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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).^[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 $n$ . 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 $n=2$ , 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 $\chi ^{(3)}$ .^[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 $n=4$ . Reported around the year 2000,^[8] 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.^[9]

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.

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

^ 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.
^ Sutherland, Richard L. (2003). Handbook of Nonlinear Optics (2nd ed.). CRC Press. ISBN 9780824742430.
^ Boyd, R.W. (2007). Nonlinear optics (third ed.). Elsevier. ISBN 9780123694706.
^ 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.
^ 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.
^ 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.
^ Pavone, Francesco S.; Campagnola, Paul J. (2016). Second Harmonic Generation Imaging, 2nd edition. CRC Taylor&Francis. ISBN 978-1-4398-4914-9.
^ 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.
^ "BBO for FHG". raicol.com. Retrieved 2019-12-01.

[Boyd2007-1] 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.

[handbookNL-2] Sutherland, Richard L. (2003). Handbook of Nonlinear Optics (2nd ed.). CRC Press. ISBN 9780824742430.

[boyd-3] Boyd, R.W. (2007). Nonlinear optics (third ed.). Elsevier. ISBN 9780123694706.

[MoreauxSandre2001-4] 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.

[KajzarMessier1985-5] 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.

[ChengXie2002-6] 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.

[KojimaKonno2000-8] 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.

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