Magnetic reflection of constant-radius annular electron beams

The conditions that have to be satisfied by an axially symmetric field, when the scalar-potential distribution on a cylinder coaxial with the field is specified, are determined. It is shown that an infinite number of fields may be found to satisfy these conditions. A method is then given whereby this result may be used to synthetise magnetic fields that would reflect a constant-radius annular electron beam, the retardation of the beam taking place according to a prescribed function. An application of this result to the design of a reflex-klystron oscillator operating at millimetre wavelengths is discussed.