Dirichlet-to-Neumann Map Method with Boundary Cells for Photonic Crystals Devices

Abstract

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.

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

“…One approach is to truncate the domain by including only one layer of empty unit cells in the surrounding homogeneous media and use special DtN maps for the boundary and corner unit cells. In [18], the boundary and corner DtN maps are constructed using plane waves, but the accuracy is limited. In the following, we construct new DtN maps for the boundary and corner unit cells based on the method of fictitious sources and show improved accuracy in numerical examples.…”

“…We note that p 0 may be different from p. Based on the DtN maps, we can set up a system of equations for all interior edges [18]. For each interior edge, an equation is obtained by equating the normal derivative (x or y derivative for a vertical or horizontal edge, respectively) of the wave field evaluated by the DtN maps of the two neighboring unit cells.…”

“…The Dirichlet-to-Neumann (DtN) map method [16][17][18] is a recently developed method for analyzing 2D PhC devices. It uses a cylindrical wave expansion in each unit cell to establish a relation (the so-called DtN map) between the wave field and its normal derivative on the cell boundary then solves the wave fields on the boundaries of the unit cells only.…”

“…The DtN map method can also be used to analyze PhC devices involving finite number of cylinders, but a different domain truncation technique is needed. The approach used in [18] is to surround the cylinders by one layer of empty unit cells and construct special DtN maps for the boundary and corner unit cells using expansions in plane waves.…”

“…The DtN maps constructed using plane waves [18] seem to have limited accuracy. We construct more accurate DtN maps for boundary and corner unit cells based on the method of fictitious sources [19] and illustrate the DtN map method using a number of examples including a structure with a layered background.…”