1. Introduction

Nonsequential double ionization (NSDI) is one of the most fundamental processes in intense laser-atom interactions [1–4]. It is widely accepted that NSDI occurs by an inelastic recollision of the first ionized electron with the parent ion [5,6]. Depending on the recollision energy, after recollision the system may immediately release the second electron or form a transition excited state [7–11] (ionic excited state or doubly excited state) with subsequent ionization by the laser electric field. Because of the recollision, the two ionized electrons exhibit a highly correlated behavior which has drawn much attention [12–25].

In the past decades, numerous studies for near 800 nm laser pulses have provided deep insight into the electron correlation behavior and underlying ultrafast dynamics in NSDI. For example, at the laser intensity above the recollision threshold, the fingerlike structure in the correlated electron momentum distribution [26] reveals the critical role of the nuclear attraction at recollision on final electron correlation [27, 28]. The V-like structure [29] in the correlated electron momentum distribution provides a hint of the asymmetric energy sharing at recollision [30]. At the laser intensity below the recollision threshold, the correlated electron spectra exhibit the dominant anticorrelation behavior [31]. Considerable contribution from multiple-recollision events on NSDI are confirmed at low intensities [32,33]. For few-cycle laser pulses, the correlated electron momentum spectra exhibit a cross-shaped structure [34] and a two-line structure [8] parallel to the diagonal at relatively high and low laser intensities respectively.

Because the long-wavelength pulse (>1000 nm) can easily make the system deep into tunneling regime and produce high-energy recolliding electrons, in recent years the long-wavelength pulse gained more and more attention. It played a decisive role in observation and discovery of the low-energy structure (LES) [35, 36] and the photoelectron holography [37] in above-threshold ionization. It is also demonstrated that the long-wavelength pulse can not only produce much more energetic harmonic photons but also reduce harmonic chirps [38]. For NSDI, several preliminary studies for the long-wavelength pulses (1300 nm in [39], 1313 nm and 2016 nm in [40,41] and 1600 nm in [42]) have been reported. Their results show that the ion momentum distributions exhibit more pronounced double-hump structure with respect to 800 nm which is attributed to more significant contribution from the recollision ionization to NSDI because of the higher recollision energy for the longer wavelength. Very recently, the study of NSDI is extended to 3100 nm by Wolter et al. [43], where the correlated electron momentum spectrum shows a prominent near-axes V-shaped structure in the first and third quadrants.

In this paper, we investigate the recollision dynamics and electron correlation behavior in NSDI of atoms for several long wavelengths (1200 nm, 1500 nm, 2300 nm and 3000 nm, the result of 750 nm is also shown for comparison) with the 3D classical ensemble model. Our results indicate that with increasing laser wavelength the contribution from the single-return events on NSDI decreases sharply, which is because that the spread of the electron wave packet during its excursion is more serious for longer wavelengths. The contribution from the multiple-return events on NSDI is dominant for the four wavelengths considered here with the help of Coulomb focusing effect. Moreover, with increasing laser wavelength the NSDI events move from the diagonal to the two axes in the correlated electron momentum distributions, and finally form an experimentally observed near-axes V-shaped structure in the first and third quadrants. Back analysis indicates that the asymmetric energy sharing between the two electrons at recollision is responsible for the formation of the near-axes V-shaped structure in the correlated electron momentum distribution of 3000 nm.

2. The fully classical ensemble model

Due to the huge computational demand of numerically solving the time-dependent Schröinger equation for multielectron systems in strong laser fields, in the past decade numerous studies have resorted to the semiclassical model [44] and the fully classical model [45] which have been widely recognized as reliable and useful approaches in exploring electron dynamics in NSDI. In this paper, we employ the 3D classical ensemble model [46] proposed by Eberly et al. to study the recollision dynamics of NDSI. In this model the evolution of the three-particle system is determined by the Newton’s equations of motion (atomic units are used throughout until stated otherwise):

