Results for Crystal optics
On this page:
 
Sci-Tech Dictionary:

crystal optics

(¦krist·əl ′äp·tiks)

(optics) The study of the propagation of light, and associated phenomena, in crystalline solids.

 
 
Sci-Tech Encyclopedia: Crystal optics

The study of the propagation of light, and associated phenomena, in crystalline solids. For a simple cubic crystal the atomic arrangement is such that in each direction through the crystal the crystal presents the same optical appearance. The atoms in anisotropic crystals are closer together in some planes through the material than in others. In anisotropic crystals the optical characteristics are different in different directions. In classical physics the progress of an electromagnetic wave through a material involves the periodic displacement of electrons. In anisotropic substances the forces resisting these displacements depend on the displacement direction. Thus the velocity of a light wave is different in different directions and for different states of polarization. The absorption of the wave may also be different in different directions. See also Dichroism; Trichroism.

In an isotropic medium the light from a point source spreads out in a spherical shell. The light from a point source embedded in an anisotropic crystal spreads out in two wave surfaces, one of which travels at a faster rate than the other. The polarization of the light varies from point to point over each wave surface, and in any particular direction from the source the polarization of the two surfaces is opposite. The characteristics of these surfaces can be determined experimentally by making measurements on a given crystal.

In the most general case of a transparent anisotropic medium, the dielectric constant is different along each of three orthogonal axes. This means that when the light vector is oriented along each direction, the velocity of light is different. One method for calculating the behavior of a transparent anisotropic material is through the use of the index ellipsoid, also called the reciprocal ellipsoid, optical indicatrix, or ellipsoid of wave normals. This is the surface obtained by plotting the value of the refractive index in each principal direction for a linearly polarized light vector lying in that direction (see illustration). The different indices of refraction, or wave velocities associated with a given propagation direction, are then given by sections through the origin of the coordinates in which the index ellipsoid is drawn. These sections are ellipses, and the major and minor axes of the ellipse represent the fast and slow axes for light proceeding along the normal to the plane of the ellipse. The length of the axes represents the refractive indices for the fast and slow wave, respectively. The most asymmetric type of ellipsoid has three unequal axes. It is a general rule in crystallography that no property of a crystal will have less symmetry than the class in which the crystal belongs.

Index ellipsoid, showing construction of directions of vibrations of <i>D</i> vectors belonging to a wave normal <i>s</i>. (<i>After M. Born and E. Wolf, Principles of Optics, 7th ed., Cambridge University Press, 1999</i>)
Index ellipsoid, showing construction of directions of vibrations of D vectors belonging to a wave normal s. (After M. Born and E. Wolf, Principles of Optics, 7th ed., Cambridge University Press, 1999)

Accordingly, there are many crystals which, for example, have four- or sixfold rotation symmetry about an axis, and for these the index ellipsoid cannot have three unequal axes but is an ellipsoid of revolution. In such a crystal, light will be propagated along this axis as though the crystal were isotropic, and the velocity of propagation will be independent of the state of polarization. The section of the index ellipsoid at right angles to this direction is a circle. Such crystals are called uniaxial and the mathematics of their optical behavior is relatively straightforward.

In crystals of low symmetry the index ellipsoid has three unequal axes. These crystals are termed biaxial and have two directions along which the wave velocity is independent of the polarization direction. These correspond to the two sections of the ellipsoid which are circular. See also Crystallography.

The normal to a plane wavefront moves with the phase velocity. The Huygens wavelet, which is the light moving out from a point disturbance, will propagate with a ray velocity. Just as the index ellipsoid can be used to compute the phase or wave velocity, so can a ray ellipsoid be used to calculate the ray velocity. The length of the axes of this ellipsoid is given by the velocity of the linearly polarized ray whose electric vector lies in the axis direction. See also Phase velocity.

The refraction of a light ray on passing through the surface of an anisotropic uniaxial crystal can be calculated with Huygens wavelets in the same manner as in an isotropic material. For the ellipsoidal wavelet this results in an optical behavior which is completely different from that normally associated with refraction. The ray associated with this behavior is termed the extraordinary ray. At a crystal surface where the optic axis is inclined at an angle, a ray of unpolarized light incident normally on the surface is split into two beams: the ordinary ray, which proceeds through the surface without deviation; and the extraordinary ray, which is deviated by an angle determined by a line drawn from the center of one of the Huygens ellipsoidal wavelets to the point at which the ellipsoid is tangent to a line parallel to the surface. The two beams are oppositely linearly polarized.

 
Wikipedia: 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.

Isotropic media

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:

\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.

Electric susceptibility

In an isotropic and linear medium, this polarisation field P is proportional to and parallel to the electric field E:

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

where χ is the electric susceptibility of the medium. The relation between D and E is thus:

\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

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:

\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:

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:

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.

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.

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.

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.

(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.)

External links


This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

 
 

Join the WikiAnswers Q&A community. Post a question or answer questions about "Crystal optics" at WikiAnswers.

 

Copyrights:

Sci-Tech Dictionary. McGraw-Hill Dictionary of Scientific and Technical Terms. Copyright © 2003, 1994, 1989, 1984, 1978, 1976, 1974 by McGraw-Hill Companies, Inc. All rights reserved.  Read more
Sci-Tech Encyclopedia. McGraw-Hill Encyclopedia of Science and Technology. Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.  Read more
Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Crystal optics".  Read more

Search for answers directly from your browser with the FREE Answers.com Toolbar!  
Click here to download now. 

Get Answers your way! Check out all our free tools and products.

On this page:   E-mail   print Print  Link  

 

Keep Reading

Mentioned In:

 |  More >More >