Analysis of recombination in high-order harmonic generation in molecules

We show that the dependence of high-order harmonic generation (HHG) on the molecular orientation can be understood within a theoretical treatment that does not involve the strong field of the laser. The results for ${\mathrm{H}}_2$ show excellent agreement with time-dependent strong-field calculations for model molecules, and this motivates a prediction for the orientation dependence of HHG from the ${\mathrm{N}}_2$ $3{\sigma }_g$ valence orbital. For both molecules, we find that the polarization of recombination photons is influenced by the molecular orientation. The variations are particularly pronounced for the ${\mathrm{N}}_2$ valence orbital, which can be explained by the presence of atomic $p$ orbitals.

Figure 1

The recombination geometry is shown schematically. The molecule with a bond length $r_0$ is oriented relative to the electron momentum ${\mathbf{k}}_e$ at an angle $\theta$. The photon is emitted perpendicular to the electron momentum.

Figure 2

Dependence of the recombination photon intensity on the angle between molecular orientation and the electron momentum for ${\mathrm{H}}_2$. The molecule lies in the plane spanned by electron momentum and photon direction (geometry I). The solid curves are for an electron wavelength ${\lambda }_e$ of 1.4 a.u., the dotted curves for ${\lambda }_e=1.6\mathrm{a.u.}$, and the dashed curves for ${\lambda }_e=1.8\mathrm{a.u.}$ Molecular bond length $r_0=$ (a) 1.4 a.u.; (b) 2.0 a.u. The positions of minima and maxima are marked by arrows. Obviously, the positions of the extrema depend on the bond length and on the wavelength of the electron. This behavior can be explained by a two-center interference model (see text). Clearly, the minima are very pronounced.

Figure 3

Recombination in geometry I for ${\mathrm{H}}_2$. For the extrema in the photon-intensity orientation dependence, the projection of the bond length on the electron momentum direction, $r_0\cos \left({\theta }_{\mathrm{ex}}\right)$, is plotted against the electron wavelength ${\lambda }_e$. The dotted (dashed) curve marks minima (maxima) due to two-center emittance (see text). The solid curve represents minima for $r_0=1.4\mathrm{a.u.}$, whereas the long-dashed (dash-dotted) curves represent minima (maxima) for $r_0=2\mathrm{a.u.}$ For comparison the extrema in the orientation dependence in HHG for time-dependent strong-field calculations for model molecules [12] are plotted as well: interference minima for (▿) ${\mathrm{H}}_2^{+}$ at $r_0=2\mathrm{a.u.}$, laser intensity $I={10}^{15}\mathrm{W}∕{\mathrm{cm}}^2$; (●) ${\mathrm{H}}_2^{+}$ at $r_0=2\mathrm{a.u.}$, $I=5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$; (◻) ${\mathrm{H}}_2$ at $r_0=1.4\mathrm{a.u.}$, $I=5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$; (+) interference maxima for ${\mathrm{H}}_2^{+}$ at $r_0=2\mathrm{a.u.}$, $I={10}^{15}\mathrm{W}∕{\mathrm{cm}}^2$. Note that the present definition of ${\lambda }_e$ differs from that of 12.

Figure 4

The ${\mathrm{H}}_2$ molecule lies in the plane perpendicular to the photon propagation direction which is parallel to the $z$ axis in the Stokes parameters frame (geometry II). The electron is here chosen to move along the $x$ axis. Shown as a solid curve is the recombination photon intensity for ${\lambda }_e=1.6\mathrm{a.u.}$, $r_0=1.4\mathrm{a.u.}$ against the angle between the molecular axis and the propagation direction of the electron. Dotted, dash-dotted, and dashed curves are the Stokes parameters $p_1$, $p_2$, and $p_3$, respectively. These show strong variations in the vicinity of the minima.

Figure 5

Projection $r_0\cos \left({\theta }_{\mathrm{ex}}\right)$ versus electron wavelength ${\lambda }_e$ for molecular orientations ${\theta }_{\mathrm{ex}}$ under which minima $(\times )$ and maxima (+) occur in the ${\mathrm{N}}_2$ $3{\sigma }_g$ recombination photon intensity. The bond length $r_0$ is 1.768 a.u. in (a) and (d), 2.068 a.u. in (b) and (e) (ground-state bond length), and 2.368 a.u. in (c) and (f), respectively. The upper plots [(a),(b),(c)] show the positions of extrema when the molecule lies in the plane spanned by the electron and photon direction (geometry I), whereas the molecule rotates in the plane perpendicular to the photon direction in the lower plots (d),(e),(f) (geometry II). The dotted (dashed) curve is the one for minima (maxima) according to the two-center interference model for $p$ orbital contributions. Note that minima and maxima are interchanged as compared to the $s$ orbital contributions. The dash-dotted line marks minima according to $r_0\cos \left({\theta }_{\mathrm{ex}}\right)=0.75{\lambda }_e$.

Figure 6

The ${\mathrm{N}}_2$ molecule lies in the plane perpendicular to the photon propagation direction which is parallel to the $z$ axis in the Stokes parameters frame (geometry II). The electron is here chosen to move along the $x$ axis. Shown as a solid curve is the recombination photon intensity for ${\lambda }_e=1.6\mathrm{a.u.}$, $r_0=2.068\mathrm{a.u.}$ against the angle between the molecular axis and the propagation direction of the electron. Dotted, dash-dotted, and dashed curves are the Stokes parameters $p_1$, $p_2$, and $p_3$, respectively.

Figure 7

Same as Fig. 5 when only the component polarized perpendicular to the electron momentum is measured in geometry II. (The molecular axis lies in a plane perpendicular to the photon propagation direction.) The dotted (dashed) curve is the one for minima (maxima) according to the two-center interference model for $p$ orbital contributions.