Magneto-optic effect
Encyclopedia
A magneto-optic effect is any one of a number of phenomena in which an electromagnetic wave propagates through a medium that has been altered by the presence of a quasistatic magnetic field
. In such a material, which is also called gyrotropic or gyromagnetic, left- and right-rotating elliptical polarizations can propagate at different speeds, leading to a number of important phenomena. When light is transmitted through a layer of magneto-optic material, the result is called the Faraday effect
: the plane of polarization can be rotated, forming a Faraday rotator
. The results of reflection from a magneto-optic material are known as the magneto-optic Kerr effect
(not to be confused with the nonlinear
Kerr effect
).
In general, magneto-optic effects break time reversal symmetry locally (i.e. when only the propagation of light, and not the source of the magnetic field, is considered) as well as Lorentz reciprocity, which is a necessary condition to construct devices such as optical isolators (through which light passes in one direction but not the other). (The other, less useful, way to break time reversal symmetry is to rely upon absorption loss.)
Two gyrotropic materials with reversed rotation directions of the two principal polarizations, corresponding to complex-conjugate ε tensors for lossless media, are called optical isomers.
) can cause a change in the permittivity
tensor ε of the material. The ε becomes anisotropic, a 3×3 matrix, with complex
off-diagonal components, depending of course on the frequency ω of incident light. If the absorption losses can be neglected, ε is a Hermitian matrix. The resulting principal axes become complex as well, corresponding to elliptically-polarized light where left- and right-rotating polarizations can travel at different speeds (analogous to birefringence
).
More specifically, for the case where absorption losses can be neglected, the most general form of Hermitian ε is:
or equivalently the relationship between the displacement field
D and the electric field
E is:
where
is a real symmetric matrix and 
is a real pseudovector
called the gyration vector, whose magnitude is generally small compared to the eigenvalues of
. The direction of g is called the axis of gyration of the material. To first order, g is proportional to the applied magnetic field
:
where
is the magneto-optical susceptibility (a scalar
in isotropic media, but more generally a tensor
). If this susceptibility itself depends upon the electric field, one can obtain a nonlinear optical
effect of magneto-optical parametric generation (somewhat analogous to a Pockels effect
whose strength is controlled by the applied magnetic field).
The simplest case to analyze is the one in which g is a principal axis (eigenvector) of
, and the other two eigenvalues of 
are identical. Then, if we let g lie in the z direction for simplicity, the ε tensor simplifies to the form:
Most commonly, one considers light propagating in the z direction (parallel to g). In this case the solutions are elliptically polarized electromagnetic waves with phase velocities
(where μ is the magnetic permeability). This difference in phase velocities leads to the Faraday effect.
For light propagating purely perpendicular to the axis of gyration, the properties are known as the Cotton-Mouton effect
and used for a Circulator
.
polarized light on the plane through which it propagates. Changes in the orientation of polarized incident light can be quantified using these two properties.
According to classical physics, the speed of light varies with the permittivity of a material:
where
is the velocity of light through the material, 
is the material permittivity, and 
is the material permeability. Because the permittivity is anisotropic, polarized light of different orientations will travel at different speeds.
This can be better understood if we consider a wave of light that is circularly polarized (seen to the right). If this wave interacts with a material at which the horizontal component (green sinusoid) travels at a different speed than the vertical component (blue sinusoid), the two components will fall out of the 90 degree phase difference (required for circular polarization) changing the Kerr Ellipticity.
A change in Kerr Rotation is most easily recognized in linearly polarized light, which can be separated into two Circularly polarized
components: Left-Handed Circular Polarized (LCP) light and Right-Handed Circular Polarized (RCP) light. The anisotropy of the Magneto Optic material permittivity causes a difference in the speed of LCP and RCP light, which will cause a change in the angle of polarized light. Materials that exhibit this property are known as Birefringent
From this rotation, we can calculate the difference in orthogonal velocity components, find the anisotropic permittivity, find the gyration vector, and calculate the applied magnetic field
.
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
. In such a material, which is also called gyrotropic or gyromagnetic, left- and right-rotating elliptical polarizations can propagate at different speeds, leading to a number of important phenomena. When light is transmitted through a layer of magneto-optic material, the result is called the Faraday effect
Faraday 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...
: the plane of polarization can be rotated, forming a Faraday rotator
Faraday rotator
A Faraday rotator is an optical device that rotates the polarization of light due to the Faraday effect, which in turn is based on a magneto-optic effect....
. The results of reflection from a magneto-optic material are known as the magneto-optic Kerr effect
Magneto-optic Kerr effect
Magneto-optic Kerr effect is one of the magneto-optic effects. It describes the changes of light reflected from magnetized media.-Definition:The light that is reflected from a magnetized surface can change in both polarization and reflected intensity...
