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{{Short description|Physical magneto-optical phenomenon}}
The '''Faraday effect''' or '''Faraday rotation''', sometimes referred to as the '''magneto-optic Faraday effect''' ('''MOFE'''),<ref name="UrsMozooni2016">{{cite journal|last1=Urs|first1=Necdet Onur|last2=Mozooni|first2=Babak|last3=Mazalski|first3=Piotr|last4=Kustov|first4=Mikhail|last5=Hayes|first5=Patrick|last6=Deldar|first6=Shayan|last7=Quandt|first7=Eckhard|last8=McCord|first8=Jeffrey|year=2016|title=Advanced magneto-optical microscopy: Imaging from picoseconds to centimeters - imaging spin waves and temperature distributions (invited)|journal=AIP Advances|volume=6|issue=5|pages=055605|bibcode=2016AIPA....6e5605U|doi=10.1063/1.4943760|issn=2158-3226|doi-access=free|hdl=10044/1/34544|hdl-access=free}}</ref> is a [[physics|physical]] [[magneto-optic]]al phenomenon. The Faraday effect causes a [[polarization (waves)|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.<ref name="Prati2003">{{cite journal |last1=Prati |first1=E. |date=2003 |title=Propagation in gyroelectromagnetic guiding systems |journal= [[Journal of Electromagnetic Waves and Applications]]|volume=17 |issue=8 |pages=1177–1196 |doi=10.1163/156939303322519810 |bibcode=2003JEWA...17.1177P |s2cid=121509049 }}</ref> This effect occurs in most optically [[Transparency (optics)|transparent]] [[dielectric]] materials (including liquids) under the influence of [[magnetic field]]s.
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]].
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The Faraday effect is caused by left and right [[Circular polarization|circularly polarized]] waves propagating at slightly different speeds, a property known as [[Optical rotation|circular birefringence]]. Since a linear polarization can be decomposed into the [[Superposition principle|superposition]] of two equal-amplitude circularly polarized components of opposite handedness and different phase, the effect of a relative [[Phase (waves)|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 sensor]]s). The Faraday effect is used in [[spintronics]] research to study the polarization of electron spins in semiconductors. [[Faraday rotator]]s can be used for amplitude modulation of light, and are the basis of [[optical isolator]]s and [[optical circulators]]; such components are required in optical telecommunications and other laser applications.<ref>See https://www.rp-photonics.com/regenerative_amplifiers.html</ref>
==History==
[[Image:Faraday
By 1845, it was known through the work of [[Augustin-Jean
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.<ref name=mansuripur>{{cite journal|last1=Mansuripur|first1=Masud|title=The Faraday Effect|journal=Optics and Photonics News|issue=10|pages=32–36|url=http://www.mmresearch.com/articles/article3/|access-date=15 June 2014}}</ref> 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 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.<ref name="crowther-1920"/>
The discovery is well documented in Faraday's daily notebook.<ref>
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| location = London
| isbn = 978-0-7503-0570-9
| url=https://archive.org/details/faradaysdiarybei00fara_2
}} 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.</ref> On 13 Sept. 1845, in paragraph #7504, under the rubric ''Heavy Glass'', he wrote:
{{Blockquote|
He summarized the results of his experiments on 30 Sept. 1845, in paragraph #7718, famously writing:
{{Blockquote|
==Physical interpretation==
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Formally, the magnetic [[Permeability (electromagnetism)|permeability]] is treated as a non-diagonal tensor as expressed by the equation:<ref>{{Cite journal | last1 = Kales | first1 = M. L. | title = Modes in Wave Guides Containing Ferrites | doi = 10.1063/1.1721335 | journal = Journal of Applied Physics | volume = 24 | issue = 5 | pages = 604–608 | year = 1953 |bibcode = 1953JAP....24..604K }}</ref>
