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Encyclopedia > Polarization

In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. More generally, the polarization of a transverse wave describes the direction of oscillation in the plane perpendicular to the direction of travel. Longitudinal waves such as sound waves do not exhibit polarization, because for these waves the direction of oscillation is along the direction of travel. Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ... Electromagnetic radiation is a propagating wave in space with electric and magnetic components. ... Prism splitting light Light is electromagnetic radiation with a wavelength that is visible to the eye (visible light) or, in a technical or scientific context, electromagnetic radiation of any wavelength[1]. The elementary particle that defines light is the photon. ... It has been suggested that optical field be merged into this article or section. ... A light wave is an example of a transverse wave. ... Longitudinal waves are waves that have vibrations along or parallel to their direction of travel. ... This article is about compression waves. ...

Contents

Theory

Basics — plane waves

The simplest manifestation of polarization to visualize is that of a plane wave, which is a good approximation to most light waves (a plane wave is a wave with infinitely long and wide wavefronts). All electromagnetic waves propagating in free space or in a uniform material of infinite extent have electric and magnetic fields perpendicular to the direction of propagation. Conventionally, when considering polarization, the electric field vector is described and the magnetic field is ignored since it is perpendicular to the electric field and proportional to it. The electric field vector may be arbitrarily divided into two perpendicular components labelled x and y (with z indicating the direction of travel). For a simple harmonic wave, where the amplitude of the electric vector varies in a sinusoidal manner, the two components have exactly the same frequency. However, these components have two other defining characteristics that can differ. First, the two components may not have the same amplitude. Second, the two components may not have the same phase, that is they may not reach their maxima and minima at the same time. The shape traced out in a fixed plane by the electric vector as such a plane wave passes over it (a Lissajous figure), is a description of the polarization state. The following figures show some examples of the evolution of the electric field vector (blue) with time (the vertical axes), along with its x and y components (red/left and green/right), and the path traced by the tip of the vector in the plane (purple): In the physics of wave propagation (especially electromagnetic waves), a plane wave (also spelled planewave) is a constant-frequency wave whose wavefronts (surfaces of constant amplitude and phase) are infinite parallel planes normal to the propagation direction. ... In optics, a wavefront is the locus (a line or surface in an electromagnetic wave) of points having the same phase. ... It has been suggested that optical field be merged into this article or section. ... Current (I) flowing through a wire produces a magnetic field () around the wire. ... In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... Fig. ... Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. ... Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Lissajous figure on an oscilloscope- the shape of the ABC logo Lissajous figure in three dimensions In mathematics, a Lissajous curve (Lissajous figure or Bowditch curve) is the graph of the system of parametric equations which describes complex harmonic motion. ...

Linear
Circular
Elliptical


Gnuplot source: set term png crop set output Linear_polarization_schematic. ... Gnuplot source: set term png crop set output Circular_polarization_schematic. ... Gnuplot source: set term png crop set output Elliptical_polarization_schematic. ...


In the figure on the left, the two orthogonal (perpendicular) components are in phase. In this case the ratio of the strengths of the two components is constant, so the direction of the electric vector (the vector sum of these two components) is constant. Since the tip of the vector traces out a single line in the plane, this special case is called linear polarization. The direction of this line depends on the relative amplitudes of the two components. 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. ...


In the middle figure above, the two orthogonal components have exactly the same amplitude and are exactly ninety degrees out of phase. In this case one component is zero when the other component is at maximum or minimum amplitude. There are two possible phase relationships that satisfy this requirement: the x component can be ninety degrees ahead of the y component or it can be ninety degrees behind the y component. In this special case the electric vector traces out a circle in the plane, so this special case is called circular polarization. The direction the field rotates in depends on which of the two phase relationships exists. These cases are called right-hand circular polarization and left-hand circular polarization, depending on which way the electric vector rotates. In electrodynamics, circular polarization of electromagnetic radiation is a polarization such that the tip of the electric field vector, at a fixed point in space, describes a circle as time progresses. ...


All other cases, that is where the two components are not in phase and either do not have the same amplitude and/or are not ninety degrees out of phase are called elliptical polarization because the electric vector traces out an ellipse in the plane (the polarization ellipse). In electrodynamics elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. ... For other uses, see Ellipse (disambiguation). ...


The "cartesian" decomposition of the electric field into x and y components is, of course, arbitrary. Plane waves of any polarization can be described instead by combining waves of opposite circular polarization, for example. The cartesian polarization decomposition is natural when dealing with reflection from surfaces, birefringent materials, or synchrotron radiation. The circularly polarized modes are a more useful basis for the study of light propagation in stereoisomers. A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on... Synchrotron radiation is electromagnetic radiation, similar to cyclotron radiation, but generated by the acceleration of ultrarelativistic (i. ... Stereoisomerism is the arrangement of atoms in molecules whose connectivity remains the same but their arrangement in space is different in each isomer. ...


Incoherent radiation

In nature, electromagnetic radiation is often produced by a large number of individual radiators, producing waves independently of each other. This type of light is described as incoherent. In general there is no single frequency but rather a spectrum of different frequencies present, and even if filtered to an arbitrarily narrow frequency range, there may not be a consistent state of polarization. However, this does not mean that polarization is only a feature of coherent radiation. Incoherent radiation may show statistical correlation between the components of the electric field, which can be interpreted as partial polarization. In general it is possible to describe an observed wave field as the sum of a completely incoherent part (no correlations) and a completely polarized part. One may then describe the light in terms of the degree of polarization, and the parameters of the polarization ellipse. Coherence is the property of wave-like states that enables them to exhibit interference. ... In most modern usages of the word spectrum, there is a unifying theme of between extremes at either end. ... In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables. ...


