zdenek@heathcliff.columbia.edu (Zdenek Radouch) (10/15/86)
>First, you mention in passing that the apparant velocity of X-Rays in >glass is greater than c. I expect this is a reference to the group vs >phase velocity problem, otherwise an X-Ray laser driving fiber optics >would provide a faster than light channel. Would you please elaborate? Yes, it was phase velocity. We have been talking about refraction and we concluded that it is possible to introduce apparent velocity (phase velocity) to account for the phase shifts, caused by scattered light. When the source is oscillating, the phase of the wave observed at O is advanced with respect to the phase of the source. BUT THAT DOESN'T MEAN THAT THE WAVE AT O STARTED EARLIER!!!!! Let's look back at our little universe | 0 0 vacuum | vacuum O | * oscillating charge atom an observer (source) and ask a question. Are we sending an information from our source to the observer? The answer is NO! If we want to send a signal to the observer, we have to CHANGE something in our transmitter. It doesn't matter what we do - modulate, turn off, turn on - as long as we cause some change in the wave. This change then propagates to the observer and it's velocity (group velocity - always smaller than c) determines, how fast our communication channel really is. It's really important to understand the distinction between the phase and group velocities, so we shall look at them more closely. Let's consider a simple case of modulation. Say, amplitude modulation. We are broadcasting a carrier cos(wt) and changing its amplitude according to a signal cos(wst). The broadcast signal is then cos(wst)*cos(wt). We still remember from the school, that cos(A)cos(B) = 1/2[cos(A+B) + cos(A-B)]. If we use this identity and analyse the broadcast signal, we get: cos(wt) cos(wst) = 1/2[cos(wt+wst) + cos(wt-wst)] = 1/2[cos(w1t) + cos(w2t)] where w1 = w + ws, and w2 = w - ws. This is actually a very simple case, but we can see, that if we modulate a wave it's an equivalent to sending more waves at the same time! We have figured out, that we can send a signal by generating two waves with frequencies w1 and w2. We can simplify our life a little bit by introducing "wave number" k=w/v. 1. VACUUM In a vacuum, both waves propagate with phase velocity = c, so k=w/c, ks=ws/vg (vg=group (signal) velocity) and k1=w1/c, k2=w2/c are wave numbers for the two broadcast waves. The sum of the two waves is: 1/2 [cos(w1t-k1x) + cos(w2t-k2x)] = cos(wt-(k1+k2)x/2) cos(wst-(k1-k2)x/2)= [ (k1+k2)/2 = w/c, (k1-k2)/2 = ws/c ] = cos(wt-kx) cos(wst-ksx) if we put vg=c! So there was no surprise at all. Both waves are propagating together; the phase between them doesn't change, and - as we know from adding two sine waves - the modulated signal "sits" on the waves and moves with them. Conlusion: In a vacuum, phase velocity and group velocity are same and equal to c! 2. GLASS As usual, the more interesting case is when the waves travel in a material. We understand now, that the phase velocity in a material depends on the material itself and on the frequency of the wave. Therefore the wave numbers are going to be different k1=w1/vp1, and k2=w2/vp2, where vp1 and vp2 are the phase velocities of the two broadcast waves. Let's add the waves again: 1/2 [cos(w1t-k1x) + cos(w2t-k2x)] = cos(wt-(k1+k2)x/2) cos(wst-(k1-k2)x/2) That's about as much as we can do. We can see, that the waves travel from the source to destination with different velocities, which means, the phase between them is constantly changing. But we still want to get at least a feeling for the group velocity. It's time for a careful simplification. The frequency of our signal is much lower than the carrier, which implies that (w1-w2)->0. In this limit case k1~k2~k = w/vp, so k1+k2/2 -> k. Nothing can be done with k1-k2/2 so we just call it ks. The sum is now: cos(wt-(k1+k2)x/2) cos(wst-(k1-k2)x/2) =~ cos(wt-kx) cos(wst-ksx) The first term is again our carrier wave, propagating with mean phase velocity, the second term is our signal and its obvious that its velocity i.e. the group velocity is different! There is another way of representing the two waves. The following are "pictures" of two waves, traveling from left to right. The pictures were taken at times t0, t0+1, t0+2, t0+3. Wave A has wavelength 8 and phase velocity 11, wave B has wavelength 7 and phase velocity 10. The signal (one full beat) is marked sssssssss. +...A...