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IEEE Journal of Quantum Electronics

Theory of photorefractive phase conjugators with mutually incoherent beams

He Qi-Chi

Published Dec 1, 1988• 28 citations
Publisher

Abstract

Plane-wave analysis of the recently demonstrated photorefractive phase conjugators with mutually incoherent beams is presented. Based on four-wave mixing theory, two important configurations, the bird-wing phase conjugator and the mutually incoherent beam coupler are investigated by solving coupled nonlinear equations with beam depletion and absorption. Various intensity performances are theoretically analyzed. Different configurations of this kind of mutually incoherent phase conjugators are compared. Competition of different processes is discussed. Experimental data are well fitted by the theory.>

DOI
10.1109/3.14382

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Using perturbation theory, an examination is made of the fidelity of the photorefractive degenerate four-wave mixing phase conjugator. The general case is considered in which the probe is spatially varying (a beam) and temporally varying (a pulse), and the pumps are allowed to deplete. The probe wave is expanded as a sum of monochromatic plane-wave components. Carrying out the perturbation procedure up to the third order, the authors found that the components of the probe interact in pairs, so that the reflectance of each component of the conjugate wave is a combination of a linear term, a self-nonlinear term, and a term representing crosstalk with all other components of the probe. The crosstalk effect as a function of the frequency difference exhibits resonance behavior.>

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We develop a numerical method to solve the coupled-wave equations for four-wave mixing phase conjugation in a photorefractive crystal with unequal complex coupling constants; by this method we can obtain both the intensities and the phases of various beams. We discuss the intensities and the phases of the interacting beams in a photorefractive crystal for the cases in which the phase shifts between the refractive-index gratings and the interference pattern are all π/2 and in which they are not all π/2.

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The demonstration of optical phase conjugation in photorefractive materials has been accomplished by numerous methods, including i) externally-pumped four-wave mixing,1ii) self-pumped (passive) phase conjugation,24iii) stimulated backscattering,5,6and iv) mutually-pumped phase conjugation.7-10Stimulated backscattering and self-pumped phase conjugators have only a single incident beam. On the other hand, externally-pumped four-wave mixing has two auxiliary pumping beams in addition to the incident probe beam. Different still is the cross-conjugation of the two incident beams in mutually pumped phase conjugators. To use these conjugators in a given application, the compatibility between the processes generating the phase-conjugate wave(s) and the application must be considered. The distinguishing features of the different conjugators translate to advantages or disadvantages for certain applications.

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Plane-wave analysis of the recently demonstrated photorefractive phase conjugators with mutually incoherent beams is presented. Based on four-wave mixing theory, two important configurations, the bird-wing phase conjugator and the mutually incoherent beam coupler are investigated by solving coupled nonlinear equations with beam depletion and absorption. Various intensity performances are theoretically analyzed. Different configurations of this kind of mutually incoherent phase conjugators are compared. Competition of different processes is discussed. Experimental data are well fitted by the theory.>
Using perturbation theory, an examination is made of the fidelity of the photorefractive degenerate four-wave mixing phase conjugator. The general case is considered in which the probe is spatially varying (a beam) and temporally varying (a pulse), and the pumps are allowed to deplete. The probe wave is expanded as a sum of monochromatic plane-wave components. Carrying out the perturbation procedure up to the third order, the authors found that the components of the probe interact in pairs, so that the reflectance of each component of the conjugate wave is a combination of a linear term, a self-nonlinear term, and a term representing crosstalk with all other components of the probe. The crosstalk effect as a function of the frequency difference exhibits resonance behavior.>
While the interaction of optical beams in photorefractive crystals can often be accurately described in terms of 1-D plane-wave approximation, 2-D models are sometimes more suitable, as suggested by recent work.1 Here we discuss some aspects of photorefractive interaction of finite size optical beams. We have solved the system of partial differential equations governing two- and four-wave mixing numerically by using iterative procedures. Simulations of beam amplification in two-beam coupling show strong beam deformation even without any spatial information imposed on them. Spatially nonuniform energy transfer is responsible for this effect. The pump beam loses energy during propagation and at the end of its passage through the beam intersection region (which was completely inside the crystal) it is strongly depleted and cannot contribute significantly to beam amplification. As the pump beam crosses through the probe, it amplifies most strongly that side of the probe that it meets first. By the time the pump has reached the far side of the probe, it is strongly depleted and less effective at probe amplification. In the case of phase conjugation through four-wave mixing, we found that the conjugate beam is strongly deformed only when the coupling constant is negative (corresponding to deamplification of the probe input beam). For positive coupling constants the output conjugate beam is a good replica of the input probe beam.
We investigate the effect of beam temporal coherence on four-wave mixing via transmission gratings in photorefractive media. We take into account the propagation of mutual coherence and self-coherence in space and the evolution of mutual coherence and self-coherence with time. Analytical solutions for nondepleted-pump cases and numerical results for general cases are obtained. We find that the mutual coherence of the signal beam and the pump beam can be enhanced or reduced, depending on the coupling constant and the beam ratio of the signal beam and the pump beam. A decrease of the initial mutual coherence can lead to a lower phase-conjugate reflectivity in the nondepleted-pump case. Even with low initial mutual coherence, high phase-conjugate reflectivity can still be obtained in the case with pump depletion. As a result of the shift of the grating profile, partial coherence can lead to an enhancement of the phase-conjugate reflectivity when the signal–pump-beam ratio is small. We also found that when the initial value of the phase-conjugate beam is zero, the phase-conjugate beam and the pump beam are in phase during propagation. Results for beam intensity and mutual coherence are presented and discussed to provide an insight into four-wave mixing with partially temporally coherent waves.
We develop a numerical method to solve the coupled-wave equations for four-wave mixing phase conjugation in a photorefractive crystal with unequal complex coupling constants; by this method we can obtain both the intensities and the phases of various beams. We discuss the intensities and the phases of the interacting beams in a photorefractive crystal for the cases in which the phase shifts between the refractive-index gratings and the interference pattern are all π/2 and in which they are not all π/2.
The demonstration of optical phase conjugation in photorefractive materials has been accomplished by numerous methods, including i) externally-pumped four-wave mixing,1ii) self-pumped (passive) phase conjugation,24iii) stimulated backscattering,5,6and iv) mutually-pumped phase conjugation.7-10Stimulated backscattering and self-pumped phase conjugators have only a single incident beam. On the other hand, externally-pumped four-wave mixing has two auxiliary pumping beams in addition to the incident probe beam. Different still is the cross-conjugation of the two incident beams in mutually pumped phase conjugators. To use these conjugators in a given application, the compatibility between the processes generating the phase-conjugate wave(s) and the application must be considered. The distinguishing features of the different conjugators translate to advantages or disadvantages for certain applications.
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.