A self-consistent nonlinear theory of the current modulation in a relativistic electron beam propagating through a klystron amplifier is developed. The modulation amplitude increases, reaching peak and decreases slowly, as the beam propagates through the amplifier. A simple expression of scaling law for maximum current modulation is obtained. This scaling law could be useful in the design and fabrication of high-performance klystron. Nonlinear mode evolution in current profile is also investigated by Fourier-decomposing the current modulation obtained from the integro-differential equation. The mode evolution in a long-range propagation of electron beam exhibits various interesting physical properties. For example, the maximum amplitude of the fundamental mode (l equals 1) occurs at the propagation distance, where all other modes vanish or have a very small amplitude. This property ensures a monochromatic frequency.
Han Sup Uhm,
"Nonlinear mode evolution of current modulation in relativistic klystron amplifiers", Proc. SPIE 1872, Intense Microwave Pulses, (8 July 1993); doi: 10.1117/12.147482; https://doi.org/10.1117/12.147482