Exact solutions of the linearized transient envelope equations for degenerate four-wave mixing in a finite-length cell show that reflection coefficients near unity severely restrict the conditions under which chirp reversal occurs, and can lead to nulls in the amplitude of the reflected and transmitted fields.
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.
The exact theory of degenerate four-wave mixing in a Kerr-like medium in the Bragg scattering regime is used to study the reflectivity of phase-conjugate mirrors pumped by intense laser beams. The analysis takes into account pump depletion, nonlinear phase modulation and the relaxation time of the nonlinearity. The steady-state response exhibits multivalued reflectivity, hysteresis and potential bistability; however, the transient response shows that at most one value of the phase-conjugate reflectivity is stable. Periodic or chaotic pulsations are predicted beyond threshold values of pump intensity. The onset of a natural oscillation in the Kerr-like medium is also predicted in the absence of a continuous probe beam; the resulting field amplitude is stable and weakly sensitive upon the presence of external noise.
We present analytical results for the steady-state signal and phase-conjugate fields interacting with smoothly varying nondepleted pump waves in a nearly degenerate four-wave-mixing geometry. Signal depletion is retained in the equations and we study the effect that detuning on the input signal from the degenerate pump amplitudes has on the reflected conjugate wave and the transmitted signal wave. A general result is derived for the appearance of spontaneous amplification under weak restrictions on the profile of the degenerate pump fields. The conjugation of the phase front of the signal is studied for exponential and hyperbolic secant functions multiplying the coupling coefficients and comparison is made with corresponding results for square profile pump fields. These results are discussed and their relevance to phase conjugation of pulses is examined.
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.