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

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