Phase and Group Velocity

If $\epsilon$ and $\mu$ are constant, then from (3.10), the dispersion relation for Maxwell's Equations has the form

$$\omega^{2}\left(\epsilon\mu\omega^{2}-\Vert\mbox{\boldmath$\beta$}\Vert _{2}^{2}\right)^{2} = 0$$

in terms of frequencies $\omega$ and wavenumber magnitudes $\Vert$$\beta$$\Vert _{2}$. This equation has solutions

$$\omega = 0\hspace{1.0in}\omega = \pm\frac{\Vert\mbox{\boldmath$\beta$}\Vert _{2}}{\sqrt{\epsilon\mu}}$$

Leaving aside the non-propagating mode with $\omega = 0$, the phase and group velocities will then be given by

$$\gamma_{Maxwell}^{p} = \gamma_{Maxwell}^{g} = \pm\frac{1}{\sqrt{\epsilon\mu}}$$

For spatially inhomogeneous problems, the maximum group velocity will be

$$\gamma_{Maxwell,max}^{g} = \frac{1}{\sqrt{(\epsilon\mu)_{min}}}$$