Output power and degree of transverse coherence of X-ray free electron lasers

The explicit solution of the initial value problem is obtained for a SASE FEL (self-amplified spontaneous emission free electron laser) operating with a large ratio of electron beam emittance to the reduced wavelength, ϵˆ=2πϵ/λ≫1. The output power and the degree of transverse coherence are explicitly calculated for a high-gain linear regime. The degree of coherence is shown to be dependent on the ratio of the number of FEL gain lengths to the parameter ϵˆ. In particular, in the multi-mode limit the radiation from a SASE FEL has by the squared number of gain lengths higher degree of transverse coherence than a synchrotron radiation generated by a beam with the same emittance. An estimate for the degree of coherence of the radiation from X-ray SASE FEL at saturation is presented.