High-order-harmonic generation in atomic and molecular systems

High-order-harmonic generation (HHG) results from the interaction of ultrashort laser pulses with matter. It configures an invaluable tool to produce attosecond pulses, moreover, to extract electron structural and dynamical information of the target, i.e., atoms, molecules, and solids. In this contribution, we introduce an analytical description of atomic and molecular HHG, that extends the well-established theoretical strong-field approximation (SFA). Our approach involves two innovative aspects: (i) First, the bound-continuum and rescattering matrix elements can be analytically computed for both atomic and multicenter molecular systems, using a nonlocal short range model, but separable, potential. When compared with the standard models, these analytical derivations make possible to directly examine how the HHG spectra depend on the driven media and laser-pulse features. Furthermore, we can turn on and off contributions having distinct physical origins or corresponding to different mechanisms. This allows us to quantify their importance in the various regions of the HHG spectra. (ii) Second, as reported recently [N. Suárez et al., Phys. Rev. A 94, 043423 (2016)], the multicenter matrix elements in our theory are free from nonphysical gauge- and coordinate-system-dependent terms; this is accomplished by adapting the coordinate system to the center from which the corresponding time-dependent wave function originates. Our SFA results are contrasted, when possible, with the direct numerical integration of the time-dependent Schrödinger equation in reduced and full dimensionality. Very good agreement is found for single and multielectronic atomic systems, modeled under the single active electron approximation, and for simple diatomic molecular systems. Interference features, ubiquitously present in every strong-field phenomenon involving a multicenter target, are also captured by our model.

Figure 1

HHG spectra $I_{1\mathrm{N}}\left(\omega \right)$ (in logarithmic scale) of hydrogen driven by a strong few-cycle pulse at different wavelengths. ${\lambda }_1=800$ nm (red asterisk line), ${\lambda }_2=1200$ nm (blue square line), and ${\lambda }_3=1400$ nm (green circle line). (a) Quasiclassical SFA model; (b) 3D-TDSE. The arrows in all the panels indicate the position of the classical HHG cutoff (see the text for details).

Figure 2

HHG spectra $I_{1\mathrm{N}}\left(\omega \right)$ (in logarithmic scale) of Ar driven by a strong few-cycle pulse with $\lambda =800$ nm, at different laser peak intensities. (a) Our quasiclassical SFA at $I_1=1.58\times {10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$ (red square line) and $I_2=2.08\times {10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$ (blue cross line); (b) same as in (a) but solving the 3D-TDSE. Note that in this case the minimum in the efficiency around the $27\mathrm{th}$ harmonic is the Cooper minimum in Ar. The arrows in all the panels indicate the position of the classical HHG cutoff (see the text for details).

Figure 3

Total harmonic spectra $I_{2\mathrm{N}}\left(\omega \right)$ (in logarithmic scale) [Eq. (3)] of an ${\mathrm{H}}_2{}^{+}$ molecule driven by a strong few-cycle pulse as a function of the harmonic order computed using our quasiclassical SFA. (a) HHG for an ${\mathrm{H}}_2{}^{+}$ molecule aligned with the laser-pulse polarization axis, i.e., $\theta =0^{\circ }$; (b) the same as (a) but for $\theta ={20}^{\circ }$; (c) the same as (a) but for $\theta ={40}^{\circ }$; (d) the same as (a) but for $\theta ={45}^{\circ }$. The vertical lines indicate the position of the interference minima of our quasiclassical SFA model and the arrows in all the panels the position of the classical HHG cutoff (see the text for details), respectively.

Figure 4

Harmonic spectra $I_{2\text{N}}\left(\omega \right)$ (in logarithmic scale) of an ${\mathrm{H}}_2{}^{+}$ molecule [Eq. (3)] as a function of the harmonic order calculated using our quasiclassical SFA and for an orientation angle $\theta ={20}^{\circ }$. (a) Local, cross, and total contributions to the HHG spectrum; (b) contributions depending on the recombination atom. Green circle line: recombination at ${\mathbf{R}}_1$ and light green line: recombination at ${\mathbf{R}}_2$. The vertical lines indicate the position of the interference minima (see the text for details).

Figure 5

Gabor transformation of the time-dependent dipole moment of an ${\mathrm{H}}_2{}^{+}$ diatomic molecule oriented $\theta ={20}^{\circ }$ with the laser field. (a) Local processes at ${\mathbf{R}}_1$ using the time-dependent dipole moment $\overrightarrow{{\mu }_{11}}\left(t\right)$; (b) the same as (a) but for the cross processes $\overrightarrow{{\mu }_{21}}\left(t\right)$; (c) the same as (a) but for all the recombination processes at ${\mathbf{R}}_1$, i.e., $\overrightarrow{{\mu }_{11}}\left(t\right)+\overrightarrow{{\mu }_{21}}\left(t\right)$; (d) Gabor transformation for the total time-dependent dipole moment ${\vec{\mu }}_{2\text{N}}\left(t\right)$.

Figure 6

Different contributions to the molecular HHG spectrum (in logarithmic scale) for an ${\mathrm{H}}_2$ molecule. (a) Total, local, and cross contributions for a molecule oriented parallel ($\theta =0^{\circ }$) to the laser field polarization; (b) the same as in (a) but for $\theta ={90}^{\circ }$ (perpendicular orientation).

Figure 7

Total ${\mathrm{H}}_2$ molecular HHG spectra (in logarithmic scale) for $\theta =0^{\circ }, {45}^{\circ }, {90}^{\circ }$ and averaged over nine different molecular orientations (see the text for more details).

Figure 8

Gabor transformation of the time-dependent dipole. (a) H atom driven by a laser pulse with a peak intensity of $I_0=5\times {10}^{14}\mathrm{W}$ ${\mathrm{cm}}^{-2}$; (b) same as (a) but for an ${\mathrm{H}}_2$ molecule.

Figure 9

${\mathrm{CO}}_2$ molecular HHG spectra $I_{3\text{N}}\left(\omega \right)$ (in logarithmic scale) as a function of the harmonic order calculated by using our quasiclassical SFA. In panels (a)–(c) the ${\mathrm{CO}}_2$ molecule is oriented perpendicular to the laser polarization, i.e., $\theta ={90}^{\circ }$ (see text for more details).

Figure 10

HHG spectra $I_{3\mathrm{N}}\left(\omega \right)$ (in logarithmic scale) of an ${\mathrm{H}}_2\mathrm{O}$ molecule, as a function of the harmonic order, computed using our quasiclassical SFA model. (a) HHG spectra for $\theta =[0^{\circ },{20}^{\circ },{45}^{\circ },{60}^{\circ },{90}^{\circ }]$ and averaged over eight orientations in the range $\theta =[0^{\circ }–{360}^{\circ }]$; (b) different contributions to the averaged HHG spectra.

Figure 11

Different contributions to the ${\mathrm{H}}_2\mathrm{O}$ molecular HHG spectra. (a) $\theta =0^{\circ }$; (b) $\theta ={90}^{\circ }$.

Figure 12

Three-center molecular system aligned a $\theta$ angle with respect to the laser electric field polarization. The red line represents the molecular axis that forms an angle of $\alpha /2$ between ${\mathbf{R}}_1=[0,\frac{R}{2}sin(\frac{\alpha }{2}+\theta ),\frac{R}{2}cos(\frac{\alpha }{2}+\theta )]$ and ${\mathbf{R}}_3=[0,-\frac{R}{2}sin(\frac{\alpha }{2}-\theta ),\frac{R}{2}cos(\frac{\alpha }{2}-\theta )]$.