A differential geometry approach to asymmetric transmission of light

TL;DR

This paper extends transformation optics using differential geometry to describe non-impedance matched materials, introducing a second tensor to model asymmetric light transmission, enabling the design of novel birefringent optical materials.

Contribution

It generalizes geometric methods of transformation optics to non-impedance matched materials by incorporating a second tensor, the W-tensor, to explain asymmetric light transmission.

Findings

Introduces the W-tensor to describe non-metric properties of materials.

Shows how the W-tensor relates to asymmetric light transmission.

Designs materials with direction-dependent birefringence.

Abstract

In the last ten years, the technology of differential geometry, ubiquitous in gravitational physics, has found its place in the field of optics. It has been successfully used in the design of optical metamaterials, through a technique now known as "transformation optics". This method, however, only applies for the particular class of metamaterials known as impedance matched, that is, materials whose electric permittivity is equal to their magnetic permeability. In that case, the material may be described by a spacetime metric. In the present work we will introduce a generalization of the geometric methods of transformation optics to situations in which the material is not impedance matched. In such situation, the material -or more precisely, its constitutive tensor- will not be described by a metric only. We bring in a second tensor, with the local symmetries of the Weyl tensor, the "$W$-tensor". In the geometric optics approximation we show how the properties of the $W$-tensor are related to the asymmetric transmission of the material. We apply this feature into the design of a particularly interesting set of asymmetric materials. These materials are birefringent when light rays approach the material in a given direction, but behave just like vacuum when the rays have the opposite direction with the appropriate polarization (or, in some cases, independently of the polarization).