Self-consistent nonlinear theory of a premodulated electron beam propagating through a drift tube

A self-consistent nonlinear theory of the current and energy modulations in an electron beam propagating through a grounded drift tube is developed. The theoretical analysis is based on the assumption that each beam segment has a prescribed current profile at the drift tube entrance. A closed integrodifferential equation for beam current is obtained in terms of time and propagation distance. Properties of the current and energy modulations are investigated from the integrodifferential equation for a broad range of system parameters. The current modulation amplitude decreases, reaches its minimum value, and increases as the beam propagates downstream. By linearizing the integrodifferential equation for beam current for small modulation, it is shown that the current modulation in the linear regime is a sum of the forward and backward density waves. On the other hand, the energy modulation in the linear regime is a difference between the forward and backward density waves. Several points are noteworthy from the current and energy modulations in the linear regime. First, the maximum current modulation occurs at the propagation distance, where the forward and backward density waves have the same intensity and sign. The corresponding energy modulation is zero. Second, the maximum energy modulation occurs at the propagation distance, where the forward and backward density waves almost cancel each other and where the corresponding current modulation is least. Third, wavelength of the amplitude oscillation in modulations increases with beam energy and decreases with beam intensity. copyright 1996 American Institute of Physics