Tutorial 2.0: computing topological invariants in 3D photonic crystals

Abstract

The field of topological photonics has been on the rise due to its versatility in manufacturing and its applications as topological lasers or unidirectional waveguides. Contrary to 1D or 2D photonic crystals, the transversal and vectorial nature of light in 3D precludes using standard methods for diagnosing topology. This tutorial describes the problems that emerge in computing topological invariants in 3D photonic crystals and the diverse strategies for overcoming them. Firstly, we introduce the fundamentals of light propagation in 3D periodic media and expose the complications of directly implementing the usual topological diagnosis tools. Secondly, we describe the properties of electromagnetic Wilson loops and how they can be used to diagnose topology and compute topological invariants in 3D photonic crystals. Finally, we apply the previously described methods to several examples of 3D photonic crystals showing different topological phases, such as Weyl nodes and walls, 3D photonic Chern insulators, and photonic axion insulators.