Extraordinary phase-index angular dependence

For a lossless uniaxial dielectric, $n_{\mathrm{ex}}^{-2}={\cos }^2\vartheta /n_o^2+{\sin }^2\vartheta /n_e^2$, where $\vartheta$ is the angle between the wave vector and optic axis. Along the axis $n_{\mathrm{ex}}=n_o$, while perpendicular propagation gives $n_{\mathrm{ex}}=n_e$.