Harmonic generation in free-electron laser with circularly polarized wiggler and ion-channel guiding

A detailed analysis of the trajectory of an electron injected axially in a free-electron laser (FEL), having a circularly polarized wiggler and ion-channel focusing, has been carried out using single particle dynamics. We show that along with transverse oscillations at the wiggler frequency, betatron oscillations due to the ion-channel field are also setup. This leads to axial oscillations in the electron motion. It is found that spontaneous emission is possible at the upshifted ion-channel betatron frequency and its odd harmonics. Further, we evaluate the gain in radiation at these frequencies with the help of the pendulum equation, in the low-gain-per-pass limit.