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Optics Letters

Contradirectional two-wave mixing with partially coherent waves in photorefractive crystals

Xianmin Yi, Ken Y. Hsu, Shiuan Huei Lin, Pochi Yeh

Published Aug 1, 1996• 8 citations
Publisher

Abstract

We investigate contradirectional two-wave mixing with partially coherent waves in photorefractive crystals in the nondepleted pump regime. Equations governing the propagation of the self-coherence function and the mutual-coherence function of the signal wave and the pump wave are derived and simulated numerically. Numerical solutions of these equations are in excellent agreement with the experimental measurements.

DOI
10.1364/ol.21.001123

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We investigate contradirectional two-wave mixing with partially coherent waves in photorefractive crystals in the nondepleted pump regime. Equations governing the propagation of the self-coherence function and the mutual-coherence function of the signal wave and the pump wave are derived and simulated numerically. Numerical solutions of these equations are in excellent agreement with the experimental measurements.
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.
The effect of partial coherence on four-wave mixing in photorefractive media is studied by taking into account the evolution and propagation of the mutual coherence via a theoretical model. This model is applied to four-wave mixing and two-wave mixing with fully or partially coherent waves. We studied the effect of partial coherence with the presence of all the gratings, and compared the results with those of the transmission grating approximation and of the reflection grating approximation, respectively.
In a biased photorefractive crystal, the process of two one-dimensional waves mixing, i.e., the dynamical evolution of both pump beam and signal beam, is traced by numerically solving the coupled-wave equation. Direct simulations show that the propagation and stability of the two beams are completely determined by the system parameters, such as the external bias field, the intensity and the beam waist of the pump beam. By adjusting these parameters, one can control the state of two Gaussian waves mixing. The numerical results are helpful for performing a two-wave mixing experiment.
A theoretical analysis is presented of two-wave mixing in photorefractive materials in which the complex amplitude of a weak signal beam has slowly varying but otherwise arbitrary spatial dependence and arbitrary time variation. The set of coupled partial differential equations that couple the dynamics of the optical fields to the photorefractive medium is solved analytically to give the complex amplitude of the transmitted signal in the undepleted pump regime. The results are compared with known results for the transient two-wave mixing of plane waves.
It is known that two-wave coupling in photorefractive crystal may result in an intensity-dependent phase shift as well as energy transfer between the two waves without phase crosstalk. In a sense, this kind of phase shift is similar to the intensity-dependent refractive-index effects in Kerr media. Consequently, optical bistability can be established in an optical cavity with a photorefractive crystal. We formulate the model and get the expected bistable transmission with numerical simulations. Several cases, including that of the codirectional coupling and the contradirectional coupling, are considered. The effects of the photorefractive crystal’s distribution absorption and the spatial phase shift between the index grating and the light interference pattern in the crystal on the nonlinear transmission characteristics have also been studied. It is interesting to note that the output of the pumping wave is dependent on the cavity intensity and thus shows bistable behavior with respect to the cavity input as well.