Editing Crystal optics

{{Short description|Sub-branch of Optical Physics}}
{{more footnotes needed|date=January 2013}}
'''Crystal optics''' is the branch of [[optics]] that describes the behaviour of [[light]] in ''[[anisotropic]] media'', that is, media (such as [[crystal]]s) in which light behaves differently depending on which direction the light is [[Wave propagation|propagating]]. The index of refraction depends on both composition and crystal structure and can be calculated using the [[Gladstone–Dale relation]]. Crystals are often naturally anisotropic, and in some media (such as [[liquid crystal]]s) it is possible to induce anisotropy by applying an external electric field.

==Isotropic media==

Typical transparent media such as [[glass]]es 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''':

:<math> \mathbf{D} = \varepsilon_0  \mathbf{E} + \mathbf{P}  </math>

where ε<sub>0</sub> is the [[permittivity]] of free space and '''P''' is the electric [[polarization (electrostatics)|polarization]] (the [[vector field]] corresponding to [[electric dipole moment]]s present in the medium). Physically, the polarization 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 polarization field '''P''' is proportional and parallel to the electric field '''E''':

:<math> \mathbf{P}  = \chi \varepsilon_0 \mathbf{E}  </math>

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

:<math> \mathbf{D}  =   \varepsilon_0 \mathbf{E}  +  \chi \varepsilon_0 \mathbf{E}   
=  \varepsilon_0  (1 + \chi)  \mathbf{E}   =  \varepsilon   \mathbf{E}  </math>

where

:<math> \varepsilon =  \varepsilon_0  (1 + \chi) </math>

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

:<math>  n = \sqrt{ 1 + \chi}  </math>

==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:

:<math> \mathbf{P} = \varepsilon_0 \boldsymbol{\chi} \mathbf{E} .</math>

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:

<math>\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}
</math>

or using the summation convention:

:<math> P_i = \varepsilon_0 \sum_{j\in\{x,y,z\}}\chi_{ij} E_j \quad.</math>

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

In nonmagnetic and transparent materials, χ<sub>''ij''</sub> = χ<sub>''ji''</sub>, i.e. the '''χ''' tensor is real and [[symmetric tensor|symmetric]].<ref>Amnon Yariv, Pochi Yeh. (2006). Photonics optical electronics in modern communications (6th ed.). Oxford University Press. pp. 30-31.</ref> In accordance with the [[spectral theorem]], it is thus possible to [[Matrix diagonalization|diagonalise]] the tensor by choosing the appropriate set of coordinate axes, zeroing all components of the tensor except χ<sub>xx</sub>, χ<sub>yy</sub> and χ<sub>zz</sub>. This gives the set of relations:

:<math> P_x = \varepsilon_0 \chi_{xx} E_x</math>
:<math> P_y = \varepsilon_0 \chi_{yy} E_y</math>
:<math> P_z = \varepsilon_0 \chi_{zz} E_z</math>

The directions x, y and z are in this case known as the ''principal axes'' of the medium. Note that these axes will be orthogonal if all entries in the '''χ''' tensor are real, corresponding to a case in which the refractive index is real in all directions.

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

:<math> \mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P} = \varepsilon_0 \mathbf{E} + \varepsilon_0 \boldsymbol{\chi} \mathbf{E} = \varepsilon_0 (I + \boldsymbol{\chi}) \mathbf{E} = \varepsilon_0 \boldsymbol{\varepsilon} \mathbf{E} .</math>

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 [[Polarization (waves)|polarised]] such the electric field of the wave is parallel to the x-axis. The wave experiences a susceptibility χ<sub>xx</sub> and a permittivity ε<sub>xx</sub>. The refractive index is thus:

:<math>n_{xx} = (1 + \chi_{xx})^{1/2} = (\varepsilon_{xx})^{1/2} .</math>

For a wave polarised in the y direction:

:<math>n_{yy} = (1 + \chi_{yy})^{1/2} = (\varepsilon_{yy})^{1/2} .</math>

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 χ<sub>xx</sub> = χ<sub>yy</sub> ≠ χ<sub>zz</sub>, the crystal is known as '''uniaxial'''.  (See [[Optic axis of a crystal]].) If χ<sub>xx</sub> ≠ χ<sub>yy</sub> and χ<sub>yy</sub> ≠ χ<sub>zz</sub> the crystal is called '''biaxial'''. A uniaxial crystal exhibits two refractive indices, an "ordinary" index (''n''<sub>o</sub>) for light polarised in the x or y directions, and an "extraordinary" index (''n''<sub>e</sub>) for polarisation in the z direction. A uniaxial crystal is "positive" if n<sub>e</sub> > n<sub>o</sub> and "negative" if n<sub>e</sub> < n<sub>o</sub>. 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 optics|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 axis (crystallography)|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 isolator]]s, 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.

==See also==
*[[Birefringence]]
*[[Optic crystals]]
*[[Index ellipsoid]]
*[[Optical rotation]]
*[[Prism (optics)|Prism]]

==References==
<references />

== External links ==
* [http://gerdbreitenbach.de/crystal/crystal.html A virtual polarization microscope]

{{Authority control}}

[[Category:Condensed matter physics]]
[[Category:Crystallography]]
[[Category:Nonlinear optics]]