# Polarization of light: Wikis

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Polarization (also polarisation) is a property of certain types of waves that describes the orientation of their oscillations. Electromagnetic waves such as light exhibit polarization; acoustic waves (sound waves) in a gas or liquid do not have polarization because the direction of vibration and direction of propagation are the same.

By convention, the polarization of light is described by specifying the orientation of the wave's electric field at a point in space over one period of the oscillation. When light travels in free space, in most cases it propagates as a transverse wave—the polarization is perpendicular to the wave's direction of travel. In this case, the electric field may be oriented in a single direction (linear polarization), or it may rotate as the wave travels (circular or elliptical polarization). In the latter cases, the oscillations can rotate rightward or leftward in the direction of travel, and which of those two rotations is present in a wave is called the wave's chirality or handedness. In general the polarization of an electromagnetic (EM) wave is a complex issue. For instance in a waveguide such as an optical fiber, or for radially polarized beams in free space,[1] the description of the wave's polarization is more complicated, as the fields can have longitudinal as well as transverse components. Such EM waves are either TM or hybrid modes.

For longitudinal waves such as sound waves in fluids, the direction of oscillation is by definition along the direction of travel, so there is no polarization. In a solid medium, however, sound waves can be transverse. In this case, the polarization is associated with the direction of the shear stress in the plane perpendicular to the propagation direction. This is important in seismology.

Polarization is significant in areas of science and technology dealing with wave propagation, such as optics, seismology, telecommunications and radar science. The polarization of light can be measured with a polarimeter.

## Theory

### Basics: plane waves

The simplest manifestation of polarization to visualize is that of a plane wave, which is a good approximation of most light waves (a plane wave is a wave with infinitely long and wide wavefronts). For plane waves Maxwell's equations, specifically Gauss's laws, impose the transversality requirement that the electric and magnetic field be perpendicular to the direction of propagation and to each other. 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 of a plane wave may be arbitrarily divided into two perpendicular components labeled 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 in time, 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. Mathematically, the electric field of a plane wave can be written as,

$\vec{E}(\vec{r},t) = \mathrm{Re} \left[\left(A_{x}, A_{y}\cdot e^{i\phi}, 0 \right) e^{i(kz - \omega t)} \right]$

or alternatively,

$\vec{E}(\vec{r},t) = (A_{x}\cdot \cos(kz - \omega t), A_{y}\cdot \cos(kz - \omega t + \phi), 0)$

where Ax and Ay are the amplitudes of the x and y directions and φ is the relative phase between the two components.

### Polarization state

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), at a particular point in space, 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): The same evolution would occur when looking at the electric field at a particular time while evolving the point in space, along the direction opposite to propagation.

Linear
Circular
Elliptical

In the leftmost figure above, 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 the middle figure, 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 and the chosen convention.

Another case is when the two components are not in phase and either do not have the same amplitude or are not ninety degrees out of phase, though their phase offset and their amplitude ratio are constant.[2] This kind of polarization is called elliptical polarization because the electric vector traces out an ellipse in the plane (the polarization ellipse). This is shown in the above figure on the right.

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 any two orthogonally polarized waves, for instance waves of opposite circular polarization. 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.

Though this section discusses polarization for idealized plane waves, all the above is a very accurate description for most practical optical experiments which use TEM modes, including Gaussian optics.

### Unpolarized light

Most sources of electromagnetic radiation contain a large number of atoms or molecules that emit light. The orientation of the electric fields produced by these emitters may not be correlated, in which case the light is said to be unpolarized. If there is partial correlation between the emitters, the light is partially polarized. If the polarization is consistent across the spectrum of the source, partially polarized light can be described as a superposition of a completely unpolarized component, and a completely polarized one. One may then describe the light in terms of the degree of polarization, and the parameters of the polarization ellipse.

### Parameterization

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 tilt angle or azimuth angle, ψ (the angle between the major semi-axis of the ellipse and the x-axis) and the ellipticity, ε (the major-to-minor-axis ratio), also known as the axial ratio.[3][4][5][6] An ellipticity of zero or infinity corresponds to linear polarization and an ellipticity of 1 corresponds to circular polarization. The arccotangent of the ellipticity, χ = arccot 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).

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):

$\mathbf{e} = \begin{bmatrix} a_1 e^{i \theta_1} \\ a_2 e^{i \theta_2} \end{bmatrix} .$

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.

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 component of the electric field parallel to this plane is termed p-like (parallel) and the component perpendicular to this plane is termed s-like (from 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.

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:

$\mathbf{\Psi} = \left\langle\mathbf{e} \mathbf{e}^\dagger \right\rangle\,$
$=\left\langle\begin{bmatrix} e_1 e_1^* & e_1 e_2^* \ e_2 e_1^* & e_2 e_2^* \end{bmatrix} \right\rangle$
$=\left\langle\begin{bmatrix} a_1^2 & a_1 a_2 e^{i (\theta_1-\theta_2)} \ a_1 a_2 e^{-i (\theta_1-\theta_2)}& a_2^2 \end{bmatrix} \right\rangle$

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.

The coherency matrix contains all second order statistical information about the polarization. This matrix 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.

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.

Poincaré sphere diagram
$S_0 = I \,$
$S_1 = I p \cos 2\psi \cos 2\chi\,$
$S_2 = I p \sin 2\psi \cos 2\chi\,$
$S_3 = I p \sin 2\chi\,$

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.

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:

$\mathbf{\Psi} = \frac{1}{2}\sum_{j=0}^3 S_j \mathbf{\sigma}_j,\;\mbox{where}$
$\begin{matrix} \mathbf{\sigma}_0 &=& \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} & \mathbf{\sigma}_1 &=& \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \ \ \mathbf{\sigma}_2 &=& \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} & \mathbf{\sigma}_3 &=& \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \end{matrix}$

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

$\mathbf{u} = \frac{1}{S_0}\begin{bmatrix} S_1\\S_2\\S_3\end{bmatrix}.$

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.

### 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 and time while the polarization state does not. That is,

$\mathbf{e}(z+\Delta z,t+\Delta t) = \mathbf{e}(z, t) e^{i k (c\Delta t - \Delta z)},$

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.

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:

$\mathbf{e'} = \mathbf{J}\mathbf{e}.$

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

$\mathbf{J} = \mathbf{T} \begin{bmatrix} g_1 & 0 \\ 0 & g_2 \end{bmatrix} \mathbf{T}^{-1},$

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.

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).

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.

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 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

$\mathbf{J} = Je^{\mathbf{D}},$

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

$\begin{matrix} \mathbf{J_b} &=& J_be^{\beta \mathbf{\sigma}\cdot\mathbf{\hat{n}}} & \mbox{and} & \mathbf{J_r} &=& J_re^{\phi i\mathbf{\sigma}\cdot\mathbf{\hat{m}}}, \end{matrix}$

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 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. 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.

## 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 specularly reflected sunlight is blocked.
The effects of a polarizing filter on the sky in a photograph. The picture on the right uses the filter.

Light reflected by shiny transparent materials is partly or fully polarized, except when the light is normal (perpendicular) to the surface. It was through this effect that polarization was first discovered in 1808 by the mathematician Etienne Louis Malus. A polarizing filter, such as a pair of polarizing sunglasses, can be used to observe this effect by rotating the filter while looking through it at the reflection off of a distant horizontal surface. At certain rotation 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, there is minimal light transmission.

