Nonlinear Effects in Fiber
Calculate the nonlinear coefficient γ, SPM and XPM phase, four-wave-mixing efficiency and SBS/SRS thresholds of a fiber span.
Nonlinear Coefficient γ
1.317 /(W·km)
2πn₂/(λ·A_eff)
Effective Length L_eff
21.17 km
SPM Phase φ_SPM
0.279 rad
0.089π
SPM RMS Broadening
× 1.030
unchirped Gaussian, no dispersion
XPM Phase, max (7 other ch.)
3.905 rad
co-polarized, no walk-off
FWM Efficiency η
3.206e-5
nearest product, 100 GHz
FWM Product / Channel
-56.0 dB
degenerate; non-degenerate +6.0 dB
SBS / SRS Threshold
2.0 / 29.7 dBm
K = 1 / 1, narrow CW line; modulated channels have a much higher threshold · launch exceeds the SBS threshold
Nonlinear Phase vs Launch Power
φ_SPM vs Distance
FWM Product vs Channel Spacing
Key Formulas
γ = 2π·n₂ / (λ·A_eff), L_eff = (1 − e^(−αL)) / α
φ_SPM = γ·P·L_eff, φ_XPM ≤ 2γ·P·L_eff·(N − 1)
Δω_rms/Δω₀ = √(1 + 4φ²/(3√3)) [unchirped Gaussian]
Δβ = −Δω²(β₂ ± β₃Δω), η = α²/(α² + Δβ²)·[1 + 4e^(−αL) sin²(ΔβL/2)/(1 − e^(−αL))²]
P_FWM / P_ch = η·(d/3)²·(γ·P·L_eff)², d = 3 degenerate, 6 non-degenerate
P_SBS = 21·K·A_eff·(1 + Δν_s/Δν_B) / (g_B·L_eff), g_B = 5×10⁻¹¹ m/W; P_SRS = 16·K·A_eff / (g_R·L_eff), g_R ∝ 1/λ
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
γ = 2πn₂/(λA_eff), SPM phase and Gaussian rms broadening, XPM upper bound (no walk-off), Inoue FWM efficiency (β₂, β₃) for the nearest product, Smith SBS/SRS thresholds with a polarization factor K and a Lorentzian source linewidth (SBS). Co-polarized γ: in randomly birefringent fibre the effective γ is 8/9 of it (Manakov). Not a Q penalty.
- Agrawal G. P. (2019). Nonlinear Fiber Optics, 6th ed., §2.3 (γ), §4.1 (SPM), §7.1 (XPM). Academic Press. doi:10.1016/C2018-0-01168-8
- Inoue K. (1992). Four-wave mixing in an optical fiber in the zero-dispersion wavelength region. J. Lightwave Technol. 10, 1553. FWM efficiency η. doi:10.1109/50.184893
- Smith R. G. (1972). Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering. Appl. Opt. 11, 2489. Critical powers 16 (SRS) and 21 (SBS). doi:10.1364/AO.11.002489