  Above-threshold ionization (ATI) results from strong field laser-matter
interaction and it is one of the fundamental processes that may be used to
extract electron structural and dynamical information about the atomic or
molecular target. Moreover, it can also be used to characterize the laser field
itself. Here, we develop an analytical description of ATI, which extends the
theoretical Strong Field Approximation (SFA), for both the direct and
re-scattering transition amplitudes in atoms. From a non-local, but separable
potential, the bound-free dipole and the re-scattering transition matrix
elements are analytically computed. In comparison with the standard approaches
to the ATI process, our analytical derivation of the re-scattering matrix
elements allows us to study directly how the re-scattering process depends on
the atomic target and laser pulse features -we can turn on and off
contributions having different physical origins or corresponding to different
physical mechanisms. We compare SFA results with the full numerical solutions
of the time-dependent Schroedinger equation (TDSE) within the few-cycle pulse
regime. Good agreement between our SFA and TDSE model is found for the ATI
spectrum. Our model captures also the strong dependence of the photoelectron
spectra on the carrier envelope phase of the laser field.
