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Journal of the Optical Society of America A

Degenerate four-wave mixing in the presence of nonuniform pump wave fronts

B. Crosignani, Amnon Yariv

Published Oct 1, 1984• 21 citations
Publisher

Abstract

We derive the set of coupled equations that describes the process of degenerate four-wave mixing in the presence of spatially nonuniform pump-beam wave fronts. We investigate the influence of phase mismatch between plane-wave pump beams on the efficiency and on the fidelity of the phase-conjugation process, and we furnish, in the near-collinear geometry, the expression of the spatial degree of coherence of the phase-conjugate wave in terms of those of the input signal and of the pump beams.

DOI
10.1364/josaa.1.001034

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Annual Meeting Optical Society of America•Jan 1, 1985•0 citations
We derive the set of coupled equations that describes the process of degenerate four-wave mixing in the presence of spatially nonuniform pump-beam wave fronts. We investigate the influence of phase mismatch between plane-wave pump beams on the efficiency and on the fidelity of the phase-conjugation process, and we furnish, in the near-collinear geometry, the expression of the spatial degree of coherence of the phase-conjugate wave in terms of those of the input signal and of the pump beams.
Phase-conjugate beam fidelity is studied in degenerate four-wave mixing with spatially varying pump beams. The theory includes long-cell-length (or high-gain) effects, diffraction, and nonlinear phase variation, focusing of the probe beam, and, finally, losses. Relatively simple algebraic expressions for the reflectivity are given that cover the cases of collinear or near-collinear beam geometries. An easily calculated parameter is defined that characterizes phase-conjugate fidelity and is applied to study focusing effects. It is shown that focusing effects may be useful in greatly reducing the principal distortion of the phase-conjugate beam over a considerable range of variation of beam parameters.
A study of the influence of the spatial structure and statistics of the pump beams in the phase-conjugation process is made, showing that the reflectivity varies along the phase-conjugation mirror when the phase factor in the coupling equations that describe the process is not constant. The mutual coherence function of the generated field is calculated from the mutual reflectivity of the mirror for two commonly used statistics for the partially coherent pump beams. Finally, the influence of diffractional effects is also analyzed, showing that, in general, the fidelity of the process is degraded.
We present analytical results for the steady-state signal and phase-conjugate fields interacting with smoothly varying nondepleted pump waves in a nearly degenerate four-wave-mixing geometry. Signal depletion is retained in the equations and we study the effect that detuning on the input signal from the degenerate pump amplitudes has on the reflected conjugate wave and the transmitted signal wave. A general result is derived for the appearance of spontaneous amplification under weak restrictions on the profile of the degenerate pump fields. The conjugation of the phase front of the signal is studied for exponential and hyperbolic secant functions multiplying the coupling coefficients and comparison is made with corresponding results for square profile pump fields. These results are discussed and their relevance to phase conjugation of pulses is examined.
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.