Kerr Effect

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The Kerr effect or the '''quadratic electro-optic effect''' ('''QEO effect''') is a change in the Refractive Index of a material in response to an Electric Field . It is distinct from the Pockels Effect in that the induced index change is Directly Proportional to the ''square'' of the electric field instead of to the magnitude of the field. All materials show a Kerr effect, but certain liquids display the effect more strongly than other materials do. The Kerr effect was discovered in 1875 by John Kerr , a Scottish physicist. Two special cases of the Kerr effect are normally considered: the Kerr electro-optic effect, or '''DC Kerr effect''', and the '''optical Kerr effect''', or '''AC Kerr effect'''. KERR ELECTRO-OPTIC EFFECT The Kerr electro-optic effect, or '''DC Kerr effect''', is the special case in which the electric field is a slowly varying external field applied by, for instance, a Voltage on electrodes across the material. Under the influence of the applied field, the material becomes Birefringent , with different indexes of refraction for light Polarized parallel to or perpendicular to the applied field. The difference in index of refraction, ''Δn'', is given by :$\backslash Delta\; n\; =\; \backslash lambda\; K\; E^2\backslash \; ,$ where ''λ'' is the wavelength of the light, ''K'' is the ''Kerr constant'', and ''E'' is the amplitude 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. 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 Cell s, which can operate at much lower voltages. Another disadvantage of Kerr cells is that the best available material, nitrobenzene, is both poisonous and explosive. Some transparent crystals have also been used for Kerr modulation, although they have smaller Kerr constants. OPTICAL KERR EFFECT 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 Optical effects of Self-focusing and Self-phase Modulation , and is the basis for Kerr-lens Modelocking . This effect only becomes significant with very intense beams such as those from Laser s. MAGNETO-OPTIC KERR EFFECT See Also: Magneto-optic Kerr effect 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. THEORY DC Kerr effect For a nonlinear material, the Electric Polarization field P will depend on the electric field '''E''': : where ε0 is the vacuum Permittivity and χ(''n'') is the ''n''-th order component of the Electric Susceptibility of the medium. 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: :$P\_i\; =$ arepsilon_0 \sum_{j=1}^{3} \chi^{(1)}_{i j} E_j + arepsilon_0 \sum_{j=1}^{3} \sum_{k=1}^{3} \chi^{(2)}_{i j k} E_j E_k + arepsilon_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 + \dots 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: :$\backslash mathbf\{E\}\; =\; \backslash mathbf\{E\}\_0\; +\; \backslash mathbf\{E\}\_\backslash omega\; \backslash cos(\backslash omega\; t),$ where Eω is the vector amplitude of the wave.
	:<math> \Delta N	\lambda_0 K \mathbf{E}_0^2, </math>

\simeq n_0 \left( 1 + rac{1}{2 {n_0}^2} \chi_{\mathrm{NL}} ight)

where ''n''₀=(1+χ_LIN)^1/2 is the linear refractive index. Using a Taylor Expansion since χ_NL << ''n''₀², this give an ''intensity dependent refractive index'' (IDRI) of: