Decoupling the complex amplitudes of partial waves in two-photon ionization

Taking the two-photon ionization process in hydrogen atoms as an example, we propose a scheme to decouple the partial-wave information in ultrafast ionization. By analyzing the interference in the photoelectron angular distributions obtained from the solution of the time-dependent Schr\"odinger equation, the phase difference and amplitude ratio for different partial waves can be extracted. In the perturbative regime, the one-photon ionization by the 2ω pulse is taken as a reference to distinguish the contributions from the s and d waves in two-photon ionization. In the nonresonant cases, the phases of different partial waves in two-photon ionization are essentially the Coulomb phase shifts, which are independent of the pulse width. In the resonant case, an additional phase is accumulated during the resonant transition. The resulting time delays for both partial waves are proportional to the pulse duration. When the above-threshold ionization occurs, the continuum-continuum transition introduces an additional phase. The amplitudes of both partial waves are strongly related to the photon energy. They can be used to reveal hidden structures, such as the nonlinear Cooper minimum of different partial waves. This method of decoupling the partial-wave information can be extended to more complex atoms with photons across a wide range of energy.