The nonlinear interaction between electrons and high-frequency fields with discrete phase changes in a periodic structure

The interaction between an electron beam and high-frequency fields is discussed in terms of the general theory of the excitation of periodic structures, with allowance made for all the spatial harmonics of the forward and backward waves. A method is proposed for the fast numerical integration of equations with rapidly changing coefficients, which makes it possible to solve the interaction equations with allowance for the distribution function of the high-frequency field. It is shown that the high-frequency field distribution function must be fully taken into account for systems that are less than ten periods long.