Harmonic generation

times the energy of the initial photons (equivalently,

In a medium having a substantial nonlinear susceptibility, harmonic generation is possible.

[1] 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

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).

[2] A special case in which the number of photons in the interaction is

A special case in which the number of photons in the interaction is

Also a special case of sum-frequency generation in which both photons are at the same frequency

A special case in which the number of photons in the interaction is

If they have different frequency, the general term of four-wave mixing is preferred.

This process involves the 3rd order nonlinear susceptibility

[3] Unlike SHG, it is a volumetric process[4] and has been shown in liquids.

Reported around the year 2000,[8] powerful lasers now enable efficient FHG.

This process involves the 4th order nonlinear susceptibility

(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.

and so on), the substantially different process of high harmonic generation can be used.