Synchronism conditions for a hybrid weakly-relativistic FEL

We propose an analytical technique of twofold perturbation expansion to solve the nonlinear equations of motion of weakly-relativistic electrons moving in the magnetostatic pump field of a hybrid planar free electron laser (FEL) and apply it to obtain linear corrections to the electron trajectories caused by interaction with given propagating modes of a circular waveguide. In a single-particle approximation conditions of the most efficient interaction between an electron beam and propagating modes of a circular waveguide (resonant conditions) are derived analytically using small-signal theory in the microwave field.