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The non-canonical Lie perturbation method for analyzing relativistic electron beams in free electron lasers [Y. Kishimoto et al., Phys. Plasmas 2, 1316 (1995)] is extended to the case with harmonic gyro-resonance due to the coexistence of a focusing wiggler and an axial guiding field, which allow the maximum beam current to be increased. By using non-canonical guiding-center variables, we have solved the particle motion not only far from the harmonic gyro-resonance but also near the resonance. Far from the resonance, the maximum beam current is found to increase in proportion to (Bg/Bw)2 (Bw and Bg are the strength of the wiggler and guiding fields, respectively). On the other hand, near the resonance, the beam is found to be confined in a finite radial region and then transmitted because of higher order secular perturbations.

Stability Analysis of Relativistic Electron Beams in a Wiggler with Harmonic Gyro-Resonance Using the Lie Perturbation Method