View source for Magneto-optic effect

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A '''magneto-optic effect''' is any one of a number of phenomena in which an [[electromagnetic wave]] propagates through a medium that has been altered by the presence of a quasistatic [[magnetic field]].   In such a medium, which is also called '''gyrotropic''' or '''gyromagnetic''', left- and right-rotating elliptical polarizations can propagate at different speeds, leading to a number of important phenomena.  When light is transmitted through a layer of magneto-optic material, the result is called the [[Faraday effect]]: the plane of [[Polarization (waves)|polarization]] can be rotated, forming a [[Faraday rotator]].  The results of reflection from a magneto-optic material are known as the [[magneto-optic Kerr effect]] (not to be confused with the [[nonlinear optics|nonlinear]] [[Kerr effect]]).

In general, magneto-optic effects break [[time reversal symmetry]]  locally (i.e. when only the propagation of light, and not the source of the magnetic field, is considered) as well as [[Lorentz reciprocity]], which is a necessary condition to construct devices such as [[Faraday isolator|optical isolator]]s (through which light passes in one direction but not the other).

Two gyrotropic materials with reversed rotation directions of the two principal polarizations, corresponding to complex-conjugate ε tensors for lossless media, are called [[optical isomer]]s.

==Gyrotropic permittivity==

In particular, in a magneto-optic material the presence of a  magnetic field (either externally applied or because the material itself is [[ferromagnetism|ferromagnetic]]) can cause a change in the [[permittivity]] tensor ε of the material. The ε becomes anisotropic, a 3×3 matrix, with [[complex number|complex]] off-diagonal components,  depending on the frequency ω of incident light. If the absorption losses can be neglected, ε is a [[Hermitian matrix]]. The resulting  [[Optical axis|principal axes]] become complex as well, corresponding to elliptically-polarized light where left- and right-rotating polarizations can travel at different speeds (analogous to [[birefringence]]).

More specifically, for the case where absorption losses can be neglected, the most general form of Hermitian ε is:

:<math>\varepsilon = \begin{pmatrix}
\varepsilon_{xx}' & \varepsilon_{xy}' + i g_z & \varepsilon_{xz}' - i g_y \\
\varepsilon_{xy}' - i g_z & \varepsilon_{yy}' & \varepsilon_{yz}' + i g_x \\
\varepsilon_{xz}' + i g_y & \varepsilon_{yz}' - i g_x & \varepsilon_{zz}' \\
\end{pmatrix}</math>

or equivalently the relationship between the [[Electric displacement field|displacement field]] '''D''' and the [[electric field]] '''E''' is:

:<math>\mathbf{D} = \varepsilon \mathbf{E} = \varepsilon' \mathbf{E} + i \mathbf{E} \times \mathbf{g}</math>

where <math>\varepsilon'</math> is a real [[symmetric matrix]] and <math>\mathbf{g} = (g_x,g_y,g_z)</math> is a real [[pseudovector]] called the '''gyration vector''', whose magnitude is generally small compared to the eigenvalues of <math>\varepsilon'</math>. The direction of '''g''' is called the '''axis of gyration''' of the material.  To first order, '''g''' is proportional to the applied [[magnetic field]]:

:<math>\mathbf{g} = \varepsilon_0 \chi^{(m)} \mathbf{H}</math>

where <math>\chi^{(m)} \!</math> is the [[magneto-optical susceptibility]] (a [[scalar (physics)|scalar]] in isotropic media, but more generally a [[tensor]]).  If this susceptibility itself depends upon the electric field, one can obtain a [[nonlinear optics|nonlinear optical]] effect of [[magneto-optical parametric generation]] (somewhat analogous to a [[Pockels effect]] whose strength is controlled by the applied magnetic field).

The simplest case to analyze is the one in which '''g''' is a principal axis (eigenvector) of <math>\varepsilon'</math>, and the other two eigenvalues of <math>\varepsilon'</math> are identical.  Then, if we let '''g''' lie in the ''z'' direction for simplicity, the ε tensor simplifies to the form:

:<math>\varepsilon = \begin{pmatrix}
\varepsilon_1 & + i g_z & 0 \\
 - i g_z & \varepsilon_1 & 0 \\
0 & 0 & \varepsilon_2 \\
\end{pmatrix}</math>

Most commonly, one considers light propagating in the ''z'' direction (parallel to '''g''').  In this case the solutions are elliptically polarized electromagnetic waves with [[phase velocity|phase velocities]] <math>1 / \sqrt{\mu(\varepsilon_1 \pm g_z)}</math> (where μ is the [[magnetic permeability]]).  This difference in phase velocities leads to the Faraday effect.

