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. 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. If D is Hermitian the effect is dichroism, while a unitary matrix models birefringence.
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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. 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 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. Certain effects do not produce linear transformations of the Jones vector, and thus cannot be described with constant Jones matrices. Such matrices were first used by Paul Soleillet in , 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.
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.
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.
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. The role played by polarization in the operation of liquid crystal displays LCD's is also frequently apparent to the wearer of polarizing sunglasses, which may reduce the contrast or even make the display unreadable.
The photograph on 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.
Fundamentals of Polarized Light: A Statistical Optics Approach
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. 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. 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. 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. 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. 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. 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. 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 inherently polarised. Technological applications of polarization are extremely widespread. Perhaps the most commonly encountered examples are liquid crystal displays and polarized sunglasses.
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 elliptical 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. Third Edition. This is the third edition of Modern Optical Engineering. The first edition appeared in ; the second in Strictly by coincidence, this third General Principles.
This book deals with certain phenomena associated with a relatively narrow slice of the electromagnetic Dec 29, The superposition principle in quantum physics is not only valid for all orders of interference with photons in classical It has. Wolf, Principles of Optics 7th ed. Cambridge University Press, Cambridge, Cam- bridge University Press. Buchdahl H. Buchdahl, An Introduction to Hamiltonian Optics. University Press.
Catalog Record: Fundamentals of polarized light : a | HathiTrust Digital Library
Within the Heisenberg's uncertainty principle it is explicitly discussed the impact of these inequalities on the theory of Born, E. Wolf, Principles of Optics, 7th ed. Wilkinson, T. Fromhold, R. Taylor, and A. Jul 21, Allen, M. Beijersbergen, R. Spreeuw, and J. Wo- erdman, Phys. A 45, Milione, S. Evans, D. Nolan, and R. Alfano, Phys. OCIS codes: Leipzig 25, —. However, this does not mean that polarization is only a feature of coherent radiation.
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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. For ease of visualization, polarization states are often specified in terms of the polarization ellipse, specifically its orientation and elongation.
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An ellipticity of zero corresponds to linear polarization and an ellipticity of 1 corresponds to circular polarization. An example is shown in the diagram to the right. 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 :.
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.
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