+.......+.......+.......+.......+.......+.......+.......+.......+.... .+..B...+......+......+......+......+......+......+......+......+......+...... sssssssssssssssssssssssssssssssssssssssssssssssssssssssss ...+.......+...A...+.......+.......+.......+.......+.......+.......+.......+... ....+......+..B...+......+......+......+......+......+......+......+......+.... sssssssssssssssssssssssssssssssssssssssssssssssssssssssss ......+.......+.......+...A...+.......+.......+.......+.......+.......+.......+. .......+......+......+..B...+......+......+......+......+......+......+......+. sssssssssssssssssssssssssssssssssssssssssssssssssssssssss .+.......+.......+.......+.......+...A...+.......+.......+.......+.......+..... ...+......+......+......+......+..B...+......+......+......+......+......+..... sssssssssssssssssssssssssssssssssssssssssssssssssssssssss There is no need for a commentary - the group velocity = 3! > >Second, what about Cerenkov(sp?) radiation? If the slower than c >apparant velocity is an effect of the induced radiation from electrons, >why should a particle moving through the intervening vacuum cause >radiation? Doesn't make sense, does it! Well, if you send a particle through the vacuum there won't be any radiation (assuming no other forces). Cerenkov radiation actually refers to a special case of a large group of phenomena. They all have in common, that an object is moving in a MEDIUM and it's velocity, is greater than the phase velocity of the wave, it generates. The result is a conical wave (as opposed to a spherical, which would be generated if the source was at rest) with the source (moving object) at the tip of the cone. You know the effect very well. If you drop a stone into the water, you can see circular waves propagating from the point where it hit the water. That's the "rest" case, and you probably remember, that the waves don't propagate too fast. It should be then possible to see the conical (or in this planar case - triangular) wave, if you could move the stone on the water faster than the waves. And surely, if you look at the waves behind you when you are waterskiing - the triangle is there! We understand now, that wave can propagate in a piece of glass with phase velocity smaller than c. We also know, that it's possible to accelerate an electron so that it's velocity is close to c. If we shoot such an electron into a piece of glass we are going to get the conical wave again, this time only light wave. The light generated (or more precisely the cone angle) can be used to calculate the velocity of the electron. The phenomenon is referred to as Cerenkov radiation. For more interesting details, get "Free Electron Lasers" by T. Marshall (Macmillan 1985). zdenek ------------------------------------------------------------------------- Men are four: He who knows and knows that he knows, he is wise - follow him; He who knows and knows not that he knows, he is asleep - wake him; He who knows not and knows that he knows not, he is simple - teach him; He who knows not and knows not that he knows not, he is a fool - shun him! zdenek@CS.COLUMBIA.EDU or ...!seismo!columbia!cs!zdenek Zdenek Radouch, 457 Computer Science, Columbia University, 500 West 120th St., New York, NY 10027
guy@slu70.UUCP (10/17/86)
In article <3494@columbia.UUCP>, zdenek@heathcliff.columbia.edu (Zdenek Radouch) writes: > 2. GLASS > > As usual, the more interesting case is when the waves travel in a material. If you want an even more interesting case, look at crystalline materials, which are not only dispersive (i.e., group and phase velocity differ) but are also anisotropic. This means that the polarization of the wave is a factor as well as the velocity. This behavior is used extensively by geologists to identify minerals using a polarizing microscope as each mineral passes polarized light in a somewhat different fashion (yes, I know I've oversimplified).