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.
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I like how you don't shy away from using the maths.. but do you not find it lowers your potential readership?
I've gotten given out to for leaving equations in my posts lol!
You might find this guy interesting: http://ftlfactor.com/2011/04/23/how-does-special-relativity-allow-effec…
He apparently has a PhD in Space Physics, I can't quite make out what he is trying to say. So I am passing it around =D haha
Well of course it lowers the potential readership. If I wanted lots of readers, I'd replace the equations with Victoria's Secret models. ;)
That article you posted is an interesting one. In fact, I wrote something similar a while back. The short version is that speed is distance/time, and the distance/time < c limit only applies when distance and time are being measured in the same frame.
Nice and interesting, thanks.
Matt: What equation are you using for the group velocity?
I ask because I'm surprised by your graph: if I'm reading it correctly, your group velocity takes a value very different from c when the derivative of the index of refraction (with respect to frequency) is zero. That's not what I'd normally expect.
Good catch, looks like I got bit by the way Mathematica handles derivatives of functions involving Re[stuff]. The corrected graph doesn't show the spikes, but it still (as expected) goes above c. I'll put it in the article.
Incidentally though, negative group velocity is possible and shows up in so-called slow light experiments. But it doesn't happen for this n(ω) in particular, and I'll have to revise the text a bit.
"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..."
Why are they difficult questions ? after all, the time-dependent Maxwell equations coupled to the Lorentz polarization dynamics constitute a hyperbolic system with characteristic speed c (more precisely, with speed equal to c divided by the square root of the relative frequency-dependent permittivity evaluated at infinite frequency).
Any comment on how this affects OTH radar? I think that the radar wave enters a medium (ionosphere) that slows group propagation and should cause it to deflect upwards by a naive version of Snell's Law. But because phase propagation is >c, the wave deflects back down to the ground.
You must be joking! I don't understand nor do I pretend too. But it looks good. It could be the equation for buttermilk for all I know.
Thanks
Guys!
Tom @7: I'm not sure what frequencies are used for OTH radar, but AM and shortwave radio are strongly affected by the ionosphere. The frequency ω stays fixed (because the wave is being driven at that frequency), and if it is in the right range the wavelength predicted by the dispersion relation becomes infinite. (Obviously the real situation is a bit more complicated because geometric optics is no longer valid in this regime.) Above this point the wave would be evanescent (i.e., it decays instead of oscillating). But the wave can be deflected downward. That's why you can sometimes hear radio broadcasts from long distances away, especially at night when the ionosphere generally has a lower density of unbound electrons (no photoionization to offset recombination) and the waves can reach higher altitudes before reflecting. It's also one source of static on AM radio: if you're close enough to receive the directly propagating wave, these reflected waves can interfere with it.
At higher frequencies (above 10 MHz or so, depending on the maximum density of unbound electrons) this trick no longer works, and you mainly depend on line-of-sight. You can still get wave reflection off of density gradients--this is how weather radar works (raindrops are denser than the surrounding air), and ionospheric research radars also exploit this fact.
#6 They're difficult questions because it turns out that he wavefront ahead of the main signal behaves in a complicated but quite specific way. The details comprise the theory of optical precursors first developed by Sommerfeld and Brillouin. Next post will be about that theory.
Matt@ #5
Thank you much for the fix on the graph. I'm still having a little bit of a hard time understanding it.
If I'm looking at the graph, it appears that the extrema for the group velocity occur at the extrema of the index of refraction. That's what I would expect for phase velocity, but for group velocity I would expect the group velocity extrema would be where the frequency-derivative of the index of refraction is maximum.
But maybe my expectations are off because I'm used to situations where the effect of the derivative term usually dominates the "n" term. Perhaps your graph is in the opposite regime, where the effect of the magnitude of "n" dominates the effect from the "dn/df"?
[Continuation of above]
Thinking about it a bit more, maybe that's why you don't have the negative group velocities anymore? If you want negative group velocities, try making your entire resonant absorption feature narrower in frequency.
Thank you much for the fix on the graph. I'm still having a little bit of a hard time understanding it.
If I'm looking at the graph, it appears that the extrema for the group velocity occur at the extrema of the index of refraction. That's what I would expect for phase velocity, but for group velocity I would expect the group velocity extrema would be where the frequency-derivative of the index of refraction is maximum.
But maybe my expectations are off because I'm used to situations where the effect of the derivative term usually dominates the "n" term. Perhaps your graph is in the opposite regime, where the effect of the magnitude of "n" dominates the effect from the "dn/df"?
That's why you can sometimes hear radio broadcasts from long distances away, especially at night when the ionosphere generally has a lower density of unbound electrons (no photoionization to offset recombination) and the waves can reach higher altitudes before reflecting. It's also one source of static on AM radio: if you're close enough to receive the directly propagating wave, these reflected waves can interfere with it.
I'm used to situations where the effect of the derivative term usually dominates the "n" term. Perhaps your graph is in the opposite regime, where the effect of the magnitude of "n" dominates the effect from the
Ama dalga aÅaÄı doÄru bükülmesi olabilir. Bu iyonosfer genellikle yansıtan önce yüksek rakımlı ulaÅabilirsiniz iliÅkisiz elektronlar (rekombinasyon ofset hiçbir photoionization) ve dalgalar daha düÅük bir yoÄunluÄa sahiptir zaman bazen özellikle geceleri, uzak uzak mesafelerden radyo yayınları duymak verebilirler. Aynı zamanda AM radyo statik bir kaynaÄıdır: EÄer yakın doÄrudan yayılan dalga, bu yansıyan dalgaların onu engelleyebilir almak için yeterli olsalar bile.
"If I wanted lots of readers, I'd replace the equations with Victoria's Secret models. ;)"
Aye, you could "model" the wave as a ripple of boobies, the speed of light as how a chain of such boobies propogates the wave, and so on.
Hey, if they can have financial news presented by a woman stripping, why can't science be done like this?
Tora Bora in particular was a cluster fuck and should simply not have happened.