Abstract
Simple connections between structure and optical response empower us with essential intuition to engineer complex optical functionalities. In this thesis, I study photonic crystals, quasicrystals and amorphous materials to quantify the structural properties that give rise to photonic band gaps (PBGs). Along the way, I develop two novel perspectives on the optical analysis of arbitrarily-structured media: generalised photonic band structure and local self-uniformity (LSU). Generalised photonic band structures reproduce the Bloch-wave band structure of photonic crystals but, crucially, also yield naturally unfolded complete dispersion relations for aperiodic materials. Using generalised band structures, I demonstrate that the overall form of a material's dispersion relation is determined by the non-zero momentum transfers of its structure factor. I observe in great detail the fractal-like PBG spectra of a pair of Penrose photonic quasicrystals. Here, I demonstrate that, in most cases, the Penrose PBGs form through a mechanism of Bragg scattering-induced standing wave formation. The fundamental gap of each structure, however, is attributed to spatially localised scattering resonances. I also fabricate and characterise hyperuniform gold metasurfaces. Fluorescence emission characterisation reveals a statistically isotropic distribution of momentum states within the light cone; this property is shown to result from the metasurface structure factor. In the second part of this thesis I introduce LSU as a continuous measure of the extent to which a network possesses an optimal PBG-forming structure. Specifically, LSU measures the geometrical and topological similarities of the local vertex environments in a network of uniform valency. I demonstrate that both known optimal photonic crystal structures and disordered PBG-forming networks possess significant LSU. Further, I produce the first known designs of amorphous gyroid networks. Amorphous gyroids possess significant LSU and can exhibit a sizeable complete PBG; these PBGs are validated experimentally by performing microwave transmission experiments on centimetre-scale alumina prototypes (εr = 9:5 ± 0.3 at 22 GHz). Using ensembles of both amorphous gyroids and planar hyperuniform networks, I demonstrate the striking correlation between LSU and PBG-forming ability. I rationalise the success of LSU by advancing a picture of photonic tight binding in high index connected networks. This picture explains the origin of PBGs in both ordered and disordered connected networks, and suggests why the diamond architecture possesses the largest known PBG. To conclude, I explore the possibility that amorphous gyroids exist in the wing scales of butteries. I reveal that the microstructure in the scales of Pseudolycaena marsyas possesses substantial amorphous gyroid character and demonstrate that the buttery's reflectance spectrum can be effectively reproduced by amorphous gyroid microstructures.