  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 (2,2)-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 isotropic medium the Fresnel surface is at each
point a Lorentz light cone. In a recent paper, I. Lindell, A. Favaro and L.
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 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 of uniaxial medium like
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.
