Geometrical approach to light in inhomogeneous media

TL;DR

This paper presents a differential geometric approach to describe light propagation in inhomogeneous media, simplifying the analysis by treating light rays as geodesics in a curved, effective space with constant refractive index.

Contribution

It introduces a three-dimensional metric transformation that simplifies electromagnetic field analysis in inhomogeneous media by making light rays geodesics and the refractive index effectively constant.

Findings

Light rays follow geodesics in the effective metric.

The approach simplifies the description of light in inhomogeneous media.

The method relates the medium parameters to a conformally transformed metric.

Abstract

Electromagnetism in an inhomogeneous dielectric medium at rest is described using the methods of differential geometry. In contrast to a general relativistic approach the electromagnetic fields are discussed in three-dimensional space only. The introduction of an appropriately chosen three-dimensional metric leads to a significant simplification of the description of light propagation in an inhomogeneous medium: light rays become geodesics of the metric and the field vectors are parallel transported along the rays. The new metric is connected to the usual flat space metric diag[1,1,1] via a conformal transformation leading to new, effective values of the medium parameters leading to an effective constant value of the index of refraction n=1. The corresponding index of refraction is thus constant and so is the effective velocity of light. Space becomes effectively empty but curved. All deviations from straight line propagation are now due to curvature. The approach is finally used for a discussion of the Riemann-Silberstein vector, an alternative, complex formulation of the electromagnetic fields.