## Light breaking its own speed limit: how ‘superluminal’ shenanigans work

In a previous optics basics post, we discussed challenges associated with trying to define the velocity of a localized wave or ‘pulse’ of light. Traditional measurements of the velocity of an object involve measuring how far Δd an object travels in a certain amount of time Δt; then the velocity is simply

velocity = distance/time = Δd/Δt.

But a wave is an extended disturbance, not definitely associated with any particular point in space, and so measuring Δd becomes tricky. If there is a definite feature of the wave (such as a peak), we can define the velocity by measuring how fast the peak moves. If the wave changes shape (i.e. the peak disappears), as happens when waves propagate in matter, it is not immediately clear how one defines wave velocity.

The answer, as discussed previously, seems to be to define a ‘group’ velocity: we can mathematically characterize the velocity of the overall wave signal by

$\Delta \omega/\Delta k$,

where Δω is the range of temporal frequencies in the wave pulse and Δk is the range of spatial wavenumbers in the pulse. This measure seemed quite good: under most circumstances, the quantity was less than the vacuum speed of light c, and therefore didn’t violate Einstein’s relativity, and those cases where the group velocity was greater than c seemed to always involve a significant attenuation or distortion of the wave.

However, in 2000 researchers Wang, Kuzmich and Dogariu from the NEC Research Institute shocked the physics and optics community by demonstrating* that materials exist for which the group velocity is greater than c, sometimes much greater than c, and the pulse travels at this higher speed without any obvious distortion or attenuation. What was going on?

The response from the physics community was initially complete chaos. Some scientists derided the results as flawed because they violated Einstein’s relativity (not true). Others said that these pulses did not violate relativity because they couldn’t carry information (not true). Even more scientists derided the results as mere ‘pulse shaping’, something the authors of the paper explicitly rejected, adding to the confusion.

For my money, this effect is in fact the result of pulse shaping, but a very special and previously unobserved form of pulse shaping. The key is that the dispersive medium which is used to achieve superluminal speeds is a gain medium, i.e. a material which can release previously stored energy in the form of light when illuminated. Atoms in the gain medium are excited to a higher-energy state by some technique (optically, electrically, chemically), and the atoms ‘settle’ into a different excited state in which their energy can be released by an illuminating light beam, in a process first suggested by Einstein known as stimulated emission. Such media are an essential component of lasers.

A good analogy to light emission from a gain medium is given by a rockslide. Along the face of a steep mountain slope, rocks are very precariously balanced. These rocks are analogous to trapped energy in an atom. If a single rock from the top of the slope comes loose, it will bounce down the slope and knock more rocks loose along the way. Even though a single rock started moving at the top of the slope, many rocks arrive at the bottom. Similarly, in a gain medium, a single photon entering the medium can result in many photons leaving the other end of the medium. This process is illustrated below:

How can a gain medium make a pulse of light appear to move at velocity greater than c? We can roughly demonstrate this with our rockslide. Let us imagine a ‘pulse’ of rocks are tumbling down the hill:

Let us say that a rock moving down slope moves at a constant speed c. Our rock pulse starts to dislodge ‘cliff rocks’ from the side of the mountain, but these cliff rocks are dislodged at the front of the rock pulse! When the pulse arrives at the bottom of the hill, because new rocks have been continually added to the front of the pulse, and the peak of the pulse has in effect moved faster than c! It is important to note, though, that no individual rock has broken the speed limit c.

The overall pulse is now bigger than what we started with, however, because we have constantly added rocks to it. The shape is typically also be highly distorted and shifted forwards (as schematically shown), because rocks are only added to the front of the pulse. We can make a more ‘realistic’ picture, however, by noting that when rocks are freed from the cliff-face, they leave a hole which can be filled by a rock further back in the pulse. Similarly, when an atom releases its energy into a light pulse, it may reabsorb energy from further back in the pulse:

If the circumstances are chosen just right, two things happen: (a) the rock pulse comes out with exactly the same shape as it went in with, and (b) the rock pulse is exactly the same size as it went in with.

Odd effects can be associated with such superluminal propagation. One of the most dramatic and seemingly counterintuitive of these is that the peak of a pulse can exit the medium before it has entered! This is illustrated crudely and schematically below:

Even though the peak hasn’t yet entered the medium, it already seems to be exiting! This isn’t, despite appearances, some weird form of time travel. We note that the front edge of the input pulse has already entered the medium; that front edge can stimulate the emission of light ‘stored’ in the gain medium, and that emitted light can then mimic the shape of the original, complete input pulse.

