Fig. 1. Generation of a phase-nested beam for prolonging the filament. (a) Experimental setup, (b) phase distribution of the generated phase-nested beam behind lens L3, (c) simulated interference pattern of the phase-nested beam behind lens L3 with the Gaussian beam, and (d) the experimentally measured result corresponding to (c).

Fig. 2. Equivalent focal length F of the phase-nested beam for different γ, ρc, and ρ1 in BK7 glass, as a function of the normalized radial coordinate.

Fig. 3. Simulated nonlinear propagation behaviors of different beams under the conditions of P=10Pc, ϵ∼10−4, f=0.0125Ld, γ=γ0, ρc=0.35, ρ1=0.32, and ϕ0=0. Simulated on-axis intensity of nonlinear propagation of (a) the whole phase-nested beam, (b) the central part (ρ≤ρc) of the phase-nested beam, (c) the annular part (ρ>ρc) of the phase-nested beam, and (d) the Gaussian beam with the same input power and size as the phase-nested beam. (e)–(h) show the intensity in cross-sections as a function of propagation distance corresponding to (a)–(d), respectively.

Fig. 4. Experimental results of the filament extended by the phase-nested beam. (a) Filament produced by the phase-nested beam. (b) Filament produced by the central part of only the phase-nested beam. (c) No filament is formed by the annular part of the phase-nested beam. (d) Filament generated by the Gaussian beam with same pulse energy and beam width as the whole phase-nested beam. The corresponding input beams are shown in the left side, while the corresponding far-field patterns are shown in the right side. (e) The output spectra of the above beams. The ruler is also captured simultaneously as the filaments to measure the filamentation length.

Fig. 5. Filaments produced by the phase-nested beams with different phase shifts. The parameters remain unchanged as r0=1.2  mm, γ=1/18, ρc=0.35, ρ1=0.32, and E=13  μJ. The exposure time of the digital camera stays at 20 s.

Fig. 6. Filaments produced by the phase-nested beams with the different parameter sets γ, ρc, ρ1, and ϕ0. The pulse energy of all of the input fields is kept at E=13  μJ, and the exposure time of the digital camera stays at 20 s.

Fig. 7. Filaments produced by the partially blocked phase-nested beams. The intensity profiles of the input beams are shown on the left. (a), (c), (e) The filaments produced by the phase-nested beams blocked by a specific ring-like band-stop filter, whose normalized internal radius and width are ρa and d, respectively. (b), (d), (f) The filaments produced by the phase-nested beams blocked by a low-passing circular aperture with a normalized radius of ρb.