Smith-Purcell Radiation from A Charge Moving above A Finite-Length Grating

Smith-Purcell radiation (SPR), generated by an electron beam traveling above a grating, is characterized by a broad range of frequencies in which the radiated wavelength depends on the angle of observation according to the SPR resonance relationship. A rigorous theoretical model of SPR from a three-dimensional bunch of relativistic electrons passing above a grating of finite-length is presented by an electric-field integral equation method. The finite length grating results are compared with an infinitely-long grating assumption in which periodic boundary conditions are rigorously applied and with a model based on the image charge approximation. While the SPR resonance relationship is the same in all three formalisms, significant errors up to an order of magnitude in the strength of the radiated energy are introduced by the two approximations. Numerical examples are calculated for an ∼ 18 MeV bunch traveling above different finite length gratings with a period of 2.5 mm.