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 (see Brewster's angle), 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.

Polarizing sunglasses reveal stress in car window (see text for explanation.)

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. 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.

### Biology

Many animals are apparently capable of perceiving some of the components of the polarization of light, e.g. linear horizontally-polarized light. This 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. In the latter case, one species measures all six orthogonal components of polarization, and is believed to have optimal polarization vision.[7] 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.[8]

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.

### Geology

Photomicrograph of a volcanic sand grain; upper picture is plane-polarized light, bottom picture is cross-polarized light, scale box at left-center is 0.25 millimeter.

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 optical mineralogy for more details.

### Chemistry

Polarization is principally of importance in chemistry due to circular dichroism and "optical rotation" (circular birefringence) exhibited by optically active (chiral) molecules. It may be measured using polarimetry.

The term "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. This concept is based on the formation of an electric dipole within a molecule, which is generally not related to the polarization of electromagnetic waves.

Polarized light does interact with anisotropic materials, which is the basis for birefringence. This is usually seen in crystalline materials and is especially useful in geology (see above). The polarized light is "double refracted", as the refractive index is different for horizontally and vertically polarized light in these materials. This is to say, the polarizability of anisotropic materials is not equivalent in all directions. This anisotropy causes changes in the polarization of the incident beam, and is easily observable using cross-polar microscopy or polarimetry. The optical rotation of chiral compounds (as opposed to achiral compounds that form anisotropic crystals), is derived from circular birefringence. Like linear birefringence described above, circular birefringence is the "double refraction" of circular polarized light.[9]

### 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 inherently polarised.

### 3D movies

Polarization is also used for some 3D movies, in which the images intended for each eye are either projected from two different projectors with orthogonally oriented polarizing filters or, more typically, from a single projector with time multiplexed polarization (a fast alternating polarization device for successive frames). Polarized 3D glasses with suitable polarized filters ensure that each eye receives only the intended image. Historical stereoscopic projection displays used linear polarization encoding because it was inexpensive and offered good separation. Circular polarization makes left-eye/right-eye separation insensitive to the viewing orientation; circular polarization is used in typical 3-D movie exhibition today, such as the system from RealD. Polarized 3-D only works on screens that maintain polarization (such as silver screens); a normal projection screen would cause depolarization which would void the effect.

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 use vertical polarization, while television uses horizontal polarization. Alternating vertical and horizontal polarization is used on satellite communications (including television satellites), to allow the satellite to carry two separate transmissions on a given frequency, thus doubling the number of customers a single satellite can serve. Electronically controlled birefringent devices are used in combination with polarizing filters as modulators in fiber optics.

### Materials science

Strain in plastic glasses

In engineering, the relationship between strain and birefringence motivates the use of polarization in characterizing the distribution of stress and strain in prototypes.

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.

### Photography

In photography, polarizing filters are used, mostly to improve the appearance of the sky (deeper blue, and clouds more visible):

A polarizer filters out the polarized component of light from the sky in a color photograph, increasing contrast with the clouds (right).

### Art

Several visual artists have worked with polarized light and birefringent materials to create colorful, sometimes changing images. One example is contemporary artist Austine Wood Comarow, whose "Polage" art works have been exhibited at the Museum of Science, Boston,[1] the New Mexico Museum of Natural History and Science in Albuquerque, NM, and the Cité des Sciences et de l'Industrie (the City of Science and Industry) in Paris.[10][11] The artist works by cutting hundreds of small pieces of cellophane and other birefringent films and laminating them between plane polarizing filters.

## 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.

## 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.
• Polarized Light, second edition, Dennis Goldstein, Marcel Dekker, 2003, ISBN 0-8247-4053-X
• 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.
• Ellipsometry and Polarized Light, R. M. A. Azzam and N. M. Bashara, North-Holland, 1977, ISBN 0-444-87016-4
• 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. ^ Dorn, R. and Quabis, S. and Leuchs, G. (dec 2003). "Sharper Focus for a Radially Polarized Light Beam". Physical Review Letters 91 (23,): 233901-+.
2. ^ Subrahmanyan Chandrasekhar (1960) Radiative transfer, p.27
3. ^ Merrill Ivan Skolnik (1990) Radar Handbook, Fig. 6.52, sec. 6.60.
4. ^ Hamish Meikle (2001) Modern Radar Systems, eq. 5.83.
5. ^ T. Koryu Ishii (Editor), 1995, Handbook of Microwave Technology. Volume 2, Applications, p. 177.
6. ^ John Volakis (ed) 2007 Antenna Engineering Handbook, Fourth Edition, sec. 26.1. Note: in contrast with other authors, this source initially defines ellipticity reciprocally, as the minor-to-major-axis ratio, but then goes on to say that "Although [it] is less than unity, when expressing ellipticity in decibels, the minus sign is frequently omitted for convenience", which essentially reverts back to the definition adopted by other authors.
7. ^ Sonja Kleinlogel, Andrew White (2008). "The secret world of shrimps: polarisation vision at its best". PLoS ONE 3: e2190. doi:10.1371/journal.pone.0002190.
8. ^ "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.
9. ^ Hecht, Eugene (1998). Optics (3rd ed.). Reading, MA: Addison Wesley Longman.
10. ^ "Austine Studios Polarized Light Art". Retrieved January 31, 2009.
11. ^ Mann, James (2005). Austine Wood Comarow: Paintings in Polarized Light. Las Vegas, NV: Wasabi Pub.. ISBN 0-9768198-0-5.

# 1911 encyclopedia

Up to date as of January 14, 2010

### From LoveToKnow 1911

POLARIZATION OF LIGHT. A stream of light coming directly from a natural source has no relation to space except that concerned in its direction of propagation, round which its properties are alike on all sides. That this is not a necessary characteristic of light was discovered by Christian Huygens, who found that, whereas a stream of sunlight in traversing a rhomb of spar in any but one direction always gives rise to two streams of equal brightness, each of these emergent streams is divided by a second rhomb into two portions having a relative intensity dependent upon the position with respect to one another of the principal planes of the faces of entry into the rhombs - the planes through the axes of the crystals perpendicular to the refracting surfaces. In certain cases, indeed, one portion vanishes entirely: thus the stream ordinarily refracted in the first rhomb gives an ordinary or an extraordinary stream alone in the second, according as the principal planes are parallel or perpendicular, the reverse being the case with the extraordinary stream of the first rhomb. In intermediate cases the intensities of the two beams are proportional to the squares of the cosines of the angles that the principal plane of the second rhomb makes with the positions in which they have the greatest intensity.

On the other hand, if the emergent streams overlap and the common part be examined, it is found to have all the properties of common light. To this phenomenon E. T Malus gave the name of polarization, as he attributed it, on the emission theory of light, to a kind of polarity of the light-corpuscles. This term has been retained and the ordinary stream is said to be plane polarized in the principal plane of the face of entry into the rhomb, and the extraordinary stream to be plane polarized in the perpendicular plane.

