Interference effects in high-order harmonic generation by homonuclear diatomic molecules

A theory of high-order harmonic generation by diatomic molecules is introduced. Various versions (with or without the dressing of the initial and/or final molecular state) of the molecular strong-field approximation are investigated. Using examples of homonuclear diatomic molecules such as ${\mathrm{H}}_2$, ${\mathrm{N}}_2$, and ${\mathrm{O}}_2$, it is shown that clear two-center interference minima in the harmonic spectra as a function of the molecular orientation appear only if the final molecular state is undressed. For ${\mathrm{H}}_2$ the positions of these minima are in agreement with the ab initio numerical results. Physically, the returned electron wave packet recombines into a molecular orbital, which is a linear combination of the atomic orbitals having different parities. The interference minima in the harmonic spectrum are caused by the destructive interference of the corresponding partial recombination amplitudes. In accordance with this, we have derived an interference minima condition which is valid for arbitrary homonuclear diatomic molecules.

Figure 1

(Color online) Comparison of the high-order harmonic yields of the ${\mathrm{H}}_2$ molecule obtained using the four possible combinations of the dressed (undressed) initial (final) molecular bound states. The corresponding results are denoted by the letters dd, du, ud, and uu, where the first letter denotes whether the initial state is dressed (d) or undressed (u), while the second letter characterizes the final state. The laser field intensity is $5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ and the wavelength $780\mathrm{nm}$. The molecular axis is parallel to the polarization axis of the linearly polarized laser field $({\theta }_{\mathrm{L}}=0°)$. The yield for the uu case is divided by 10. All results are obtained using the saddle-point method with the first four saddle-point solutions.

Figure 2

(Color online) Projected internuclear separation vs effective electron wavelength for HHG by the ${\mathrm{H}}_2$ molecule. Orientation angle ${\theta }_{\mathrm{L}}$ changes from 0 to 90°, while the other molecular and laser parameters are as in Fig. 1. The results presented by the curve with red circles are obtained by solving Eq. (24) over $n_{\min }$ for fixed $R_0$ and ${\theta }_{\mathrm{L}}$ and presenting $R_0\cos {\theta }_{\mathrm{L}}$ as a function of $\lambda =2\pi ∕\sqrt{2n_{\min }\omega }$. The dashed line represent the destructive interference condition $R_0\cos {\theta }_{\mathrm{L}}=(2m+1)\lambda ∕2$ for $m=0$.

Figure 3

(Color online) High-order harmonic spectra of the ${\mathrm{N}}_2$ molecule obtained using a linearly polarized laser field having the intensity $4\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ and photon energy $\omega =1.55\mathrm{eV}$. (a)–(d) illustrate HHG spectra for the four possible combinations where the initial (final) molecular bound states are dressed (undressed). The angle between the laser field polarization axis and the molecular axis is plotted along the horizontal axis, while the harmonic order is along the vertical axis. The curves $n_{\min }^{\left(\mathrm{u}\right)}=n_{\min }^{\left(\mathrm{u}\right)}\left({\theta }_{\mathrm{L}}\right)$, which express the interference minima condition as the solution of the nonlinear equation defined by Eq. (D10) for $j=0$, are presented by white lines in (b) and (d).

Figure 4

(Color online) As in Fig. 3 but taking into account only the $s$ orbitals (left panel) and only the $p$ orbitals (right panel). Only the combination with the dressed initial state and the undressed final state is presented. The curve given by Eq. (D2) [Eq. (D3)] is depicted in the left (right) panel by a white line.

Figure 5

(Color online) As in Fig. 3 but for the ${\mathrm{O}}_2$ molecule and the laser intensity $6\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$. The interference minima curves $n_{\min }^{\left(\mathrm{u}\right)}=n_{\min }^{\left(\mathrm{u}\right)}\left({\theta }_{\mathrm{L}}\right)$ in (b) and (d) are the solutions of Eq. (D10) for $j=-1$.

Figure 6

(Color online) Comparison of the high-order harmonic spectra of the ${\mathrm{N}}_2$ molecule obtained using the saddle-point method with the first four saddle-point solutions (circles and diamonds denoted by “SP4”) and the “exact” spectra obtained using numerical integration (squares and triangles). The laser field is linearly polarized having the intensity $4\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ and photon energy $\omega =1.55\mathrm{eV}$. For the black curve with circles and red curve with squares the angle between the internuclear axis and the laser polarization axis is ${\theta }_{\mathrm{L}}=0°$, while for the green curve with diamonds and blue curve with triangles it is ${\theta }_{\mathrm{L}}=45°$, and these curves are shifted down by four orders of magnitude.