Theory of periodic electrostatic focusing of electron beams

A general theory of periodic electrostatic focusing structures in micowave tubes is presented. A general method for evaluating the properties of focusing systems is developed. The method is based on the use of the Action Function and is applicable to structures whose potential distribution can be found by the method of separable variables. It is shown that this approach can yield results similar to, but more exact, than those obtained by the classical paraxial ray equation. The basic and double-ring structures have been examined in detail. Parameters describing these structures are presented in graphical and tabular form. It is shown that for irrotational flow, a specific current distribution is required. The use of computer techniques for the solution of electron flow problems is discussed. A computer analysis has been developed and results obtained are discussed. The accuracy of the Action Function and the computer analyses has been verified experimentally. It has been shown that the focusing potentials predicted by the Action Function analysis are in good agreement with the experimental values. The computer analysis has also been shown to give results comparable with those obtained practically. Hollow beams of 3.85, 5.4, and 15.6 µP have been focused using the basic ring structure. In order to investigate the effect of unbalance of focusing and defocusing forces at the inner beam boundary, pinhole measurements have been made. It is shown that the ring structure focuses a hollow beam with some change in the beam cross section. The effect of the unbalance of forces is to transform the current density profile. Results indicate that the transformation is desirable since a beam of constant current density may be transformed to a distribution which approaches that required for irrotational flow.