Effect of dressing on high-order harmonic generation in vibrating ${\text{H}}_2$ molecules

We develop the strong-field approximation for high-order harmonic generation in hydrogen molecules, including the vibrational motion and the laser-induced coupling of the lowest two Born-Oppenheimer states in the molecular ion that is created by the initial ionization of the molecule. We show that the field dressing becomes important at long laser wavelengths $\left(\approx 2\mu \text{m}\right)$, leading to an overall reduction of harmonic generation and modifying the ratio of harmonic signals from different isotopes.

Figure 1

(Color online) Schematic view of the harmonic generation process in ${\text{H}}_2$. Shown are the ground-state BO potential of the ${\text{H}}_2$ and the two lowest BO potentials of ${\text{H}}_2^{+}$. The three-step process consists of (1) ionization, (2) evolution of the remaining molecular core as prescribed by the field-dressed ionic ${\sigma }_g$ and ${\sigma }_u$ potential-surfaces while the active electron is driven by the external field, and (3) recombination into the molecular ground state.

Figure 2

(Color online) Harmonic generation using 800 nm laser pulses. Left-hand column, 1L case; right-hand column, 2L case. The upper row shows the harmonic intensities for the ${\text{H}}_2$ molecule. The lower row shows the harmonic ratio ${\text{D}}_2/{\text{H}}_2$ (black continuous curve), and the ratio predicted if one takes into account the short trajectory only (red dashed curve) or the long trajectory only (blue dotted curve).

Figure 3

(Color online) Trajectory weights for 800 nm laser pulses (${\text{H}}_2$ only). Left-hand panel: The trajectory weights for the 1L calculation. Right-hand panel: The trajectory weights for the 2L calculation. The red dashed curves are used for the short trajectory and the blue dotted curves are for the long trajectory. The black circles correspond to the longer trajectories. The trajectories shown by the dashed and the dotted lines are used to calculate the ratios depicted by the same curves in Fig. 2.

Figure 4

(Color online) Harmonic generation using 1500 nm laser pulses. Left-hand column, 1L case; right-hand column, 2L case. The upper row shows the harmonic intensities for the ${\text{H}}_2$ molecule. The lower row shows the harmonic ratio ${\text{D}}_2/{\text{H}}_2$ (black continuous curve), and the ratio predicted if one takes into account the short trajectory only (red dashed curve) or the long trajectory only (blue dotted curve).

Figure 5

(Color online) Trajectory weights for 1500 nm wavelength (${\text{H}}_2$ only). Left-hand panel: The trajectory weights for the 1L calculation. Right-hand panel: The trajectory weights for the 2L calculation. The red dashed curves are used for the short trajectory and the blue dotted curves are used for the long trajectory. The black circles correspond to the longer trajectories. The trajectories shown by the dashed and the dotted lines are used to calculate the ratios depicted by the same curves in Fig. 4.

Figure 6

(Color online) Harmonic generation using 2000 nm laser pulse with $2.5\times {10}^{14}\text{W}/{\text{cm}}^2$ intensity. Left-hand column, 1L case; right-hand column, 2L case. The upper row shows the harmonic intensities for the ${\text{H}}_2$ molecule. The lower row shows the harmonic ratio ${\text{D}}_2/{\text{H}}_2$ (black continuous curve), and the ratio predicted if one takes into account the short trajectory only (red dashed curve) or the long trajectory only (blue dotted curve).

Figure 7

(Color online) Trajectory weights for 2000 nm wavelength and $2.5\times {10}^{14}\text{W}/{\text{cm}}^2$ intensity (${\text{H}}_2$ only). Left-hand panel: The trajectory weights for the 1L calculation. Right-hand panel: The trajectory weights for the 2L calculation. The red dashed curves are used for the short trajectory and the blue dotted curves are for the long trajectory. The black circles correspond to the longer trajectories. The trajectories shown by the dashed and the dotted lines are used to calculate the ratios depicted by the same curves in Fig. 6.

Figure 8

(Color online) Harmonic generation using 2000 nm laser pulses with $5\times {10}^{14}\text{W}/{\text{cm}}^2$ intensity. Left-hand column, 1L case; right-hand column, 2L case. The upper row shows the harmonic intensities for the ${\text{H}}_2$ molecule. The lower row shows the harmonic ratio ${\text{D}}_2/{\text{H}}_2$ (black continuous curve), and the ratio predicted if one takes into account the short trajectory only (red dashed curve) or the long trajectory only (blue dotted curve).

Figure 9

(Color online) Trajectory weights for 2000 nm wavelength and $5\times {10}^{14}\text{W}/{\text{cm}}^2$ intensity. Left-hand panel: The trajectory weights for the 1L calculation. Right-hand panel: The trajectory weights for the 2L calculation. The red dashed curves are used for the short trajectory and the blue dotted curves are for the long trajectory. The black circles correspond to the longer trajectories. The trajectories shown by the dashed and the dotted lines are used to calculate the ratios depicted by the same curves in Fig. 8.

Figure 10

(Color online) Harmonic generation using 2000 nm laser with $5\times {10}^{14}\text{W}/{\text{cm}}^2$ interacting with molecules perpendicular to the laser polarization direction $\left(\theta =90°\right)$. Top row: Harmonic ratio ${\text{D}}_2/{\text{H}}_2$ compared to the SM ratios for shortest trajectories (left-hand panel) and the trajectory weights for ${\text{H}}_2$ from the simple-man’s model (right-hand panel). The red dashed curves are used for the short trajectory and the blue dotted curves are for the long trajectory. The black circles in the top-right-hand panel correspond to the longer trajectories. Bottom row: Harmonic spectrum for ${\text{H}}_2$.

Figure 11

(Color online) Comparison between the full calculation (black curves) based on Eq. (4) and by using the saddle-point approximation (red dotted curves). The calculations are for the 1L case.