Parameterizing polarization

For ease of visualization, polarization states are often specified in terms of the polarization ellipse, specifically its orientation and elongation. A common parameterization uses the azimuth angle, ψ (the angle between the major semi-axis of the ellipse and the x-axis) and the ellipticity, ε (the ratio of the two semi-axes). An ellipticity of zero corresponds to linear polarization and an ellipticity of 1 corresponds to circular polarization. The arctangent of the ellipticity, χ = tan−1 ε (the "ellipticity angle"), is also commonly used. An example is shown in the diagram to the right. An alternative to the ellipticity or ellipticity angle is the eccentricity, however unlike the azimuth angle and ellipticity angle, the latter has no obvious geometrical interpretation in terms of the Poincaré sphere (see below). Image File history File links Information_icon. ... Rasterized version of Image:Polarization_ellipse. ... (This page refers to eccentricity in mathematics. ...


Full information on a completely polarized state is also provided by the amplitude and phase of oscillations in two components of the electric field vector in the plane of polarization. This representation was used above to show how different states of polarization are possible. The amplitude and phase information can be conveniently represented as a two-dimensional complex vector (the Jones vector): In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In optics one can describe polarization using the Jones calculus, invented by R. C. Jones in 1941. ...

Here a1 and a2 denote the amplitude of the wave in the two components of the electric field vector, while θ1 and θ2 represent the phases. The product of a Jones vector with a complex number of unit modulus gives a different Jones vector representing the same ellipse, and thus the same state of polarization. The physical electric field, as the real part of the Jones vector, would be altered but the polarization state itself is independent of absolute phase. The basis vectors used to represent the Jones vector need not represent linear polarization states (i.e. be real). In general any two orthogonal states can be used, where an orthogonal vector pair is formally defined as one having a zero inner product. A common choice is left and right circular polarizations, for example to model the different propagation of waves in two such components in circularly birefringent media (see below) or signal paths of coherent detectors sensitive to circular polarization. In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... Absolute phase refers to the phase of a waveform relative to some standard (strictly speaking, phase is always relative). ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ... Please refer to Real vs. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

Reflection of a plane wave from a surface perpendicular to the page. The p-components of the waves are in the plane of the page, while the s components are perpendicular to it.
Reflection of a plane wave from a surface perpendicular to the page. The p-components of the waves are in the plane of the page, while the s components are perpendicular to it.

Regardless of whether polarization ellipses are represented using geometric parameters or Jones vectors, implicit in the parameterization is the orientation of the coordinate frame. This permits a degree of freedom, namely rotation about the propagation direction. When considering light that is propagating parallel to the surface of the Earth, the terms "horizontal" and "vertical" polarization are often used, with the former being associated with the first component of the Jones vector, or zero azimuth angle. On the other hand, in astronomy the equatorial coordinate system is generally used instead, with the zero azimuth (or position angle, as it is more commonly called in astronomy to avoid confusion with the horizontal coordinate system) corresponding to due north. Another coordinate system frequently used relates to the plane made by the propagation direction and a vector normal to the plane of a reflecting surface. This is known as the plane of incidence. The rays in this plane are illustrated in the diagram to the right. The components of the electric field parallel and perpendicular to this plane are termed p-like (parallel) and s-like (senkrecht, German for perpendicular). Light with a p-like electric field is said to be p-polarized, pi-polarized, tangential plane polarized, or is said to be a transverse-magnetic (TM) wave. Light with an s-like electric field is s-polarized, also known as sigma-polarized or sagittal plane polarized, or it can be called a transverse-electric (TE) wave. Rasterized version of Image:Reflection_Polarization. ... A giant Hubble mosaic of the Crab Nebula, a supernova remnant Astronomy is the science of celestial objects (such as stars, planets, comets, and galaxies) and phenomena that originate outside the Earths atmosphere (such as auroras and cosmic background radiation). ... The equatorial coordinate system is probably the most widely used celestial coordinate system, whose equatorial coordinates are: declination () right ascension () -also RA-, or hour angle () -also HA- It is the most closely related to the geographic coordinate system, because they use the same fundamental plane, and the same poles. ... The horizontal coordinate system is a celestial coordinate system that uses the observers local horizon as the fundamental plane. ...


In the case of partially polarized radiation, the Jones vector varies in time and space in a way that differs from the constant rate of phase rotation of monochromatic, purely polarized waves. In this case, the wave field is likely stochastic, and only statistical information can be gathered about the variations and correlations between components of the electric field. This information is embodied in the coherency matrix: Stochastic, from the Greek stochos or goal, means of, relating to, or characterized by conjecture; conjectural; random. ... For Wikipedia statistics, see m:Statistics Statistics is the science and practice of developing human knowledge through the use of empirical data expressed in quantitative form. ... For the square matrix section, see square matrix. ...

where angular brackets denote averaging over many wave cycles. Several variants of the coherency matrix have been proposed: the Wiener coherency matrix and the spectral coherency matrix of Richard Barakat measure the coherence of a spectral decomposition of the signal, while the Wolf coherency matrix averages over all time/frequencies. Norbert Wiener Norbert Wiener (November 26, 1894 - March 18, 1964) was a U.S. mathematician and applied mathematician, especially in the field of electronics engineering. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ... Emil Wolf (July 30, 1922- ) made advancements in Physical Optics, including diffraction, coherence properties of optical fields, spectroscopy of partially coherent radiation, and the theory of direct scattering and inverse scattering. ...


