Waveguide phase and group velocities ($v_{\mathrm{ph}}{v}_g=c^2$)

Above a mode's waveguide cutoff frequency ${\omega }_c$, its phase velocity is $c/[1-({\omega }_c/\omega {)}^2{]}^{1/2}$ and its group velocity is $c[1-({\omega }_c/\omega {)}^2{]}^{1/2}$. Differentiate ${\omega }^2=c^2{k}^2+{\omega }_c^2$ to obtain the latter. Phase travels faster than $c$ while the wave packet travels slower, so the phase speed does not imply superluminal information transport.