of the equation (n = 2 for SHG). No phase-matching A non-perfect phase-matching is a more realistic condition in practice, especially in biological samples. The paraxial approximation is however supposed still valid: , and in the harmonic expression, is now . In the special case of SHG (n = 2), in a medium of length L and a focus position , the intensity writes: where is the speed of light in vacuum, the vacuum permittivity, the optical index of the medium at and the waist size of excitation. Thus, the SHG intensity quickly decays in the bulk (), due to the Gouy phase-shift of the Gaussian beam. In conformity with experiments, the SHG signal vanishes in the bulk (if the medium thickness is too large), and the SHG must be generated at the surface of the material: the conversion therefore does not strictly scales with the square of the number of scatterers, contrary to what the plane wave model indicates. Interestingly, the signal also vanishes in bulk for higher orders, like THG. Materials used for second-harmonic generation Materials capable of generating a second harmonic are crystals without inversion symmetry. This eliminates water, cubic symmetry crystals and glass. Here are some crystals used with certain types of laser for SHG conversion: Fundamental excitation at 600–1500 nm: BiBO (BiB3O6) Fundamental excitation at 570–4000 nm: Lithium iodate LiIO3. Fundamental excitation at 800–1100, often 860 or 980 nm: Potassium niobate KNbO3 Fundamental excitation at 410–2000 nm : BBO (β-BaB2O4) Fundamental excitation at 984 nm–3400 nm: KTP (KTiOPO4) or KTA, Fundamental excitation at 1 064 nm : monopotassium phosphate KDP (KH2PO4), lithium triborate (LiB3O5), CsLiB6O10 and Barium borate BBO(β-BaB2O4). Fundamental excitation at 1 319 nm : KNbO3, BBO (β-BaB2O4), monopotassium phosphate KDP (KH2PO4), LiIO3, LiNbO3, and potassium titanyl phosphate KTP (KTiOPO4). Fundamental excitation at ~1000–2000 nm : periodically-poled crystals, like PPLN. Notably, filamentous biological proteins with a cylindrical symmetric such as collagen, tubulin or myosin, but also certain carbohydrates (such as starch or cellulose) are also quite good converters of SHG (fundamental in the near infrared). See also Half-harmonic generation Nonlinear optics Optical frequency multiplier Second-harmonic imaging microscopy Spontaneous parametric down-conversion Surface second harmonic generation Harmonic generation References External links Articles Second harmonic generation