The propagation dynamics of light in optical waveguide arrays is characteristic of that encountered in discrete systems. As a result, it is possible to engineer the diffraction properties of such structures, which leads to the ability to control the flow of light in ways that are impossible in continuous media. In this work, a detailed theoretical investigation of both linear and nonlinear optical wave propagation in one- and two-dimensional waveguide lattices is presented. The ability to completely overcome the effects of discrete diffraction through the mutual trapping of two orthogonally polarized coherent beams interacting in Kerr nonlinear arrays of birefringent waveguides is discussed. The existence and stability of such highly localized vector discrete solitons is analyzed and compared to similar scenarios in a single birefringent waveguide. This mutual trapping is also shown to occur within the first few waveguides of a semi-infinite array leading to the existence of vector discrete surface waves. Interfaces between two detuned semi-infinite waveguide arrays or waveguide array heterojunctions and their possible applications are also considered. It is shown that the detuning between the two arrays shifts the dispersion relation of one array with respect to the other. Consequently, these systems provide spatial filtering functions that may prove useful in future all-optical networks. In addition by exploiting the unique diffraction properties of discrete arrays, diffraction compensation can be achieved in a way analogous to dispersion compensation in dispersion managed optical fiber systems. Finally, it is demonstrated that both the linear (diffraction) and nonlinear dynamics of two-dimensional waveguide arrays are significantly more complex and considerably more versatile than their one-dimensional counterparts. As is the case in one-dimensional arrays, the discrete diffraction properties of these two-dimensional lattices can be effectively altered depending on the propagation Bloch k-vector within the first Brillouin zone. In general, this diffraction behavior is anisotropic and as a result, allows the existence of a new class of discrete elliptic solitons in the nonlinear regime. Moreover, such arrays support two-dimensional vector soliton states, and their existence and stability are also thoroughly explored in this work.

Soliton states in three coupled optical waveguide systems were studied: two linearly coupled waveguides with quadratic nonlinearity, two linearly coupled waveguides with cubic nonlinearity and Bragg gratings, and a quadratic nonlinear waveguide with resonant gratings, which enable three-wave interaction. The methods adopted to tackle the problems were both analytical and numerical. The analytical method mainly made use of the variational approximation. Since no exact analytical method is available to find solutions for the waveguide systems under study, the variational approach was proved to be very useful to find accurate approximations. Numerically, the shooting method and the relaxation method were used. The numerical results verified the results obtained analytically. New asymmetric soliton states were discovered for the coupled quadratically nonlinear waveguides, and for the coupled waveguides with both cubic nonlinearity and Bragg gratings. Stability of the soliton states was studied numerically, using the Beam Propagation Method. Asymmetric couplers with quadratic nonlinearity were also studied. The bifurcation diagrams for the asymmetric couplers were those unfolded from the corresponding diagrams of the symmetric couplers. Novel stable two-soliton bound states due to three-wave interaction were discovered for a quadratically nonlinear waveguide equipped with resonant gratings. Since the coupled optical waveguide systems are controlled by a larger number of parameters than in the corresponding single waveguide, the coupled systems can find a much broader field of applications. This study provides useful background information to support these applications.

published_or_final_version / abstract / Mechanical Engineering / Master / Master of Philosophy

The waveguiding effect of spatial solitary waves in nonlinear optical media has been suggested as a potential basis for future all-optical devices, such as optical interconnects. It has been shown that low power (∼ mW) beams, which can encode information, can be optically steered using external electric fields or through interactions with other beams. This opens up the possibility of creating reconfigurable optical interconnects. Nematic liquid crystals are a potential medium for such future optical interconnects, possessing many advantageous properties, including a “huge” nonlinear response at comparatively low input power levels. Consequently, a thorough understanding of the behaviour of spatial optical solitary waves in nematic liquid crystals, termed nematicons, is needed. The investigation of multiple beam interaction behaviour will form an essential part of this understanding due to the possibility of beam-on-beam control. Here, the interactions of two nematicons of different wavelengths in nematic liquid crystals, and the optical steering of nematicons in dye-doped nematic liquid crystals will be investigated with the aim of achieving a broader understanding of nematicon interaction and steering. The governing equations modelling nematicon interactions are nonintegrable, which means that nematicon collisions are inelastic and radiative losses occur during and after collision. Consequently numerical techniques have been employed to solve these equations. However, to fully understand the physical dynamics of nematicon interactions in a simple manner, an approximate variational method is used here which reduces the infinite-dimensional partial differential equation problem to a finite dynamical system of comparatively simple ordinary differential equations. The resulting ordinary differential equations are modified to include radiative losses due to beam evolution and interaction, and are then quickly solved numerically, in contrast to the original governing partial differential equations. N¨other’s Theorem is applied to find various conservation laws which determine the final steady states, aid in calculating shed radiation and accurately compute the trajectories of nematicons. Solutions of the approximate equations are compared with numerical solutions of the original governing equations to determine the accuracy of the approximation. Excellent agreement is found between full numerical solutions and approximate solutions for each physical situation modelled. Furthermore, the results obtained not only confirm, but explain theoretically, the interaction phenomena observed experimentally. Finally, the relationship between the nature of the nonlinear response of the medium, the trajectories of the beams and radiation shed as the beams evolve is investigated.

