Harmonic generation

Harmonic generation (HG, also called multiple harmonic generation) is a nonlinear optical process in which $$n$$ photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with $$n$$ times the energy of the initial photons (equivalently, $$n$$ times the frequency and the wavelength divided by $$n$$).

General process
In a medium having a substantial nonlinear susceptibility, harmonic generation is possible. Note that for even orders ($$n = 2,4,\dots$$), 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 $$n$$. 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 $$n = 2$$, but with two different photons at frequencies $$\omega_1$$ and $$\omega_2$$.

Second-harmonic generation (SHG)
A special case in which the number of photons in the interaction is $$n = 2$$. Also a special case of sum-frequency generation in which both photons are at the same frequency $$\omega$$.

Third-harmonic generation (THG)
A special case in which the number of photons in the interaction is $$n = 3$$, if all the photons have the same frequency $$\omega$$. If they have different frequency, the general term of four-wave mixing is preferred. This process involves the 3rd order nonlinear susceptibility $$\chi^{(3)}$$.

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-BaB2O4) 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 $$n = 4$$. Reported around the year 2000, powerful lasers now enable efficient FHG. This process involves the 4th order nonlinear susceptibility $$\chi^{(4)}$$.

Materials used for FHG
Some BBO (β-BaB2O4) are used for FHG.

Harmonic generation for $$n > 4$$
Harmonic generation for $$n = 5$$ (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 $$n = 30$$ and so on), the substantially different process of high harmonic generation can be used.