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Phase and Group Velocity

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

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

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

$\displaystyle \omega = 0\hspace{1.0in}\omega = \pm\frac{\Vert\mbox{\boldmath$\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

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

For spatially inhomogeneous problems, the maximum group velocity will be

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

Stefan Bilbao 2002-01-22