Additionally to the "vivo" online discussion, a post-discussion is opened here:

@ Anton BABAEV

The formula for luminosity in slide 22 is exact for Gaussian beams only. So, it is interesting what beams are used in a practice (the remark is more relevant for laser beam than for electron one, I guess).

This formula can be used for estimation of the order of the effect, but it should be modified taking into account real beam profiles for more or less precise calculations.

The formula in slide 22 is valid, but it is just the result of a wise collection og physical quantities. 

In slide (35) I have reported what that formula misses. A very accurate and thoghtful calculation is reported in the article Hartemann, F. V. ; Brown, W. J.; Gibson, D. J.; Anderson, S. G. ; Tremaine, A. M. ; Springer, P. T.; Wootton, A. J. ; Hartouni, E.P. and Barty, C. P. J., High-energy scaling of Compton scattering light sources., Phys. Rev. Accel. Beams,vol. 21(8), pp. 100702 (2005) - https://doi.org/10.1103/PhysRevSTAB.8.100702, In which the various steps for a complete account of the electron and laser beam profiles is included. 

The calculations are however limited to the spatial parts, including the 4-D phase space, in terms of Gaussian distributions.

A more general calculation should however be based on the use of the Wigner transform with the inclusion of the whole 6-D phase space structure. 

The analysis with non-gaussian profiles has not yet been done (as far I know), I have the impression that long tail electron beams as those from plasma acceleration, could provide significant differences. However if distortion are not significant an expansion in terms of Gauss Hermite beams, could, in principle be used, to infer possible deviations.