Kerr effect
The Kerr effect is a change in the optical properties of a material produced by an applied electric field or by the electric field of light itself. In its classical electro-optic form, the effect consists of birefringence proportional to the square of an externally applied electric field. In nonlinear optics, the same term denotes an intensity-dependent contribution to the refractive index. Both manifestations arise from third-order optical susceptibility, although their experimental regimes and customary mathematical descriptions differ.
The term is also used in the distinct expression magneto-optic Kerr effect, which describes changes in the polarization of light reflected from a magnetized surface. The magneto-optic phenomenon depends on magnetization rather than on the quadratic electro-optic response and is therefore treated separately.
Classical electro-optic Kerr effect
In an initially isotropic medium exposed to a static or slowly varying electric field (E), the induced difference between the principal refractive indices is conventionally written as
[ \Delta n = \lambda K E^2, ]
where (\Delta n) is the induced birefringence, (\lambda) is the vacuum wavelength of the probing light, and (K) is the Kerr constant under the adopted sign convention. Because the response is quadratic, reversing the direction of the applied field leaves the induced birefringence unchanged.
For a uniform field extending over an optical path of length (L), the corresponding phase retardation is
[ \Delta \phi = \frac{2\pi}{\lambda}\Delta n L = 2\pi KLE^2. ]
The applied field establishes principal optical axes parallel and perpendicular to its direction. Light polarized along these axes accumulates different phases, so a general incident polarization becomes elliptically polarized after propagation through the medium. The result is commonly analyzed through Jones calculus when the light remains fully polarized, or through the Mueller calculus when depolarization must also be represented.
The Kerr constant depends on the material, temperature, optical wavelength, and frequency of the applied field. Its sign identifies which field-defined principal axis has the larger refractive index, while its magnitude describes the strength of the quadratic response. Molecular liquids can exhibit a comparatively strong effect because the field alters both electronic polarization and the statistical orientation of anisotropic molecules. At sufficiently high modulation frequencies, molecular reorientation no longer follows the field, leaving the faster electronic contribution as the dominant response.
Microscopic and tensor description
The induced polarization of a dielectric may be expanded in powers of the electric field:
[ P_i
\varepsilon_0 \left( \chi^{(1)}{ij}E_j + \chi^{(2)}{ijk}E_jE_k + \chi^{(3)}_{ijkl}E_jE_kE_l +\cdots \right), ]
where (\chi^{(1)}) is the linear susceptibility and (\chi^{(3)}) is the third-order susceptibility responsible for the Kerr response. Repeated Cartesian indices denote summation. The quadratic electro-optic effect is associated with a third-order interaction because one optical field couples with two factors of the applied low-frequency field.
In a centrosymmetric material, electric-dipole symmetry requires the bulk second-order susceptibility (\chi^{(2)}) to vanish. The third-order susceptibility need not vanish, so the Kerr effect occurs in gases, liquids, glasses, and centrosymmetric crystals that do not exhibit a bulk linear Pockels effect. In anisotropic media, the response cannot generally be reduced to a single scalar Kerr constant because different components of (\chi^{(3)}) couple the applied and optical fields.
The quadratic dependence on field distinguishes the Kerr effect from the Pockels effect, for which the induced refractive-index change is linear in the applied field. The two effects may coexist in non-centrosymmetric crystals. Their relative contribution then depends on crystal symmetry, electrode geometry, field strength, and the orientation of the optical polarization.
Optical Kerr effect
The optical Kerr effect is the intensity-dependent refractive index generated by the electric field of an optical wave. For a nearly monochromatic field in a medium with an instantaneous third-order response, the refractive index is commonly expressed as
[ n = n_0 + n_2 I, ]
where (n_0) is the linear refractive index, (I) is optical intensity, and (n_2) is the nonlinear refractive-index coefficient. The coefficient (n_2) is related to the appropriate tensor component of (\chi^{(3)}), with the precise numerical factor depending on field normalization, polarization, and unit conventions.
Because optical intensity can vary across a beam and over the duration of a pulse, the induced refractive index can also vary in space and time. A beam with greater intensity near its axis produces a transverse refractive-index profile that acts as an intensity-dependent lens. This phenomenon is known as self-focusing when the sign of (n_2) causes the central region to acquire greater optical phase delay, while the opposite sign produces self-defocusing.
