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Optics Letters

Degenerate-four-wave mixing as a Sturm–Liouville problem

Milivoj R. Belić

Published Feb 1, 1987• 23 citations
Publisher

Abstract

The problem of steady-state degenerate holographic four-wave mixing in transmission geometry in nonlinear dynamic media is reduced to a Sturm-Liouville or a one-dimensional quantum-mechanical scattering problem, which is treated exactly. Linear absorption in the medium is accounted for, and pump depletion is allowed. No restrictions are placed on the spatial phase mismatch between light-interference fringes and refractive-index gratings. Energy and phase transfer are considered simultaneously.

DOI
10.1364/ol.12.000105

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The problem of steady-state degenerate holographic four-wave mixing in transmission geometry in nonlinear dynamic media is reduced to a Sturm-Liouville or a one-dimensional quantum-mechanical scattering problem, which is treated exactly. Linear absorption in the medium is accounted for, and pump depletion is allowed. No restrictions are placed on the spatial phase mismatch between light-interference fringes and refractive-index gratings. Energy and phase transfer are considered simultaneously.
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.
We consider degenerate four-wave mixing in a Kerr medium with a standard noncollinear geometry and the polarization directions of all the beams parallel to one another. The two pump beams, the probe beam, and the phase-conjugate beam are denoted by the subscripts 1,2,4 and 3, respectively. The nonlinear medium is placed between z = 0 and z = 1; the values of the intensities of the beams at z = 0 and 1 are denoted by Ii0 andIi1, where i = 1,2,3,4. Beams 2 and 4 are incident at z = 1; and beam 1 is incident at z = 0. We have determined the dependence of the phase-conjugate reflectivity R = I31/I41 and the pump transmission T = I11/I10 on the following parameters: the probe ratio, q=I41/(I10 + I21); the pump ratio, r = I21/I 1 0; the absorption coefficient–path length product, al; and the coupling constant of the medium, q. In the absence of nonlinear refraction, pump depletion, and linear absorption in the medium, R = tan2q. When all of these effects are taken into account, the maximum value of R is given by Rmax = exp (-2α1)/(q(1 + r)). The minimum value that T can reach is proportional to q for q < 1. Even for small values of q, multistable solutions for R and T are obtained when q is greater than a threshold value, which we denote by qth. We have determined the dependence of qth on the parameters q and r.