Higher harmonic generation in a FEL with an axisymmetric undulator of the induction type

It is shown that the amplitudes of transverse components of velocities of electrons in an axisymmetric undulator depend resonantly on the value of an axial guiding magnetic field. They increase substantially when the cyclotron frequency is approximately equal to the frequency of the undulator oscillations of the electrons or its higher harmonic, which corresponds to resonance pumping of the undulator motion. Such an effect was observed experimentally in a free electron laser with a hollow kiloampere electron beam of microsecond duration and an axisymmetric undulator consisting of alternating conducting and dielectric rings in an axial magnetic field of a solenoid. The maximum generated microwave power in a FEL reached several megawatts in the centimeter and millimeter ranges of wavelenghs.