Conservation Laws of an Interacting Electron Beam

Abstract

The linearized theory of electron beams interacting with a circuit is studied. Such systems may be considered as coupling the modes of, for example, the beam and the circuit. It has been found, previously, that if these modes are properly defined, then there is a conservation law which states the constancy of the total power in these individual modes, independent of the extent of coupling. In single‐frequency beam‐type devices, this conservation law is Chu's kinetic power theorem. In multiple‐frequency parametric systems, it is the linearized Manley‐Rowe relation. The problem of such an interpretation is to determine what is an appropriate modal description that will lead to a convenient and useful conservation law. To obtain a systematic means of finding such a representation, we observe that any linear homogeneous system with n degrees of freedom has n quadratic invariances. The kinetic power theorem and the linearized Manley‐Rowe relation are examples of such quadratic invariances, which are distinguished from the others by being independent of certain of the parameters of the system. Methods are obtained for determining, from the differential equations of the system expressed in any appropriate variables, the quadratic invariance, if it exists, which is independent of a given parameter. If the basis is now changed to one which diagonalizes this invariance, then we will have determined a modal, or wave, description in terms of which the invariance becomes a kinetic power law, as it is usually written. This method of analysis is illustrated by application to the infinite‐plane parallel beam, the interaction of a longitudinal thin beam with an incompletely coupled circuit, and the interaction of a weakly constrained filamentary beam with a transverse‐field circuit.