Small-scale self-focusing of intense laser beams in the presence of vector effect

We extend Bespalov-Talanov (B-T) theory on small-scale self-focusing (SSSF) to include vector effect of a very narrow intense laser beam with application of the vector self-focusing model. The gain spectrum for perturbations is obtained by using the standard linear instability analysis. It is shown that the influence on SSSF of vector effect is closely related to the beam width. For a very narrow beam, the role played by vector effect becomes significant, it reduces the fastest growing frequency and the maximum growth rate, and shortens the frequency range for perturbation growing, and thus deviates the gain spectrum from that of B-T theory.