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IEEE Journal of Quantum Electronics

Nonlinear response of the photorefractive phase conjugator: a perturbation analysis

G. Li, Bahaa E. A. Saleh

Published Apr 1, 1990• 2 citations
Publisher

Abstract

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.>

DOI
10.1109/3.53398

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IEEE Journal of Quantum Electronics•Sep 1, 1985•2 citations
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.
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.
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.
The effect of biasing illumination on the degenerate four-wave mixing (DFWM) in absorbing photorefractive (PR) media has been investigated. The input beams are considered to be spatially uniform. The analysis has been made both for transmission and reflection geometry. Considering pump beam depletion and the slowly varying amplitude approximation, nonlinear coupled wave equations are obtained and have been solved numerically by a shooting method. The effect of different parameters such as input beam intensities ratio, thickness of the interacting medium on the phase conjugate reflectively (PCR) has been investigated. It has been observed that PCR can be controlled to a large extent by controlling the biasing intensity.
A mean field theory of nearly-degenerate four-wave mixing is developed for the optical bistability, which has recently been found in a phase-conjugate resonator consisting of a normal mirror and a phase-conjugate mirror under relatively strong pumping. The bistability in transmission of the probe beam occurs as the mirror separation or the probe offset frequency is changed, but for a limited range of probe beam intensity. This behavior is explained as occurring under injection-locked conditions because of combined effects of tilting and broadening, or narrowing of the resonance curves due to nonlinearities in the phase-conjugate reflectivity arising from pump depletion.