Editing Nonlinear optics (section)

==Example uses==

===Frequency doubling===
One of the most commonly used frequency-mixing processes is '''frequency doubling''', or second-harmonic generation. With this technique, the 1064&nbsp;nm output from [[Nd-YAG laser|Nd:YAG lasers]] or the 800&nbsp;nm output from [[Ti-sapphire laser|Ti:sapphire lasers]] can be converted to visible light, with wavelengths of 532&nbsp;nm (green) or 400&nbsp;nm (violet) respectively.<ref>{{Cite journal |last1=Bai |first1=Zhenxu |last2=Wang |first2=Yulei |last3=Lu |first3=Zhiwei |last4=Yuan |first4=Hang |last5=Jiang |first5=Li |last6=Tan |first6=Tan |last7=Liu |first7=Zhaohong |last8=Wang |first8=Hongli |last9=Cui |first9=Can |last10=Hasi |first10=Wuliji |date=2016-10-01 |title=Efficient KDP frequency doubling SBS pulse compressed 532nm hundred picosecond laser |url=https://www.sciencedirect.com/science/article/pii/S0030402616307914 |journal=Optik |volume=127 |issue=20 |pages=9201–9205 |doi=10.1016/j.ijleo.2016.07.021 |bibcode=2016Optik.127.9201B |issn=0030-4026}}</ref>

Practically, frequency doubling is carried out by placing a nonlinear medium in a laser beam. While there are many types of nonlinear media, the most common media are crystals. Commonly used crystals are BBO ([[β-barium borate]]), KDP ([[potassium dihydrogen phosphate]]), KTP ([[potassium titanyl phosphate]]), and [[lithium niobate]]. These crystals have the necessary properties of being strongly [[birefringence|birefringent]] (necessary to obtain phase matching, see below), having a specific crystal symmetry, being transparent for both the impinging laser light and the frequency-doubled wavelength, and having high damage thresholds, which makes them resistant against the high-intensity laser light.

===Optical phase conjugation===<!-- This section is linked from [[Holography]] -->
It is possible, using nonlinear optical processes, to exactly reverse the propagation direction and phase variation of a beam of light. The reversed beam is called a ''conjugate'' beam, and thus the technique is known as '''optical phase conjugation'''<ref>{{cite journal |first1=Vladimir |last1=Shkunov |first2=Boris |last2=Zel'dovich |title=Phase Conjugation |journal=Scientific American |volume=253 |issue=6 |pages=54–59 |date=December 1985 |doi=10.1038/scientificamerican1285-54 |jstor=24967871 }}</ref><ref>{{cite journal |first=David M. |last=Pepper |title=Applications of Optical Phase Conjugation |journal=Scientific American |volume=254 |issue=1 |pages=74–83 |date=January 1986 |doi=10.1038/scientificamerican0186-74 |jstor=24975872 }}</ref> (also called ''time reversal'', ''wavefront reversal'' and is significantly different from ''[[retroreflector|retroreflection]]'').

A device producing the phase-conjugation effect is known as a '''phase-conjugate mirror''' (PCM).

====Principles====
[[Image:wiki_perfect_PCM_photon_recoil.jpg|thumb|right| Vortex photon (blue) with linear momentum <math>\mathbf{P} = \hbar \mathbf{k}</math> and angular momentum <math>L =\pm\hbar\ell</math> is reflected from perfect phase-conjugating mirror. Normal to mirror is <math>\vec{n}</math> , propagation axis is <math>\vec{z}</math>. Reflected photon (magenta) has opposite linear momentum <math>\mathbf{P} = - \hbar \mathbf{k}</math> and angular momentum <math>L = \mp\hbar\ell</math>. Because of conservation laws PC mirror experiences recoil: the vortex phonon (orange) with doubled  linear momentum <math>\mathbf{P} = 2 \hbar \mathbf{k}</math> and angular momentum <math>L = \pm 2 \hbar \ell</math> is excited within mirror.]]

One can interpret optical phase conjugation as being analogous to a [[holography#Dynamic holography|real-time holographic process]].<ref>{{cite journal |first1=David M. |last1=Pepper |first2=Jack |last2=Feinberg |first3=Nicolai V. |last3=Kukhtarev |title=The Photorefractive Effect |journal=Scientific American |volume=263 |issue=4 |pages=62–75 |date=October 1990 |doi=10.1038/scientificamerican1090-62 |jstor=24997062 }}</ref>  In this case, the interacting beams simultaneously interact in a nonlinear optical material to form a dynamic hologram (two of the three input beams), or real-time diffraction pattern, in the material.  The third incident beam diffracts at this dynamic hologram, and, in the process, reads out the ''phase-conjugate'' wave.  In effect, all three incident beams interact (essentially) simultaneously to form several real-time holograms, resulting in a set of diffracted output waves that phase up as the "time-reversed" beam.  In the language of nonlinear optics, the interacting beams result in a nonlinear polarization within the material, which coherently radiates to form the phase-conjugate wave.

