Self-Phase Modulation (SPM)
Self-phase modulation of a Gaussian or sech² pulse: peak nonlinear phase k₀n₂I₀L, largest frequency shift, rms broadening and the SPM spectrum by FFT, with a self-focusing check.
Peak power P₀
9.394 MW
0.9394 E/FWHM; on-axis I₀ = 2P₀/(πw²) = 2.392e11 W/cm²
Peak nonlinear phase φ_max
1.202 rad
k₀n₂I₀L = 0.383π; the B-integral of this pass
Rms spectral broadening
1.454×
√(1 + 4φ²/(3√3))
Largest frequency shift
±2.733 THz
0.858 φ_max/T₀, ±5.835 nm at λ
Critical power P_cr
1.705 MW
3.77 λ²/(8π n₀ n₂); P₀/P_cr = 5.511
Self-focusing distance
4.261 mm
Marburger, collimated beam at the entrance
The pulse's own intensity changes the index, so it gains the phase φ(T) = φ_max |U(T)|² with φ_max = k₀n₂I₀L. The time-varying phase shifts the instantaneous frequency by −dφ/dT: down on the leading edge, up on the trailing edge. The spectrum below is the Fourier transform of U(T)e^(iφ(T)). It breaks into about φ_max/π + ½ peaks while the pulse shape stays the same. The values are on axis (the beam's edge sees less phase) and leave out dispersion, self-steepening and the Raman response, so they hold while L is much shorter than the dispersion length.
P₀ > P_cr: the beam already self-focuses inside the medium (L/z_sf = 0.47), so the on-axis intensity grows and φ_max is a lower bound.
Spectrum before and after SPM
- Input (transform limited)
- After SPM
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
Dispersion-free SPM on the beam axis: φ(T) = φ_max|U(T)|², φ_max = (2π/λ)n₂I₀L with I₀ = 2P₀/(πw²). The spectrum is the exact FFT of U(T)e^(iφ(T)); rms broadening √(1 + 4φ²/(3√3)) (Gaussian) or √(1 + 32φ²/35) (sech²). Dispersion, self-steepening, Raman and the beam's radial phase are left out.
- Stolen R. H., Lin C. (1978). Self-phase-modulation in silica optical fibers. Phys. Rev. A 17, 1448–1453. doi:10.1103/PhysRevA.17.1448
- Agrawal G. P. (2019). Nonlinear Fiber Optics, 6th ed. Academic Press. SPM phase, frequency chirp, rms broadening and the peak count φ_max ≈ (M − ½)π.
- 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