The phenomenon of polarization observed by Huygens remained an isolated fact for over a century, until Malus in 1808 discovered that polarization can be produced independently of double refraction, and must consequently be something closely connected with the nature of light itself. Examining the light reflected from the windows of the Luxemburg palace with a doubly refracting prism, he was led to infer (though more refined experiments have shown that this is not strictly the case) that light reflected at a certain angle, called the polarizing angle, from the surface of transparent substances has the same properties with respect to the plane of incidence as those of the ordinary stream in Iceland spar with respect to the principal plane of the crystal. Thus in accordance with the definition, it is polarized in the plane of incidence. Further, if polarized light fall at the polarizing angle on a reflecting surface, the intensity of the reflected stream depends upon the azimuth of the plane of incidence, being proportional to the square of the cosine of the angle between this plane and the plane of the polarization. At angles other than the polarizing angle common light gives a reflected stream .that behaves as a mixture of common light with light polarized in the plane of incidence, and is accordingly said to be partially polarized in that plane. The refracted light, whatever be the angle of incidence, is found to be partially polarized in a plane perpendicular to the plane of incidence, and D. F. J. Arago showed that at all angles of incidence the reflected and refracted streams contain equal quantities of polarized light. The polarizing angle varies from one transparent substance to another, and Sir David Brewster in 1815 enunciated the law that the tangent of the polarizing angle is equal to the refractive index of the substance. It follows then that if a stream of light be incident at the polarizing angle on a pile of parallel transparent plates of the same nature, each surface in turn will be met by the light at the polarizing angle and will give rise to a reflected portion polarized in the plane of incidence. Hence the total reflected light will be polarized in this plane and will of necessity have a greater intensity than that produced by a single surface. Fhe polarization of the light transmitted by the pile is never complete, but tends to become more nearly so as the number of the plates is increased and at the same time the angle of incidence for which the polarization is a maximum approaches indefinitely the polarizing angle (Sir. G. G. Stokes, Math. and Phys. Papers, iv. 145).

In order to isolate a polarized pencil of rays with a rhomb of Iceland spar, it is necessary to have a crystal of such a thickness that the emergent streams are separated, so that one may be stopped by a screen. There are, however, certain crystals that with a moderate thickness give an emergent stream of light that is more or less completely polarized. The polarizing action of such crystals is due to the unequal absorption that they exert on polarized streams. Thus a plate of tourmaline of from I mm. to amm. in thickness with its faces perpendicular to the optic axis is nearly opaque to light falling normally upon it, and a plate of this thickness parallel to the axis permits of the passage of a single stream polarized in a plane perpendicular to the principal section. Such a plate acts in the same way on polarized light, stopping it or allowing it to pass, according as the plane of polarization is parallel or perpendicular to the principal section. Certain artificial salts, e.g. iodo-sulphate of quinine, act in a similar manner.

From the above instances we see that an instrumental appliance that polarizes a beam of light may be used as a means of detecting and examining polarization. This latter process is termed analysation, and an instrument is called a polarizer or an analyser according as it is used for the first or the second of these purposes.

In addition to the above facts of polarization mention may be made of the partial polarization, in a plane perpendicular to that of emission, of the light emitted in an oblique direction from a white-hot solid, and of the polarization produced by diffraction. Experiments with gratings have been instituted by Sir G. Gabriel Stokes, C. H. A. Holtzmann, F. Eisenlohr and others, with the view of determining the direction of the vibrations in polarized light (vide infra), but the results have not been consistent, and H. Fizeau and G. H. Quincke have shown that they depend upon the size and form of the apertures and upon the state of the surface on which they are traced. The polarization of the light reflected from a glass grating has also been investigated by I. Frohlich, while L. G. Gouy has studied the more simple case of diffraction at a straight' edge. The polarization of the light scattered by small particles has been examined by G. Govi, J. Tyndall, L. Soret and A. Lallemand, and in the case of ultramicroscopic particles by H. Siedentopf and R. Zsigmondy (Drude Ann. 1903, x. r); an interesting case of this phenomenon is the polarization of the light from the sky - a subject that has been treated theoretically by Lord Rayleigh in an important series of papers (See SKY, Colour Of, and Rayleigh, Scientific Works, i. 87, 104, 518; iv. 397).

An important addition to the knowledge of polarization was made in 1816 by Augustin J. Fresnel and D. F. J. Arago, 'who summed up the results of a searching series of experiments in the following laws of the interference of polarized light: (r) Under the same conditions in which two streams of common light interfere, two streams polarized at right angles are without mutual influence. (2) Two streams polarized in parallel planes give the same phenomena of interference as common light.

(3) Two streams polarized at right angles and coming from a stream of common light can be brought to the same plane of polarization without thereby acquiring the faculty of interfering.

(4) Two streams polarized at right angles and coming from a stream of polarized light interfere as common light, when brought to the same plane of polarization. (5) In calculating the conditions of interference in the last case, it is necessary to add a half wave-length to the actual difference of path of the streams, unless the primitive and final planes of polarization lie in the same angle between the two perpendicular planes. The lateral characteristics of a polarized stream lead at once to the conclusion that the stream may be represented by a vector, and since this vector must indicate the direction in which the light travels as well as the plane of polarization, it is natural to infer that it is transverse to the direction of propagation. That this is actually the case is proved by experiments on the interference of polarized light, from which it may be deduced that the polarization-vector of a train of plane waves of plane polarized light executes rectilinear vibrations in the plane of the waves. By symmetry the polarization-vector must be either parallel or perpendicular to the plane of polarization: which of these directions is assumed depends upon the physical characteristic that is attributed to the vector. In fact, whatever theory of light be adopted, there are two vectors to be considered, that are at right angles to one another and connected by purely geometrical relations.

The general expressions for the rectangular components of a vector transverse to the direction of propagation (z) in the case of waves of length X travelling with speed v are: - u= a cos (T - a), v=b cos (T - (3), where T= 27r(vt - z)/h. The path of the extremity of the vector is then in general an ellipse, traversed in a right-handed direction to an observer receiving the light when a - (3 is between o and 7r, or between - 7r and - air, and in a left-handed direction if this angle be between 7r and 27, or between o and - 7r. In conformity with the form of the path, the light is said to be elliptically polarized, rightor left-handedly as the case may be, and the axes of the elliptic path are determined by the planes of maximum and minimum polarization of the light. In the particular case in which a= b and a - ,(3 = (2n + 1) 7r/2, the vibrations are circular and the light is said to be circularly polarized.

These different types of polarization may be obtained from a plane polarized stream by passing it through a quarter-wave plate, i.e. a crystalline plate of such a thickness that it introduces a relative retardation of a quarter of a wave between the component streams within it. Such plates are generally made of mica or selenite, and the normal to the plane of polarization of the most retarded stream is called "the axis of the plate." If this axis be parallel or perpendicular to the primitive plane of polarization, the emergent beam remains plane polarized; it is circularly polarized if the axis be at 45° to the plane of polarization, and in other cases it is elliptically polarized with the axes of the elliptic path parallel and perpendicular to the axis of the plate. Conversely a quarter-wave plate may be employed for reducing a circularly or elliptically polarized stream to a state of plane polarization.

Two streams are said to be oppositely polarized when the one is, so far as relates to its polarization, what the other becomes when it is turned through an azimuth of 90° and has its character reversed as regards right and left hand. An analytical investigation of the conditions of interference of polarized streams of the most general type leads to the result that there will be no interference only when the two streams are oppositely polarized, and that when the polarizations are identical the interference will be perfect, the fluctuations of intensity being the greatest that the difference of intensity of the streams admits (Sir G. G. Stokes, Math. and Phys. Papers, iii. 233).

