Nonlinear Propagation of Light in One Dimensional Periodic Structures

We consider the nonlinear propagation of light in an optical fiber waveguide as modeled by the anharmonic Maxwell-Lorentz equations (AMLE). The waveguide is assumed to have an index of refraction which varies periodically along its length. The wavelength of light is selected to be in resonance with the periodic structure (Bragg resonance). The AMLE system considered incorporates the effects non-instantaneous response of the medium to the electromagnetic field (chromatic or material dispersion), the periodic structure (photonic band dispersion) and nonlinearity. We present a detailed discussion of the role of these effects individually and in concert. We derive the nonlinear coupled mode equations (NLCME) which govern the envelope of the coupled backward and forward components of the electromagnetic field. We prove the validity of the NLCME description and give explicit estimates for the deviation of the approximation given by NLCME from the {\it exact} dynamics, governed by AMLE. NLCME is known to have gap soliton states. A consequence of our results is the existence of very long-lived {\it gap soliton} states of AMLE. We present numerical simulations which validate as well as illustrate the limits of the theory. Finally, we verify that the assumptions of our model apply to the parameter regimes explored in recent physical experiments in which gap solitons were observed.Comment: To appear in The Journal of Nonlinear Science; 55 pages, 13 figure

Light propagation in tuneable nonlinear periodic photonic structures

The action of light in periodic structures can be quite different to that in a homogenous medium. For example, while a nonlinear beam will spread out in a medium with a negative nonlinearity, in a periodic structure the beam is focused and a localised state is formed. In this thesis I will show my work on light propagation in tuneable nonlinear periodic photonic structures. Nature provides us with dazzling displays of periodic photonic structures in the form of butterfly wings, peacock feathers, and opals. How these magnificent natural spectacles work has been a source of great scientific interest since we mastered the modern scientific method. With new technologies we can utilise periodic photonic structures to control how light propagates, which wavelengths are transmitted or reflected, and how light moves between waveguiding structures. Coupled waveguides provide a platform in which to study the linear and nonlinear light propagation and interaction in periodic photonic structures. Nonlinearity in optics provides a feedback mechanism which allows one beam of light to influence the propagation of another, or even itself. Advancements in our understanding of how light propagates and interacts in nonlinear periodic photonic structures is leading us to new and interesting areas of Physics. It is hoped that one day photons and photonic components can be used in place of electrons in electronic components widely used today. This will propel our computing power and further advance our understanding of the physical universe. In order to fully understand how light behaves in photonic structures and to make use of nonlinear features to allow light to control light, we first must understand the fundamental interactions of light in linear and nonlinear periodic photonic structures. We must be able to tune the properties of the system to investigate the fundamental behaviour of nonlinear beam propagation. In this thesis I investigate light propagation in tuneable nonlinear periodic photonic structures. I begin by introducing relevant concepts and ideas necessary to understand my work (Chapter 1). Included in this introduction is theoretical and experimental work I conducted with two interacting beams in a bulk nonlinear liquid (Sec. 1.4.6). I discover that a high power pump beam influences the nonlinear medium in a way which locally alters its refractive index. This alteration occurs due to a change in temperature of the medium caused by absorption of the pump beam and results in the reflection of a probe beam from the pump beam. I then present my research on the development of two platforms in which liquid is used to guide light in a one-dimensional (1D) periodic array. The first platform is made from photolithographically defined air-filled channels in SU8 polymer (Sec. 2.1). These channels are infiltrated with an index matching oil and the linear diffraction is observed as the temperature of the platform is changed. I find that the discrete diffraction observed matches very well with an accompanying theoretical model of the system, and I am able to estimate the temperature of the liquid in the channels. The second platform for light propagation in a 1D periodic array is developed using selectively infiltrated Photonic Crystal Fibres (Sec. 2.2). I use a simple method of blocking an inverse pattern with oil on one side of the fibre. The other end of the fibre is then submersed in a reservoir of the infiltrating liquid to fill any unblocked holes. I produce a 1D periodic array in a of coupled waveguides and demonstrate temperature tuneable linear diffraction, and nonlinear defocusing. I then move on to present my observation of truncated nonlinear Bloch waves in Lithium Niobate waveguide arrays (Sec. 2.3). Such states are excited with a broad Gaussian input beam in a 1D array of coupled nonlinear waveguides. This state is different from well known solitons and nonlinear Bloch modes because it contains features of both: a constant phase across all guiding waveguides characteristic of a nonlinear Bloch wave, with sharp edges otherwise seen in gap solitons. This work is supported by theoretical modelling, and I am able to show that the width of the soliton is dependant only on the width of the input beam, in contrast to discrete or gap solitons who's width depends on the nonlinearity. Chapter 3 then exhibits my work with liquid infiltrated Photonic Crystal Fibres as a two-dimensional (2D) periodic array of nonlinear waveguides. Firstly I show the existence and excitation conditions of nonlocal gap solitons (Sec. 3.1), where the properties of the system far from the light field influence soliton formation. I find that below a certain refractive index contrast these solitons are no longer excitable and the beam only defocuses. I then present my work on this crossover from focusing to defocusing in nonlinear periodic systems (Sec. 3.2). I show that the bandgap closes before the index contrast reaches zero, and that the system crosses from focusing to defocusing before the bandgap is fully closed. I will finally discuss my theoretical and experimental work on vortex beams propagating around a surface in a nonlinear hexagonal array (Sec. 3.3). I use liquid infiltrated Photonic Crystal Fibres and propagate a vortex beam around the core defect of the fibre. I find that nonlinear vortex modes of charge one are unstable and will focus to occupy a single waveguide on the surface of the core using the discrete model. A continuous model shows that linear and nonlinear charge one vortex modes are unstable and result in an asymmetric output. Linear charge three vortex modes show greater stability due to the staggered phase profile of the input beam, while nonlinear charge three vortex modes lose symmetry at the output due to a loss of this phase profile. I will finish this thesis with conclusions about my work and ideas for future directions this work could take, including specific experimental ideas directly related to this work. I will include some ideas as to the future direction these ideas may provide

