Geometry and Topological photonics

TL;DR

This paper explores the deep connection between topological photonics and the mathematical topology of surfaces, revealing how topological indices relate to surface genus and implications for wave phenomena.

Contribution

It establishes a novel geometrical framework linking topological photonics to the topological theory of surfaces, enhancing understanding of wave behavior in complex systems.

Findings

Topological band theory can be viewed as a generalization of surface topology.

The genus of a surface corresponds to a Chern number in topological operators.

Topology influences radiation problems and bulk-edge correspondence in photonics.

Abstract

Topological photonics provides a powerful framework to describe and understand many nontrivial wave phenomena in complex electromagnetic platforms. The topological index of a physical system is an abstract global property that depends on the family of operators that describes the propagation of Bloch waves. Here, we highlight that there is a profound geometrical connection between topological physics and the topological theory of mathematical surfaces. We show that topological band theory can be understood as a generalization of the topological theory of surfaces and that the genus of a surface can be regarded as a Chern number of a suitable operator defined over the surface. We point out some nontrivial implications of topology in the context of radiation problems and discuss why for physical problems the topological index is often associated with a bulk-edge correspondence.