Electrodynamics of rotating relativistic electron beams

On the basis of the Minkowski formulation, this paper discusses the basic laws governing the small-signal fields propagated along an electron beam which is rotating around its axis with constant angular frequency of rotation and drifting in the axial direction at constant relativistic velocity. In the first preliminary section are described the dc conditions for getting a stable relativistic electron beam in the presence of neutralizing ions with arbitrary number density and externally applied static magnetic field. Then, after a brief discussion of the ac-field equations, the constitutive relations for small-signal fields are obtained in both the laboratory frame, which is assumed to be an inertial frame, and the rest frame of the electron beam, which is not an inertial but a rotating frame. The rotating relativistic electron beam is found to be a nonuniformly moving dispersive medium or, more specifically, an inhomogeneous bianisotropic medium with space and time dispersion. With the use of the constitutive relations derived above, the following sections consider the energy and momentum for the small-signal fields, and their conservation relations, together with their transformation laws between the laboratory frame and the rest frame of the electron beam. Our discussion includes, as the special cases, all the important cases of an ion-neutralized or axially confined beam and the Brillouin beam.