Periodic electrostatic focusing of a hollow electron beam

A method of focusing a long hollow electron beam is described in which beam space-charge fields are balanced against uniform radial electric fields and periodic radial and longitudinal electric fields. There is no magnetic field associated with this focusing method. Elimination of the usual large axial magnetic-field requirement of "confined flow" focusing allows the tube system employing the electrostatically focused hollow beam to be lighter in weight, more compact, and to enjoy an enhancement of over-all efficiency. A physical description of the focusing mechanism is given, followed by a mathematical treatment which yields conditions for stable beam flow. Expressions are derived for electrode voltages required to focus a given hollow beam, as well as for velocity variation and beam deflection. A comparison of beam "stiffness" is made with a conventional "confined flow" beam. Beam stability is investigated under various "overfocusing" conditions and imperfect entrance conditions. Finally, this focusing method is compared with other purely electrostatic methods. An experimental program was carried out which demonstrated the usefulness of this type of focusing. Results indicate that good beam definition and transmission coefficients can be obtained with some insensitivity to entrance conditions.