Fast-Light Tutorial

Contrary to popular belief, Einstein's special theory of relativity does not forbid everything from traveling faster than the speed of light in vacuum, c. Relativity only restricts information to speeds of c or less (and of course anything that would take information with it). As an example of something that is not restricted by relativity, imagine the spot created by shining a laser pointer at the moon1. If you (standing here on Earth) simply flick your wrist, the spot on the moon will travel from one side to the other. Because the moon is so far away, it's easy to make the spot move very very fast. It can travel faster than c because no information is being transferred from one side of the moon to the other.

With optical pulses, it gets a little tricky. Obviously, optical pulses can carry information. Most likely, the very information you're reading now came to you on optical pulses through an optical fiber! But they can go faster than c? Yes... sortof.

Pulses come from interference

 Interference of two comb patterns with slightly different frequencies

We tend to think of pulses as single objects with a fixed shape, but they're really a little more complicated than that. A pulse is composed of many single-frequency components, each of which has a constant amplitude, or strength. When these components are combined, the places where they interfere constructively (add up) corresponds to a large pulse amplitude, near the center. Where they interfere destructively (cancel each other out) corresponds to small pulse amplitudes near the front and back of the pulse.

An example of interference is shown in the figure to the right. This is an image of two partially overlapping patterns. It is simply the overlap of two striped regions. One stripe pattern is slightly larger than the other, corresponding to the different single-frequency components in a pulse. In the center (vertically), you can see the result of this overlap. In regions where the two patterns line up with each other (we say they are in phase) the combined pattern is bright. Where they don't line up (out of phase) one is black where the other is white, and so the combination is almost all black. This leads to structure--a repeating pulse pattern--that is not present in either component alone. The pulse pattern repeats because there are only two frequency components. In a real optical pulse, there are an infinite number of components and so the repetition doesn't occur.

Phase and group velocities

 Animation of a fast group velocity. The two phase velocities are only slightly different, and both much slower than the group velocity

Each of the frequency components that make up a pulse moves with its own velocity, called the phase velocity and represented by the symbol vp = c/n, where n is the index of refraction. If all of the components have the same phase velocity, then the whole pattern simply moves together and the pulse also moves at the phase velocity. However, if the components have different phase velocities, the pulse motion depends on the relationship between those velocities and then moves at the group velocity vg.

The group velocity effect is demonstrated in the figure to the left. This figure is simply an animated version of the figure from the previous section. This time, the two patterns are moving to the right with slightly different speeds or phase velocities. However, the larger light/dark pattern (the pulses) are moving much faster, at the group velocity. By making relatively small changes to the relationship between the phase velocities, it is possible to adjust the group velocity dramatically, it can be made faster than c or very slow (or it can go backward). This is what is meant by fast or slow light.

So, where is the information, you ask? That's a very good question. Why can the pulse go faster than c while the information can't? That's a good one, too. A partial answer is that the pulse doesn't preserve its exact shape, but is distorted in just such a way that the information doesn't move faster than c. However, the full answer is still being hotly debated!

Other resources

If you want to experiment with this a little more, print out this PDF file onto a transparency (or photocopy it onto a transparency). You can then cut the two patterns apart and see how the group velocity and phase velocity are related2.

The Budker group at Berkeley has put together an excellent fast light (or superluminal) tutorial in Mathematica. It is more advanced than this, but very detailed.

Footnotes and References

1. Don't rush to your backyard yet; lasers bright enough to make a visible spot on the moon are not available over-the-counter.
2. For you educators out there, this makes an excellent display on an overhead projector, but it's even better to print one out for everyone!