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Most bright sources of the radiation in hard x-ray and gamma--ray regions areundulator sources and Compton based ones. These sources are ultimate forproduction of polarized positrons necessary for future linear colliders ILC,CLIC. We developed a novel method for evaluating the energy spectrum ofelectrons emitting the undulator- and the inverse Compton radiation. The methodbased on Poisson weighted superposition of electron states is applicable forwhole range of the emission intensity per electron pass through the drivingforce, from much less than unity emitted photons (Compton sources) to manyphotons emitted (undulators), and for any energy of the photons. The methodallows for account contributions in the energy spread both from the Poissonstatistics and diffusion due to recoils. The theoretical results were confirmedby simulations. The electron energy spectrum was used for evaluation of theon-axis density of photons and their coherency making use of the`carrier--envelope' presentation for the emitting photons. The evaluatedmaximum coherency degree of single--electron radiation is evaluated to beproportional to the undulator spatial period and inversely to the energy ofelectrons, the number of coherently emitted undulator periods almostindependent of the undulator deflection parameter. The results of our study areapplicable both for the classical limit of classical undulator and for thequantum limit of Compton gamma--ray sources.

Electron spectra and coherence of radiation in undulators