The angular divergence and spectral bandwidth of a spatially incoherent laser beam propagating in an underdense plasma are shown to correspond to additional spatial and temporal incoherence in the regime where the average laser intensity in a speckle approaches the self-focusing threshold. The transverse and longitudinal sizes of laser speckles inside the plasma are related in a way that makes it possible to define a local effective beam f-number. The effective f-number decreases as light propagates through the plasma, and the effective f-number at the outgoing boundary is consistent with the transmitted light angular spreading. The spectral broadening and red shift of the transmitted light are interpreted as a result of multiple near-forward stimulated Brillouin scattering of the propagating laser beam. The reduction of backward stimulated Brillouin scattering follows from decrease of the effective speckle length and from the plasma induced temporal incoherence. (C) 2001 American Institute of Physics.