In this study, the relativistic electron beam with an elliptical cross-section is introduced instead of the relativistic electron beam with a circular cross-section in free-electron laser (FEL). Influences of ion-channel guiding, axial magnetic field, Planar Wiggler Magnetic Field (PWMF) and self-fields on trajectories of the elliptical relativistic electron beam have also been investigated. At first, it was assumed that the PWMF is applied along the minor axis of the beam cross-section ellipse. In the second case, the PWMF was applied along the major axis of beam cross-section ellipse, the results of which were compared for the two cases. Effects of elliptical cross-section demotions and elliptical electron beam density on electron trajectory have been studied. The results were compared in the limit that the elliptical electron beam is converted to a circular electron beam, which has also been graphically presented. Moreover, a limit case in which the elliptical beam is converted into a sheet beam was studied. Furthermore, a formula was obtained for the small signal gain in a FEL with elliptical electron beam. The obtained gain for a FEL, together with a relativistic electron beam with an elliptical cross section was compared to the maximum gain for a FEL together with a relativistic electron beam with a circular cross section. It is shown that increasing the ratio of the semi-major to semi-minor axes of the beam cross-section causes the gain rate of the FEL to increases. We investigated a study of electron trajectories and dispersion equation in relativistic electron beam with elliptical cross section under the influence of electromagnetic wave wiggler and axial magnetic field considering ion-channel guiding and self fields in Appendix A.
