Dispersion equation of FEL with combined undulator and uniform magnetic field

Abstract

A general quintic dispersion equation, describing the linear stage of the amplification of an electromagnetic wave in FEL-ubitrons with a combined spiral undulator and longitudinal uniform magnetic field, is derived. It is shown that the relativistic electron beam in such an undulator has four normal waves: two longitudinal waves (fast and slow waves of space charge) and two transverse waves, which in the limiting case of vanishingly small amplitude of the undulator field transform into fast and slow cyclotron waves. The instability increments are found in the case when the electromagnetic wave is synchronized with the longitudinal or transverse beam waves both separately and at the same time. It is shown that for low beam current the increment is all the higher the larger the number of electron waves participating in the interaction with the signal wave.