Revisiting beam filamentation formation conditions in high power lasers

The Bespalov-Talanov gain (BT-gain) and IL-rule (i.e., the product of input intensity and self-focusing length is constant) expressions are examined and generalized for filamentation under realistic conditions associated with high power lasers: filamentation seeded by both amplitude and phase perturbations on a large, flat-top beam, and the impact of cross-phase modulation from unconverted light in UV frequency-converted lasers. The validity of these models is examined with NLSE numerical calculations, which show that there are parameters beyond the commonly-used IL rule, such as the perturbation amplitude and period content. The BT-gain model presents a fair description of the tendency of spatial periods to filament, but not of the quantitative self-focusing length. Spatial filtering of short periods is shown to suppress filamentation, due to both, the removal of the more prone to filament periods, as well as the reduction of the spatial intensity amplitude root-mean-square. At the edge of a top hat beam we find that the IL product reduces in the roll-off regions, even though the self-focusing length increases. When adding a co-propagating harmonic, we find that the cross-phase modulation (XPM) could enhance or inhibit the filamentation formation, depending on the perturbation period.