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Journal of the Optical Society of America B

Generalized phase-matching condition for nondegenerate four-wave mixing

Phillip R. Peterson, A. Gavrielides, David A. Cardimona

Published Mar 1, 1987• 3 citations
Publisher

Abstract

The coupled-probe, conjugate, and pump differential equations are solved in the undepleted pump approximation when the pumps are of unequal frequencies. In the usual analysis the two pump wave vectors satisfy k1 + k2 = 0, so that the conjugate and probe wave vectors satisfy kp + kc = 0. The vector phase matching then consists of two one-dimensional problems. However, the unequal pump frequencies require a true two-dimensional analysis, since the pump wave vectors do not sum to zero separately from the probe/conjugate fields. This then requires the inclusion of a transverse dimension in the differential equations. The net result is that the direction of the maximum conjugate intensity, usually determined by the phase-match condition Δk = 0, is instead determined by the generalized condition kc · Δk = 0, where Δk = k1 + k2 − kp − kc. This new phase-matching condition is quadratic in kc, leading to two peaks in the conjugate reflection coefficient.

DOI
10.1364/josab.4.000375

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The coupled-probe, conjugate, and pump differential equations are solved in the undepleted pump approximation when the pumps are of unequal frequencies. In the usual analysis the two pump wave vectors satisfy k1 + k2 = 0, so that the conjugate and probe wave vectors satisfy kp + kc = 0. The vector phase matching then consists of two one-dimensional problems. However, the unequal pump frequencies require a true two-dimensional analysis, since the pump wave vectors do not sum to zero separately from the probe/conjugate fields. This then requires the inclusion of a transverse dimension in the differential equations. The net result is that the direction of the maximum conjugate intensity, usually determined by the phase-match condition Δk = 0, is instead determined by the generalized condition kc · Δk = 0, where Δk = k1 + k2 − kp − kc. This new phase-matching condition is quadratic in kc, leading to two peaks in the conjugate reflection coefficient.
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 coupled-wave equations describing the optical phase conjugation by degenerate four-wave mixing are usually solved in the so-called slowly varying amplitude approximation, wherein only the first derivatives of the amplitudes of the probe and the conjugate waves are taken into account. Here we solve the coupled-wave equations without this approximation for nonresonance interactions. This leads to an oscillatory dependence of both reflectivity and transmittance on the normalized coupling constant, κL. Also, the intensity distributions of the probe and the conjugate waves show oscillatory beatlike behaviors around the corresponding slowly varying amplitude approximation distributions. These behaviors can be interpreted as the diffraction of the conjugate (probe) wave from a grating resulting from the superposition of the pump–pump and the pump–probe gratings.
We present a theoretical treatment of four-wave mixing (FWM) in a Brillouin-active medium for the case in which the pump waves differ in frequency by approximately twice the Brillouin frequency shift of the medium and in which the probe-wave frequency is approximately the arithmetic mean of the frequencies of the two pump waves. Under these conditions, the conjugate wave produced by the FWM process has the desirable property of being at the same frequency as the probe. We derive the coupled amplitude equations describing this interaction. We solve these equations analytically in the limit of negligible pump depletion and find that large phase conjugate reflectivities are readily achievable. The coupled amplitude equations are solved numerically for the general case, and it is found that large power transfer from the pumps to the output wave is possible. The output wave is shown to be a nearly perfect phase conjugate of the probe wave, even far into the regime where pump depletion effects are important. Our formalism predicts the existence of a parametric instability in the propagation of the pump waves, but good performance is predicted before the onset of this instability.
The coupled wave equations describing the optical phase conjugation by degenerate four wave mixing DFWM is usually, solved in the so called slowly varying amplitude approximation (SVA), where only the first derivatives of the amplitudes of the probe and of conjugate waves are taken into account. Here, we solve the coupled equations exactly for nonresonance ]interactions. This leads to an oscillatory dependence of reflectivity and of transmittance on the normalized coupling constant, κL. Also the intensity distributions of the probe and conjugate waves show oscillatory beat-like behaviors around the corresponding SWA distributions. The latter can be interpreted as the diffraction of the conjugate (probe) waves from the super-position of the pump-pump and pump-probe gratings.
Our semiclassical theory of two-photon multiwave mixing1 is a generalization of the degenerate four-wave mixing case2 to treat nondegenerate pump/signal/conjugate frequencies. We apply that theory to four-wave mixing in two-photon media. We present explicit formulas for the nondegenerate signal/conjugate absorption and coupling coefficients for cases for which the two-photon Stark shifts are sufficiently small compared to the homogeneous linewidths. More general cases require that the spatial hole burning averages be performed numerically. We find intricate dependencies of the transmission and reflection spectra on the pump intensity and on dynamic Stark shifts. In particular, the reflection coefficients do not necessarily bleach to zero as the pump intensity is increased indefinitely, and interferences occur between pure index and mixed absorption/index contributions
We have derived an analytic expression for the intensity distribution of two counterpropagating waves of arbitrary input intensity in a saturable absorber. We use this distribution to extend the theory of degenerate four-wave mixing to include the effects of pump wave absorption. In particular, the spatial variation of the nonlinear absorption and coupling constants that appear in the coupled-amplitude equations for the probe and the signal (i.e., conjugate) waves are determined. These coupled-amplitude equations are solved numerically in a noniterative manner, leading to a prediction for the phase-conjugate reflectivity. The nature of the predictions differs qualitatively from those1,2 derived from theories that ignore pumpwave absorption, especially for the case of off-line-center excitation. At large values of the input pump intensities, the phase conjugate reflectivity can be larger than that predicted by theories that ignore pump-wave absorption.