Thomson backscattering X-rays from ultra-relativistic electron bunches and temporally shaped laser pulses

The process of Thomson scattering of an ultra-intense laser pulse by a relativistic electron bunch has been proposed as a way to obtain a bright source of short, tunable and quasi-monochromatic X-ray pulses. The real applicability of such a method depends crucially on the electron-beam quality, the angular and energetic distributions playing a relevant role. In this paper we present the computation of the Thomson-scattered radiation generated by a plane-wave, linearly polarized and flat-top laser pulse, incident on a counterpropagating electron bunch having a sizable angular divergence and a generic energy distribution. Both linear and nonlinear Thomson-scattering regimes are considered and the impact of the rising front of the pulse on the scattered-radiation distribution has been taken into account. Simplified relations valid for long laser pulses and small values of both scattering angle and bunch divergence are also reported. Finally, we apply the results to the cases of backscattering with electron bunches typically produced with both standard radio-frequency-based accelerators and laser-plasma accelerators.