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Journal of Optics

Photorefractive four-wave mixing cannot be treated perturbatively. Revisiting the undepleted-pump approximation

Germán J. de Valcárcel, Fernando Silva, A. Esteban-Martín, Eugenio Roldán

Published Apr 24, 2023• 0 citations
Publisher

Abstract

Abstract We theoretically consider photorefractive degenerate four-wave mixing oscillators and show that, for this particular system, the linear stability analysis technique leads to incorrect results. The reason for this astonishing failure lies in the unphysical predictions of the undepleted-pump approximation, which appear naturally during the linearization process of the full model. As a consequence, photorefractive four-wave mixing does not seem to permit a perturbative treatment that allows the derivation of mean-field models, and one is forced to use the full model in order to make sensible predictions.

DOI
10.1088/2040-8986/accfab

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Abstract We theoretically consider photorefractive degenerate four-wave mixing oscillators and show that, for this particular system, the linear stability analysis technique leads to incorrect results. The reason for this astonishing failure lies in the unphysical predictions of the undepleted-pump approximation, which appear naturally during the linearization process of the full model. As a consequence, photorefractive four-wave mixing does not seem to permit a perturbative treatment that allows the derivation of mean-field models, and one is forced to use the full model in order to make sensible predictions.
The stability of the steady-state solutions of four-wave mixing in photorefractive crystals has been studied. Two complementary methods have been used: linear stability analysis and time integration of the nonlinear equations. A critical dependence of the stability on the phase of the perturbation has been revealed. The existence of two stable solutions for certain values of some parameters has been found. Also, regimes of unstable behavior in cases with a unique steady-state solution for large values of the pump ratio have been found.
The undepleted-pump photorefractive four-wave mixing equations for the case of orthogonally polarized pumps are solved taking into account the linear absorption of the medium. A phenomenon of the enhancement of the phase conjugate reflectivity due to linear absorption is found to occur for a certain range of the phase conjugator parameters. This phenomenon is shown to occur as a consequence of the existence of the self-oscillatory modes in the presence of linear absorption. \textcopyright{} 1996 The American Physical Society.
The analysis of pump-ratio dependences of the output intensity for a semilinear photorefractive coherent oscillator reveals two domains of multivalued solutions for sufficiently large coupling strength ensured by the crystal. We show that even in a strictly degenerate case the nonzero output intensity can be reached in a broad range of pump ratios r from 10−6 to infinity, including the interval where both pump intensities coincide or are very close to each other. This does not contradict the existence of the known gap in the oscillation threshold near the equal intensities of two pump waves: in this particular region the oscillation is not self-starting. The output intensities for frequency degenerate oscillation are calculated analytically in the standard plane-wave approximation of backward-wave four-wave mixing and numerical simulations are used to analyze the stability of multivalued solutions.
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.