Wave propagation in complex coordinates

TL;DR

This paper explores the analytic continuation of electromagnetic wave equations into complex coordinates, revealing how reflectionless media can be engineered by controlling branch cuts and poles in the complex plane, with implications for wave manipulation.

Contribution

It introduces a physical interpretation of complex coordinate continuation for wave equations and derives new reflectionless media by eliminating branch cuts, extending to generalized Poschl Teller potentials.

Findings

Reflection can be linked to branch cuts from permittivity poles.

Reflectionless media are achievable by removing branch cuts.

Extended to all angles, leading to generalized Poschl Teller potentials.

Abstract

We investigate the analytic continuation of wave equations into the complex position plane. For the particular case of electromagnetic waves we provide a physical meaning for such an analytic continuation in terms of a family of closely related inhomogeneous media. For bounded permittivity profiles we find the phenomenon of reflection can be related to branch cuts in the wave that originate from poles of the permittivity at complex positions. Demanding that these branch cuts disappear, we derive a large family of inhomogeneous media that are reflectionless for a single angle of incidence. Extending this property to all angles of incidence leads us to a generalized form of the Poschl Teller potentials. We conclude by analyzing our findings within the phase integral (WKB) method.