On Simulating a Medium with Special Reflecting Properties by Lobachevsky Geometry

Abstract

Lobachevsky geometry simulates a medium with special constitutive relations, Di = ϵ0ϵ ikEk ,Bi = μ0μikHk where two matrices coincide: ϵik(x) = μik(x). The situation is specified in quasi-cartesian coordinates (x, y, z). Exact solutions of the Maxwell equations in complex 3-vector E+iB form, extended to curved space models within the tetrad formalism, have been found in Lobachevsky space. The problem reduces to a second order differential equation which can be associated with an 1-dimensional Schröodinger problem for a particle in the external potential field U(z) = U0e2z. In quantum mechanics, curved geometry acts as an effective potential barrier with reflection coefficient R = 1; in electrodynamic context results similar to quantum-mechanical ones arise: the Lobachevsky geometry simulates a medium that effectively acts as an ideal mirror. Penetration of the electromagnetic field into the effective medium, depends on the parameters of an electromagnetic wave, frequency ω,k21 + k22, and the curvature radius ρ.