Modulation instability of a coherent-incoherent mixture

We examine the nonlinear coupling and modulation instability of a coherent beam with one that is partially spatially incoherent. Using a mutual coherence approach, we derive the growth rate for perturbations and show that the presence of any amount of coherent component eliminates the nonlinear threshold for instability. The fraction of coherent light is shown to determine the gain and characteristic period of the resulting patterns. Theoretical considerations are confirmed by numerical simulation and by experimental observations in a self-focusing photorefractive crystal.