  We present ionization probability and line shape calculations for the
two-step 3-photon ionization process, $1S \stackrel{2\hbar
\omega}{\longrightarrow}2S \stackrel{\hbar \omega}{\longrightarrow}\epsilon P
$, of the ground state of hydrogenic atoms in a non-monochromatic laser field
with a time--dependent amplitude. Within the framework of a three--level model,
the {\it AC Stark} shifts and non-zero ionization rates of all states involved
were taken into account together with spatial and temporal inhomogeneities of
the laser signal. In contrast with the usual perturbative technique, the time
evolution of the atomic states was simulated by direct numerically solving the
system of coupled time--dependent inhomogeneous differential equations, being
equivalent to the appropriate non-stationary Schr\"{o}dinger equation.
Particular numerical results were obtained for typical parameters of the pulsed
laser field that are employed in a new experiment to measure the $1S-2S$ energy
separation in muonium at the Rutherford Appleton Laboratory. The shifts and
asymmetries of the photoionization line shapes revealed may be of relevance for
ultra-high precision experiments in hydrogen in CW laser fields.
