Characterisation and representation of non-dissipative electromagnetic medium with two Lorentz null cones

Abstract

We study Maxwell's equations on a 4-manifold N with a medium that is non-dissipative and has a linear and pointwise response. In this setting, the medium can be represented by a suitable $\scriptsize{\big({\begin{array}{l}{2}\\ {2}\end{array}}\big)}$22-tensor on the 4-manifold N. Moreover, in each cotangent space on N, the medium defines a Fresnel surface. Essentially, the Fresnel surface is a tensorial analogue of the dispersion equation that describes the response of the medium for signals in the geometric optics limit. For example, in an isotropic medium the Fresnel surface is at each point a Lorentz null cone. In a recent paper, Lindell, Favaro, and Bergamin introduced a condition that constrains the polarisation for plane waves. In this paper we show (under suitable assumptions) that a slight strengthening of this condition gives a complete pointwise characterisation of all medium tensors for which the Fresnel surface is the union of two distinct Lorentz null cones. This is, for example, the behaviour in uniaxial media such as calcite. Moreover, using the representation formulas from Lindell et al. we obtain a closed form representation formula that pointwise parameterises all medium tensors for which the Fresnel surface is the union of two distinct Lorentz null cones. Both the characterisation and the representation formula are tensorial and do not depend on local coordinates.