Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice

An efficient semi-analytic method is developed for computing the band structures of two-dimensional photonic crystals which are triangular lattices of circular cylinders. The problem is formulated as an eigenvalue problem for a given frequency using the Dirichlet-to-Neumann (DtN) map of a hexagon unit cell. This is a linear eigenvalue problem even if the material is dispersive, where the eigenvalue depends on the Bloch wave vector. The DtN map is constructed from a cylindrical wave expansion, without using sophisticated lattice sums techniques. The eigenvalue problem can be efficiently solved by standard linear algebra programs, since it involves only matrices of relatively small size.