations set by publisher requirements.

In the case of this chapter: Propagation of converging low-intensity Gaussian fields in photorefractive media with real and imaginary intensity-dependent refractive index components M. W. Jones, E. Jaatinen, G. W. Michael, Applied Physics B. doi: 10.1007/s00340-010-4309-y

Contributor Statement of contribution*

Michael W. Jones

Signature

Date

Undertook experiments, conducted analysis (experimental

and numerical), paper writing, data analysis, discussion

Esa Jaatinen Conducted analysis (numerical and analytical), discussion,

paper writing

Greg W. Michael Data analysis

Principal Supervisor Confirmation

I have sighted email or other correspondence from all Co-authors confirming their

certifying authorship

______________________ _____________________________ ____________

Name Signature Date

79

Appl Phys BDOI 10.1007/s00340-010-4309-y

Propagation of low-intensity Gaussian fields in photorefractivemedia with real and imaginary intensity-dependent refractiveindex components

M.W. Jones · E. Jaatinen · G.W. Michael

Received: 23 June 2010 / Revised version: 11 October 2010© Springer-Verlag 2010

Abstract A theoretical analysis of a low power Gaussianfield propagating in unbiased self-defocussing photorefrac-tive media that includes both real and imaginary componentsof the intensity-dependent refractive index is presented. Theanalysis, which is in excellent agreement with experimen-tal observations, shows that the imaginary component of theintensity-dependent refractive index can have a focussing ef-fect independent of the focussing or defocussing effect ofthe real component of the intensity-dependent refractive in-dex. These findings suggest that the imaginary componentof the intensity-dependent refractive index is the cause ofthe previously observed apparent self-focussing in unbiasedself-defocussing photorefractive media.

6. Soliton-like interactions of bright, low power, low-

divergence Gaussian fields in unbiased self-

defocusing photorefractive BaTiO3 crystal

Michael W. Jones1, Esa Jaatinen1, G. W. Michael

1School of Physical and Chemical Sciences, Queensland University of Technology, 2

George Street, Brisbane QLD 4000, Australia

Published in: Optical Engineering 50, 019701-1 (2011)

Corresponding Author

Michael W. Jones

School of Physical and Chemical Sciences

Queensland University of Technology

2 George Street

Brisbane QLD 4000, Australia

Fax: +61 7 3138 9079

Email: [email protected]

Keywords: Photorefractive optics, spatial solitons

87

The authors listed below have certified* that; 1. they meet criteria for authorship in that they have participated in the

conception, execution, or interpretation, of at least that part of the publication in their field of expertise;

2. they take public responsibility for their part of the publication, except for the responsible author who accepts overall responsibility for the publication;

3. there are no other authors according to these criteria; 4. potential conflicts of interest have been disclosed to (a) granting bodies, (b)

the editor or publisher of journals and their publications, and (c) the head of the responsible academic unit, and

5. they agree to the use of the publication in the student’s thesis and its publication on the Australasian Digital Thesis database consistent with any limitations set by publisher requirements.

In the case of this chapter: Soliton-like interactions of bright, low power, low-divergence Gaussian fields in unbiased self-defocusing photorefractive BaTiO3 crystal M. W. Jones, E. Jaatinen, G. W. Michael, Optical Engineering 50, 019071-1 (2001)

Contributor Statement of contribution*

Michael W. Jones

Signature

Date

Original idea, undertook experiments, conducted analysis

(experimental and numerical), paper writing, data analysis,

discussion

Esa Jaatinen Conducted numerical analysis, discussion, paper writing

Greg W. Michael Data analysis

Principal Supervisor Confirmation

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certifying authorship

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88

C O M M U N I C A T I O N S

Soliton-like interactions of bright,low-power, low-divergenceGaussian fields in an unbiasedself-defocusing photorefractiveBaTiO3 crystal

Michael W. JonesEsa JaatinenGreg W. MichaelQueensland University of TechnologyFaculty of Science and TechnologyApplied Optics and Nanotechnology2 George StreetBrisbane, 4000, AustraliaE-mail: [email protected]

