Topology and the optical Dirac equation

Through understanding Maxwell's equations as an effective Dirac equation (the optical Dirac equation), we reexamine the relationship between electromagnetic interface states and topology. We illustrate a simple case where electromagnetic material parameters play the roles of mass and energy in an equivalent Dirac equation. The modes trapped between a gyrotropic medium and a mirror are then the counterpart of those at a domain wall, where the mass of the Dirac particle changes sign. Considering the general case of arbitrary electromagnetic media, we provide an analytical proof relating the integral of the Berry curvature (the Chern number) to the number of interface states. We show that this reduces to the usual result for periodic media and also that the Chern number can be computed without knowledge of how the material parameters depend on frequency.