  We revisit the description of inverse Compton scattering sources and the
photon beams generated therein, emphasizing the behavior of their phase space
density distributions and how they depend upon those of the two colliding beams
of electrons and photons. Main objective is to provide practical formulas for
bandwidth, spectral density, brilliance, which are valid in general for any
value of the recoil factor, i.e. both in the Thomson regime of negligible
electron recoil, and in the deep Compton recoil dominated region, which is of
interest for gamma-gamma colliders and Compton Sources for the production of
multi-GeV photon beams. We adopt a description based on the center of mass
reference system of the electron-photon collision, in order to underline the
role of the electron recoil and how it controls the relativistic Doppler/boost
effect in various regimes. Comparisons with Monte Carlo simulations of inverse
Compton scattering in various scenarios are presented, showing very good
agreement with the analytical formulas: in particular we find that the
bandwidth dependence on the electron beam emittance, of paramount importance in
Thomson regime, as it limits the amount of focusing imparted to the electron
beam, becomes much less sensitive in deep Compton regime, allowing a stronger
focusing of the electron beam to enhance luminosity without loss of
mono-chromaticity. A similar effect occurs concerning the bandwidth dependence
on the frequency spread of the incident photons: in deep recoil regime the
bandwidth comes out to be much less dependent on the frequency spread.
