Rotation measure

The Faraday effect or Faraday rotation, sometimes referred to as the magneto-optic Faraday effect (MOFE), is a physical magneto-optical phenomenon. The Faraday effect causes a polarization rotation which is proportional to the projection of the magnetic field along the direction of the light propagation. Formally, it is a special case of gyroelectromagnetism obtained when the dielectric permittivity tensor is diagonal. This effect occurs in most optically transparent dielectric materials (including liquids) under the influence of magnetic fields.

Discovered by Michael Faraday in 1845, the Faraday effect was the first experimental evidence that light and electromagnetism are related. The theoretical basis of electromagnetic radiation (which includes visible light) was completed by James Clerk Maxwell in the 1860s. Maxwell's equations were rewritten in their current form in the 1870s by Oliver Heaviside.

The Faraday effect is caused by left and right circularly polarized waves propagating at slightly different speeds, a property known as circular birefringence. Since a linear polarization can be decomposed into the superposition of two equal-amplitude circularly polarized components of opposite handedness and different phase, the effect of a relative phase shift, induced by the Faraday effect, is to rotate the orientation of a wave's linear polarization.

The Faraday effect has applications in measuring instruments. For instance, the Faraday effect has been used to measure optical rotatory power and for remote sensing of magnetic fields (such as fiber optic current sensors). The Faraday effect is used in spintronics research to study the polarization of electron spins in semiconductors. Faraday rotators can be used for amplitude modulation of light, and are the basis of optical isolators and optical circulators; such components are required in optical telecommunications and other laser applications.

History

By 1845, it was known through the work of Fresnel, Malus, and others that different materials are able to modify the direction of polarization of light when appropriately oriented, making polarized light a very powerful tool to investigate the properties of transparent materials. Faraday firmly believed that light was an electromagnetic phenomenon, and as such should be affected by electromagnetic forces. He spent considerable effort looking for evidence of electric forces affecting the polarization of light through what are now known as electro-optic effects, starting with decomposing electrolytes. However, his experimental methods were not sensitive enough, and the effect was only measured thirty years later by John Kerr.

Faraday then attempted to look for the effects of magnetic forces on light passing through various substances. After several unsuccessful trials, he happened to test a piece of "heavy" glass, containing equal proportions of silica, boracic acid and lead oxide, that he had made during his earlier work on glass manufacturing. Faraday observed that when a beam of polarized light passed through the glass in the direction of an applied magnetic force, the polarization of light rotated by an angle that was proportional to the strength of the force. He was later able to reproduce the effect in several other solids, liquids, and gases by procuring stronger electromagnets.

The discovery is well documented in Faraday's daily notebook. On 13 Sept. 1845, in paragraph #7504, under the rubric ''Heavy Glass'', he wrote:

He summarized the results of his experiments on 30 Sept. 1845, in paragraph #7718, famously writing:

Physical interpretation

The linear polarized light that is seen to rotate in the Faraday effect can be seen as consisting of the superposition of a right- and a left- circularly polarized beam (this superposition principle is fundamental in many branches of physics). We can look at the effects of each component (right- or left-polarized) separately, and see what effect this has on the result.

In circularly polarized light the direction of the electric field rotates at the frequency of the light, either clockwise or counter-clockwise. In a material, this electric field causes a force on the charged particles that compose the material (because of their large charge to mass ratio, the electrons are most heavily affected). The motion thus effected will be circular, and circularly moving charges will create their own (magnetic) field in addition to the external magnetic field. There will thus be two different cases: the created field will be parallel to the external field for one (circular) polarization, and in the opposing direction for the other polarization direction – thus the net B field is enhanced in one direction and diminished in the opposite direction. This changes the dynamics of the interaction for each beam and one of the beams will be slowed more than the other, causing a phase difference between the left- and right-polarized beam. When the two beams are added after this phase shift, the result is again a linearly polarized beam, but with a rotation of the polarization vector.

The direction of polarization rotation depends on the properties of the material through which the light is shone. A full treatment would have to take into account the effect of the external and radiation-induced fields on the wave function of the electrons, and then calculate the effect of this change on the refractive index of the material for each polarization, to see whether the right- or left-circular polarization is slowed more.

Mathematical formulation

Formally, the magnetic permeability is treated as a non-diagonal tensor as expressed by the equation:

:$\mathbf(\omega) = \begin \mu_ & -i \mu_ & 0 \\ i \mu_ & \mu_ & 0 \\ 0 & 0 & \mu_ \\ \end \mathbf(\omega)$

The relation between the angle of rotation of the polarization and the magnetic field in a transparent material is:

:$\beta = \mathcalBd$

where

:β is the angle of rotation (in radians)
:''B'' is the magnetic flux density in the direction of propagation (in teslas)
:''d'' is the length of the path (in meters) where the light and magnetic field interact
:$\scriptstyle \mathcal$ is the Verdet constant for the material. This empirical proportionality constant (in units of radians per tesla per meter) varies with wavelength and temperature and is tabulated for various materials.

A positive Verdet constant corresponds to L-rotation (anticlockwise) when the direction of propagation is parallel to the magnetic field and to R-rotation (clockwise) when the direction of propagation is anti-parallel. Thus, if a ray of light is passed through a material and reflected back through it, the rotation doubles.

Some materials, such as terbium gallium garnet (TGG) have extremely high Verdet constants (≈ for 632 nm light). By placing a rod of this material in a strong magnetic field, Faraday rotation angles of over 0.78 rad (45°) can be achieved. This allows the construction of Faraday rotators, which are the principal component of Faraday isolators, devices which transmit light in only one direction. The Faraday effect can, however, be observed and measured in a Terbium-doped glass with Verdet constant as low as (≈ for 632 nm light). Similar isolators are constructed for microwave systems by using ferrite rods in a waveguide with a surrounding magnetic field. A thorough mathematical description can be foun
here

Examples

Interstellar medium

The effect is imposed on light over the course of its propagation from its origin to the Earth, through the interstellar medium. Here, the effect is caused by free electrons and can be characterized as a difference in the refractive index seen by the two circularly polarized propagation modes. Hence, in contrast to the Faraday effect in solids or liquids, interstellar Faraday rotation (β) has a simple dependence on the wavelength of light (λ), namely:

:$\beta = \mathrm \, \lambda^2$

where the overall strength of the effect is characterized by RM, the rotation measure. This in turn depends on the axial component of the interstellar magnetic field ''B_{, ,} '', and the number density of electrons ''n_e'', both of which vary along the propagation path. In Gaussian cgs units the rotation measure is given by:
:$\mathrm = \frac\int_0^d n_e(s) B_\parallel(s) \;\mathrms$

or in SI units:
:$\mathrm = \frac \int_0^d n_e(s) B_(s) \;\mathrms \approx \left(2.62 \times 10^\, T^\right)\, \int_0^d n_e(s) B_\parallel(s)\; \mathrms$

where
:''n_e(s)'' is the density of electrons at each point ''s'' along the path
:''B_‖(s)'' is the component of the interstellar magnetic field in the direction of propagation at each point ''s'' along the path
:''e'' is the charge of an electron;
:''c'' is the speed of light in a vacuum;
:''m'' is the mass of an electron;
:$\scriptstyle\epsilon_0$ is the vacuum permittivity;

The integral is taken over the entire path from the source to the observer.

Faraday rotation is an important tool in astronomy for the measurement of magnetic fields, which can be estimated from rotation measures given a knowledge of the electron number density. In the case of radio pulsars, the dispersion caused by these electrons results in a time delay between pulses received at different wavelengths, which can be measured in terms of the electron column density, or di