Conformal mapping in hyperbolic metamaterial: Application of two-dimensional Clifford algebra

In this paper, we begin with the metric which describes the propagation of the extraordinary ray in the hyperbolic metamaterials as the following: equation in which ε
o
> 0 and ε
e
< 0. We are interested in the z = const and also rescale the radial coordinate by ρ = √ε
o
r such that equation which γ
²
= |ε
e
/ε
o
|. By considering the coordinate transformation X = ρ cosh γφ and Y = ρ sinh γφ, we can rewrite the above metric as: equation which is so-called ultrahyperbolic metric. We indicate that, by applying the two-dimensional Clifford algebra, z = X + jY, which the hyperbolic number j saturates j
²
= 1, we can rewrite the ultrahyperbolic metric as the following: equation in which z* = X−jY, in the Kline disk. By defining the conformal mapping, which indeed is the map between two Kline disk we can investigate behaviour of light in the conformal hyperbolic metamaterial, in the level of ray optics. Also, in the framework of the wave optics, which the Klein-Gordon equation describes the behaviour of light, by applying the abovementioned conformal mapping and using the Clifford algebra, we can obtain the light propagation in the framework of the wave optics for conformal hyperbolic metamaterials. Moreover, by using this concept, we stablish a connection between the conformal field theory and classical optics.