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Annual Meeting Optical Society of America

Effects of absorption, pump depletion, and nonlinear refraction on thermal degenerate four-wave mixing

Shekhar Guha, Wenpeng Chen, Philip W. Conner

Published Jan 1, 1987• 0 citations
Publisher

Abstract

Degenerate four-wave mixing, arising from the thermal grating effect or other mechanisms such as the Kerr or photorefractive effect, can be described by a set of four coupled differential equations involving the four fields. Analytical solutions to these equations have been presented for cases where either the pump depletion or the effects of absorption were ignored.1 When both the pump depletion and absorption effects were included, a numerical result involving an iterative solution of the differential equations was presented.2 We present here a solution of the coupled-wave equations in which absorption as well as pump depletion effects are taken into account along with nonlinear refraction. The DFWM reflectivity is reduced to a quadrature using a method proposed by Kessel and Musin.3 For special cases of small absorption and no pump depletion, the solution reduces to the expressions presented earlier.4 The method of solution does not involve finding first integrals of the set of differential equations and can be extended to the description of DFWM using other physical mechanisms.

DOI
10.1364/oam.1987.wi2

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Degenerate four-wave mixing, arising from the thermal grating effect or other mechanisms such as the Kerr or photorefractive effect, can be described by a set of four coupled differential equations involving the four fields. Analytical solutions to these equations have been presented for cases where either the pump depletion or the effects of absorption were ignored.1 When both the pump depletion and absorption effects were included, a numerical result involving an iterative solution of the differential equations was presented.2 We present here a solution of the coupled-wave equations in which absorption as well as pump depletion effects are taken into account along with nonlinear refraction. The DFWM reflectivity is reduced to a quadrature using a method proposed by Kessel and Musin.3 For special cases of small absorption and no pump depletion, the solution reduces to the expressions presented earlier.4 The method of solution does not involve finding first integrals of the set of differential equations and can be extended to the description of DFWM using other physical mechanisms.
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.
Co-polarised degenerate four-wave mixing (DFWM), in which all the four interacting beams have the same direction of polarisation, has been utilised to generate high phase conjugate reflectivities (~ 2500) recently [1]. In the co-polarised geometries six gratings are written in the material in which each beam forms a grating with another beam. Several theoretical models have been used to describe this co-polarised DFWM process in gain media [2]-[3]. However, all of them involve the use of small probe approximation and hence not all gratings are present in these models. Moreover, the effect of pump depletion has been neglected. A novel approach using the underlying symmetry is described recently to solve the co-polarised DFWM problem [4]. This approach allows us to take into account the effect of pump depletion, although the two standing gratings between the pumps and between the probe and the conjugate beams are ignored.
We have derived an analytic expression for the intensity distribution of two counterpropagating waves of arbitrary input intensity in a saturable absorber. We use this distribution to extend the theory of degenerate four-wave mixing to include the effects of pump wave absorption. In particular, the spatial variation of the nonlinear absorption and coupling constants that appear in the coupled-amplitude equations for the probe and the signal (i.e., conjugate) waves are determined. These coupled-amplitude equations are solved numerically in a noniterative manner, leading to a prediction for the phase-conjugate reflectivity. The nature of the predictions differs qualitatively from those1,2 derived from theories that ignore pumpwave absorption, especially for the case of off-line-center excitation. At large values of the input pump intensities, the phase conjugate reflectivity can be larger than that predicted by theories that ignore pump-wave absorption.
The effect of biasing illumination on the degenerate four-wave mixing (DFWM) in absorbing photorefractive (PR) media has been investigated. The input beams are considered to be spatially uniform. The analysis has been made both for transmission and reflection geometry. Considering pump beam depletion and the slowly varying amplitude approximation, nonlinear coupled wave equations are obtained and have been solved numerically by a shooting method. The effect of different parameters such as input beam intensities ratio, thickness of the interacting medium on the phase conjugate reflectively (PCR) has been investigated. It has been observed that PCR can be controlled to a large extent by controlling the biasing intensity.