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], 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?