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

TL;DR

This paper demonstrates that Fresnel wave surfaces in certain linear media can have up to 16 singular points, exceeding the known 4, and provides a metamaterial design to realize such media, with implications for wave surface geometry.

Contribution

It introduces a highly symmetric linear medium with 16 singular points on its Fresnel surface, establishing 16 as the maximum number of isolated singularities for such media.

Findings

Fresnel surfaces can have up to 16 singular points in certain media.

A metamaterial design with specific components can realize this media.

The singularities follow Kummer's (16,6)-configuration.

Abstract

It is known that the Fresnel wave surfaces of transparent biaxial media have 4 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.