Faraday effect
The Faraday effect or Faraday rotation, sometimes referred to as the magneto-optic Faraday effect, 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 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 was completed by James Clerk Maxwell in the 1860s.
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, for remote sensing of magnetic fields and for magneto-optical imaging. The Faraday effect is used in spintronics research to study the polarization of electron spins in semiconductors. In the superconducting field, it is used to study the dynamic of fluxons in thin films. 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 Augustin-Jean Fresnel, Étienne-Louis 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.Faraday experiments
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 field, the polarization of light rotated by an angle that was proportional to the strength of the force. He used a Nicol prism to measure the polarization. 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:
Other experiments
A year after Faraday, Edmond Becquerel discovered that the rotation depends on the wavelength. George Biddell Airy developed a theoretical model about the same time.From 1854 to 1863, Émile Verdet carried out an extensive investigation of the effect. Verdet verified quantitatively the proportionality relation between rotation and magnetic field. The proportionality constant is now called the Verdet constant. Verdet suggested that the origin of the effect was the difference in velocity between right and left-handed polarized light, based on Airy's theory. He also discovered negative Faraday rotation in iron salts.
In 1876, the magneto-optic Kerr effect was discovered by John Kerr, changes in light polarization when reflected from a magnetic surface.
In 1878, August Kundt and Wilhelm Röntgen demonstrated the Faraday effect in a gas at high pressures. The same year, Augusto Righi devised an experiment to test the hypothesis that the right- and left-handed components traveled at different speeds.
In 1897, Henri Becquerel wrote the formula for the angle of rotation of the Faraday effect. He also demonstrated the Faraday effect in gases at ambient pressure and also tested the speed dependence on polarization. In 1906, his son Jean Becquerel, discovered a specific type of Faraday effect in paramagnetic materials. The dispersion curve was symmetric in paramagnetic materials and asymmetric in diamagnetic ones.
Microscopic theory
Using the Heisenberg model of ferromagnetism in 1932, and Henry Rainsford Hulme related different selection rules of right and left-handed polarized light in the presence of spin–orbit interaction to the Faraday effectA more detailed theory of magneto-optic effects was developed in 1955 later by Petros N. Argyres who included spin-orbit coupling and spin interactions.
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. We can look at the effects of each component 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. The motion thus effected will be circular, and circularly moving charges will create their own 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 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:The relation between the angle of rotation of the polarization and the magnetic field in a transparent material is:
where
A positive Verdet constant corresponds to L-rotation when the direction of propagation is parallel to the magnetic field and to R-rotation 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 have extremely high Verdet constants. By placing a rod of this material in a strong magnetic field, Faraday rotation angles of over 0.78 rad 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. 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 found .
Examples
In plasma
In plasma, 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, the Faraday rotation angle has a simple dependence on the wavelength of light, namely:where the overall strength of the effect is characterized by, the rotation measure. This in turn depends on the projection of the magnetic field along the line of sight and the number density of electrons ne, both of which vary along the propagation path. In the ideal-plasma approximation, the rotation measure is given by
or, in the SI units, by
where
and the integral is taken over the entire path from the source to the observer.
Electron Coulomb collisions and plasma instabilities may significantly alter this simple expression, however.