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IEEE Journal of Quantum Electronics

Two- and four-wave mixing with saturable absorption and gain

Dejan V. Timotijević, Milivoj R. Belić, Robert W. Boyd

Published Jan 1, 1992• 9 citations
Publisher

Abstract

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.>

DOI
10.1109/3.144484

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Optics Letters•Apr 1, 1986•25 citations
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.>
We have investigated 2WM and 4WM processes in photorefractive crystals in the nonsaturated regime, i.e. when the coupling between beams and the absorption depend on the light intensity. Different models of saturable absorption in 2WM via reflection grating are considered, and the corresponding wave equations are solved exactly. The solutions are written in terms of hypergeometric functions or in quadratures. From the solutions it was found that the energy transfer between the waves is less effective in the nonsaturated stage of the process.
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.
There is presently a considerable amount of interest in wave conjugation via four-wave mixing in saturable absorbing media. Typically, the four-wave process in volves the interaction, of two counter-propagating pump waves and an Input probe wave via a third-order nonlinearity in the polarization of the medium. Introduction of the probe wave produces a reflected signal wave in the retrodirection. Abrams and Lind have determined the magnitude of the reflected wave obtained from a two-level saturable absorber by solving a set of coupled equations for the probe and signal waves. To obtain this solution, they assumed that the probe and signal waves are weak compared with the pump waves and that the amplitudes of the pumps are constant. In this paper we shall consider the effect of pump attenuation caused by absorption and pump depletion caused by the transfer of energy to the probe and signal waves.
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.
The coupled equations satisfied by the pump, probe, and signal waves in a four-wave interaction in a two-level saturable absorber are derived, taking into account the effects of absorption and depletion. Numerical solutions of these equations are presented in two limits; one in which the probe-wave amplitude is small compared with that of the pump waves and another in which the probe- and pump-wave amplitudes are comparable. The results indicate that absorption effects are important when the wave amplitudes are less than the saturation amplitude of the absorption process. The importance of the depletion effects depends on the type of four-wave interaction. The most significant effects occur when the undepleted theory predicts probe-wave reflectivities exceeding unity.