On the near axis approximation for thermal self-focusing of intense Gaussian laser beams in a collisional plasma

The propagation and self-focusing of a Gaussian laser beam in a collisional plasma is discussed in a paraxial solution of the wave equation in Wentzel Kramers Brillouin (WKB) approximation. The equation is solved using a radial expansion of properties up to the order of r4. A model for the dielectric constant of the plasma, which is based on the redistribution of density around the axis due to a balance of inverse Bremsstrahlung heating and radial losses due to heat conduction and electron-ion collisions, is presented without the necessity of assuming one of the loss processes being dominant. The validity of the beam width equation with expansion only to the order of r2 is discussed in a direct comparison with the r4-approximation solution for selected examples. A parameter region where the r4-approximation does not lead to a non-diverging solution in contrast to the r2-approximation is identified. The results are compared with a numerical solution of the nonlinear wave equation in WKB-approximation.