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Journal of the Optical Society of America B

Signal enhancement in collinear four-wave mixing

Colin J. McKinstrie, G. G. Luther, Steven H. Batha

Published Mar 1, 1990• 18 citations
Publisher

Abstract

The solitary-wave solutions of the four-wave equations are studied, and their relevance to four-wave mixing in finite media is discussed. In general, the transfer of action from the pump waves to the probe and signal waves is limited by nonlinear phase shifts that detune the interaction. However, by controlling the linear phase mismatch judiciously, it is often possible to effect a complete transfer of action from the pump waves to the probe and signal waves.

DOI
10.1364/josab.7.000340

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The solitary-wave solutions of the four-wave equations are studied, and their relevance to four-wave mixing in finite media is discussed. In general, the transfer of action from the pump waves to the probe and signal waves is limited by nonlinear phase shifts that detune the interaction. However, by controlling the linear phase mismatch judiciously, it is often possible to effect a complete transfer of action from the pump waves to the probe and signal waves.
This paper presents an exact solution to the slowly varying envelope equations for the probe and signal fluxes in degenerate four-wave mixing in Kerr media. The analysis is performed for a noncollinear geometry, and all beams have the same linear polarization in the nonlinear medium. It is assumed that the angle separating the directions of the counterpropagating pumps and probe and signal beams is large enough so that cross-coupled waves do not contribute to the envelope equations describing the spatial evolution of the pump, probe, and signal beams. Multiple values of the transmitted probe are obtained for equal pump fluxes above a threshold determined by the nonlinear phase shift in the medium. Significant pump depletion occurs in this regime, but it can also occur without the appearance of multiple solutions. Since exact optical phase conjugation occurs only in the limit where the pumps are undepleted in Kerr media, this result implies that there are fundamental upper limits on the pump intensity that can be used to increase signal reflectivities in optical phase conjugation.
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.
We describe the steady-state enhancement of the phase-conjugate reflectivity that occurs with linear absorption near self-oscillation in four-wave mixing systems. Using a simplified general model, we show that this effect can arise in a variety of phase conjugators including those based on photorefractive and Kerr media. A new expression is introduced for the optimal reflectivity condition, which occurs if the transfer of energy from the strong forward pump to the weak probe is maximized at the entrance to the nonlinear medium.
We derive an exact solution for both amplitude and phase of a signal that is back-reflected by collinear nonlinear degenerate mixing of four circularly polarized waves of arbitrary intensity in a Kerr-like medium. Our analysis suggests that taking pump depletion into account may spoil the ideal phase-conjugation condition, even in the limit of weak nonlinear coupling, and indicates that substantial deterioration of the probe phase-reversal property may occur for strongly interacting waves in a manner that is highly sensitive to the anisotropy of the nonlinear susceptibility.
A systematic study of degenerate two-wave mixing in the stationary regime in thin nonlinear local-response media is presented. The model is based on the coherent interaction of four waves, the two incident beams, and the two main diffracted waves. Analytical results are obtained in all cases investigated under the assumption of weak-beam amplification. It is demonstrated that the energy transfer arises only from the forward degenerate four-wave mixing terms. In particular, the theoretical results show two new features: (1) the gain factor of the interaction is substantially reduced because of the phase change associated with the two-wave mixing terms and (2) a phase mismatch can compensate for this phase change, leading to an appreciable gain enhancement. Moreover, the general case is solved numerically. Particular attention is given to the influence of separate two-wave and forward degenerate four-wave mixing terms. Self-diffraction effects and the gain dependence versus the ratio of the pump-beam intensity to the signal-beam intensity are investigated.