Abstract: By using 1-D fluid model, linear dispersion equations for the free-electon laser (FEL) with a slow wave medium are derived. It is shown that the eigen wave interacting with a right-hand helical electron beam is a right-hand circularly polarized wave when the drifting velocity of the beam is smaller than that of light in the medium, and the eigen wave interacting with the same beam is a left-hand circularly polarized wave when the drifting velocity is larger than the medium light velocity. An appropriate choice of the medium refractive index not only leads to a shorter wavelength but also gives rise to enhancing the growth. Based on the conception of eigen waves existing in the meddium-filled FEL, some new physical explanations of the interaction of an electron beam with radiation waves are given, and both parallel and inverse cyclotron synchronisms are presented. Analyses indicate that for the FEL with a helical magnetic filed longitudinal bunching and azimuthal bunching appear at the same time, and that the inverse cyclotron synchronism can take place only in the medium-filled FEL.