Extracting photoelectron spectra from the time-dependent wave function. II. Validation of two methods: Projection on plane waves and time-dependent surface flux

Extraction of the photoelectron spectrum (PES) from the time-dependent wave-packet calculations is a fundamental requirement for studying strong-field ionization. To analyze the PES, one needs to extract the physical observables from the time-dependent wave function at the end of the laser pulse. The exact PES can be obtained by projecting this wave function onto the continuum states of the same binding potential, which have to obey the incoming boundary condition. This means that the continuum states at large distances from the atomic or molecular targets have to be localized in momentum space so that asymptotically the continuum states can be approximated by plane waves. In this paper, we extend our previous analysis [B. Fetic??, W. Becker, and D. B. Milo??evic??, Extracting photoelectron spectra from the time-dependent wave function: Comparison of the projection onto continuum states and window-operator methods, Phys. Rev. A 102, 023101 (2020)] and investigate two alternative methods of the PES extraction: projecting the time-dependent wave function onto plane waves as well as the so-called time-dependent surface flux (tSURFF) method. A thorough analysis is performed to check the reliability of these methods in comparison with the exact method. The time integral, which appears in the tSURFF ionization amplitude, is divided into smaller time intervals. Analyzing the corresponding partial amplitudes and their interference we relate these time intervals to particular parts of the PES. The method is applied to analyze the high-energy PES as well as the low-energy and very-low-energy structures in the PES.