What is group velocity and how does it relate to the speed of light?

[KingNothing]

What is group velocity? I have found a few small definitions, but for some reason I'm not quite understanding how it works.

Here's one bit I have read:
"Recent experimental evidence shows that it is possible for the group velocity of light to exceed c. One experiment made the group velocity of laser beams travel for extremely short distances through caesium atoms at 300 times c. However, it is not possible to use this technique to transfer information faster than c; the product of the group velocity and the velocity of information transfer is equal to the square of the normal speed of light in the material.

Exceeding the group velocity of light in this manner is comparable to exceeding the speed of sound by arranging people in a distantly spaced line of people, and asking them all to shout "I'm here!", one after another with short intervals, each one timing it by looking at their own wristwatch so they don't have to wait until they hear the last person shouting."

Another bit:
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitude (known as the envelope of the wave) progagates through space. It is often thought of as the velocity at which energy or information is conveyed along a wave. In most cases this is accurate, and the group velocity can be thought of as the signal velocity of the waveform.

It is however possible to design experiments where the group velocity of laser light pulses sent through specially prepared materials significantly exceeds the signal velocity, and even exceeds the speed of light.

Basically, what I am drawing from this is that group velocity is basically setting up events to happen, and measuring them as if they were dependent upon each other, when they really arent.

I have this feeling that I am really far off.

 

[turin]

Basically, what I am drawing from this is that group velocity is basically setting up events to happen, and measuring them as if they were dependent upon each other, when they really arent.

That is exactly the impression I got when I read both of the quotes that you gave.

 

[chroot]

The phase velocity is the velocity at which a wave of a single frequency will propagate. The phase velocity is the velocity with which most people are innately familiar; when people say "wave velocity" or "velocity of propagation," they really mean "phase velocity."

When you combine waves of several different frequencies, something different happens: you get interference. In an extreme example, you can combine several waves such that they cancel out everywhere except in a small area of space. The envelope of the resulting "wave packet" is not really a wave proper -- it's the result of the interference of several component waves. The envelope of the wave packet, the peak of constructive interference, can travel at a different speed from the phase velocity, the velocity of each of its individual component waves.

Here's a website with some examples of waves with different combinations of group and phase velocities:

http://krypton.mankato.msus.edu/~7364eb/Math113/groupvelocity.html

- Warren

 

[ZapperZ]

The phase velocity is the velocity at which a wave of a single frequency will propagate. In other words, the energy, or information, or whatever, contained in the wave propagates at the phase velocity. The phase velocity is the velocity with which most people are innately familiar; when people say "wave velocity" or "velocity of propagation," they really mean "phase velocity."

When you combine waves of several different frequencies, something different happens: you get interference. In an extreme example, you can combine several waves such that they cancel out everywhere except in a small area of space. The resulting "wave packet" is not really a wave proper -- it's the result of the interference of several component waves. The wave packet, the little peak of constructive interference, can travel at a different speed from the phase velocity, the velocity of each of its individual component waves. The velocity of the wave packet, which is called its group velocity, can be smaller, or larger, than the phase velocity. In fact, there are no physical restrictions at all about group velocity. You can easily add several waves of light, each of which moves with phase velocity c, and produce a wave packet which "moves" with a group velocity much, much larger than c.

It is important to recognize (though it might take you some thinking to make this conclusion) that no energy or information is actually transmitted by the wave packet, only by its constituent "real" waves.

- Warren

Actually, I think you may have it the other way around. It is the group velocity that typically has a c limit and the one that carries information - the modulation of the group wave is a "signal packet" essentially. The phase velocity can have no restriction and also carries no information.

http://www.mathpages.com/home/kmath210/kmath210.htm

The "apparent" superluminal group velocity in the NEC expt of a few years ago is due to the medium - it reshaped and "assisted" the pules, so that the front foot of the gaussian pulse is amplified while the peak was attenuated. This created the appearence as if the peak of the pulse moved through the medium faster than c. In reality, no part of the pulse moved faster than c.

Zz.

 

[chroot]

*rubs chin* oof.. right.

- Warren

 

[outandbeyond2004]

Group velocity is always less than c or equal in some cases including free space, according to my electromagnetism textbook; phase velocity is greater (or equal).