The coherency matrix contains all of the information on polarization that is obtainable using second order statistics. It can be decomposed into the sum of two idempotent matrices, corresponding to the eigenvectors of the coherency matrix, each representing a polarization state that is orthogonal to the other. An alternative decomposition is into completely polarized (zero determinant) and unpolarized (scaled identity matrix) components. In either case, the operation of summing the components corresponds to the incoherent superposition of waves from the two components. The latter case gives rise to the concept of the "degree of polarization", i.e. the fraction of the total intensity contributed by the completely polarized component. In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...


The coherency matrix is not easy to visualize, and it is therefore common to describe incoherent or partially polarized radiation in terms of its total intensity (I), (fractional) degree of polarization (p), and the shape parameters of the polarization ellipse. An alternative and mathematically convenient description is given by the Stokes parameters, introduced by George Gabriel Stokes in 1852. The relationship of the Stokes parameters to intensity and polarization ellipse parameters is shown in the equations and figure below. It is common to describe incoherent or partially polarized radiation in terms of its total intensity (I), (fractional) degree of polarization (p), and the shape parameters of the polarization ellipse. ... Sir George Gabriel Stokes, 1st Baronet (13 August 1819–1 February 1903) was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics (including the Navier-Stokes equations), optics, and mathematical physics (including Stokes theorem). ... 1852 was a leap year starting on Thursday (see link for calendar). ...

Here Ip, 2ψ and 2χ are the spherical coordinates of the polarization state in the three-dimensional space of the last three Stokes parameters. Note the factors of two before ψ and χ corresponding respectively to the facts that any polarization ellipse is indistinguishable from one rotated by 180°, or one with the semi-axis lengths swapped accompanied by a 90° rotation. The Stokes parameters are sometimes denoted I, Q, U and V. Rasterized version of Image:Poincare_sphere. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...


The Stokes parameters contain all of the information of the coherency matrix, and are related to it linearly by means of the identity matrix plus the three Pauli matrices: The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. ...

Mathematically, the factor of two relating physical angles to their counterparts in Stokes space derives from the use of second-order moments and correlations, and incorporates the loss of information due to absolute phase invariance.


The figure above makes use of a convenient representation of the last three Stokes parameters as components in a three-dimensional vector space. This space is closely related to the Poincaré sphere, which is the spherical surface occupied by completely polarized states in the space of the vector

All four Stokes parameters can also be combined into the four-dimensional Stokes vector, which can be interpreted as four-vectors of Minkowski space. In this case, all physically realizable polarization states correspond to time-like, future-directed vectors. It has been suggested that this article or section be merged with Stokes parameters. ... In relativity, a four-vector is a vector in a four-dimensional real vector space, whose components transform like the space and time coordinates (ct, x, y, z) under spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...


Propagation, reflection and scattering

In a vacuum, the components of the electric field propagate at the speed of light, so that the phase of the wave varies in space in time while the polarization state does not. That is: Image File history File links Information_icon. ... Look up Vacuum in Wiktionary, the free dictionary. ... A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation...

where k is the wavenumber and positive z is the direction of propagation. As noted above, the physical electric vector is the real part of the Jones vector. When electromagnetic waves interact with matter, their propagation is altered. If this depends on the polarization states of the waves, then their polarization may also be altered. Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters (m−1). ...


In many types of media, electromagnetic waves may be decomposed into two orthogonal components that encounter different propagation effects. A similar situation occurs in the signal processing paths of detection systems that record the electric field directly. Such effects are most easily characterized in the form of a complex 2×2 transformation matrix called the Jones matrix: In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In optics one can describe polarization using the Jones calculus, invented by R. C. Jones in 1941. ...

In general the Jones matrix of a medium depends on the frequency of the waves.


For propagation effects in two orthogonal modes, the Jones matrix can be written as:

where g1 and g2 are complex numbers representing the change in amplitude and phase caused in each of the two propagation modes, and T is a unitary matrix representing a change of basis from these propagation modes to the linear system used for the Jones vectors. For those media in which the amplitudes are unchanged but a differential phase delay occurs, the Jones matrix is unitary, while those affecting amplitude without phase have Hermitian Jones matrices. In fact, since any matrix may be written as the product of unitary and positive Hermitian matrices, any sequence of linear propagation effects, no matter how complex, can be written as the product of these two basic types of transformations. In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

Paths taken by vectors in the Poincaré sphere under birefringence. The propagation modes (=rotation axes) are shown with red, blue and yellow lines, the initial vectors by thick black lines, and the paths they take by colored ellipses (which represent circles in three dimensions). Rasterized version of Image:Birefringence. ... In algebraic topology, a homology sphere is a topological space X having the homology groups of an n-sphere, for some integer n ≥ 1. ...

