Beam debunching due to ISR-induced energy diffusion

One of the options for increasing longitudinal coherency of X-ray free electron lasers (XFELs) is seeding with a microbunched electron beam. Several schemes leading to significant amplitude of the beam bunching at X-ray wavelengths were recently proposed. All these schemes rely on beam optics having several magnetic dipoles. While the beam passes through a dipole, its energy spread increases due to quantum effects of synchrotron radiation. As a result, the bunching factor at small wavelengths reduces since electrons having different energies follow different trajectories in the bend. We rigorously calculate the reduction in the bunching factor due to incoherent synchrotron radiation while the beam travels in an arbitrary beamline. We apply general results to estimate reduction of harmonic current in common schemes proposed for XFEL seeding.