:<math>\mathbf{B}(\omega) = \begin{
\mu_{1} & -i \mu_{2} & 0 \\
i \mu_{2} & \mu_{1} & 0 \\
0 & 0 & \mu_{z} \\
\end{
The relation between the [[angle of rotation]] of the polarization and the magnetic field in a transparent material is:
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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 (≈ {{val|-134|u=rad/(T·m)}} for 632 nm light).<ref>{{cite web |url=http://www.northropgrumman.com/BusinessVentures/SYNOPTICS/Products/SpecialtyCrystals/Pages/TGG.aspx |title=TGG (Terbium Gallium Garnet) |access-date=2013-09-26 |archive-date=2018-07-18 |archive-url=https://web.archive.org/web/20180718162519/http://www.northropgrumman.com/BusinessVentures/SYNOPTICS/Products/SpecialtyCrystals/Pages/TGG.aspx |url-status=dead }}</ref> 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 rotator]]s, which are the principal component of [[Faraday isolator]]s, 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 (≈ {{val|-20|u=rad/(T·m)}} for 632 nm light).<ref>{{cite web|last=Dylan Bleier|title=Faraday Rotation Instructable|url=http://dylanbleier.com/faraday-rotation/|access-date=2013-09-26|archive-date=2014-12-26|archive-url=https://web.archive.org/web/20141226005822/http://dylanbleier.com/faraday-rotation/|url-status=dead}}</ref> Similar isolators are constructed for microwave systems by using [[Ferrite (magnet)|ferrite]] rods in a [[waveguide]] with a surrounding magnetic field. A thorough mathematical description can be found [http://farside.ph.utexas.edu/teaching/em/lectures/node101.html here].
== Examples ==
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or in [[SI]] units:
:<math>\mathrm{RM} = \frac{e^3}{8\pi^2 \varepsilon_0 m^2 c^3} \int_0^d n_e(s) B_{||}(s) \;\mathrm{d}s \approx
</math>
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:''B<sub>‖</sub>(s)'' is the component of the interstellar magnetic field in the direction of propagation at each point ''s'' along the path
:''e'' is the [[electric charge|charge]] of an electron;
:''c'' is the [[speed of light|speed of light in
:''m'' is the [[mass]] of an electron;
:<math>\scriptstyle\epsilon_0</math> is the [[vacuum permittivity]];
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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.<ref>{{cite book|last1=Longair|first1=Malcolm | author-link=Malcolm Longair|title=High Energy Astrophysics|publisher=Cambridge University Press|date = 1992|isbn=978-0-521-43584-0 }}</ref> In the case of [[radio pulsar]]s, the [[dispersion (optics)|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.<ref>{{cite journal |last1=Mancuso |first1=S. |last2=Spangler |first2=S. R. |title=Faraday Rotation and Models for the Plasma Structure of the Solar Corona |date=2000 |journal=[[The Astrophysical Journal]] |volume=539 |issue=1 |pages=480–491 |doi=10.1086/309205 |bibcode = 2000ApJ...539..480M |doi-access=free }}</ref>
===
[[Radio wave]]s passing through the Earth's [[ionosphere]] are likewise subject to the Faraday effect. The ionosphere consists of a [[Plasma (physics)|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
=== Semiconductors ===
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==External links==
* [http://scienceworld.wolfram.com/physics/FaradayRotation.html Faraday Rotation] ''(at Eric W. Weisstein's World of Physics)''
* [http://home.earthlink.net/~jimlux/hv/eo.htm Electro-optical measurements (Kerr, Pockels, and Faraday)] {{Webarchive|url=https://web.archive.org/web/20060510050745/http://home.earthlink.net/~jimlux/hv/eo.htm |date=2006-05-10 }}
* [https://ned.ipac.caltech.edu/level5/Sept04/Govoni/Govoni3_5.html Faraday Rotation Effect] in (radio)astronomy
* A simple {{youTube|id=XhU-nNiAgtI|title=demonstration of the effect}}
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