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When an light travels in free space, it has a velocity of propagation equal to the speed of light.

However, then the light enters a medium with a refractive index of n, the velocity of propagation changes to

$\ v_p = c / n $

Is this change in velocity instant? Or is there a gradual deceleration over distance, such as a ball rolling into a sand box (and acceleration when it exits the medium)?

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Inside a medium too photons will travel with same speed, but the "effective speed" will be low. –  user10001 Jul 23 '12 at 20:25

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Yeah, The resolution to this question is if you use a billiard ball type interpretation for atoms and light. In vacuum $c$ is always the same, so in between interactions with atoms, light travels at $c$ also. The "effective speed" noted above is a result of the decay time for excited states of atoms. So the light balls move between the atoms at $c$ and are held onto by the atoms for some time $\tau$ before released again. Thus light obtains an effective speed in a medium as a result of the finite excitation time for atoms in a medium. If you apply this situation at the interface of a medium and vacuum, the question becomes null. A change to the quanta perspective has these sort of perks. I hope this helps.

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I understand. It's not the the light slows down, the excited states of the medium "hold on" to the light for a brief amount of time, leading to an aggregated slowing of the propagation of the wave. So is this what is meant by absorption and re-emission of light through a medium? –  Michael Jul 23 '12 at 22:07
How do you explain anomalous dispersion in this picture (when the index is less than 1, the phase speed of light is bigger than c)? The holding and emitting picture is also misleading. –  Ron Maimon Jul 24 '12 at 2:22
sorry I was out of town till yesterday. @Michael this is kind of what I had in mind, except the process is the culmination of all of the absorbing and emitting atoms involved in the medium. –  kηives Jul 27 '12 at 16:42
@RonMaimon I have never thought about that... I looks like it fails. Do you recommend a better physical picture of what's going on? –  kηives Jul 27 '12 at 16:43
@kηives: The picture is ok, so long as you don't think of it as a causal picture. You have to consider emission followed by absorption too. The phase velocity greater than c is counterintuitive, but it doesn't mean any actual disturbance goes faster than c, because there is a causal formulation in terms of fields. –  Ron Maimon Jul 27 '12 at 23:19

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