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L'efecte Kerr, també anomenat efecte electro-òptic quadràtic (quadratic electro-optic, QEO), és un canvi en l'índex de refracció d'un material en resposta a un camp elèctric aplicat. L'efecte Kerr és diferent de l'efecte Pockels perquè el canvi d'índex induït és directament proporcional al quadrat del camp elèctric en lloc de variar linealment amb ell. Tots els materials mostren un efecte Kerr, però alguns líquids el mostren amb més força que altres. L'efecte Kerr va ser descobert el 1875 pel físic escocès John Kerr.

Normalment es consideren dos casos especials de l'efecte Kerr; aquests són l'efecte Kerr electro-òptic (o efecte Kerr corrent continu), i l'efecte Kerr òptic (o efecte Kerr corrent altern).

Efecte Kerr electro-òptic

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 polarized parallel to or perpendicular to the applied field. The difference in index of refraction, Δn, is given by

$\Delta n=\lambda KE^{2},\$

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 polarizers, 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.

Some polar liquids, such as nitrotoluene (C₇H₇NO₂) and nitrobenzene (C₆H₅NO₂) exhibit very large Kerr constants. A glass cell filled with one of these liquids is called a Kerr cell. These are frequently used to 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 GHz. Because the Kerr effect is relatively weak, a typical Kerr cell may require voltages as high as 30 kV to achieve complete transparency. This is in contrast to Pockels cells, 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.

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.

Efecte Kerr òptic

L'efecte Kerr òptic, o efecte Kerr corrent altern, és el cas en què el camp elèctric és degut a la pròpia llum. Això provoca una variació de l'índex de refracció que és proporcional a la irradiància local de la llum. Aquesta variació de l'índex de refracció és responsable dels efectes òptics no lineals de l'automodulació de fase, i la inestabilitat modulacional, i és la base del mode locking de lents Ker. Aquest efecte només es fa important amb feixos molt intensos com els dels làsers. També s'ha observat que l'efecte Kerr òptic altera dinàmicament les propietats d'acoblament de modes en fibra òptica multimode, una tècnica que té aplicacions potencials per a mecanismes de commutació totalment òptics, sistemes nanofotònics i dispositius de sensors fotogràfics de baixa dimensió.

Efecte Kerr magneto-òptic

L'efecte Kerr magnet-oòptic (magneto-optic Kerr effect, MOKE) és el fenomen que la llum reflectida d'un material magnetitzat té un pla de polarització lleugerament girat. És similar a l'efecte Faraday on es fa girar el pla de polarització de la llum transmesa.

Teoria

Efecte Kerr corrent continu

For a nonlinear material, the electric polarization $\mathbf {P}$ will depend on the electric field $\mathbf {E}$:

$\mathbf {P} =\varepsilon _{0}\chi ^{(1)}\mathbf {E} +\varepsilon _{0}\chi ^{(2)}\mathbf {EE} +\varepsilon _{0}\chi ^{(3)}\mathbf {EEE} +\cdots$

where $\varepsilon _{0}$ is the vacuum permittivity and $\chi ^{(n)}$ is the $n$-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:

$P_{i}=\varepsilon _{0}\sum _{j=1}^{3}\chi _{ij}^{(1)}E_{j}+\varepsilon _{0}\sum _{j=1}^{3}\sum _{k=1}^{3}\chi _{ijk}^{(2)}E_{j}E_{k}+\varepsilon _{0}\sum _{j=1}^{3}\sum _{k=1}^{3}\sum _{l=1}^{3}\chi _{ijkl}^{(3)}E_{j}E_{k}E_{l}+\cdots$

where $i=1,2,3$. It is often assumed that $P_{1}$ ∥ $P_{x}$, i.e., the component parallel to x of the polarization field; $E_{2}$ ∥ $E_{y}$ 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⁽³⁾ 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 E₀:

$\mathbf {E} =\mathbf {E} _{0}+\mathbf {E} _{\omega }\cos(\omega t),$

where E_ω 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 $\chi ^{(3)}|\mathbf {E} _{0}|^{2}\mathbf {E} _{\omega }$:

$\mathbf {P} \simeq \varepsilon _{0}\left(\chi ^{(1)}+3\chi ^{(3)}|\mathbf {E} _{0}|^{2}\right)\mathbf {E} _{\omega }\cos(\omega t),$

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:

$\Delta n=\lambda _{0}K|\mathbf {E} _{0}|^{2},$

where λ₀ 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 9.4×10⁻¹⁴ m·V⁻² for water, and 4.4×10⁻¹² m·V⁻² for nitrobenzene.Plantilla:Coelho For crystals, the susceptibility of the medium will in general be a tensor, and the Kerr effect produces a modification of this tensor.

