A rigorous solution is obtained for the problem of radiation from an electric point charge that moves, at a constant speed, parallel to an electrically perfectly conducting grating. The relevant vectorial electromagnetic problem is reduced to two two-dimensional scalar ones. A Green’s-function formulation of the two problems is employed. For both cases, an integral equation of the second kind for the remaining unknown function is derived. This integral equation is solved numerically by a method of moments. Some numerical results for the radiation from a moving point charge above a sinusoidal grating are presented; in particular, the power losses of the point charge have been estimated.
P. M. van den Berg, "Smith-Purcell radiation from a point charge moving parallel to a reflection grating," J. Opt. Soc. Am. 63, 1588-1597 (1973)