OICC Press Journal of Theoretical and Applied Physics 2251-7235 14 2 2023 11 17 Start current and growth rate in Smith–Purcell free-electron laser with dielectric-loaded cylindrical grating 10.1007/s40094-019-00358-0 EN Zahra Rezaei Department of Physics, Faculty of Science, Arak University, P.O. Box 38156‑8‑8349, Arāk, Iran 0000-0002-5910-8628 Bizhan Farokhi Department of Physics, Faculty of Science, Arak University, P.O. Box 38156‑8‑8349, Arāk, Iran Journal Article 2023 11 17 AbstractWe present an analytic theory for Smith–Purcell device in which a cylindrical metal–dielectric grating was derived by an annular electron beam propagating along the grating axis. A dispersion relation is obtained for azimuthally symmetric modes. Also, the first-order and second-order growth rates of the modes which are in phase with the beam are compared. It is shown that the second-order growth rate gives a more accurate description of beam–wave interaction for beams with larger thicknesses, as well as grating slots, with smaller depths and greater lengths. The start current for BWO operation of the SP-FEL is presented too. The importance of the minimum value of start current is that above it, the SP-FEL will operate as an oscillator, even in the absence of external feedback. In this case, the group velocity of the synchronous evanescent wave is negative, while the electron beam travels in the forward direction. It is shown in this paper that the dielectric and grating parameters affect the value of start current. So, by changing these parameters, the minimum value for the start current can be obtained.