Probing Nuclear Motion by Frequency Modulation of Molecular High-Order Harmonic Generation

Molecular high-order harmonic generation (MHOHG) in a non-Born-Oppenheimer treatment of ${{\mathrm{H}}_2}^{+}$, ${{\mathrm{D}}_2}^{+}$, is investigated by numerical simulations of the corresponding time-dependent Schrödinger equations in full dimensions. As opposed to previous studies on amplitude modulation of intracycle dynamics in MHOHG, we demonstrate redshifts as frequency modulation (FM) of intercycle dynamics in MHOHG. The FM is induced by nuclear motion using intense laser pulses. Compared to fixed-nuclei approximations, the intensity of MHOHG is much higher due to the dependence of enhanced ionization on the internuclear distance. The width and symmetry of the spectrum of each harmonic in MHOHG encode rich information on the dissociation process of molecules at the rising and falling parts of the laser pulses, which can be used to retrieve the nuclear dynamics. Isotope effects are studied to confirm the FM mechanism.

Figure 1

Illustration of pulse envelope effect on the shifts of harmonic spectra. The amplitude chirp in the pulse has a similar effect as the frequency chirp [20]. On the rising part of the laser pulse ($t<0$, solid blue line), $dI(t)/dt>0$ leads to a blueshift in HHG. On the falling part ($t>0$, red dotted line), $dI(t)/dt<0$, which leads to a redshift. At the beginning, the molecule is at its equilibrium distance $R_e$. However, during the interaction with the laser pulses, the molecule dissociates, leading to larger ionization and more intense spectra on the falling part.

Figure 2

Envelope of the laser pulse $f(t)$ and the internuclear distance $⟨R(t)⟩$ as a function of time. The duration of the laser pulse is 20 cycles in total. The wavelength is 800 nm, and the intensity is $I=3\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 3

MHOHG spectra of ${{\mathrm{H}}_2}^{+}$ and ${{\mathrm{D}}_2}^{+}$ in NBOA calculations and in fixed-nuclei approximation. The laser parameters are the same as those in Fig. 2. In (a), the intensities of harmonics are shifted by 2 orders for clarity. In (b), the intensities of harmonics are not shifted.

Figure 4

Time profile of MHOHG with order $N>20$. The upper panel is the time profile of MHOHG from ${{\mathrm{H}}_2}^{+}$ in fixed-nuclei approximation, the middle and the lower panels are the time profiles of MHOHG from ${{\mathrm{H}}_2}^{+}$ and ${{\mathrm{D}}_2}^{+}$ in NBOA, respectively.