Nonlinear theory of free-electron lasers with an oppositely directed signal wave

Energy exchange between relativistic electron beams and electromagnetic waves is studied for the case of FEL ubitrons and scattrons in which the electrons move parallel to the pumping wave and antiparallel to the signal wave. For a fast pumping wave whose phase velocity exceeds the translational velocity of the electrons, it is shown that some of the wave energy is absorbed by the electron beam, in which a fast space charge wave is excited. For slow pumping waves, the signal and pumping wave amplitudes both grow at the expense of the electron beam, in which a slow space charge wave is excited. Although the latter case occurs in ubitrons, the signal amplitude does not change significantly during the interaction because the pump field is static.