Modulation instability of incoherent beams revisited

We examine the spatial modulation instability (MI) of a partially incoherent laser beam. We show that the P < (a/r_c)²P₀ criterion of beam stability, with a laser peak power P, beam radius a, correlation radius r_c, and critical power of self-focusing P₀, is applicable only to a limited class of MIs, viz., MIs that can be described as instabilities of a pertinent transverse correlation function found as a solution to the evolution equation, where the expectation of the four-field-product nonlinear source term is factorized as a product of the field intensity and a two-point transverse correlation function. When extended to a more general class of MIs, field evolution analysis of partially coherent beams suggests that MIs can be attenuated, but never completely suppressed. We show that spatial incoherence can lower the MI-buildup rate, thus helping avoid MI-induced beam breakup in physical settings where the MI-buildup length l_MI can be kept longer than the length of the nonlinear medium L. Because the l_MI > L condition sets a limitation on the field intensity rather than the laser peak power, MI-induced beam breakup can be avoided, even at laser peak powers well above the critical power of self-focusing P₀.