1. Introduction

As an outstanding manifestation of electron-electron correlations [1] in strong field, nonsequential double ionization (NSDI) [2, 3] of atoms and molecules in strong laser field has continued to attract experimental and theoretical attention [4–11]. The physical mechanism of NSDI has been profoundly investigated by means of the correlated electron momentum spectrum [6–8, 11–13]. Nowadays, the widely accepted picture for NSDI is the quasiclassical recollision model [14]. Here, the first ionized electron is driven back to its parent ion by the oscillating laser field, causing the ionization of the second electron by inelastic recollisions [15]. The previous investigations for atoms indicated that the electron’s final emission direction depends on its ionization phase. The electron escapes over the barrier into either the backward or forward direction, depending on how many half-cycles elapse before it ionizes and whether it escapes before or after the field maximum [11, 16]. Based on the general recollision scenario, more detailed dynamics processes are proposed to explain these novel characteristics of correlated electron momentum spectra in NSDI [6–8, 17, 18].

The investigations [19–21] have shown that the molecular structure decisively influences double ionization dynamics in the strong field. Besides, the molecular alignment has an significant effect on double ionization yield and electron-electron momentum correlations along the laser polarization axis in NSDI of diatomic molecules [22]. The experiment for N ₂ has revealed that the two electrons involved in NSDI more likely escape into the opposite hemispheres for perpendicular molecules than for parallel molecules [22]. In this paper, we employ the 3D fully classical ensemble model to investigate the effect of molecular alignment on the electron-electron correlation in NSDI by linearly polarized 800 nm laser pulses. The result shows that more electron pairs are emitted into the opposite hemispheres for perpendicular molecules compared with parallel molecules, which agrees well with the experimental result [22]. Back analysis reveals that the most likely recollision scenario involves the production of a short-lived doubly bound state and then the two electrons escape one after the other. The more energetic electron often escapes before the first field maximum after recollision, and most of the second electrons ionize either at the first or the second field maximum for both alignments. However, compared with parallel molecules, for perpendicular molecules the second ionization has longer time delay and has a larger probability to occur before the second field maximum, which results in more anticorrelated emissions. This alignment effect is attributed to the difference between the suppressed potential barriers of parallel molecules and perpendicular molecules. Additionally, the intensity dependence of the alignment effect is also explored.

2. The classical ensemble model

The 3D fully classical ensemble model has been successful in understanding of NSDI before [11, 16, 23] and it has been described in detail in [23]. The electron-nuclear interaction and the electron-electron interaction for N ₂ are represented by a two-center 3D soft-Coulomb potential (in atomic units) $V({\mathbf{\text{r}}}_1,\hspace{0.17em}{\mathbf{\text{r}}}_2)=-1/\sqrt{{({\mathbf{\text{r}}}_1+\mathbf{\text{R}}/2)}^2+a^2}-1/\sqrt{{({\mathbf{\text{r}}}_1-\mathbf{\text{R}}/2)}^2+a^2}-1/\sqrt{{({\mathbf{\text{r}}}_2+\mathbf{\text{R}}/2)}^2+a^2}-1/\sqrt{{({\mathbf{\text{r}}}_2-\mathbf{\text{R}}/2)}^2+a^2}+1/\sqrt{{({\mathbf{\text{r}}}_1-{\mathbf{\text{r}}}_2)}^2+b^2}$. r ₁, r ₂ represent the electronic coordinates, and R is the internuclear vector. a, b are shielding parameters for the attraction of N ⁺ and the electron-electron interaction respectively. The evolution of the two-electron system is determined by the Newton’s classical motion equations: d ² r _i/dt ² = −E(t) − ∇_ri V (r ₁, r ₂), where the subscript i=1,2 is the electron label. E(t) is a linearly polarized laser field. To avoid autoionization, we set the screening parameter a ² to be 1.25. b ² is set to be 0.0025. To obtain the initial values, the ensemble is populated starting from a classically allowed position for the N ₂ ground-state energy of −1.67 a.u. The available kinetic energy is distributed between the two electrons randomly in momentum space. Then the electrons are allowed to evolve a sufficient long time (600 a.u.) in the absence of the laser field to obtain stable position and momentum distribution [24–26]. In present calculations, the internuclear distance R is set to be 2.0 a.u. corresponding to N ₂ molecule. The electric field E(t) is an 800 nm linearly polarized laser pulse with a polarization direction along the z axis and a total duration of 10 optical cycles (two-cycle turn on, six cycles at full strength, and two-cycle turn off).