Media in which the two modes accrue a differential delay are called birefringent. Well known manifestations of this effect appear in optical wave plates/retarders (linear modes) and in Faraday rotation/optical rotation (circular modes). An easily visualized example is one where the propagation modes are linear, and the incoming radiation is linearly polarized at a 45° angle to the modes. As the phase difference starts to appear, the polarization becomes elliptical, eventually changing to purely circular polarization (90° phase difference), then to elliptical and eventually linear polarization (180° phase) with an azimuth angle perpendicular to the original direction, then through circular again (270° phase), then elliptical with the original azimuth angle, and finally back to the original linearly polarized state (360° phase) where the cycle begins anew. In general the situation is more complicated and can be characterized as a rotation in the Poincaré sphere about the axis defined by the propagation modes (this is a consequence of the isomorphism of SU(2) with SO(3)). Examples for linear (blue), circular (red) and elliptical (yellow) birefringence are shown in the figure on the left. The total intensity and degree of polarization are unaffected. If the path length in the birefringent medium is sufficient, plane waves will exit the material with a significantly different propagation direction, due to refraction. For example, this is the case with macroscopic crystals of calcite, which present the viewer with two offset, orthogonally polarized images of whatever is viewed through them. It was this effect that provided the first discovery of polarization, by Erasmus Bartholinus in 1669. In addition, the phase shift, and thus the change in polarization state, is usually frequency dependent, which, in combination with dichroism, often gives rise to bright colors and rainbow-like effects. A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on... A wave plate is an optical device that alters the polarization state of a light wave travelling through it. ... In physics, the Faraday effect or Faraday rotation is an interaction between light and a magnetic field. ... When polarized light is passed through a substance containing chiral molecules (or nonchiral molecules arranged asymmetrically), the direction of polarization can be changed. ... In linear algebra and geometry, a coordinate rotation is a type of transformation from one system of coordinates to another system of coordinates such that distance between any two points remains invariant under the transformation. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... The straw seems to be broken, due to refraction of light as it emerges into the air. ... Quartz crystal Copper(II) sulfate and iodine crystal Synthetic bismuth crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid... Doubly refracting Calcite from Iceberg claim, Dixon, New Mexico. ... Rasmus Bartholin (Latinized Erasmus Bartholinus; August 13, 1625 - November 4, 1698) was a Danish scientist and physician. ... // Events Samuel Pepys stopped writing his diary. ...


Media in which the amplitude of waves propagating in one of the modes is reduced are called dichroic. Devices that block nearly all of the radiation in one mode are known as polarizing filters or simply "polarizers". In terms of the Stokes parameters, the total intensity is reduced while vectors in the Poincaré sphere are "dragged" towards the direction of the favored mode. Mathematically, under the treatment of the Stokes parameters as a Minkowski 4-vector, the transformation is a scaled Lorentz boost (due to the isomorphism of SL(2,C) and the restricted Lorentz group, SO(3,1)). Just as the Lorentz transformation preserves the proper time, the quantity det Ψ = S02-S12-S22-S32 is invariant within a multiplicative scalar constant under Jones matrix transformations (dichroic and/or birefringent). In optics, the term dichroic has two related but distinct meanings. ... A polarizer is a device that converts an unpolarized or mixed-polarization beam of electromagnetic waves (e. ... The Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, which has been introduced to remove contradictions between the theories of electromagnetism and classical mechanics. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. ...


In birefringent and dichroic media, in addition to writing a Jones matrix for the net effect of passing through a particular path in a given medium, the evolution of the polarization state along that path can be characterized as the (matrix) product of an infinite series of infinitesimal steps, each operating on the state produced by all earlier matrices. In a uniform medium each step is the same, and one may write

where J is an overall (real) gain/loss factor. Here D is a traceless matrix such that αDe gives the derivative of e with respect to z. If D is Hermitian the effect is dichroism, while a unitary matrix models birefringence. The matrix D can be expressed as a linear combination of the Pauli matrices, where real coefficients give Hermitian matrices and imaginary coefficients give unitary matrices. The Jones matrix in each case may therefore be written with the convenient construction: In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...

where σ is a 3-vector composed of the Pauli matrices (used here as generators for the Lie group SL(2,C)) and n and m are real 3-vectors on the Poincaré sphere corresponding to one of the propagation modes of the medium. The effects in that space correspond to a Lorentz boost of velocity parameter 2β along the given direction, or a rotation of angle 2φ about the given axis. These transformations may also be written as biquaternions (quaternions with complex elements), where the elements are related to the Jones matrix in the same way that the Stokes parameters are related to the coherency matrix. They may then be applied in pre- and post-multiplication to the quaternion representation of the coherency matrix, with the usual exploitation of the quaternion exponential for performing rotations and boosts taking a form equivalent to the matrix exponential equations above (See: Quaternion rotation). In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In mathematics, a biquaternion (or complex quaternion) is an element of the (unique) quaternion algebra over the complex numbers. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... Quaternions are used in computer graphics and related fields because they allow for compact representations of rotations in 3D space. ...


In addition to birefringence and dichroism in extended media, polarization effects describable using Jones matrices can also occur at (reflective) interface between two materials of different refractive index. These effects are treated by the Fresnel equations. Part of the wave is transmitted and part is reflected, with the ratio depending on angle of incidence and the angle of refraction. In addition, if the plane of the reflecting surface is not aligned with the plane of propagation of the wave, the polarization of the two parts is altered. In general, the Jones matrices of the reflection and transmission are real and diagonal, making the effect similar to that of a simple linear polarizer. For unpolarized light striking a surface at a certain optimum angle of incidence known as Brewster's angle, the reflected wave will be completely s-polarized. The refractive index (or index of refraction) of a material is the factor by which the phase velocity of electromagnetic radiation is slowed in that material, relative to its velocity in a vacuum. ... The Fresnel equations, deduced by Augustin-Jean Fresnel, describe the behaviour of light when moving between media of differing refractive indices. ... In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ... An illustration of the polarization of light which is incident on an interface at Brewsters angle. ...


