Multiphoton ionization distributions beyond the dipole approximation: Retardation versus recoil corrections

We study nondipole effects in multiphoton ionization of a two-dimensional hydrogen-like atom by a flat-top laser pulse of varied intensity. For this purpose, we solve numerically a two-dimensional Schrödinger equation treating a propagating laser pulse exactly. The resulting distributions are then compared to those calculated in the dipole approximation. A directional dependence of the energy-angular photoelectron distributions is demonstrated numerically in the case of a propagating laser pulse of a moderate and a high intensity. It is analytically interpreted based on the leading order relativistic expansion of the electron Volkov state, showing a significant contribution of the electron recoil to that behavior. In contrast, the retardation correction originating from the space- and time-dependence of the laser field leads to a tiny redshift of the photoelectron energy spectra. Other features of ionization distributions are also analyzed, including the sidelobes and the double-hump structures of multiphoton peaks, or their disappearance for intense propagating laser pulses.