Editing Kerr effect (section)

=== 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.