# Group, phase and signal velocity of light

• PWiz
In summary: The signal velocity is the speed of light in a medium when the light is traveling in a straight line.
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?

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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 traveling 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.

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?

## 1. What is the difference between group and phase velocity of light?

The group velocity of light refers to the speed at which the envelope or amplitude of a light wave travels, while the phase velocity refers to the speed at which the phase or position of the wave travels. In general, the group velocity is slower than the phase velocity.

## 2. How are group and phase velocity related?

The group and phase velocity are related by the dispersion relationship, which is a mathematical formula that describes the relationship between the wavelength and the frequency of a wave. In most cases, the group velocity is equal to the phase velocity when the light wave is propagating in a vacuum.

## 3. What factors affect the group and phase velocity of light?

The group and phase velocity of light can be affected by the medium through which it travels. In materials with a high refractive index, the group velocity can be significantly slower than the phase velocity. Additionally, the wavelength and frequency of the light wave can also affect the group and phase velocity.

## 4. How is the signal velocity of light different from the group and phase velocity?

The signal velocity of light refers to the speed at which information is transmitted through a medium. This can be affected by factors such as the absorption and scattering of light. It is different from the group and phase velocity, which only describe the movement of the light wave itself.

## 5. Can the group or phase velocity of light exceed the speed of light in a vacuum?

No, according to the theory of relativity, the maximum speed at which any object, including light, can travel is the speed of light in a vacuum. While the group and phase velocity of light can change depending on the medium, they can never exceed the speed of light in a vacuum.

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