Kerr effect
Encyclopedia
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
, a Scottish physicist.
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.
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
where λ is the wavelength of the light, K is the Kerr constant, and E is the strength of the electric field. The Kerr constant typically ranges from 10-18 to 10-14 for crystals and 10-22 to 10-19 for liquids. 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 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.
of the light. This refractive index variation is responsible for the 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.
where the plane of polarization of the transmitted light is rotated.
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:
where
. It is often assumed that 
, i.e. the component parallel to x 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 field E produced by a light wave of frequency ω together with an external electric field E0:
where Eω 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
:
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 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
.
For crystal
s, the susceptibility of the medium will in general be a tensor
, and the Kerr effect produces a modification of this tensor.
where Eω 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:
As before, this looks like a linear susceptibility with an additional non-linear term:
and since:
where n0=(1+χLIN)1/2 is the linear refractive index. Using a Taylor expansion since χNL << n02, this gives an intensity dependent refractive index (IDRI) of:
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 (irradiance
s) 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 soliton
s.
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.
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
of a material in response to an applied electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
. The Kerr effect is distinct from the Pockels effect
Pockels effect
The Pockels effect , or Pockels electro-optic effect, produces birefringence in an optical medium induced by a constant or varying electric field. It is distinguished from the Kerr effect by the fact that the birefringence is proportional to the electric field, whereas in the Kerr effect it is...
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
John Kerr (physicist)
John Kerr FRS was a Scottish physicist and a pioneer in the field of electro-optics. He is best known for the discovery of what is now called the Kerr effect.-Life and work:...
, a Scottish physicist.
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.
Kerr electro-optic 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, a voltageVoltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
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
where λ is the wavelength of the light, K is the Kerr constant, and E is the strength of the electric field. The Kerr constant typically ranges from 10-18 to 10-14 for crystals and 10-22 to 10-19 for liquids. 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
Polarizer
A polarizer is an optical filter that passes light of a specific polarization and blocks waves of other polarizations. It can convert a beam of light of undefined or mixed polarization into a beam with well-defined polarization. The common types of polarizers are linear polarizers and circular...
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
Nitrobenzene
Nitrobenzene is an organic compound with the chemical formula C6H5NO2. It is a water-insoluble pale yellow oil with an almond-like odor. It freezes to give greenish-yellow crystals. It is produced on a large scale as a precursor to aniline. Although occasionally used as a flavoring or perfume...
(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
Modulation
In electronics and telecommunications, modulation is the process of varying one or more properties of a high-frequency periodic waveform, called the carrier signal, with a modulating signal which typically contains information to be transmitted...
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
Pockels effect
The Pockels effect , or Pockels electro-optic effect, produces birefringence in an optical medium induced by a constant or varying electric field. It is distinguished from the Kerr effect by the fact that the birefringence is proportional to the electric field, whereas in the Kerr effect it is...
. The Kerr effect is still present, however, and in many cases can be detected independently of Pockels effect contributions.
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 irradianceIrradiance
Irradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...
of the light. This refractive index variation is responsible for the nonlinear optical
Nonlinear optics
Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...
effects of self-focusing
Self-focusing
Self-focusing is a non-linear optical process induced by the change in refractive index of materials exposed to intense electromagnetic radiation. A medium whose refractive index increases with the electric field intensity acts as a focusing lens for an electromagnetic wave characterised by an...
, self-phase modulation
Self-phase modulation
Self-phase modulation is a nonlinear optical effect of light-matter interaction.An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect...
and modulational instability
Modulational instability
In the field of nonlinear optics, modulational instability is a phenomenon whereby deviations from an optical waveform are reinforced by nonlinearity, leading to the generation of spectral-sidebands and the eventual breakup of the waveform into a train of pulses.-Initial instability and...
, and is the basis for Kerr-lens modelocking
Kerr-lens modelocking
Kerr-lens modelocking is a method of modelocking lasers via a nonlinear optical process known as the optical Kerr effect. This method allows the generation of pulses of light with a duration as short as a few femtoseconds....
. This effect only becomes significant with very intense beams such as those from laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
s.
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 effectFaraday effect
In physics, the Faraday effect or Faraday rotation is a Magneto-optical phenomenon, that is, an interaction between light and a magnetic field in a medium...
where the plane of polarization of the transmitted light is rotated.
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
Permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
and χ(n) is the n-th order component of the electric susceptibility
Electric susceptibility
In electromagnetism, the electric susceptibility \chi_e is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field...
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:
where
. It is often assumed that , i.e. the component parallel to x 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 field E produced by a light wave of frequency ω together with an external electric field E0:
where Eω 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
: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 vacuum wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
and K is the Kerr constant for the medium. The applied field induces birefringence
Birefringence
Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite...
in the medium in the direction of the field. A Kerr cell with a transverse field can thus act as a switchable wave plate
Wave plate
A wave plate or retarder is an optical device that alters the polarization state of a light wave travelling through it.- Operation :A wave plate works by shifting the phase between two perpendicular polarization components of the light wave. A typical wave plate is simply a birefringent crystal...
, 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
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
-2 for water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
, and 4.4×10-12 m V-2 for nitrobenzene
Nitrobenzene
Nitrobenzene is an organic compound with the chemical formula C6H5NO2. It is a water-insoluble pale yellow oil with an almond-like odor. It freezes to give greenish-yellow crystals. It is produced on a large scale as a precursor to aniline. Although occasionally used as a flavoring or perfume...
.
For crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
s, the susceptibility of the medium will in general be a tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
, and the Kerr effect produces a modification of this tensor.
AC Kerr effect
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:where Eω 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:
As before, this looks like a linear susceptibility with an additional non-linear term:
and since:
where n0=(1+χLIN)1/2 is the linear refractive index. Using a Taylor expansion since χNL << n02, this gives an intensity dependent refractive index (IDRI) of:
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 (irradiance
Irradiance
Irradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...
s) 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
Dispersion (optics)
In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...
, can produce optical soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
s.
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
Gaussian beam
In optics, a Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity distributions are well approximated by Gaussian functions. Many lasers emit beams that approximate a Gaussian profile, in which case the laser is said to be operating on the fundamental...
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
Self-focusing
Self-focusing is a non-linear optical process induced by the change in refractive index of materials exposed to intense electromagnetic radiation. A medium whose refractive index increases with the electric field intensity acts as a focusing lens for an electromagnetic wave characterised by an...
.
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.
See also
- Faraday effectFaraday effectIn physics, the Faraday effect or Faraday rotation is a Magneto-optical phenomenon, that is, an interaction between light and a magnetic field in a medium...
- Jeffree cellJeffree cellThe Jeffree cell was an early acousto-optic modulator, best known for its use in the Scophony system of mechanical television. It was invented by J.H...
-- An early acousto-optic modulator - Filament propagationFilament propagationIn nonlinear optics, filament propagation is propagation of a beam of light through a medium without diffraction. This is possible because the Kerr effect causes an index of refraction change in the medium, resulting in self-focusing of the beam....
- Rapatronic cameraRapatronic cameraThe rapatronic camera is a high-speed camera capable of recording a still image with an exposure time as brief as 10 nanoseconds ....
, which used a Kerr cell to take sub-millisecond photographs of nuclear explosions
External links
- Kerr cells in early television (Scroll down the page for several early articles on Kerr cells.)