Abstract. The soliton-like interactions of two low divergencefields propagating in unbiased self-defocusing photorefractivemedia are examined. The interactions appear to be a resultof factors that are not dissimilar to those between two steadystate screening solitons. Analysis of the conditions for theinteractions show a cutoff for the degree of beam divergencerequired for the soliton-like interactions to occur. The analysisalso indicates that the strength of the interaction increases asthe beam divergence decreases. C!2011 Society of Photo-OpticalInstrumentation Engineers (SPIE). [DOI: 10.1117/1.3526683]

Subject terms: self-defocusing photorefractive media; unbiased pho-torefractive media; low divergence interactions.

Paper 100538CR received Jun. 29, 2010; revised manuscript re-ceived Nov. 2, 2010; accepted for publication Nov. 10, 2011; pub-lished online Jan. 13, 2011.

1 IntroductionOne of the key motivations for the study of photorefractive(PR) optical spatial solitons lies in the properties of their in-teractions, which allow for the creation of all optical switches.Two main types of steady state PR spatial solitons exist: PRscreening solitons and PR photovoltaic soliton. The formerrequires a large external biasing electric field on the orderof kilovolts, whereas the latter is formed due to the photo-voltaic space-charge field and is dependent on the intensityof the incident light. Bright PR screening solitons and theirinteractions are well known, having been studied intensivelyfor a number of years.1–5 Bright photovoltaic PR spatial soli-tons however are a more recent observation, having beenformed in both self-focusing6 and in self-defocusing PR me-dia with the aid of a high-powered background beam in bothcases.7 Interactions of these types of PR spatial solitons havealso been investigated and found to behave similarly to PRscreening solitons.5 However, the requirement of an exter-nal biasing field in the form of either an external electrical

0091-3286/2011/$25.00 C! 2011 SPIE

field or high-power background illumination “is the hurdlefor realisation of applications” (see p. 1923 of Ref. 5). In astep toward overcoming this hurdle, the recent observation offocusing in unbiased self-defocusing PR crystals8, 9 has leadto the prediction of the formation of bright, low-divergencephotovoltaic soliton-like fields in self-defocusing PR mediawithout an external field or background illumination.10 Inthis approach, the real and imaginary components of the re-fractive index required for the formation of these fields is10

n2R = n0

I0

!1

k2w2 (z)! w2 (z)

4R2 (z)

", (1)

n2I = n0

I0k R (z), (2)

where n0 is equal to the background refractive index, I0 isequal to the incident beam intensity, k is the modulus ofthe wave vector, and w(z) and R(z) represent the beam ra-dius and wavefront curvature at a distance z from the beamwaist w0. However, in order to be applicable for use asall optical switches, these low-divergence soliton-like fieldsmust be capable of interactions. Therefore, this letter seeksto experimentally determine if two low-divergence soliton-like fields created from initially converging beams in unbi-ased self-defocusing Ce:BaTiO3 are capable of soliton-likeinteractions.

2 MethodTwo beams, ! = 532 nm, propagating with a relative angleof 2" = 0.2 deg and radius at the beam waist, w0 = 30 µm,were created from reflections off the front and back faces ofa thin wedge. Under this arrangement, the beam waists ofeach beam were at approximately the same z distance, andthe two beams intersected "10 cm before the beam waists.The PR Ce:BaTiO3 crystal with 50 ppm cerium concentrationwas translated in a z-scan arrangement at distances betweenthe intersection point of the two beams and the Rayleighlength, zR = 5.3 mm, with the optical c-axis perpendicularto the direction of translation. Therefore, within the crystal,the beams were diverging from each other at an angle of0.2 deg, while each individual beam was converging, as canbe seen in Fig. 1. The beam power for each beam was set inturn to 0.8, 1.0, and 1.3 mW per beam.

After the beams had propagated through the PR crystal,images of the far field were taken where the Gaussian beamscan be approximated as plane waves. Because the divergenceof each individual beam was greater than the angle betweenthem, the beams were superimposed in the far field, resultingin an interference pattern. The spatial intensity profile of theinterference pattern formed by the two beams in the far fieldis described by

I = 4I0 cos2 [k sin (" )] , (3)

where I0 is the intensity of each beam, k is the modulus of thewave vector of each beam, and " is the half angle betweenthe two beams (i.e., the beams intersect at an angle of 2" ).

