One dimensional models for relativistic electron beam diode design

Most gyrolron electron gun designs are based on the use of electron trajectory codes; however, the complexity of the problem is such that a design procedure using only a simulation code can be very tedious and expensive. For these reasons one dimensional approximations to the flow are extremely useful. Such one dimensional flows can be used to synthesize first approximations to the electrode shapes. The purpose of this paper is to show how previous work on this technique for non-relativistic beams can be extended to the relativistic intense electron beam regime. The analysis is based on four one-dimensional planar or conical models for electron flow in diodes. The planar models are relativistic and can be used to obtain electrode shapes for space charge limited flow. The conical models shed light on important geometric effects. Principally, it is shown that the space charge limited current is larger in realistic geometries than in the corresponding planar configuration. A design procedure based on these flow models is illustrated by application to a diode for a 1 GW beam for a 35 GHz gyrotron.