We consider electron-atom scattering in a circularly polarized laser field at sufficiently high electron energies, permitting to describe the scattering process by the first order Born approximation. Assuming the radiation field has sufficiently moderate intensities, the laser-dressing of the hydrogen target atom in its ground state will be treated in second order perturbation theory. Within this approximation scheme, it is shown that the nonlinear differential cross sections of free-free transitions do neither depend on the {\it dynamical phase} $\phi$ of the radiative process nor on the {\it helicity} of the circularly polarized laser light. Relations to the corresponding results for linear laser polarization are established

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Floquet close-coupling (FCC) analysis of stimulated electron-hydrogen scattering in the field of a low-frequency CO 2 laser field is carried out. Results are obtained for one-, two-and three-photon absorption and emission processes in terms of the differential scattering cross sections as a function of the scattering angle. Two critical geometries are considered in which the argument of the Bessel function in the well known low-frequency formula becomes negligibly small. It is found that the ratios of the cross sections for small-angle scattering in both geometries are many orders of magnitude larger than predicted by the low-frequency formula (LFF). The result supports an analogous finding of many orders of magnitude larger values of the cross section measured in Ar and He by Wallbank and Holmes, and confirms the failure of LFF for small-angle scattering.

We consider the one-photon excitation of atomic hydrogen by electron impact in the presence of a homogeneous monochromatic linearly polarized electric field as a model for the radiation field of the laser. In the framework of the approximative approach of Faisal and Rahman [in Fifth International Conference on Atomic Physics, edited by R. Marrus, M. H. Prior, and H. A. Stutgart (University of California, Berkeley, 1976), p. 49], suited for high-energy electrons, we present an analytic expression for the transition amplitude to an arbitrary nlm state, in terms of hypergeometric Gauss and Appell I'I functions, together with the numerical results for excitation to the n =4 states. PACS number(s): 32.90.+a, 34.80.Qb