Compton inverse radiation emitted due to backscattering of laser pulses off the relativistic electrons possesses high spectral density and high energy of photons - in hard x-ray up to gamma-ray energies - because of short wavelength of laser radiation as compared with the classical electromagnetic devices such as undulators. In this report, the possibility of such radiation to monochromatization by means of collimation is studied. Two approaches have been considered for the description of the spectral-angular density of Compton radiation based on the classical field theory and on the quantum electrodynamics. As is shown, both descriptions produce similar total spectra. On the contrary, angular distribution of the radiation is different: the classical approach predicted a more narrow radiation cone. Also proposed and estimated is a method of the `electronic monochromatization based on the electronic subtraction of the two images produced by the electron beams with slightly different energies. A `proof-of-principle experiment of this method is proposed for the LUXC facility of KEK (Japan).
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