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Encyclopedia > Crystal optics

Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media (such as crystals) in which light behaves differently depending on which direction the light is propagating. Crystals are often naturally anisotropic, and in some media (such as liquid crystals) it is possible to induce anisotropy by applying e.g. an external electric field. Optical redirects here. ... Prism splitting light Light is electromagnetic radiation with a wavelength that is visible to the eye (visible light) or, in a technical or scientific context, electromagnetic radiation of any wavelength. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... Quartz crystal In chemistry and mineralogy, a crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. ... Bold text Schlieren texture of Liquid Crystal nematic phase Liquid crystals are substances that exhibit a phase of matter that has properties between those of a conventional liquid, and those of a solid crystal. ...

## Isotropic media GA_googleFillSlot("encyclopedia_square");

Typical transparent media such as glasses are isotropic, which means that light behaves the same way no matter which direction it is travelling in the medium. In terms of Maxwell's equations in a dielectric, this gives a relationship between the electric displacement field D and the electric field E: Glass can be made transparent and flat, or into other shapes and colours as shown in this ball from the Verrerie of Brehat in Brittany. ... Isotropic means independent of direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. ... Maxwells equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... A dielectric, or electrical insulator, is a substance that is highly resistant to the flow of electric current and has a relative permittivity greater than unity. ... The factual accuracy of this article is disputed. ... In physics, an electric field or E-field is an effect produced by an electric charge (or a time-varying magnetic field) that exerts a force on charged objects in the field. ...

$mathbf{D} = varepsilon_0 mathbf{E} + mathbf{P}$

where ε0 is the permittivity of free space and P is the electric polarisation (the vector field corresponding to electric dipole moments present in the medium). Physically, the polarisation field can be regarded as the response of the medium to the electric field of the light. Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ... In electrostatics, the polarization is the vector field that results from permanent or induced electric dipole moments in a dielectric material. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... The Earths magnetic field, which is approximately a dipole. ...

## Electric susceptibility

In an isotropic and linear medium, this polarisation field P is proportional to and parallel to the electric field E: Isotropic means independent of direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. ... The word linear comes from the Latin word linearis, which means created by lines. ...

$mathbf{P} = chi varepsilon_0 mathbf{E}$

where χ is the electric susceptibility of the medium. The relation between D and E is thus: It has been suggested that this article or section be merged into permittivity. ...

$mathbf{D} = varepsilon_0 mathbf{E} + chi varepsilon_0 mathbf{E} = varepsilon_0 (1 + chi) mathbf{E} = varepsilon mathbf{E}$

where

$varepsilon = varepsilon_0 (1 + chi)$

is the dielectric constant of the medium. The value 1+χ is called the relative permittivity of the medium, and is related to the refractive index n, for non-magnetic media, by The dielectric constant Îµr (represented as or K in some cases) is defined as the ratio: where Îµs is the static permittivity of the material in question, and Îµ0 is the vacuum permittivity. ... The refractive index (or index of refraction) of a material is the factor by which the phase velocity of electromagnetic radiation is slowed in that material, relative to its velocity in a vacuum. ...

$n = sqrt{ 1 + chi}$

## Anisotropic media

In an anisotropic medium, such as a crystal, the polarisation field P is not necessarily aligned with the electric field of the light E. In a physical picture, this can be thought of as the dipoles induced in the medium by the electric field having certain preferred directions, related to the physical structure of the crystal. This can be written as:

$mathbf{P} = varepsilon_0 boldsymbol{chi} times mathbf{E} .$

Here χ is not a number as before but a tensor of rank 2, the electric susceptibility tensor. In terms of components in 3 dimensions: In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

$begin{pmatrix} P_x P_y P_z end{pmatrix} = varepsilon_0 begin{pmatrix} chi_{xx} & chi_{xy} & chi_{xz} chi_{yx} & chi_{yy} & chi_{yz} chi_{zx} & chi_{zy} & chi_{zz} end{pmatrix} begin{pmatrix} E_x E_y E_z end{pmatrix}$

or using the summation convention:

$P_i = varepsilon_0 chi_{ij} E_j quad.$

Since χ is a tensor, P is not necessarily colinear with E.

From thermodynamic arguments it can be shown that χij = χji, i.e. the χ tensor is symmetric. In accordance with the spectral theorem, it is thus possible to diagonalise the tensor by choosing the appropriate set of coordinate axes, zeroing all components of the tensor except χxx, χyy and χzz. This gives the set of relations: â€¹ The template below has been proposed for deletion. ... A tensor A, with components Aij, is said to be symmetric if Aij = Aji for all i, j. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ... The term diagonalization is used in two different senses in mathematics: The process of finding a diagonal matrix similar to a given square matrix or representing a given linear map. ...

