Non-Hermitian photonic band winding and skin effects: a tutorial

Non-Hermitian band structures have gained considerable attention due to the novel phenomena not present in their Hermitian counterparts and their connection to various branches of mathematics such as topology and complex analysis. The study of such band structures may also find applications in laser design and in sensing. The spectra and eigenmode characteristics of extended non-Hermitian systems depend strongly on the boundary conditions. With periodic boundary conditions, the spectra can become complex, leading to band winding on the complex frequency plane. With open boundary conditions, the eigenmodes have spatial profiles that are localized at the boundary, an effect known as the non-Hermitian skin effect. Here we provide an overview of the band winding and skin effects in non-Hermitian photonics bands, focusing on one-dimensional cases and photonic applications. We aim to provide a detailed, consistent, and unifying treatment of various phenomena associated with non-Hermitian band structures.