Ultra-intense and ultra-short laser pulses may be generated up to the exawatt-zetawatt regime due to parametric processes in plasmas. The minimization of unwanted plasma processes leads to operational limits which are discussed here with respect to filamentation. Transverse filamentation, which originally was derived for plane waves, is being investigated for seed pulse propagation in the so called π-pulse limit. A three-dimensional (3D) three-wave-interaction model is the basis of the present investigation. To demonstrate the applicability of the three-wave-interaction model, the 1D pulse forms are compared with those obtained from 1D particle in cell and Vlasov simulations. Although wave-breaking may occur, the kinetic simulations show that the leading pumped pulse develops a form similar to that obtained from the three-wave-interaction model. In the main part, 2D and 3D filamentation processes of (localized) pulses are investigated with the three-wave-interaction model. It is shown that the leading pulse front can stay filamentation-free, whereas the rear parts show transverse modulations.
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