Optical Parametric Oscillator
Parametric gain, walk-off-limited interaction length and singly-resonant OPO threshold from pump wavelength, d_eff, beam radius and cavity loss.
Key Equations
ωₚ = ωₛ + ωᵢ | λᵢ = λₚ·λₛ / (λₛ − λₚ)
g² = 2ωₛωᵢ d_eff² Iₚ / (nₚnₛnᵢ ε₀ c³), Iₚ = P/(πw²)
L_eff² = ∬ exp(−ρ²[(u+v)² + 2(u−v)²]/4w²) du dv over the crystal → 2^¼·√π·w/ρ | single-pass signal gain cosh²(g·L_eff) − 1
SRO threshold: cosh²(g·L_eff)·(1 − α) = 1 (g·L_eff ≈ √α)
Δk = kₚ − kₛ − kᵢ | QPM period Λ = 2π/|Δk|
Plane-wave estimate (Boyd, Nonlinear Optics §2.8–2.9): perfect phase matching assumed, no pump depletion, no focusing optimisation (Boyd–Kleinman). Walk-off: pump and signal mode of equal radius in the near field, pump centred on the signal axis at mid-crystal. α is the signal's round-trip power loss, output coupling included; the idler is not resonated.
Parameters
Refractive Indices
BBO type I cut (e → o + o)
θ = 22.12 °; nₚ = 1.65550 (e), nₛ = 1.66040, nᵢ = 1.64576 (o); pump walk-off 3.114 °; d_eff = 2.068 pm/V.
Indices from the Tamošauskas et al. 2018 Sellmeier fits (Opt. Mater. Express 8, 1410); d_eff = d₃₁ sinθ + d₂₂ cosθ with the EKSMA d₂₂ = 2.2 and d₃₁ = 0.08 pm/V (1064 nm, no Miller scaling). The defaults are this cut at 532 → 800 nm.
PPLN type 0 (e → e + e), quasi-phase-matched
nₚ = 2.23357, nₛ = 2.17547, nᵢ = 2.13647; grating period Λ = 7.48 µm; d_eff = 2d₃₃/π = 12.41 pm/V; no walk-off.
Extraordinary indices of congruent LiNbO₃ from Zelmon et al. 1997 (J. Opt. Soc. Am. B 14, 3319) at 21 °C; PPLN OPOs often run hot against photorefractive damage, which shifts Λ. d₃₃ = 19.5 pm/V (Shoji et al. 1997, SHG of 1.313 µm), 50 % duty cycle; d₃₃ rises towards shorter wavelengths (25.2 pm/V at 1.064 µm), so d_eff is low for a visible pump. The indices are used as they are, so Δk below is the mismatch the grating cancels.
Results
| Idler λ | 1588.1 nm |
| Pump intensity P/(πw²) | 0.064 MW/cm² |
| Gain coefficient g | 3.75 m⁻¹ |
| Walk-off aperture length √π·w/ρ | 1.63 mm |
| Effective length L_eff | 1.94 mm |
| g·L_eff | 0.0073 |
| Single-pass signal gain | 0.01 % |
| SRO threshold (g·L_eff) | 0.1750 |
| SRO threshold pump power | 2.89 kW |
| Above threshold? | No (0.002× threshold) |
| Phase mismatch Δk (with these indices) | 0.05 mm⁻¹ |
| First-order QPM period 2π/|Δk| | 120909.09 µm |
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 estimate: perfect phase matching, no pump depletion, no focusing optimisation, idler not resonated. Presets: BBO type I (angle cut) and PPLN type 0 (Zelmon e-indices, d_eff = 2d₃₃/π, Δk gives the grating period). Pump walk-off enters through a near-field overlap length (pump and signal mode of equal radius, pump centred at mid-crystal), our own derivation: L for no walk-off, 2^¼·√π·w/ρ for a long crystal.
- Boyd R. W. (2008). Nonlinear Optics, 3rd ed., §2.8 and §2.9. Academic Press. Parametric gain coefficient and singly resonant threshold.
- Boyd G. D., Kleinman D. A. (1968). Parametric Interaction of Focused Gaussian Light Beams. J. Appl. Phys. 39, 3597–3639. Walk-off aperture length l_a = √π w/ρ. doi:10.1063/1.1656831
- 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. PPLN e-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), used for the PPLN preset. doi:10.1364/JOSAB.14.002268
- Myers L. E. et al. (1995). Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO₃. J. Opt. Soc. Am. B 12, 2102. Wavelength form of the gain coefficient, equal to Boyd's. doi:10.1364/josab.12.002102
- Tamošauskas G., Beresnevičius G., Gadonas D., Dubietis A. (2018). Transmittance and phase matching of BBO crystal in the 3−5 μm range and its application for the characterization of mid-infrared laser pulses. Opt. Mater. Express 8, 1410. BBO Sellmeier fits (0.188–5.2 µm) for the type I cut and the defaults. doi:10.1364/OME.8.001410