Faraday effect

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The Faraday effect or Faraday rotation, sometimes referred to as the magneto-optic Faraday effect (MOFE), [1] is a physical magneto-optical phenomenon. The Faraday effect causes a 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. [2] This effect occurs in most optically transparent dielectric materials (including liquids) under the influence of magnetic fields.

Contents

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 remote sensing of magnetic fields (such as fiber optic current sensors). The Faraday effect is used in spintronics research to study the polarization of electron spins in semiconductors. Faraday rotators can be used for amplitude modulation of light, and are the basis of optical isolators and optical circulators; such components are required in optical telecommunications and other laser applications. [3]

History

Michael Faraday holding a piece of glass of the type he used to demonstrate the effect of magnetism on polarization of light, c. 1857. Faraday with glass bar crop2.jpg
Michael Faraday holding a piece of glass of the type he used to demonstrate the effect of magnetism on polarization of light, c. 1857.

By 1845, it was known through the work of Augustin-Jean Fresnel, Étienne-Louis Malus, and others that different materials are able to modify the direction of polarization of light when appropriately oriented, [4] making polarized light a very powerful tool to investigate the properties of transparent materials. Faraday firmly believed that light was an electromagnetic phenomenon, and as such should be affected by electromagnetic forces. He spent considerable effort looking for evidence of electric forces affecting the polarization of light through what are now known as electro-optic effects, starting with decomposing electrolytes. However, his experimental methods were not sensitive enough, and the effect was only measured thirty years later by John Kerr. [5]

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]

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

Polarization rotation due to the Faraday effect Faraday-effect.svg
Polarization rotation due to the Faraday effect

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
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 terbium gallium garnet (TGG) have extremely high Verdet constants (≈ −134 rad/(T·m) for 632 nm light). [12] 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 rotators, which are the principal component of Faraday isolators, 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 (≈ −20 rad/(T·m) for 632 nm light). [13] Similar isolators are constructed for microwave systems by using ferrite rods in a waveguide with a surrounding magnetic field. A thorough mathematical description can be found here.

Examples

Interstellar medium

The effect is imposed on light over the course of its propagation from its origin to the Earth, through the interstellar medium. Here, the effect is caused by free electrons and can be characterized as a difference in the refractive index seen by the two circularly polarized propagation modes. Hence, in contrast to the Faraday effect in solids or liquids, interstellar Faraday rotation (β) has a simple dependence on the wavelength of light (λ), namely:

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

or in SI units:

where

ne(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;
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 astronomy for the measurement of magnetic fields, which can be estimated from rotation measures given a knowledge of the electron number density. [14] In the case of radio pulsars, the 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. [15]

Ionosphere

Radio waves passing through the Earth's ionosphere are likewise subject to the Faraday effect. The ionosphere consists of a 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 Earth's magnetic field, rotation of the polarization of radio waves thus occurs. Since the density of electrons in the ionosphere varies greatly on a daily basis, as well as over the sunspot cycle, the magnitude of the effect varies. However the effect is always proportional to the square of the wavelength, so even at the UHF television frequency of 500 MHz (λ = 60 cm), there can be more than a complete rotation of the axis of polarization. [16] A consequence is that although most radio transmitting antennas are either vertically or horizontally polarized, the polarization of a medium or short wave signal after reflection by the ionosphere is rather unpredictable. However the Faraday effect due to free electrons diminishes rapidly at higher frequencies (shorter wavelengths) so that at microwave frequencies, used by satellite communications, the transmitted polarization is maintained between the satellite and the ground.

Semiconductors

GaAs-Faraday rotation spectrum GaAs-Faraday rotation spectrum.png
GaAs-Faraday rotation spectrum

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 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

Optical cavity created by plasmonic materials.png

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

Related Research Articles

<span class="mw-page-title-main">Refractive index</span> Ratio of the speed of light in vacuum to that in the medium

In optics, the refractive index of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.