Certain effects do not produce linear transformations of the Jones vector, and thus cannot be described with (constant) Jones matrices. For these cases it is usual instead to use a 4×4 matrix that acts upon the Stokes 4-vector. Such matrices were first used by Paul Soleillet in 1929, although they have come to be known as Mueller matrices. While every Jones matrix has a Mueller matrix, the reverse is not true. Mueller matrices are frequently used to study the effects of the scattering of waves from complex surfaces or ensembles of particles. 1929 (MCMXXIX) was a common year starting on Tuesday (link will take you to calendar). ... Mueller calculus is a matrix method for manipulating Stokes vectors, which represent the polarization of incoherent light. ... In particle physics, scattering is a class of phenomena by which particles are deflected by collisions with other particles. ...


Polarization in nature, science, and technology

Polarization effects in everyday life

Effect of a polarizer on reflection from mud flats. In the picture on the left, the polarizer is rotated to transmit the reflections as well as possible; by rotating the polarizer by 90° (picture on the right) almost all reflected sunlight is blocked.
Effect of a polarizer on reflection from mud flats. In the picture on the left, the polarizer is rotated to transmit the reflections as well as possible; by rotating the polarizer by 90° (picture on the right) almost all reflected sunlight is blocked.

Light reflected by shiny transparent materials is partly or fully polarized, except when the light is normal (perpendicular) to the surface. A polarizing filter, such as a pair of polarizing sunglasses, can be used to observe this by rotating the filter while looking through. At certain angles, the reflected light will be reduced or eliminated. Polarizing filters remove light polarized at 90° to the filter's polarization axis. If two polarizers are placed atop one another at 90° angles to one another, no light passes through. Image File history File linksMetadata Download high resolution version (900x300, 45 KB) Summary Mudflats, showing effect of rotating polarising filter Licensing I, the creator of this work, hereby release it into the public domain. ... Image File history File linksMetadata Download high resolution version (900x300, 45 KB) Summary Mudflats, showing effect of rotating polarising filter Licensing I, the creator of this work, hereby release it into the public domain. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ... This article or section does not adequately cite its references or sources. ...

The effects of a polarizer on the sky in a color photograph. The right picture has the polarizer, the left does not. Some photographic polarizers are called circular polarizers. These actually select a linear polarization state from the scene, but then convert it into circularly-polarized light for the camera.[citation needed]
The effects of a polarizer on the sky in a color photograph. The right picture has the polarizer, the left does not. Some photographic polarizers are called circular polarizers. These actually select a linear polarization state from the scene, but then convert it into circularly-polarized light for the camera.[citation needed]

Polarization by scattering is observed as light passes through the atmosphere. The scattered light produces the brightness and color in clear skies. This partial polarization of scattered light can be used to darken the sky in photographs, increasing the contrast. This effect is easiest to observe at sunset, on the horizon at a 90° angle from the setting sun. Another easily observed effect is the drastic reduction in brightness of images of the sky and clouds reflected from horizontal surfaces, which is the main reason polarizing filters are often used in sunglasses. Also frequently visible through polarizing sunglasses are rainbow-like patterns caused by color-dependent birefringent effects, for example in toughened glass (e.g. car windows) or items made from transparent plastics. The role played by polarization in the operation of liquid crystal displays (LCDs) is also frequently apparent to the wearer of polarizing sunglasses, which may reduce the contrast or even make the display unreadable. ImageMetadata File history File links Download high resolution version (2484x946, 874 KB) The effects of a circular polarizer in a color photograph. ... ImageMetadata File history File links Download high resolution version (2484x946, 874 KB) The effects of a circular polarizer in a color photograph. ... In photography, a filter is a camera accessory consisting of an optical filter that can be inserted in the optical path. ... Layers of Atmosphere - not to scale (NOAA)[2] Earths atmosphere is a layer of gases surrounding the planet Earth and retained by the Earths gravity. ... Rayleigh scattering causing a reddened sky at sunset Rayleigh scattering (named after Lord Rayleigh (RAY-lee)) is the scattering of light, or other electromagnetic radiation, by particles much smaller than the wavelength of the light. ... A composite image showing the terminator dividing night from day, running across Europe and Africa. ... Full featured rainbow in Wrangell-St. ... Glass can be made transparent and flat, or into other shapes and colors as shown in this sphere from the Verrerie of Brehat in Brittany. ... This article or section does not adequately cite its references or sources. ... Reflective twisted nematic liquid crystal display. ...

A view through polarized sunglasses
A view through polarized sunglasses

The photograph at the right was taken through polarizing sunglasses and through the rear window of a car. Light from the sky is reflected by the windshield of the other car at an angle, making it mostly horizontally polarized. The rear window is made of tempered glass. Stress in the glass, left from its heat treatment, causes it to alter the polarization of light passing through it, like a wave plate. Without this effect, the sunglasses would block the horizontally polarized light reflected from the other car's window. The stress in the rear window, however, changes some of the horizontally polarized light into vertically polarized light that can pass through the glasses. As a result, the regular pattern of the heat treatment becomes visible. Image File history File links 06_03_14_IMG_0405_polarization. ... Image File history File links 06_03_14_IMG_0405_polarization. ... The materials definition of a glass is a uniform amorphous solid material, usually produced when a suitably viscous molten material cools very rapidly, thereby not giving enough time for a regular crystal lattice to form. ... A wave plate is an optical device that alters the polarization state of a light wave travelling through it. ...


