Optical concentrator from a hyperbolic liquid crystal metamaterial

We examine the optical properties of two different configurations of a cylindrical device made from a hyperbolic metamaterial with optical axis defined by circular and radial director fields. The hyperbolic metamaterial is an uniaxial anisotropic medium for which the ratio between ordinary and extraordinary permittivities is negative, leading to a particular effective geometry with two timelike coordinates in the metric (Kleinian signature). By using differential geometry tools we are able to perform a comparison between a simple geometrical optics treatment and the wave formalism that shows the concentration of light along the cylinder axis for the case of the circular field configuration, whatever the injection conditions are.