Harmonic generation
Harmonic generation (HG, also called multiple harmonic generation) is a nonlinear optical process in which formula_1 photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with formula_1 times the energy of the initial photons (equivalently, formula_1 times the frequency and the wavelength divided by formula_1). General process. In a medium having a substantial nonlinear susceptibility, harmonic generation is possible. Note that for even orders (formula_5), the medium must have no center of symmetry (non-centrosymmetrical). Because the process requires that many photons are present at the same time and at  the same place, the generation process has a low probability to occur, and this probability decreases with the order formula_1. To generate efficiently, the symmetry of the medium must allow the signal to be amplified (through phase matching, for instance), and the light source must be intense and well-controlled spatially (with a collimated laser) and temporally (more signal if the laser has short pulses). Sum-frequency generation (SFG). A special case in which the number of photons in the interaction is formula_7, but with two different photons at frequencies formula_8 and formula_9. Second-harmonic generation (SHG). A special case  in which the number of photons in the interaction is formula_7. Also a special case of sum-frequency generation in which both photons are at the same frequency formula_11. Third-harmonic generation (THG). A special case in which the number of photons in the interaction is formula_12, if all the photons have the same frequency formula_11. If they have different frequency, the general term of four-wave mixing is preferred. This process involves the 3rd order nonlinear susceptibility formula_14. Unlike SHG, it is a volumetric process and has been shown in liquids. However, it is enhanced at interfaces. Materials used  for THG. Nonlinear crystals such as BBO (β-BaBO) or LBO can convert THG, otherwise THG can be generated from membranes in microscopy. Fourth-harmonic generation (FHG or 4HG). A special case in which the number of photons in interaction is formula_15. Reported around the year 2000, powerful lasers now enable efficient FHG. This process involves the 4th order nonlinear susceptibility formula_16. Materials used for FHG. Some BBO (β-BaBO) are used for FHG. Harmonic generation for formula_17. Harmonic generation for formula_18 (5HG) or more is theoretically possible, but the interaction requires a very high number of photons to interact  and has therefore a low probability to happen: the signal at higher harmonics will be very low, and requires very intense lasers to be generated. To generate high harmonics (like formula_19 and so on), the substantially different process of high harmonic generation can be used.