Biology

Many animals are apparently capable of perceiving the polarization of light, which is generally used for navigational purposes, since the linear polarization of sky light is always perpendicular to the direction of the sun. This ability is very common among the insects, including bees, which use this information to orient their communicative dances. Polarization sensitivity has also been observed in species of octopus, squid, cuttlefish, and mantis shrimp. The rapidly changing, vividly colored skin patterns of cuttlefish, used for communication, also incorporate polarization patterns, and mantis shrimp are known to have polarization selective reflective tissue. Sky polarization was thought to be perceived by pigeons, which was assumed to be one of their aids in homing, but research indicates this is a popular myth.[1] Digimon, the only known animals. ... Orders See taxonomy Insects (Class Insecta) are a major group of arthropods and the most diverse group of animals on the Earth, with over a million described species — more than all other animal groups combined [1]. Insects may be found in nearly all environments on the planet, although only a... Families Andrenidae Apidae Colletidae Halictidae Megachilidae Melittidae Stenotritidae Bee collecting pollen Bees are flying insects, closely related to wasps and ants. ... Honey bees learn and communicate in order to find food sources and for other means. ... Suborders †Pohlsepia (incertae sedis) †Proteroctopus (incertae sedis) †Palaeoctopus (incertae sedis) Cirrina Incirrina Synonyms Octopoida Leach, 1817 The octopus (Greek , eight-legs) is a cephalopod of the order Octopoda that inhabits many diverse regions of the ocean, especially coral reefs. ... Suborders Myopsina Oegopsina Squid are a large, diverse group of marine cephalopods. ... Families Sepiadariidae Sepiidae Cuttlefish are marine animals of the order Sepiida belonging to the Cephalopoda class (which also includes squid, octopuses and nautiluses). ... Superfamilies and families Bathysquillidoidea Bathysquillidae Indosquillidae Erythrosquilloidea Erythrosquillidae Gonodactyloidea Alainosquillidae Eurysquillidae Gonodactylidae Hemisquillidae Odontodactylidae Protosquillidae Pseudosquillidae Takuidae Lysiosquilloidea Coronididae Heterosquillidae Lysiosquillidae Nannosquillidae Tetrasquillidae Squilloidea Harpiosquillidae Squillidae incertae sedis Parasquillidae Sculdidae Mantis shrimp or stomatopods are marine crustaceans belonging to the order Stomatopoda, one part of the class Malacostraca, the largest... Pigeon redirects here. ... // Some research has been performed with the intention of discovering how birds can find their way back from distant places they have never visited before. ...


The naked human eye is weakly sensitive to polarization, without the need for intervening filters. Polarized light creates a very faint pattern near the center of the visual field, called Haidinger's brush. This pattern is very difficult to see, but with practice one can learn to detect polarized light with the naked eye. This article refers to the sight organ. ... Simulated appearance of Haidingers brush for vertically polarized light. ...


Geology

The property of (linear) birefringence is widespread in crystalline minerals, and indeed was pivotal in the initial discovery of polarization. In mineralogy, this property is frequently exploited using polarization microscopes, for the purpose of identifying minerals. See pleochroism. Minerals are natural compounds formed through geological processes. ... Mineralogy is an earth science that involves the chemistry, crystal structure, and physical (including optical) properties of minerals. ... Robert Hookes microscope (1665) - an engineered device used to study living systems. ... Pleochroism is an optical phenomenon where due to double refraction of light by a colored gem or crystal, the light is divided into two paths which are polarized at a 90° angle to each other. ...


Chemistry

Polarization is principally of importance in chemistry due to the circular dichroism and "optical rotation" (circular birefringence) exhibited by optically active (chiral) molecules. It may be measured using a polarimeter. Chemistry - the study of atoms, made of nuclei (center particles) and electrons (outer particles), and the structures they form. ... Circular dichroism (CD), is the differential absorption of left- and right-handed circularly polarized light. ... When polarized light is passed through a substance containing chiral molecules (or nonchiral molecules arranged asymmetrically), the direction of polarization can be changed. ... The term chiral (pronounced ) is used to describe an object which is non-superimposable on its mirror image. ... In science, a molecule is the smallest particle of a pure chemical substance that still retains its chemical composition and properties. ... A polarimeter is a scientific instrument for measuring the rotation of the plane of polarized light as it passes through a sample of a compound which exhibits optical activity. ...


Polarization may also refer to the through-bond (inductive or resonant effect) or through-space influence of a nearby functional group on the electronic properties (e.g. dipole moment) of a covalent bond or atom. The inductive effect is associated with the dipole moment of the compound R-X. If X is at the negative end of a dipole moment, it will draw electrons from the ring and produce -I effect. ... Resonance structures of Benzene Resonance structures are diagrammatic tools used predominately in organic chemistry to symbolize resonant bonds between atoms in molecules. ... This article is about the electromagnetic phenomenon. ... Covalent bonding is a description of chemical bonding that is characterized by the sharing of pairs of electrons between atoms. ...


