Speed of information in a medium

[neobaud]

TL;DR Summary

It is well established that the speed of light slows down in a medium. Does the speed of information also slow down? Does the speed of causality slow down? Have there been experiments that show this?

I am aware of the explanation for light slowing down. From what I understand the EM field causes electrons in the medium to oscillate interfering with the wave and slowing its progress through the medium. The question I am asking is about a light source (or any source) that is turned on. If I turned on the light and measured the time it takes to enter and then exit the medium, and then did d/t, would this be less than the speed of light? Most of the explanations for light slowing down consider a standing wave that has been on since the beginning of time. I cannot understand how the front of the EM wave would be slowed down given the common explanation.

I would appreciate the theory and possibly a link to a good experiment where this has actually been measured.

Also I was wondering if the speed of causality slowed down in a medium. I would think not just the speed of light but I was hoping to confirm this as well.

 

[Dale]

The invariant speed remains $c$ even in a medium where light travels slower than $c$. So you can, for example, get particles that travel faster than light in the medium, but not faster than $c$. The invariant speed is the speed that you are calling the speed of causality.

If I turned on the light and measured the time it takes to enter and then exit the medium, and then did d/t, would this be less than the speed of light?

Yes. It will be slower.

See this undergraduate lab experiment. This was already understood before the Fizeau experiment in 1851. But I don't know what the first experiment to measure this, pre-Fizeau.

 

[neobaud]

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Yes. It will be slower.
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Thanks. I can't really understand this. Take a pulse of light. The explanation that I have read is the speed of light is slowed down because the vibrating electrons move in response to the EM field and this causes another EM field that interferes with the first. But for a pulse, wouldn't the original EM field always be ahead of the secondary field produced by the electrons?

 

[Dale]

Thanks. I can't really understand this. Take a pulse of light. The explanation that I have read is the speed of light is slowed down because the vibrating electrons move in response to the EM field and this causes another EM field that interferes with the first. But for a pulse, wouldn't the original EM field always be ahead of the secondary field produced by the electrons?

I don’t feel that is a particularly helpful explanation. It is not exactly wrong, but the fields it describes fall off exponentially. So it is only important at quantum mechanically short distances. So yes, it is always ahead of a classical wave, but anywhere more than a few atoms ahead it is essentially 0.

 

[anuttarasammyak]

If the information you say is one transmitted by light, e.g. TV broadcast, it would be transmitted by light speed in the medium.
If the information speed you say is about the principle of maximum speed, which happen to coincide with light speed in vacuum, it keeps being invariant. For an example, particles run through water emitting Cerenkov light would transmit information by its impulse to water molecule with speed faster that light speed in water. If we succeed neutrino transmission in future, we might achieve (almost) light speed in vacuum c information tranmission in the medium.

 

[PeroK]

Thanks. I can't really understand this. Take a pulse of light. The explanation that I have read is the speed of light is slowed down because the vibrating electrons move in response to the EM field and this causes another EM field that interferes with the first. But for a pulse, wouldn't the original EM field always be ahead of the secondary field produced by the electrons?

The speed of light in a medium is determined by Maxwell's equations and depends on the refractive index of the medium. You can't solve Maxwell's equations by waving your hands. A derivation of the speed of light in a homogeneous medium is given in most EM texbooks. E.g. section 9.3.1. of Introduction to Electrodynamics by Griffiths.

 

[neobaud]

The speed of light in a medium is determined by Maxwell's equations and depends on the refractive index of the medium. You can't solve Maxwell's equations by waving your hands. A derivation of the speed of light in a homogeneous medium is given in most EM texbooks. E.g. section 9.3.1. of Introduction to Electrodynamics by Griffiths.

Not true. Feynman managed it in his lectures https://www.feynmanlectures.caltech.edu/I_31.html See section 31-1. This is a more intuitive explanation for the slowing of light in a medium. Anyway I am asking a very specific question about the wave front of the light.

