Amplification of current density modulation in a FEL with an infinite electron beam

We show that the paraxial field equation for a free electron laser (FEL) in an infinitely wide electron beam with κ-2 energy distribution can be reduced to a fourth ordinary differential equation (ODE). Its solution for arbitrary initial phase space density modulation has been derived in the wave-vector domain. For initial current modulation with Gaussian profile, close form solutions are obtained in space-time domain. In developing an analytical model for a FEL-based coherent electron cooling system, an infinite electron beam has been assumed for the modulation and correction processes. While the assumption has its limitation, it allows for an analytical close form solution to be obtained, which is essential for investigating the underlying scaling law, benchmarking the simulation codes and understanding the fundamental physics. 1D theory was previously applied to model a CeC FEL amplifier. However, the theory ignores diffraction effects and does not provide the transverse profile of the amplified electron density modulation. On the other hand, 3D theories developed for a finite electron beam usually have solutions expanded over infinite number of modes determined by the specific transverse boundary conditions. Unless the mode with the largest growth rate substantially dominates other modes, both evaluation and extracting scaling laws can be complicated. Furthermore, it is also preferable to have an analytical FEL model with assumptions consistent with the other two sections of a CeC system. Recently, we developed the FEL theory in an infinitely wide electron beam with κ-1 (Lorentzian) energy distribution. Close form solutions have been obtained for the amplified current modulation initiated by an external electric field with various spatial-profiles. In this work, we extend the theory into κ-2 energy distribution and study the evolution of current density induced by an initial density modulation.