Astronomy

In many areas of astronomy, the study of polarized electromagnetic radiation from outer space is of great importance. Although not usually a factor in the thermal radiation of stars, polarization is also present in radiation from coherent astronomical sources (e.g. hydroxyl or methanol masers), and incoherent sources such as the large radio lobes in active galaxies, and pulsar radio radiation (which may, it is speculated, sometimes be coherent), and is also imposed upon starlight by scattering from interstellar dust. Apart from providing information on sources of radiation and scattering, polarization also probes the interstellar magnetic field via Faraday rotation. The polarization of the cosmic microwave background is being used to study the physics of the very early universe. Synchrotron radiation is highly polarised. Polarization is an important phenomenon in astronomy. ... A giant Hubble mosaic of the Crab Nebula, a supernova remnant Astronomy is the science of celestial objects (such as stars, planets, comets, and galaxies) and phenomena that originate outside the Earths atmosphere (such as auroras and cosmic background radiation). ... Layers of Atmosphere - not to scale (NOAA)[1] Outer space, also simply called space, refers to the relatively empty regions of the universe outside the atmospheres of celestial bodies. ... Thermal radiation is electromagnetic radiation emitted from the surface of an object which is due to the objects temperature. ... This article is about the astronomical object. ... A Hydrogen RF discharge, the first element inside a Hydrogen Maser (see description below), courtesy NASA/JPL-Caltech. ... Interstellar cloud is the generic name given to accumulations of gas and dust in our galaxy. ... Current (I) flowing through a wire produces a magnetic field () around the wire. ... In physics, the Faraday effect or Faraday rotation is an interaction between light and a magnetic field. ... In cosmology, the cosmic microwave background radiation (most often abbreviated CMB but occasionally CMBR, CBR or MBR, also referred as relic radiation) is a form of electromagnetic radiation discovered in 1965 that fills the entire universe. ... Synchrotron radiation is electromagnetic radiation, similar to cyclotron radiation, but generated by the acceleration of ultrarelativistic (i. ...


Technology

Technological applications of polarization are extremely widespread. Perhaps the most commonly encountered examples are liquid crystal displays and polarized sunglasses. Reflective twisted nematic liquid crystal display. ... This article or section does not adequately cite its references or sources. ...


All radio transmitting and receiving antennas are intrinsically polarized, special use of which is made in radar. Most antennas radiate either horizontal, vertical or circular polarization although eliptical polarization also exists. The electric field or E-plane determines the polarization or orientation of the radio wave. Vertical polarization is most often used when it is desired to radiate a radio signal in all directions such as widely distributed mobile units. AM and FM radio uses vertical polarization. Television uses horizontal polarization. Alternating vertical and horizontal polarization is used on satellite communications (including television satellites), to reduce interference between programs on the same frequency band transmitted from adjacent satellites (one uses vertical, the next horizontal, and so on), allowing for reduced angular separation between the satellites. A Yagi-Uda beam antenna Short Wave Curtain Antenna (Moosbrunn, Austria) A building rooftop supporting numerous dish and sectored mobile telecommunications antennas (Doncaster, Victoria, Australia) An antenna or aerial is an arrangement of aerial electrical conductors designed to transmit or receive radio waves which is a class of electromagnetic waves. ... This long range Radar antenna, known as ALTAIR, is used to detect and track space objects in conjunction with ABM testing at the Ronald Reagan Test Site on the Kwajalein atoll[1]. Radar is a system that uses radio waves to determine and map the location, direction, and/or speed... E-Plane E-plane: For a linearly polarized antenna, the plane containing the electric field vector and the direction of maximum radiation. For a vertically-polarized antenna, the E-plane usually coincides with the vertical/elevation plane. ... U.S. military MILSTAR communications satellite A communications satellite (sometimes abbreviated to comsat) is an artificial satellite stationed in space for the purposes of telecommunications using radio at microwave frequencies. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ...

Strain in Glass
Strain in Glass

In engineering, the relationship between strain and birefringence motivates the use of polarization in characterizing the distribution of stress and strain in prototypes. Electronically controlled birefringent devices are used in combination with polarizing filters as modulators in fiber optics. Polarizing filters are also used in photography. They can deepen the color of a blue sky and eliminate reflections from windows and standing water. Image File history File links Strain_in_Glass. ... Image File history File links Strain_in_Glass. ... Engineering is the design, analysis, and/or construction of works for practical purposes. ... Look up strain in Wiktionary, the free dictionary. ... Stress is the internal distribution of force per unit area that balances and reacts to external loads applied to a body. ... Fiber Optic strands An optical fiber in American English or fibre in British English is a transparent thin fiber for transmitting light. ... Photography is the process of making pictures by means of capturing light on a light-sensitive medium, such as a sensor or film. ...


Sky polarization has been exploited in the "sky compass", which was used in the 1950s when navigating near the poles of the Earth's magnetic field when neither the sun nor stars were visible (e.g. under daytime cloud or twilight). It has been suggested, controversially, that the Vikings exploited a similar device (the "sunstone") in their extensive expeditions across the North Atlantic in the 9th - 11th centuries, before the arrival of the magnetic compass in Europe in the 12th century. Related to the sky compass is the "polar clock", invented by Charles Wheatstone in the late 19th century. // Recovering from World War II and its aftermath, the economic miracle emerged in West Germany and Italy. ... The magnetosphere shields the surface of the Earth from the charged particles of the solar wind. ... The Sun is the star at the center of the Solar System. ... This article is about the astronomical object. ... Cumulonimbus capillatus incus floating over Swifts Creek, Victoria in Australia A cloud is a visible mass of condensed droplets or frozen crystals suspended in the atmosphere above the surface of the Earth or another planetary body. ... Twilight in Denmark, just after sunset Twilight in the midwestern US featuring Venus as a brilliant evening star and the crescent moon Finland - Lapland at midnight in July Twilight in Acapulco with Long time Exposure Twilight is the time before sunrise or after sunset when sunlight scattered in the upper... The term Viking commonly denotes the ship-borne explorers, traders, and warriors of the Norsemen (literally, men from the north) who originated in Scandinavia and raided the coasts of the British Isles, France and other parts of Europe as far east as the Volga River in Russia from the late... Sunstone, a feldspar exhibiting in certain directions a brilliant spangled appearance, which has led to its use as an ornamental stone. ... For other uses, see Atlantic (disambiguation) The Atlantic Ocean is Earths second-largest ocean, covering approximately one-fifth of its surface. ... As a means of recording the passage of time the 9th century was that century that lasted from 801 to 900. ... As a means of recording the passage of time, the 11th century was that century which lasted from 1001 to 1100. ... This article is about the navigational tool. ... (11th century - 12th century - 13th century - other centuries) As a means of recording the passage of time, the 12th century was that century which lasted from 1101 to 1200. ... Charles Wheatstone Sir Charles Wheatstone (February 6, 1802 - October 19, 1875) was the British inventor of many innovations including the English concertina the Stereoscope an early form of microphone the Playfair cipher (named for Lord Playfair, the person who publicized it) He was a major figure in the development of... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ...