To obtain the initial conditions for Eq. (1), the ensemble is populated starting from a classically allowed position for the energy of −2.9035 a.u., corresponding to the sum of the first and second ionization potentials of He. The available kinetic energy is distributed between the two electrons randomly, and the directions of the momentum vectors of both electrons are also randomly assigned. Then the two-electron system is allowed to evolve a sufficient long time (400 a.u.) in the absence of the laser field to obtain stable position and momentum distributions. Once the initial ensemble is obtained, the laser field is turned on and all trajectories are evolved in the combined Coulomb and laser fields. We check the energies of the two electrons at the end of the laser pulse, and a DI event is determined if both electrons achieve positive energies. In this work, we have carried out numerical calculations for five different wavelengths. As we know, the longer the wavelength is, the more significant the transverse spread of the free electron is. Thus for relatively longer wavelength the recollision probability and the NSDI probability are both smaller when the free electron returns to the parent ion in the longitudinal direction. In order to obtain the comparable NSDI events for each wavelength, we increase the ensemble size with increasing laser wavelength. The ensemble sizes in our calculations are 3.6×10⁷ for 750 nm, 1.1×10⁸ for 1200 nm, 1.7×10⁸ for 1500 nm, 3.6×10⁸ for 2300 nm and 1.3×10⁹ for 3000 nm, respectively.

3. Results and discussions

Figure 1 shows correlated longitudinal electron momentum distributions for NSDI of He by a linearly polarized laser pulse at the intensity of 4.0×10¹⁴ W/cm² for the wavelengths of 750 nm (a), 1200 nm (b), 1500 nm (c), 2300 nm (d) and 3000 nm (e). All of the five distributions exhibit the dominant correlation behavior, i.e., the two electrons are mainly emitted into the same hemisphere. But the distributions of these correlated electron pairs have notable difference. For 750 nm the correlated electron pairs mainly distribute near the diagonal and exhibit a faint fingerlike structure which has been more clearly shown at a slightly longer wavelength and higher intensity in experiment (800 nm, 4.5×10¹⁴ W/cm²) [26]. For 1200 nm the correlated electron pairs still mainly distribute near the diagonal. For 1500 nm the distribution becomes slightly broader. When the wavelength increases further to 2300 nm, the correlated electron pairs are almost uniformly distributed in the first and third quadrants. For 3000 nm, the correlated electron pairs deviate from the diagonal and are mainly clustered near the two axes exhibiting a prominent V-shaped structure, which is well in agreement with the experimental data by Wolter et al. [43]. These results indicate that with increasing laser wavelength the NSDI events move from the diagonal to the two axes in the correlated electron momentum distributions, and finally form an experimentally observed prominent near-axes V-shaped structure in the first and third quadrants at 3000 nm. In addition, Fig. 1(f) shows NSDI probability as a function of the wavelength. One can see that the NSDI probability decreases sharply with increasing laser wavelength. For 750 nm the NSDI probability is about 4×10⁻⁴ and it is only 1.4×10⁻⁵ for 3000 nm.

 

Fig. 1 (a)–(e) Correlated longitudinal electron momentum distributions for NSDI of He by a linearly polarized laser pulse at the intensity of 4.0×10¹⁴ W/cm² for the wavelengths of 750 nm, 1200 nm, 1500 nm, 2300 nm and 3000 nm. (f) NSDI probability as a function of the laser wavelength.

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By tracing the NSDI trajectories from the long wavelengths, we find that for many NSDI events the free electron misses the parent ion in the transverse direction when it returns to the parent ion at the first time in the longitudinal direction. For these trajectories no energy exchange (recollision) occurs at the first return of the free electron in the longitudinal direction. Then, the free electron continues oscillating by the laser electric field. After several returns in the longitudinal direction, the free electron can recollide with the parent ion and release the other bound electron. These NSDI trajectories can be called multiple-return trajectories. For NSDI by the linearly polarized near-800 nm pulse at the moderate intensity, single-return events are dominant and multiple-return events are quite rare.

In order to obtain a deep understanding for multiple-return events and its dependence on the laser wavelength, we trace the classical NSDI trajectories and perform statistical analysis. We find out the single ionization time t_SI, the return time t_ret, the recollision time t_rec, and the double ionization time t_DI. Here, the single ionization time is defined as the instant when one electron achieves positive energy or is outside the nuclear well [41]. The double ionization time is defined as the instant when both electrons achieve positive energies. In our calculations, almost all of the NSDI trajectories experienced a significant energy exchange (i.e., once recollision). The recollision time is defined as the instant of the closest approach after the first departure of one electron from the parent ion. The return times are defined as those instants when the free electron passes through the zero point in longitudinal direction (z=0) before the recollision occurs.

