# Topology and the optical Dirac equation

## TL;DR

This paper explores the topological nature of electromagnetic interface states by modeling Maxwell's equations as an optical Dirac equation, establishing a link between material parameters, Berry curvature, and interface modes.

## Contribution

It introduces a framework relating electromagnetic media to topological invariants, providing analytical proofs connecting Berry curvature and interface states without frequency dependence.

## Key findings

- Electromagnetic interface states correspond to topological invariants.
- The Chern number can be computed independently of frequency dependence.
- Analytical proof links Berry curvature integral to interface mode count.

## Abstract

Through understanding Maxwell's equations as an effective Dirac equation (the `optical Dirac equation'), we re--examine 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.