Bulk-edge correspondence in finite photonic structure

TL;DR

This paper rigorously proves the bulk-edge correspondence in finite 2D photonic structures, linking topological invariants to boundary energy circulation, thus supporting the design of robust topological photonic devices.

Contribution

It establishes a rigorous proof of bulk-edge correspondence for finite photonic systems, connecting the gap Chern number to an edge index via trace formulas.

Findings

Proves the equality between bulk Chern number and edge index in finite photonic structures.

Demonstrates the edge index characterizes electromagnetic energy circulation along boundaries.

Provides a theoretical foundation for designing robust topological photonic devices.

Abstract

In this work, we establish the bulk-edge correspondence principle for finite two-dimensional photonic structures. Specifically, we focus on the divergence-form operator with periodic coefficients and prove the equality between the well-known gap Chern number (the bulk invariant) and an edge index defined via a trace formula for the operator restricted to a finite domain with Dirichlet boundary conditions. We demonstrate that the edge index characterizes the circulation of electromagnetic energy along the system's boundary, and the BEC principle is a consequence of energy conservation. The proof leverages Green function techniques and can be extended to other systems. These results provide a rigorous theoretical foundation for designing robust topological photonic devices with finite geometries, complementing recent advances in discrete models.