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The extraordinary wave's electric displacement field lies in that plane and is perpendicular to the wavevector; its electric field need not be. For angle $\theta$ between the wavevector and optic axis,Thus $n_{\mathrm{ex}}(0)=n_o$ and $n_{\mathrm{ex}}(\pi /2)=n_e$.^[ref]

A short derivation starts from the existing plane wave in an anisotropic dielectric equation. Put $\mathbf{k}=(k_x,0,k_z)$ and $k_0=\omega /c$, with angular frequency $\omega$ and vacuum speed of light $c$. The $y$ component gives the ordinary wave relation $k_x^2+k_z^2=n_o^2{k}_0^2$. The determinant of the $x,z$ system givesSubstituting $k_x=n_{\mathrm{ex}}{k}_0\sin \theta$ and $k_z=n_{\mathrm{ex}}{k}_0\cos \theta$ yields the displayed directional refractive index.

Substitution of the sinusoidal plane waves into Faraday's law givesThe source-free Ampère-Maxwell equation, with $\mathbf{H}=\mathbf{B}/\mu$ and $D_i={\epsilon }_{ij}{E}_j$, givesEliminating $\mathbf{b}=(\mathbf{k}\times \mathbf{e})/\omega$ yields the wave equation for a plane wave in an anisotropic dielectric:

For a uniform linear medium with isotropy and local constitutive response, Maxwell equations give $k^2={\omega }^2\epsilon (\omega )\mu (\omega )$. Here $\epsilon$ is permittivity and $\mu$ is magnetic permeability. Since $c^{-2}={\epsilon }_0{\mu }_0$, the relative permittivity ${\epsilon }_r=\epsilon /{\epsilon }_0$ and the relative magnetic permeability ${\mu }_r=\mu /{\mu }_0$ obeyIf both responses are real and positive with negligible loss, and the ordinary positive-index branch is intended, then $n=+\sqrt{{\epsilon }_r{\mu }_r}$. If additionally ${\mu }_r\simeq 1$, then $n\simeq \sqrt{{\epsilon }_r}$. All these responses must be evaluated at the electromagnetic wave's frequency; a static relative permittivity cannot simply be substituted at an optical frequency. ^[ref]

With loss, use a complex index $\tilde{n}=n^{\prime }+i\kappa$ and explicitly fix the phase convention for complex amplitudes. For $E\propto \exp [i(kz-\omega t)]$ with $k=\tilde{n}\omega /c$, its wave amplitude has factor $\exp (-\kappa \omega z/c)$. A forward attenuating branch has $\kappa \ge 0$, while its wave phase is controlled by $n^{\prime }$. The complex relation ${\tilde{n}}^2={\epsilon }_r{\mu }_r$ alone leaves a choice of sign: the physical solution also needs causality and the appropriate radiation or decay condition. In negative-index media the physically selected real part can be negative. The positive square root is therefore not a universal rule for complex or negative-index media. Direction and polarization of an electromagnetic wave also matter in a plane wave in an anisotropic dielectric, for which one scalar index need not describe every propagation branch.