Reversal of wavefront means a perfect reversal of photons' linear momentum and angular momentum. The reversal of [[angular momentum]] means reversal of both polarization state and orbital angular momentum.<ref name=Okulov2008/>  Reversal of orbital angular momentum of optical vortex is due to the perfect match of helical phase profiles of the incident and reflected beams.  [[Optical phase conjugation]] is implemented via stimulated Brillouin scattering,<ref name=Okulov2008J/> four-wave mixing, three-wave mixing, static linear holograms and some other tools.

[[Image:PhaseConjugationPrinciple.en.svg|thumb|right|350px|Comparison of a phase-conjugate mirror with a conventional mirror. With the phase-conjugate mirror the image is not deformed when passing through an aberrating element twice.<ref>[http://www.osa-opn.org/home/articles/volume_6/issue_3/features/the_fascinating_behavior_of_light_in_photorefracti/ The Fascinating Behavior of Light in Photorefractive Media | Optics & Photonics News<!-- Bot generated title -->] {{Webarchive|url=https://web.archive.org/web/20150402093011/http://www.osa-opn.org/home/articles/volume_6/issue_3/features/the_fascinating_behavior_of_light_in_photorefracti/ |date=2015-04-02 }}.</ref>]]

The most common way of producing optical phase conjugation is to use a four-wave mixing technique, though it is also possible to use processes such as stimulated Brillouin scattering.

====Four-wave mixing technique====
For the four-wave mixing technique, we can describe four beams (''j'' = 1, 2, 3, 4) with electric fields:

:<math>\Xi_j(\mathbf{x},t) = \frac{1}{2} E_j(\mathbf{x}) e^{i \left(\omega_j t - \mathbf{k} \cdot \mathbf{x}\right)} + \text{c.c.},</math>

where ''E<sub>j</sub>'' are the electric field amplitudes. Ξ<sub>1</sub> and Ξ<sub>2</sub> are known as the two pump waves, with Ξ<sub>3</sub> being the signal wave, and Ξ<sub>4</sub> being the generated conjugate wave.

If the pump waves and the signal wave are superimposed in a medium with a non-zero χ<sup>(3)</sup>, this produces a nonlinear polarization field:

:<math>P_\text{NL} = \varepsilon_0 \chi^{(3)} (\Xi_1 + \Xi_2 + \Xi_3)^3,</math>

resulting in generation of waves with frequencies given by ω = ±ω<sub>1</sub> ± ω<sub>2</sub> ± ω<sub>3</sub> in addition to third-harmonic generation waves with ω = 3ω<sub>1</sub>, 3ω<sub>2</sub>, 3ω<sub>3</sub>.

As above, the phase-matching condition determines which of these waves is the dominant. By choosing conditions such that ω = ω<sub>1</sub>  + ω<sub>2</sub> − ω<sub>3</sub> and '''k''' = '''k'''<sub>1</sub> + '''k'''<sub>2</sub> − '''k'''<sub>3</sub>, this gives a polarization field:

:<math>P_\omega = \frac{1}{2} \chi^{(3)} \varepsilon_0 E_1 E_2 E_3^* e^{i(\omega t - \mathbf{k} \cdot \mathbf{x})} + \text{c.c.}</math>

This is the generating field for the phase-conjugate beam, Ξ<sub>4</sub>. Its direction is given by '''k'''<sub>4</sub> = '''k'''<sub>1</sub> + '''k'''<sub>2</sub> − '''k'''<sub>3</sub>, and so if the two pump beams are counterpropagating ('''k'''<sub>1</sub> = −'''k'''<sub>2</sub>), then the conjugate and signal beams propagate in opposite directions ('''k'''<sub>4</sub> = −'''k'''<sub>3</sub>). This results in the retroreflecting property of the effect.

