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

General process

In a medium having a substantial nonlinear susceptibility, harmonic generation is possible. Note that for even orders (${\displaystyle n=2,4,\dots }$), the medium must have no center of symmetry (non-centrosymmetrical).^[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 ${\displaystyle n}$. 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).^[2]

Sum-frequency generation (SFG)

A special case in which the number of photons in the interaction is ${\displaystyle n=2}$, but with two different photons at frequencies ${\displaystyle \omega _{1))$ and ${\displaystyle \omega _{2))$.

Second-harmonic generation (SHG)

A special case in which the number of photons in the interaction is ${\displaystyle n=2}$. Also a special case of sum-frequency generation in which both photons are at the same frequency ${\displaystyle \omega }$.

Third-harmonic generation (THG)

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

Unlike SHG, it is a volumetric process^[4] and has been shown in liquids.^[5] However, it is enhanced at interfaces.^[6]

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.^[7]

Fourth-harmonic generation (FHG or 4HG)

A special case in which the number of photons in interaction is ${\displaystyle n=4}$. Reported around the year 2000,^[8] powerful lasers now enable efficient FHG. This process involves the 4th order nonlinear susceptibility ${\displaystyle \chi ^{(4)))$.

Materials used for FHG

Some BBO (β-BaB₂O₄) are used for FHG.^[9]

Harmonic generation for ${\displaystyle n>4}$

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

Sources

• Boyd, R.W. (2007). Nonlinear optics (third ed.). ISBN 9780123694706.
• Sutherland, Richard L. (2003). Handbook of Nonlinear Optics (2nd ed.). ISBN 9780824742430.
• Hecht, Eugene (2002). Optics (4th ed.). Addison-Wesley. ISBN 978-0805385663.
• Zernike, Frits; Midwinter, John E. (2006). Applied Nonlinear Optics. Dover Publications. ISBN 978-0486453606.