Chern numbers for the index surfaces of photonic crystals: conical refraction as a basis for topological materials

The classification of bandstructures by topological invariants provides a powerful tool for understanding phenomena such as the quantum Hall effect. This classification was originally developed in the context of electrons, but can also be applied to photonic crystals. In this paper we study the topological classification of the refractive index surfaces of two-dimensional photonic crystals. We consider crystals formed from birefringent materials, in which the constitutive relation provides an optical spin-orbit coupling. We show that this coupling, in conjunction with optical activity, can lead to a gapped set of index surfaces with non-zero Chern numbers. This method for designing photonic Chern insulators exploits birefringence rather than lattice structure, and does not require band crossings originating from specific lattice geometries.