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Experiments such as Focault's measure speed of light in matter. Focault's experimental set-up is based on the idea that it takes more time for light to travel through matter, which will result in the light hitting a rotating mirror at a different angle from the angle it hit it previously, which will result in a shifted image.

However, as explained e.g. in [Feynman's Lectures on Physics I.31][1], the incident electromagnetic wave actually travels through the material at the speed of light, and the slower "speed" of light in matter is just the phase velocity of the superposition of that incident wave and the wave emitted in response from the material.

Now, I know Focault's experiment worked and gets the right speed of light in matter. I don't understand why it worked. How can an experiment built on the premise that light actually slows down in matter work, when light actually travels through the apparatus at the speed of light in a vacuum and only the phase-velocity of the superposed wave is lowered?

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  • $\begingroup$ It is the group velocity not the phase velocity that gets smaller en.wikipedia.org/wiki/Group_velocity $\endgroup$
    – anna v
    Commented Jun 10, 2021 at 5:43
  • $\begingroup$ please see my answer to this related phenomenon physics.stackexchange.com/questions/643540/… $\endgroup$
    – anna v
    Commented Jun 10, 2021 at 5:46
  • $\begingroup$ In fact the phase velicity in matter can exceed $c$. $\endgroup$
    – my2cts
    Commented Jun 10, 2021 at 7:19
  • $\begingroup$ group velocity in matter can also exceed $c$, when there is anomalous dispersion. $\endgroup$
    – Math Keeps Me Busy
    Commented Jun 10, 2021 at 7:23

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Suppose an EM wave is emitted by some source in a step fashion. That is, ideally, prior to some time $t_0$ there is no emitted wave, and after that time there is an emitted wave with some constant non-zero amplitude.

An initial wave will leave the emitter at the speed of light in a vacuum, even if the light is passing through matter.

This initial "front" of the wave will be strongly attenuated as it passes through the matter, as it acts upon that matter.

The acted upon matter, under the influence of the EM field will re-emit the EM radiation, but with some delay. If the material is transparent at the frequency of interest, the delayed radiation will have nearly the same amplitude as the emitted radiation.

What a person measuring "downstream" would observe from the step function emitter is

  • a low intensity signal arriving at the speed of light in a vacuum, (a precursor) followed by

  • a much higher intensity signal arriving at the speed of light for that material (less than $c$.)

In most cases, the first signal is too weak to detect. However, in X-ray crystallography, it can be significant.

The physics of this was explained by Leon Brillouin in "Wave propagation and Group Velocity" (1960). However any defects in the presentation are my own.*

*[For simplification, I omitted the fact that there are actually two precursor wave fronts before the main "body" of radiation arrives. For a more complete exposition, see "Electromagnetic Theory" by Stratton page 338 and following, available online here.]

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  • $\begingroup$ Thanks, that's a very cool explanation. I still don't understand something - how is the attenuation-then-delayed-emission achieved? Is this a quantum phenomenon like stimulated emission, or a classical one like a driven oscillator (a la Lorentz model)? $\endgroup$
    – PhysicsTeacher
    Commented Jun 10, 2021 at 4:49
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    $\begingroup$ @PhysicsTeacher The same principle works whether the EM radiation is treated as photons that are absorbed and re-emitted, or as a waves in a classical electromagnetic field. Solutions to the Telegrapher's Equation, which models waves in a transmission line, reveal this phenomenon, but I believe the first observation of this phenomenon was made with X-ray crystallography, which is at the "quantum" end of the spectrum. $\endgroup$
    – Math Keeps Me Busy
    Commented Jun 10, 2021 at 5:10
  • $\begingroup$ Thanks again. Sticking to the classical picture, I guess my problem is that on the one hand we say the response and emission is instantaneous as the incident wave is attenuated, and on the other hand say that the emitted main-response is delayed. I'm sure this is all due to transient dynamics, I am just having trouble understanding it. $\endgroup$
    – PhysicsTeacher
    Commented Jun 10, 2021 at 5:14
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    $\begingroup$ I have to point out that the common conception of photon absorption and re-emission quantum mechanically cannot work in transparent media, (where the experiments are done) because the direction and intensity of the incoming beam will be lost, re-emission is random at 360 degrees. It has to be an elastic scattering at the center of mass of the interaction, be it with the field of the lattice, with the field of atoms, or molecules for frequency to change very little. $\endgroup$
    – anna v
    Commented Jun 10, 2021 at 5:53
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    $\begingroup$ I am am not commenting on the classical electromagnetic interactions, but on the handwavingly assumed photon interaction . Photons inteact with quantum electrodynamics. Classical electromagnetic waves are a superposition of a large number of photons, but photons are not classical electromagnetic waves, the are quantm mechanical entitities. They build up classical waves in a complicated QED manne, but their single photon interactions follow QED.. $\endgroup$
    – anna v
    Commented Jun 10, 2021 at 6:26
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I don't have Feynman's lectures available, but I am not sure if you have interpreted him correctly. The fact that the speed of light is different in matter (and varies with wavelength) is the foundation for such profound effects as refraction and dispersion. Just take any prism and see the effects, and you'll know that the speed of light in matter is real.

