Last time we took a pulse of light and shot it through a medium with a frequency-dependent refractive index. The particular form of the refractive index was sort of interesting – for some frequencies, it was less than 1. That implied that the phase velocity of a sine wave would be faster than the speed of light. But pulses of light contain a band of frequencies so normally we’d invoke the group velocity to show that the pulse as a whole propagates with a speed less than c. For reference, here’s the refractive index curve we postulated:
We can’t do that group velocity jazz here. Take a look. Here’s the group velocity as a function of frequency for the refractive index I posted last time (c = 300 nm/fs):
Away from that little wiggly feature in the refractive index, the group velocity is a nice and sedate sub-300 figure. It slows down and slows down as it approaches the wiggly feature – which we might as well call by its dignified proper name: the region of anomalous dispersion – but when it gets there it goes well above c. Surely this won’t work. [Paragraph and figure revised per correction in comments. A Mathematica syntax issue gave me some problems and the previous graph was incorrect. Tip o' the hat to my commenters, the best in the web!]
It doesn’t. Group velocity is only meaningful when the pulse behaves as a group. When that pulse contains frequencies in or near a region of anomalous dispersion, it doesn’t behave as a group. The pulse becomes badly distorted, but it does so in such a way as to absolutely prevent a signal from propagating faster than the speed of light. Here’s why. Refraction and absorption result from the action of the atoms which make up the dispersive medium. Intuitively, the electromagnetic field of the light causes the electrons of the atoms to wiggle and emit their own fields. This emission interferes with the original field so as to produce the macroscopic dispersive behavior of glass, diamond, plastic, etc. If you make the one reasonable assumption that the electrons do not start wiggling before the electromagnetic wave first arrives, it turns out to be the equivalent of requiring that the complex permittivity be analytic on the upper half of the complex plane.
Huh? Well, never mind the details. Physicists, mathematicians and masochists can find it worked out in Byron & Fuller or equivalent. What it boils down to is the fact that a material doesn’t have separate refractive index curves and absorption curves. The are linked one-to-one by the Kramers-Kronig relation:
Where α is the distance attenuation coefficient, i.e., a wave of frequency ω will have its intensity attenuated by a factor of e^-α(ω)x after traveling a distance x through the material. This is quite useful practically. It’s hard to directly measure a frequency-dependent refractive index, but usually frequency-dependent absorption curves are easy to measure. For completeness, though in practice it’s much less useful for calculations, here’s the expression in reverse:
(The P means the integral is a Cauchy principle value integral, which is a way to cancel the divergence of the integrand at Ω = ω). Systems which behave in a causal way produce dispersion curves that satisfy the Kramers-Kronig relation. Dispersion curves that satisfy the Kramers-Kronig relation, it turns out, do not and cannot allow a signal to propagate faster than the speed of light. Whatever weird speeds the individual Fourier components of a pulse might have, they will always and everywhere cancel each other out identically ahead of the wavefront, which propagates no faster than c.
For our refractive index curve, we can calculate the absorption*:
The spike in the absorption corresponds to the region of anomalous dispersion. This is a general feature of dispersion curves. If we create a pulse which has a frequency of 7.2 fs^-1, which is right in the middle of the anomalous dispersion, it’ll initially look like this:
After 100 nm of propagation, it looks like this:
It’s badly distorted and greatly attenuated, but it sure isn’t propagating faster than c.
Ok, great as far as it goes. But what speed is it propagating at? Or what if we have a signal that’s exactly zero before t = 0 but finite after t = 0? What speed does that sharp discontinuity propagate at – c exactly, or some other speed? These are actually difficult questions, and their answer was a hot topic in physics after special relativity was developed by Einstein and others. Related questions continue to be topics of active research, including some of my own. (Which is why I’m bloviating about this to such an extent!)
*This isn’t actually the absorption curve I’m plotting, it’s the imaginary part of the complex index of refraction. But they’re the same thing up to a proportionality factor.