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 Faraday effect causes a rotation of the plane of polarization which is linearly proportional to the component of the magnetic field in the direction of propagation.

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 and 1870s. This effect occurs in most optically transparent dielectric materials (including liquids) under the influence of magnetic fields.

The Faraday effect causes left and right circularly polarized waves to propagate 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 a few applications in measuring instruments. For instance, the Faraday effect has been used to measure optical rotatory power and for remote sensing of magnetic fields. 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.^[1]

1 Discovery
2 Mathematical formulation
3 Faraday rotation in the interstellar medium
4 Faraday rotation in the ionosphere
5 See also
6 References
7 External links

Discovery

The discovery is well documented, because Faraday's daily notebook has been published.^[2] In 1845 he undertook a series of experiments explicitly intended to find some effect on light from electric and magnetic fields, and succeeded.

On 13 Sept. 1845, in paragraph #7504, under the rubric Heavy Glass, he wrote: -

...BUT, when the contrary magnetic poles were on the same side, there was an effect produced on the polarized ray, and thus magnetic force and light were proved to have relation to each other....

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

...Still, I have at last succeeded in illuminating a magnetic curve or line of force, and in magnetizing a ray of light....

Mathematical formulation

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

$\beta = \mathcal{V}Bd$

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

$\mathcal{V}$ 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 (≈ −40 rad T⁻¹ m⁻¹). 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.

Similar isolators are constructed for microwave systems by using ferrite rods in a waveguide with a surrounding magnetic field.

Faraday rotation in the 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 has a simple dependence on the wavelength of light (λ), namely:

$\beta = \mathrm{RM} \lambda^2 \,$

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 cgs units the rotation measure is given by:

$\mathrm{RM} = \frac{e^3}{2\pi m^2c^4}\int_0^d n_e(s) B_{||}(s) \;\mathrm{d}s,$

or in SI units:

$\mathrm{RM} = \frac{e^3}{8\pi^2 \varepsilon_0 m^2c^3} \int_0^d n_e(s) B_{||}(s) \;\mathrm{d}s \approx (2.62 \times 10^{-13} \, T^{-1} ) \, \int_0^d n_e(s) B_{||}(s) \;\mathrm{d}s,$

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;

$\epsilon_0$ ' 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.^[3] 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 dispersion measure. A measurement of both the dispersion measure and the rotation measure therefore yields the weighted mean of the magnetic field along the line of sight. The same information can be obtained from objects other than pulsars, if the dispersion measure can be estimated based on reasonable guesses about the propagation path length and typical electron densities. In particular, Faraday rotation measurements of polarized radio signals from extragalactic radio sources occulted by the solar corona can be used to estimate both the electron density distribution and the direction and strength of the magnetic field in the coronal plasma.^[4]

Faraday rotation in the ionosphere

Radio waves passing through the Earth's ionosphere are likewise subject to the Faraday effect. The ionosphere consists of a plasma containing free electrons which contribute to Faraday rotation according to the above equation, whereas the positive ions are relatively massive and have little influence. In conjunction with the earth's magnetic field, rotation of the polarization of radio waves thus occurs. Since the density of electrons in the ionosphere varies greatly on a daily basis, as well as over the sunspot cycle, the magnitude of the effect varies. However the effect is always proportional to the square of the wavelength, so even at the UHF television frequency of 500 MHz (λ= 60 cm), there can be more than a complete rotation of the axis of polarization. A consequence is that although most radio transmitting antennas are either vertically or horizontally polarized, the polarization of a medium or short wave signal after reflection by the ionosphere is rather unpredictable. However the Faraday effect due to free electrons diminishes rapidly at higher frequencies (shorter wavelengths) so that at microwave frequencies, used by satellite communications, the transmitted polarization is maintained between the satellite and the ground.

References

^ See http://www.rp-photonics.com/regenerative_amplifiers.html
^ Faraday, Michael (1933). Faraday's Diary. Volume IV, Nov. 12, 1839 - June 26, 1847 (Thomas Martin ed.). London: George Bell and Sons, Ltd.. ISBN 0-7503-05703. The diary is indexed by Faraday's original running paragraph numbers, not by page. For this discovery see #7504, 13 Sept. 1845 to #7718, 30 Sept. 1845. The complete seven volume diary is now in print again.
^ Longair, Malcolm (1992). High Energy Astrophysics. Cambridge University Press. ISBN 0521435846.
^ Mancuso S. and Spangler S. R. "Faraday Rotation and Models for the Plasma Structure of the Solar Corona" (2000), The Astrophysical Journal, 539, 480–491

External links

Faraday Rotation (at Eric W. Weisstein's World of Physics)
Electro-optical measurements (Kerr, Pockels, and Faraday)