Solving Maxwell eigenvalue problems for three dimensional isotropic photonic crystals with fourteen Bravais lattices

Abstract

In this paper, we present a unified finite difference framework to efficiently compute band structures of three dimensional linear non-dispersive isotropic photonic crystals with any of 14 Bravais lattice structures to a reasonable accuracy. Specifically, we redefine a suitable orthogonal coordinate system, and meticulously reformulate the Bloch condition for oblique Bravais lattices, and clearly identify the hierarchical companion matrix structure of the resulting discretized partial derivative operators. As a result, eigen-decompositions of discretized partial derivative operators and notably the discretized double-curl operator of any size, become trivial, and more importantly, the nullspace free method for the Maxwell’s equations holds naturally in all 14 Bravais lattices. Thus, the great difficulty arising from high multiplicity of zero eigenvalues has been completely overcome. On the basis of these results, we perform calculations of band structures of several typical photonic crystals to demonstrate the efficiency and accuracy of our algorithm.