Figure 2 displays the traveling time distribution for those NSDI trajectories shown in Fig. 1, where the traveling time is defined as the time difference between the recollision and the single ionization. For comparing NSDI yields from the different wavelengths, all ensembles are normalized to 1.3×10⁹. For 750 nm the traveling time distribution exhibits a main peak located near 0.5T (T is the laser period), followed by several small peaks at 1.1T, 1.6T and so on. They correspond to the recollisions that occur at the first return, the second return, the third return and so on, respectively. In this case, for most of NSDI events the recollisions occur at the first return of the free electron, i.e., single-return trajectory is dominant in NSDI of 750 nm. However, for the long wavelength we focus on in this work, the situation is quite different. From Fig. 2 we can see that for the long wavelength the contribution of those peaks with the traveling time longer than 1.0T becomes much more significant compared to the first peak. Moreover, with increasing laser wavelength, the first peak becomes lower and lower, i.e., the contribution of the single-return events becomes smaller and smaller. Correspondingly, the contribution of the multiple-return events increases gradually with the laser wavelength increasing.

 

Fig. 2 Traveling time distribution for those NSDI trajectories shown in Fig. 1. For comparing NSDI yields from the different wavelengths, all ensembles are normalized to 1.3×10⁹.

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Figure 3(a) shows NSDI yields as a function of number of returns of the free electron before the recollision for the five different wavelengths. For comparing NSDI yields from the different wavelengths, all ensembles are normalized to 1.3×10⁹. One can see that except even-order return at 750 nm, with increasing laser wavelength the NSDI yield decreases regardless of the number of returns. But the NSDI yield from single-return trajectory shows a much faster decrease than that from multiple-return trajectory. It means that the contribution from multiple-return trajectory is more significant for longer wavelength. In order to quantitatively understanding the dependence of the contribution of multiple-return trajectory on the laser wavelength, we further present the proportion of each return in the total NSDI yield for the different wavelengths. One can see that for 750 nm the recollisions of 76% NSDI events occur at first return. It means that the single-return events are dominant in NSDI for 750 nm. With increasing laser wavelength, the proportion of the single-return events decrease rapidly. For 1200 nm, the proportion of the single-return events is 40%. It indicates that the multiple-return trajectory (60%) exceeds the single-return trajectory and becomes the dominant NSDI channel. When the wavelength reaches to 3000 nm, the proportion of the single-return events is only 5% and the other 95% NSDI are from the multiple-return trajectory. That is to say, for all of the four long wavelengthes (1200 nm, 1500 nm, 2300 nm and 3000 nm) considered in this work, the multiple-return events are dominant. In addition, it can be found that with increasing laser wavelength, except the first return, all of the contributions from other returns to total NSDI yield increase.

 

Fig. 3 (a) NSDI yields as a function of number of returns of the free electron before the recollision. For comparing NSDI yields from the different wavelengths, all ensembles are normalized to 1.3×10⁹. (b) Proportion of each return in the total NSDI yield.

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It is noteworthy that for 750 nm, it is obvious that the NSDI yield at the even-order return is smaller than that at odd-order return. It is because that the recollision electron possesses lower energy at the even-order return than at the odd-order return. However, for the long wavelength the NSDI proportion decreases monotonously with increasing number of returns from the two returns. It indicates that the recollision energy is more important for NSDI of the short wavelength.

