Speed of information in a medium

[PeterDonis]

the extinction length

Ah, I see there is a section on that at the end of the article.

There must be a limit to the scale at which purely classical EM can be used and make sense here.

I agree, but I would expect that scale to be far smaller than a typical extinction length for a medium. What you quote from Feynman basically says that even at the scale of a single atom, for the purposes of analyzing the effects of EM radiation passing through a medium, a classical model of electrons as charged oscillators works fine. Of course one ultimately has to justify this by looking at the underlying QM, but Feynman is saying that when you do that, yes, the justification works.

 

[anuttarasammyak]

So what do you think? Will any part of the original light exit the medium at time d/c? Dale seems to think it won't (or at least a measurable amount) and everyone else seems uncommitted. Maybe it's too complicated to know for sure without a really good experiment?

Please find below illustrated my idea of experiment .

 

[vanhees71]

So what do you think? Will any part of the original light exit the medium at time d/c? Dale seems to think it won't (or at least a measurable amount) and everyone else seems uncommitted. Maybe it's too complicated to know for sure without a really good experiment?

The "wave front" propagates with the speed of light in standard dispersion theory (no matter whether you use a crude classical "Drude-like model" or quantum-mechanical or even full-fledged in-medium relativsitic QED). That's because the causality constraints are very robust, i.e., all you need are the analytical properties of the propagator, and for a hyperbolic differential equation as the (relativistic) wave equation that implies Einstein causality.

All this is well-known since 1907, when Sommerfeld answered the question by Willy Wien about waves in the frequency regime of "anomalous dispersion" of a medium, i.e., close to a resonance of the bound charged particles making up the dielectric. In such a region both the phase velocity and the group velocity are >c, which however does not mean a violation of relativistic causality, because both velocities/speeds do not describe a causal signal propagation velocity/speed. The phase velocity simply describes the dispersion relation between $\omega$ and $\vec{k}$ for a plane-wave solution, i.e., a solution, which describes and em. wave field that's "switched on" for a very long time and the medium is oscillating with the frequency of the em. wave, i.e., all transient states have damped out.

The group velocity as the velocity with which the "center" of a wave packet moves makes only sense when the stationary-phase approximation of the corresponding Fourier integral from $\vec{k}$ to position space is applicable, which it is not in the region of anomalous dispersion.

As has been shown by Sommerfeld in 1907 by using an elegant analytical argument (theorem of residues) one can show that for arbitrary waves with compact spatial support the boundary of the support moves with the speed of light in vacuum inside the medium. That's understandable, because the medium can only be disturbed by and react to the incoming wave when this wave reaches it. Only then the medium emits its own electromagnetic waves which superimposes with the incoming wave.

In 1914 these considerations have been worked out in 2 famous papers by Sommerfeld and Brillouin in great detail, where the onset of the propagation of the wave front in the medium has been described reaching the "stationary state" only after some time, and particularly without ever violating relativistic causality.

As already shown by Sommerfeld in 1907, this is due to pretty weak analytical properties related to the choice of the retarded Green's function. This in turn has been worked out in more detail, also in connection with more general wave equations and in connection with quantum (field) theory by Kramers and Kronig. These socalled Kramers-Kronig relations can be found in any textbook dealing with wave phenomena. The QFT analogue is the celebrated Källen-Lehmann representation of the (interacting) propagator of various relativistic wave fields and their generalizations for finite temperature and density in the many-body context.

 

[gentzen]

The "wave front" propagates with the speed of light in standard dispersion theory (no matter whether you use a crude classical "Drude-like model" or quantum-mechanical or even full-fledged in-medium relativsitic QED). That's because the causality constraints are very robust, i.e., all you need are the analytical properties of the propagator, and for a hyperbolic differential equation as the (relativistic) wave equation that implies Einstein causality.

That sounds complicated. But more down to earth, my guess would be that in the end, the frequency dependent complex dielectric constant will simply converge to 1 for extremely high frequencies (i.e. hard gamma radiation). And now quantum mechanics kicks-in, and tells me that I will only have a reasonable change of observing 'the "wave front" propagate with the speed of light', if I put in enough energy to generate at least a few of those hard gamma radiation photons.

And since I am already at it, I guess the following YouTube videos which 3Blue1Brown and Looking Glass Universe published simultaneously on 30.11.2023 are part of the reason why this topic is currently discussed:

Explaining prisms fully requires understanding springs | Optics puzzles 3


I didn't believe that light slows down in water (part 1)


The reason light slows down in water is complicated (part 2)

 

[neobaud]

That sounds complicated. But more down to earth, my guess would be that in the end, the frequency dependent complex dielectric constant will simply converge to 1 for extremely high frequencies (i.e. hard gamma radiation). And now quantum mechanics kicks-in, and tells me that I will only have a reasonable change of observing 'the "wave front" propagate with the speed of light', if I put in enough energy to generate at least a few of those hard gamma radiation photons.

And since I am already at it, I guess the following YouTube videos which 3Blue1Brown and Looking Glass Universe published simultaneously on 30.11.2023 are part of the reason why this topic is currently discussed:

Explaining prisms fully requires understanding springs | Optics puzzles 3I didn't believe that light slows down in water (part 1)The reason light slows down in water is complicated (part 2)

Right. I am skeptical of her experiment because it is hard to say if it was a limitation of the sensors. She is using her cell phone and some cheap distance finder from a hardware store. What if the SNR was just not high enough to detect the return beam. A detector like this would most likely just set a threshold for detection and who is to say if there wasn't some small amount of signal that made it through.

Anyway this thread was really good and cleared it up for me. Thanks everyone.

 

[Vanadium 50]

There is no such thing as "the initial EM wavefront"

True, in the sense one can never set foot in the same stream twice.

One can always split the material EM wave solution into two: the new solution is the sum of the orignial wave and the "response". There's no physics here, just subtraction.

So the real question is whether there is utility in thinking of things this way. Well, "sometimes" and "it depends". I can think of only one example (transition radiation) so its probably not helpful all that often.

 

[wnvl2]

As has been shown by Sommerfeld in 1907 by using an elegant analytical argument (theorem of residues) one can show that for arbitrary waves with compact spatial support the boundary of the support moves with the speed of light in vacuum inside the medium. That's understandable, because the medium can only be disturbed by and react to the incoming wave when this wave reaches it. Only then the medium emits its own electromagnetic waves which superimposes with the incoming wave.

Are you refering here to precursors?

https://en.wikipedia.org/wiki/Precursor_(physics)

Do you perhaps have a reference where it is explained as simple as possible?

So this implies that if you send an EM pulse wave through water a first very little signal arrives with the speed of light in vacuum at the other side of the water.

 

[tech99]

Are you refering here to precursors?

https://en.wikipedia.org/wiki/Precursor_(physics)

Do you perhaps have a reference where it is explained as simple as possible?

So this implies that if you send an EM pulse wave through water a first very little signal arrives with the speed of light in vacuum at the other side of the water.

I don't think this can be correct or else we would see a leading echo in optical fibre communication, which we don't.

 

[wnvl2]

But if I understand it correctly it will be very small as Dale said in post #26 because of the extinction theorem. That is probably why we do not detect it in practice.