by Li Kam Shun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaf 65). / Chapter CHAPTER 1 --- Soliton Solutions of the Nonisospectral and Variable Coefficient Korteweg-de Vries Equation / Chapter §1.1 --- Introduction --- p.4 / Chapter §1.2 --- Inverse Scattering --- p.6 / Chapter §1.3 --- N-Soliton Solution --- p.11 / Chapter §1.4 --- One-Soliton Solutions --- p.15 / Chapter §1.5 --- Two-Soliton Solutions --- p.18 / Chapter §1.6 --- Oscillating and Asymptotically Standing Solitons --- p.23 / Chapter CHAPTER 2 --- Asymptotic Behaviour of Nonsoliton Solutions of the Nonisospectral and Variable Coefficient Korteweg-de Vries Equation / Chapter §2.1 --- Introduction --- p.31 / Chapter §2.2 --- Main Results --- p.36 / Chapter §2.3 --- Lemmas --- p.39 / Chapter §2.4 --- Proof of the Main Results --- p.59 / References --- p.65

Yuen Chi Yan = 有關光折變晶體中空間孤子、偏振轉動以及霍爾效應的研究 / 阮志仁. / Thesis submitted in: July 2004. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 81-82). / Text in English; abstracts in English and Chinese. / Yuen Chi Yan = You guan guang zhe bian jing ti zhong kong jian gu zi, pian zhen zhuan dong yi ji Huoer xiao ying de yan jiu / Ruan Zhiren. / Acknowledgments --- p.i / Abstract --- p.ii / Table of Contents --- p.v / Chapter Chapter 1 --- Photorefractive Spatial Soliton --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- Bright Spatial Soliton --- p.3 / Chapter 1.2.1 --- Experiment --- p.4 / Chapter 1.2.2 --- Results and Discussion --- p.6 / Chapter 1.2.2.1 --- Expansion --- p.6 / Chapter 1.2.2.2 --- Contraction --- p.10 / Chapter 1.3 --- Dark Spatial Soliton --- p.15 / Chapter 1.3.1 --- Experiment --- p.15 / Chapter 1.3.2 --- Results and Discussion --- p.20 / Chapter Chapter 2 --- Polarization Rotation --- p.23 / Chapter 2.1 --- Introduction --- p.23 / Chapter 2.2 --- Experiment --- p.24 / Chapter 2.3 --- Results and Discussion --- p.30 / Chapter 2.3.1 --- Effect of varying pump beam power --- p.30 / Chapter 2.3.2 --- Effect of different polarizations of signal beam --- p.41 / Chapter 2.3.3 --- Effect of signal beam size --- p.43 / Chapter 2.3.4 --- Effect of applied E-field --- p.46 / Chapter 2.3.5 --- Effect of signal beam and pump beam separation and perpendicularly --- p.52 / Chapter 2.3.6 --- Investigation of Δne using interferometer --- p.60 / Chapter 2.3.7 --- Computer Simulation --- p.69 / Chapter Chapter 3 --- Hall Effect --- p.72 / Chapter 3.1 --- Introduction --- p.72 / Chapter 3.2 --- Experiment --- p.75 / Chapter 3.3 --- Results and Discussion --- p.76 / Conclusion and Possible Further Works --- p.79 / References --- p.81

Quando os núcleons ligados num núcleo se sobrepõem, a sua estrutura interna influencia as propriedades nucleares. Em especial, a estatística dos quanta elementares que constituem o núcleon se torna relevante. No intuito de investigar estes efeitos no contexto dos modelos de sóliton, considera-se sistemas simples unidimensionais. No modelo de sine-Gordon, o operador de Mandelstam, que cria sólitons topológicos p ~.mtuais, é modificado de maneira a levar em conta a estrutura do sóliton. O operador resultante cria férmions de carga topológica unitária e a sua aplicação sobre o vácuo de Fock produz um estado coerente no qual o campo médio é dado pela solução solitônica clássica. Pela aplicação sucessiva dos operadores, criando sólitons centrados em pontos diferentes, obtem-se um estado coerente no qual o valor esperado do campo é dado pela soma dos campos médios individuais. No estado de dois sólitons, a energia média de interação possui um comportamento de barreira de potencial, evidenciando a repulsão entre sólitons neste modelo. Consideramos também um modelo simplificado de sóliton não - topológico constituído de um único férmion confinado por um campo escalar bosônico. Tanto no modelo topológico como no não - topológico, a variação da norma de um estado de dois sóli tons com a distância entre seus centros revela a competição entre as estatísticas fermiônicas e bosônicas. Na região de separação pequena, o aspecto fermiônico prevalece fazendo a norma do estado se aproximar de zero. Na região de separação média a norma excede a unidade devido à superposição dos quanta bosônicos. / When the nucleons, bounded inside a nucleus, overlap, their internai structure affects the nuclear properties. In particular, the statistics of the elementary quanta that constitute the nucleon become relevant. In order to investigate such effects in the context of soliton models, we consider simple one-dimensional sistems. In the sine-Gordon model, the Mandelstam operator, which creates point-like topological solitons, is modifided in such a way as to account for the soliton structure. The resulting operator creates fermions with unitary topological charge and its application on the Fock vacuum produces a coherent state in which the mean field is given by the classical solitonic solution. By successive application of soliton creation operators, centered at different points, one obtains a coherent state in which the mean field is given by the sum of the individual mean fields. In the two-soliton state the mean interaction energy possesses a potential barrier behavior, displaying the repulsion between solitons in this model. Vve also consider a simplified non-topological soliton model, composed of one fermion confined in a scalar bosonic field. In the topological soliton modelas well as in the non-topological one, the variation of the norm of the two-soliton state, with the distance between their centers reveals a competition between the fermionic and bosonic statistics. In the small separation region the fermionic aspect prevails, making the norm go to zero. In the intermediate separation region, the norm exceeds unity due to the overlap of the bosonic quanta.