3. Results and discussions

The top row of Fig. 1 shows the correlated electron momentum spectra along the laser polarization axis for NSDI of nitrogen molecules at λ = 800 nm, I = 1.0 × 10¹⁴ W/cm ². Figures 1(a) and 1(b) correspond to the molecules aligned parallel and perpendicular to the laser polarization axis, respectively. For parallel molecules [see Fig. 1(a)], the double ionization events are mainly clustered in the first and third quadrants, i.e., the electron pairs involved in NSDI mainly escape into the same hemispheres along the laser polarization axis. However, these electron pairs from perpendicular molecules [see Fig. 1(b)] are uniformly distributed in the four quadrants, i.e., the two electrons involved in NSDI are almost equally likely emitted into the same or opposite hemispheres. In other words, more electron pairs are emitted into the opposite hemispheres for perpendicular molecules than for parallel molecules. That is well consistent with the experimental result [22]. In our calculation, for the same number of events, there are 19.8% more anticorrelated electron pairs for perpendicular molecules than for parallel molecules. The second row of Fig. 1 highlights this difference between Figs. 1(a) and 1(b). To obtain Fig. 1(c), we normalize the spectra of Figs. 1(a) and 1(b) to the same number of events, then subtract the perpendicular from the parallel spectrum, and set all negative values to zero [22]. For Fig. 1(d), we proceed in the same manner, but subtract the parallel from the perpendicular spectrum. In this way, the increase of anticorrelated emissions for perpendicular molecules is more clearly visible [see Fig. 1(d)].

 

Fig. 1 The top row shows correlated electron momentum spectra in the direction parallel to the laser polarization for NSDI of nitrogen molecules at λ = 800 nm, I = 1.0 × 10¹⁴ W/cm ². (a) molecules aligned parallel to the laser polarization, (b) molecules aligned perpendicular to the laser polarization. The second row displays the difference between the correlated electron spectra shown in the top row. Before taking their difference, the spectra have been normalized to the same number of events. (c) parallel minus perpendicular, (d) perpendicular minus parallel, and all negative values have been set to zero.

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In order to explore the responsible dynamics process for the increase of anticorrelated emissions for perpendicular molecules, we take advantage of back analysis. Tracing the classical DI trajectories allows us easily to determine the recollision time and ionization time. Here, the recollision time is defined to be the instant of the closest approach after the first departure of one electron from the parent ion. The ionization time is defined as the instant when the energy of the electron becomes positive for the first time, where the energy of each electron contains the kinetic energy, potential energy of the electron-ion interaction and half electron-electron repulsion. Back analysis reveals that the most likely recollision scenario involves the production of a short-lived doubly bound state and then the two electrons escape one after the other. Figure 2 shows that final momenta of the first electron to ionize vs the second one for the same populations as in Fig. 1. There we define the recollision direction (the longitudinal direction of motion of the returning electron just before recollision) as positive [23]. Figures 2(a) and 2(b) correspond to the molecules aligned parallel and perpendicular to the laser field polarization, respectively. It is found that the first electron always escapes into the backward direction (relative to the recollision direction) for both molecular alignments in Fig. 2. This is because the first electron often ionizes within a quarter cycle, i.e., the first electron is often emitted before the first laser maximum after recollision. As a result, the first electron drifts out into the backward direction relative to the recollision direction. However, the second electron has a high probability to drift out into the forward direction (relative to the recollision direction) for both molecular alignments. Comparing the two spectra of Fig. 2, the second electron more likely escapes into the forward direction for perpendicular molecules than for parallel molecules, which corresponds to more anticorrelated emissions for perpendicular molecules.

