Abstract: The behavior of self- focusing of the ultra- intensity rotation symmetric super- Gaussian beam ( RSSGB) during propagation in the plasmas is studied, with the aid of the numerical solution to the nonlinear paraxial wave equation.The result implies for a RSSGB with initial undulations, the self- focusing ring may split after traveling certain distance inside the plasmas. And because of the effect of the defocusing free- electron, the self- focusing ring is limited within a certain scales.

Key words: plasmas, self- focus;Kerr effect, super- Gaussian beam