<span class="mw-page-title-main">Optical rotation</span> Rotation of the plane of linearly polarized light as it travels through a chiral material

Optical rotation, also known as polarization rotation or circular birefringence, is the rotation of the orientation of the plane of polarization about the optical axis of linearly polarized light as it travels through certain materials. Circular birefringence and circular dichroism are the manifestations of optical activity. Optical activity occurs only in chiral materials, those lacking microscopic mirror symmetry. Unlike other sources of birefringence which alter a beam's state of polarization, optical activity can be observed in fluids. This can include gases or solutions of chiral molecules such as sugars, molecules with helical secondary structure such as some proteins, and also chiral liquid crystals. It can also be observed in chiral solids such as certain crystals with a rotation between adjacent crystal planes or metamaterials.

<span class="mw-page-title-main">Circular polarization</span> Polarization state

In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to the direction of the wave.

In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. The term linear polarization was coined by Augustin-Jean Fresnel in 1822. See polarization and plane of polarization for more information.

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 can be rotated, forming a Faraday rotator. The results of reflection from a magneto-optic material are known as the magneto-optic Kerr effect.

<span class="mw-page-title-main">Optical isolator</span> Optical component allowing the transmission of light in only one direction

An optical isolator, or optical diode, is an optical component which allows the transmission of light in only one direction. It is typically used to prevent unwanted feedback into an optical oscillator, such as a laser cavity.

<span class="mw-page-title-main">Polarization (waves)</span> Property of waves that can oscillate with more than one orientation

Polarization is a property of transverse waves which specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string (see image); for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, and transverse sound waves in solids.

Circular dichroism (CD) is dichroism involving circularly polarized light, i.e., the differential absorption of left- and right-handed light. Left-hand circular (LHC) and right-hand circular (RHC) polarized light represent two possible spin angular momentum states for a photon, and so circular dichroism is also referred to as dichroism for spin angular momentum. This phenomenon was discovered by Jean-Baptiste Biot, Augustin Fresnel, and Aimé Cotton in the first half of the 19th century. Circular dichroism and circular birefringence are manifestations of optical activity. It is exhibited in the absorption bands of optically active chiral molecules. CD spectroscopy has a wide range of applications in many different fields. Most notably, UV CD is used to investigate the secondary structure of proteins. UV/Vis CD is used to investigate charge-transfer transitions. Near-infrared CD is used to investigate geometric and electronic structure by probing metal d→d transitions. Vibrational circular dichroism, which uses light from the infrared energy region, is used for structural studies of small organic molecules, and most recently proteins and DNA.

<span class="mw-page-title-main">Radio wave</span> Type of electromagnetic radiation

Radio waves are a type of electromagnetic radiation with the lowest frequencies and the longest wavelengths in the electromagnetic spectrum, typically with frequencies below 300 gigahertz (GHz) and wavelengths greater than 1 millimeter, about the diameter of a grain of rice. Like all electromagnetic waves, radio waves in a vacuum travel at the speed of light, and in the Earth's atmosphere at a slightly slower speed. Radio waves are generated by charged particles undergoing acceleration, such as time-varying electric currents. Naturally occurring radio waves are emitted by lightning and astronomical objects, and are part of the blackbody radiation emitted by all warm objects.

<span class="mw-page-title-main">Birefringence</span> Property of materials whose refractive index depends on light polarization and direction

Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are described as birefringent or birefractive. The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress.

The Verdet constant is an optical property named after the French physicist Émile Verdet. It describes the strength of the Faraday effect for a particular material. For a constant magnetic field parallel to the path of the light, it can be calculated by:

<span class="mw-page-title-main">Faraday rotator</span>

A Faraday rotator is a polarization rotator based on the Faraday effect, a magneto-optic effect involving transmission of light through a material when a longitudinal static magnetic field is present. The state of polarization is rotated as the wave traverses the device, which is explained by a slight difference in the phase velocity between the left and right circular polarizations. Thus it is an example of circular birefringence, as is optical activity, but involves a material only having this property in the presence of a magnetic field.

