Filamentation Dynamics
Where a femtosecond filament starts: critical power for self-focusing, peak power against it, and the Marburger collapse distance of a collimated or lens-focused Gaussian beam.
Critical power P_cr
2.759 MW
3.77 λ²/(8π n₀ n₂); 0.2937 µJ at this duration
Peak power
9.394 MW
0.9394 E/FWHM; P/P_cr = 3.405
Collapse distance
5.319 mm
Marburger, collimated beam
Rayleigh range z_R
14.24 mm
π n₀ w²/λ inside the medium
Above the critical power the Kerr lens beats diffraction and a Gaussian beam self-focuses. A collimated beam collapses after z_sf = 0.367 z_R / √((√(P/P_cr) − 0.852)² − 0.0219), Dawes and Marburger's fit to numerical solutions. Near the collapse the intensity climbs until multiphoton ionisation makes a plasma that defocuses the beam and absorbs energy. The intensity then clamps, at about 4×10¹³ W/cm² in air (Kasparian, Sauerbrey & Chin 2000), and a filament forms. The filament's length and plasma density need a full propagation model; this page gives the threshold and where the filament starts. P uses the pulse's peak power, which self-focuses first. Far above P_cr, small irregularities in the beam make it break into several filaments.
Collapse distance against peak power
- Collimated (Marburger)
- This pulse
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Gaussian beam, instantaneous Kerr effect. P_cr = 3.77 λ²/(8π n₀ n₂) and the Dawes–Marburger fit z_sf = 0.367 z_R/√((√(P/P_cr) − 0.852)² − 0.0219), with a lens 1/z′ = 1/z_sf + 1/f. Only the onset: plasma defocusing, multiphoton absorption and the filament length aren't modelled. Default n₂ and n₀ are fused silica at 800 nm.
- Fibich G., Gaeta A. L. (2000). Critical power for self-focusing in bulk media and in hollow waveguides. Opt. Lett. 25, 335–337. doi:10.1364/OL.25.000335
- Marburger J. H. (1975). Self-focusing: theory. Prog. Quantum Electron. 4, 35–110. doi:10.1016/0079-6727(75)90003-8
- Dubietis A., Tamošauskas G., Šuminas R., Jukna V., Couairon A. (2017). Ultrafast supercontinuum generation in bulk condensed media. Lith. J. Phys. 57(3). Lens-shifted collapse distance and n₂ of fused silica (Table 1). doi:10.3952/physics.v57i3.3541
- Couairon A., Mysyrowicz A. (2007). Femtosecond filamentation in transparent media. Phys. Rep. 441, 47–189. doi:10.1016/j.physrep.2006.12.005
- Kasparian J., Sauerbrey R., Chin S. L. (2000). The critical laser intensity of self-guided light filaments in air. Appl. Phys. B 71, 877–879. doi:10.1007/s003400000463