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Fig. 1 Experimental setup showing the width of the two beams(shaded area) as they intersect. The solid lines within the shadedarea represents beam axis of each beam and inset (a) represents atypical far field profile recorded from the CCD camera.

Therefore, in air the period of maxima in the interferencepattern is equal to

! = "

2 sin #. (4)

Intensity profiles of the far-field interference pattern wereleast squares fitted to Eq. (3). From this, the period of themaxima [given by Eq. (4)] and therefore the angle that thetwo beams intersected the plane of the CCD array, # , wasdetermined.

3 Results and DiscussionA change in the angle that the beam exits the PR crystal forsome positions under a z-scan would indicate that the twobeams are interacting under those specific incident-beam pa-rameters. Figure 2 shows the angle that the beams exit thecrystal increases by 3–5% for a narrow range of incidentbeam areas, with the position of the interaction occurringover a similar range of incident beam areas for each of thetwo cases. The maximum relative increase in the exit angleof between the two beams was 5% for the 1.0 mW case and4% for the 1.3 mW case. No interaction was evident for thecase of two 0.8-mW beams, and the results are therefore notpresented. These results suggest that the two beams interact,with a resulting divergence between the two beams. This in-dicates that the two low-divergence fields are repelled fromeach other. In terms of bright-soliton interactions, this resultcan be interpreted as the refractive index between the twobeams decreases (self-defocusing PR media) and the beamsare steered away from this area. The interactions in Fig. 2 areobserved when the required changes in the real and imagi-nary parts of the refractive index are between –9.6!10" 6 <n2R < –1.5!10" 5 and –3.4!10" 11 < n2I < –4.7!10" 11,respectively, as given by Eqs. (1) and (2). Although this rangeof values for the change in the real part of the refractive indexis typical for this material at this wavelength and intensity,Ce:BaTiO3 has a positive change in the imaginary part of therefractive index at this wavelength11, 12 and thus will resultin a beam with divergence greater than the ideal case, wherebeam divergence is equal to zero.10

Numerical simulations of the divergence of each beamwere undertaken and found that at a power of 1.0 mW perbeam, the beam diverges by 1.77% over the length of the

Fig. 2 Results for the relative change in exit angle between the twobeam as a function of the incident beam area for individual beampowers of (a) 1.0 mW and (b) 1.3 mW. The solid line in each caserepresents a relative change equal to 1.0 (i.e., no change).

medium in the area that the interactions were observed. Forthe beams at 1.3 mW each, the beam divergence is predictedat 1.84%. Figures 2(a) and 2(b) show that at the higher beampower, the beam interaction is weaker, and this could be dueto the greater divergence of the beam. For beams at 0.8 mWeach, numerical simulations predict that the beam divergesby 1.96% over the length of the medium. For comparison,in a homogeneous medium, the same beam will diverge by#17% over the same length. Table 1 shows the beam powers,theoretical divergences, and whether or not an interaction wasobserved. From Table 1, it is clear that under these circum-stances, beam divergence must be <1.96% over the lengthof the medium (5 mm) in order for a soliton-like interac-tion to occur. The relatively small change in the interactionstrength between the two case where soliton-like interactionsoccurs suggest there must be a beam divergence limit for theobservation of soliton-like interactions.

Table 1 Beam powers, theoretical divergences, and interactions forthe three cases, ordered by beam divergence.

Beam power (mW) Theoretical divergence (%) Interaction (%)

1.0 1.77 5

1.3 1.84 4

0.8 1.96 0*

*No change in the relative exit angle between the two beams wasevident.

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4 ConclusionThese results show that soliton-like collisions are possible forlow power Gaussian beams propagating with low divergence.The divergence cutoff for the interaction has been found tobe between 1.84 and 1.96%, which is far greater that the idealcase of divergence outlined in the literature.10 This suggeststhat, although the interactions observed were weak, strongerinteraction should be possible for beams propagating withdivergence closer to the ideal theoretical value.

