Sum-Frequency Generation Microscopy Calculator
SFG wavelength, focal spot, coherence lengths and thin-layer signal power for sum-frequency generation microscopy.
Results
ω₃ = ω₁ + ω₂, so 1/λ₃ = 1/λ₁ + 1/λ₂ (energy conservation).
I₃ = 8π² d_eff² I₁ I₂ L² sinc²(ΔkL/2)/(ε₀ c n² n₃ λ₃²), Δk = −2πΔn/λ₃ (Boyd, Nonlinear Optics, eq. 2.2.10 integrated), over the overlap ∫I₁I₂ dA = 2P₁P₂/(π(w₁² + w₂²)).
Thin uniform layer (L ≪ b), synchronised Gaussian pulses of equal width: ⟨P₃⟩ = K·0.664·⟨P₁⟩⟨P₂⟩/(f τ). Thicker samples need the focused-beam (Boyd–Kleinman) treatment, where the Gouy phase suppresses the forward signal of a thick uniform medium; see the SHG calculator.
Focus: Gaussian fit to each beam (Zipfel et al. 2003); w = 2ω_xy is the 1/e² intensity radius, not the Airy radius.
SFG Wavelength vs Beam 2 Wavelength
SFG Power vs Layer Thickness (up to b)
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
Wavelength from energy conservation (exact). Power: plane-wave L² sinc²(ΔkL/2) over the overlap of two Gaussian foci, valid for a uniform layer much thinner than the confocal parameter; paraxial, NA < n; synchronised Gaussian pulses of equal width. No walk-off or absorption, undepleted inputs.
- Boyd R. W. (2008). Nonlinear Optics, 3rd ed., §2.2, eq. 2.2.10. Academic Press. Coupled-amplitude equation of sum-frequency generation.
- Zipfel W. R., Williams R. M., Webb W. W. (2003). Nonlinear magic: multiphoton microscopy in the biosciences. Nat. Biotechnol. 21, 1369–1377. Gaussian fit to the focus at each wavelength. doi:10.1038/nbt899