For light propagating purely perpendicular to the axis of gyration, the properties are known as the [[Cotton-Mouton effect]] and used for a [[Circulator]].

===Kerr rotation and Kerr ellipticity===
Kerr rotation and Kerr ellipticity are changes in the polarization of incident light which comes in contact with a gyromagnetic material. Kerr rotation is a rotation in the plane of polarization of transmitted light, and Kerr ellipticity is the ratio of the major to minor axis of the ellipse traced out by [[Elliptical polarization|elliptically]] polarized light on the plane through which it propagates. Changes in the orientation of polarized incident light can be quantified using these two properties.

[[Image:Circular.Polarization.Circularly.Polarized.Light With.Components Right.Handed.svg|thumb|Circular Polarized Light]]
According to classical physics, the speed of light varies with the permittivity of a material:

<math>v_p = \frac{1}{\sqrt{\epsilon \mu}} </math>

where <math>v_p</math> is the velocity of light through the material, <math>\epsilon</math> is the material permittivity, and <math>\mu</math> is the material permeability.  Because the permittivity is anisotropic, polarized light of different orientations will travel at different speeds.

This can be better understood if we consider a wave of light that is circularly polarized (seen to the right). If this wave interacts with a material at which the horizontal component (green sinusoid) travels at a different speed than the vertical component (blue sinusoid), the two components will fall out of the 90 degree phase difference (required for circular polarization) changing the Kerr ellipticity.

A change in Kerr rotation is most easily recognized in linearly polarized light, which can be separated into two [[Circular polarization|circularly polarized]] components: Left-handed circular polarized (LHCP) light and right-handed circular polarized (RHCP) light. The anisotropy of the magneto-optic material permittivity causes a difference in the speed of LHCP and RHCP light, which will cause a change in the angle of polarized light. Materials that exhibit this property are known as [[Birefringence|birefringent]].

From this rotation, we can calculate the difference in orthogonal velocity components, find the anisotropic permittivity, find the gyration vector, and calculate the applied magnetic field<ref>{{cite journal |last1=Garcia-Merino |first1=J. A. |title=Magneto-conductivity and magnetically-controlled nonlinear optical transmittance in multi-wall carbon nanotubes |journal=Optics Express |year=2016 |volume=24 |issue=17 |pages=19552–19557 |doi=10.1364/OE.24.019552 |pmid=27557232 |bibcode=2016OExpr..2419552G |url=https://www.osapublishing.org/oe/abstract.cfm?uri=oe-24-17-19552|doi-access=free }}</ref> <math>\mathbf{H}</math>.

==See also==
*[[Zeeman effect]]
*[[QMR effect]]
*[[Magneto-optic Kerr effect]]
*[[Faraday effect]]
*[[Voigt Effect]]
*[[Photoelectric effect]]

==References==
{{Reflist}}
* [[Federal Standard 1037C]] and from [[MIL-STD-188]]
* {{cite book|author1=Lev Davídovich Landau|author2=Evgeniĭ Mikhaĭlovich Lifshit︠s︡|title=Electrodynamics of continuous media|url=https://books.google.com/books?id=sxAJAQAAIAAJ|access-date=3 June 2012|year=1960|publisher=Pergamon Press|page=82|isbn=9780080091051 }}
* {{cite book|last=Jackson|first=John David|title=Classical electrodynamics|year=1998|publisher=Wiley|location=New York|isbn=978-0471309321|edition=3rd|pages=6–10}}
* {{cite journal|last=Jonsson|first=Fredrik|author2=Flytzanis, Christos|title=Optical parametric generation and phase matching in magneto-optic media|journal=Optics Letters|date=1 November 1999|volume=24|issue=21|pages=1514–1516|doi=10.1364/OL.24.001514|pmid=18079850 |bibcode = 1999OptL...24.1514J }}
* {{cite journal|last=Pershan|first=P. S.|title=Magneto-Optical Effects|journal=Journal of Applied Physics|date=1 January 1967|volume=38|issue=3|pages=1482–1490|doi=10.1063/1.1709678|bibcode = 1967JAP....38.1482P }} 
* {{cite journal|last=Freiser|first=M.|title=A survey of magnetooptic effects|journal=IEEE Transactions on Magnetics|date=1 June 1968|volume=4|issue=2|pages=152–161|doi=10.1109/TMAG.1968.1066210|bibcode = 1968ITM.....4..152F }} 
* [http://magnetooptics.phy.bme.hu/research/topics/broad-band-magneto-optical-spectroscopy/ Broad band magneto-optical spectroscopy]

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[[Category:Optical phenomena]]
[[Category:Electric and magnetic fields in matter]]
[[Category:Magneto-optic effects| ]]

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