Conformal hyperbolic optics

Hyperbolic metamaterials have attracted considerable interest in the research community for their peculiar ability to enhance control of electromagnetic waves' propagation. Although conformal transformation optics provides a unique platform for metamaterials design, this method has not been used for hyperbolic metamaterials yet. This comes from the lack of a well-defined mathematical structure. We extend conformal transformation optics to hyperbolic metamaterials, by applying Clifford algebra to analyze light propagation. We will show that the effective line element of a trajectory of light propagation conforms to ultrahyperbolic---not Euclidean---geometry. We also, by using conformal hyperbolic mapping, obtain all conformal spaces with Minkowski space-time. Finally, we employ this theory to study the electric-field pattern of dipoles.