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# Harmonic generation

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.). Elsevier. ISBN 9780123694706.
• Sutherland, Richard L. (2003). Handbook of Nonlinear Optics (2nd ed.). CRC Press. 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.

## References

1. ^ Boyd, R. (2007). "The Nonlinear Optical Susceptibility". Nonlinear optics (third ed.). pp. 1–67. doi:10.1016/B978-0-12-369470-6.00001-0. ISBN 9780123694706. S2CID 15660817.
2. ^ Sutherland, Richard L. (2003). Handbook of Nonlinear Optics (2nd ed.). CRC Press. ISBN 9780824742430.
3. ^ Boyd, R.W. (2007). Nonlinear optics (third ed.). Elsevier. ISBN 9780123694706.
4. ^ Moreaux, Laurent; Sandre, Olivier; Charpak, Serge; Blanchard-Desce, Mireille; Mertz, Jerome (2001). "Coherent Scattering in Multi-Harmonic Light Microscopy". Biophysical Journal. 80 (3): 1568–1574. Bibcode:2001BpJ....80.1568M. doi:10.1016/S0006-3495(01)76129-2. ISSN 0006-3495. PMC 1301348. PMID 11222317.
5. ^ Kajzar, F.; Messier, J. (1985). "Third-harmonic generation in liquids". Physical Review A. 32 (4): 2352–2363. Bibcode:1985PhRvA..32.2352K. doi:10.1103/PhysRevA.32.2352. ISSN 0556-2791. PMID 9896350.
6. ^ Cheng, Ji-Xin; Xie, X. Sunney (2002). "Green's function formulation for third-harmonic generation microscopy". Journal of the Optical Society of America B. 19 (7): 1604. Bibcode:2002JOSAB..19.1604C. doi:10.1364/JOSAB.19.001604. ISSN 0740-3224.
7. ^ Pavone, Francesco S.; Campagnola, Paul J. (2016). Second Harmonic Generation Imaging, 2nd edition. CRC Taylor&Francis. ISBN 978-1-4398-4914-9.
8. ^ Kojima, Tetsuo; Konno, Susumu; Fujikawa, Shuichi; Yasui, Koji; Yoshizawa, Kenji; Mori, Yusuke; Sasaki, Takatomo; Tanaka, Mitsuhiro; Okada, Yukikatsu (2000). "20-W ultraviolet-beam generation by fourth-harmonic generation of an all-solid-state laser". Optics Letters. 25 (1): 58–60. Bibcode:2000OptL...25...58K. doi:10.1364/OL.25.000058. ISSN 0146-9592. PMID 18059781.
9. ^ "BBO for FHG". raicol.com. Retrieved 2019-12-01.