Dirichlet-to-Neumann Map Method with Boundary Cells for Photonic Crystals Devices (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 devicescan be created by removing or modifying some cylinders. Most existing numericalmethods 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) mapmethod was developed for general 2D PhC devices with an infinite background PhCto take full advantage of the underlying lattice structure. The DtN map of a unit cell isan operator that maps the wave field to its normal derivative on the cell boundary andit allows one to avoid computing the wave field in the interior of the unit cell. In thispaper, 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 techniquefor truncating the domain is needed. We enclose the finite structure with a layerof empty boundary and corner unit cells, and approximate the DtN maps of these cellsbased on expanding the scattered wave in outgoing plane waves. Our method givesrise to a relatively small and sparse linear systems that are particularly easy to solve.