Theory of free-electron lasers with focusing pumping field

A nonlinear theory of ubitrons and scatterons with focusing pumping field is investigated for the case of Doppler synchronism. Average electron movement equations are derived. Scattering under normal and anamalous Doppler effect conditions is analyzed. Since all electron exchange energy with the magnetic field in exactly the same way, there are a number of possibilities for improving the electron efficiency, such as varying the phase velocity of the combination waves along the instrument so that the electrons are always in the retarding phase, and exploiting recuperation of the energy of the electron beam on a collector with lower voltage.