Microscopic theory of Smith-Purcell radiation from 2D photonic crystal

The theory of Smith-Purcell effect for a 2D photonic crystal is constructed from the first principles proceeding from Maxwell's equations and microscopic characteristics of particles the crystal consists of. Two-dimensionality 2D is thought in two ways: i) the photonic crystal is an arranged system of particles disposed in a monolayer, ii) the periodicity is in two different directions. We derive the expression for the spectral and angular distribution of arising Smith-Purcell radiation. We analyse the features distinctive for the 2D photonic crystal and show that its spatial distribution differs drastically from the distribution of the radiation from conventional diffraction gratings.