Zhou Yong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 51-53). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Chapter 1 --- Introduction --- p.2 / Chapter 2 --- Preliminaries --- p.10 / Chapter 3 --- Periodic Case --- p.22 / Chapter 4 --- Non-periodic Case --- p.35 / Bibliography --- p.51

This thesis is about some extensions of the ideas and techniques used in integrable field theories to deal with non-integrable theories. It is presented in two parts. The first part deals with gauge theories in 3 and 4 dimensional space-time; we propose what we call the integral formulation of them, which at the end give us a natural way of defining the conserved charges that are gauge invariant and do not depend on the parametrisation of space-time. The definition of gauge invariant conserved charges in non-Abelian gauge theories is an open issue in physics and we think our solution might be a first step into its full understanding. The integral formulation shows a deeper connection between different gauge theories: they share the same basic structure when written in the loop space. Moreover, in our construction the arguments leading to the conservation of the charges are dynamical and independent of the particular solution. In the second part we discuss the recently introduced concept called quasi-integrability: one observes soliton-like configurations evolving through non-integrable equations having properties similar to those expected for integrable theories. We study the case of a model which is a deformation of the non-linear Schr¨odinger equation consisting of a more general potential, connected in a way with the integrable one. The idea is to develop a mathematical approach to treat more realistic theories, which is in particular very important from the point of view of applications; the NLS model appears in many branches of physics, specially in optical fibres and Bose-Einstein condensation. The problem was treated analytically and numerically, and the results are interesting. Indeed, due to the fact that the model is not integrable one does not find an infinite number of conserved charges but, instead, a set of infinitely many charges that are asymptotically conserved, i.e., when two solitons undergo a scattering process the charges they carry before the collision change, but after the collision their values are recovered. / Essa tese discute algumas extensões de ideias e técnicas usadas em teorias de campos integráveis para tratar teorias que não são integráveis. Sua apresentação é feita em duas partes. A primeira tem como tema teorias de calibre em 3 e 4 dimensões; propomos o que chamamos de equação integral para uma tal teoria, o que nos permite de maneira natural a construção de suas cargas invariantes de calibre, e independentes da parametrização do espaço-tempo. A definição de cargas conservadas in variantes de calibre em teorias não-Abelianas ainda é um assunto em aberto e acreditamos que a nossa solução pode ser um primeiro passo em seu entendimento. A formulação integral mostra uma conexão profunda entre diferentes teorias de calibre: elas compartilham da mesma estrutura básica quando formuladas no espaço dos laços. Mais ainda, em nossa construção os argumentos que levam `a conservação das cargas são dinâmicos e independentes de qualquer solução particular. Na segunda parte discutimos o recentemente introduzido conceito de quasi-integrabilidade: em (1 + 1) dimensões existem modelos não integráveis que admitem soluções solitonicas com propriedades similares `aquelas de teorias integráveis. Estudamos o caso de um modelo que consiste de uma deformação (não-integrável) da equação de Schrödinger não-linear (NLS), proveniente de um potencial mais geral, obtido a partir do caso integrável. O que se busca é desenvolver uma abordagem matemática sistemática para tratar teorias mais realistas (e portanto não integráveis), algo bastante relevante do ponto de vista de aplicações; o modelo NLS aparece em diversas áreas da física, especialmente no contexto de fibra ótica e condensação de Bose-Einstein. O problema foi tratado de maneira analítica e numérica, e os resultados se mostram interessantes. De fato, sendo a teoria não integrável não é encontrado um conjunto com infinitas cargas conservadas, mas, pode-se encontrar um conjunto com infinitas cargas assintoticamente conservadas, i.e., quando dois solitons colidem as cargas que eles tinham antes tem os seus valores alterados, mas após a colisão, os valores inicias, de antes do espalhamento, são recobrados.

Thesis (Ph. D.)--University of Hong Kong, 2007. / Title proper from title frame. Also available in printed format.