Single-pass high-gain tapered free-electron laser with transverse diffraction in the postsaturation regime

It has been well known that the resonant interaction of an ultrarelativistic electron beam and the radiation field in the single-pass high-gain free electron laser (FEL) amplifier leads to the optical gain guiding. The transverse Laplacian term of the slowly varying wave equation in the linear regime can be approximated as a constant detuning parameter, i.e., |∇$\frac{2}{\text{⟂}}$ |~ k_R/z_R where k_R is the resonant wave number and z_R is the Rayleigh range of the laser. In the post-saturation regime, the radiation power begins to oscillate about an equilibrium for the untapered case while continues to grow by undulator tapering. Moreover, in this regime the gain guiding decreases and the simple constant detune is no longer valid. In this paper we study the single-pass high-gain FEL performance in the post-saturation regime with inclusion of diffraction effect and undulator tapering. Our analysis relies upon two constants of motion, one from the energy conservation and the other from the adiabatic invariant of the action variable. By constructing a two-dimensional axisymmetric wave equation and the coupled one-dimensional electron dynamical equations, the performance of a tapered FEL in the postsaturation regime can be analyzed, including the fundamental mode profile, the power efficiency and the scaled energy spread. We begin the analytical investigation with two different axisymmetric electron beam profiles, the uniform and bounded parabolic ones. It is found that the tapered FEL power efficiency can be smaller but close to the taper ratio provided the resonant phase remains constant and the beam-wave is properly matched. Such a tapered efficiency is nearly independent of transverse electron beam size before significant electron detrapping occurs. This is essentially different from the untapered case, where the power extraction efficiency is around the essential FEL gain bandwidth (or ρ , the Pierce or FEL parameter) and depends on the beam size. In conclusion it is also found that the power enhancement due to undulator tapering is attributed more by the field increase outside the transverse electron beam than that inside the transverse electron beam. Several scaling properties on the taper ratio and the transverse electron beam size are also discussed in this paper. Authors: