Harmonic generation#Third-harmonic generation (THG)

In a medium having a substantial nonlinear susceptibility, harmonic generation is possible. Note that for even orders ($n\; =\; 2,4,\backslash dots$), the medium must have no center of symmetry (non-centrosymmetrical).{{cite book|language=en |title= Nonlinear optics |url=https://archive.org/details/nonlinearopticst00boyd |url-access=limited | chapter = The Nonlinear Optical Susceptibility |year=2007|last1=Boyd|first1= R. |pages=1–67| doi = 10.1016/B978-0-12-369470-6.00001-0|isbn= 9780123694706 |s2cid= 15660817 |edition= third }}

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).{{cite book|language=en |title= Handbook of Nonlinear Optics|year=2003|last1=Sutherland|first1= Richard L.|publisher= CRC Press|isbn= 9780824742430 |edition= 2nd}}

{{Main|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 $\backslash omega\_1$ and $\backslash omega\_2$.

{{Main|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 $\backslash omega$.

A special case in which the number of photons in the interaction is $n\; =\; 3$, if all the photons have the same frequency $\backslash omega$. If they have different frequency, the general term of four-wave mixing is preferred. This process involves the 3rd order nonlinear susceptibility $\backslash chi^\{(3)\}$.{{cite book|language=en |title= Nonlinear optics|year=2007|last1=Boyd|first1= R.W.|publisher= Elsevier|isbn= 9780123694706 | url=https://books.google.com/books?id=uoRUi1Yb7ooC&q=Nonlinear+optics|edition= third}}

Unlike SHG, it is a volumetric process{{cite journal|last1=Moreaux|first1=Laurent|last2=Sandre|first2=Olivier|last3=Charpak|first3=Serge|last4=Blanchard-Desce|first4=Mireille|last5=Mertz|first5=Jerome|title=Coherent Scattering in Multi-Harmonic Light Microscopy|journal=Biophysical Journal|volume=80|issue=3|year=2001|pages=1568–1574|issn=0006-3495|doi=10.1016/S0006-3495(01)76129-2|pmid=11222317|pmc=1301348|bibcode=2001BpJ....80.1568M}} and has been shown in liquids.{{cite journal|last1=Kajzar|first1=F.|last2=Messier|first2=J.|title=Third-harmonic generation in liquids|journal=Physical Review A|volume=32|issue=4|year=1985|pages=2352–2363|issn=0556-2791|doi=10.1103/PhysRevA.32.2352|pmid=9896350|bibcode=1985PhRvA..32.2352K}} However, it is enhanced at interfaces.{{cite journal|last1=Cheng|first1=Ji-Xin|last2=Xie|first2=X. Sunney|title=Green's function formulation for third-harmonic generation microscopy|journal=Journal of the Optical Society of America B|volume=19|issue=7|year=2002|pages=1604|issn=0740-3224|doi=10.1364/JOSAB.19.001604|bibcode=2002JOSAB..19.1604C}}

Nonlinear crystals such as BBO (β-BaB₂O₄) or LBO can convert THG, otherwise THG can be generated from membranes in microscopy.{{cite book|title= Second Harmonic Generation Imaging, 2nd edition|surname1= Pavone|first1= Francesco S. |surname2= Campagnola|first2= Paul J. | year=2016|isbn=978-1-4398-4914-9|publisher= CRC Taylor&Francis|url=https://books.google.com/books?id=EiTOBQAAQBAJ&q=Second+Harmonic+Generation+Imaging+Pavone+Francesco}}

A special case in which the number of photons in interaction is $n\; =\; 4$.

Reported around the year 2000,{{cite journal|last1=Kojima|first1=Tetsuo|last2=Konno|first2=Susumu|last3=Fujikawa|first3=Shuichi|last4=Yasui|first4=Koji|last5=Yoshizawa|first5=Kenji|last6=Mori|first6=Yusuke|last7=Sasaki|first7=Takatomo|last8=Tanaka|first8=Mitsuhiro|last9=Okada|first9=Yukikatsu|title=20-W ultraviolet-beam generation by fourth-harmonic generation of an all-solid-state laser|journal=Optics Letters|volume=25|issue=1|year=2000|pages=58–60|issn=0146-9592|doi=10.1364/OL.25.000058|pmid=18059781|bibcode=2000OptL...25...58K}} powerful lasers now enable efficient FHG. This process involves the 4th order nonlinear susceptibility $\backslash chi^\{(4)\}$.

Some BBO (β-BaB₂O₄) are used for FHG.{{cite web |url= http://raicol.com/bbo/bbo-for-fhg |title=BBO for FHG |author= |date= |website= raicol.com|publisher= |access-date=2019-12-01 |quote=}}

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.

• {{cite book|language=en |title= Nonlinear optics|year=2007|last1=Boyd|first1= R.W.|publisher= Elsevier|isbn= 9780123694706 | url=https://books.google.com/books?id=uoRUi1Yb7ooC&q=Nonlinear+optics|edition= third}}
• {{cite book|language=en |title= Handbook of Nonlinear Optics|year=2003|last1=Sutherland|first1= Richard L.|publisher= CRC Press|isbn= 9780824742430 | url=https://books.google.com/books?id=ccXo3WrHp2UC&q=Handbook+of+Nonlinear+Optics|edition= 2nd}}
• {{cite book|language=en |title= Optics|year=2002|last1=Hecht|first1= Eugene|isbn= 978-0805385663|publisher = Addison-Wesley|edition= 4th}}
• {{cite book|language=en |title= Applied Nonlinear Optics |year=2006|last1=Zernike|first1= Frits|last2=Midwinter|first2= John E.|isbn= 978-0486453606|publisher = Dover Publications|url=https://books.google.com/books?id=Y3Wsyo7TgdkC&q=Applied+Nonlinear+Optics}}

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

{{Lasers}}

Category:Nonlinear optics