Transformation optics, isotropic chiral media and non-Riemannian geometry

Abstract

The geometrical interpretation of electromagnetism in transparent media (transformation optics) is extended to include chiral media that are isotropic but inhomogeneous. It was found that such media may be described through introducing the non-Riemannian geometrical property of torsion into the Maxwell equations, and it is shown how such an interpretation may be applied to the design of optical devices.

GENERAL SCIENTIFIC SUMMARY Introduction and background. On a macroscopic scale, electromagnetism in a transparent material can be understood in terms of an average polarization and magnetization of the medium. This has been known for a long time, but it certainly isn't well adapted for the design of optical devices. Transformation optics is a recent development that makes use of the finding that Maxwell's equations in a curved space–time geometry take the same form as they do in a material. An effective permittivity and permeability can be derived from the measure of length and, in short, we can design optical devices with geometry. This approach has famously brought us designs for cloaking devices and 'sub-diffraction' lenses, as well as the investigation of analogues of astrophysical effects, such as Hawking radiation, in the laboratory.

Main results. In this paper it is found that the relationship between transparent materials and geometry goes further than was previously thought. Not only are curved space–times equivalent to materials, but some materials can be understood in terms of geometrical ideas that go beyond just curvature (i.e. beyond the mathematics of general relativity). In particular, it is found that light propagation in a geometry exhibiting space–time torsion (see figure) is equivalent to propagation through a chiral medium (space–time torsion can be related to optical activity). Such a finding can be applied to the design of optical devices. It allows us to incorporate chiral media into transformation optics, and has an immediate use—allowing for the independent control of polarization within transformation optics.

Wider implications. Incorporating chiral media into transformation optics may well reinvigorate the investigation of useful optical devices based on chiral media, which attracted attention in the past. There is also a fundamental interest in the close investigation of materials that mimic space–time torsion, as it is expected to play a prominent role in the physics of gravity beyond general relativity.