If a light beam passes through an anisotropic dielectric crystal whose electrical permittivity is ε≠1 and direction-dependent, the electrical flux density D of the electric field component E of the light via the material equation

D=εE

depends on its direction of propagation relative to the crystal, where ε is the permittivity vector. This indicates the value of the permittivity along the respective main axis(x,y,z). Using the relationship

n_i=√(ε_i / ε_0), i∈{x,y,z}

where ε_0 is the permittivity of the vacuum, the three main refractive indices n_i of the respective crystal follow from this.

The refractive index n generally influences the phase velocity of the light:

c=c_0/n

where c_0 is the speed of light in a vacuum. As direction-dependent refractive indices occur in the anisotropic dielectric crystals, the phase velocity of the light is therefore direction-dependent when travelling through these crystals. In practice, this effect becomes interesting as soon as the polarisation plane of the electric field of the light beam is not parallel to one of the main axes of the crystal. In this case, the plane of polarisation is split into two components parallel to two perpendicular crystal axes, the so-called normal modes, and is delayed non-uniformly due to their different refractive indices. When the light leaves the crystal, its polarisation changes.
This effect is particularly easy to understand by looking at the propagation of linearly polarised light in the direction of one of the main axes of the crystal. In this case, there are only two possibilities: Either the plane of polarisation of the light is parallel to one of the other two main axes or it is not. In the first case, there is no change in the polarisation of the light, as there is only one normal mode. In the second case, on the other hand, the two normal modes created by the above-mentioned projection are delayed against each other, so that the light is generally elliptically polarised after passing through the crystal.

In the general case, where the light propagates in any direction, the two normal modes must be determined via the index ellipsoid. This three-dimensional geometric object is defined by the three main refractive indices:

1 = x²/n_x²+ y²/n_y²+ z²/n_z²

The refractive indices n_a and n_b of the two normal modes result from the index ellipsoid in the following way:

Firstly, the plane perpendicular to the direction of propagation k of the light is determined by the origin of the index ellipsoid. The intersection of this plane with the index ellipsoid is the index ellipse. The two main axes of this ellipse are then even n_a and n_b.

A simplifying special case is given by uniaxial crystals in which two refractive indices are equal.

For uniaxial crystals, the index ellipsoid becomes an index rotational ellipsoid, which is rotationally symmetrical to the optical axis z, i.e. n_x=n_y=n_0. The light, whose direction of propagation k encloses an angle θ with the optical axis z, then decomposes into the two normal modes with the refractive indices n_0 and n(θ). The normal mode with n_0 is referred to as an ordinary wave and the one with n(θ) as an extraordinary wave. n(θ) is obtained from the ellipsoid equation by inserting spherical coordinates:

1/n²(θ) = cos²(θ)/n_0²+ sin²(θ)/n_z²

Depending on θ, the refractive index of the extraordinary wave thus moves between n_0 and n_z.

The following video shows the dynamics of the refractive index n_0 (pink) of the ordinary wave and n(θ) of the extraordinary wave (purple) as a function of the direction of propagation of the light k (red). The index ellipsoid is shaded grey and the index ellipse blue.
The constant refractive index n_0 of the ordinary wave is clearly visible and the shape of the index ellipsoid shows the property n_0 > n_z of the crystal shown here.