References1. A. Keshavarz and Z. Khalifeh, “One-dimensional optical bright screen-

ing photovoltaic photorefractive solitons, soliton pairs and incoher-ent interaction between them in BaTiO3 crystal,” Optik 120, 535–542(2009).

2. G. S. Garcia-Quirino, M. D. Iturbe Casitillo, V. A. Vysloukh,J. J. Sanchez-Mondragon, S. I. Stepanov, G. Lugo-Martinez, andG. E. Torres-Cisneros, “Observation of interaction forces between one-dimensional spatial solitons in photorefractive crystals,” Opt. Lett. 22,154–156 (1997).

3. H. X. Meng, G. Salamo, M. Shih, and M. Segev, “Coherent collisonsof photorefractive solitons,” Opt. Lett. 22, 448–450 (1997).

4. J. A. Andrade-Lucio, B. Alvarado-Mendez, R. Rojas-Laguna, O. G.Ibarra-Manzano, G. E. Torres-Cisneros, R. Jaime-Rivas, and E. A.Kuzin, “Optical switching by coherent collison of spatial solitons,”Electron. Lett. 36, 1403–1405 (2000).

5. W. K. Lee and T. S. Chan, “Experimental investigation of phase depen-dent interactions of photovoltaic bright spatial solitons in photorefrac-tive Fe:LiNbO3,” J. Opt. Soc. Am. B 23, 1920–1924 (2006).

6. W. L. She, K. K. Lee, and W. K. Lee, “Observation of two-dimensionalbright photovoltaic spatial solitons,” Phys. Rev. Lett. 83, 3182–3185(1999).

7. W. L. She, C. C. Xu, B. Guo, and W. K. Lee, “Formation of photovoltaicbright spatial soliton in photorefractive LiNbO3 crystal by a defocusedlaser beam induced by a background laser beam,” J. Opt. Soc. Am. B23, 2121–2126 (2006).

8. A. Rausch, A. Kiessling, and R. Kowarschik, “Self-focusing withoutexternal electric field in BaTiO3,” Opt. Express 14, 6207–6212 (2006).

9. M. W. Jones, E. Jaatinen, and G. Michael, “Propagation of low in-tensity Gaussian fields in photorefractive media with real and imagi-nary intensity-dependent refractive index components,” Appl. Phys. B(2010).

10. E. Jaatinen and M. W. Jones, “Thoeretical model of non-divergingGaussian beam propagation in self-defocusing photorefractive media,”Opt. Commun. 281, 3201–3207 (2008).

11. M. W. Jones and E. Jaatinen, “Transition from intensity dependentabsorption to transperancy in Ce:BaTiO3,” Opt. Mater. 31, 122–125(2008).

12. C. Yang, Y. Zhang, X. Yi, P. Yeh, Y. Zhu, M. Hui, and X. Wu, “Intensity-dependent absorption and photorefractive properties in cerium-dopedBaTiO3 crystals,” J. Appl. Phys. 78, 4323–4330 (1995).

Optical Engineering January 2011/Vol. 50(1)019701-3

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7. Conclusion

This PhD thesis studied the propagation of low-power Gaussian fields in unbiased

self-defocusing photorefractive media with low spatial divergence. The main goals

were to theoretically investigate and experimentally verify the existence of these

fields, and to produce soliton-like interactions between two such fields. This research

is important for the application of rewritable all optical interconnects in nonlinear

media, where current alternatives require large applied voltages or high power

background illumination, thus making them unsuitable for replacing existing

electrical circuitry. This research outlines a case where this additional voltage or

background illumination is not required, opening up the possibility to create

rewritable optical circuitry.

To realise these aims, a variety of methods were utilised. Far field images

were used to determine the evolution of the complex beam parameter, q, of the

Gaussian beam after it had passed through the PR medium. By comparing the

complex beam parameter before and after the beam had propagated through the

medium, the effect that the medium had on the beam was determined. By translating

the PR medium in a z-scan arrangement the beam is incident on the medium with a

different complex beam parameter without altering the optical power of the beam.

