Unidirectional wave propagation in media with complex principal axes

TL;DR

This paper explores wave behavior in complex anisotropic media, revealing conditions for unidirectional propagation linked to analyticity in a complex variable and topology, with implications for transformation optics and topological photonics.

Contribution

It introduces the concept of wave analyticity in complex directions in gyrotropic media, connecting topology and complex coordinate transformations to unidirectional wave propagation.

Findings

Wave becomes an analytic function of a complex variable z.

Unidirectional propagation occurs in simply connected media.

Topology influences wave propagation, with holes disrupting unidirectionality.

Abstract

In an anisotropic medium, the refractive index depends on the direction of propagation. Zero index in a fixed direction implies a stretching of the wave to uniformity along that axis, reducing the effective number of dimensions by one. Here we investigate two dimensional gyrotropic media where the refractive index is zero in a complex valued direction, finding that the wave becomes an analytic function of a single complex variable z. For simply connected media this analyticity implies unidirectional propagation of electromagnetic waves, similar to the edge states that occur in photonic 'topological insulators'. For a medium containing holes the propagation is no longer unidirectional. We illustrate the sensitivity of the field to the topology of the space using an exactly solvable example. To conclude we provide a generalization of transformation optics where a complex coordinate transformations can be used to relate ordinary anisotropic media to the recently highlighted gyrotropic ones supporting one-way edge states.