Forward- and backward-scattering quantum orbits in above-threshold ionization
High-order above-threshold ionization (ATI) is characterized by the rescattering of the ionized electron off the parent ion. The ATI amplitude can be calculated applying the saddle-point (SP) method and the uniform approximation to a five-dimensional integral over the ionization time, intermediate electron momentum, and rescattering time. The so-obtained amplitude is a coherent sum of amplitudes associated to the two classes of the SP solutions. The low-energy structures in the ATI spectra are determined by the forward-scattering SP solutions, while the high-energy spectra (plateau and cutoff) are determined by the backward-scattering SP solutions. In order to properly describe the intermediate-electron-energy spectra, additional SP solutions should be taken into account. For these solutions, the imaginary part of the rescattering time can be large and, for their explanation, we introduce complex-time quantum orbits.