A theory for multimode oscillation in an optical ring resonator that has photorefractive gain is derived by using a self-consistent model for the nonlinear interaction of the resonator field with an external pump field through the photorefractive agency. The theory is developed in the weak-field limit so that pump depletion can be neglected and the intensity spatial modulation index remains small. The characteristics of single-mode, two-mode, and three-mode behavior are analyzed in detail. The slow time response of the photorefractive effect has great leverage on the frequencies of the modes of oscillation; as a consequence, oscillation occurs very near the pump frequency. In the weak-field limit, the modulation index corresponding to the intermode gratings is negligibly small when compared with the modulation index for the gratings formed from the interference of the pump beam with each mode. As a result, in this limit, there appear no terms leading to mode locking. However, as in a laser, mode competition for the gain does occur, and under the proper conditions the presence of one mode can suppress the oscillation of one or more other modes. We show that the competition between modes having similar transverse distributions is stronger than that between modes having dissimilar transverse distributions.
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