It remains to consider the constitution of common unpolarized light. Since a beam of common light can be resolved into plane polarized streams and these on recomposition give a stream with properties indistinguishable from those of common light, whatever their relative retardation may be, it is natural to assume that an analytical representation of common light can be obtained in which no longitudinal vector occurs. On the other hand a stream of strictly monochromatic light with a polarization-vector that is entirely transversal must be (in general elliptically) polarized. Consequently it follows that common light cannot be absolutely monochromatic. The conditions that are necessary in order that a stream of light may behave as natural light have been investigated by Sir G. Gabriel Stokes (loc. cit.) and by E. Verdet (Oeuvres, i. 281), and it may be shown that two polarized streams of a definite character are analytically equivalent to common light provided that they are of equal intensity and oppositely polarized and that there is no common phase relation between the corresponding monochromatic constituents. Further a stream of light of the most general character is equivalent to the admixture of common and polarized light, the polarization being elliptical, circular or plane.

We see then that there are seven possible types of light: common light, polarized light and partially polarized light; the polarization in the two latter cases being elliptical, circular or plane. Common light, circularly polarized and partially circularly polarized light all have the characteristic of giving two streams of equal intensity on passing through a rhomb of Iceland spar, however it may be turned. They may, however, be distinguished by the fact that on previous transmission through a quarter-wave plate this property is retained in the case of common light, while with the two other types the relative intensity of the streams depends upon the orientation of the rhomb, and with circularly polarized light one stream may be made to vanish. Plane polarized light gives in general two streams of unequal intensity when examined with a rhomb, and for certain positions of the crystal there isonly one emergent stream. Elliptically polarized, partially elliptically polarized and partially plane polarized light give with Iceland spar two streams of, in general, unequal intensity, neither of which can be made to vanish. They may be differentiated by first passing the light through a quarter-wave plate with its axis parallel or perpendicular to the plane of maximum polarization: for elliptically polarized light thereby becomes plane polarized and one of the streams is extinguished on rotating the rhomb; but with the other two kinds of light this is not the case, and the light is partially plane or partially elliptically polarized according as the plane of maximum polarization remains the same or is changed.

## Colours of Crystalline Plates

It was known to E. T. Malus that the interposition of a doubly refracting plate between a polarizer and an analyser regulated for extinction has the effect of partially restoring the light, and he used this property to discover double refraction in cases in which the separation of the two refracted streams was too slight to be directly detected. D. F. J. Arago in 1811 found that in the case of white light and with moderately thin plates the transmitted light is no longer white but coloured, a variation of brightness but not of tint being produced when the polarizer and analyser being crossed are rotated together, while the rotation of the analyser alone produces a change of colour, which passes through white into the complementary tint. This phenomenon was subjected to a detailed investigation by Jean Baptiste Biot during the years 1812 to 1814, and from the results of his experiments Thomas Young, with his brilliant acumen, was led to infer that the colours were to be attributed to interference between the ordinary and extraordinary streams in the plate of crystal. This explanation is incomplete, as it leaves out of account the action of the polarizer and analyser, and it was with the purpose of removing this defect that Fresnel and Arago undertook the investigations mentioned above and thus supplied what was wanting in Young's explanation. In Biot's earlier experiments the beam of light employed was nearly parallel: the phenomena of rings and brushes that are seen with a conical pencil of light were discovered by Sir David Brewster in the case of uniaxal crystals in 1813 and in that of biaxal crystals in 1815.

Let a, 0, be the angles that the primitive and final planes of polarization and the plane of polarization of the quicker wave within the plate make with a fixed plane, and let be the relative retardation of phase of the two streams on emergence from the plate for light of period T. On entry into the crystal the original polarized stream is resolved into components represented by a cos(- a) cos T, a sin (1P - a)cos T, T =27rt/r, and on emergence we may take as the expression of the waves cos (p - a) cos T, sin (4, - a) cos (T - p).

Finally after traversing the analyser the sum of the two resolved components is a cos (l i t - a) cos 41 3) cos T +a sin (p - a) sin (>G - a) cos (T - p), of which the intensity is {a cos (1k - a) cos (,f,-0) sin sin (-0) cos a 2 sin 2 (tP - a) sin' (-0) sin 2 a 2 cos 2 (13 - a) - a 2 sin 24 - a) sin 2(>! ' - sine 2p When the primitive light is white, this expression must be summed for the different monochromatic constituents. In strictness the angle is dependent upon the frequency, but if the dispersion be weak relatively to the double refraction, the product sin 24 - a)sin 2Ni - (3) has sensibly the same value for all terms of the summation, and we may write I=cos 2 (1 3 - a)/a 2 - sin 2 (1 ' - a) sin 2 (t ' - a 2 sin 2 2 This formula contains the whole theory of the colours of crystalline plates in polarized light. Since the first term represents a stream of white light, the plate will appear uncoloured whenever the plane of polarization of either stream transmitted by it coincides with either the primitive or final plane of polarization. In intermediate cases the field is coloured, and the tint changes to its complementary as the plate passes through one of these eight positions, since the second term in the above expression then changes sign. If, however, the primitive and final planes of polarization be parallel or crossed, the field exhibits only one colour during a complete revolution of the plate. The crystalline plate shows no colour when it is very thin, and also when its thickness exceeds a moderate amount. In the former case the retardation of phase varies so little with the period that the intensity is nearly the same for all colours; in the latter case it alters so rapidly that for a small change in the period the intensity passes from a maximum or a minimum, and consequently so many constituents of the light are weakened and these are so close to one another in frequency, that the light presents to the eye the appearance of being white. The true character of the light in this case may be revealed by analysing it with a spectroscope, when a spectrum is obtained traversed by dark bands corresponding to the constituents that are weakened or annulled. The phenomenon of colour may, however, be obtained with thick plates by superposing two of them in a suitable manner, the combination acting as a thicker or a thinner plate according as the planes of polarization of the quicker waves within them are parallel or crossed. In this way a delicate test for slight traces of double refraction is obtained. When the retardation of phase for light of mean period is it or a small multiple of it a crystalline plate placed between a crossed polarizer and analyser exhibits in white light a distinctive greyish violet colour, known as a sensitive tint from the fact that it changes rapidly to blue or red, when the retardation is very slightly increased or diminished. If then the sensitive plate be cut in half and the two parts be placed side by side after the one has been turned through 90° in its own plane, the tint of the one half will be raised and that of the other will be lowered when the compound plate is associated with a second doubly refracting plate.

