Editing Four-wave mixing (section)

== Mechanism ==
[[File:FWM energy levels.png|alt=FWM energy level diagram|thumb|Energy level diagram for a non-degenerate four-wave mixing process. The top energy level could be a real atomic or molecular level (resonant four-wave mixing) or a virtual level, far detuned off-resonance. This diagram describes the four-wave mixing interaction between frequencies f<sub>1</sub>, f<sub>2</sub>, f<sub>3</sub> and f<sub>4</sub>.]]
When three frequencies (f<sub>1</sub>, f<sub>2</sub>, and f<sub>3</sub>) interact in a nonlinear medium, they give rise to a fourth frequency (f<sub>4</sub>) which is formed by the scattering of the incident photons, producing the fourth photon.

Given inputs ''f<sub>1</sub>, f<sub>2</sub>,'' and ''f<sub>3</sub>'', the nonlinear system will produce

:<math> \pm f_{1} \pm f_{2} \pm f_{3}</math>

From calculations with the three input signals, it is found that 12 interfering frequencies are produced, three of which lie on one of the original incoming frequencies. Note that these three frequencies which lie at the original incoming frequencies are typically attributed to [[self-phase modulation]] and [[cross-phase modulation]], and are naturally phase-matched unlike FWM.

===Sum- and difference-frequency generation===
Two common forms of four-wave mixing are dubbed [[sum-frequency generation]] and difference-frequency generation.  In sum-frequency generation three fields are input and the output is a new high frequency field at the sum of the three input frequencies.  In difference-frequency generation, the typical output is the sum of two minus the third.

A condition for efficient generation of FWM is phase matching: the associated k-vectors of the four components must add to zero when they are plane waves.  This becomes significant since sum- and difference-frequency generation are often enhanced when resonance in the mixing media is exploited.  In many configurations the sum of the first two photons will be tuned close to a resonant state.<ref name=straussfunk />  However, close to resonances the index of refraction changes rapidly and makes addition four co-linear k-vectors fail to add exactly to zero—thus long mixing path lengths are not always possible as the four component lose phase lock.  Consequently, beams are often focused both for intensity but also to shorten the mixing zone.

In gaseous media an often overlooked complication is that light beams are rarely plane waves but are often focused for extra intensity, this can add an addition pi-phase shift to each k-vector in the phase matching condition.<ref name=cardoso2000four>
 {{Cite journal| last1 = Cardoso
 | first1 = GC
 | last2 = Tabosa
 | first2 = JWR
 | title = Four-wave mixing in dressed cold cesium atoms
 | journal = Optics Communications
 | volume = 185
 | issue = 4–6
 | page = 353
 |date=2000
 | doi=10.1016/S0030-4018(00)01033-6
 | bibcode = 2000OptCo.185..353C
 }}</ref><ref name=cardoso2002saturated>
 {{Cite journal| last1 = Cardoso
 | first1 = GC
 | last2 = Tabosa
 | first2 = JWR
 | title = Saturated lineshapes and high-order susceptibilities of cold cesium atoms observed via a transferred population grating
 | journal = Optics Communications
 | volume = 210
 | issue = 3–6
 | page = 271
 |date=2002
 | doi=10.1016/S0030-4018(02)01820-5
 | bibcode = 2002OptCo.210..271C
 }}</ref>  It is often very hard to satisfy this in the sum-frequency configuration but it is more easily satisfied in the difference-frequency configuration (where the pi phase shifts cancel out).<ref name=straussfunk>
 {{Cite journal| last1 = Strauss
 | first1 = CEM
 | last2 = Funk
 | first2 = DJ
 | title = Broadly tunable difference-frequency generation of VUV using two-photon resonances in H2 and Kr 
 | journal = Optics Letters
 | volume = 16
 | issue = 15
 | pages = 1192–4
 |date=1991
 | url = https://www.osapublishing.org/ol/fulltext.cfm?uri=ol-16-15-1192&id=10705
 | doi=10.1364/ol.16.001192
 | pmid = 19776917
 | bibcode = 1991OptL...16.1192S
 | url-access = subscription
 }}</ref> As a result, difference-frequency is usually more broadly tunable and easier to set up than sum-frequency generation, making it preferable as a light source even though it's less [[quantum efficiency|quantum efficient]] than sum-frequency generation.

The special case of sum-frequency generation where all the input photons have the same frequency (and wavelength) is [[Harmonic generation#Third-harmonic generation (THG)|Third-Harmonic Generation (THG)]].

=== Degenerate four-wave mixing ===

Four-wave mixing is also present if only two components interact. In this case the term

:<math> f_{0} = f_{1} + f_{1} - f_{2} </math>

couples three components, thus generating so-called '''degenerate four-wave mixing''', showing identical properties to the case of three interacting waves.<ref>{{Cite book|title=Advanced Optical Communication Systems and Networks|last=Cvijetic, Djordjevic|first=Milorad, Ivan B.|publisher=Artech House|year=2013|isbn=978-1-60807-555-3|pages=314 to 217}}</ref>