Role of blue-shift length in macroscopic properties of high-harmonic generation

Abstract: The production of brighter coherent XUV radiation by intense laser pulses through the process of high-harmonic generation (HHG) is a central challenge in contemporary nonlinear optics. We study the generation and spatial propagation of high harmonics analytically and via ab initio simulations. We focus on the length scales defining the growth of the harmonic signal with propagation distance and show that the well-known coherence length limits HHG only for relatively low driving intensities. For higher intensities, the photoionisation of the medium, naturally accompanying HHG, leads to essentially transient phase matching and laser frequency blue shift. By systematically taking both of these factors into account, we demonstrate that the behaviour of the harmonic signal at higher intensities is defined by another length scale -- the blue-shift length. In this generation regime the XUV intensity at a given frequency first grows quadratically and then saturates passing the blue-shift length, but the total harmonic efficiency continues growing linearly due to the linear increase of the harmonic line bandwidth. The changeover to this generation regime takes place for all harmonic orders roughly simultaneously. The rate of the efficiency growth is maximal if the static dispersion is compensated by photoelectrons near the centre of the laser pulse. Our theory offers a robust way to choose the generation conditions that optimise the growth of the harmonic signal with propagation.