 

Fig. 2 Final momenta of the first electron to ionize vs the second one for the same populations as in Fig. 1, with the forward direction (relative to the recollision direction) defined as positive. (a) molecules aligned parallel to the laser polarization, (b) molecules aligned perpendicular to the laser polarization.

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To the lowest approximation, the electron’s final direction depends on its ionization phase. Thus the electron escapes over the barrier into either the backward or forward direction, depending on how many half-cycles elapse before it ionizes and whether it escapes before or after the field maximum. Those electrons which escape before the odd field maximum or after the even field maximum after recollision will drift into the backward direction, whereas ones emitted after the odd field maximum or before the even field maximum after recollision will drift into the forward direction. Because the first electron often escapes before the first field maximum after recollision and drifts out into the backward direction for both molecular alignments [see Fig. 2], now we need only consider ionization phase and final direction of the second electron.

Figure 3 displays the pulses of final ionization for DI trajectories, with solid blue and dashed red curves for the parallel and perpendicular molecules respectively. Figures 3(a) and 3(b) correspond to all DI trajectories and anticorrelated trajectories respectively. To generate Fig. 3 we grouped all recollision times into half-cycle bins extending from 2.1 to 2.6c, 2.6 to 3.1c, etc. We then determined the time interval Δt from the preceding laser zero to the final ionization for each trajectory [11] (t_i−2.0 for collisions occurring from 2.1c to 2.6c, t_i−2.5 for collisions occurring from 2.6c to 3.1c, and so forth). Thus the first field maximum after recollision is at 0.75c in Fig. 3. Examining Fig. 3(a), one can find that most of final ionizations occur either at the first or the second field maximum for both molecular alignments, and more final ionizations are delayed for perpendicular molecules (red curve) than for parallel molecules (blue curve). For anticorrelated emissions in Fig. 3(b), the heights of the first two peaks are much higher than the subsequent ones for both molecular alignments, i.e., the anticorrelated emissions mainly origin from those events of which final ionizations occur after the first field maximum and before the second field maximum after recollision for both molecular alignments. Moreover, in Fig. 3(b) it is obvious that the increase of anticorrelated emissions for perpendicular molecules mainly results from more trajectories finally ionized just before the second field maximum after recollision for perpendicular molecules than parallel molecules. Final ionizations of more trajectories are delayed to the second field maximum for perpendicular molecules. That results in more anticorrelated emissions in NSDI for perpendicular molecules than for parallel molecules.

 

Fig. 3. Pulses of final ionization for DI trajectories vs time since beginning of the half-cycle bin in which recollision occurred. (t_i − 2.0 for collisions occurring from 2.1c to 2.6c, t_i − 2.5 for collisions occurring from 2.6c to 3.1c, and so forth). The solid blue curves: parallel molecules, dashed red curves: perpendicular molecules. (a) all DI trajectories, (b) anticorrelated trajectories. The green lines (0.75c and 1.25c) are drawn to guide the eye.

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According to analysis above, more trajectories are delayed to finally ionize just before the second field maximum for perpendicular molecules. That is responsible for the increase of anticorrelated emissions for perpendicular molecules relative to parallel molecules. To explore why final ionization is more likely delayed for perpendicular molecules, we examine combined Coulomb and laser field potential energy curves along the laser polarization for perpendicular molecules (red curve) and parallel molecules (blue curve) with the laser intensity I = 1.0 × 10¹⁴ W/cm ² in Fig. 4. Compared with parallel molecules, the suppressed potential barrier for perpendicular molecules is higher, and the second electron is more difficult to ionize. Consequently, for perpendicular molecules the second ionization has longer time delay and has larger probability to occur before the second field maximum, which results in more anticorrelated emissions. A previous study [27] did not reveal the alignment effect on electron-electron momentum correlation using a semiclassical model in which unsoftened Coulomb potential is used. We suggest that the soft-Coulomb potential can more accurately represents the attraction of N ⁺ to electron than unsoftened Coulomb potential. This is because of the influence of inner bound electrons.

