Modeling two-dimensional anisotropic photonic crystals by Dirichlet-to-Neumann maps

For photonic crystals (PhCs) and related devices, it is useful to calculate the Dirichlet-to-Neumann (DtN) map of a unit cell, which maps the wave field to its normal derivative on the boundary. The DtN map can be used to avoid further calculations in the interiors of the unit cells and formulate mathematical problems on the cell boundaries. We develop a method to approximate the DtN map for two-dimensional PhCs involving anisotropic media, and we calculate band structures for PhCs involving liquid crystals. For band structures of triangular lattice PhCs, we also develop new eigenvalue problem formulations involving smaller matrices.