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

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.EI052899-29055