Theory of Smith-Purcell radiation from a 2D array of small noninteracting particles

We construct the first-principles theory for the scattering of the Coulomb field of a fast electron traveling along an array of nano- or microparticles. The electron's trajectory is arbitrarily oriented in the plane parallel to the surface. We show that Smith-Purcell radiation accompanying this process results in very rich diffraction patterns, which differ dramatically from the ones for conventional diffraction gratings; a numerical analysis was made for THz frequencies. The generalized Smith-Purcell dispersion relation has been derived; it describes the link between frequency, two periods of the structure, the electron's velocity, and the angle of radiation with maximal intensity. The unique spatial distribution of the generated light can form the basis for the generation of structured-light electron-driven photon sources.