 

[neobaud]

If the information you say is one transmitted by light, e.g. TV broadcast, it would be transmitted by light speed in the medium.
If the information speed you say is about the principle of maximum speed, which happen to coincide with light speed in vacuum, it keeps being invariant. For an example, particles run through water emitting Cerenkov light would transmit information by its impulse to water molecule with speed faster that light speed in water. If we succeed neutrino transmission in future, we might achieve (almost) light speed in vacuum c information tranmission in the medium.

Thanks makes sense.

 

[neobaud]

I don’t feel that is a particularly helpful explanation. It is not exactly wrong, but the fields it describes fall off exponentially. So it is only important at quantum mechanically short distances. So yes, it is always ahead of a classical wave, but anywhere more than a few atoms ahead it is essentially 0.

So do you think that light at a high enough intensity could make it through a medium at speed C?

You said "essentially 0" which I am interpreting to mean "not zero".

 

[DaveC426913]

So do you think that light at a high enough intensity could make it through a medium at speed C?

Sure. If a medium is rarified enough that some of the light never encounters any atoms in its path, then I see no reason why some of it couldn't pass through at c. Of course it also wouldn't be refracted. And we're being pretty generous with what constitutes a "medium" at this point.

You could say "light from stars passes through the interstellar medium (atoms) without being slowed or refracted".

But this seems like it's kind of bifurcating bunnies at this point.

 

[PeroK]

Not true. Feynman managed it in his lectures https://www.feynmanlectures.caltech.edu/I_31.html See section 31-1. This is a more intuitive explanation for the slowing of light in a medium. Anyway I am asking a very specific question about the wave front of the light.

Section 31-1 is a heuristic argument, also known as advanced hand waving! From section 31-1:

You can see that it would take a complicated set of equations to get the complete and exact formula. It is so complicated that we postpone this problem until next year.

I guess that's the point at which hand-waving runs out of steam and even Feynman would need to introduce Maxwell's equations.

 

[Vanadium 50]

Not true. Feynman managed it in his lectures https://www.feynmanlectures.caltech.edu/I_31.html See section 31-1.

Sure, if by "intuitive" you mean "about seven pages and 20 numbered equations" plus Feynman's comment that this is the most complicated topic in the entire course. Me, I am not sure that's the word I would use to mean "A Nobel prize winner can do this with only 20 euations."

It's like the old joke: a professor writes down an equation and says its obvious. A student asks "Is it obvious?" The professor stares at the equation for a few minutes, and then leaves. He comes back with a book, flips through it for a few minutes, looking back and forth between the book and the blackboard.

He then leaves and comes back a few minutes later with an even thicker book, and again flips through the book, looking alternately at the book and the blackboard. Finally he sets the book down, turns to the class and says "Yes. it's obvious."

 

[Dale]

So do you think that light at a high enough intensity could make it through a medium at speed C?

No. A high enough intensity would destroy the medium, which I would not consider to be light “making it through a medium”

You said "essentially 0" which I am interpreting to mean "not zero".

Go ahead and interpret it that way if you wish. There is no way to experimentally distinguish between this particular meaning of “not zero” and “zero”. Which is why I call it “essentially zero”.

 

[PeterDonis]

I am asking a very specific question about the wave front of the light.

Feynman's answer to the second question you pose in your OP, namely this...

I was wondering if the speed of causality slowed down in a medium.

...is "no". He explicitly says that retardation effects are evaluated with the speed $c$, i.e., the speed of light in vacuum, even in cases where there are materials present.

 

[PeterDonis]

I cannot understand how the front of the EM wave would be slowed down given the common explanation.

The Feynman lecture you linked to also answers this, in two ways:

First, the effect of the medium on the electric field is not to "slow down" anything, but to add an additional phase (Eq. 31.8).

Second, in section 31.3 on dispersion, Feynman discusses signal transmission in the case where the wave is turned on at some particular time. The effect of the medium is shown in Fig. 31.4. A key point from this discussion is the distinction between phase velocity (how fast the nodes of the wave "move") and signal velocity (how fast information is actually transmitted). Your question about the "front of the EM wave" has to do with phase velocity, which is less than $c$ for $n > 1$. But that phase velocity is not the same as signal velocity. (And it is also not the same as the "speed of causality", i.e., the "speed" you use when evaluating retardation effects; that is always $c$, as noted in my previous post.)

