Analytical description of free-electron-laser oscillations in a perfectly synchronized optical cavity

We present an analytical description of free-electron-laser (FEL) oscillations in a perfectly synchronized optical cavity by solving the one-dimensional FEL equations. It is shown that the radiation stored in the cavity eventually evolves into an intense few-cycle optical pulse in the high-gain and low-loss regime despite the lethargy effect. The evolution of the leading slope of the optical pulse, which is defined from the front edge toward the primary peak, is found to play an important role in generating the intense few-cycle optical pulse. The phase space evolution of electrons which interact with the leading slope is solved analytically in a perturbation method, leading to an analytical solution for the optical pulse evolution. The peak amplitude and the pulse length at saturation are found to scale with the electron beam density and optical cavity loss. Those scalings agree well with the intense few-cycle pulses recently observed in a high-power FEL.