Grand initial value problem of high gain free electron lasers

Initial value problem is one of the cornerstones in the framework of high gain free electron laser (FEL) theory. It determines the startup of FEL interaction from initial signal or noise in either laser field or electron beam. Yet, this problem was solved only for the cases without emittance and betatron oscillations. In this paper, we present the first solution to the initial value problem in a grand scale by expanding the startup theory into the six-dimensional phase space. In particular, with Gaussian beam distribution, explicit calculations of mode coupling from input field and noise power for self-amplified spontaneous emission (SASE) are presented. One of the major results is the discovery of excessively large noise power for SASE.