Computation and visualization of photonic quasicrystal spectra via Bloch's theorem

Previous methods for determining photonic quasicrystal (PQC) spectra have relied on the use of large supercells to compute the eigenfrequencies and/or local density of states. In this paper, we present a method by which the energy spectrum and the eigenstates of a PQC can be obtained by solving Maxwell's equations in higher dimensions for any PQC defined by the standard cut-and-project construction, to which a generalization of Bloch's theorem applies. In addition, we demonstrate how one can compute band structures with defect states in the same higher-dimensional superspace. As a proof of concept, these general ideas are demonstrated for the simple case of one-dimensional quasicrystals, which can also be solved by simple transfer-matrix techniques.