Three-Dimensional Theory of the Smith—Purcell Effect

TLDR: In this article, a perturbation theory is used to derive explicit expressions for the electromagnetic field generated by a beam of electrons passing over the surface of a perfectly conducting metal, where the electrons constituting the beam all move with the same velocity in the same straight line but the charge density along the beam may vary in an arbitrary manner.

Abstract: The radiation emitted when a beam of electrons passes over the surface of a perfectly conducting metal is discussed. The electrons constituting the beam all move with the same velocity in the same straight line but the charge density along the beam may vary in an arbitrary manner. The metallic surface considered is any surface which may be obtained by a small perturbation of a plane. Perturbation theory is used to derive explicit expressions for the electromagnetic field generated. The mathematical techniques employed involve the use of the angular-spectrum-of-plane-waves representation of the electromagnetic field. The results obtained go far beyond those of previous (two-dimensional) theories. The special case of a sinusoidally varying surface is treated in detail and a numerical calculation yields the physically important quantity---the number of optical photons emitted per charge per unit path length---for a particular arrangement.