Waves on a Filamentary Electron Beam in a Transverse-Field Slow-Wave Circuit

Abstract

The waves on a filamentary electron beam in a longitudinal dc magnetic field, and their interaction with a transverse‐field slow‐wave circuit, are studied in detail. All quantities are expressed in terms of circular polarization, with the circuit fields having arbitrary polarization. The beam is found to carry four waves: a positively polarized negative‐energy slow cyclotron wave, a negatively polarized positive‐energy fast cyclotron wave, and two synchronous (β=β_e) waves, one with positive polarization and positive energy, one with negative polarization and negative energy. The coupling of these waves to the circuit is described both in the manner of Pierce's longitudinal traveling wave tube (TWT) analysis and in a coupled‐mode description. For the two special cases of positive or negative circularly polarized fields, only the appropriately polarized beam waves couple. A third special case of linear polarization is more complicated, but essentially only the two cyclotron waves couple. In each of the three cases one positive‐energy and one negative‐energy beam wave is involved. In each case the equations can be made identical with the longitudinal TWT equations. As one example, a positively polarized circuit can be used as a fast‐wave coupler for an Adler‐type parametric amplifier, and its design becomes formally identical with the longitudinal Kompfner dip problem. Published Kompfner dip data can be used. The beam‐circuit interaction is found to be correctly described by considering only the apparent transverse motion of the beam's position, although the actual interaction mechanism involves both the true transverse interaction with the actual transverse electron velocities, and the longitudinal interaction of the dc beam velocity with the longitudinal ac fields off the dc beam axis. The latter leads to changes in longitudinal dc velocity of the electrons and thus accounts for the negative rf energies of two of the waves.