Spontaneous and amplified radiation at the initial stage of a SASE FEL

At the initial stage of a self-amplified spontaneous emission (SASE) free-electron laser (FEL), spontaneous undulator radiation in certain experimental configurations can dominate the amplified signal over an extended undulator distance. In this paper we study both the spontaneous and the amplified radiation in the framework of the paraxial wave equation and determine the transition from the dominance of spontaneous emission to exponential amplification. We compare theoretical expectations with SASE simulation codes GINGER and GENESIS.