摘要： In this paper, the energy-extraction efficiencies of the electron beam are studied for several cases of free-electron laser (FEL). For an initially cold electron beam, the optimal initial detuning for the maximum energy-extraction efficiency and the corresponding saturation length are given. A scheme of the ‘top up’ is proposed to enhance the efficiency: after the electron energy being modulated somewhat, a phase shift and a step-down of the phase bucket are introduced, so that all electrons are located near the upper separatrix of the bucket in the phase space. Finally, the energy-extraction efficiency can be increased by 30% compared with that of the normal undulator case. For a linear tapering undulator with an initially cold electron beam, the simple scaling laws for the maximum energy-extraction efficiency and the corresponding optical power gain are obtained. For a tapered undulator with the pre-bunched electron beam, our analysis gives that the energy-extraction efficiency reaches the maximum when the phase bucket height of the tapered undulator is equal to the amplitude of the detuning modulation. The numerical results validated this reasoning and show that the efficiency has a large increase compared with the case of the unbunched electron beam. The single stair-step undulator is investigated, the optimal step and the corresponding saturation power and saturation length are given by analysis, they agree well with the numerical simulation results.