Mathematical Theory for Photonic Hall Effect in Honeycomb Photonic Crystals

TL;DR

This paper develops a mathematical framework to analyze the photonic Hall effect in honeycomb photonic crystals, proving the existence of topologically protected guided waves at their interfaces, similar to electronic edge states.

Contribution

It introduces a mathematical theory linking topological perturbations in honeycomb photonic crystals to the existence of interface-guided electromagnetic waves.

Findings

Guided waves exist at the interface of two honeycomb photonic crystals.

Topological phase differences induce edge states similar to electronic systems.

Spectral analysis confirms the relationship between perturbations and guided mode existence.

Abstract

In this work, we develop a mathematical theory for the photonic Hall effect and prove the existence of guided electromagnetic waves at the interface of two honeycomb photonic crystals. The guided wave resembles the edge states in electronic systems: it is induced by the topological Hall effect, and the wave propagates along the interface but not in the bulk media. Starting from a symmetric honeycomb photonic crystal that attains Dirac points at the high-symmetry points of the Brillouin zone, $K$ and $K'$, we introduce two classes of perturbations for the periodic medium. The perturbations lift the Dirac degeneracy, forming a spectral band valley at the points $K$ and $K'$ with well-defined topological phase that depends on the sign of the perturbation parameters. By employing the layer potential techniques and spectral analysis, we investigate the existence of guided wave along an interface when two honeycomb photonic crystals are glued together. In particular, we elucidate the relationship between the existence of the interface mode and the nature of perturbations imposed on the two periodic media separated by the interface.