 

[tech99]

I cannot understand how the front of the EM wave would be slowed down given the common explanation.

I think that in essence, when a pulse approaches the medium, a standing wave is set up in the vicinity of the boundary. This standing wave contains reactive, or evanescent, energy. The incoming wave has to build this up as an energy store, and it then releases energy into the new medium at the new velocity. When the pulse ends, the store empties by radiating into both media.

 

[neobaud]

No. A high enough intensity would destroy the medium, which I would not consider to be light “making it through a medium”

Go ahead and interpret it that way if you wish. There is no way to experimentally distinguish between this particular meaning of “not zero” and “zero”. Which is why I call it “essentially zero”.

The high intensity statement was just questioning if this was a problem with the signal to noise ratio. If I had a light beam of high enough intensity, would I be able to detect the light exiting the material at time d/c.

I am just thinking of a material with a really high refractive index. It's hard to imagine the secondary fields from the material exactly cancelling the original light as its front exits the boundary. So you are saying that the intensity of the front would be so small that it would be immeasurable regardless of the intensity of the light or the sensitivity of the detector?

 

[neobaud]

The Feynman lecture you linked to also answers this, in two ways:

First, the effect of the medium on the electric field is not to "slow down" anything, but to add an additional phase (Eq. 31.8).

Second, in section 31.3 on dispersion, Feynman discusses signal transmission in the case where the wave is turned on at some particular time. The effect of the medium is shown in Fig. 31.4. A key point from this discussion is the distinction between phase velocity (how fast the nodes of the wave "move") and signal velocity (how fast information is actually transmitted). Your question about the "front of the EM wave" has to do with phase velocity, which is less than $c$ for $n > 1$. But that phase velocity is not the same as signal velocity. (And it is also not the same as the "speed of causality", i.e., the "speed" you use when evaluating retardation effects; that is always $c$, as noted in my previous post.)

Ya that figure shows exactly what I was wondering about. At t_start the original wave starts ramping up. The transmitted wave doesn't actually get going until a bit later right? So it seems like some of the original wave should get to the edge of the material before the cancelling phase can have an affect. Maybe it is like dale is saying and the effect is really small and immeasurable.

 

[PeterDonis]

The transmitted wave doesn't actually get going until a bit later right?

That's right; it can't until at least time $d / c$, where $d$ is the thickness of the medium.

So it seems like some of the original wave should get to the edge of the material before the cancelling phase can have an affect.

The "cancelling phase" doesn't wait until the "original wave" has gone through the material. As soon as the original wave enters the material, it starts affecting the charges inside the material (the ones nearest the surface that the original wave enters at get affected first), and the charges start moving and generating their own fields. As soon as the first charge inside the material starts moving and generating its own field, the overall "wave" inside the material won't be the same as the original wave.

 

[neobaud]

That's right; it can't until at least time $d / c$, where $d$ is the thickness of the medium.The "cancelling phase" doesn't wait until the "original wave" has gone through the material. As soon as the original wave enters the material, it starts affecting the charges inside the material (the ones nearest the surface that the original wave enters at get affected first), and the charges start moving and generating their own fields. As soon as the first charge inside the material starts moving and generating its own field, the overall "wave" inside the material won't be the same as the original wave.

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?

 

[PeterDonis]

Will any part of the original light exit the medium at time d/c?

The question isn't even well defined, because, as I said, as soon as the light enters the medium and charges in the medium start moving and generating their own fields, there is no such thing as "the original light". There is just the overall field inside the medium. Similar remarks apply to the field just outside the medium on the other side: you can't measure "the original field" as a separate thing. You can only measure the overall field, or more precisely the change in the field at that point from what it was before the experiment started. (Usually things are set up so the field on the other side of the medium before the experiment started is zero, so any detection of a nonzero field counts as a change.)

