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| {{about|the Kerr nonlinear optical effect|the magneto-optic phenomenon of the same name|magneto-optic Kerr effect}}
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| The '''Kerr effect''', also called the '''quadratic electro-optic effect''' (QEO effect), is a change in the [[refractive index]] of a material in response to an applied [[electric field]]. The Kerr effect is distinct from the [[Pockels effect]] in that the induced index change is [[directly proportional]] to the ''square'' of the electric field instead of varying linearly with it. All materials show a Kerr effect, but certain liquids display it more strongly than others. The Kerr effect was discovered in 1875 by [[John Kerr (physicist)|John Kerr]], a Scottish physicist.<ref>{{cite journal | author = Weinberger, P. | title = John Kerr and his Effects Found in 1877 and 1878 | journal = Philosophical Magazine Letters | volume = 88 | issue = 12 | pages = 897–907 | url = http://www.computational-nanoscience.de/Weinberger/Famous-Papers/PML-2008.pdf | doi = 10.1080/09500830802526604 | year = 2008|bibcode = 2008PMagL..88..897W }}</ref>
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| Two special cases of the Kerr effect are normally considered, these being the Kerr electro-optic effect, or DC Kerr effect, and the optical Kerr effect, or AC Kerr effect.
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| ==Kerr electro-optic effect==
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| The Kerr electro-optic effect, or DC Kerr effect, is the special case in which a slowly varying external electric field is applied by, for instance, a [[voltage]] on electrodes across the sample material. Under this influence, the sample becomes [[birefringent]], with different indices of refraction for light [[Polarization (waves)|polarized]] parallel to or perpendicular to the applied field. The difference in index of refraction, ''Δn'', is given by
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| :<math>\Delta n = \lambda K E^2,\ </math>
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| where ''λ'' is the wavelength of the light, ''K'' is the ''[[Kerr constant]]'', and ''E'' is the strength of the electric field. This difference in index of refraction causes the material to act like a [[waveplate]] when light is incident on it in a direction perpendicular to the electric field. If the material is placed between two "crossed" (perpendicular) linear [[polarizer]]s, no light will be transmitted when the electric field is turned off, while nearly all of the light will be transmitted for some optimum value of the electric field. Higher values of the Kerr constant allow complete transmission to be achieved with a smaller applied electric field.
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| Some [[Polar molecule|polar]] liquids, such as [[nitrotoluene]] (C<sub>7</sub>H<sub>7</sub>NO<sub>2</sub>) and [[nitrobenzene]] (C<sub>6</sub>H<sub>5</sub>NO<sub>2</sub>) exhibit very large Kerr constants. A glass cell filled with one of these liquids is called a ''Kerr cell''. These are frequently used to [[modulation|modulate]] light, since the Kerr effect responds very quickly to changes in electric field. Light can be modulated with these devices at frequencies as high as 10 [[gigahertz|GHz]]. Because the Kerr effect is relatively weak, a typical Kerr cell may require voltages as high as 30 [[kilovolt|kV]] to achieve complete transparency. This is in contrast to [[Pockels cell]]s, which can operate at much lower voltages. Another disadvantage of Kerr cells is that the best available material, nitrobenzene, is poisonous. Some transparent crystals have also been used for Kerr modulation, although they have smaller Kerr constants.
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| In media that lack [[inversion symmetry]], the Kerr effect is generally masked by the much stronger [[Pockels effect]]. The Kerr effect is still present, however, and in many cases can be detected independently of Pockels effect contributions.<ref>{{cite journal |journal=Phys. Rev. A |volume=82 |page=013821 |year=2010 |title=Direct Kerr electro-optic effect in noncentrosymmetric materials |first1=Mike |last1=Melnichuk |first2=Lowell T. |last2=Wood |url=http://link.aps.org/doi/10.1103/PhysRevA.82.013821 |doi=10.1103/PhysRevA.82.013821|bibcode = 2010PhRvA..82a3821M }}</ref>
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| ==Optical Kerr effect==
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| The optical Kerr effect, or AC Kerr effect is the case in which the electric field is due to the light itself. This causes a variation in index of refraction which is proportional to the local [[irradiance]] of the light. This refractive index variation is responsible for the [[nonlinear optics|nonlinear optical]] effects of [[self-focusing]], [[self-phase modulation]] and [[modulational instability]], and is the basis for [[Kerr-lens modelocking]]. This effect only becomes significant with very intense beams such as those from [[laser]]s.
