Spectral splitting and phase matching of the macroscopic high-order harmonic generation in intense laser fields

We theoretically investigate the macroscopic high-order harmonic generation in argon gaseous medium by numerically solving the three-dimensional macroscopic propagation equation, and the results show that the harmonic spectral and spatial profiles of harmonics are gradually splitting with the increasing of the driving laser intensity. This splitting is mainly due to the distortions which the driving field suffers during propagation and the consequence is also on phase matching. To illustrate the physical mechanisms of harmonic splitting, a theoretical analysis of the phase matching is presented. The harmonic spectra from different focus positions reveal that the split is dependent on the focus-gas-jet relative position. Moreover, we demonstrate that the spectral splitting of high harmonics can hardly be observed for neon gas jet due to the high ionization energy.