Harmonic generation explained

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 , ... ), 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)

See main article: Sum-frequency generation. 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)

See main article: Second-harmonic generation. 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

• Book: en . Nonlinear optics. 2007. Boyd. R.W.. Elsevier. 9780123694706 . third.
• Book: en . Handbook of Nonlinear Optics. 2003. Sutherland. Richard L.. CRC Press. 9780824742430 . 2nd.
• Book: en . Optics. 2002. Hecht. Eugene. 978-0805385663. Addison-Wesley. 4th.
• Book: en . Applied Nonlinear Optics . 2006. Zernike. Frits. Midwinter. John E.. 978-0486453606. Dover Publications.

See also

• Nonlinear optics
• Second-harmonic generation
• High harmonic generation
• Optical frequency multiplier

References

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