  The advent of powerful laser sources has made it possible to observe a
relatively large cross section of the excited state of Hydrogen atom. This is
due to the effect of joint collisions of a linearly polarized $N$-photon and
high-energy electron. For such a process, we evaluate the excitation cross
section for geometries, in which the laser field is perpendicular or parallel
to the initial momentum of the electron. The second-order, time-dependent
perturbation theory together with Bethe-Born approximation suitable for an
electron with a large incident energy is employed to obtain the transition
amplitude. The amplitudes are calculated for the S-S and S-D transitions of the
Hydrogen atom in the Sturmian representation of the non-relativistic Green's
function. In particular, we investigate the excitation cross sections for
transitions, which have an initial state $1S$ and final state nS with $n\in
\lbrace 2,3,4,5\rbrace$. The characteristic dependence of the excitation cross
section on the momentum of the projectile is shown and discussed. Our
investigation indicates that the Bethe-Born approximation yields reasonable
results for the excitation cross section of the simultaneous electron photon
excitation process when a high energy projectile is treated.
