Light propagation in local and linear media: Fresnel-Kummer wave surfaces with 16 singular points

It is known that the Fresnel wave surfaces of transparent biaxial media have four singular points, located on two special directions. We show that, in more general media, the number of singularities can exceed 4. In fact, a highly symmetric linear material is proposed whose Fresnel surface exhibits 16 singular points. Because for every linear material the dispersion equation is quartic, we conclude that 16 is the maximum number of isolated singularities. The identity of Fresnel and Kummer surfaces, which holds true for media with a certain symmetry (zero skewon piece), provides an elegant interpretation of the results. We describe a metamaterial realization for our linear medium with 16 singular points. It is found that an appropriate combination of metal bars, split-ring resonators, and magnetized particles can generate the correct permittivity, permeability, and magnetoelectric moduli. Lastly, we discuss the arrangement of the singularities in terms of Kummer's ${16}_6$ configuration of points and planes. An investigation parallel to ours, but in linear elasticity, is suggested for future research.

Figure 1

Cross section of the Fresnel surface for the biaxial medium (1). The remaining half of the surface is obtained via symmetry.

Figure 2

The Fresnel surface of the magnetoelectric medium (4) with (5) has 12 singularities in the finite (and four at infinity).

Figure 3

The Fresnel surface given by the dispersion equation (9). Some readers will note that the plot illustrates a Kummer surface.