In this paper, we present a general theory of the double-undulator free-electron laser, in which an additional undulator with period close to the beam electron betatron oscillation period in the main undulator is introduced.

FEL small signal gain reduction due to phase error of undulator;

It will use the femto-second electron beam passing through a variable elliptically polarized undulator to produce the high bright THz radiation with various linear,elliptical or circular polarization.