A new approach for modeling aperiodic photonic bandgap crystals

Abstract

Photonic Bandgap Crystals (PBGs) have begun to see use in optical communication networks and photonic device coupling due to their high, selective transmission bandgaps and novel optical filtering properties. Understanding how PBGs can be designed for varying purposes is crucial in continuing to find practical applications of these crystals. PBGs are typically periodic as the math is well understood and the transmission or reflection of periodic optical structures is generally well-known, even in closed form. Crystal lattices in position space can be shown to correlate to an “inverse lattice” in momentum space through Fourier analysis. With further description, we can express any form of crystal as a sum of its fundamental, periodic lattices. For simplicity, we analyze only 1-dimensional structures but generate the groundwork for descriptions in higher dimensions. This description of aperiodic lattices allows us to extract information about effective lattice periods and effective lattice wave numbers supported by the crystal lattice. Using the Transfer Matrix Method (TMM) we further examine the optical properties of aperiodic structures by generating dispersion diagrams of general structures. By comparing the results of the TMM and the results of the effective lattice parameters, we seek to determine the efficacy of the claim that disordered structures are sums of ordered structures. To do so, we will apply novel disordered functions such as the logistic map to generate layer thicknesses of an aperiodic crystal and compare results from both the TMM and Lumerical FDTD simulations.