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Optics Letters

Exact solution to four-wave mixing with complex couplings: reflection geometry

Milan S. Petrović, Milivoj R. Belić, Dejan V. Timotijević, Marko V. Jarić

Published Mar 1, 1996• 3 citations
Publisher

Abstract

An exact solution to photorefractive four-wave mixing equations with complex couplings in ref lection geometry is obtained. It is shown that the efficiency of the process of phase conjugation can be enhanced by introduction of a frequency shift between the pumps and the signal, similar to the case of transmission geometry. However, to obtain an improved agreement with experiment, the inclusion of transverse effects is found to be necessary.

DOI
10.1364/ol.21.000321

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An exact solution to photorefractive four-wave mixing equations with complex couplings in ref lection geometry is obtained. It is shown that the efficiency of the process of phase conjugation can be enhanced by introduction of a frequency shift between the pumps and the signal, similar to the case of transmission geometry. However, to obtain an improved agreement with experiment, the inclusion of transverse effects is found to be necessary.
Here we present four wave mixing configuration as one of the most important methods of generation and explanation of process of generation phase conjugation light the coupled equation was solved by three methods: undepleted pump approximation exact solution and two wave mixing approximation. Although exact solutions can be derived for four-wave mixing, the results are difficult to use because of the complicated transcendental equation involved. Solved the coupled equations and introduce an approximation in which two-wave mixing is the dominant process.
A method for exact solution of the stationary holographic degenerate four-wave mixing in reflection geometry in photorefractive media is presented. The effects of pump depletion and light absorption in the nonlinear dynamic medium are rigorously taken into account. Numerical steps in the procedure are reduced to the solution of one simple nonlinear first-order differential equation, or are absent altogether when the equation is linearized, or for the case of no absorption. The method of solution is applicable to other wave-mixing geometries as well.
The solution of the scalar four-mixing equations in the transmission geometry in the presence of material absorption has been found to represent a difficult problem when dealing with photorefractive materials. A new matrix approach is taken here which allows the identification of an underlying symmetry based upon the SU(2) group. This symmetry is used to uncouple the wave equations, and solutions are then found in terms of hypergeometric functions. This represents an analytical solution to the coupled wave equations, but a numerical solution to the boundary value problem is required. The full numerical solution is then performed in terms of the Euler angles that parameterize the SU(2) group. The symmetry introduced by the group reduces the computation to the evaluation of two real variables, instead of the evaluation of four complex beam amplitudes. Results are shown for parallel and orthogonally polarized pump four-wave mixing with shifted gratings. The novel result of absorption enhanced phase conjugate reflectivities is shown, and results are presented for the first time that show the effect of absorption on arbitrarily shifted gratings.
We present an eight-dimensional matrix formulation of vector four-wave mixing. We study the case of a nonoptically active photorefractive for the transmission grating. The coupled equations are represented in a group formalism and irreducible forms are found. We show that when cross-polarization coupling is neglected, the system of equations is governed by two coupled SU(2) groups. We develop a theory that allows for pump depletion and electric-field-induced bulk birefringence. Analytical solutions are found when cubic 4¯3m materials are considered in two standard experimental geometries, i.e., electric fields perpendicular to the (110) crystal faces and electric fields perpendicular to the (001) crystal faces. Results are presented that show the effects of linear and circular polarized beams, and the use of these materials for vector phase conjugation is discussed. Solution routes to other geometries and noncubic photorefractive materials are also presented.