Index ellipsoid

Remarks

For a unit vector $\mathbf{s}$, the plane through the origin orthogonal to $\mathbf{s}$ cuts the ellipsoid in an ellipse. The lengths of its semi-axes are the refractive indices of the two plane waves with wave normal $\mathbf{s}$, and the axes give the directions of their $\mathbf{D}$ fields.

Example: for $\varepsilon = \mathrm{diag}(n_o^2, n_o^2, n_e^2)$ the index ellipsoid is a spheroid about the $z$-axis. The plane orthogonal to $z$ cuts it in a circle of radius $n_o$, so light along $z$ has one index. The plane orthogonal to $x$ cuts it in an ellipse with semi-axes $n_o$ along $y$ and $n_e$ along $z$.

Non-example: if $\varepsilon$ has one or two negative eigenvalues, as in some artificial media, the quadric $\mathbf{r}^T \varepsilon^{-1} \mathbf{r} = 1$ is a hyperboloid, not an ellipsoid. The construction needs $\varepsilon$ to be positive definite.

A general biaxial ellipsoid has exactly two central circular sections; the normals to them are the two optic axes.