When light from an extended source is made to converge upon the crystal, the phenomenon of rings and brushes localized at infinity is obtained.The exact calculation of the intensity in this case is very complicated and the resulting expression is too unwieldy to be of any use, but as an approximation the formula for the case of a parallel beam may be employed, the quantities and p therein occurring being regarded as functions of the angle and plane of incidence and consequently as variables. In monochromatic light, then, the interference pattern is characterized by three systems of curves: the curves of constant retardation p = const.; the lines of like polarization = const.; the curves of constant intensity I = const. When p = 2nir and also when 4, = a or a-1-7/2 or Ili = 1 3 or 0+7r/2, that is at points for which the streams within the plate are polarized in planes parallel and perpendicular to the planes of primitive and final polarization, the intensity (called the fundamental intensity) is the same as when the plate is removed. These conditions define two systems of curves called respectively the principal curves of constant retardation and the principal lines of like polarization, these latter lines dividing the field into regions in which the intensity is alternately greater and less than the fundamental intensity. When, however, the planes of polarization and analysation are parallel or crossed, the two pairs of principal lines of like polarization coincide, and the intensity is at all points in the former case not greater than, and in the latter case not less than, it was before the introduction of the plate. The determination of the curves of constant retardation depends upon expressing the retardation in terms of the optical constants of the crystal, the angle of incidence and the azimuth of the plane of incidence. P. A. Bertin has shown that a useful picture of the form of these curves may be obtained by taking sections, parallel to the plate, of a surface that he calls the "isochromatic surface," and that is the locus of points on the crystal at which the relative retardation of two plane waves passing simultaneously through a given point and travelling in the same direction has an assigned value. But as this surface is obtained by assuming that the interfering streams follow the same route in the crystal, and by neglecting the refraction out of the crystal, it does not lend itself to accurate numerical calculations. To the same degree of accuracy as that employed in obtaining the expression for the intensity, the form of the lines of like polarization is given by the section, parallel to the plate, of a cone, whose generating lines are the directions of propagation of waves that have their planes of polarization parallel and perpendicular to a given plane: the cone is in general of the third degree and passes through the optic axes of the crystal. We must limit ourselves in this article to indicating the chief features of the phenomenon in the more important cases. (Reference should be made to the article Crystallography for illustrations, and for applications of these phenomena to the determination of crystal form.) With an uniaxal plate perpendicular to the optic axis, the curves of constant retardation are concentric circles and the lines of like polarization are the radii: thus with polarizer and analyser regulated for extinction, the pattern consists of a series of bright and dark circles interrupted by a black cross with its arms parallel to the planes of polarization and analysation. In the case of a biaxal plate perpendicular to the bisector of the acute angle between the optic axes, the curves of constant retardation are approximately Cassini's ovals, and the lines of like polarization are equilateral hyperbolae passing through the points corresponding to the optic axes. With a crossed polarizer and analyser the rings are interrupted by a dark hyperbolic brush that cuts the plane of the optic axes at right angles, if this plane be at 45° to the planes of polarization and analysation - the so-called diagonal position - and that becomes a rectangular cross with its arms parallel and perpendicular to the plane of the optic axes when this plane coincides with the plane of primitive or final polarization - the normal position.

When white light is employed coloured rings are obtained, provided the relative retardation of the interfering streams be not too great. The isochromatic lines, unless the dispersion be excessive, follow in the main the course of the curves of constant retardation, and the principal lines of like polarization are with a crossed polarizer and analyser dark brushes, that in certain cases are fringed with colour. This state of things may, however, be considerably departed from if the axes of optical symmetry of the crystal are different for the various colours. The examination of dispersion of the optic axes in biaxal crystals (see Refraction, § Double) may be conveniently made with a plate perpendicular to the acute bisectrix placed in the diagonal position for light of mean period between a crossed polarizer and analyser. When the rings are coloured symmetrically with respect to two perpendicular lines the acute bisectrix and the plane of the optic axes are the same for all frequencies, and the colour for which the separation of the axes is the least is that on the concave side of the summit of the hyperbolic brushes.

Crossed, inclined and horizontal dispersion are characterized respectively by a distribution of colour that is symmetrical with respect to the centre alone, the plane of the optic axes, and the perpendicular plane.

The phenomenon of interference produced by crystalline plates is considerably modified if the light be circularly or elliptically polarized or analysed by the interposition of a quarter-wave between the crystal and the polarizer or analyser. Thus in the two cases described above the brushes disappear and the rings are continuous when the light is both polarized and analysed circularly. But the most important case, on account of its practical application to determining the sign of a crystal, is that in which the light is plane polarized and circularly analysed or the reverse. Let us suppose that the light is circularly analysed and that the primitive and final planes of polarization are at right angles. Then with an uniaxal plate perpendicular to the optic axis, the black cross is replaced by two lines, on crossing which the rings are discontinuous, expansion or contraction occurring in the quadrants that contain the axis of the quarter-wave plate, according as the crystal is positive or negative. With a biaxal plate perpendicular to the optic axis in the diagonal position, the hyperbolic brush becomes an hyperbolic line and the rings are expanded or contracted on its concave side, with a positive plate, according as the plane of the optic axes is parallel or perpendicular to the axis of the quarter-wave plate, the reverse being the case with a negative plate.

With a combination of plates in plane-polarized and plane-analysed light the interference pattern with monochromatic light is generally very complicated, the dark curves when polarizer and analyser are crossed being replaced by isolated dark spots or segments of lines. When, however, the field is very small, or when the primitive light is white so that interference is only visible for small relative retardations, the problem becomes in many cases one of far less complexity. An instance of considerable importance is afforded by the combination known as Savart's plate. This consists of two plates of an uniaxal crystal of equal thickness, cut at the same inclination of about 45° to the optic axis and superposed with their principal planes at right angles. The interference pattern produced by this combination is, when the field is small, a system of parallel straight lines bisecting the angle between the principal planes of its constituents. These attain their maximum visibility when the plane of analysation is at 45° to these planes, and vanish when the plane of polarization is parallel to either of the principal planes.

The phenomena of chromatic polarization afford a ready means of detecting doubly refracting structure in cases, such as that produced in isotropic bodies by strain, in which its effects are very minute. Thus a bar of glass of sufficient thickness, placed in the diagonal position between a crossed polarizer and analyser and bent in a plane perpendicular to that of vision, exhibits two sets of coloured bands separated by a neutral line, the double refraction being positive on the dilated and negative on the compressed side. Again, a system of rings, similar to those of an uniaxal plate perpendicular to the axis, may be produced with a glass cylinder by transmitting heat from its surface to its axes by immersion in heated oil, and glass that has been raised to a red heat and then cooled rapidly at its edges gives in polarized light an interference pattern of a regular form dependent upon the shape of the contour.

## Rotary Polarization

In general a stream of plane-polarized light undergoes no change in traversing a plate of an uniaxal crystal in the direction of its axis, and when the emergent stream is analysed, the light, if originally white, is found to be colourless and to be extinguished when the polarizer and analyser are crossed. When, however, a plate of quartz is used in this experiment, the light is coloured and is in no case cut off by the analyser, the tint, however, changing as the analyser is rotated. This phenomenon may be explained, as D. F. J. Arago pointed out, by supposing that in passing through the plate the plane of polarization of each monochromatic constituent is rotated by an amount dependent upon the frequency - an explanation that may be at once verified either by using monochromatic light or by analysing the light with a spectroscope, the spectrum in the latter case being traversed by one or more dark bands, according to the thickness of the plate, that pass along the spectrum from end to end as the analyser is rotated. J. B. Biot further ascertained that this rotation of the plane of polarization varies as the distance traversed in the plate and very nearly as the inverse square of the wave-length, and found that with certain specimens of quartz the rotation is in a clockwise or right-handed direction to an observer receiving the light, while in others it is in the opposite direction, and that equal plates of the rightand lefthand varieties neutralize one another's effects.

