Computing equifrequency contours of two-dimensional photonic crystals by Dirichlet-to-Neumann maps

Abstract

Equifrequency contours provide important information for designing special photonic crystal devices. In this paper, we present an efficient method to compute equifrequency contours for two-dimensional photonic crystals with triangular and honeycomb lattices. Our method is based on the Dirichlet-to-Neumann (DtN) operator of a unit cell in the photonic crystal. The DtN operator maps the wave field on the boundary of the unit cell to its normal derivatives. For photonic crystals with a triangular or honeycomb lattice, a small linear eigenvalue problem is formulated to calculate the dispersion relation. The formulation is based on the DtN map of the unit cell for a given angular frequency, and the eigenvalue is related to the wave vector. Our method is especially suitable for calculating the equifrequency contours, if a relatively small number of frequencies are involved.