Editing Crystal optics (section)

==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]].