Propagation in a crossed field periodic structure

A field theory of a crossed field periodic structure is given. Taking into account the effect of space charge, an integral equation method is employed to obtain a general dispersion equation for a structure supporting any space harmonic. The case of pure scalloping is treated in detail and the results are applied to forward and backward-wave amplifiers and backward-wave oscillators. Criteria for gain and oscillations are derived and corresponding expressions for the gain and start oscillation current are obtained. These results include the effect of all space harmonics.