Let v_g be the group speed and v_p be the phase speed. (speed is the proper term, tho 'group velocity' and 'phase velocity' are the usual terms) Then

v_g * v_p = c*c

If v_g did exceed c, that would violate not only Maxwell's equations but Special Relativity as well. One should be cautious about taking such experiment reports as reliable. People have been fooled by phase speeds before. The distinction between group speeds and phase speeds is poorly held by far too many scientists. One thing to keep in mind is that there is no such thing as a perfect train of cycles (infinite in length and monochromatic) in nature, though you could have a single photon, I suppose. Another thing is that normally a carrier wave conveys no information. As soon as you 'imprint' info on the wave, the phase speed may exceed c, but that would only mean v_g < c.

It may seem that we should ignore the distinction between v_g and v_p as being too confusing and unnecessary, but in reality some experiments confuse some people who just don't analyze them carefully enough.

 

[KingNothing]

I'm trying my best to understand...but now I'm not quite sure what part of chroot's post were true and what parts were false. I seemed to get a fairly good idea from reading that, but then a few things started getting backwards...and I am in the fried egg state you see here.

 

[chroot]

I just edited my post -- maybe it will make more sense now. :)

- Warren

 

[outandbeyond2004]

Decker, don't be hard on yourself. It is not clear to me either, and I believe most people would feel like scratching their head, too.

You do know that in a single frame of reference things often appear to separate at speeds exceeding c? We have a similar situation wrt mixtures of monochromatic waves. Remember there is no such thing as a monochromatic wave train. It would have to be infinitely long to stay monochromatic. Each finite wave is in reality a mixture - or could be analyzed as such of monochromatic waves (heard of Fourier transform?)

Perhaps this will help. http://www.physics.nmt.edu/~raymond/classes/ph13xbook/node17.html

 

[outandbeyond2004]

The article that ZapperZ linked to is my favorite explanation.

 

[KingNothing]

Remember there is no such thing as a monochromatic wave train. It would have to be infinitely long to stay monochromatic. Each finite wave is in reality a mixture - or could be analyzed as such of monochromatic waves (heard of Fourier transform?)

I really don't knwo much about waves or trains. But it seems like it relates to something I used to ponder in my elementary school - when we were doing patterns. We'd have something in the book like 363636...and we'd have to continue the pattern. Of coruse the answer was just more 3's and 6's, but one class period I brought up that the 363636 could also just be a 'member' of a larger pattern, that is to say - the pattern could also be 363636TTTT363636TTTT or something else that just hasn't been shown yet.

It's probably really far off, but that's what it seems like when talking about a 'mixture' and a 'train' and such. The connection that I make is that the entire pattern (infinitely long) would be like a truly 'monochromatic wave' however its technically a 'mixture' in that it can't be infinitely long...just like the pages in our textbooks, they can't literally write out the pattern infinitely.

The whole subjects getting a little bit clearer - this helps me a lot when I do it this way. I like to try to draw my own conclusions and be corrected when I'm wrong.

 

[outandbeyond2004]

Oh, oh, the article that ZapperZ linked to may be a bit advanced for you. I hope not, it made things more clear for me.

 

[turin]

I've always understood group velocity to mean what it sounds like, i.e. the herd of light rays moving down an optical fiber. Of course, this is slower than c, and even slower than u (= c / index of refraction), because they don't all travel straight down the fiber (only the TE₀₀ travels straight down the fiber).

I've always understood phase velocity to be what the first post says about group velocity. My favorite illustrative examples are the following:

1) Imagine a wave coming into a straight shoreline at a slight angle. It will hit the shoreline at one point, and this point will move along the shoreline. The speed at which the point of contact moves is the phase velocity. It is always greater than the speed at which the wave is traveling towards the shore. In the extreme that the wave is itself moving along the shore (angle of incidence = 90^o), the point moves with the wave at the wave velocity. In the other extreme that the wave is heading straight for the shore (angle of incidence = 0^o), the point of contact exists everywhere along the shore at the same time, which is the infinite phase velocity case.

2) A radio pulse is sent from α-centari towards the solar system. It reaches the sun first, then the Earth about 1 min. later. Thus, the signal reaches the Earth only 1 min. after it was at the sun. This does not violate SR because the signal did not causally come from the sun. The phase of the signal traveled from the sun to the Earth in 1 min., not the signal itself.