A Collection of Formulae for the Design of Compton Back-Scattering X-ray Sources

Along with the development of FEL X-ray sources, the design and the construction of Compton Back-Scattering (CBS) devices for different applications are being pursued. These sources are designed to provide monochromatic, high peak brightness tunable light beams, with photon energies above 10 KeV. The source brightness is the most significant figure of merit, which specifies the suitability of X-ray beams for their application in a specific field of interest and also reflects the quality of the electron beam itself, which in turn depends on the charge, emittance, energy spread, etcetera. This article is intended as an introductory review to the field and we embed a few theoretical considerations and analytical formulae to develop simple tools for the design of CBS X-ray sources. The present attempt is analogous to previous efforts, put forward to model Free Electron Laser devices, using scaling relations accounting for the entanglement between the different physical quantities contributing to the output beam performances. We comment on the reliability of our analysis by a comparison with the wealth of results (either numerical and analytical) reported in the dedicated literature.