# What really is the speed of light in a medium/vacuum, group or phase velocity?

While reading about refractive index 2 terms popped up, group velocity which alway slows down in a medium and phase velocity which may exceed speed of light. Say in a complete vacuum and using laser with only 1 frequency as an example, which of the two kinds of velocity is defined as standard $$c$$ in physics?

The answer to your question is that in a vacuum for light, the two are equivalent. If you solve Maxwell's equations in a vacuum what you will find is that

$$E(x,t) = E_0 \cos(kx - \omega t)$$

likewise for the magnetic field (in magnitude). You will also find that these solve maxwell's equations only when

$$\frac{\omega}{k} \equiv v_p = \frac{1}{\sqrt{\mu_0\epsilon_0}} = c$$

That is, in a vacuum $$\omega = c k$$. From which is follows that

$$\frac{d\omega}{dk} \equiv v_g = c$$

as well.

To answer your question directly, what we usually mean by "speed of light" is the group velocity, as this is the one that Einstein's postulate says should be constant (the max speed at which information can be transmitted). So while we could define the number either way I think it's convention to define the speed of light in vacuum to be referring to the group velocity.

Having said that, with respect to your question about what really is the speed of light in a medium/vacuum, that's not ontologically the right question to ask. That is, it's not that one exists in any more fundamental sense than the other. There are simply two different velocities and we always need to specify which one we are talking about.

Even though the question in the text has a simple answer (phase and group velocity in vacuum are the same, due to the linear dispersion), the general question put in the title ( What really is the speed of light in a medium/vacuum, group or phase velocity? ) has a much more complex answer in the case of a medium.

The case of anomalous dispersion makes clear that in some cases even the group velocity may become higher than $$c$$. In such cases, the relevant velocity (the speed of energy and information transfer) is something else. Quite often, people refer to the signal velocity as the appropriate generalization. However one has to notice that, under such extreme dispersion condition, the deformation with time of a wavepacket may be significant and up to nine different velocities have been identified. In the classical Jackson's textbook on electrodynamics you may find the definition of a couple of these additional velocities (the forerunners velocities).

• Yes, you are absolutely right about these subtleties, and I do want to make that clear. The signal velocity is the appropriate generalization, and I just didn't want to get into Jackson level stuff given the level of the question :) – InertialObserver Dec 26 '18 at 9:22