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  • $\begingroup$ There is supposed to be a link to the text there, there is an editing problem. feynmanlectures.caltech.edu/I_31.html $\endgroup$
    – PhysicsTeacher
    Commented Jun 9, 2021 at 23:22
  • $\begingroup$ Note in particular: " In Fig. 31–4 we give a schematic idea of how the waves might look for a case where the wave is suddenly turned on (to make a signal). You will see from the diagram that the signal (i.e., the start of the wave) is not earlier for the wave which ends up with an advance in phase." $\endgroup$
    – PhysicsTeacher
    Commented Jun 9, 2021 at 23:23
  • $\begingroup$ Refraction and dispersion et al are perfectly explained by the superposed wave, as these are steady-state phenomena. Focault's experiment, however, is a signalling-phenomena, so should not work. $\endgroup$
    – PhysicsTeacher
    Commented Jun 9, 2021 at 23:29
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How can an experiment built on the premise that light actually slows down in matter work, when light actually travels through the apparatus at the speed of light in a vacuum and only the phase-velocity of the superposed wave is lowered?

Indeed. The correct functioning of the experiment is certainly good evidence supporting the premise upon which the experiment is based.

I would suggest that the argument against the experiment is the thing that is suspect here. The key error in the argument is the implied decision that “the speed of light in matter” should be associated with some wave traveling at c instead of the “superposed wave”. Since the “superposed wave” is a measurable thing, it is reasonable to give it a name and describe its speed.

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  • $\begingroup$ The physics is quite clear that the signal arrives at the speed of light, as Feynman and many others explain. The issue cannot possibly be that we only measure the superposed wave (which of course we do). It has to be that somehow the experiment actually measures a property of the stead-state phase-velocity rather than the arrival (signal) time it appears to. $\endgroup$
    – PhysicsTeacher
    Commented Jun 9, 2021 at 23:33
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    $\begingroup$ @PhysicsTeacher I can assure you that the arrival time of a pulse is also slowed down by matter. A pulse sent through 1 km of optical cover will take about 50% longer to arrive at its receiver than one Sent through 1 km of air or vacuum. $\endgroup$
    – The Photon
    Commented Jun 10, 2021 at 0:33
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    $\begingroup$ @photon Some of the light does indeed travel at the speed of light Feynman is not wrong here. But what he didn't account for is that the initial "wavefront" is very rapidly attenuated, and it is generally too weak to be detected. See my answer. $\endgroup$
    – Math Keeps Me Busy
    Commented Jun 10, 2021 at 1:07
  • $\begingroup$ @PhysicsTeacher it is not a question of physics so much as terminology. The phase velocity of the superposed wave is what we call “the speed of light in matter” $\endgroup$
    – Dale
    Commented Jun 10, 2021 at 1:08
  • $\begingroup$ @Dale I understand that, what I don't understand is why then does Focault's experiment work. $\endgroup$
    – PhysicsTeacher
    Commented Jun 10, 2021 at 4:43
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In a medium there in general is dispersion, so that the group velocity depends on the frequency. Note that the phase velocity can even exceed $c$, namely when $n<1$ such as occurs for x-rays. A pulse of light contains many frequencies, each with their own amplitude and phase. These components all travel at their own velocity and with their own extinction. Therefore the speed of a pulse in a medium is nor precisely defined. It will deform while travelling through the medium by losing its high frequency components and acquiring phase shifts between its frequency components.

The Foucault experiment is based on continuous waves so these effects are insignificant. It should work fine as a method to measure the (group) speed of light in a medium.

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