Editing Kerr effect (section)

== Theory ==

=== DC Kerr effect ===
For a nonlinear material, the [[polarization (electrostatics)|electric polarization]] <math> \mathbf{P} </math> will depend on the electric field <math> \mathbf{E} </math>:<ref name="New, Intro to Nonlinear Optics">{{cite book|first=Geoffery|last=New|title=Introduction to Nonlinear Optics|publisher=Cambridge University Press|year=2011|isbn=978-0-521-87701-5}}</ref>

:<math> \mathbf{P} = \varepsilon_0 \chi^{(1)}\mathbf{E} + \varepsilon_0 \chi^{(2)}\mathbf{E E} + \varepsilon_0 \chi^{(3)}\mathbf{E E E} + \cdots </math>

where <math>\varepsilon_0</math> is the vacuum [[permittivity]] and <math>\chi^{(n)}</math> is the <math>n</math>-th order component of the [[electric susceptibility]] of the medium.
We can write that relationship explicitly; the ''i-''th component for the vector ''P'' can be expressed as:<ref name="Moreno, Kerr Effect">{{cite web|url=https://www.ifsc.usp.br/~strontium/Teaching/Material2018-1%20SFI5708%20Eletromagnetismo/Monografia%20-%20Michelle%20-%20Kerr.pdf|title=Kerr Effect|last=Moreno|first=Michelle|date=2018-06-14|access-date=2023-11-17}}</ref>
:<math>P_i =
\varepsilon_0 \sum_{j=1}^{3} \chi^{(1)}_{i j} E_j +
\varepsilon_0 \sum_{j=1}^{3} \sum_{k=1}^{3} \chi^{(2)}_{i j k} E_j E_k +
\varepsilon_0 \sum_{j=1}^{3} \sum_{k=1}^{3} \sum_{l=1}^{3} \chi^{(3)}_{i j k l} E_j E_k E_l + \cdots
</math>

where <math>i = 1,2,3</math>.  It is often assumed that <math>P_1</math> ∥ <math>P_x</math>, i.e., the component parallel to ''x'' of the polarization field; <math>E_2</math> ∥ <math>E_y</math> and so on.

For a linear medium, only the first term of this equation is significant and the polarization varies linearly with the electric field.

For materials exhibiting a non-negligible Kerr effect, the third, χ<sup>(3)</sup> term is significant, with the even-order terms typically dropping out due to inversion symmetry of the Kerr medium. Consider the net electric field '''E''' produced by a light wave of frequency ω together with an external electric field '''E'''<sub>0</sub>:

:<math> \mathbf{E} = \mathbf{E}_0 + \mathbf{E}_\omega \cos(\omega t), </math>

where '''E'''<sub>ω</sub> is the vector amplitude of the wave.

Combining these two equations produces a complex expression for '''P'''. For the DC Kerr effect, we can neglect all except the linear terms and those in <math>\chi^{(3)}|\mathbf{E}_0|^2 \mathbf{E}_\omega</math>:

:<math>\mathbf{P} \simeq \varepsilon_0  \left( \chi^{(1)} + 3 \chi^{(3)} |\mathbf{E}_0|^2 \right) \mathbf{E}_\omega \cos(\omega
t),</math>

which is similar to the linear relationship between polarization and an electric field of a wave, with an additional non-linear susceptibility term proportional to the square of the amplitude of the external field.

For non-symmetric media (e.g. liquids), this induced change of susceptibility produces a change in refractive index in the direction of the electric field:

:<math> \Delta n = \lambda_0 K |\mathbf{E}_0|^2, </math>

where λ<sub>0</sub> is the vacuum [[wavelength]] and ''K'' is the ''Kerr constant'' for the medium. The applied field  induces [[birefringence]] in the medium in the direction of the field. A Kerr cell with a transverse field can thus act as a switchable [[wave plate]], rotating the plane of polarization of a wave travelling through it. In combination with polarizers, it can be used as a shutter or modulator.

The values of ''K'' depend on the medium and are about 9.4×10<sup>−14</sup> m·[[volt|V]]<sup>−2</sup> for [[water]],{{Citation needed|date=March 2013}} and 4.4×10<sup>−12</sup> m·V<sup>−2</sup> for [[nitrobenzene]].<ref>{{cite book |url=https://books.google.com/books?id=oT4wH0E5wbUC&q=kerr+constant+table+dielectrics&pg=PA51 |first=Roland |last=Coelho |title=Physics of Dielectrics for the Engineer |publisher=[[Elsevier]] |year=2012 |isbn=978-0-444-60180-3 |page=52}}</ref>

For [[crystal]]s, the susceptibility of the medium will in general be a [[tensor]], and the Kerr effect produces a modification of this tensor.

