The phase-conjugate reflectivity produced by degenerate four-wave mixing in a Kerr medium in the Bragg scattering regime is shown to be unstable when the intensity of the external pump beams is greater than a threshold value that can be predicted analytically. Below this threshold, a single branch of the reflectivity curve is stable, eliminating any possibility for bistability. A numerical simulation indicates that the unstable regime is characterized by fast oscillations with a period related to the transit time in the nonlinear medium. A finite relaxation time of the nonlinearity eliminates the high-frequency oscillations but can increase the amplitude of the low-frequency components. The insertion of a phase-conjugate mirror into an optical cavity leads to an instability whose threshold can be evaluated analytically. The temporal evolution of the field intensity in the cavity is shown to change from a stable regime to periodic oscillations and then to quasi-periodic oscillations as the pump intensity is increased.
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