The index of refraction depends on both composition and crystal structure and can be calculated using the Gladstone–Dale relation.
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.
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 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.
or using the summation convention: Since χ is a tensor, P is not necessarily colinear with E. In nonmagnetic and transparent materials, χij = χji, i.e. the χ tensor is real and symmetric.
[1] 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: The directions x, y and z are in this case known as the principal axes of the medium.
This phenomenon is known as birefringence and occurs in some common crystals such as calcite and quartz.
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.
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.
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.
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.
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.