An investigation of lattice dynamics using waveguide arrays
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The propagation of light through a periodic array of evanescently coupled optical waveguides is described by the paraxial equation. This Schrodinger-like equation closely resembles the Schr¨odinger equation describing the motion of a particle in a periodic potential. This close correspondence means that the evolution of the light mimics the complex dynamics of a quantum particle in a lattice. This mapping combined with the powerful capabilities of ultrafast laser inscription to precisely control the properties of the simulated lattice makes coupled optical waveguides a potent probe of solid-state phenomena. In this thesis we theoretically investigate and experimentally observe, using the photonic platform, various single-particle effects from solid-state physics. These include a new type of particle localisation due to flat energy bands and a novel type of topological edge mode which is unique to slowly-driven lattices. In addition theoretical results are presented which show how a particle subject to an artificial magnetic flux can be simulated using optical waveguides. This result paves the way for the use of photonic lattices to investigate the paradigmatic Hofstadter-Harper model and its associated topological properties. Moving beyond single-particle effects photonic lattices are capable of investigating certain phenomena associated with particle interactions, such as the dynamics of two interacting particles in one-dimension. In this thesis the dynamics of two interacting particles in two quasi one-dimensional lattices, the cross-stitch and diamond lattices, are theoretically investigated. The single-particle energy spectrums of these lattices both feature a flat band which implies that the particle’s dynamics within this band are determined solely by the interaction. The theoretical work conducted in this thesis is mainly focused around the experimental platform of photonic lattices. The results, however, are derived from a Schrodinger-like equation which implies that they will be relevant to a wide community of researchers.