Relativistic self-focusing and its effect on stimulated Raman and stimulated Brillouin scattering in laser plasma interaction Phys. Plasmas 8, 3419 (2001); 10.1063/1.1377048 Stimulated Brillouin and Raman scattering from a randomized laser beam in large inhomogeneous collisional plasmas. II. Model description and comparison with experiments Phys. Plasmas 8, 1636 (2001); 10.1063/1.1357218 Effect of pump depletion and self-focusing on stimulated Brillouin scattering process in laser-plasma interactions Phys. Plasmas 6, 927 (1999); 10.1063/1.873332Effect of the speckle self-focusing on the stationary stimulated Brillouin scattering reflectivity from a randomized laser beam in an inhomogeneous plasma The origin of the low level of stimulated Brillouin scattering (SBS) observed in laser-plasma experiments carried out with a single laser speckle is investigated by means of three-dimensional simulations and modeling in the limit when the laser beam power P is well above the critical power for ponderomotive self-focusing We find that the order of magnitude of the time averaged reflectivities, together with the temporal and spatial SBS localization observed in our simulations, are correctly reproduced by our modeling. It is observed that, after a short transient stage, SBS reaches a significant level only (i) as long as the incident laser pulse is increasing in amplitude and (ii) in a single self-focused speckle located in the low-density front part of the plasma. In order to describe self-focusing in an inhomogeneous expanding plasma, we have derived a new Lagrangian density describing this process. Using then a variational approach, our model reproduces the position and the peak intensity of the self-focusing hot spot in the front part of the plasma density profile as well as the local density depletion in this hot spot. The knowledge of these parameters then makes it possible to estimate the spatial amplification of SBS as a function of the laser beam power and consequently to explain the experimentally observed SBS reflectivity, considerably reduced with respect to standard theory in the regime of large laser beam power. V C 2014 AIP Publishing LLC.