Group and Phase velocity

[Gear300]

Group and Phase velocity of Light

I read that in cases of anomalous dispersion, the group velocity of an electromagnetic wave can exceed the speed of light c in a vacuum. How does this fit in relativistically?

 

[Dr Lots-o'watts]

Group velocity is not the speed of any photon.

 

[Gear300]

Though it still correlates to packets of information, doesn't it (I read that this does not imply superluminal communication, but I'm not too knowledgeable of the argument)?

 

[Dr Lots-o'watts]

In free space, no velocity (group, phase etc.) will excede c.

As I understands it, when group velocity exceeds c, it is necessarily in special matter (amplified crystal), and the final pulse is not the same as the initial one. It can be "engineered" to look like the initial one, but it does not contain the same photons.

Imagine a camel with its head lowered, entering a tunnel. Its hump is the "peak". As it walks inside the tunnel (amplification medium), it raises its head (amplification of the faster wavelengths). When it comes out, the head is now the "peak". So the peak has traveled faster (group velocity) that the walking speed (average phase velocity) of the camel.

OMG! The peak of the camel is traveling faster than the camel! We brought causality to its knees! ;-)

So my question is which part of the camel is information?

 

[collinsmark]

Many "things" can and do greatly exceed the speed of light (some of them even in a vacuum). By a lot. And this happens all the time. (Seriously, this whole group velocity thing is nothing new at all [in terms of something moving faster than c], and way overblown.) But when taken by themselves, these things cannot have mass or momentum. They are real, but not tangible. A shadow is common example.

Imagine a movie projector and a screen some distance away. Suppose an insect flies in the path of the projector and the screen. The insect's shadow on the screen not only appears bigger, but it flies faster than the actual insect.

Now shine the projector at the Andromeda galaxy. Allow an insect to fly in front of the projector. In about two million, five hundred thousand years, the shadow of that insect will whiz from one side of the Andromeda galaxy to the other in mere moments.

In a sense the shadow contains information (if you wish, substitute the insect with a transparency containing a message). And some might say that information passes from one side of the Andromeda galaxy to the other much, much faster than the speed of light. And technically that's true. But the important point is that nowhere in this scenario is it possible to communicate any information faster than the speed of light. An alien on one side of the Andromeda galaxy, seeing the shadow first, can do nothing to piggyback upon the shadow to send a message to an alien on the other side of the galaxy. So even though the information itself is moving faster than light, it is still totally impossible to communicate information faster than light.

Group velocity is something like a shadow. Yes, in a sense it does contain some phase information about the photons making up the light. So it is associated with information, yes. So one might state that since group velocity moves faster than light (in certain setups), and it does involve information, then some sort of information is moving faster than light. While technically true, that's sort of misleading. It's not the same type of information that is useful for communication. Group velocity's information is no more useful than the knowledge that a shadow is moving. Ya' can't do anything with it. And most certainly, ya' can't influence anything with it such that the original cause to subsequent effect travels faster than light.

 

[Dr Lots-o'watts]

Another analogy that came to mind:

Suppose a group of marathon runners. Their distribution along the course can be gaussian i.e. a pulse. If for some reason (perhaps the passing of the first runner ahead of the gaussian group), all the spectators between the group and the finish line decide to join in the marathon, the "group" has suddenly shifted forward and will collectively arrive at the finish line sooner than would allow the average speed of the original runners. The new group can form continuously but it is mostly not composed of the same element as the original. The peak of the group has traveled much faster than that of any individual runner. In addition, some of the original runners could have abandoned the group, which shifts the average, or peak position, forward even faster.

In this case, the final signal, received by the finish line, originates mostly from the spectator stands, not from the starting line (except for the first original runner, who may have triggered the influx of new runners). The shape of the group, or pulse may not have changed, and its peak may have gone faster than light for a while, but no runner has exceeded the speed of light.

Similarly, in a light pulse scenario, most of the photons coming out ahead of of the amplified medium, are not from the original pulse. The final pulse is simply engineered to have the same shape as the original (through anomalous dispersion relation, essentially triggering of many new photons by a few photons from the front edge of the original pulse).

Interpretation inspired by:

http://www.nature.com/nature/journal/v406/n6793/full/406277a0.html