Wave propagation in complex coordinates

S A R Horsley,C G King,T G Philbin

doi:10.1088/2040-8978/18/4/044016

Abstract

We give an interpretation for the use of complex spatial coordinates in electromagnetism, in terms of a family of closely related inhomogeneous media. Using this understanding we find that the phenomenon of reflection can be related to branch cuts in the wave that originate from poles of at complex positions. Demanding that these branch cuts disappear, we derive a new 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 Pöschl Teller potentials that in general include regions of loss and gain. We conclude by analyzing our findings within the phase integral (WKB) method, and find another very large family of isotropic planar media that from one side have a transmission of unity and reflection of zero, for all angles of incidence.

Highlights

We provide a physical meaning for the use of complex spatial coordinates in electromagnetism— given in terms of a family of closely related inhomogeneous media—and investigate the behaviour of wave in the whole complex position plane
In particular we find that when a medium is not analytic in one half of the complex position plane reflection can be related to branch cuts in the wave
Given that the branch cuts in φ are intimately connected to the phenomenon of reflection, we investigate under what circumstances they disappear and thereby determine a large set of inhomogeneous media from which the reflection is zero

Summary

A PHYSICAL MEANING FOR COMPLEX COORDINATES

Interpreting γ as a new position variable, this is equivalent to propagation through a new inhomogeneous anisotropic medium where zz(γ) = (γ)z (γ) μxx(γ) = μ(γ)(z (γ))−1 μyy(γ) = μ(γ)z (γ) In general these material parameters are complex functions of position, and the medium exhibits some combination of dissipation and gain. In these two examples we show how to use this to design a reflectionless absorber and a periodic medium with a combination of loss and gain, both on the basis of the solution to free space propagation in.

POLES IN THE PERMITTIVITY AND BRANCH CUTS IN THE WAVE

RELATION TO THE WKB METHOD

EXAMPLES

SUMMARY AND CONCLUSIONS