Birefringence

Remarks

In a birefringent medium, for each direction of propagation there are two plane-wave modes with orthogonal directions of $\mathbf{D}$ and, in general, different refractive indices. The index ellipsoid gives both indices and both polarizations. In a uniaxial medium with $\varepsilon = \mathrm{diag}(n_o^2, n_o^2, n_e^2)$, the $z$-axis is the optic axis, $n_o$ is the ordinary index and $n_e$ the extraordinary index, and $\Delta n = n_e - n_o$ is the birefringence.

Example: take $\varepsilon = \mathrm{diag}(n_o^2, n_o^2, n_e^2)$ with $n_o \ne n_e$ and light travelling along the $x$-axis. Light polarized along $y$ sees the index $n_o$, and light polarized along $z$ sees $n_e$. For light travelling along the optic axis $z$, both polarizations see $n_o$, so there is no splitting in that one direction.

Non-example: ordinary glass has $\varepsilon = n^2 I$, a multiple of the identity, so it is not birefringent.