Chapter One Computation and visualization of photonic quasicrystal spectra

In this chapter, we describe various approaches to computing the spectra of photonic qua-sicrystals (PQC). We focus on a recently developed superspace method for computing the spectra and eigenstates of PQCs defined by the standard cut-and-project construction, that involves directly solving Maxwell’s equations in higher dimensions, where a generalization of Bloch’s theorem applies. We emphasize the conceptual and practical differences between superspace and supercell methods, which involve computing the eigenfrequencies and/or local density of states to an approximation of the true aperiodic structure. 1.1. Introduction and Background Photonic quasicrystals (PQCs) have a number of unique properties compared to ordinary periodic structures,1–28 especially in two or three dimensions where they can have greater rotational symmetry and therefore offer some hope of achieving complete photonic band gaps with lower index contrast8,26,27,29–31 (lower than th