Ming Xie
Lawrence Berkeley National Laboratory
Initial value problem is one of the most important issues in the theory of high gain free electron lasers. It determines the start-up of amplification of initial signal or fluctuation in either laser field or electron beam. Yet, this problem was solved only in 1D case and 3D case without emittance and betatron oscillations. For 14 years since 1986, there has not been any theoretical progress on the solution of this problem. As a result, there has been no rigorous theory permitting a calculation in the presence of emittance and betatron oscillations of even the most essential quantities, for instance, power coupling from an input transverse field to an exponentially growing FEL mode, and noise power for SASE. In this paper, we present the first solution of the initial value problem in a grand scale, by taking into account simultaneously the following effects: diffraction and optical guiding of laser field, energy spread, emittance and betatron oscillation of electron beam. In particular, for the case of Gaussian beam distribution in energy and in transverse phase space, explicit calculations of input power coupling for coherent amplification and noise power for SASE are presented.
