Wave Propagation Along a Magnetically-Focused Cylindrical Electron Beam

T h e theory of the helix traveling-wave has been treated in previous papers, 1 - 4 for eases in which the electrons move along straight lines parallel to the axis of the helix, as though immersed in an infinitely strong magnetic field. In practice, however, the electron beam is focused by a magnetic field of finite intensity, 0 ' fi such that the electrons follow spiral paths about the common axis. The purpose of this paper is to extend traveling-wave tube theory to the case of such focused beairs, and to compare the gain constants for the two types of electron motion. The motion of the beam in an infinite field is usually described as rectilinear flow; that in a finite focusing field, as Brillouin flow. The gain constant of the dominant mode in a traveling-wave tube may be computed from the field solution for the electron beam in the presence of its circuit structure. This procedure, however, requires the solution of cumbersome transcendental equations for each particular set of dimensions and operating conditions. A more flexible method of analysis has been provided by Pierce, 1 based on an expansion in terms of