Recently, topological materials and topological effects have elicited a great interest in the photonics community [1]. While condensed-matter phenomena are traditionally described by Hermitian operators, the same is not true in the context of macroscopic electrodynamics where a dissipative response is the rule, not the exception. In this talk, I will discuss how to determine the topological phases of dissipative (non-Hermitian) photonic structures from first principles using a gauge-independent Green function [2, 3]. It is shown that analogous to the Hermitian case, the Chern number can be expressed as an integral of the system Green function over a line parallel to the imaginary-frequency axis. The approach introduces in a natural way the "band-gaps" of non-Hermitian systems as the strips of the complex-frequency plane wherein the system Green function is analytical. I apply the developed theory to nonreciprocal electromagnetic continua and photonic crystals, with lossy and or gainy elements. Furthermore, I discuss the validity of the bulk-edge correspondence in the non-Hermitian case.

1. L. Lu, J. D. Joannopoulos, M. Soljačić, Topological photonics, Nat. Photonics, 8, 821, (2014).
2. M. G. Silveirinha, Topological theory of non-Hermitian photonic systems, Phys. Rev. B, 99, 125155, 2019.
3. F. R. Prudêncio, M. G. Silveirinha, First Principles Calculation of Topological Invariants of non-Hermitian Photonic Crystals.