Optimization of energy conversion during locking of electron bunches by electromagnetic wave in longitudinal electrostatic field

The variational method is used for theoretically optimizing the energy conversion during locking of long electron bunches by the longitudinal field component of a traveling electromagnetic wave with large amplitude E in a longitudinal electrostatic field of intensity E sub st. The governing three nonlinear equations and one inequality for such a distributed system are formulated so as to allow for a widely arbitrary variation of the electrostatic field, with stable motion of the oscillators (electrons) as the only constraint. The optimum law of this variation, which will yield the most efficient conversion of electrostatic field energy to traveling wave energy, is then sought on the basis of Pontryagin's maximum principle and a solution of the Euler equation for an appropriately introduced auxiliary functional. Calculations reveal that the space width of an electron bunch decreases and the amplitude of a traveling wave increases in the forward axial direction. Increasing the amplitude of the traveling wave will result in decreasing the phase width of the electron bunch and shifting of its center toward the peak of the wave's retarding field. As the phase width of the electron bunch is decreased, the ratio E/e sub st increases toward its limit of unity. The results of this analysis are applicable to the design of traveling-wave tubes.