The following thesis is comprised of four main areas of work. These are centred around the experimental observation of phenomena associated with both linear and non-linear optics in silicon photonic-wires. As a comparison, I also discuss a similar coupled-waveguide system; dual-core hollow-core photonic crystal fibre. To introduce the reader to this work, the first chapter will recap some undergraduate level theory; a general introduction to optical waveguides. It is not intended to be a complete theoretical picture, as many beautiful texts on optics already exist [1–3]. This chapter concerns itself only with the aspects of optics with which the author was intimately aware of throughout the completion of this thesis. Thereafter, the chapters become specific to the particular experiments undertaken. Each one follows a simple framework: examination of the relevant theory, extending upon that already discussed in the first chapter, a literature review and finally a discussion of the work I completed within this thesis. Chapter 2 is the only chapter not related to silicon based photonics. Here I discuss dual-core hollow-core photonic crystal fibres; including guidance mechanisms, fabrication methods and the numerical modelling techniques employed in my work. I will compare these numerical results to experimental results taken by colleagues at the university of Bath. Chapter 3 analyses linear propagation in arrays of silicon photonic wires. I extend the simple picture of light propagating in waveguides to discuss the di↵erent types of dispersion inherent in this system and how dispersion tailoring can be achieved; with reference to the other literature on this topic. Experimental results are examined and discussed. Chapters 4 and 5 discuss non-linear propagation in silicon photonic wire arrays; modulation instability and spatio-temporal solitons respectively. In each case I extend the ideas on non-linearity presented in Chapter 1 to explain both modulation instability and optical solitons. Detailed descriptions of the experiments undertaken, and associated numerical modelling completed are then discussed. Whilst the work I present is incomplete, I will discuss subsequent work performed by my colleagues at the University of Bath based on my initial work. Finally, Chapter 6 draws together my conclusions.

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.

Propagation of coherent waves through disordered media, whether optical, acoustic, or radio waves, results in a spatially redistributed random intensity pattern known as speckle -- a statistical phenomenon. The subject of this dissertation is the statistics of monochromatic coherent light traversing disordered photonic lattices and its dependence on the disorder class, the level of disorder and the excitation configuration at the input. Throughout the dissertation, two disorder classes are considered, namely, diagonal and off-diagonal disorders. The latter exhibits disorder-immune chiral symmetry -- the appearance of the eigenmodes in skew-symmetric pairs and the corresponding eigenvalues in opposite signs. When a disordered photonic lattice, an array of evanescently coupled waveguides, is illuminated with an extended coherent optical field, discrete speckle develops. Numerical simulations and analytical modeling reveal that discrete speckle shows a set of surprising features, that are qualitatively indistinguishable in both disorder classes. First, the fingerprint of transverse Anderson localization -- associated with disordered lattices, is exhibited in the narrowing of the spatial coherence function. Second, the transverse coherence length (or speckle grain size) freezes upon propagation. Third, the axial coherence depth is independent of the axial position, thereby resulting in a coherence voxel of fixed volume independently of position. When a single lattice site is coherently excited, I discovered that a thermalization gap emerges for light propagating in disordered lattices endowed with disorder-immune chiral symmetry. In these systems, the span of sub-thermal photon statistics is inaccessible to the input coherent light, which -- once the steady state is reached -- always emerges with super-thermal statistics no matter how small the disorder level. An independent constraint of the input field for the chiral symmetry to be activated and the gap to be observed is formulated. This unique feature enables a new form of photon-statistics interferometry: by exciting two lattice sites with a variable relative phase, as in a traditional two-path interferometer, the excitation-symmetry of the chiral mode pairs is judiciously broken and interferometric control over the photon statistics is exercised, spanning sub-thermal and super-thermal regimes. By considering an ensemble of disorder realizations, this phenomenon is demonstrated experimentally: a deterministic tuning of the intensity fluctuations while the mean intensity remains constant. Finally, I examined the statistics of the emerging light in two different lattice topologies: linear and ring lattices. I showed that the topology dictates the light statistics in the off-diagonal case: for even-sited ring and linear lattices, the electromagnetic field evolves into a single quadrature component, so that the field takes discrete phase values and is non-circular in the complex plane. As a consequence, the statistics become super-thermal. For odd-sited ring lattices, the field becomes random in both quadratures resulting in sub-thermal statistics. However, this effect is suppressed due to the transverse localization of light in lattices with high disorder. In the diagonal case, the lattice topology does not play a role and the transmitted field always acquires random components in both quadratures, hence the phase distribution is uniform in the steady state.