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. Formally, it is a special case of gyroelectromagnetism obtained when the dielectric permittivity tensor is diagonal.
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 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 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 History
- 2 Physical Interpretation of the Faraday effect
- 3 Mathematical formulation
- 4 Faraday rotation in the interstellar medium
- 5 Faraday rotation in the ionosphere
- 6 Faraday rotation of semiconductors
- 7 Faraday rotation of organic materials
- 8 Faraday rotation in plasmonic/magnetic materials
- 9 See also
- 10 References
- 11 External links
The discovery is well documented, because Faraday's daily notebook has been published. 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. …
— Faraday, Paragraph #7504, Daily notebook
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. …
— Faraday, Paragraph #7718, Daily notebook
Physical Interpretation of the Faraday effect
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 a 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 anticlockwise. In a material, this electric field causes a force on the charged particles comprising the material (because of their light mass the electrons are most heavily affected). The effected motion 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 (circular) polarization, and in the opposing direction for the other polarization direction. This changes the dynamics of the interaction for each beam and one of the beams will be slowed down more than the other, causing a phase difference between the left- and right-polarized beam. When you add the two beams after this phase shift, the result is again a linearly polarized beam, but with a rotation in the polarization direction.
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 wavefunction 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 down more.
The relation between the angle of rotation of the polarization and the magnetic field in a transparent material is:
- 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
- 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 (≈ −134 rad T−1 m−1) (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 (≈ −20 rad T−1 m−1) (for 632 nm light).
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:
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 ne, both of which vary along the propagation path. In Gaussian cgs units the rotation measure is given by:
or in SI units:
- ne(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;
- ' 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 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.
Faraday rotation in the ionosphere
|This section does not cite any references or sources. (June 2013)|
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.
Faraday rotation of semiconductors
Due to spin-orbit coupling, undoped GaAs single crystal exhibits much larger Faraday rotation than glass (SiO2). Considering the atomic arrangement is different along the (100) and (110) plane, one might think the Faraday rotation is polarization dependent. However, experimental work revealed an immeasurable anisotropy in the wavelength range from 880–1,600 nm. Based on the large Faraday rotation, one might be able to use GaAs to calibrate the B field of the terahertz electromagnetic wave which requires very fast response time. Around the band gap, the Faraday effect shows resonance behavior.
More generally, (ferromagnetic) semiconductors return both electro-gyration and a Faraday response in the high frequency domain. The combination of the two is described by gyroelectromagnetic media, for which gyroelectricity and gyromagnetism (Faraday effect) may occur at the same time.
Faraday rotation of organic materials
In organic materials, Faraday rotation is typically small, with a Verdet constant on the order of a few hundred degrees per Tesla per meter, decreasing proportional to . While the Verdet constant of organic materials does increase around electronic transitions in the molecule, the associated light absorption makes most organic materials bad candidates for applications. There are however also isolated reports of large Faraday rotation in organic liquid crystals without associated absorption.
Faraday rotation in plasmonic/magnetic materials
One of the pioneer works in the field of surface plasmon resonance enhanced magneto-optical was published in Nano Letter in 2009 from Adam E. Cohen in Harvard. They synthesized the γ-Fe2O3-Au core-shell nanostructure to integrate the magnetic (γ-Fe2O3) and plasmonic(Au) properties into one composite. They tested the Faraday rotation with and without the plasmonic materials involved and reported the rotation enhancement under light irradiation peaked at 530 nm. they claim the magnitude of the magneto-optical enhancement is governed primarily by the spectral overlap of the magneto-optical transition and the plasmon resonance.
The composite magnetic/plasmonic nanostructure can be visualized to be a magnetic particle embedded in a resonant optical cavity. Because of the large density of photon states in the cavity, the interaction between the electromagnetic field of the light and the electronic transitions of the magnetic material is enhanced, resulting in a larger difference between the velocities of the right- and left-hand circularized polarization, therefore an enhanced Faraday rotation.
- Magneto-optic Kerr effect
- Electro-optic Kerr effect
- Faraday rotator
- Scientific phenomena named after people
- Inverse Faraday effect
- Optical rotation
- QMR effect
- Voigt effect
- Polarization spectroscopy
- Magnetic circular dichroism
- Faraday Cage
- Prati, E. (2003). "Propagation in gyroelectromagnetic guiding systems". J. of Electr. Wav. and Appl. 17 (8): 1177–1196. doi:10.1163/156939303322519810.
- 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 0-7503-0570-3. 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.
- Kales, M. L. (1953). "Modes in Wave Guides Containing Ferrites". Journal of Applied Physics 24 (5): 604–601. Bibcode:1953JAP....24..604K. doi:10.1063/1.1721335.
- http://www.northropgrumman.com/BusinessVentures/SYNOPTICS/Products/SpecialtyCrystals/Pages/TGG.aspx. Missing or empty
- Dylan Bleier. "Faraday Rotation Instructable".
- Longair, Malcolm (1992). High Energy Astrophysics. Cambridge University Press. ISBN 0-521-43584-6.
- Mancuso, S.; Spangler, S. R. (2000). "Faraday Rotation and Models for the Plasma Structure of the Solar Corona". The Astrophysical Journal 539 (1): 480–491. Bibcode:2000ApJ...539..480M. doi:10.1086/309205.
- G. X., Du. "Fast Magneto-optical Spectrometry by Spectrometer". Rev. Sci. Instrum. Retrieved 1 February 2012.
- Vandendriessche, Stefaan et al. "Faraday rotation and its dispersion in the visible region for saturated organic liquids". Physical Chemistry Chemical Physics 14 (6): 1860–1864. doi:10.1039/C2CP23311H.
- Vandendriessche, Stefaan et al. "Giant Faraday Rotation in Mesogenic Organic Molecules". Chemistry of Materials 25 (7): 1139–1143. doi:10.1021/cm4004118.
- Cohen, Adam. "Surface Plasmon Resonance Enhanced Magneto-optics(SuPREMO): Faraday Rotation Enhancement in Gold-Coated Iron Oxide Nanocrystals". American Chemical Society. Retrieved 23 April 2013.
- Faraday Rotation (at Eric W. Weisstein's World of Physics)
- Electro-optical measurements (Kerr, Pockels, and Faraday)