| When matter (atoms, clusters and molecular) is exposed to intense laser fields, a wealth of exciting phenomena can be observed, including high-order harmonic generation (HHG), above-threshold ionization, atomic stabilization, X-ray lasing, laser-induced damage of dielectrics and molecular dissociation and so on. And high-order harmonic generation is all through one of the primary researches. High-order harmonic generation in gases is a useful source of coherent radiation, and higher order harmonics can be used to generate shorter wavelength, especially for soft X-rays radiation in the 'water window' region of the spectrum, which is characteristic of good spatial and temporal coherence. In particular recent years, with the rapid development of laser technology, especially femto-second technology, the light with shorter pulse duration and higher energy can be produced and the resulting intensity is comparable to (or higher than) the binding atomic Coulomb field, which makes a promising way to study the un-perturbative nonlinear optics. At present, the conversion efficiency for generating harmonic radiation is rather low. Therefore, how to produce higher order harmonics and more efficient conversion efficiency is the most important problem to the utility of HHG. The main jobs of this article are as follows:Firstly, the research on the one-dimension multi-atom molecular ions exposed in the ultra-short laser pulse. In the past years, HHG from molecular was focused on the laser wavelength of 800nm or longer. It is known to all, harmonic spectrum is sensitive to the laser frequency, which has been discussed in atom systems. Is the effect of laser frequency on HHG in molecular system same or analogous with that in atom systems? Beginning with the time-dependent Schrodinger equation, thehigh-order harmonic generation from one-dimension multi-atom molecular ions in an ultra-short laser pulse is calculated by Crank-Nicolson algorithm. That the dependence of high-order harmonic generation on the inter-nuclear distance R and the laser carrier frequency ω is discussed for the first time, and the conclusion is drawn that the optimal range of inter-nuclear distance should be changed for theextended harmonic generation for different laser frequency, and the lower frequency laser pulse is favorable to higher order harmonic generation as the inter-nuclear distance R increasing..Secondly, quasi-phase matched high-order harmonic generation in a gas medium in a hollow-core periodically modulated fiber at high gas pressure is described theoretically by self-consistent solution of three equations including time-dependent Schrodinger equation and wave equations for the fundamental and the harmonicfields. Higher harmonic orders and more efficient harmonics are produced with corresponding photon energies of hundreds of eV, and different modulation periods are also investigated. The simulated results are identical with the experimental observations published in Nature 2003. It is predicted to be possible to generate enhanced higher order harmonics in the 'water window' spectrum by this technique. For HHG in a gas jet the conversion efficiency for each harmonic order istypically l(Tn due to the short interaction region and the Guoy phase shift. The use ofa guiding geometry in a hollow-fiber waveguide makes it possible both to increase the interaction length and also to phase match the process by matching the phase velocity of the laser with that of the generated harmonics. This increases the conversion efficiency by factors of 100-1000 for mid-plateau harmonics around 30-70eV. However, the conversion efficiency for higher energy photons (>100eV) issignificantly lower falling to about 10"" per harmonic at photon energies of ~500eVclose to the cut-off for HHG using helium. This is because the higher energy photons are generated at high levels of ionization, where plasma-induced variation in the refractive index destroys the phase matching. So, techniques are put forward to improving the efficiency of HHG such as...
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