Complex Brillouin Zone for Localised Modes in Hermitian and Non-Hermitian Problems

We develop a mathematical and numerical framework for studying evanescent waves in subwavelength band gap materials. By establishing a link between the complex Brillouin zone and various Hermitian and non-Hermitian phenomena, including defect localisation in band gap materials and the non-Hermitian skin effect, we provide a unified perspective on these systems. In two-dimensional structures, we develop analytical techniques and numerical methods to study singularities of the complex band structure. This way, we demonstrate that gap functions effectively predict the decay rates of defect states. Furthermore, we present an analysis of the Floquet transform with respect to complex quasimomenta. Based on this, we show that evanescent waves may undergo a phase transition, where local oscillations drastically depend on the location of corresponding frequency inside the band gap.