I post in the QM section as an answer in terms of photons rather than just EM waves would be better.
I think there might be a complicated path integral-like formalism for photons, but a good way of looking at it is with EM waves. The crystal structure of the dielectric is composed of a bunch of dipoles, which oscillate in the E field of the EM wave. And then, an oscillating dipole gives off its own radiation. The values of epsilon in either medium should determine the direction of that radiation in those media.
You may want to start by reading this thread:
Your question was moved to the solid state forum because I'm assuming that you're asking about specular reflection off ordinary mirrors. Mirrors are typically metal films, and thus, it is a topic in solid state/condensed matter physics. The quantum mechanical picture of optical phenomena in solids is complex and extensive, and it involves just just the photon, but also the quantum mechanical description of the solid itself. This means that the nature of the solid as crucial, since that is what is interacting with the photons.
As far as I remember, Feynman has a very nice discussion, in terms of dipoles, in Vol. 1 of his lectures. He also discusses refraction, interference and diffraction.
Feynman goes into a Non-mathematical explanation in QED (for the layman) of photons reflected off glass surfaces, its a nice read, even if you are mathematically inclined.
Classical Electrodynamics by JD Jackson will give you a detailed mathematical overview in terms of photons I believe
As someone has already pointed out,that the reflection and refraction from a metal surface depends on plasma frequency.if frequency of light is above the plasma frequency ,it will transmit otherwise it will reflect. Plasmons the quantum excitation of plasma state will explain it.see here
Have you read Jackson? There is very little in it on photons.
Chapter 7 titled: Plane Electrodynamic Waves, and Wave Propagation. Your right, it deals with relflection and refraction of wave based light, but not photons, so your right Whoops!
Separate names with a comma.