Solution to the initial value problem for a high-gain FEL via Van Kampenʹs method

Using Van Kampenʹs normal mode expansion, we solve the initial value problem for a high-gain free-electron laser described by the three-dimensional Maxwell–Klimontovich equations. An expression of the radiation spectrum is given for the process of coherent amplification and self-amplified spontaneous emission. It is noted that the input coupling coefficient for either process increases with the initial beam energy spread. The effective start-up noise is identified as the coherent fraction of the spontaneous undulator radiation in one field gain length, and is larger with increasing energy spread and emittance mainly because of the increase in gain length.