Quantum-path analysis and phase matching of high-order harmonic generation and high-order frequency mixing processes in strong laser fields

We study phase-matching conditions for high-order harmonic generation as well as high-order sum-and difference-frequency mixing processes in strong laser fields, using a graphical approach described in Balcou et al (1997 Phys. Rev. A 55 3204-10). This method is based on the analysis of the different quantum paths that contribute, with different phase properties, to the single-atom response. We propose a simple numerical method to disentangle the quantum paths contributing to the generation process. We present graphical maps of the phase matching around the laser focus, which allow one to predict the geometries that optimize the conversion efficiency of the process considered. The method is applied to the study of sum-and difference-frequency mixing processes. The qualitative predictions of the graphical phase-matching approach are confirmed by numerical propagation calculations.