Continuum-continuum dipole transitions in femtosecond-laser-pulse excitation of atomic hydrogen

We study in detail the electron energy spectrum produced by the interaction of atomic hydrogen initially in the 2s state with a femtosecond laser pulse. As a first step, we treat the problem within second-order perturbation theory. For that purpose, we analyze in the first instance the off-diagonal continuum-continuum transition matrix elements and derive asymptotic expressions when both the initial- and final-state energies are very close to the ionization threshold. We also consider the diagonal divergence and extend a recent calculation of Madajczyk and Trippenbach [J. Phys. A 22, 2369 (1989)]. By means of a proper analytical regularization of these matrix elements, we show that they may be considered as a distribution, which, in addition to a function and its first derivative, involves three principal parts. When this distribution is acting on a given function of energy, it is shown that the calculation of these three principal parts may be carried out fully analytically. We then present results for the electron energy spectrum. Various pulse shapes are considered, and a detailed discussion is given on the contribution of the intermediate continuum states to the second-order probability amplitude.