Beam-shape effects in nonlinear Compton and Thomson scattering

We discuss intensity effects in collisions between beams of optical photons from a high-power laser and relativistic electrons. Our main focus is on the modifications of the emission spectra due to realistic finite-beam geometries. By carefully analyzing the classical limit we precisely quantify the distinction between strong-field QED Compton scattering and classical Thomson scattering. A purely classical, but fully covariant, calculation of the bremsstrahlung emitted by an electron in a plane-wave laser field yields radiation into harmonics, as expected. This result is generalized to pulses of finite duration and explains the appearance of line broadening and harmonic substructure as an interference phenomenon. The ensuing numerical treatment confirms that strong focusing of the laser leads to a broad continuum while higher harmonics become visible only at moderate focusing, and hence lower intensity. We present a scaling law for the backscattered photon spectral density which facilitates averaging over electron beam phase space. Finally, we propose a set of realistic parameters such that the observation of intensity-induced spectral red shift, higher harmonics, and their substructure becomes feasible.