Nonlinear Phenomena in Complex Systems, 2013, Vol.16, No.4, pp.331-344

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2013, Vol.16, No.4, pp.331-344

On Simulating a Medium with Special Reflecting Properties by Lobachevsky Geometry
E. M. Ovsiyuk, O. V. Veko, V. M. Red'kov

Lobachevsky geometry simulates a medium with special constitutive relations, Dⁱ = ϵ₀ϵ ^ikE^k ,Bⁱ = μ₀μ^ikH_k where two matrices coincide: ϵ^ik(x) = μ^ik(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) = U₀e^2z. 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.

Key words: spaces of constant negative curvature, Maxwell equations, medium reflecting properties

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