Contribution of the pre-ionized H${}_2$ and the ionized H${}_2^{+}$ subsystems to the HHG Spectra of H${}_2$ in intense laser fields

Contributions of the pre-ionized H${}_2$ (PI-H${}_2$) and ionized H${}_2^{+}$ subsystems of the two-electron H${}_2$ system to its high-order harmonic generation in 8-cycle ${\sin }^2$-like ultrafast intense laser pulses are calculated and analyzed based on the solution of the time-dependent Schr"{o}dinger equation (TDSE) for the one-dimensional two-electronic H${}_2$ system with fixed nuclei. The laser pulses have $\lambda =390\mathrm{\&}532$ nm wavelengths and $I=1\times {10}^{14}$, 5$\times {10}^{14}$, 1$\times {10}^{15}$ &
5$\times {10}^{15}$ Wcm${}^{-2}$ intensities. It is found that at the two lower intensities, the PI-H${}_2$ subsystem dominantly produces the HHG spectra. While, at the two higher intensities, both PI-H${}_2$ and ionized H${}_2^{+}$ subsystems contribute comparably to the HHG spectra. In the H${}_2^{+}$ subsystem, the symmetry of the populations of H${}_2^{+}$(I) and H${}_2^{+}$(II) regions (left and right regions of H${}_2^{+}$ subsystem) is broken by increasing the laser intensity. Complex patterns and even harmonics also appear at these two higher intensities. For instance, at $1\times {10}^{15}$ Wcm${}^{-2}$ intensity and $\lambda$ = 532 nm, the even harmonics are appeared near cut-off region. Interestingly, at $5\times {10}^{15}$ Wcm${}^{-2}$ intensity and $\lambda$ = 390 nm, the even harmonics replaced by the odd harmonics with red shift. At $\lambda$ = 390 &\ 532 nm and $I=1\times {10}^{15}$ intensity, the two-electron cutoffs corresponding to nonsequential double-recombination (NSDR) with maximum return kinetic energy of 4.70$U_p$ are detected. The HHG spectra of the whole H${}_2$ system obtained with and without nuclear dynamics treated classically are approximately similar. However, at $1\times {10}^{15}$ Wcm${}^{-2}$ intensity and $\lambda$ = 532 nm, if we take into account nuclear dynamics, the even harmonics which are appeared near cutoff region, replaced by the odd harmonics with blue shift.