# Leaky Space-Charge Waves I: Čerenkov Radiation

## Abstract

In this paper a new theoretical approach is presented (or the analysis of the Čerenkov effect The radiation is associated with the presence of leaky space-charge waves on a modulated electron beam. The new theory, which is rigorous within the customary linearized regime, takes into account the effect of finite beam thickness and the actual space-charge waves on the beam. Of greater significance is the fact that it permits a self-consistent type of description, i.e., it includes the influence of the radiation back on the beam. A linear perturbation calculation is also employed to obtain useful expressions in simple but accurate form. This leak wave approach not only rapidly reproduces the known results relating to Čerenkov radiation, but offers some new information as well. Among other results, it is found that there exists an optimum beam thickness and velocity, and that a discrete infinite set of leaky space-charge waves occur so that the radiation does not emerge ideally at a single angle, although the angular spread is found to be very small. Interesting differences arise between the fast and slow space-charge waves; the fast waves decay along the beam while the slow waves are growing waves. The transverse field dependences of these two wave types are also different Numerical results are presented for certain typical parameter values. Copyright © 1965, Institute of Electrical and Electronics Engineers, Inc.