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Because of the Meissner effect, no magnetic fields can pierce through the body of a superconductor. Since EM waves need both their electric and magnetic field components, it cannot pierce through the body of the superconductor.

Is this correct, or is there something I'm missing?

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At the frequencies (=energy) of visible light, there is no superconductivity. The band gap due to Cooper pairs is much smaller. Radiation at higher frequencies (energies) breaks Cooper pairs, creating essentially normal electrons. These electrons behave like in normal metals, and so are not entirely opaque.

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    $\begingroup$ Neutron star material is superconducting with a band gap of order MeV. But I take your point. $\endgroup$
    – ProfRob
    Jan 17, 2015 at 19:02

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