In order to make clear why the contribution from multiple-return events is larger than single-return events for long-wavelength pulses, we further examine the history of those NSDI trajectories. It is found that most of ionized electrons have a non-zero transverse momentum. After the excursion of the ionized electron, when it returns to the parent ion at the first time there is a distance in the transverse direction between the free electron and the parent ion. For a certain initial transverse momentum, this distance depends on the time of the excursion of the free electron, i.e., the laser wavelength. Thus for longer wavelength, the free electron more likely misses the parent ion at the first return and can not induce the recollision. This is why the contribution of the single-return events decreases with increasing laser wavelength. Since the large transverse distance between the free electron and the parent ion at the first return make it miss the parent ion, many free electrons collide with the parent ion at subsequent returns and result in NSDI. It can be attributed to the focusing effect [36, 47, 48] of the Coulomb potential to the transverse motion of the free electron when it returns to the vicinity of the parent ion. Figures 4 (1500 nm) and 5 (3000 nm) show two sample NSDI trajectories to illustrate the Coulomb focusing effect. Panels (a)–(f) show, respectively, the energies of the two electrons, their distances from the ion, their longitudinal momenta, their longitudinal position coordinates z, their transverse momenta $\sqrt{p_x^2+p_y^2}$ and their transverse positions $\sqrt{x^2+y^2}$. Figure 4 shows a three-return trajectory from 1500 nm. The single ionization occurs at t=4.04T. After ionization, the free electron has a initial transverse momentum of 0.056 a.u. [see the red curve in Fig. 4(e)] and thus it gradually moves away from the parent ion in the transverse direction [see Fig. 4(f)]. In the longitudinal direction the free electron is driven back by the laser electric field at the first time near t=4.84T. But at this time the transverse position at 9.4 a.u. and thus the free electron misses the parent ion and no recollision occurs. However, during the first return its transverse momentum is first decreased and then reversed by the Coulomb potential of the parent ion. The free electron begins to move to the parent ion in the transverse direction after the first return [see Fig. 4(f)]. When the free electron returns to the parent ion at the second time near t=5.20T, it is accelerated by the Coulomb potential in the transverse direction and moves faster to the parent ion. When the free electron returns to the parent ion in the longitudinal direction at the third time (t=5.70T), it is very close to the parent ion and the distance between them is only 0.1 a.u. At this time the significant energy exchange happens between the two electrons [see Fig. 4(a)], i.e., the recollision occurs at the third return. The success of the recollision is due to the Coulomb focusing effect of the parent ion to the transverse motion of the free electron when it returns to the parent ion in the longitudinal direction at the first two times. Similarly, Fig. 5 shows a seven-return trajectory from 3000 nm. At the first six returns in the longitudinal direction, the free electron misses the parent ion because of the large transverse distance [see Fig. 5(f)]. But at each return the transverse momentum of the free electron is changed significantly. Firstly, the transverse velocity moving away is gradually decreased to zero and then the transverse velocity approaching the parent ion is increased, which is clearly shown in Fig. 5(e). Correspondingly, in the transverse direction the free electron firstly moves away from the parent ion (the largest distance is 22 a.u.), and then reverses and moves back to the parent ion, finally encountering the parent ion at the seventh return. Figures 5(e) and 5(f) is a very intuitive exhibition of the Coulomb focusing effect of the parent ion.

 

Fig. 4 A sample NSDI trajectory for 1500 nm. Panels (a)–(f) show, respectively, the energies of the two electrons, their distances from the ion, their longitudinal momenta p_z, their longitudinal position coordinates z, their transverse momenta $\sqrt{p_x^2+p_y^2}$ and their transverse positions $\sqrt{x^2+y^2}$, as functions of time. The solid gray lines show the laser electric field in arbitrary units. In panel (a), the black and magenta arrows indicate the instant of single ionization (t_SI) and the recollision (t_r) (see text for details), respectively. The double-headed arrow denotes the traveling time. The green vertical lines indicate the return times of the free electron in the longitudinal direction (z-axis direction).

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Fig. 5 A sample NSDI trajectory, repeating Fig. 4, but for the laser wavelength 3000 nm.

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The discussion above demonstrates the importance of the multiple-return trajectory for NSDI by long-wavelength pulses and the decisive role of the Coulomb focusing effect in the multiple-return trajectory. Now we turn our attention to the electron correlations in the Fig. 1. The NSDI events move from the diagonal to the two axes in the correlated electron momentum distributions with increasing laser wavelength and finally form an experimentally observed near-axes V-shaped structure in the first and third quadrants. That is to say that for shorter wavelength the two electrons are emitted with similar final longitudinal momenta, whereas for longer wavelength the two electrons are emitted with a considerable momentum difference.

