Staircase approximation of oblique boundaries to compute band structures of photonic crystals

Abstract

The paper deals with band-structure calculations of photonic crystals with hexagonal lattice where the unit cell has the shape of a parallelogram. The boundary of this parallelogram, which is identical to the computational window, is modeled with a staircase approximation. In this way Cartesian coordinates can be used to compute the electromagnetic fields in the structure. When using this approach one must relate fields outside the computational window to those inside this area. This is done by utilizing the periodicity of the fields in photonic crystals. The procedure was used to compute the band structures of hexagonal photonic crystals with the Method of Lines and with a Finite Difference Method. The determined results agree very well with those from the literature.