Reconstruction of ionization probabilities from spatially averaged data in N-dimensions

We present an analytical inversion technique which can be used to recover ionization probabilities from spatially averaged data in an N-dimensional detection scheme. The solution is given as a power series in intensity. For this reason, we call this technique a multiphoton expansion (MPE). The MPE formalism was verified with an exactly solvable inversion problem in 2D, and probabilities in the postsaturation region, where the intensity-selective scanning approach breaks down, were recovered. In 3D, ionization probabilities of Xe were successfully recovered with MPE from simulated (using the ADK tunneling theory) ion yields. Finally, we tested our approach with intensity-resolved benzene ion yields showing a resonant multiphoton ionization process. By applying MPE to this data (which was artificially averaged) the resonant structure was recovered-suggesting that the resonance in benzene may have been observable in spatially averaged data taken elsewhere.