Nonlinear Compton scattering with a laser pulse

We investigate the scattering of intense laser radiation on free electrons using a semiclassical relativistic approach. The laser field is described as an ideal pulse with a finite duration, a fixed direction of propagation, and indefinitely extended in the plane perpendicular to it. This allows the use of Volkov solutions and leads to a transition amplitude which is a product of a three-dimensional delta function with a linear combination of three one-dimensional integrals that we evaluate numerically. We give the general expression of the emitted photon spectrum as a function of frequency and direction valid for any initial geometry of the electron-laser beam scattering and for arbitrary shape, duration, and polarization of the laser pulse averaged over the initial electron spin and summed over the emitted photon and ejected electron polarizations. At a fixed photon scattering angle, one obtains a continuous frequency distribution with a succession of maxima located near the discrete values corresponding to the monochromatic case. We present results for head-on collisions and circularly polarized laser pulses. Our figures illustrate the dependence of the photon spectrum on pulse parameters (duration, shape, and maximum intensity) and the role of the initial electron energy. For a few-cycle linearly polarized pulse we also explore the effect of the carrier-envelope phase.