(untitled) (Communications in Computational Physics, 2011)
In a two-dimensional (2D) photonic crystal (PhC) composed of circular cylinders (dielectric rods or air holes) on a square or triangular lattice, various PhC devices can be created by removing or modifying some cylinders. Most existing numerical methods for PhC devices give rise to large sparse or smaller but dense linear systems, all of which are expensive to solve if the device is large. In a previous work [Z. Hu et al., Optics Express, 16 (2008), 17383-17399], an efficient Dirichlet-to-Neumann (DtN) map method was developed for general 2D PhC devices with an infinite background PhC to take full advantage of the underlying lattice structure. The DtN map of a unit cell is an operator that maps the wave field to its normal derivative on the cell boundary and it allows one to avoid computing the wave field in the interior of the unit cell. In this paper, we extend the DtN map method to PhC devices with a finite background PhC. Since there is no bandgap effect to confine the light in a finite PhC, a different technique for truncating the domain is needed. We enclose the finite structure with a layer of empty boundary and corner unit cells, and approximate the DtN maps of these cells based on expanding the scattered wave in outgoing plane waves. Our method gives rise to a relatively small and sparse linear systems that are particularly easy to solve.