Difference Frequency Generation (DFG)

Idler power, wavelength and quasi-phase-matching period of χ⁽²⁾ difference-frequency generation (ωp − ωs → ωi) from pump and signal powers, beam radius, d_eff, indices and phase mismatch.

Model: Textbook approximation

Pump λ_p (nm)Signal λ_s (nm)d_eff (pm/V)Pump power P_p (mW)Signal power P_s (mW)Beam radius w, 1/e², both beams (µm)Crystal length L (mm)n_p at λ_pn_s at λ_sn_i at λ_iResidual Δk after phase matching (rad/mm)

Idler power P_i

1.172 µW

η·2P_pP_s/(π(w_p² + w_s²)), low gain

Idler wavelength λ_i

3393.4 nm

1/λ_i = 1/λ_p − 1/λ_s; 2946.9 cm⁻¹

Normalized efficiency

0.0234 %/W

P_i/(P_pP_s); on-axis ΓL = 0.0227

First-order QPM period

29.95 µm

2π/|Δk| from the indices; coherence length 14.98 µm

A strong pump at ω_p and a weaker signal at ω_s generate the idler at ω_i = ω_p − ω_s, usually in the mid-infrared. With both inputs undepleted and the signal not yet amplified, I_i = 8π² d_eff² L² sinc²(ΔkL/2) I_pI_s/(ε₀ c n_pn_sn_i λ_i²): the sum-frequency law with the idler wavelength. Integrated over two Gaussian beams of equal radius w it gives P_i = η P_pP_s/(πw²). Each idler photon also adds a signal photon. Once ΓL approaches 1 the signal grows too, and the process becomes parametric amplification. The defaults are a 1064 nm − 1550 nm → 3.39 µm 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). Miller's rule scales d by 0.87 for the 3.39 µm idler, to ≈ 10.8 pm/V.

Idler power vs crystal length

P_i(L)

Idler power vs residual phase mismatch

P_i(Δk)
Set 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, low parametric gain (ΓL ≪ 1), no focusing, walk-off or absorption; the page flags crystals longer than the smallest confocal parameter, the idler's (radius w/√2) included. 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.
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

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