Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice

Abstract

An efficient numerical method is developed for computing the transmission and reflection spectra of finite two-dimensional photonic crystals composed of circular cylinders in a triangular lattice. Our method manipulates a pair of operators defined on a set of curves, and it remains effective when the radius of the cylinders is larger than sqrt 3/4 of the lattice constant—a condition where different arrays of cylinders cannot be separated by planes without intersecting the cylinders. The method is efficient since it never calculates the wave field in the interiors of the (hexagon) unit cells and it approximates the operators by small matrices. This is achieved by using the Dirichlet-to-Neumann (DtN) maps of the unit cells, which map the wave field on the boundaries of the unit cells to its normal derivative.