(not to be confused with the nonlinear
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...
Kerr effect
Kerr effect
The Kerr effect, also called the quadratic electro-optic 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...
).
In general, magneto-optic effects break time reversal symmetry locally (i.e. when only the propagation of light, and not the source of the magnetic field, is considered) as well as Lorentz reciprocity, which is a necessary condition to construct devices such as optical isolators (through which light passes in one direction but not the other). (The other, less useful, way to break time reversal symmetry is to rely upon absorption loss.)
Two gyrotropic materials with reversed rotation directions of the two principal polarizations, corresponding to complex-conjugate ε tensors for lossless media, are called optical isomers.
Gyrotropic permittivity
In particular, in a magneto-optic material the presence of a magnetic field (either externally applied or because the material itself is ferromagneticFerromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...
) can cause a change in the 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...
tensor ε of the material. The ε becomes anisotropic, a 3×3 matrix, with complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
off-diagonal components, depending of course on the frequency ω of incident light. If the absorption losses can be neglected, ε is a Hermitian matrix. The resulting principal axes become complex as well, corresponding to elliptically-polarized light where left- and right-rotating polarizations can travel at different speeds (analogous to 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...
).
More specifically, for the case where absorption losses can be neglected, the most general form of Hermitian ε is:
or equivalently the relationship between the displacement field
Displacement field
Displacement field may refer to:*Displacement field *Electric displacement field...
D and the 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...
E is:
where
is a real symmetric matrix and is a real pseudovectorPseudovector
In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...
called the gyration vector, whose magnitude is generally small compared to the eigenvalues of
. The direction of g is called the axis of gyration of the material. To first order, g is proportional to the applied magnetic fieldMagnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
:
where
is the magneto-optical susceptibility (a scalarScalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...
in isotropic media, but more generally 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...
). If this susceptibility itself depends upon the electric field, one can obtain a 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...
effect of magneto-optical parametric generation (somewhat analogous to a 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...
whose strength is controlled by the applied magnetic field).
The simplest case to analyze is the one in which g is a principal axis (eigenvector) of
, and the other two eigenvalues of are identical. Then, if we let g lie in the z direction for simplicity, the ε tensor simplifies to the form:Most commonly, one considers light propagating in the z direction (parallel to g). In this case the solutions are elliptically polarized electromagnetic waves with phase velocities
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...
(where μ is the magnetic permeability). This difference in phase velocities leads to the Faraday effect.For light propagating purely perpendicular to the axis of gyration, the properties are known as the Cotton-Mouton effect
Cotton-Mouton effect
In physical optics, the Cotton–Mouton effect refers to the double refraction of light in a liquid in the presence of a constant transverse magnetic field. It is a similar but stronger effect than the Voigt effect...
and used for a Circulator
Circulator
A circulator is a passive non-reciprocal three- or four-port device, in which microwave or radio frequency power entering any port is transmitted to the next port in rotation...
.
Kerr Rotation and Kerr Ellipticity
Kerr Rotation and Kerr Ellipticity are changes in the polarization of incident light which comes in contact with a gyromagnetic material. Kerr Rotation is a rotation in the angle of transmitted light, and Kerr Ellipticity is the ratio of the major to minor axis of the ellipse traced out by ellipticallyElliptical polarization
In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation...
polarized light on the plane through which it propagates. Changes in the orientation of polarized incident light can be quantified using these two properties.
According to classical physics, the speed of light varies with the permittivity of a material:
where
is the velocity of light through the material, is the material permittivity, and is the material permeability. Because the permittivity is anisotropic, polarized light of different orientations will travel at different speeds.This can be better understood if we consider a wave of light that is circularly polarized (seen to the right). If this wave interacts with a material at which the horizontal component (green sinusoid) travels at a different speed than the vertical component (blue sinusoid), the two components will fall out of the 90 degree phase difference (required for circular polarization) changing the Kerr Ellipticity.
A change in Kerr Rotation is most easily recognized in linearly polarized light, which can be separated into two Circularly polarized
Circular polarization
In electrodynamics, circular polarization of an electromagnetic wave is a polarization in which the electric field of the passing wave does not change strength but only changes direction in a rotary type manner....
components: Left-Handed Circular Polarized (LCP) light and Right-Handed Circular Polarized (RCP) light. The anisotropy of the Magneto Optic material permittivity causes a difference in the speed of LCP and RCP light, which will cause a change in the angle of polarized light. Materials that exhibit this property are known as Birefringent
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...
From this rotation, we can calculate the difference in orthogonal velocity components, find the anisotropic permittivity, find the gyration vector, and calculate the applied magnetic field
.