# A query on group velocity of light in a vacuum of 30c

I have read that the speed of information propagation in a light packet is equal to the group velocity. But here I wonder if that is always correct.

This article below that claims to have experimentally attained group velocities in light in a vacuum of up to 30c, where c is the speed of light in a vacuum

https://www.nature.com/articles/s41467-019-08735-8

They do it by controlling the spatial-frequency spectrum. Specifically they set

$$\frac{\omega}{c}=\frac{\omega_{0}}{c}+\left ( k_{z} -k_{0}\right )tan\left ( \theta \right )$$

with the wave vector satisfying the light cone condition:

$$k_{x}^2+k_{x}^2=(\frac{\omega}{c})^2$$

They create the superposition:

$$E(x,z,t)=e^{i\left ( k_{0} z- \omega_{0} t\right )}\int dk_{x} \tilde{\psi\left ( k_{x} \right )} e^{ i(k_{x}x+[k_{z}-k_{0}][z-ct\cdot tan(\theta) ])}$$

which equals

$$E(x,z,t)=e^{i\left ( k_{0} z- \omega_{0} t\right )}\psi(x,z-v_{g})$$

Thus they create a wave with group velocity:

$$v_g=\frac{d \omega}{dk_{z}}=c \cdot tan(\theta)$$

which they can make go up to 30c by suitable choice of projection angle $$\theta$$

I assume that the speed of propagation of information here is actually c. So my questions is:

When is the speed of information propagation equal to the group velocity, and when is it not?

• You may find valuable information for your question in the answers and links connected to another question on the same subject: physics.stackexchange.com/questions/5326/… – GiorgioP Jun 16 '19 at 20:45
• To answer to your direct question about the speed of information, I think that you should make explicit what is your definition of physical information. – GiorgioP Jun 16 '19 at 20:48
• Ok. I will try to state what I mean by information. Information propagation as in the ability to encode a "bit" of information, i.e. interpret a pattern as a 0 or 1. Then encode that pattern in the waveform, have it propagate from point A to point B and then be read at point B. I assume that in this case, the fastest you could send such a signal would be c. – Rory Cornish Jun 16 '19 at 21:03
• GiorgioP. Thanks. That is exactly what I was thinking. That you have a packet moving with a wavefront speed c. And that crucially, the vg> c is contained in that localized wave packet. Thus information cannot get from A to B faster than c. That is exactly the picture that I had in my mind. But I couldn't find any resources confirming it. – Rory Cornish Jun 16 '19 at 21:08

When is the speed of information propagation equal to the group velocity, and when is it not?

Waves in a non-dispersive medium behave in a relatively simple way. In 1D, the basic solution of wave equation is a profile which travels without modification in one direction. In more than 1D, some changes of the wave profile are required by the conservation of energy, but it remains possible to measure the speed of propagation of the wave profile with a unique parameter: the phase velocity which directly enters in the wave equation. In such a simple case, the energy of a pulse like a gaussian packet travels with the phase velocity.

As soon as dispersion enters into play, i.e. as soon as the frequency is a non-linear function of the wavenumber, things become immediately more complicate. However, in the presence of weak dispersion, which in practice means when the deformation of the wave induced by dispersion is not strong enough, it is possible to introduce a second velocity, the group velocity, as a measure of the speed the center of a localized pulse travels. The usual expression for group velocity $$v_g = \frac{\partial{\omega}}{\partial{k}}$$ comes from an asymptotic analysis of the motion in the space of the important part of a wave-packet made by plane waves of different wavelengths. In the process of derivation of the above expression, explicit use is made of the hypothesis that the non-linear terms in a Taylor series expansion of the frequency as a function of wavenumber are negligible.

In the case of strong dispersion, for example in the case of the so-called anomalous dispersion, the underlying hypotheses behind the introduction of group velocity as a measure of the speed of the main part of the wave-packet, and then of its energy, cease to be valid. A signal of such a condition is clearly the presence of a group velocity larger than the speed of light in the vacuum.

In such conditions, a much more refined analysis is required. First steps in this directions were pioneered in the last century by Sommerfeld who introduced the concept of velocity of the signal. This is the velocity the important part of the wave travels. It coincides with group velocity in the case of weak dispersion, but in general differs from the group velocity because it can never be larger than $$c$$. The full story is actually even more complex, because, in addition to the "signal" it is possible to introduce the so called precursors which do travel at speed $$c$$. The final stage of this story of asymptotic analysis is that, depending on how one defines the speed of a wave, there could be up to nine different speeds! However, speed which are associable with transfer of information are all smaller than $$c$$.

A first, simplified accound of this story can be found in Jackson's book on Electrodynamics and maybe it is enough for your needs. I read more technical information in a review paper appeared in Optics around 2000. If I could find the exact reference, I'll add here.

• Thanks GiorgioP, for a brilliant answer. It is a fascinating subject. The subtitles in wave signal analysis never cease to impress me. I will look your references up as I would love to know more about this. – Rory Cornish Jun 17 '19 at 13:04