Further, it can be shown that for a medium with refractive index ''n'' and a beam interaction length ''l'', the electric field amplitude of the conjugate beam is approximated by

:<math>E_4 =  \frac{i \omega l}{2 n c} \chi^{(3)} E_1 E_2 E_3^*, </math>

where ''c'' is the speed of light. If the pump beams ''E''<sub>1</sub> and ''E''<sub>2</sub> are plane (counterpropagating) waves, then

:<math>E_4(\mathbf{x}) \propto E_3^*(\mathbf{x}),</math>

that is, the generated beam amplitude is the complex conjugate of the signal beam amplitude. Since the imaginary part of the amplitude contains the phase of the beam, this results in the reversal of phase property of the effect.

Note that the constant of proportionality between the signal and conjugate beams can be greater than 1. This is effectively a mirror with a reflection coefficient greater than 100%, producing an amplified reflection. The power for this comes from the two pump beams, which are depleted by the process.

The frequency of the conjugate wave can be different from that of the signal wave. If the pump waves are of frequency ω<sub>1</sub> = ω<sub>2</sub> = ω, and the signal wave is higher in frequency such that ω<sub>3</sub> = ω + Δω, then the conjugate wave is of frequency ω<sub>4</sub> = ω&nbsp;− Δω. This is known as ''frequency flipping''.

===Angular and linear momenta in optical phase conjugation===

====Classical picture ====

In ''classical Maxwell electrodynamics'' a phase-conjugating mirror performs reversal of the [[Poynting vector]]:

:<math>\mathbf{S}_\text{out}(\mathbf{r},t) = -\mathbf{S}_\text{in}(\mathbf{r},t),</math>

("in" means incident field, "out" means reflected field) where

:<math>\mathbf{S}(\mathbf{r},t) = \epsilon_0 c^2 \mathbf{E}(\mathbf{r},t) \times \mathbf{B}(\mathbf{r},t),</math>
 
which is a linear momentum density of electromagnetic field.<ref name="Okulov2008">{{cite journal |first=A. Yu. |last=Okulov |title=Angular momentum of photons and phase conjugation |journal=J. Phys. B: At. Mol. Opt. Phys. |volume=41 |issue=10 |pages=101001 |date=2008 |doi=10.1088/0953-4075/41/10/101001 |arxiv=0801.2675}}</ref>
In the same way a phase-conjugated wave has an opposite angular momentum density vector <math> \mathbf{L}(\mathbf{r},t) = \mathbf{r} \times \mathbf{S}(\mathbf{r},t) </math> 
with respect to incident field:<ref name="Okulov2008J">{{cite journal |first=A. Yu. |last=Okulov |title=Optical and Sound Helical structures in a Mandelstam–Brillouin mirror |journal=JETP Lett. |volume=88 |issue=8 |pages=561–566 |date=2008 |doi=10.1134/S0021364008200046 }}</ref>

:<math>\mathbf{L}_\text{out}(\mathbf{r},t) = -\mathbf{L}_\text{in}(\mathbf{r},t).</math>

The above identities are valid ''locally'', i.e. in each space point <math>\mathbf{r}</math> in a given moment <math>t</math> for an ''ideal phase-conjugating mirror''.

====Quantum picture ====
<!-- Deleted image removed: [[Image:Pointing Oam wiki okulov.jpg|thumb|right|350px|Reversal of Linear Momentum '''''<math>\vec \mathbf{P} </math>''''' and Angular Momentum '''<math>\vec \mathbf{L} </math>''' in Phase Conjugating Mirror.]]
-->
In ''quantum electrodynamics'' the photon with energy <math>\hbar \omega</math> also possesses linear momentum <math>\mathbf{P} = \hbar \mathbf{k}</math> and angular momentum, whose projection on propagation axis is <math>L_\mathbf{z} = \pm \hbar \ell</math>, where <math>\ell</math> is ''topological charge'' of photon, or winding number, <math>\mathbf{z}</math> is propagation axis. The angular momentum projection on propagation axis has ''discrete values'' <math>\pm \hbar \ell</math>.

In ''quantum electrodynamics'' the interpretation of phase conjugation is much simpler compared to ''classical electrodynamics''.  The photon reflected from phase conjugating-mirror (out) has opposite directions of linear and angular momenta with respect to incident photon (in):

:<math>\begin{align}
  \mathbf{P}_\text{out} &= -\hbar \mathbf{k} = -\mathbf{P}_\text{in} = \hbar\mathbf{k}, \\
       {L_\mathbf{z}}_\text{out} &= -\hbar \ell = -{L_\mathbf{z}}_\text{in} = \hbar \ell.
\end{align}</math>