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).
In a medium having a substantial nonlinear susceptibility, harmonic generation is possible. Note that for even orders (n=2,4,...), 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).
A special case where the number of photons in interaction is n=2, but with two different photons at and .
A special case where the number of photons in interaction is n=2. Also a special case of sum-frequency generation where both photons are at the same frequency .
A special case where the number of photons in interaction is n=3, if all the photons have the same frequency . If they have different frequency, the general term of four-wave mixing is preferred. This process involves the 3rd order nonlinear susceptibility .
A special case where 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 .
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), a substantially different process can be used : High harmonic generation.