Numerical analysis of radiation by a relativistic electron beam propagating parallel to a reflection grating

Two-dimensional radiation by a relativistic sheet electron beam propagating parallel to a reflection grating composed of a sinusoidally corrugated conducting surface is studied rigorously using the mode-matching method. Accurate dispersion curves for the eigenmodes that govern the Smith-Purcell radiation and the instability of the electromagnetic surface wave are presented. The imaginary parts of the eigenwavenumber and eigenfrequency give the leakage coefficient of space-charge waves and the growth rate of electromagnetic surface waves. Their dependences on the dimensions of the radiating structure are discussed. Optimum parameters for achieving the maximum leakage coefficient and the growth rate are determined. The method is general and can be applied to any two-dimensional system of electron beam coupled to an open periodic structure of arbitrary profile