AbstractsPhysics

Paraxial light propagation in media with a periodically modulated refractive index - photonic lattices - shares many similarities with condensed matter and quantum systems, such as electron dynamics in crystals, Bose-Einstein condensates in optical lattices, and polaritons in structured microcavities. Analogies between these different physical settings are both fundamentally interesting and have practical applications, providing novel ways to control the flow of light in optical devices. For example, the concept of a photonic band gap was inspired by electronic band gaps in semiconductors. Following seminal advances in condensed matter physics in the past decade including the isolation of graphene and the discovery of topological insulators, it is now crucially important to explore the opportunities offered by photonic analogues of these exotic systems. This thesis studies theoretically and experimentally the linear and nonlinear singular optics of photonic lattices, with emphasis on the interplay between wave singularities including optical vortices, and singularities in their energy-momentum spectrum such as conical intersections and flat bands. Beginning with ring lattices governed by the discrete nonlinear Schr¨odinger equation, we study in detail the existence, stability, and nonlinear dynamics of discrete vortex solitons, establishing novel mechanisms for all-optical switching of their topological charge, including the use of a parity time-symmetric defect with balanced gain and loss to discriminate between left- and right-handed vortices. We demonstrate experimentally a powercontrolled vortex switch in a ring lattice optically-induced in a photorefractive crystal. Next we study connections between optical vortices, orbital angular momentum, and conical intersections. We show analytically that the orbital angular momentum of light is sensitive to the Bloch bands’ Berry curvature and the conical intersection’s pseudospin, confirming this result with numerical simulations of wavepacket propagation in honeycomb and kagome lattices. We present a detailed study of an integer pseudospin conical intersection appearing in the Lieb lattice, focusing on how its additional flat band influences linear and nonlinear conical diffraction, before verifying our predictions experimentally in a femtosecond laser-written lattice in fused silica glass. Generalising the Lieb lattice to other flat band lattices, we examine the important question of their robustness to disorder within the Anderson model of localisation. The singular, divergent flat band density of states leads to sensitivity to perturbations such as disorder, resulting in anomalous scaling of the Anderson localisation length and heavytailed linear and nonlinear transport statistics which we explain using an analogy with Fano resonances. We find correlated disorder can transform the singular density of states into weaker square root or logarithmic singularities. Finally, we analyse spontaneous parametric down-conversion in one…