Relation of harmonic radiative energy and laser phase in high-order harmonic generation

Abstract

The behavior of an atom in strong laser field and the dynamic process of high-order harmonic generation (HHG) are analyzed with the quantum theory and saddle-point method. Within a laser optical cycle of a long duration laser, the relations between the radiative energy and the laser phase of HHG show a Gaussian-like distribution which peaks at 7180°. A parameterized formula is deduced to calculate the high harmonic emission duration and the bandwidth of the HHG energy-phase distribution. The peak position, maximum and the bandwidth of a selected energy-phase distribution pulse depend on the laser parameters. Calculations show that a pure single (double) energy-phase distribution pulse(s) can be generated by a 3-cycle laser with a carrier-envelope phase of 175° (105°). A single symmetric energy-phase distribution pulse peaks at the carrier phase of 5386° and spans 5334°. The double energy-phase distribution pulses separated by 17533° have the same height. They peak at -3151° and 14381°, and span 7219° and 7334°, respectively. These results may be useful for researches of ultra-short X-ray generation, pulse separation, measurement and its application.