# Magneto-optics

The magneto-optics is a field of the optical system that deals with the interaction of light with matter in the magnetic field employed. In short, the interaction consists in the fact that a material is made birefringent by an externally applied magnetic field .

## history

The history of magneto-optics began in 1845 when Michael Faraday observed a rotation of the plane of polarization of linearly polarized light after it had passed through a transparent medium parallel to an externally applied magnetic field ( Faraday effect ).

In 1876 John Kerr discovered a rotation of the plane of polarization of light that is reflected on ferromagnetic metal surfaces ( magneto-optical Kerr effect ).

Another important discovery for magneto-optics is the Zeeman effect (1896), which could only be explained with the development of quantum mechanics .

In 1908 Woldemar Voigt presented a comprehensive theory of magneto-optics in the context of classical electrodynamics .

## Basics

Optical effects that are limited to reflection and absorption are described in the context of classical electrodynamics with the help of the complex-valued dielectric tensor  ε.

In the case of an isotropic medium in the absence of external fields, ε has identical diagonal terms ( ), the off-diagonal elements are all zero. This special form of the tensor allows ε to be treated like a scalar . ${\ displaystyle \ varepsilon _ {xx} = \ varepsilon _ {yy} = \ varepsilon _ {zz}}$

In the presence of an external magnetic field, the diagonal elements change in such a way that they are no longer all identical to one another, and off-diagonal elements appear which are also asymmetrical. These off-diagonal terms of the tensor, which are also called magneto-optical constants , are of great importance for the magneto-optical effects .

In a magnetic field in the z-direction, the dielectric tensor of a body that is isotropic in the absence of a magnetic field has the following appearance:

${\ displaystyle \ varepsilon = {\ begin {pmatrix} \ varepsilon _ {xx} & \ varepsilon _ {xy} & 0 \\ - \ varepsilon _ {xy} & \ varepsilon _ {yy} & 0 \\ 0 & 0 & \ varepsilon _ { zz} \\\ end {pmatrix}}}$,

where based on the isotropy assumption applies. has the value of the scalar dielectric constant that the body has in the absence of a magnetic field. The off-diagonal elements are generally small compared to the diagonal elements and are linearly dependent on the magnetic field. depends on the square of the magnetic field and is small against . ${\ displaystyle \ varepsilon _ {xx} = \ varepsilon _ {yy}}$${\ displaystyle \ varepsilon _ {zz}}$${\ displaystyle \ varepsilon _ {xy}}$${\ displaystyle (\ varepsilon _ {xx} - \ varepsilon _ {zz})}$${\ displaystyle \ varepsilon _ {zz}}$

With the help of the mathematical formalism described in crystal optics , the refractive indices and the polarization character in this medium are obtained by solving the wave equation for anisotropic solids :

• for waves that propagate parallel to the magnetic field, one obtains two circularly polarized waves with a refractive index ( is the imaginary unit )${\ displaystyle n _ {\ pm} = {\ sqrt {\ varepsilon _ {xx} \ pm i \ varepsilon _ {xy}}}}$${\ displaystyle i}$
• for waves that propagate perpendicular to the magnetic field, the solution is two linearly polarized waves:
• the first wave, polarized parallel to the magnetic field, has the refractive index ${\ displaystyle n_ {||} = {\ sqrt {\ varepsilon _ {zz}}}}$
• the second wave, polarized perpendicular to the magnetic field, has the refractive index .${\ displaystyle n _ {\ perp} = {\ sqrt {\ varepsilon _ {xx} + {\ frac {\ varepsilon _ {xy} ^ {2}} {\ varepsilon _ {xx}}}}}}$

## Magneto-optical effects

### Effects in absorption

#### Circular magnetic dichroism

In circular magnetic dichroism (MCD), the magnetization is parallel to the direction of propagation of the light, which is circularly polarized. A distinction is made between a polar and a longitudinal geometry. In the polar geometry, the magnetization is perpendicular to the surface, in the longitudinal the magnetization is parallel to the surface in the plane of incidence. The different absorption for the two polarization directions is used here. This is proportional to the imaginary part of the refractive index. The measured effect thus corresponds to:

${\ displaystyle \ operatorname {Im} (n _ {+} - n _ {-}) = \ operatorname {Im} \ left ({\ sqrt {\ varepsilon _ {xx} + \ mathrm {i} \, \ varepsilon _ { xy}}} - {\ sqrt {\ varepsilon _ {xx} - \ mathrm {i} \, \ varepsilon _ {xy}}} \ right) \ approx \ operatorname {Re} \ left ({\ frac {\ varepsilon _ {xy}} {\ sqrt {\ varepsilon _ {xx}}}} \ right)}$

#### Voigt effect and linear magnetic dichroism

In the Voigt effect discovered in 1898 , the linear magnetic dichroism ( English magnetic linear dichroism , MLD) and the Cotton-Mouton effect , the direction of the magnetic field is parallel to the surface that is hit perpendicularly by the incident wave. The Cotton-Mouton effect, which occurs mainly in liquids, is based on the electrical and magnetic anisotropy of the molecules. By the applied field, the molecules are aligned and produce a square dependent on the field change in the diagonal terms of the tensor, . The Voigt effect, which is measured in metal vapor, and the MLD, which is measured on the solid, are caused by the alignment of the electron shells. ${\ displaystyle \ varepsilon}$${\ displaystyle \ varepsilon _ {xx} = \ varepsilon _ {yy} \ neq \ varepsilon _ {zz}}$

${\ displaystyle \ operatorname {Im} (n _ {\ perp} -n_ {||}) = \ operatorname {Im} \ left ({\ sqrt {\ varepsilon _ {xx} + {\ frac {\ varepsilon _ {xy } ^ {2}} {\ varepsilon _ {xx}}}}} - {\ sqrt {\ varepsilon _ {zz}}}) \ right)}$

### Effects in reflection

The most important effect is the magneto-optical Kerr effect (MOKE), which exists in three different geometries:

• Polar Magneto-Optic Kerr Effect (PMOKE)
• Longitudinal magneto-optic Kerr effect (LMOKE)
• Transverse Magneto-Optic Kerr Effect (TMOKE)

there is also the surface magneto-optical Kerr effect (SMOKE)

The magneto-optical Kerr effect should not be confused with the electro-optical Kerr effect , in which the plane of polarization is rotated by applying electric fields.

## Technical applications

The best known application of magneto-optics is found in the Magneto Optical Disk (MOD). This is read out with the help of the magneto-optical Kerr effect .