Phase Properties of High Harmonics and Spatial Structure of the Harmonic Beam

The harmonic phase dependence on the laser field intensity is one of the key properties of high-harmonic generation (HHG) in intense laser field. We have studied this dependence both analytically and numerically. Our method, in contrast to other approaches, does not use any phenomenological parameters. We suggest an analytical equation describing the dependence of the harmonic phase on the laser intensity. Analytical results agree with the results of numerical integration of the 3D time-dependent Schrödinger equation (TDSE) for a model one-electron atom in external laser field. We compare the analytical and numerical results obtained for two different fundamental wavelengths and various fundamental intensities. The investigated dependence of the harmonic phase leads to curved harmonic wavefront in the case of generation in the laser beam focus. We suggest an equation for this curvature, whose solution agrees well with the numerical results. Finally, we present an equation for the radius of curvature of the harmonic wavefront generated at some distance from the focus of the generating beam.