Optical Parametric Amplifier
Signal gain, idler output and gain bandwidth of an optical parametric amplifier from pump power and radius, d_eff, indices, phase mismatch and signal–idler group-velocity mismatch.
Signal gain G
25.09 dB
G = 323.2; 1 + (Γ/g)² sinh²(gL), g² = Γ² − (Δk/2)²
Gain coefficient Γ
179.1 m⁻¹
ΓL = 3.58; on-axis pump 35.37 MW/cm²
Output signal / idler
323.2 W / 147.2 W
idler at 3393.4 nm, (λ_s/λ_i)(G − 1)P_s
Gain bandwidth (FWHM)
591.1 GHz
4.74 nm at λ_s; low-gain limit 0.886/(|δ|L) = 453.4 GHz
A strong pump at ω_p amplifies a signal at ω_s and generates the idler at ω_i = ω_p − ω_s. With the pump undepleted, Γ² = 2ω_sω_i d_eff² I_p/(n_pn_sn_i ε₀c³) and the signal grows by G = 1 + (Γ/g)² sinh²(gL), which is ≈ e^(2ΓL)/4 at high gain. A phase mismatch lowers g; beyond |Δk| = 2Γ the gain only oscillates. The gain is computed at the pump's on-axis intensity 2P/(πw²), so it is the peak gain of a Gaussian beam. Detuning the signal by Δν changes Δk by −2πΔν δ_si, where δ_si is the signal–idler group-velocity mismatch. That sets the gain bandwidth: 0.886/(|δ_si|L) at low gain, widening as √(ΓL) towards (2√ln2/π)√(Γ/L)/|δ_si| at high gain (Cerullo & De Silvestri 2003). The spectrum stops at degeneracy 2λ_p, where signal and idler swap roles. The defaults are a 1064 nm-pumped PPLN amplifier for 1550 nm: indices and δ_si from the Zelmon Sellmeier (21 °C, extraordinary), first-order QPM period for the entered indices (when you change a wavelength, the line under the inputs offers PPLN's indices and δ_si) 29.95µm, and d_eff = 2d₃₃/π = 12.4 pm/V from d₃₃ = 19.5 pm/V of congruent LiNbO₃ for SHG of 1.313 µm (Shoji et al. 1997; 25.2 pm/V at 1.064 µm). Miller's rule scales d by 0.87 for the 3.39 µm idler, to ≈ 10.8 pm/V.
Gain vs crystal length
- Gain G(L)
Gain spectrum (group-velocity mismatch)
- Gain spectrum
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
Plane-wave, undepleted-pump gain at the pump's on-axis intensity; the bandwidth comes from first-order group-velocity mismatch only (no group-velocity dispersion). No focusing, walk-off or pump depletion; the depletion check compares the on-axis gain with the total pump power, so a seed narrower than the pump depletes it locally sooner. d_eff is an input: 12.4 pm/V is 2d₃₃/π of congruent PPLN, d₃₃ for SHG of 1.313 µm (Shoji et al. 1997); Miller's rule scales it by 0.87 (≈ 10.8 pm/V) for a 3.39 µm idler.
- Boyd R. W. (2008). Nonlinear Optics, 3rd ed., §2.2 and §2.8. Academic Press. Coupled-amplitude equations for sum- and difference-frequency generation and parametric amplification.
- Cerullo G., De Silvestri S. (2003). Ultrafast optical parametric amplifiers. Rev. Sci. Instrum. 74, 1–18. Gain bandwidth from group-velocity mismatch. doi:10.1063/1.1523642
- Zelmon D. E., Small D. L., Jundt D. (1997). Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol% magnesium oxide-doped lithium niobate. J. Opt. Soc. Am. B 14, 3319. Default extraordinary indices. doi:10.1364/josab.14.003319
- Shoji I., Kondo T., Kitamoto A., Shirane M., Ito R. (1997). Absolute scale of second-order nonlinear-optical coefficients. J. Opt. Soc. Am. B 14, 2268. Table 10: congruent LiNbO₃ d₃₃ = 19.5 pm/V (SHG of 1.313 µm), 25.2 pm/V (1.064 µm); default d_eff = 2d₃₃/π. doi:10.1364/JOSAB.14.002268