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Journal of the Optical Society of America B

Degenerate four-wave mixing without slowly varying amplitude approximation

Mahmood Hosseini Farzad, Mohammad Taghi Tavassoly

Published Jul 1, 1997• 5 citations
Publisher

Abstract

The coupled-wave equations describing the optical phase conjugation by degenerate four-wave mixing are usually solved in the so-called slowly varying amplitude approximation, wherein only the first derivatives of the amplitudes of the probe and the conjugate waves are taken into account. Here we solve the coupled-wave equations without this approximation for nonresonance interactions. This leads to an oscillatory dependence of both reflectivity and transmittance on the normalized coupling constant, κL. Also, the intensity distributions of the probe and the conjugate waves show oscillatory beatlike behaviors around the corresponding slowly varying amplitude approximation distributions. These behaviors can be interpreted as the diffraction of the conjugate (probe) wave from a grating resulting from the superposition of the pump–pump and the pump–probe gratings.

DOI
10.1364/josab.14.001707

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The coupled-wave equations describing the optical phase conjugation by degenerate four-wave mixing are usually solved in the so-called slowly varying amplitude approximation, wherein only the first derivatives of the amplitudes of the probe and the conjugate waves are taken into account. Here we solve the coupled-wave equations without this approximation for nonresonance interactions. This leads to an oscillatory dependence of both reflectivity and transmittance on the normalized coupling constant, κL. Also, the intensity distributions of the probe and the conjugate waves show oscillatory beatlike behaviors around the corresponding slowly varying amplitude approximation distributions. These behaviors can be interpreted as the diffraction of the conjugate (probe) wave from a grating resulting from the superposition of the pump–pump and the pump–probe gratings.
The coupled wave equations describing the optical phase conjugation by degenerate four wave mixing DFWM is usually, solved in the so called slowly varying amplitude approximation (SVA), where only the first derivatives of the amplitudes of the probe and of conjugate waves are taken into account. Here, we solve the coupled equations exactly for nonresonance ]interactions. This leads to an oscillatory dependence of reflectivity and of transmittance on the normalized coupling constant, κL. Also the intensity distributions of the probe and conjugate waves show oscillatory beat-like behaviors around the corresponding SWA distributions. The latter can be interpreted as the diffraction of the conjugate (probe) waves from the super-position of the pump-pump and pump-probe gratings.
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.
In this study, a semiclassical version of the degenerate four-wave mixing (DFWM) solution without slowly varying amplitude (SVA) approximation is investigated. The quadrature fluctuations of the output modes exhibit an oscillatory behavior with respect to that of the corresponding SVA modes. Furthermore, the variation of the two quadrature fluctuations of the combined mode, plotted against the normalized coupling constant κL, oscillates. Beyond a certain value of κL, the two quadratures oscillate in opposite directions, which implies that the squeezing effect of DFWM without the SVA approximation oscillates around that of the SVA. In addition, the squeezing effect in DFWM without SVA approximation depends on the refractive index of the medium and increases sharply upon the increase of the latter. This study also shows that the super-Poissonian behavior of the conjugate mode is preserved but is oscillatory. PACS No.: 42.65.Hw
This paper presents an exact solution to the slowly varying envelope equations for the probe and signal fluxes in degenerate four-wave mixing in Kerr media. The analysis is performed for a noncollinear geometry, and all beams have the same linear polarization in the nonlinear medium. It is assumed that the angle separating the directions of the counterpropagating pumps and probe and signal beams is large enough so that cross-coupled waves do not contribute to the envelope equations describing the spatial evolution of the pump, probe, and signal beams. Multiple values of the transmitted probe are obtained for equal pump fluxes above a threshold determined by the nonlinear phase shift in the medium. Significant pump depletion occurs in this regime, but it can also occur without the appearance of multiple solutions. Since exact optical phase conjugation occurs only in the limit where the pumps are undepleted in Kerr media, this result implies that there are fundamental upper limits on the pump intensity that can be used to increase signal reflectivities in optical phase conjugation.
An analytic solution for the intensity distribution of two counterpropagating pump waves within a saturable absorber is derived. From this distribution, 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 results of the exact theory are compared with those of previously published theories. It is found that at large values of the input-pump intensities, the predicted phase-conjugate reflectivity is larger when pump-absorption effects are included in the theory.
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.