It is important to note that various schemes for performing this superluminal’ (and now we hopefully see the quotation marks are justified) light propagation before Wang et al. performed their specific experiment. Raymond Chiao, arguably the ‘father’ of modern superluminal light theory, proposed a pair of schemes to produce relatively undistorted superluminal pulses. In 1993 he introduced** the propagation of light in a gain medium which is resonant with a single resonant frequency (single gain line). This scheme would produce a superluminal pulse which is nearly, but not perfectly, undistorted and unamplified. Chiao also introduced the physical argument described above (though not in such a hand-wavy way). In 1994 he and a collaborator*** suggested an even better scheme in which a medium is used with two gain lines. The frequency response of the medium between the two gain frequencies results in almost perfect dispersionless superluminal light propagation.

This latter result by Chiao is essentially the technique used by Wang et. al in their experiment; curiously, however, they specifically argue against Chiao’s interpretation of amplified-front/absorbed tail:

Here we note that the physical mechanism that governs the observed superluminal light propagation differs for the previously studied anomalous dispersion associated with an absorption or a gain resonance… The probe pulse thus contains essentially no spectral components that are resonant with the Raman gain lines to be amplified. Therefore, the argument that the probe pulse is advanced by amplification of its front edge does not apply. The superluminal light propagation observed here is the result only of the anomalous dispersion region created with the assistance of two nearby Raman gain resonances. We emphasize that the observed superluminal light propagation is a result of the wave nature of light. It can be understood by the classical theory of wave propagation in an anomalous dispersion region where interference between different frequency components produces this rather counterintuitive effect.

This explanation is confusing and doesn’t really explain anything. ‘Interference between different frequency components’ is really a mathematical explanation of the effect, not a physical explanation. Their argument centers around the fact that their light signal is not directly affected by the gain material; however, there would be no superluminal effect without the gain medium in the first place! The dependence of the entire effect upon the presence of a gain medium strongly supports the interpretation given above, at least as a rough qualitative explanation of the superluminal effect.

So what implications does this superluminal travel have for Einstein’s theory of relativity, which states that nothing can go faster than the speed of light c? Well, none, actually. No matter what happens within the gain medium, nothing will come out of the medium ahead of the front end of the input pulse traveling at c. The picture above should really be redrawn more accurately as:

Just like our rockslide, nothing ever really gets ahead of the input pulse. The ‘center of energy’ of the pulse shifts forward, giving the illusion of faster motion, but the front edge of the input pulse will never advance faster than the free-space speed of light, regardless of the medium it is traveling in. (Though some people suggest that there may be more subtle loopholes to Einstein’s speed limit, as yet undiscovered.)

Just in case the reader isn’t convinced yet that the peak of a pulse can travel faster than any of its individual components, we present another simple analogy. Suppose we have a busload of passengers, and most of the passengers start out sitting at the rear of the bus. At every stop, one passenger gets out of the rear of the bus and one more passenger gets on at the front of the bus. After a few stops, the center of mass of the passengers will have shifted to the front of the bus, implying that the bulk of the passengers has been moving faster on average than the bus itself! This is illustrated below:

In conclusion, superluminal light is not in fact ‘faster-than-light light’: the experiments which demonstrated its existence, however, challenged previous assumptions about the meaning of the velocity of a wave and furthered the understanding of optical wave propagation in materials.

###### *** A.M. Steinberg and R.Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49 (1994), 2071.

This entry was posted in Optics. Bookmark the permalink.

### 3 Responses to Light breaking its own speed limit: how ‘superluminal’ shenanigans work

1. Tom says:

Very good explanation. I like the rockslide analogy of a gain medium.

2. Tom: Thanks! I find it also works well, with slight modification, for discussions of ‘critical mass’ in describing nuclear fission.

3. Richard Lieu says:

Hmm…but how does the medium know’ the shape of the rest of the pulse that it emitted `ahead of time’? The forward tail may be Gaussian in shape, but that does not mean the peak has to be part of the same Gaussian. In terms of your bus analogy, the bus actually unloaded 5 passengers from its back end, when only 1 boarded from the front end. Yes, there are 4 more to board, but only if the shape of the pulse that has not entered the medium at this moment is also Gaussian.

This site uses Akismet to reduce spam. Learn how your comment data is processed.