A Unified Theory of Electron Beam Interaction with Slow Wave Structures, with Application to Cut-off Conditions†

We present a new analysis of slow wave structures coupled to electron beams, which is based on their resonant properties rather than their travelling wave properties. Thus the analysis is particularly appropriate for narrow band slow wave structures. We expand the fields of a general symmetrical cavity chain in a set of solenoidal and irrotational short-circuit modes, defined within electric shorting planes on the coupling surfaces. The amplitudes of the modes are evaluated in terms of the beam current and tangential electric field on the coupling surfaces. The latter field is expanded in a set of open-circuit modes which amplitudes are related back to those of the short-circuit modes with the aid of a Variational Principle. Alternately, we expand the fields of a cavity in a set of open-circuit modes, ' driven ' by the beam current and by tangential magnetic field on the coupling surfaces. This latter field is then expanded in a set of short-circuit modes, which amplitudes are related to the open-circuit mode amplitudes with the aid of the Variational Principle. The accuracy of this procedure is verified for a narrow pass-band by proving the relation power flow equals stored energy times group velocity. It is possible to prove the equivalence of the dispersion relations for a 'cold' structure obtained from the short- and from the open-circuit mode expansions. We then obtain the small signal transmission line equations for a longitudinally confined beam, including relativistic effects, under the action of cavity solenoidal electric field and space charge irrotational field. We apply our analysis to a study of the beam-circuit interaction when the slow space charge beam mode is nearly synchronous with the ' cold ' circuit near the latter's cut-off. This is the situation which often leads to oscillations in high-power structures, interfering with their operation at a lower frequency. On a small-signal basis, four waves exist, two of constant amplitude and a pair of growing and decaying waves. We attribute the cut-off oscillations, which exist in a relatively wide voltage range such that the slow beam mode is in synchronism with the ' cold ' circuit near the cut-off, to the presence of the growing wave of forward circuit power and negative beam power. As the voltage is lowered (in a structure of forward group velocity from zero to πphase shift), the negative power of the growing wave changes to positive power at about the condition marking the boundary of the cut-off oscillations. As the voltage is lowered further, two constant amplitude waves of negative beam power appear, enabling backward wave oscillations to exist within a narrow voltage range. This distinction between cut-off oscillations and backward wave oscillations should be made in the general class of slow wave structures.