Faraday effect

The Faraday effect or Faraday rotation, sometimes referred to as the magneto-optic Faraday effect (MOFE),

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.

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 applications in measuring instruments. For instance, the Faraday effect has been used to measure optical rotatory power and for

By 1845, it was known through the work of

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.[6] 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.^[5]

The discovery is well documented in Faraday's daily notebook.^[7] 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

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 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 counter-clockwise. In a material, this electric field causes a force on the charged particles that compose the material (because of their large charge to mass ratio, the electrons are most heavily affected). The motion thus effected will be circular, and circularly moving charges will create their own (magnetic) 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 – thus the net B field is enhanced in one direction and diminished in the opposite direction. This changes the dynamics of the interaction for each beam and one of the beams will be slowed more than the other, causing a phase difference between the left- and right-polarized beam. When the two beams are added after this phase shift, the result is again a linearly polarized beam, but with a rotation of the polarization vector.

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 wave function 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 more.

Mathematical formulation

Formally, the magnetic permeability is treated as a non-diagonal tensor as expressed by the equation:^[8]

${\displaystyle \mathbf {B} (\omega )={\begin{vmatrix}\mu _{1}&-i\mu _{2}&0\\i\mu _{2}&\mu _{1}&0\\0&0&\mu _{z}\\\end{vmatrix}}\mathbf {H} (\omega )}$

The relation between the

${\displaystyle \beta ={\mathcal {V}}Bd}$

where

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

${\displaystyle \scriptstyle {\mathcal {V}}}$ is the Verdet constant for the material. This empirical proportionality constant (in units of radians per tesla per meter) varies with wavelength and temperature^[9][10][11] 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

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 n_e, both of which vary along the propagation path. In Gaussian cgs units the rotation measure is given by:

${\displaystyle \mathrm {RM} ={\frac {e^{3}}{2\pi m^{2}c^{4}}}\int _{0}^{d}n_{e}(s)B_{\parallel }(s)\;\mathrm {d} s}$

or in

${\displaystyle \mathrm {RM} ={\frac {e^{3}}{8\pi ^{2}\varepsilon _{0}m^{2}c^{3}}}\int _{0}^{d}n_{e}(s)B_{||}(s)\;\mathrm {d} s\approx \left(2.62\times 10^{-13}\,T^{-1}\right)\,\int _{0}^{d}n_{e}(s)B_{\parallel }(s)\;\mathrm {d} s}$

where

n_e(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 vacuum;

m is the mass of an electron;

${\displaystyle \scriptstyle \epsilon _{0}}$ 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

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.^[17]

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,^[2] for which gyroelectricity and gyromagnetism (Faraday effect) may occur at the same time.

Organic materials

In organic materials, Faraday rotation is typically small, with a Verdet constant in the visible wavelength region on the order of a few hundred degrees per Tesla per meter, decreasing proportional to ${\displaystyle \lambda ^{-2}}$ in this region.^[18] 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.^[19][20]

Plasmonic and magnetic materials

In 2009 ^[21] γ-Fe₂O₃-Au core-shell nanostructures were synthesized to integrate magnetic (γ-Fe₂O₃) and plasmonic (Au) properties into one composite. Faraday rotation with and without the plasmonic materials was tested and rotation enhancement under 530 nm light irradiation was observed. Researchers claim that the magnitude of the magneto-optical enhancement is governed primarily by the spectral overlap of the magneto-optical transition and the plasmon resonance.

The reported 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 enhancing Faraday rotation.

See also

References