Topological photonics of generalized and nonlinear eigenvalue equations

TL;DR

This paper reviews topological photonics involving generalized and nonlinear eigenvalue equations, highlighting complex band structures, exceptional points, and the robustness of edge modes in advanced photonic systems.

Contribution

It introduces a comprehensive topological band theory for systems described by generalized and nonlinear eigenvalue equations, extending beyond standard models.

Findings

Complex band structures explain hyperbolic metamaterials dispersion.

Prediction of symmetry-protected exceptional points in photonic crystals.

Nonlinear bulk-edge correspondence shows robustness of chiral edge modes.

Abstract

Topological photonics is developed based on the analogy of Schr\"{o}dinger equation which is mathematically reduced to a standard eigenvalue equation. Notably, several photonic systems are beyond the standard topological band theory as they are described by generalized or nonlinear eigenvalue equations. In this article, we review the topological band theory of this category. In the first part, we discuss topological photonics of generalized eigenvalue equations where the band structure may take complex values even when the involved matrices are Hermitian. These complex bands explain the characteristic dispersion relation of hyperbolic metamaterials. In addition, our numerical analysis predicts the emergence of symmetry-protected exceptional points in a photonic crystal composed of negative index media. In the second part, by introducing auxiliary bands, we establish the nonlinear bulk-edge correspondence under ``weak" nonlinearity of eigenvalues. The nonlinear bulk-edge correspondence elucidates the robustness of chiral edge modes in photonic systems where the permittivity and permeability are frequency dependent.