Feasibility study of a hard x-ray FEL oscillator at 3 to 4 GeV based on harmonic lasing and transverse gradient undulator

We studied the feasibility of a hard x-ray FEL oscillator (XFELO) based on a 3 to 4 GeV storage ring considered for the low-emittance upgrade of NSLS-II. We present a more detailed derivation of a formula for the small-gain gain calculation for 3 GeV XFELO published in the proceedings of IPAC'21 [1]. We modified the small-signal low-gain formula developed by K.J. Kim, et.al. [4{6] so that the gain can be derived without taking the \no focusing approximation" and a strong focusing can be applied. In this formula, the gain is cast in the form of a product of two factors with one of them depending only on the harmonic number, undulator period, and gap. Using this factor, we show that it is favorable to use harmonic lasing to achieve hard x-ray FEL working in the small-signal low-gain regime with the medium-energy electron beam (3-4 GeV). Our formula also allows FEL optimization by varying the vertical gradient of the undulator, the vertical dispersion, and the horizontal and vertical focusing, independently. Since a quite high peak current is required for the FEL, the collective effects of beam dynamics in medium-energy synchrotrons significantly affect the electron beam parameters. We carried out a multiple-parameter optimization taking collective effects into account and the result indicates the XFELO is feasible for storage ring energy as low as 3 GeV, with local correction of betatron coupling.