  The interaction of a fast electron with a photonic crystal is studied by
solving the Maxwell equations exactly for the external field provided by the
electron in the presence of the crystal. The polarization currents and charges
produced by the passage of the electron give rise to the emission of the
so-called Smith-Purcell radiation. The emitted light probability is obtained by
integrating the Poynting vector over planes parallel to the crystal at a large
distance from the latter. Both reflected and transmitted light components are
analyzed and related to the photonic band structure of the crystal. Emission
spectra are compared with the energy loss probability and also with the
reflectance and transmittance of the crystal.
