A type-I superconductor can expel almost all magnetic flux (below some critical value $H_c$) from its interior when superconducting. Light as we know is an electromagnetic wave. So what would happen if we shine a light deep into a type-I superconductor? Suppose the magnetic field of the light is significantly lower than $H_c$.
And what about type-II superconductors?
Answer
In basic electrodynamics it is stated that light cannot "get inside" a metal. Therefore one just thinks of piece of metal as of boundary conditions for any light-related problems. I don't see any difference with that approach when it comes to superconductors. You can just think of superconductor as of ordinary metal with absolutiely the same conclusion about reflection of light off it's surface.
On the other hand it is obvious that some atoms or the metal must somehow interact with the fields. When we talk about electrodynamics of continuous media we deal with scales that are much larger than atomic scale. The statement about non-penetration of magnetic field inside of superconductor is valid for large scales as well, while it actually gets inside the media to the depth around $10^{-5}$ centimetres. In comparison to inter-atomic scales this is quite large. The same holds for "light not getting inside metal".
When it comes to x-rays, I don't think that one can use classical electrodynamics at all, because wavelengths are starting to be comparable to the atomic sizes (1 nm for soft x-ray and 0.01 for hard x-ray against 0.05nm for Bohr radius).
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