Dale seems to think it won't

@Dale doesn't "seem to think" anything. He gave you an explicit description of an experiment that undergraduates can run that measures the time it takes for light to pass through a given thickness of a medium, as compared to air (you could in principle run the experiment in a vacuum chamber and compare the times to the time taken in vacuum, but doing that is beyond the capability of most undergraduate physics labs). But that experiment is not measuring "the original light". It's just measuring the time it takes to detect any light on the other side of the medium.

 

[PeroK]

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?

I guess your idea is that a short pulse of light could get through a dielectric material at the speed of light before the atoms in the material have had time to react to the initial pulse? And that it's this reaction that would subsequently slow down the next wave of light?

 

[PeterDonis]

I guess your idea is that a short pulse of light could get through a dielectric material at the speed of light before the atoms in the material have had time to react to the initial pulse?

If this is the idea, it's wrong. The electrons in the medium (which are the charges that react) react to the light pulse as it passes them. There is no waiting time.

 

[PeroK]

If this is the idea, it's wrong. The electrons in the medium (which are the charges that react) react to the light pulse as it passes them. There is no waiting time.

Yes, but the idea is that the initial EM wavefront is not affected by that. So, a couple of wavelengths, perhaps, get through unhindered! And the rest are subsequenty slowed down.

 

[neobaud]

Yes, but the idea is that the initial EM wavefront is not affected by that. So, a couple of wavelengths, perhaps, get through unhindered!

Yes this is exactly what I am wondering. Is this just a very small amount (immeasurable?)

 

[Dale]

The question isn't even well defined, because, as I said, as soon as the light enters the medium and charges in the medium start moving and generating their own fields, there is no such thing as "the original light".

@neobaud to further emphasize this point you might consider the extinction theorem. In air at optical frequencies there is no more "original light" after 1 mm. In denser material that distance will be even shorter. The original light gets extinguished very shortly.

 

[PeterDonis]

Yes, but the idea is that the initial EM wavefront is not affected by that. So, a couple of wavelengths, perhaps, get through unhindered! And the rest are subsequenty slowed down.

As I said, this idea is wrong. There is no such thing as "the initial EM wavefront" once it enters the medium and charges in the medium start moving. There is just the overall field.

 

[PeterDonis]

@neobaud to further emphasize this point you might consider the extinction theorem.

If I'm reading correctly the article you link to here, there is no distance into the medium at which the "original light" is still present. Inside the medium, the field generated by the charges in the medium exactly cancels the "original light" and leaves behind only the "slower" field traveling at $c / n$.

 

[Dale]

If I'm reading correctly the article you link to here, there is no distance into the medium at which the "original light" is still present. Inside the medium, the field generated by the charges in the medium exactly cancels the "original light" and leaves behind only the "slower" field traveling at $c / n$.

The extinction theorem is often used to criticize SR experiments, the idea being that the light that was emitted was replaced en route so the experiment doesn't measure the emitted light but rather the replaced light.

Some experiments cannot be criticized by the extinction theorem because the experimental light path is far shorter than the extinction length. So I think that the usual interpretation is that for lengths well below the extinction length the light emitted by the source is the same light as received by the detector.

 

[PeroK]

If I'm reading correctly the article you link to here, there is no distance into the medium at which the "original light" is still present. Inside the medium, the field generated by the charges in the medium exactly cancels the "original light" and leaves behind only the "slower" field traveling at $c / n$.

There must be a limit to the scale at which purely classical EM can be used and make sense here. And that the OP is pushing a classical heuristic too far. In Feynman's notes above he says:

Each of the electrons in the atoms of the plate will feel this electric field and will be driven up and down ... To find what motion we expect for the electrons, we will assume that the atoms are little oscillators, that is, that the electrons are fastened elastically to the atoms, which means that if a force is applied to an electron its displacement from its normal position will be proportional to the force.

You may think that this is a funny model of an atom if you have heard about electrons whirling around in orbits. But that is just an oversimplified picture. The correct picture of an atom, which is given by the theory of wave mechanics, says that, so far as problems involving light are concerned, the electrons behave as though they were held by springs.

So that, in considering the response of atoms to an electric field, some QM thinking must ultimately be brought to bear.

 

[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.