A similar rotary property is possessed by other uniaxal crystals, such as cinnabar and the thiosulphates of potassium, lead and calcium, and as H. C. Pocklington (Phil. Mag., 1901 [6], ii. 361) and J. H. Dufet (Journ. de phys., 1904 [4], iii. 757) have shown by a few biaxal crystals, such as sugar and Rochelle salt, the rotation produced by a given thickness being in general different, and in some cases of opposite sign for the two optic axes. Further, certain cubic crystals, such as sodium chlorate and bromate, and also some liquids and even vapours, rotate the plane of polarization of the light that traverses them, whatever may be the direction of the stream.

In crystals the rotary property appears to be sometimes inherent in the crystalline arrangement of the molecules, as it is lost on fusion or solution, and in several cases belongs to enantiomorphous crystals, the two correlated forms of which are the one right-handed and the other left-handed optically as well as crystallographically, this being necessarily the case if the property be retained when the crystal is fused or dissolved. In organic bodies the rotary property, as the researches of J. A. Le Bel, J. H. van't Hoff and others have established, corresponds to the presence of one or more asymmetric atoms of carbon - that is, atoms directly united to elements or radicles all different from one another - and in every case there exists an isomer that rotates the plane of polarization to the same degree in the opposite direction. Absence of rotary power when asymmetric carbon atoms are present, may be caused by an internal compensation within the molecule as with the inactive tartaric acid (mesotartaric acid), or may be due to the fact that the compound is an equimolecular mixture of leftand right-hand varieties, this being the case with racemic acid that was broken by Louis Pasteur into laevoand dextro-tartaric acid (see Stereo-Isomerism).

Substances that by reason of the structure or arrangement of their molecules rotate the plane of polarization are said to be structurally active, and the rotation produced by unit length is called their rotary power. If unit mass of a solution contain m grammes of an active substance and if o be the density and p be the rotary power of the solution, the specific rotary power is defined by p/m8, and the molecular rotary power is obtained from this by multiplying by the hundredth part of the molecular mass. This quantity is not absolutely constant, and in many cases varies with the concentration of the solution and with the nature of the solvent. A mixture of two active substances, or even of an active and an inactive substance, in one solution sometimes produces anomalous effects.

Fresnel showed that rotary polarization could be explained kinematically by supposing that a plane-polarized stream is resolved on entering an active medium into two oppositely circularly polarized streams propagated with different speeds, the rotation being rightor left-handed according as the rightor left-handed stream travels at the greater rate: The polarization-vector of the primitive - stream being = a cos nt, the first circularly polarized stream after traversing a distance z in the medium may be represented by = a cos (nt - k i z), ni = a sin (nt - kiz), and the second b z = a cos (nt - k 2 z), n2= - a sin (nt - k2z).

The resultant of these is = 2a cos 2 (k 2 - k i)z cos {nt - 1(k2 -Fk2)z}, = 2a sin 2 (k 2 - ki)z cos {nt - z (k i + k2)z}, which shows that for any fixed value of z the light is plane polarized in a plane making an angle 1(k 2 - ki)z = ir(X i - X7 1)z, with the initial plane of polarization, X 1 and being the wave-lengths of the circular components of the same frequency.

Since the two circular streams have different speeds, Fresnel argued that it would be possible to separate them by oblique refraction, and though the divergence is small, since the difference of their refractive indices in the case of quartz is only about o 00007, he succeeded by a suitable arrangement of alternately rightand left-handed prisms of quartz in resolving a plane-polarized stream into two distinct circularly polarized streams. A similar arrangement was used by Ernst v. Fleischl for demonstrating circular polarization in liquids. This result is not, however, conclusive; for an application of Huygens's principle shows that it is a consequence of the rotation of the plane of polarization by an amount proportional to the distance traversed, independently of the state of affairs within the active medium. Not more convincing is a second experiment devised by Fresnel. If in the interference experiment with Fresnel's mirrors or biprism the slit be illuminated with white light that has passed through a polarizer and a quartz plate cut perpendicularly to the optic axis, it is found on analysing the light that in addition to the ordinary central set of coloured fringes two lateral systems are seen, one on either side of it. According to Fresnel's explanation the light in each of the interfering streams consists of two trains of waves that are circularly polarized in opposite direction and have a relative retardation of phase, introduced by the passage through the quartz: the central fringes are then due to the similarly polarized waves; the lateral systems are produced by the oppositely polarized streams, these on analysation being capable of interfering. A. Righi has, however, pointed out that this experiment may be explained by the fact that the function of the quartz plate and analyser is to eliminate the constituents of the composite stream of white light that mask the interference actually occurring at the positions of the lateral systems of fringes, and that any other method of removing them is equally effective. In fact, the lateral systems are obtained when a plate of selenite is substituted for the quartz.

Sir G. B. Airy extended Fresnel's hypothesis to directions inclined to the axis of uniaxal crystals by assuming that in any such direction the two waves, that can be propagated without alteration of their state of polarization, are oppositely elliptically polarized with their planes of maximum polarization parallel and perpendicular to the principal plane of the wave, these becoming practically plane polarized at a small inclination to the optic axis. Several investigations have been made to test the correctness of Airy's views, but it must be remembered that it is only possible to experiment on waves after they have left the crystal, and L. G. Gouy (Journ. de phys., 1885 [2], iv. 149) has shown that the results deduced from Airy's waves of permanent type may be obtained by regarding the action of the medium as the superposition of the effects of ordinary double refraction and of an independent rotary power. As regards the course - of the streams on refraction into the crystal, it is found that it is determined by the Huygenian law (see Refraction, § Double); as, however, the two streams in the direction of the axis have different speeds, the spherical and the spheroidal sheets of the wavesurface do not touch as in the case of inactive uniaxal crystals. On these principles Airy, by an elaborate mathematical investigation, successfully explained the interference patterns obtained with plates of quartz perpendicular to the optic axis. When the polarizer and analyser are parallel or crossed, the pattern is the same as with inactive plates, with the exception that the brushes do not extend to the centre of the field; but as the analyser is rotated a small cross begins to appear at the centre of the field, while the rings change their form and become nearly squares with rounded corners, when the planes of polarization and analysation are at 45°. With two plates of equal thickness and of opposite rotations, the pattern consists of a series of circles and of four similar spirals starting from the centre, each spiral being turned through 90° from that adjacent to it. When the light is circularly polarized or circularly analysed, a single plate gives two mutually enwrapping spirals, and similar spirals in circularly polarized light are obtained with plates of an active biaxal crystal perpendicular to one of the optic axes. It was in this way that the rotary property of certain biaxal crystals was first established by Pocklington.

F. E. Reusch has shown that a packet of identical inactive plates arranged in spiral fashion gives an artificial active system, and the behaviour of certain pseudosymmetric crystals indicates a formation of this character. On these results L. Sohncke (Math. Ann., 1876, ix. 504) and E. Mallard (Traite de cristallographie, vol. ii. ch. ix.) have built up a theory of the structure of active media, but in the instances in which static spirality has been shown to be effective in producing optical rotation the coarse-grainedness of the structure is comparable with the wave-length of the radiation affected.