Temporal intensity variation produces self-phase modulation. For propagation over a distance (L), the nonlinear phase accumulated in a simplified uniform medium is
[ \phi_{\mathrm{NL}}
\frac{2\pi}{\lambda}n_2 I L. ]
A time-dependent nonlinear phase changes the instantaneous optical frequency and broadens the pulse spectrum. In an optical fiber, the interaction between this nonlinearity and chromatic dispersion governs pulse evolution described by the nonlinear Schrödinger equation. Under anomalous dispersion, the opposing effects can support optical solitons.
Third-order mixing between distinct optical frequencies also produces cross-phase modulation and four-wave mixing. These processes share the susceptibility underlying the optical Kerr effect, but they differ in the arrangement of participating frequencies and in whether optical energy is transferred between spectral components.
Historical development
In 1875, John Kerr and You Watanabe documented electrically induced birefringence in transparent isotropic media. Their experiments placed dielectric samples between charged electrodes and determined the resulting phase retardation through changes in transmitted polarization. The observed dependence on the square of the electric field established the defining relation of the classical Kerr effect.
The early apparatus used electrode arrangements that produced strong fields within liquids or glass while limiting electrical conduction through the optical region. Polarizers converted the electrically induced phase difference into an intensity variation, allowing the response to be detected before electronic photodetectors became available. Subsequent measurements refined the wavelength and temperature dependence of the Kerr constant and separated rapid electronic polarization from slower molecular orientation.
The linear electro-optic effect was later systematized by Friedrich Pockels, whose analysis connected field-induced optical changes with crystal symmetry. In the study of magnetically induced birefringence, Aimé Cotton and Henri Mouton characterized the quadratic magnetic-field response now called the Cotton–Mouton effect. That effect is the magnetic analogue of classical Kerr birefringence in transmission, rather than the reflected-light phenomenon designated as the magneto-optic Kerr effect.
Kerr cells
A Kerr cell contains a Kerr-active material within an electrode structure that generates a transverse or longitudinal electric field. The field-induced phase retardation allows the cell to function as an electrically controlled wave plate. When positioned between polarization elements, it converts the phase modulation into modulation of transmitted intensity.
The voltage producing a phase retardation of (\pi) satisfies
[ 2\pi K L E^2 = \pi. ]
For an approximately uniform field between electrodes separated by (d), with (E=V/d), the corresponding voltage is
[ V_{\pi}
d\sqrt{\frac{1}{2KL}}. ]
This quadratic voltage relation differs from the linear voltage dependence of a Pockels cell. Kerr cells historically served as fast optical shutters and modulators, particularly when their switching speed exceeded that of mechanical systems. Their performance was constrained by the high electric fields required and by the electrical, thermal, and chemical properties of strongly Kerr-active liquids.
Modern electro-optic modulation more commonly employs linear electro-optic crystals where their symmetry permits a substantial Pockels response. Kerr modulation remains relevant in centrosymmetric media and in integrated structures where optical confinement increases the nonlinear interaction over a small effective area.
Relation to the magneto-optic Kerr effect
The magneto-optic Kerr effect occurs when polarized light is reflected from a magnetized material. Magnetization introduces off-diagonal components into the optical response tensor, producing a rotation of the polarization axis and, generally, an accompanying ellipticity. The geometry is classified according to the orientation of magnetization relative to the reflecting surface and the plane of incidence.
This reflected-light effect is closely related to the Faraday effect, which describes polarization rotation during transmission through a magnetized medium. Its name derives from Kerr’s later investigation of reflection from magnetized surfaces, but its microscopic basis and field dependence are not those of electro-optic Kerr birefringence. Magneto-optic Kerr measurements provide information about magnetic domains, hysteresis, and surface-sensitive magnetization without constituting a quadratic electric-field measurement.
Applications in optical physics
The optical Kerr effect contributes to pulse compression, nonlinear spectral broadening, and intensity-dependent switching. In resonators, the nonlinear index shifts the resonance frequency in proportion to the intracavity intensity. This coupling can generate optical bistability, in which a single driving frequency corresponds to more than one stable intracavity intensity over a restricted parameter range.
In frequency-domain systems, Kerr-mediated four-wave mixing can produce evenly spaced spectral components known as optical frequency combs. In waveguides and microresonators, dispersion and confinement determine whether the participating modes satisfy the required phase relations. The resulting behavior follows from the combined influence of cavity loss, external driving, third-order nonlinearity, and frequency-dependent propagation.
The same nonlinearity can limit optical transmission when self-focusing raises the local intensity enough to enhance absorption or damage. Its physical role therefore depends on the spatial scale, pulse duration, material response time, and propagation geometry rather than on intensity alone.