# Classification of Anisotropic Media

In the previous sections, we considered the surface of wave normals and the index ellipsoid, which both had refractive indices associated with the $x$, $y$ and $z$ directions of $n_x$, $n_y$, and $n_z$. If these three indices are different, then the crystal is called biaxial because, as shown in the normal surface plot, there are two optical axes, hence “bi”. If two of the refractive indices are equivalent, i.e. $n_x = n_y \neq n_z$, then the crystal is called uniaxial. In this case, the normal surface equation can be further factored to give:

\begin{equation} (\frac{k_x^2+k_y^2}{n_e^2} + \frac{k_z^2}{n_o^2} – \frac{\omega^2}{c^2})(\frac{k^2}{n_o^2} – \frac{\omega^2}{c^2}) = 0 \end{equation}

where the terms in the first parenthesis give the equation for an ellipse and the terms in the second parenthesis give the equation for a circle. Note also that the $e$ stands for “extraordinary” and $o$ stands for “ordinary”; these will be used to describe the two possible eigenwaves in uniaxial crystals. Since the equation for the circle in the uniaxial case only includes $n_o$, this is the equation for the ordinary wave. As can be seen from this, the k-vectors would then be equivalent in all directions, meaning that the ordinary wave refractive index does not change for any direction of propagation.

We will make some notes here about convention. Typically, in a biaxial system, the principle coordinate axes are labeled such that the following relation is true:

\begin{equation} n_x < n_y < n_z \end{equation}

which leads to the case that intersection of the normal surface with the plane perpendicular to the propagation only occurs with the xz-plane; the other planes do not have optical axes. We will get to what that means in the biaxial crystals section. For uniaxial crystals, the crystal is said to be positive uniaxial if $n_e > n_o$; the crystal is said to be negative uniaxial if $n_o > n_e$.