Complex-angle analysis of electromagnetic waves on interfaces

TL;DR

This paper introduces a complex-angle analysis of electromagnetic wave behavior at interfaces, extending traditional laws to account for dissipation and total internal reflection, revealing phenomena like Lifshitz transitions and anomalous spin.

Contribution

It develops a complex-angle framework to analyze electromagnetic waves at various interfaces, including dissipative and metallic systems, highlighting new wave behaviors and transitions.

Findings

Iso-frequency curves open or close depending on material parameters.

Lifshitz transition of electromagnetic waves is identified.

Excess momentum and anomalous spin are discussed.

Abstract

Electromagnetic wave is reflected and refracted at interfaces, satisfying Fresnel-Snell law which is required by conservations of energy and momentum. If the incident angle is lower than the critical angle, we can use this Fresnel-Snell law, but the Fresnel-Snell law is modified in the case of existence of dissipation ($\tilde{n} = n + i\kappa, \kappa > 0$) or in the condition of total internal reflection. In the cases, we have to extend the angle of refraction from real number to complex number ($\theta \rightarrow \tilde{\theta} = \theta + i\psi, \psi \neq 0$). In this paper, by using complex-angle approach, we analyse the behaviour of electromagnetic waves in various kind of interfaces: dielectric - dielectric system, dissipative dielectric - dielectric system, and metal - dielectric system. We show that iso-frequency curves in wavenumber space is opened in the case where $n, \kappa > 0$, and closed either when $n \rightarrow 0$ or when $\kappa \rightarrow 0$ ("Lifshitz transition" of electromagnetic waves). Excess momentum (wavenumber) and anomalous spin (circular polarization) emerging with the transition are also discussed.