How can the speed of light in matter be measured? 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?
 A: 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.]
A: 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.
A: 
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.
A: 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.
