Multiple filament’s formation on a long atmospheric path was studied for an amplitude modulation of the terawatt femtosecond beam with the four and six hole masks. Filaments appeared at the distance that is determined by the Fresnel lens formed by a hole, not by the self focusing theory for the unrestricted (Gaussian like) beam. This lens focuses 1/3 of the energy passing through the hole to the beam spot ≈1 mm in diameter. The robust multifilament array 10–15 m in length was observed if the energy per hole was ≈1-1.5 mJ (pulse duration 50 fs), while each hole produces bundle of a few filaments at higher energies.

Focused femtosecond beam filamentation after amplitude masks has been studied experimentally and numerically. We deduced conditions (energy per hole, diameter and geometrical composition of holes, focal length) providing for the formation of the regularized bundle of filaments or single on-axis filament at the given pulse duration and beam diameter. We showed that a light channel with small diameter (∼200 μm) and overcritical peak power may be formed well before both the focal distance and the Marburger length, and this channel collapses due to self-focusing and forms the filament. The start position of such a filament can be predicted based on the linear propagation equation, while a more sophisticated non-linear approach that takes into account the Kerr nonlinearity, plasma effects, etc., helps to describe the temporal structure of a filament, its frequency, and its angular spectrum.