Editing Faraday effect (section)

==Mathematical formulation==

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{bmatrix}
    \mu_{1} & -i \mu_{2} &       0 \\
  i \mu_{2} &    \mu_{1} &       0 \\
          0 &          0 & \mu_{z} \\
\end{bmatrix} \mathbf{H}(\omega)</math>

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

[[File:Faraday-effect.svg|thumb|upright=1.35|Polarization rotation due to the Faraday effect]]

:<math> \beta = \mathcal{V}Bd </math>

where

:β is the angle of rotation (in [[radian]]s)
:''B'' is the magnetic flux density in the direction of propagation (in [[tesla (unit)|tesla]]s)
:''d'' is the length of the path (in meters) where the light and magnetic field interact
:<math>\scriptstyle \mathcal{V}</math> is the [[Verdet constant]] for the material. This empirical proportionality constant (in units of radians per tesla per meter) varies with wavelength and temperature<ref>{{cite journal |last1=Vojna |first1=David |last2=Slezák |first2=Ondřej |last3=Lucianetti |first3=Antonio |last4=Mocek |first4=Tomáš |title=Verdet Constant of Magneto-Active Materials Developed for High-Power Faraday Devices |journal=Applied Sciences |date=2019 |volume=9 |issue=15 |page=3160 |doi=10.3390/app9153160 |doi-access=free }}</ref><ref>{{cite journal |last1=Vojna |first1=David |last2=Slezák |first2=Ondřej |last3=Yasuhara |first3=Ryo |last4=Furuse |first4=Hiroaki |last5=Lucianetti |first5=Antonio |last6=Mocek |first6=Tomáš |title=Faraday Rotation of Dy2O3, CeF3 and Y3Fe5O12 at the Mid-Infrared Wavelengths |journal=Materials |date=2020 |volume=13 |issue=23 |page=5324 |doi=10.3390/ma13235324 |pmid=33255447 |pmc=7727863 |bibcode=2020Mate...13.5324V |doi-access=free }}</ref><ref>{{cite journal |last1=Vojna |first1=David |last2=Duda |first2=Martin |last3=Yasuhara |first3=Ryo |last4=Slezák |first4=Ondřej |last5=Schlichting |first5=Wolfgang |last6=Stevens |first6=Kevin |last7=Chen |first7=Hengjun |last8=Lucianetti |first8=Antonio |last9=Mocek |first9=Tomáš |title=Verdet constant of potassium terbium fluoride crystal as a function of wavelength and temperature |journal=Opt. Lett. |date=2020 |volume=45 |issue=7 |pages=1683–1686 |doi=10.1364/ol.387911 |pmid=32235973 |bibcode=2020OptL...45.1683V |s2cid=213599420 |url=https://www.osapublishing.org/ol/fulltext.cfm?uri=ol-45-7-1683&id=429076|url-access=subscription }}</ref> 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 (≈ {{val|-134|u=rad/(T·m)}} for 632&nbsp;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&nbsp;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].