=== AC Kerr effect ===
In the optical or AC Kerr effect, an intense beam of light in a medium can itself provide the modulating electric field, without the need for an external field to be applied. In this case, the electric field is given by:

:<math> \mathbf{E} = \mathbf{E}_\omega \cos(\omega t), </math>

where '''E'''<sub>ω</sub> is the amplitude of the wave as before.

Combining this with the equation for the polarization, and taking only linear terms and those in χ<sup>(3)</sup>|'''E'''<sub>ω</sub>|<sup>3</sup>:<ref name="New, Intro to Nonlinear Optics"></ref>{{rp|81–82}}

:<math> \mathbf{P} \simeq \varepsilon_0  \left( \chi^{(1)} + \frac{3}{4} \chi^{(3)} |\mathbf{E}_\omega|^2 \right) \mathbf{E}_\omega \cos(\omega t).</math>

As before, this looks like a linear susceptibility with an additional non-linear term:

:<math> \chi = \chi_{\mathrm{LIN}} + \chi_{\mathrm{NL}} = \chi^{(1)} + \frac{3\chi^{(3)}}{4} |\mathbf{E}_\omega|^2,</math>

and since:

:<math> n = (1 + \chi)^{1/2} =
\left( 1+\chi_{\mathrm{LIN}} + \chi_{\mathrm{NL}} \right)^{1/2}
\simeq n_0 \left( 1 + \frac{1}{2 {n_0}^2} \chi_{\mathrm{NL}} \right)</math>

where ''n''<sub>0</sub>=(1+χ<sub>LIN</sub>)<sup>1/2</sup> is the linear refractive index. Using a [[Taylor expansion]] since χ<sub>NL</sub> ≪ ''n''<sub>0</sub><sup>2</sup>, this gives an ''intensity dependent refractive index'' (IDRI) of:

:<math> n = n_0 + \frac{3\chi^{(3)}}{8 n_0} |\mathbf{E}_{\omega}|^2 = n_0 + n_2 I</math>

where ''n''<sub>2</sub> is the second-order nonlinear refractive index, and ''I'' is the intensity of the wave. The refractive index change is thus proportional to the intensity of the light travelling through the medium.

The values of ''n''<sub>2</sub> are relatively small for most materials, on the order of 10<sup>−20</sup> m<sup>2</sup> W<sup>−1</sup> for typical glasses. Therefore, beam intensities ([[irradiance]]s) on the order of 1 GW cm<sup>−2</sup> (such as those produced by lasers) are necessary to produce significant variations in refractive index via the AC Kerr effect.

The optical Kerr effect manifests itself temporally as self-phase modulation, a self-induced phase- and frequency-shift of a pulse of light as it travels through a medium. This process, along with [[dispersion (optics)|dispersion]], can produce optical [[soliton]]s.

Spatially, an intense beam of light in a medium will produce a change in the medium's refractive index that mimics the transverse intensity pattern of the beam. For example, a [[Gaussian beam]] results in a Gaussian refractive index profile, similar to that of a [[gradient-index lens]]. This causes the beam to focus itself, a phenomenon known as [[self-focusing]].

As the beam self-focuses, the peak intensity increases which, in turn, causes more self-focusing to occur. The beam is prevented from self-focusing indefinitely by nonlinear effects such as [[multiphoton ionization]], which become important when the intensity becomes very high. As the intensity of the self-focused spot increases beyond a certain value, the medium is ionized by the high local optical field. This lowers the refractive index, defocusing the propagating light beam. Propagation then proceeds in a series of repeated focusing and defocusing steps.<ref>{{Cite journal|doi=10.1007/s00340-008-3317-7|title=Visualization of focusing–refocusing cycles during filamentation in BaF<sub>2</sub>|journal=Applied Physics B|volume=94|issue=2|pages=259|year=2008|last1=Dharmadhikari|first1=A. K.|last2=Dharmadhikari|first2=J. A.|last3=Mathur|first3=D.|bibcode=2009ApPhB..94..259D|s2cid=122865446}}</ref>