The rotary property may be induced in substances naturally inactive. Thus A. W. Ewell (Amer. Jour. of Science, 18 99 [4], viii. 89) has shown the existence of a rotational effect in twisted glass and gelatine, the rotation being opposite to the direction of the twist. But a far more important instance of induced activity is afforded by Michael Faraday's discovery of the rotary polarization connected with a magnetic field. There is, however, a marked difference between this magnetic rotation and that of a structurally active medium, for in the latter it is always right-handed or always left-handed with respect to the direction of the ray, while in the former the sense of rotation is determined by the direction of magnetization and therefore remains the same though the ray be reversed. This subject is treated in the article Magneto-Optics, to which the reader is also referred for John Kerr's discovery of the effect on polarization produced by reflection from a magnetic pole, and for the action of a magnetic field on the radiation of a source - the "Zeeman effect." Reflection and Refrciction. - Huygens satisfactorily explained the laws of reflection and refraction on the principles of the wave theory, so far as the direction of the waves is concerned, but his explanation gives no account of the intensity and the polarization of the reflected light. This was supplied by Fresnel, who, starting from a mechanical hypothesis, showed by ingenious but not strictly dynamical reasoning that if the incident stream have unit amplitude, that of the reflected stream will be - sin (i - r) /sin (i -{- r) or tan (i - r) /tan (i -{- r), according as the incident light is polarized in or perpendicularly to the plane of incidence i, r, being the angles of incidence and refraction connected by the formula sin i =,u sin r. At normal incidence the intensity of the reflected light, measured by the square of the amplitude, is { (µ -1) /(µ+ I) } 2 in both cases; but whereas in the former the intensity increases uniformly with i to the value unity for i =90°, in the latter the intensity at first decreases as i increases, until it attains the value zero when i -Fr = 90°, or tan i =,u - the polarizing angle of Brewster - and then increases until it becomes unity at grazing incidence. If the incident light be polarized in a plane, making an angle a with the plane of incidence, the stream may be resolved into two that are polarized in the principal azimuths, and these will be reflected in accordance with the above laws. Hence if 1 3 be the angle between the plane of incidence and that in which the reflected light is polarized tan (3= - tan a cos (i+r)/cos (i - r).

The expressions for the intensity of the refracted light may be obtained from those relating to the reflected light by the principle of energy. In order to avoid the question of the measurements of the intensity in different media, it is convenient to suppose that the refracted stream emerges into a medium similar to the first by a transition so gradual that no light is lost by reflection. The intensities of the incident, reflected and refracted streams are then measured in the same way, and we have merely to express that the square of the amplitude of the incident vibrations is equal to the sum of the squares of the amplitudes of the reflected and refracted vibrations.

Fresnel obtained his formulae by assuming that the optical difference of media is due to a change in the effective density of the ether, the elasticity being the same - an assumption inconsistent with his theory of double refraction - and was led to the result that the vibrations are perpendicular to the plane of polarization. Franz Neumann and James MacCullagh, starting from the opposite assumption of constant density and different elasticities, arrived at the same formulae for the intensities of the reflected light polarized in the principal azimuths, but in this case the vibrations must be regarded as parallel to the plane of polarization. The divergence of these views has led to a large number of experimental investigations, instituted with the idea of deciding between them. In the main such investigations have only an academic interest, as, whatever theory of light be adopted, we have to deal with two vectors that are parallel and perpendicular respectively to the plane of polarization. Thus certain experiments of Otto H. Wiener (Wied. Ann., 1890, xl. 203) show that chemical action is to be referred to the latter of these vectors, but whether Fresnel's or Neumann's hypothesis be correct is only to be decided when we know if it be the mean kinetic energy or the mean potential energy that determines chemical action. Similarly on the electromagnetic theory the electric or the magnetic force will be perpendicular to the plane of polarization, according as chemical action depends upon the electric or the magnetic energy. Lord Rayleigh (Scientific Papers, i. 104) has, however, shown that the polarization of the light from the sky can only be explained on the elastic solid theory by Fresnel's hypothesis of a different density, and from the study of Hertzian oscillations, in which the direction of the electric vibrations can be a priori assigned, we learn that when these are in the plane of incidence there is no reflection at a certain angle, so that the electric force is perpendicular to the plane of polarization.

It has been supposed in the above that the medium into which the light enters at the reflecting-surface is the more refracting. In the contrary case, total reflection commences as soon as sin i =µ 1, µ being still the relative refractive index of the more highly refracting medium; and for greater angles of incidence r becomes imaginary. Now Fresnel's formulae were obtained by assuming that the incident, reflected and refracted vibrations are in the same or opposite phases at the interface of the media, and since there is no real factor that converts cos T into cos (T+p), he inferred that the occurrence of imaginary expressions for the coefficients of vibration denotes a change of phase other than 7r, this being represented by a change of sign. If this be so, it is clear that the factor A / - 1 denotes a change of phase of 42, since this twice repeated converts cos T into cos (T+ir) = - cos T, and hence that the factor a+b A l - I represents a change of phase of tan1 (b/a). Applying this interpretation to the formulae given above, it follows that when the incident light is polarized at an azimuth a to the plane of incidence and the second medium is the less refracting, the reflected light at angles of incidence exceeding the critical angle is elliptically polarized with a difference of phase A between the components polarized in the principal azimuths that is given by tan (A/2) =cot i l l (1 - µ 2 cosec 2 i). Thus A is zero at grazing incidence and at the critical angle, and attains its maximum value 7r-4 tan1 (I/p) at an angle of incidence given by sin e i=2/(p.2+I).

It is of some interest to determine under what conditions it is possible to obtain a specified difference of phase. Solving for cot e i we obtain 2 cot 2 i= (p 2 -1) 2 {(p2 - tan2 (7r - A)/4) {p - cot2 - A)/411, and since tan { (ir - A)/4-} is less than unity, p must exceed cot {Or - A)/41 if cot 2 i is to be real. Thus if A = 42, p. must exceed 7r/8 or 2.414, that is, the substance must be at least as highly refracting as a diamond: if A =7 /4, µ must be greater than 37r/16 or 1.4966, and when this is the case, it is possible by two reflections to convert into a circularly polarized stream a beam of light polarized at 45° to the plane of incidence. This is the principle of Fresnel's rhomb, that is sometimes employed instead of a quarter-wave plate for obtaining a stream of circularly polarized light. It consists of a parallelopiped glass so constructed that light falling normally on one end emerges at the other after two internal reflections at such an angle as to introduce a relative retardation of phase of 4r/4 between the components polarized in the principal azimuths.

