Improved Dirichlet-to-Neumann map method for scattering by circular cylinders on a lattice.

A simple model for two-dimensional photonic crystal devices consists of a finite number of possibly different circular cylinders centered on lattice points of a square or triangular lattice and surrounded by a homogeneous or layered background medium. The Dirichlet-to-Neumann (DtN) map method is a special method for analyzing the scattering of an incident wave by such a structure. It is more efficient than existing numerical or semianalytic methods, such as the finite element method and the multipole method, since it takes advantage of the underlying lattice structure and the simple geometry of the unit cells. The DtN map of a unit cell is a relation between a wave field component and its normal derivative on the cell boundary, and it can be used to avoid further computation inside the unit cell. In this paper, an improved DtN map method is developed by constructing special DtN maps for boundary and corner unit cells using the method of fictitious sources.