Second Harmonic Generation (SHG) Calculator
SHG signal properties, wavelength conversion, and imaging resolution for collagen and other non-centrosymmetric structures.
Results
The Gaussian-beam focus is indicative only above NA ≈ 0.5.
P_2ω = 16π² d_eff² L P_ω² h(σ, ξ)/(ε₀ c n_ω n_2ω λ³), h = |∫_−ξ^ξ e^(iστ)/(1 + iτ) dτ|²/(4ξ), ξ = L/b, σ = bΔk/2, Δk = 2k_ω − k_2ω (Boyd & Kleinman, J. Appl. Phys. 39, 3597, 1968).
For L ≪ b this is the plane-wave 8π² d² L² I sinc²(ΔkL/2)/(ε₀ c n_ω² n_2ω λ²). For L ≫ b and normal dispersion (n(2ω) > n(ω)), the Gouy phase cancels the harmonic from a uniform slab: in tissue, SHG comes from structure on the scale of the focus and the coherence lengths.
Gaussian focus w₀ = 2ω_xy (Zipfel et al. 2003), b = 2πn w₀²/λ; paraxial, so only indicative above NA ≈ 0.5. Gaussian pulses: ⟨P_2ω⟩ = K · 0.664 ⟨P⟩²/(fτ).
Average SH Power vs Sample Thickness
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
Focused Gaussian beam in a uniform slab, focus at the centre; no walk-off or absorption, undepleted pump, paraxial (indicative above NA ≈ 0.5). The Gouy phase cancels SHG from a thick uniform slab in normal dispersion.
- Boyd G. D., Kleinman D. A. (1968). Parametric Interaction of Focused Gaussian Light Beams. J. Appl. Phys. 39, 3597–3639. doi:10.1063/1.1656831
- Boyd R. W. (2008). Nonlinear Optics, 3rd ed., §2.7 and §2.10. Academic Press. Plane-wave limit and Gouy-phase cancellation.
- Xu C., Webb W. W. (1996). Measurement of two-photon excitation cross sections of molecular fluorophores with data from 690 to 1050 nm. J. Opt. Soc. Am. B 13, 481. Peak-power factor g_p of a Gaussian pulse. doi:10.1364/josab.13.000481
- Zipfel W. R., Williams R. M., Webb W. W. (2003). Nonlinear magic: multiphoton microscopy in the biosciences. Nat. Biotechnol. 21, 1369–1377. Gaussian focus w₀ = 2ω_xy. doi:10.1038/nbt899