$P_x = varepsilon_0 chi_{xx} E_x$
$P_y = varepsilon_0 chi_{yy} E_y$
$P_z = varepsilon_0 chi_{zz} E_z$

The directions x, y and z are in this case known as the principal axes of the medium. Note that these axes are not necessarily orthogonal.

It follows that D and E are also related by a tensor:

$mathbf{D} = varepsilon_0 mathbf{E} + mathbf{P} = varepsilon_0 mathbf{E} + varepsilon_0 boldsymbol{chi} times mathbf{E} = varepsilon_0 (1+ boldsymbol{chi}) times mathbf{E} = varepsilon_0 boldsymbol{varepsilon} times mathbf{E} .$

Here ε is known as the relative permittivity tensor or dielectric tensor. Consequently, the refractive index of the medium must also be a tensor. Consider a light wave propagating along the z principal axis polarised such the electric field of the wave is parallel to the x-axis. The wave experiences a susceptibility χxx and a permittivity εxx. The refractive index is thus: The refractive index (or index of refraction) of a material is the factor by which the phase velocity of electromagnetic radiation is slowed in that material, relative to its velocity in a vacuum. ... This article treats polarization in electrodynamics. ...

$n_{xx} = (1 + chi_{xx})^{1/2} = (varepsilon_{xx})^{1/2} .$

For a wave polarised in the y direction:

$n_{yy} = (1 + chi_{yy})^{1/2} = (varepsilon_{yy})^{1/2} .$

Thus these waves will see two different refractive indices and travel at different speeds. This phenomenon is known as birefringence and occurs in some common crystals such as calcite and quartz. A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on... Doubly refracting Calcite from Iceberg claim, Dixon, New Mexico. ... Quartz is amongst one of the most common minerals in the Earths continental crust. ...

If χxx = χyy ≠ χzz, the crystal is known as uniaxial. If χxx ≠ χyy and χxx ≠ χzz the crystal is called biaxial. A uniaxial crystal exhibits two refractive indices, an "ordinary" index (no) for light polarised in the x or y directions, and an "extraordinary" index (ne) for polarisation in the z direction. A uniaxial crystal is "positive" if ne > no and "negative" if ne < no. Light polarised at some angle to the axes will experience a different phase velocity for different polarization components, and cannot be described by a single index of refraction. This is often depicted as an index ellipsoid. The index ellipsoid is a diagram of an ellipsoid that depictes the orientation and relative magnitude of refractive indices in a crystal. ...

## Other effects

Certain nonlinear optical phenomena such as the electro-optic effect cause a variation of a medium's permittivity tensor when an external electric field is applied, proportional (to lowest order) to the strength of the field. This causes a rotation of the principal axes of the medium and alters the behaviour of light travelling through it; the effect can be used to produce light modulators. Nonlinear optics is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization P responds nonlinearly to the electric field E of the light. ... The electro-optic effect is a change in the optical properties of a material in response to an electric field that varies slowly compared with the frequency of light. ...

In response to a magnetic field, some materials can have a dielectric tensor that is complex-Hermitian; this is called a gyro-magnetic or magneto-optic effect. In this case, the principal axes are complex-valued vectors, corresponding to elliptically polarized light, and time-reversal symmetry can be broken. This can be used to design optical isolators, for example. Current (I) flowing through a wire produces a magnetic field (B) around the wire. ... A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...

(A dielectric tensor that is not Hermitian gives rise to complex eigenvalues, which corresponds to a material with gain or absorption at a particular frequency.)

Results from FactBites:

 Crystal - Wikipedia, the free encyclopedia (606 words) A crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. Under ideal conditions, the result may be a single crystal, where all of the atoms in the solid fit into the same lattice or crystal structure but, generally, many crystals form simultaneously during solidification, leading to a polycrystalline solid. Which crystal structure the fluid will form depends on the chemistry of the fluid, the conditions under which it is being solidified, and also on the ambient pressure.
 Crystal optics - Wikipedia, the free encyclopedia (677 words) Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media (such as crystals) in which light behaves differently depending on which direction the light is propagating. Crystals are often naturally anisotropic, and in some media (such as liquid crystals) it is possible to induce anisotropy by applying e.g. In an anisotropic medium, such as a crystal, the polarisation field P is not necessarily aligned with the electric field of the light E.
More results at FactBites »

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