Efecte Kerr corrent altern

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:

$\mathbf {E} =\mathbf {E} _{\omega }\cos(\omega t),$

where E_ω 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⁽³⁾|E_ω|³:

$\mathbf {P} \simeq \varepsilon _{0}\left(\chi ^{(1)}+{\frac {3}{4}}\chi ^{(3)}|\mathbf {E} _{\omega }|^{2}\right)\mathbf {E} _{\omega }\cos(\omega t).$

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

$\chi =\chi _{\mathrm {LIN} }+\chi _{\mathrm {NL} }=\chi ^{(1)}+{\frac {3\chi ^{(3)}}{4}}|\mathbf {E} _{\omega }|^{2},$

and since:

$n=(1+\chi )^{1/2}=\left(1+\chi _{\mathrm {LIN} }+\chi _{\mathrm {NL} }\right)^{1/2}\simeq n_{0}\left(1+{\frac {1}{2{n_{0}}^{2}}}\chi _{\mathrm {NL} }\right)$

where n₀=(1+χ_LIN)^1/2 is the linear refractive index. Using a Taylor expansion since χ_NL ≪ n₀², this gives an intensity dependent refractive index (IDRI) of:

$n=n_{0}+{\frac {3\chi ^{(3)}}{8n_{0}}}|\mathbf {E} _{\omega }|^{2}=n_{0}+n_{2}I$

where n₂ 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 n₂ are relatively small for most materials, on the order of 10⁻²⁰ m² W⁻¹ for typical glasses. Therefore, beam intensities (irradiances) on the order of 1 GW cm⁻² (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, can produce optical solitons.

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.

Bibliografia

• Coelho, Roland. Physics of Dielectrics for the Engineer (en anglès). Elsevier, 2012. ISBN 978-0-444-60180-3. 
• Dharmadhikari, A. K.; Dharmadhikari, J. A.; Mathur, D. «Visualization of focusing–refocusing cycles during filamentation in BaF₂» (en anglès). Applied Physics B, 94(2), 2008. Bibcode: 2009ApPhB..94..259D. DOI: 10.1007/s00340-008-3317-7.
• Hernández Acosta, M. A; Trejo Valdez, M.; Castro Chacón, J. H; Torres San Miguel, C. R; Martínez Gutiérrez, H «Chaotic signatures of photoconductive Cu
  2ZnSnS
  4 nanostructures explored by Lorenz attractors» (en anglès). New Journal of Physics, 20(2), febrer 2018. Bibcode: 2018NJPh...20b3048H. DOI: 10.1088/1367-2630/aaad41.
• Kerr, John «A new relation between electricity and light: Dielectrified media birefringent (1)» (en anglès). Philosophical Magazine, 50(332), 1875. DOI: 10.1080/14786447508641302.
• Kerr, John «A new relation between electricity and light: Dielectrified media birefringent (2)» (en anglès). Philosophical Magazine, 50(333), 1875b. DOI: 10.1080/14786447508641319.
• Melnichuk, Mike; Wood, Lowell T. «Direct Kerr electro-optic effect in noncentrosymmetric materials» (en anglès). Phys. Rev. A, 82(1), 2010. Bibcode: 2010PhRvA..82a3821M. DOI: 10.1103/PhysRevA.82.013821.
• New, Geoffery. Introduction to Nonlinear Optics (en anglès). Cambridge University Press, 2011. ISBN 978-0-521-87701-5. 
• Rashidian Vaziri, M. R «Comment on "Nonlinear refraction measurements of materials using the moiré deflectometry"» (en anglès). Optics Communications, 357, 2015. Bibcode: 2015OptCo.357..200R. DOI: 10.1016/j.optcom.2014.09.017.
• Weinberger, P. «John Kerr and his Effects Found in 1877 and 1878» (PDF) (en anglès). Philosophical Magazine Letters, 88(12), 2008. Bibcode: 2008PMagL..88..897W. DOI: 10.1080/09500830802526604.

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