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| ==Magneto-optic Kerr effect==
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| {{main| Magneto-optic Kerr effect}}
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| The magneto-optic Kerr effect (MOKE) is the phenomenon that the light reflected from a magnetized material has a slightly rotated plane of polarization. It is similar to the [[Faraday effect]] where the plane of polarization of the transmitted light is rotated.
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| ==Theory==
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| ===DC Kerr effect===
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| For a nonlinear material, the [[polarization (electrostatics)|electric polarization]] field '''P''' will depend on the electric field '''E''':
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| :<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>
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| where ε<sub>0</sub> is the vacuum [[permittivity]] and χ<sup>(''n'')</sup> is the ''n''-th order component of the [[electric susceptibility]] of the medium.
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| The ":" symbol represents the scalar product between matrices. We can write that relationship explicitly; the ''i-''th component for the vector ''P'' can be expressed as:
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| :<math>P_i =
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| \varepsilon_0 \sum_{j=1}^{3} \chi^{(1)}_{i j} E_j +
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| \varepsilon_0 \sum_{j=1}^{3} \sum_{k=1}^{3} \chi^{(2)}_{i j k} E_j E_k +
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| \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
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| </math>
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| where <math>i = 1,2,3</math>. It is often assumed that <math>P_1 = P_x</math>, i.e. the component parallel to ''x'' of the polarization field; <math>E_2 = E_y</math> and so on.
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| For a linear medium, only the first term of this equation is significant and the polarization varies linearly with the electric field.
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| 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>:
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| :<math> \mathbf{E} = \mathbf{E}_0 + \mathbf{E}_\omega \cos(\omega t), </math>
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| where '''E'''<sub>ω</sub> is the vector amplitude of the wave.
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| 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>:
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| :<math>\mathbf{P} \simeq \varepsilon_0 \left( \chi^{(1)} + 3 \chi^{(3)} |\mathbf{E}_0|^2 \right) \mathbf{E}_\omega \cos(\omega
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| t),</math>
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| 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.
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| 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:
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| :<math> \Delta n = \lambda_0 K |\mathbf{E}_0|^2, </math>
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| 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.
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| The values of ''K'' depend on the medium and are about 9.4×10<sup>-14</sup> m [[volt|V]]<sup>-2</sup>{{Citation needed|date=March 2013}} for [[water]], and 4.4×10<sup>-12</sup> m V<sup>-2</sup>{{Citation needed|date=March 2013}} for [[nitrobenzene]].
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| For [[crystal]]s, the susceptibility of the medium will in general be a [[tensor]], and the Kerr effect produces a modification of this tensor.
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| ===AC Kerr effect===
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| 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:
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| :<math> \mathbf{E} = \mathbf{E}_\omega \cos(\omega t), </math>
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| where '''E'''<sub>ω</sub> is the amplitude of the wave as before.
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| 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>:
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| :<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>
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| As before, this looks like a linear susceptibility with an additional non-linear term:
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| :<math> \chi = \chi_{\mathrm{LIN}} + \chi_{\mathrm{NL}} = \chi^{(1)} + \frac{3\chi^{(3)}}{4} |\mathbf{E}_\omega|^2,</math>
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| and since:
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| :<math> n = (1 + \chi)^{1/2} =
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| \left( 1+\chi_{\mathrm{LIN}} + \chi_{\mathrm{NL}} \right)^{1/2}
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| \simeq n_0 \left( 1 + \frac{1}{2 {n_0}^2} \chi_{\mathrm{NL}} \right)</math>
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| 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:
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| :<math> n = n_0 + \frac{3\chi^{(3)}}{8 n_0} |\mathbf{E}_{\omega}|^2 = n_0 + n_2 I</math> | |
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| 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.
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| 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.
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| 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.
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| 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]].
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| 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>Dharmadhikari, A.K., Dharmadhikari, J.A., and Mathur, D. (2009). ``Visualization of multiple focusing-refocusing cycles during filamentation in Barium Fluoride'', Applied Physics B, Vol. 94, p. 259.</ref>
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| == See also ==
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| * [[Faraday effect]]
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| * [[Jeffree cell]], an early acousto-optic modulator
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| * [[Filament propagation]]
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| * [[Rapatronic camera]], which used a Kerr cell to take sub-millisecond photographs of nuclear explosions
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| ==References==
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| <references />
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| {{FS1037C}}
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| ==External links==
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| * [http://www.tvhistory.tv/1935%20TV%20Today%20Part%202.htm Kerr cells in early television ] (Scroll down the page for several early articles on Kerr cells.)
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| [[Category:Nonlinear optics]]
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| [[Category:Polarization (waves)]]
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