Editing Kerr effect (section)

=== 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>