Sum Frequency Generation (SFG)
Sum-frequency power, wavelength and quasi-phase-matching period of χ⁽²⁾ upconversion (ω1 + ω2 → ω3) from the input powers, beam radius, d_eff, indices and phase mismatch.
SFG power P₃
72.07 µW
η·2P₁P₂/(π(w₁² + w₂²)), sinc²(ΔkL/2) = 1.00
SFG wavelength λ₃
630.91 nm
1/λ₃ = 1/λ₁ + 1/λ₂
Normalized efficiency
1.44 %/W
P₃/(P₁P₂); 0.0587 % of the weaker beam's photons
First-order QPM period
11.58 µm
2π/|Δk| for the entered indices; update n₁, n₂, n₃ when you change a wavelength. Coherence length 5.790 µm
Two beams at ω₁ and ω₂ drive a polarization at ω₃ = ω₁ + ω₂ in the χ⁽²⁾ crystal. With both inputs undepleted and plane waves, I₃ = 8π² d_eff² L² sinc²(ΔkL/2) I₁I₂/(ε₀ c n₁n₂n₃ λ₃²). Integrated over two Gaussian beams of equal radius w this gives P₃ = η P₁P₂/(πw²). Without phase matching Δk = 2π(n₃/λ₃ − n₁/λ₁ − n₂/λ₂) and the power only oscillates over a coherence length π/|Δk|. Birefringent phase matching or a periodically poled crystal of period 2π/|Δk| removes it. The residual Δk input is what is left after that, from tuning or temperature. The defaults are a 1064 + 1550 nm → 631 nm mixer in periodically poled LiNbO₃: extraordinary indices from the Zelmon Sellmeier (21 °C) 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).
SFG power vs crystal length
- P₃(L)
SFG power vs residual phase mismatch
- P₃(Δk)
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 waves over two Gaussian beams of equal radius, both inputs undepleted, no focusing, walk-off or absorption. 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 moves it by 1 % for 1064 + 1550 nm.
- 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.
- Boyd G. D., Kleinman D. A. (1968). Parametric Interaction of Focused Gaussian Light Beams. J. Appl. Phys. 39, 3597–3639. Focusing limit of the plane-wave result. 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. 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