Numerical wavefront studies were coupled with experimental observations to gain an

understanding of the evolution of the beam within the PR medium. A combination of

these techniques allowed a complete picture of the evolution of a low powered

Gaussian beam as it propagates through PR media.

In Chapter 3, an analytical model was developed for the conditions required

in order for a low power Gaussian beam to propagate in self-defocusing PR media

92

with low divergence. This approach relies on the finding that at low powers, a

Gaussian beam propagating in PR media maintains its Gaussian spatial profile. This

result allows the refractive index profile to be treated as a quadratic function from the

beam axis, allowing simple solutions to the wave equation. Numerical analysis of

this model showed that under ideal conditions, a Gaussian beam could propagate in

unbiased self-defocusing PR media with a divergence of approximately 0.2 % after 1

m. However, these ideal conditions require a PR medium that is self-defocusing,

whilst also displaying intensity dependent transparency.

In Chapter 4, the wavelength dependence of the intensity dependent

absorption of a Ce:BaTiO3 crystal was examined to determine if the ideal conditions

for low divergence fields outlined in Chapter 3 were physically realisable. It was

found that as the wavelength of the incident light increase, the intensity dependent

absorption decreases, with intensity dependent transparency observed at an incident

wavelength of 790 nm. Theoretical fits to the data obtained revealed that it was the

shallow traps in the two centre model that were responsible for the wavelength

dependence of the intensity dependent absorption of the Ce:BaTiO3 crystal.

Chapter 5 outlined a theoretical approach to the evolution of the complex

beam parameter in PR media with a quadratic beam parameter. Unlike the model

discussed in Chapter 3, this model allowed for the change in the real and imaginary

parts of the refractive index to be functions of the propagation distance through the

medium. A numerical wavefront study of this model was applied to several

scenarios. Of particular interest was the apparent self-focusing, and the propagation

of low divergence Gaussian beams in unbiased self-defocusing PR media. With

respect to low divergent Gaussian fields, the study showed that under the ideal

conditions, the divergence of the beam will be less that 0.003 % over a 5 mm long

93

crystal. In addition, in a self-defocusing PR medium that displays intensity

dependent absorption, the divergence will be less than 0.02 %. These results suggest

that it is possible to create a low divergence field in Ce:BaTiO3 at an incident

wavelength of 532 nm, where the medium is self-defocusing and displays intensity

dependent absorption.

Spatial optical solitons interact with each other when the interaction angle is

small. Although the interaction that occurs is dependent on a variety of factors, a

repulsive interaction of this type suggests that the field is propagating in a way that is

analogous to a spatial soliton. Therefore, Chapter 6 aimed to determine if two low

divergence fields with a wavelength of 532 nm could interact in a Ce:BaTiO3 crystal.

The results showed that although far from the ideal low divergence being realised, a

weak interaction was observed. In addition, the strength of the interaction increased

as the divergence decreased, indicating that stronger interactions should be possible

with lower divergence fields. To be of practical use, these interactions would have to

occur at an angle greater than the divergence angle to ensure the two beams could be

separated downstream.

This research outlined the possibility that low power, low divergence

Gaussian fields could propagate in unbiased self-defocusing PR media. Further to

that, this research illustrated that although these fields are not perfect spatial solitons,

i.e., they still diverge to a degree, they are capable of spatial soliton-like interactions.

In this respect, several questions are left unanswered;

1. Is it possible to achieve an interaction with sufficient strength to enable

complete separation of the exit beams?

2. Are the low divergent fields described capable of guiding a second beam at a

second wavelength?

94

3. As with spatial soliton interactions, are the interactions of two of these fields

capable of manipulation? For example, can two repulsive beams be attracted

as a result of a relative phase shift between the two beams?

4. Could a guided beam be guided through an interaction, making a true all

optical interconnect?

The answers to these questions will reveal the feasibility of low power, low divergent

Gaussian fields in quadratic nonlinear media as all optical interconnects, replacing

electronic circuitry.

95

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