We present an elementary proof based on a direct calculation of the property of completeness at constant time of the solutions of the Klein-Gordon equation for a charged particle in a plane wave electromagnetic field. We also review different forms o
f the orthogonality and completeness relations previously presented in the literature and we discuss the possibility to construct the Feynman propagator for the particle in a plane-wave laser pulse as an expansion in terms of Volkov solutions. We show that this leads to a rigorous justification for the expression of the transition amplitude, currently used in the literature, for a class of laser assisted or laser induced processes.
The subject of this paper is the scattering of a very intense laser pulse (intensity $Isim10^{21};{mathrm{W/cm^2}}$) on relativistic electrons with Lorentz factor between 10 and 45. The laser pulse is modeled by a plane wave with finite length and th
e calculations are performed within the framework of the classical electrodynamics, which is valid for the field intensity and range of electron energies we consider. For a pulse with the central wavelength $lambda=1060;{mathrm{nm}}$ and circular polarization, we study systematically the angular distribution of the emitted radiation, $dW/dOmega$, in its dependence on the electron energy for two collision geometries: the head-on collision (counterpropagating electron and laser pulse), and the 90 degrees collision (the initial electron momentum orthogonal to the laser propagation direction). We investigate the relation between $dW/dOmega$ and the trajectory followed by the electron velocity during the laser pulse and, for the case of a short laser pulse, we discuss the carrier-envelope phase effects. We also present, for the two mentioned geometries, an analysis of the polarization of the emitted radiation and a comparison of the results predicted by the exact classical formula with a high-energy approximation of it.
