A small-signal field theory analysis of crossed-field amplifiers applicable to thick beams

Various authors have presented analyses of crossed-field amplifiers. Generally, these analyses have dealt with very thin electron beams in the presence of crossed-fields or with beams in which the bulk ac charge density within the beam has been neglected. It is the purpose of this paper to develop a field-theory analysis which is applicable to both thick or thin beams and which includes the effect of space charge. In place of the usual approximate match at the input boundary plane, a rigorous variational technique is presented, which considers the effect of the cutoff waves on the amplitudes of the propagating waves. This enables one to solve the input boundary problem for the first time insofar as the amplitudes of the propagating waves are concerned, without determining the individual amplitudes of the cutoff waves implicitly. As opposed to the usual approximate match, which is only valid for a thin, unmodulated entering electron stream, this variational method is valid for both thick and thin beams, as well as unmodulated or modulated beams entering a crossed-field slow-wave interaction region or entering a crossed-field drift region. The paper presents propagation constants obtained for thick beams based on the author's model. The variational procedure, which is a valid technique for other models as well as other types of beam devices, has been applied to one numerical case involving a thin beam and the results are compared with the results of the usual approximate match. A modification of the complex amplitudes of the excited waves, even in the case of thin beams, is noted.