Theoretical analysis of dynamic chemical imaging with lasers using high-order harmonic generation

We report theoretical investigations of the tomographic procedure suggested by Itatani et al. [Nature (London) 432, 867 (2004)] for reconstructing highest occupied molecular orbitals (HOMOs) using high-order harmonic generation (HHG). Due to the limited range of harmonics from the plateau region, we found that even under the most favorable assumptions, it is still very difficult to obtain accurate HOMO wave functions using the tomographic procedure, but the symmetry of the HOMOs and the internuclear separation between the atoms can be accurately extracted, especially when lasers of longer wavelengths are used to generate the HHG. Since the tomographic procedure relies on approximating the continuum wave functions in the recombination process by plane waves, the method can no longer be applied upon the improvement of the theory. For future chemical imaging with lasers, we suggest that one may want to focus on how to extract the positions of atoms in molecules instead, by developing an iterative method such that the theoretically calculated macroscopic HHG spectra can best fit the experimental HHG data.

Figure 1

(Color online) HHG spectra with parallel polarization (with respect to the laser polarization direction) from ${\mathrm{N}}_2$ aligned at ten different angles between 0 and 90°. A laser wavelength of $800\mathrm{nm}$ and intensity of $2\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ are used.

Figure 2

(Color online) Alignment dependence of selective HHG orders for parallel (top) and perpendicular (bottom) polarizations from ${\mathrm{N}}_2$. The laser parameters are the same as for Fig. 1.

Figure 3

(Color online) Same as in Fig. 2, but for O.

Figure 4

(Color online) HHG spectra from ${\mathrm{N}}_2$ aligned at 45° for $800\text{−}\mathrm{nm}$ and $1200\text{−}\mathrm{nm}$ lasers. Laser intensities of $2\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ are used in both cases.

Figure 5

(Color online) $x$ component of the transition dipoles of $N_2$ obtained from HHG data (solid lines) in comparison with the theoretical ones (dashed lines) in the case of the $800\text{−}\mathrm{nm}$ laser (a) and $1200\text{−}\mathrm{nm}$ laser (b).

Figure 6

(Color online) Same as in Fig. 5 but for the $y$ component.

Figure 7

(Color online) The transition dipole matrix element of ${\mathrm{O}}_2$ extracted from HHG data (solid lines) in comparison with the theoretical ones (dashed lines) in the case of the $1200\text{−}\mathrm{nm}$ laser. The upper and lower panels are for $y$ and $x$ components, respectively.

Figure 8

(Color online) The contour plot of the wave function of ${\mathrm{N}}_2$ obtained from HHG data in the useful range H17–H47 of the $800\text{−}\mathrm{nm}$ laser (a), from HHG data in the useful range H27–H113 of the $1200\text{−}\mathrm{nm}$ laser (b), and from the theoretical transition dipole in the range $k^2=[0–100]$ (c).

Figure 9

(Color online) ${\mathrm{N}}_2$ wave function $\Psi (x,y=0)$. The solid (dashed) curves correspond to the extracted (theoretical) wave function, respectively. The theoretical wave functions are obtained using the tomographic procedure with the same range as for the extracted wave functions. The range is $k^2=[1.0–7.5]$ for the $1200\text{−}\mathrm{nm}$ laser (upper solid and dashed curves) and $k^2=[0.9–4.2]$ for the $800\text{−}\mathrm{nm}$ laser (lower solid and dashed-dotted curves). For comparison we also plot the converged wave function obtained with the range $k^2=[0–100]$ (dotted curve).

Figure 10

(Color online) Contour plot of the wave functions of ${\mathrm{N}}_2$ with the internuclear distance $R=2.1$, 3.0, and 4.0, extracted from HHG data (left panels), as compared with the exact ones (right panels).

Figure 11

(Color online) Same as for Fig. 10 but for ${\mathrm{O}}_2$ with $R=2.3$ (top panels), $R=3.0$ (middle panels), and $R=4.0$ (bottom panels).

Figure 12

Internuclear distance of ${\mathrm{O}}_2$ extracted from HHG data using the $1200\text{−}\mathrm{nm}$ laser.

Figure 13

(Color online) Reaction path of acetylene to vinylidene calculated by DFT with hybrid PBE1PBE functional and AUG-cc-pVTZ basis set. The values in parenthesis are from experiment [42].

Figure 14

(Color online) Predicted angular dependence of the 21st, 25th, 29th, and 31st harmonic yields in the plateau region for ${\mathrm{C}}_2{\mathrm{H}}_2$ isomers: (a) acetylene (from top to bottom), (b) TS2, (c) vinylidene, (d) TS1, and (e) LM2. For (b)–(e), from top to bottom, the curves represent the 21st, 25th, 31st, and 29th harmonic.

Figure 15

(Color online) Reaction path of hydrogen cyanide HCN to hydrogen isocyanide CNH calculated by DFT with hybrid B3LYP functional and $6\text{−}311+\mathrm{G}(2df,2pd)$ Gaussian-type basis set. The value in parentheses is from experiment [48]. The contour plot of the potential energy surface at the equilibrium distance $(R_{\mathrm{N}\mathrm{C}}=1.15\mathrm{\AA })$ of the most stable isomer HCN is shown.

Figure 16

(Color online) Predicted angular dependence of the 21st, 25th, 29th, and 31st harmonic yields in the plateau region for HCN isomers: (a) hydrogen cyanide HCN, (b) transition state CHN, and (c) hydrogen isocyanide CNH. For (a) and (b) the curves should be read from the top to the bottom. For (c) the curves (from top to bottom) represent the 21st, 25th, 31st, and 29th harmonic.