Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background

TL;DR

This paper proves that in an isotropic and homogeneous background, reflectionless transformation media require the boundary coordinates to be unchanged, combining rotation and displacement, establishing a necessary and sufficient condition.

Contribution

It establishes that boundary coordinate invariance is both necessary and sufficient for reflectionless transformation media in such backgrounds.

Findings

Boundary coordinates must be unchanged for reflectionless media.

Rotation and displacement are the only transformations preserving reflectionless conditions.

The result completes the theoretical understanding of reflectionless transformation media.

Abstract

It has been known that, keeping the outer boundary coordinates intact before and after a coordinate transformation is a sufficient condition for obtaining a reflectionless transformation medium. Here we prove that it is also a necessary condition for reflectionless transformation media in an isotropic and homogenous background. Our analytical results show that the outer boundary coordinates of a reflectionless transformation medium must be the same as the original coordinates with a combination of rotation and displacement, which is equivalent to situation that the boundary coordinates are kept intact before and after transformation.