Maxwell equations in Lobachevsky space, and modeling the medium with reflecting properties

Abstract

Lobachevsky geometry simulates a medium with special constitutive relations. The situation is specified in quasi-Cartesian coordinates (x, y, z) in Lobachevsky space, they are appropriate for modeling a medium nonuniform along the axis z . Exact solutions of the Maxwell equations in complex form of Majorana – Oppenheimer have been constructed. The problem reduces to a second order differential equation for a certain primary function which can be associated with the one-dimensional Schrödinger problem for a particle in external potential field. In the frames of the quantum mechanics, the Lobachevsky geometry acts as an effective potential barrier with reflection coefficient R =1; in electrodynamic context results are similar: this geometry simulates a medium that effectively acts as an ideal mirror distributed in space. Penetration of the electromagnetic field into the effective medium along the axis z , depends on the parameters of an electromagnetic waves , and the curvature radius of the used Lobachevsky model. The generalized quasi-plane wave solutions and the relevant system of equations are transformed the real form, which permit us to relate geometry characteristics with expressions for effective tensors of electric and magnetic permittivities.