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A theory of self-fields in a free-electron laser with electromagnetic-wave wiggler and ion-channel guiding is presented. The equations of motion for an electron have been analyzed. This equation together with its numerical solutions shows that the first part of group I and II trajectories is unstable. The effects of self-fields on the gain for groups I and II orbits have also been investigated by deriving the gain formula and numerical calculation. A gain decrement is found due to the effects of self-fields for group I orbits, and the gain enhancement for group II orbits. The gain decrement (enhancement) arises from diamagnetic (paramagnetic) generated by the self-fields. The gain decrement (enhancement) increases by increasing the beam density.

A theoretical study of electron trajectories, harmonic generation, and gain in a free-electron laser (FEL) with a linearly polarized electromagnetic-wave wiggler is presented for axial injection of electron beam. The relativistic equation of motion for a single electron has been derived and solved numerically. It is found that the trajectories consist of two regimes. The stability of these regimes has been investigated. The results show that the trajectories are stable except for some parts of the regime one. The effects of interaction on the transverse velocity of the electron are a superposition of two oscillation terms, one at the wiggler frequency and the other at the betatron ion-channel frequency. A detailed analysis of the gain equation in the low-gain-per-pass limit has been employed to investigate FEL operation in higher harmonics generation. The possibility of wave amplification at both wiggler frequency and betatron ion-channel frequency for their odd harmonics has been illustrated.

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A theory of the dispersion relation for electromagnetically pumped free-electron laser in the presence of a special tapered axial guide magnetic field is presented. An analysis of the steady-state electron trajectories is obtained by solving the equations of motion. Next an eleventh-degree polynomial equation for electromagnetic and space-charge wave is derived. Numerical solution of the polynomial equation of the dispersion relation yield the complex wave number as a function of the frequency of the waves. These results are used to illustrate the dependence of growth rate curves on the axial guide field frequency. It is found that the tapered guide field shifts electron trajectories and enhances the growth rate in the comparison of employing uniform axial magnetic field, without needing a strong guide magnetic field.