  The subject of this paper is the scattering of a very intense laser pulse
(intensity $I\sim10^{21}\;{\mathrm{W/cm^2}}$) on relativistic electrons with
Lorentz factor between 10 and 45. The laser pulse is modeled by a plane wave
with finite length and the 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, $dW/d\Omega$, 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 $dW/d\Omega$ 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.
