High-power femtosecond laser filamentation in optical media enables numerous applications, including filament-induced breakdown spectroscopy, LiDAR, lightning control, and air lasing, due to its unique nonlinear characteristics. Femtosecond vortex laser filamentation, benefiting from distinctive spiral phase distributions and annular intensity profiles, exhibits enhanced resistance to air turbulence and improved air waveguide formation, making it particularly promising for practical implementations. However, achieving stable and controllable filamentation remains a critical challenge. In practical scenarios, the initial laser power often exceeds the critical power for self-focusing by orders of magnitude, leading to uncontrolled multiple filament generation. Consequently, systematic investigation of filament number dependence under various conditions for femtosecond vortex beams becomes essential. Previous studies suggest a simplified 2m+1 relationship between filament number and topological charge (m), while subsequent research reveals additional dependence on laser power. Current understanding remains incomplete with insufficient experimental validation. This work experimentally investigates filament number evolution with laser energy and topological charge in fused silica under different focusing conditions.

Femtosecond vortex beams with topological charges m=1 and m=2 were generated using q-plates and subsequently focused into a fused silica sample using lenses with focal lengths of 300 mm, 400 mm, and 500 mm. An imaging lens was used to image the transverse filamentation distributions at different propagation distances onto a screen, with images captured by a digital camera. Filament numbers were statistically analyzed while incrementally increasing input power. Critical powers for self-focusing of both Gaussian and vortex beams were measured using the S-scan technique under three different focusing configurations. Finally, filament number evolution was studied as a function of normalized power P/Pcrm,f.

Analysis of filament distribution patterns at different propagation distances confirmed that filaments formed on the intensity ring of vortex beams, exhibiting clear symmetry. Statistical analysis showed a nonlinear increase in filament number with increasing input energy, consistent with theoretical predictions. Increasing energy amplified the differences in filament number between different focusing conditions, suggesting potential underestimation of focusing conditions in existing models. When plotted against the normalized power P/Pcrm,f, the filament number evolution curves separated into two distinct groups: beamswith m=2 consistently produced more filaments than beamswith m=1 at the same normalized power, independent of focusing conditions. This indicates that normalized power, which incorporates focusing parameters, primarily determines filament number through topological charge. Notably, the universal scaling of filament number with normalized power across different focusing conditions provides new insights for filamentation control. Additionally, topological charge determination through filament number under specific normalized power conditions offers a new measurement method, particularly when supported by filament number related databases.

This study identifies the key factors governing femtosecond vortex beam filamentation in fused silica. Experimental results demonstrate that lenses with shorter focal lengths produce significantly more filaments at high energies. Under the same normalized laser power condition, vortex beamswith m=2 generate more filaments than those with m=1, showing consistent evolution across different focusing conditions. These findings enhance the understanding of nonlinear propagation characteristics of femtosecond vortex beams, and provide a new method for topological charge measurement. Moreover, the results offer practical insights for controlling filament number in high-energy applications, which is particularly important for optimizing filamentation processes in applications that require precise filament number control.