We examine the nonlinear dynamics of the optical field that arises from two-wave mixing in a single-mode photorefractive unidirectional ring resonator. We describe the interaction between the photorefractive material and the optical field with a complex intensity-dependent time constant and a complex coupling constant. Assuming an undepleted uniform pump and in the mean field limit, the dynamic equations for the refractive-index grating and the electric-field amplitudes are shown to reduce to a complex Van der Pol equation. An analysis of the limit-cycle solutions to this equation yields new analytical expressions for the asymptotic mode intensity, the asymptotic mode frequency, and the oscillation condition and produces a stability condition for the limit cycle. In the real time constant and real coupling constant case, our results reduce to those of previous investigations that have considered an intensity-independent time constant.
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