Fresnel's formulae are sufficiently accurate for most practical purposes, but that they are not an exact representation of the facts of reflection was shown by Sir David Brewster and by Sir G. B. Airy. Detailed investigations by J. C. Jamin, G. H. Quincke, C. W. Wernicke and others have established that in general light polarized in any but the principal azimuths becomes elliptically polarized by reflection, the relative retardation of phase of the components polarized in these azimuths becoming 42 at a certain angle of incidence, called the principal incidence. In some cases it is the component polarized in the plane of incidence that is most retarded and the reflection is then said to be positive: in the case of negative reflection the reverse takes place. It was at first supposed that the defect of Fresnel's formulae was due to the neglect of the superficial undulations that, on a rigorous elastic solid theory of the ether, are called into existence at reflection and refraction. But the result of taking these into account is far from being in accordance with the facts, and experiments of Lord Rayleigh and Paul Drude make it probable that we ought to assume that the transition from one medium to another, though taking place in a distance amounting to about one fiftieth of a wave-length, is gradual instead of abrupt. The effect of such a transition-layer can easily be calculated, at least approximately; but it is of little use to take account of it except in the case of a theory of reflection that gives Fresnel's formulae as the result of an abrupt transition. Lord Rayleigh has pointed out that all theories are defective in that they disregard the fact that one at least of the media is dispersive, and that it is probable that finite reflection would result at the interface of media of different dispersive powers, even in the case of waves for which the refractive indices are absolutely the same.

A more pronounced case of elliptic polarization by reflection is afforded by metals. Formulae for metallic reflection may be obtained from Fresnel's expressions by writing the ratio sin i / sin r equal to a complex quantity, and interpreting the imaginary coefficients in the manner explained above. The optical constants (refractive index and co-efficient of extinction) of the metal may then be obtained from observations of the principal incidence and the elliptic polarization then produced. A detailed investigation of these constants has been made by Drude (Wied. Ann., 1890, xxxix. 504), who has found the remarkable result that copper, gold, magnesium and silver have refractive indices less than unity, and this has been completely confirmed by observations with metallic prisms of small refracting angle. He further showed that except in the cases of copper, lead and gold the dispersion is abnormal - the index for red light being greater than that for sodium light. The higher the co-efficient of extinction for light of a given period, the more copious will be reflection of that constituent of a mixed pencil. This fact has been employed for separating waves of large wavelength, and in this way waves of length 0 . 061 mm. have been isolated by five successive reflections from the surface of sylvite.

## The Study of Polarization

The best method of obtaining a strong beam of polarized light is to isolate one of the streams into which a beam of common light is resolved by double refraction. This is effected in polarizing prisms of the earlier type, devised by A. M. de Rochon, H. H. de Senarmont and W. H. Wollaston, by blocking off one of the streams with a screen, sufficient lateral separation being obtained by combining two equal crystalline prisms cut differently with respect to the optic axis - an arrangement that achromatizes more or less completely the pencil that is allowed to pass. In a second type, called Nicol's prisms, one stream is removed by total reflection. Theoretically the best construction for prisms of this class is the following: a rectangular block of Iceland spar, of length about four times the width and having its end and two of its side faces parallel to the optic axis, is cut in half by a plane parallel to the optic axis and making an angle of about 14° with the sides; the two halves are then reunited with a cement whose refractive index is between the ordinary and extraordinary indices of the spar and as nearly as possible equal to the latter. Thus constructed, the ' prism produces no lateral shift of the transmitted pencil; a conical pencil, incident directly, has nearly constant polarization over its extent, and consequently the error in determining the polarization of a parallel pencil, incident not quite normally, is a minimum. In a Nicol's prism it is the extraordinary stream that passes; in a prism suggested by E. Sang and sometimes called a Bertrand's prism, it is the ordinary stream that is utilized. This is made by fixing a thin crystalline plate between two glass prisms turned in opposite directions by a cement of the same refractive index as the glass. This refractive index should be equal to the greatest index of the plate, and with a biaxal plate the mean axis of optical symmetry should be parallel to its faces and in the normal section of the prisms, while with an uniaxal plate the optic axis should be in a plane perpendicular to this normal section. These prisms have the advantage of economy of material and of a greater field than the ordinary Nicol's prism, but a difficulty seems to be experienced in finding a suitable permanent cement. For an accurate determination of the plane of polarization analysers that act by extinction are not of much practical use, and a different device has to be employed. Savart's analyser consists, of a Savart's plate (see above) connected to a Nicol's prism, the principal section of which bisects the angle between the principal. planes of the plate: the plane of polarization is determined by turning the analyser until the bands, ordinarily seen, disappear,. in which case it is parallel to one of the principal planes of the plate. Half-shade analysers depend upon the facility with which the eye can distinguish slight differences in the intensities of two streams seen in juxtaposition, when the illumination is not too bright. The field is divided into two parts that for most positions of the analyser have different intensities, and the setting is effected by turning the analyser until both halves are equally dark. These instruments are very sensitive, but care must be taken to avoid errors caused by changes in the relative intensities of parts of the source of light - a precaution that is sometimes overlooked in furnishing polarimeters with these analysers. In J. H. Jellet's and M. A. Cornu's analysers xxi. 30 a formed - the one from two parts of a rhomb of spar, the other from two portions of a Nicol's prism - the two halves of the field are analysed in slightly different planes; but these, though they have certain advantages, are now seldom employed, partly on account of a difficulty in their construction and partly because their sensitiveness cannot be adjusted. The more usual half-shade analyser is available for light of only one frequency, as it depends upon the action of a half-wave plate, in traversing which the plane of polarization is turned until it makes the same angle with the principal section as at first but on the opposite side: half the field is covered with the plate, to which is attached a Nicol's prism with its principal section inclined at a small angle to that of the plate. The eye must be focussed on the edge of the plate, and the two halves of the field will only be equally dark when the principal plane of the plate is parallel to the primitive plane of polarization. Another plan, due to J. H. Poynting, is before analysation to impress unequal rotations upon the plane of polarization of the two parts of the field, either by means of an active medium, or by oblique transmission through glass plates.

Elliptically polarized light is investigated by the reduction of the pencil to a state of plane polarization, and a determination of the resulting plane of polarization. One method consists in finding directly the elliptic constants of the vibration by means of a quarterwave plate and an analyser; but the more usual plan is to measure the relative retardation of two rectangular components of the stream by a Babinet's compensator. This is a plate made of two equal wedges of quartz, that can be moved over one another so as to vary its thickness, and are cut so that the faces of the plate are parallel to the optic axis, which in the first wedge is perpendicular and in the second is parallel to the refracting edge. It is clear that direct transmission through the plate at a point where the thicknesses of the prisms are d 1 and d 2 will introduce a relative retardation of (µ,; - ,u o) (d l - d2) between streams polarized in planes parallel and perpendicular to the edges of the prisms, ,u o, and being the ordinary and the extraordinary refractive indices; and it is hence possible by an adjustment of the thickness to reduce elliptically polarized to plane polarization at an assigned point marked off by two parallel lines. A subsequent determination of the plane of polarization gives the ratio of the amplitudes of the vibrations in the component streams.

## BIBLIOGRAPHY

A bibliography of the subjects treated in this article will be found at the end of the corresponding chapters of E. Verdet's Lecons d'optique physique (1869); this work has been brought to a later date in the German translation by Karl Exner (Braunschweig, 1881); references to later papers will be found in J. Walker's The Analytical Theory of Light (1904). In addition to the above the reader may consult for the general subject of polarization the following treatises: Th. Preston (3rd. ed. by C. J. Joly), The Theory of Light (1901); A. Schuster, An Introduction to the Theory of Optics (1904); R. W. Wood, Physical Optics (1905); E. Mascart, Traite d'optique (1889); and for the phenomena exhibited by crystals F. Pockel, Lehrbuch der Kristalloptik (1906); Th. Liebisch, Physikalische Kristallographie (1891). (J. WAL.*)

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