 

Fig. 4 Combined Coulomb and laser field potential energy curves along the laser polarization with the laser intensity I = 1.0 × 10¹⁴ W/cm ². Red curve: perpendicular molecules, blue curve: parallel molecules.

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We further study the dependence of the alignment effect on the laser intensity. In our calculation the increases of anticorrelated emissions for perpendicular molecules relative to parallel molecules are 14.5%, 19.8%, 15.3%, 13.7%, 7.34% and 4.82% for laser intensities I = 0.08 PW/cm ², 0.1 PW/cm ², 0.12 PW/cm ², 0.15 PW/cm ², 0.2 PW/cm ² and 0.25 PW/cm ², respectively. Figure 5 shows the intensity dependence of the increase of anticorrelated emissions from the perpendicular molecules as compared to the parallel molecules. We can easily find that the effect of molecular alignment on electron-electron momentum correlation enhances at first and then decrease slowly with increasing laser intensity. There is a critical laser intensity (0.1 PW/cm ²) at which the increase of anticorrelated emissions for perpendicular molecules is largest among these intensities considered. At the lower intensity (0.08 PW/cm ²), for both alignments the suppressed potential barriers are higher relative to the intensity of 0.1 PW/cm ². Thus there are longer time delays of final ionizations for both alignments, and the effect of the height difference between the two suppressed potential barriers on the time delay of final ionization becomes relatively small. Consequently, the increase of anticorrelated emissions for perpendicular molecules is smaller relative to that at the intensity of 0.1 PW/cm ². For the higher intensities (0.12 PW/cm ², 0.15 PW/cm ²), more trajectories are finally ionized at the first laser maximum. Correspondingly more anticorrelated emissions occur just after the first laser maximum after recollision for both alignments. Because of the lower suppressed potential barriers, for parallel alignments this anticorrelated emissions from the first field maximum more easily occur. This to some extent offsets the increase of anticorrelated emissions from the time delay for perpendicular alignments. At the intensities above the recollision threshold (0.2 PW/cm ², 0.25 PW/cm ²), the anticorrelated behaviors are mainly attributed to two factors. One is the time delay between the recollision and final ionization, as discussed above. The other is the difference of the velocities of the two electrons just after recollision, where the electron with higher velocity after recollision can exit the parent ion into forward direction finally [23]. Because of the lower suppressed potential barrier, for parallel alignment the energetic electron more easily escapes over the barrier into forward direction which results in anticorrelated emission. At the intensities of I = 0.2 PW/cm ² and 0.25 PW/cm ², for perpendicular molecules the decrease of anticorrelated emissions from the latter is comparative with the increase from the former. Thus the increase of anticorrelated emissions for the perpendicular alignment is very small at the intensities of I = 0.2 PW/cm ² and 0.25 PW/cm ².

 

Fig. 5. The intensity dependence of the increase of anticorrelated emissions from the perpendicular molecules as compared to the parallel molecules. The data is obtained when the DI events from the two cases are normalized to the same number. N ₁ and N ₂ represent anticorrelated emission numbers from the perpendicular molecules and the parallel molecules respectively. The squares are the calculated data. The solid line is drawn to guide the eye.

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4. Conclusion

In conclusion, we have investigated the effect of molecular alignment on the electron-electron correlation. The result shows that for perpendicular molecules the two electrons involved in NSDI more likely exit the molecules into the opposite hemispheres as compared to parallel molecules, which agrees well with previous experiment. Back analysis reveals that the most likely recollision scenario involves the production of a short-lived doubly bound state and then the two electrons escape one after the other. The more energetic electron often escapes before the first field maximum after recollision, whereas most of the second electrons ionize either at the first or second field maximum for both cases. Compared with parallel molecules, the suppressed potential barrier for perpendicular molecules is higher, and the second electron is more difficult to ionize. Thus for perpendicular molecules the second ionization has longer time delay and has a larger probability to occur before the second field maximum, which results in more anticorrelated emissions. Additionally, we find that the effect of the molecular alignment on electron-electron correlation enhances at first and then decrease slowly with increasing laser intensity.