Editing Polarization density (section)

===Anisotropic dielectrics===

The class of dielectrics where the polarization density and the electric field are not in the same direction are known as ''[[anisotropic]]'' materials.

In such materials, the {{mvar|i}}-th component of the polarization is related to the {{mvar|j}}-th component of the electric field according to:<ref name="Fay64"/>

<math display="block">P_i = \sum_j \varepsilon_0 \chi_{ij} E_j ,</math>

This relation shows, for example, that a material can polarize in the x direction by applying a field in the z direction, and so on. The case of an anisotropic dielectric medium is described by the field of [[crystal optics]].

As in most electromagnetism, this relation deals with macroscopic averages of the fields and dipole density, so that one has a continuum approximation of the dielectric materials that neglects atomic-scale behaviors. The [[polarizability]] of individual particles in the medium can be related to the average susceptibility and polarization density by the [[Clausius–Mossotti relation]].

In general, the susceptibility is a function of the [[frequency]] {{mvar|ω}} of the applied field.  When the field is an arbitrary function of time {{mvar|t}}, the polarization is a [[convolution]] of the [[continuous Fourier transform|Fourier transform]] of {{math|''χ''(''ω'')}} with the {{math|'''E'''(''t'')}}.  This reflects the fact that the dipoles in the material cannot respond instantaneously to the applied field, and [[causality]] considerations lead to the [[Kramers–Kronig relation]]s.

If the polarization '''P''' is not linearly proportional to the electric field {{math|'''E'''}}, the medium is termed ''nonlinear'' and is described by the field of [[nonlinear optics]].  To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), '''P''' is usually given by a [[Taylor series]] in {{math|'''E'''}} whose coefficients are the nonlinear susceptibilities:

<math display="block">\frac{P_i}{\varepsilon_0} = \sum_j \chi^{(1)}_{ij} E_j + \sum_{jk} \chi_{ijk}^{(2)} E_j E_k + \sum_{jk\ell} \chi_{ijk\ell}^{(3)} E_j E_k E_\ell + \cdots </math>

where <math>\chi^{(1)}</math> is the linear susceptibility, <math>\chi^{(2)}</math> is the second-order susceptibility (describing phenomena such as the [[Pockels effect]], [[optical rectification]] and [[second-harmonic generation]]), and <math>\chi^{(3)}</math> is the third-order susceptibility (describing third-order effects such as the [[Kerr effect]] and electric field-induced optical rectification).

In [[ferroelectric]] materials, there is no one-to-one correspondence between '''P''' and '''E''' at all because of [[hysteresis]].