Index ellipsoid

In optics, an index ellipsoid is a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive indices in a crystal.

The equation for the ellipsoid is constructed using the electric displacement vector D and the dielectric constants. Defining the field energy W as

$8\pi W= \frac{D^2_1}{\varepsilon_1} + \frac{D^2_2}{\varepsilon_2} + \frac{D^2_3}{\varepsilon_3}.$

and the reduced displacement as

$R_i= \frac{D_i}{\sqrt{8\pi W}},$

then the index ellipsoid is defined by the equation

$\frac{R_1^2}{\varepsilon_1} + \frac{R_2^2}{\varepsilon_2} + \frac{R_3^2}{\varepsilon_3} = 1.$

The semiaxes of this ellipsoid are dielectric constants of the crystal.

This ellipsoid can be used to determine the polarization of an incoming wave with wave vector $\vec s$ by taking the intersection of the plane $\vec R \cdot \vec s =0$ with the index ellipsoid. The axes of the resulting ellipse are the resulting polarization directions.

Indicatrix

An important special case of the index ellipsoid occurs when the ellipsoid is an ellipsoid of revolution, i.e. constructed by rotating an ellipse around either the minor or major axis, when two axes are equal and a third is different. In this case, there is only one optical axis, the axis of rotation, and the material is said to be uniaxial. When all axes of the index ellipsoid are equal, the material is isotropic. In all other cases, in which the ellipsoid has three distinct axes, the material is called biaxial.