Spectral topology and edge modes for one-dimensional non-Hermitian photonic crystals

TL;DR

This paper introduces a transfer matrix-based spectral topological invariant to characterize edge modes and the skin effect in one-dimensional non-Hermitian photonic crystals, extending the understanding from discrete to continuous models.

Contribution

It develops a new spectral topological invariant for continuous wave models, bridging the gap in understanding edge modes in non-Hermitian photonic systems.

Findings

The invariant is equivalent to the winding number of the spectrum.

It provides a theoretical foundation for the skin effect in continuous models.

Edge modes are characterized by the transfer matrix eigenvalues.

Abstract

This work investigates edge modes in non-Hermitian photonic crystals with broken spectral reciprocity. In such systems, the spectra of the underlying operators generally form closed loops over the complex plane with nontrivial spectral topology, which gives rise to the so-called skin effect characterized by edge modes localized at interfaces. For discrete lattice models, the skin effect can be understood through the spectral theory of Toeplitz matrices. However, this mathematical framework no longer applies to continuous wave models, where finite-dimensional approximations break down. In this work, we employ a transfer matrix approach to describe wave propagation in one-dimensional periodic media and introduce a new spectral topological invariant based on the eigenvalues of the transfer matrix. The new topological invariant is equivalent to the winding number of the non-Hermitian spectrum and it enables the characterization of edge modes in one-dimensional non-Hermitian photonic crystals. The mathematical theory provides the theoretical foundation for the skin effect in continuous wave models.