<span class="mw-page-title-main">Metamaterial</span> Materials engineered to have properties that have not yet been found in nature

A metamaterial is a type of material engineered to have a property that is rarely observed in naturally occurring materials. They are made from assemblies of multiple elements fashioned from composite materials such as metals and plastics. These materials are usually arranged in repeating patterns, at scales that are smaller than the wavelengths of the phenomena they influence. Metamaterials derive their properties not from the properties of the base materials, but from their newly designed structures. Their precise shape, geometry, size, orientation and arrangement gives them their smart properties capable of manipulating electromagnetic waves: by blocking, absorbing, enhancing, or bending waves, to achieve benefits that go beyond what is possible with conventional materials.

<span class="mw-page-title-main">Madison Symmetric Torus</span>

The Madison Symmetric Torus (MST) is a reversed field pinch (RFP) physics experiment with applications to both fusion energy research and astrophysical plasmas.

<span class="mw-page-title-main">Polarizer</span> Optical filter device

A polarizer or polariser is an optical filter that lets light waves of a specific polarization pass through while blocking light waves of other polarizations. It can filter a beam of light of undefined or mixed polarization into a beam of well-defined polarization, known as polarized light. Polarizers are used in many optical techniques and instruments. Polarizers find applications in photography and LCD technology. In photography, a polarizing filter can be used to filter out reflections.

The Faraday effect causes the index of refractions for right and left circular polarization to be different when light is propagating along either the magnetic field or the magnetization. The inverse Faraday effect (IFE) is the effect opposite to the Faraday effect. A static magnetization is induced by circularly polarized light. One reason for the name IFE is that the amplitude of the magnetization is proportional to the same Verdet coefficient that governs the Faraday effect. The induced magnetization of the IFE is proportional to the product of the Verdet coefficient and vector product of and :

<span class="mw-page-title-main">Magneto-optic Kerr effect</span> Changes to light reflected from a magnetized surface

In physics the magneto-optic Kerr effect (MOKE) or the surface magneto-optic Kerr effect (SMOKE) is one of the magneto-optic effects. It describes the changes to light reflected from a magnetized surface. It is used in materials science research in devices such as the Kerr microscope, to investigate the magnetization structure of materials.

An optical modulator is an optical device which is used to modulate a beam of light with a perturbation device. It is a kind of transmitter to convert information to optical binary signal through optical fiber or transmission medium of optical frequency in fiber optic communication. There are several methods to manipulate this device depending on the parameter of a light beam like amplitude modulator (majority), phase modulator, polarization modulator etc. The easiest way to obtain modulation is modulation of intensity of a light by the current driving the light source. This sort of modulation is called direct modulation, as opposed to the external modulation performed by a light modulator. For this reason, light modulators are called external light modulators. According to manipulation of the properties of material modulators are divided into two groups, absorptive modulators and refractive modulators. Absorption coefficient can be manipulated by Franz-Keldysh effect, Quantum-Confined Stark Effect, excitonic absorption, or changes of free carrier concentration. Usually, if several such effects appear together, the modulator is called electro-absorptive modulator. Refractive modulators most often make use of electro-optic effect, other modulators are made with acousto-optic effect, magneto-optic effect such as Faraday and Cotton-Mouton effects. The other case of modulators is spatial light modulator (SLM) which is modified two dimensional distribution of amplitude & phase of an optical wave.

<span class="mw-page-title-main">Chiral media</span> Applied to electromagnetism

The term chiral describes an object, especially a molecule, which has or produces a non-superposable mirror image of itself. In chemistry, such a molecule is called an enantiomer or is said to exhibit chirality or enantiomerism. The term "chiral" comes from the Greek word for the human hand, which itself exhibits such non-superimposeability of the left hand precisely over the right. Due to the opposition of the fingers and thumbs, no matter how the two hands are oriented, it is impossible for both hands to exactly coincide. Helices, chiral characteristics (properties), chiral media, order, and symmetry all relate to the concept of left- and right-handedness.

<span class="mw-page-title-main">Polarization rotator</span> Optical device

A polarization rotator is an optical device that rotates the polarization axis of a linearly polarized light beam by an angle of choice. Such devices can be based on the Faraday effect, on birefringence, or on total internal reflection. Rotators of linearly polarized light have found widespread applications in modern optics since laser beams tend to be linearly polarized and it is often necessary to rotate the original polarization to its orthogonal alternative.

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