# Group, phase and signal velocity of light

1. May 13, 2015

### PWiz

According to Maxwell's equations,
$$c=\frac 1 {\sqrt{μ_0 μ_r ε_0 ε_r}}$$
in a medium with an electric permittivity of $ε_r$ and magnetic permeability of $μ_r$. This means that in any medium which has values for these properties which are greater than that of a vacuum, the speed of light should be less than what is observed in a vacuum.
What is this calculated velocity? Is it the signal velocity, group velocity or phase velocity? What is the difference between these terms? I read somewhere that certain particles can move faster than $c$ in some mediums, and for these particles the speed of light in that medium is negative. What does this mean? Do these kind of mediums have electric permittivities and magnetic permeabilities which are lower than that of a vacuum?
And according to special relativity, any object moving faster than light would violate causality. Shouldn't it be impossible for information to be transmitted faster than light?

Last edited: May 13, 2015
2. May 13, 2015

### Zarqon

The formula you posted is the phase velocity. One thing to remember is that the material dependent "constants", e.g. εr, are not constant but depend on certain variable like the frequency of the light, and this causes the other effects you mentioned.

A different group velocity for a pulse of light arises when the frequency dependence in the material is so strong that the different frequency/Fourier components of the pulse travel at different speeds. Then, in addition to each travelling with their phase velocity, they will get a phase shift relative to each other. This phase shift will cause them to be in phase (where the max of the peak of the pulse is) at the different time than without such a frequency dependence. This can cause the pulse peak to appear either before or after the time of any individual phase has reached a certain point, leading to either fast or slow light, where the fast light can even travel faster than c in vacuum (but never gain more time than the pulse duration, since the re-shifting effect obviously can't happen outside of the pulse).

The signaling velocity is not fixed, but also frequency dependent. In principle, the signaling velocity is not limited by any material properties, because for high enough frequencies, the signaling velocity of light is always c, regardless of material (as the refractive index of any material approaches 1 as the frequency goes to infintity).

This was a brief answer, but perhaps gives you enough information to google and read some yourself.

3. May 23, 2015

### PWiz

Okay, I think I get it. If the waves peak out earlier than they would in a vacuum, then the group velocity is higher than $c$ even though the phase of the individual frequencies is propagating through the medium at speeds lower than $c$ , right?
And does this means that the speed which the 2nd postulate of SR talks about is phase velocity?

I still don't understand what is signal velocity. Can you please elaborate on that?