Four-Wave Mixing (FWM)
Degenerate four-wave mixing in a fiber (2ωp = ωs + ωi): parametric signal gain, idler wavelength and power, phase mismatch κ = Δβ + 2γP, gain spectrum and peak-gain detuning from pump power, n₂, A_eff, length, β₂ and β₄.
Signal gain G_s
27.09 dB
gain coefficient g = 6.996 /km
Idler λ_i and power
1540.13 nm
P_i = 5.177 mW = (λ_s/λ_i)(G_s − 1)P_s
Nonlinear coefficient γ
9.581 /(W·km)
γP_pL = 4.791 rad
Phase mismatch κ = Δβ + 2γP_p
13.09 /km
linear Δβ = β₂Ω² + β₄Ω⁴/12 = -6.069 /km
Peak gain (κ = 0)
35.59 dB
cosh²(γP_pL) at ±2.203 THz (λ_s ≈ λ_p + 17.86 nm)
Two pump photons at ω_p become a signal and an idler photon, 2ω_p = ω_s + ω_i. With an undepleted CW pump the signal grows as G_s = 1 + [(γP_p/g) sinh(gL)]², with g² = (γP_p)² − (κ/2)² and κ = Δβ + 2γP_p: the linear mismatch Δβ = β_s + β_i − 2β_p plus the pump's self- and cross-phase modulation. Gain is exponential where |κ| < 2γP_p and peaks at cosh²(γP_pL) where κ = 0, which needs anomalous dispersion at the pump. Far outside the band G − 1 tends to (γP_pL)² sinc²(κL/2), κ = Δβ + 2γP_p; for a weak pump κ → Δβ. The defaults are a 500 m highly nonlinear fiber pumped with 1 W just on the anomalous side of its zero-dispersion wavelength. With fiber loss α the pump decays as P_p e^(−αz), so γP_p and κ change along the fiber: the coupled equations are integrated numerically (exact transfer matrices over short segments) and the gains are net of the signal's own loss; in the weak-pump limit the idler follows Inoue's (γP_pL_eff)² η. Polarization and Raman gain are not modelled.
Signal gain spectrum
- Signal gain G_s
- This signal
Signal and idler power along the fiber
- Signal
- Idler
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
Undepleted CW pump, co-polarized waves: G_s = 1 + [(γP/g) sinh(gL)]², g² = (γP)² − (κ/2)², κ = β₂Ω² + β₄Ω⁴/12 + 2γP, and P_i = (λ_s/λ_i)(G_s − 1)P_s for a lossless fiber. With loss α the pump decays as e^(−αz) and the coupled equations are solved in short segments with exact transfer matrices (net gains; Inoue's (γP L_eff)² η in the weak-pump limit). Not modelled: pump depletion (flagged when signal and idler take more than 10 % of the pump), polarization, Raman gain, longitudinal dispersion fluctuations.
- Stolen R. H., Bjorkholm J. E. (1982). Parametric amplification and frequency conversion in optical fibers. IEEE J. Quantum Electron. 18, 1062. doi:10.1109/JQE.1982.1071660
- Hansryd J., Andrekson P. A., Westlund M., Li J., Hedekvist P.-O. (2002). Fiber-based optical parametric amplifiers and their applications. IEEE J. Sel. Top. Quantum Electron. 8, 506. doi:10.1109/JSTQE.2002.1016354
- Agrawal G. P. (2019). Nonlinear Fiber Optics, 6th ed., ch. 10. Academic Press. Four-wave mixing and parametric gain. doi:10.1016/C2018-0-01168-8