Numerical Study of Three-Dimensional Photonic Crystals with Large Band Gaps

Abstract

The study presents a finite difference formulation for efficiently computing band structures of three-dimensional photonic crystals. First of all, we will show how to correctly discretize the double-curl equation for the magnetic field so that the transversality condition is exactly satisfied in the discrete sense. The first few branches of nontrivial eigenfrequencies that determine the major full band gaps of photonic crystals are computed by interlacing an inverse method with conjugate gradient projection and full multigrid acceleration. The presently developed method is applied to compute band structures of photonic crystals with modified simple cubic lattice, tetragonal square spiral structure (direct and inverse structures), and diamond structure with sp³-like configuration. The computed results for the modified simple cubic and square spiral structures are in close agreement with those obtained by previous authors. Moreover, the sp³-like configuration made of silicon and air is reported to have a large band gap which is larger than the largest reported elsewhere in the literature.