Polarization is also used for some 3D movies, in which the images intended for each eye are polarized differently. Special filter glasses ensure that each eye receives only the correct image. Typical stereoscopic projection displays use linear polarization encoding, because it is not very expensive and offers high contrast. In environments where the viewer is moving, such as in simulators, circular polarization is sometimes used. This makes the channel separation insensitive to the viewing orientation. The 3-D effect only works on a silver screen. In film, the term 3-D (or 3D) is used to describe any visual presentation system that attempts to maintain or recreate moving images of the third dimension, the illusion of depth as seen by the viewer. ... Stereo card image modified for crossed eye viewing. ... The term silver screen derives from the type of projection screen used at the start of the motion picture industry and specifically refers to the actual silver (Ag) content embedded in the material (a tightly woven fabric, either natural, such as silk, or a synthetic fiber) that made up the...


Art

Several visual artists have worked with polarized light and birefringent materials to create colorful, sometimes changing images. Most notable is contemporary artist Austine Wood Comarow, whose "Polage" works have been exhibited at the Museum of Science, Boston, the Cité des Sciences et de l'Industrie in Paris. A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the division of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on... Outside the Museum of Science, August 2005 The Museum of Science (MoS) is a Boston, Massachusetts landmark, located in Science Park, a plot of land spanning the Charles River. ... La cité des sciences, Photo by Eric Pouhier Cité des Sciences et de lIndustrie ( ) is the biggest science museum in Europe. ...


Other examples of polarization

  • Shear waves in elastic materials exhibit polarization. These effects are studied as part of the field of seismology, where horizontal and vertical polarizations are termed SH and SV, respectively.

A type of seismic wave, the S-wave, secondary wave, or shear wave, sometimes called an elastic S-wave, is one of the two main types of elastic body waves, so named because they move through the body of an object, unlike surface waves. ... In solid mechanics, elasticity is the property of materials which undergo reversible deformations under applied loads. ... Seismology (from the Greek seismos = earthquake and logos = word) is the scientific study of earthquakes and the propagation of elastic waves through the Earth. ...

See also

In biology, antenna (plural: antennae) refers to the sensing organs of several arthropods. ... A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on... Circular dichroism (CD), is the differential absorption of left- and right-handed circularly polarized light. ... Electromagnetic radiation can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... The E-plane and H-plane are reference planes for linearly polarized antennas. ... The Fresnel equations, deduced by Augustin-Jean Fresnel, describe the behaviour of light when moving between media of differing refractive indices. ... A Nicol prism A Nicol prism is an optical device used to generate a beam of polarized light. ... Table of Opticks, 1728 Cyclopaedia Optics ( appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ... Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. ... Sky Digital mini-dish Astros mini-dish. Special dish for up to 16 satellite positions (Ku-band) Satellite dish antenna for C-Band Satellite Dishes installed on a apartment complex A satellite dish is a type of parabolic antenna designed with the specific purpose of transmitting signals to and...

Notes and references

  • Principles of Optics, 7th edition, M. Born & E. Wolf, Cambridge University, 1999, ISBN 0-521-64222-1.
  • Fundamentals of polarized light : a statistical optics approach, C. Brosseau, Wiley, 1998, ISBN 0-471-14302-2.
  • Field Guide to Polarization, Edward Collett, SPIE Field Guides vol. FG05, SPIE, 2005, ISBN 0-8194-5868-6.
  • Polarization Optics in Telecommunications, Jay N. Damask, Springer 2004, ISBN 0-387-22493-9.
  • Optics, 4th edition, Eugene Hecht, Addison Wesley 2002, ISBN 0-8053-8566-5.
  • Polarized Light in Nature, G. P. Können, Translated by G. A. Beerling, Cambridge University, 1985, ISBN 0-521-25862-6.
  • Polarised Light in Science and Nature, D. Pye, Institute of Physics, 2001, ISBN 0-7503-0673-4.
  • Polarized Light, Production and Use, William A. Shurcliff, Harvard University, 1962.
  • Secrets of the Viking Navigators—How the Vikings used their amazing sunstones and other techniques to cross the open oceans, Leif Karlsen, One Earth Press, 2003.
  1. ^ "No evidence for polarization sensitivity in the pigeon electroretinogram", J. J. Vos Hzn, M. A. J. M. Coemans & J. F. W. Nuboer, The Journal of Experimental Biology, 1995.

External links


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Polarization (2351 words)
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Polarization of light by use of a Polaroid filter was is often demonstrated in a Physics class through a variety of demonstrations.
The polarized light consists of waves vibrate in a plane which is ____________ (parallel, perpendicular) to the reflecting surface.
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