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Applied Optics

Asymptotic behavior of the boundary value problem of four-wave mixing in photorefractive media

Louis Sica

Published Jun 20, 1997• 0 citations
Publisher

Abstract

Solutions to the four-wave mixing problem asymptotically approach a spatial steady state over large interaction distances. In this limit, intensities of the two-point boundary value problem may be found from solutions to quadratic equations. The different limiting behavior exhibited for positive and negative gain has important implications for applications in which linearity of system response is important. In the regime studied, positive gain leads quickly to maximally depleted pumps with increasing input signal. For negative gain, such depletion does not occur until higher signal levels are reached.

DOI
10.1364/ao.36.004142

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Solutions to the four-wave mixing problem asymptotically approach a spatial steady state over large interaction distances. In this limit, intensities of the two-point boundary value problem may be found from solutions to quadratic equations. The different limiting behavior exhibited for positive and negative gain has important implications for applications in which linearity of system response is important. In the regime studied, positive gain leads quickly to maximally depleted pumps with increasing input signal. For negative gain, such depletion does not occur until higher signal levels are reached.
An exact solution to a model of two- and four-wave mixing in photorefractive media with saturable gain and absorption is presented. Pump depletion effects are accounted for, and the procedure for matching two-point boundary conditions is given. Possibilities of multistable solutions are investigated, and procedures on how to deal with such situations are outlined. It is found that the energy transfer between waves is less effective in the nonsaturated regime as compared to the saturated regime. It is also established that the nonsaturated system is more stable than the saturated under the same conditions.>
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.
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.
The equations for plane-wave steady-state propagation of n fields interacting via four wave processes in a Raman medium have been known since at least 1962. Complete analytic solutions have only been found for the process of harmonic generation. However, solutions have been obtained in the gain regime assuming zero pump depletion. Assuming all fields satisfy the phase matching condition we have found general analytic solutions to these equations. The solutions are complete in that they describe both pump deletion and saturation. Some example problems will be studied.
A method for exact solution of the stationary holographic degenerate four-wave mixing in reflection geometry in photorefractive media is presented. The effects of pump depletion and light absorption in the nonlinear dynamic medium are rigorously taken into account. Numerical steps in the procedure are reduced to the solution of one simple nonlinear first-order differential equation, or are absent altogether when the equation is linearized, or for the case of no absorption. The method of solution is applicable to other wave-mixing geometries as well.