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The propagation of high-intensity femtosecond laser pulses in air under conditions of superposed spatial phase modulation is considered theoretically. The numerical simulations are carried out on the basis of the reduced form of a nonlinear Schrödinger equation for a time-averaged electric field envelope. Initial spatial modulations are applied to pulse wavefront profiling by a staggered ( T E M 22 ) phase plate, which is simulated numerically. The dynamics of laser pulse self-focusing, filamentation, and post-filamentation self-channeling after the phase plates with variable phase jumps is studied. We show that, with specific phase modulations, the pulse filamentation region in air can be markedly shifted further and elongated compared with a nonmodulated pulse. Moreover, during the post-filamentation propagation of spatially structured radiation, the highly localized light channels are formed, possessing enhanced intensity and reduced angular divergence, which enables post-filamentation pulse self-channeling on the distance multiple exceeding the Rayleigh range.
Geints et al. (Mon,) studied this question.