Non-frequency-shifted phase conjugation by Brillouin-enhanced four-wave mixing

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.