Electron energy loss and Smith-Purcell radiation in two- and three-dimensional photonic crystals

A theoretical description of the electron energy loss and the Smith-Purcell radiation is presented for an electron moving near a two-dimensional photonic crystal slab and a three-dimensional woodpile photonic crystal. The electron energy loss and the Smith-Purcell radiation spectra are well correlated with the photonic band structures of these crystals and thus can be used as a probe of them. In particular, there is a selection rule concerning the symmetries of the photonic band modes to be excited when the electron moves in a mirror plane of the crystals. In the woodpile, a highly directional Smith-Purcell radiation is realized by using the planar defect mode inside the complete band gap.