Growth of transverse coherence in SASE FELs

We introduce the correlation function between the electric field at two different points in the transverse plane as a parameter to quantify the degree of transverse coherence. We also propose a more realistic model for the initialization of the radiation in computer codes used to study SASE FELs. We make these modifications in the code tda and use it to study the growth of transverse coherence as a function of electron beam size, beam current and transverse emittance. Our results show explicitly that the onset of full transverse coherence in SASE takes place much before the power saturates. With the more realistic model the onset of the exponential growth regime is delayed, and to get a given power from the FEL one needs a longer undulator than would be predicted by the original tda code.