Highly nonlinear solitary waves in heterogeneous periodic granular media

We use experiments, numerical simulations, and theoretical analysis to investigate the propagation of highly nonlinear solitary waves in periodic arrangements of dimer (two-mass) and trimer (three-mass) cell structures in one-dimensional granular lattices. To vary the composition of the fundamental periodic units in the granular chains, we utilize beads of different materials (stainless steel, brass, glass, nylon, polytetrafluoroethylene, and rubber). This selection allows us to tailor the response of the system based on the masses, Poisson ratios, and elastic moduli of the components. For example, we examine dimer configurations with two types of heavy particles, two types of light particles, and alternating light and heavy particles. Employing a model with Hertzian interactions between adjacent beads, we find good agreement between experiments and numerical simulations. We also find good agreement between these results and a theoretical analysis of the model in the long-wavelength regime that we derive for heterogeneous environments (dimer chains) and general bead interactions. Our analysis encompasses previously-studied examples as special cases and also provides key insights on the influence of heterogeneous lattices on the properties (width and propagation speed) of the nonlinear wave solutions of this system

Nonlinear optics and light localization in periodic photonic lattices

We review the recent developments in the field of photonic lattices emphasizing their unique properties for controlling linear and nonlinear propagation of light. We draw some important links between optical lattices and photonic crystals pointing towards practical applications in optical communications and computing, beam shaping, and bio-sensing.Comment: to appear in Journal of Nonlinear Optical Physics & Materials (JNOPM

Nonlinear photonic lattices in anisotropic nonlocal self-focusing media

We analyze theoretically and generate experimentally two-dimensional nonlinear periodic lattices in a photorefractive medium. We demonstrate that the light-induced periodically modulated nonlinear refractive index is highly anisotropic and nonlocal, and it depends on the lattice orientation relative to the crystal axis. We discuss stability of such induced photonic structures and their guiding properties.Comment: 3 pages, 3 figure

Nonlinear Bloch modes in two-dimensional photonic lattices

We generate experimentally different types of two-dimensional Bloch waves of a square photonic lattice by employing the phase imprinting technique. We probe the local dispersion of the Bloch modes in the photonic lattice by analyzing the linear diffraction of beams associated with the high-symmetry points of the Brillouin zone, and also distinguish the regimes of normal, anomalous, and anisotropic diffraction through observations of nonlinear self-action effects.Comment: 11 pages, 8 figure

Nonlinear switching and solitons in PT-symmetric photonic systems

One of the challenges of the modern photonics is to develop all-optical devices enabling increased speed and energy efficiency for transmitting and processing information on an optical chip. It is believed that the recently suggested Parity-Time (PT) symmetric photonic systems with alternating regions of gain and loss can bring novel functionalities. In such systems, losses are as important as gain and, depending on the structural parameters, gain compensates losses. Generally, PT systems demonstrate nontrivial non-conservative wave interactions and phase transitions, which can be employed for signal filtering and switching, opening new prospects for active control of light. In this review, we discuss a broad range of problems involving nonlinear PT-symmetric photonic systems with an intensity-dependent refractive index. Nonlinearity in such PT symmetric systems provides a basis for many effects such as the formation of localized modes, nonlinearly-induced PT-symmetry breaking, and all-optical switching. Nonlinear PT-symmetric systems can serve as powerful building blocks for the development of novel photonic devices targeting an active light control.Comment: 33 pages, 33 figure