TheKerr effect, also called thequadratic electro-optic (QEO)effect, is a change in therefractive index of a material in response to an appliedelectric field. The Kerr effect is distinct from thePockels effect in that the induced index change for the Kerr effect isdirectly proportional to thesquare 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 Scottish physicistJohn Kerr.[1][2][3]
Two special cases of the Kerr effect are normally considered, these being the Kerrelectro-optic effect, or DC Kerr effect, and the optical Kerr effect, or AC Kerr effect.
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, avoltage on electrodes across the sample material. Under this influence, the sample becomesbirefringent, with different indices of refraction for lightpolarized parallel to or perpendicular to the applied field. The difference in index of refraction,Δn, is given by
whereλ is the wavelength of the light,K is theKerr constant, andE is the strength of the electric field. This difference in index of refraction causes the material to act like awaveplate when light is incident on it in a direction perpendicular to the electric field. If the material is placed between two "crossed" (perpendicular) linearpolarizers, 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.
Somepolar liquids, such asnitrotoluene (C7H7NO2) andnitrobenzene (C6H5NO2) exhibit very large Kerr constants. A glass cell filled with one of these liquids is called aKerr cell. These are frequently used tomodulate 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 toPockels 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 lackinversion symmetry, the Kerr effect is generally masked by the much strongerPockels effect. The Kerr effect is still present, however, and in many cases can be detected independently of Pockels effect contributions.[4]
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 localirradiance of the light.[5] This refractive index variation is responsible for thenonlinear optical effects ofself-focusing,self-phase modulation andmodulational instability, and is the basis forKerr-lens modelocking. This effect only becomes significant with very intense beams such as those fromlasers. The optical Kerr effect has also been observed to dynamically alter the mode-coupling properties inmultimode fiber, a technique that has potential applications for all-optical switching mechanisms, nanophotonic systems and low-dimensional photo-sensors devices.[6][7]
Themagneto-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 theFaraday effect where the plane of polarization of the transmitted light is rotated.
For a nonlinear material, theelectric polarization will depend on the electric field:[8]
where is the vacuumpermittivity and is the-th order component of theelectric susceptibility of the medium.We can write that relationship explicitly; thei-th component for the vectorP can be expressed as:[9]
where. It is often assumed that ∥, i.e., the component parallel tox of the polarization field; ∥ 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 fieldE produced by a light wave of frequency ω together with an external electric fieldE0:
whereEω is the vector amplitude of the wave.
Combining these two equations produces a complex expression forP. For the DC Kerr effect, we can neglect all except the linear terms and those in:
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:
where λ0 is the vacuumwavelength andK is theKerr constant for the medium. The applied field inducesbirefringence in the medium in the direction of the field. A Kerr cell with a transverse field can thus act as a switchablewave 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 ofK depend on the medium and are about 9.4×10−14 m·V−2 forwater,[citation needed] and 4.4×10−12 m·V−2 fornitrobenzene.[10]
Forcrystals, the susceptibility of the medium will in general be atensor, and the Kerr effect produces a modification of this tensor.
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:
whereEω 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)|Eω|3:[8]: 81–82
As before, this looks like a linear susceptibility with an additional non-linear term:
and since:
wheren0=(1+χLIN)1/2 is the linear refractive index. Using aTaylor expansion since χNL ≪n02, this gives anintensity dependent refractive index (IDRI) of:
wheren2 is the second-order nonlinear refractive index, andI 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 ofn2 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 withdispersion, can produce opticalsolitons.
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, aGaussian beam results in a Gaussian refractive index profile, similar to that of agradient-index lens. This causes the beam to focus itself, a phenomenon known asself-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 asmultiphoton 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.[11]
This article incorporatespublic domain material fromFederal Standard 1037C.General Services Administration. Archived fromthe original on 2022-01-22.
