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)P criterion of beam stability, with a laser peak power P, beam radius a, correlation radius r, and critical power of self-focusing P, 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 can be kept longer than the length of the nonlinear medium L. Because the l > 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 P.