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Scaling laws of a free electron laser based on a Thomson Source

We present a study based on a parametric optimization of a Thomson Source operated in FEL mode. This deals with the proposed scheme to use a high intensity laser pulse colliding with a high brightness electron beam of low to medium energy (around 10 MeV). Electrons undulating in the incoming laser field may emit radiation in a FEL coherent mode as far as some conditions are satisfied. A set of simple analytical formulas taking into account 3D effects is derived, in order to express these conditions in terms of three free parameters, namely the wavelength of the colliding laser pulse, the FEL ρ parameter, and the peak current carried by the electron beam. A few examples of possible operating points are compared with results of 3D numerical simulations, showing the FEL coherent emission of X-rays by high brightness electron beams colliding with high intensity laser beams carrying pulse energies of about 10 J.