Supercontinuum Generation
Length scales of supercontinuum generation in a photonic crystal fibre: soliton number, dispersion length, soliton fission length and the dispersive-wave wavelength from β₂ and β₃.
Peak power
10.00 kW
sech²: E/(2T₀), T₀ = FWHM/1.7627 = 28.36 fs
Soliton number N
8.65
√(P/P₁), fundamental soliton P₁ = |β₂|/(γT₀²) = 133.7 W
Dispersion length L_D
6.801 cm
T₀²/|β₂|
Fission length L_fiss
0.7863 cm
≈ L_D/N
Dispersive wave
633.3 nm
β₂Ω²/2 + β₃Ω³/6 = γP_s/2, first ejected soliton P_s = 35.51 kW: Ω = 2π × 114.3 THz from the pump; with P₀: 667.7 nm, leading order −3β₂/β₃: 699.3 nm. A full GNLSE puts it bluer still.
Regime
Soliton fission
Soliton fission inside the fibre (L/L_fiss = 19.1): ≈ N fundamental solitons, Raman self-frequency shift and dispersive waves.
A femtosecond pulse in the anomalous dispersion of a photonic crystal fibre is a higher-order soliton of order N = √(γP₀T₀²/|β₂|). Higher-order dispersion and the Raman response break it up near the fission length L_fiss ≈ L_D/N into about N fundamental solitons. They shift to longer wavelengths, and each one radiates a phase-matched dispersive wave on the other side of the zero-dispersion wavelength. Its offset Ω solves β₂Ω²/2 + β₃Ω³/6 = γP_s/2 with the radiating soliton's peak power. The first one ejected at fission has P_s = (2N − 1)²P₀/N², up to 4× the input; later, weaker solitons radiate between this and the input-power value. The leading order Ω ≈ −3β₂/β₃ holds only when γP_s ≪ |β₂|Ω². Higher-order dispersion, Raman and self-steepening put the simulated wave further into the blue (≈ 545 nm at the defaults against 633 nm here), so read it as the least shift. Coherence degrades from N ≈ 10; above N ≈ 16 modulation instability takes over. The spectrum itself needs a generalised NLS simulation and isn't shown; these scales tell whether the fibre is long enough for fission. Pulse shape: sech², so P₀ = E/(2T₀).
Fission length against pulse duration, at fixed energy
- Fission length
- Fibre length
- Input 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
Sech² pump pulse. Fission length L_D/N is a scaling estimate. Dispersive-wave offset from the β₂, β₃ phase-matching cubic with the wavenumber γP_s/2 of the first soliton ejected at fission, P_s = (2N − 1)²P₀/N² (Satsuma–Yajima); higher-order dispersion, Raman and self-steepening, left out, put the wave further into the blue (≈ 90 nm at the defaults in a generalised NLS run). N > 16 is labelled modulation-instability dominated (Sylvestre 2021); coherence degrades from N ≈ 10. The spectrum needs a generalised NLS simulation and is not computed. Defaults: the 835 nm PCF example of Dudley et al. 2006.
- Dudley J. M., Genty G., Coen S. (2006). Supercontinuum generation in photonic crystal fiber. Rev. Mod. Phys. 78, 1135–1184. doi:10.1103/RevModPhys.78.1135
- Akhmediev N., Karlsson M. (1995). Cherenkov radiation emitted by solitons in optical fibers. Phys. Rev. A 51, 2602. doi:10.1103/PhysRevA.51.2602
- Skryabin D. V., Gorbach A. V. (2010). Colloquium: Looking at a soliton through the prism of optical supercontinuum. Rev. Mod. Phys. 82, 1287, Eqs. 3, 7, 10: resonance with the radiating soliton's wavenumber. doi:10.1103/RevModPhys.82.1287
- Sylvestre T., Genier E., Ghosh A. N. et al. (2021). Recent advances in supercontinuum generation in specialty optical fibers. J. Opt. Soc. Am. B 38, F90: modulation instability for N > 16. doi:10.1364/JOSAB.439330
- Agrawal G. P. (2019). Nonlinear Fiber Optics, 6th ed. Academic Press. Dispersion and nonlinear lengths, soliton order, the normalised NLS. doi:10.1016/C2018-0-01168-8