Soliton Propagation
Fibre soliton order N, dispersion and nonlinear lengths, fundamental soliton power and soliton period, with a split-step Fourier simulation of a sech pulse along the fibre.
Half-width T₀
56.73 fs
FWHM / 2 ln(1 + √2) = FWHM / 1.7627
Dispersion length L_D
0.2720 m
T₀²/|β₂|; soliton period z₀ = (π/2) L_D = 0.4273 m
Fundamental soliton power P₁
2828 W
|β₂|/(γT₀²); nonlinear length 1/(γP₀) = 0.1538 m
Soliton order N
1.330
1 soliton inside; the strongest settles at 7787 W, width T₀/1.66
The nonlinear Schrödinger equation ∂A/∂z = −(iβ₂/2)∂²A/∂T² + iγ|A|²A balances dispersion against self-phase modulation. The ratio of their lengths gives the order N = √(L_D/L_NL) = √(γP₀T₀²/|β₂|). With anomalous dispersion a sech pulse with N = 1 keeps its shape. Integer N > 1 compresses, splits and recovers every soliton period z₀. A non-integer N reshapes into ⌊N + ½⌋ solitons and sheds the rest as dispersive radiation. The plots solve the NLS by the split-step Fourier method for the sech input; fibre loss, β₃, self-steepening and the Raman response are left out.
Peak power and width along the fibre
- Peak power / input peak
- RMS width / input RMS width
Pulse intensity
- Input, z = 0
- Output, z = 2.00 m
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
Scalar nonlinear Schrödinger equation with β₂ and the Kerr term, solved by the symmetric split-step Fourier method for a sech input (N ≤ 4, up to 10 L_D). Fibre loss, β₃, self-steepening and the Raman response are left out; the Raman self-frequency shift grows as T₀⁻⁴ and becomes visible within a few L_D near the 57 fs default.
- 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
- Satsuma J., Yajima N. (1974). Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media. Prog. Theor. Phys. Suppl. 55, 284–306. doi:10.1143/PTPS.55.284
- Mollenauer L. F., Stolen R. H., Gordon J. P. (1980). Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys. Rev. Lett. 45, 1095. doi:10.1103/PhysRevLett.45.1095