Laser utilizing free electrons moving in transverse gradient fields

Quantum-mechanical equations are derived for the amplitudes of the probabilities of transitions in a free-electron laser when transverse electron oscillations are due to a static potential, which is homogeneous in the direction of motion of the beam, but has a transverse gradient. The resonance frequency of the system is found and the linear gain is derived for the odd harmonics of this frequency. Averaging of the gain is carried out over the initial distribution of electrons in a transverse cross section of the beam. Estimates are obtained of the maximum lasing frequency and of the gain at this frequency.