  The backward Compton scattering is a basic process at future higher energy
photon colliders. To obtain a high probability of e->gamma conversion the
density of laser photons in the conversion region should be so high that
simultaneous interaction of one electron with several laser photons is possible
(nonlinear Compton effect). In this paper a detailed consideration of energy
spectra, helicities of final photons and electrons in nonlinear backward
Compton scattering of circularly polarized laser photons is given.
Distributions of gamma-gamma luminosities with total helicities 0 and 2 are
investigated. Very high intensity of laser wave leads to broadening of the
energy (luminosity) spectra and shift to lower energies (invariant masses).
Beside complicated exact formulae, approximate formulae for energy spectrum and
polarization of backscattered photons are given for relatively small nonlinear
parameter xi^2 (first order correction). All this is necessary for optimization
of the conversion region at photon colliders and study of physics processes
where a sharp edge of the luminosity spectrum and monochromaticity of
collisions are important.
