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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Theory Of Photorefractive Nonlinear Effects In A Self-Pumped Phase Conjugator

Paul Michael Petersen

Published Jan 4, 1990• 0 citations
Publisher

Abstract

A theoretical investigation of four-wave mixing in a photorefractive oscillator in the limit of one strong probe beam has been performed. The perturbational approach is compared with an exact numerical calculation.
DOI
10.1117/12.962145

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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.>
Using perturbation theory, we examine the fidelity of the photorefractive degenerate four-wave mixing (DFWM) phase conju- gator when the probe wave is spatially and temporally varying. Previous theories are based on linearized undepleted pump approximation and Laplace transform technique. Fully nonlinear solutions assume a single monochromatic plane wave probe. We have considered instead the more general case in which the probe is spatially varying (a beam) and/or temporally varying (a pulse), and the pumps are allowed to deplete. The probe wave is expanded as a sum of monochromatic plane waves of frequencies ω j = ω p + Ω j , wavevectors k j , and complex amplitudes A j (Ω j , k j ), where ω p is the frequency of the pumps. Using a perturbation approach up to the third order, we found that the components of the probe interact in pairs, so that the component of the conjugate wave at −Ω l −k l is H(Ω l , k l ) is the linearized response, ρ(Ω l , k l ; Ω j , k j ) represents crosstalk when j ≠ l, and self-nonlinear effect otherwise. Both functions exhibit resonance behavior. We use Eq. (1) to determine the response of the conjugate to pulsed beams.
The effects of linear absorption on the degenerate and nondegenerate self-oscillation conditions of a photorefractive phase-conjugate mirror are explored. In this study, phase-conjugation results from nonlinear mixing of two strong undepleted parallel-polarized counterpropagating pump waves with a probe wave. Numerical results are presented that illustrate that the linear absorption tends to increase the value of the coupling coefficient required for self-oscillation. It is also found that the pump intensity ratio r is almost independent of the linear absorption for positive normalized detuning Δωτ between the probe and the pump frequencies and of the zero phase shift, ϕ, between the gratings and the light interference patterns when the gratings are stationary. For ϕ = ±π/2, the variation of r with respect to absorption is symmetric for positive and negative detunings.
The physical origin of frequency shift and dynamic instability in a photorefractive self-pumped phase conjugator is analyzed by use of four-wave mixing and multi-four-wave mixing. We show that the frequency shift and the instability result from the requirement of the maximal energy transfer between the interacting beams. Two examples of numerical results in a cat self-pumped phase conjugator are given, which are in good agreement with previous experimental results. A simple method is proposed to change an unstable phase-conjugate output to a stable one.
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.
The effects of linear absorption on the degenerate and nondegenerate self-oscillation of a photorefractive phase-conjugate mirror under condition of anisotropic four-wave mixing are considered. The phase conjugation results from nonlinear mixing of two strong undepleted orthogonally polarized counterpropagating pump waves with a probe wave. We consider two types of situation for orthogonally polarized pump beams corresponding to two crystal families: one such as the Bi12SiO20 crystal family (sillenite family) and the other such as the BaTiO3 and strontium barium niobate crystal family. In both of these crystal families the degenerate and nondegenerate self-oscillations are shown to occur in the presence of linear absorption, which tends to increase the value of the coupling coefficient required for self-oscillation. Numerical results are presented that exhibit various features of degenerate and nondegenerate self-oscillation in both crystal families for different values of the phase ϕb of the backward grating.