FEL small signal gain reduction due to phase error of undulator

Effects of undulator phase errors on the Free Electron Laser small signal gain was analyzed and discussed. The gain reduction factor due to the phase error is given analytically for low-gain regimes, it shows that degradation of the gain is similar to that of the spontaneous radiation, which has a simple exponential relation with square of the rms phase error, and the linear variation part of phase error induces the position shift of maximum gain. The result also shows that the Madey's theorem still hold in the presence of phase error. The gain reduction factor due to the phase error for high-gain regimes also can be given in a simple way.