Stimulated emission by electrons in an electromagnetic wave propagating along an electrostatic field

The interaction of a beam with a slow purely electromagnetic wave in the presence of a longitudinal electrostatic field is examined where the magnetic field of the wave acts to keep the particles in equilibrium states. The stability of the equilibrium electron states is studied in the linear and nonlinear approximations with regards to the amplitude of the phase oscillations of the particles, and the limits imposed on the magnitude of the longitudinal static field by the stability requirement are derived. Stimulated emission is shown to be possible when a parallel monoenergetic electron beam enters a plane transverse electromagnetic wave propagating along an electrostatic field, and no initial phasing or grouping occurs because the wave is amplified by the potential energy of the electrons in the static field.