By tracing the NSDI trajectories, it is found that the final momentum depends on the energy sharing at the recollision. In Fig. 4 after recollision the two electrons have the similar energies and thus have similar longitudinal momenta. Subsequently, the same acceleration from the electric field is imposed on the two electrons. Finally, the two electrons have similar final longitudinal momenta and thus are located near the diagonal. Figure 5 shows a sample trajectory with asymmetric energy sharing (AES) at the recollision. The free electron returns with an energy up to 15.7 a.u. and recollides with the parent ion. But only a small part of returning energy is transferred to the second electron and the recolliding electron still have the energy of 12.1 a.u. From Fig. 5(d), it is clear that after recollision the returning electron continues traveling in the positive z direction, with a quite large residual momentum [see the red curve in Fig. 4(c)]. The struck electron is right ionized with a near-zero initial longitudinal momentum. Subsequently the two electrons obtain the same negative drift momentum by the acceleration of the laser electric field. For the struck electron, the final longitudinal momentum is approximatively equal to the acceleration of the laser electric field. However, for the recolliding electron the residual momentum and the acceleration of the laser electric field largely cancel each other, resulting in a small final longitudinal momentum. Finally, the two electrons have very different longitudinal momentum and thus deviate from the diagonal and distribute near the two axes. So the symmetric energy sharing (SES) during the recollision results in the electron pair located near the diagonal, and AES makes the electron pairs distribute near the axes.

In order to explain the dependence of the electron correlation behavior on the laser wavelength, in Fig. 6(a) we present the distributions of the energy difference of the two electrons after recollision for those long wavelengths considered in this work. The distributions of the energy difference show a linear decrease in low-energy part (0–2.5 a.u.) followed by a plateau extending to the high energy. The range of the plateau increases with the increase of the laser wavelength. It means that the events with AES are more for the longer wavelength. Finally, it results in more off-diagonal or near-axes NSDI events in the correlated electron longitudinal momentum distribution. Why are there more events with AES for the longer wavelength? Back analysis implies that the energetic recollisions often favor AES while the less energetic ones tend to SES, which is consistent with the result in NSDI for 800 nm and 2.0 PW/cm² [30]. This can be understood as follows. For the high returning-energy recollision, the recolliding electron passes the core very quickly, thus the time of the e-e interaction is so short that the recolliding electron can transfer only a small part of its energy to the bound electron, resulting in the serious AES. For the low returning-energy recollision, the recolliding electron has enough time to interact with the bound electron and finally the two electrons achieve similar energies after recollision. According to the simple-man model [5, 6], the returning energy of the free electron is proportional to the laser intensity and the square of the laser wavelength. In this work the laser intensity is fixed. If the laser wavelength is increased the returning energy must increase. We further calculated the returning energy of the free electron before the recollision for NSDI events from different wavelengths. The statistical distributions of the returning energy are shown in Fig. 6(b). The statistical result shows that with increasing laser wavelength the returning energy increases gradually, which is in agreement with the prediction of the simple-man model. In summary, for longer wavelength, the returning energy of the free electron is higher. Thus AES more likely occurs and results in the off-diagonal or near-axes momentum distribution.

 

Fig. 6 (a) Distributions of the energy difference of the two electrons after recollision for four different wavelengths. (b) Distributions of the returning energy of the free electron for four different wavelengths.

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In Fig. 7 we show the distributions of the energy difference of the two electrons after recollision for those NSDI events in and out of the two-arm regions of the V-shaped structure for 3000 nm. The regions are also shown in Fig. 1(e). It is clear that the energy difference of the two electrons after recollision for those NSDI events in the two-arm regions are mainly from 2.5 a.u. to 17 a.u., and the energy difference of the two electrons after recollision for those NSDI events out of the two-arm regions are mainly smaller than 2.5 a.u. This well exhibits that the AES between the two electrons at recollision is responsible for the formation of the prominent V-shaped structure of 3000 nm.

 

Fig. 7 Distributions of the energy difference of the two electrons after recollision for those correlated electron pairs in (magenta) and out (black) of the two-arm regions of the V-shaped structure for 3000 nm. Those electron pairs are also marked in Fig. 1(e) by magenta and black rectangle frames.

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The discussion above has indicated that AES between the two electrons at recollision can result in off-diagonal (V-shaped) momentum distribution and the high returning energy favors AES. According to the simple-man model, the returning energy of the free electron is proportional to the laser intensity and the square of the laser wavelength. Thus the near-axes V-shaped structure discussed in this work is more sensitive to the laser wavelength than to the laser intensity.

4. Conclusion

In conclusion, we have investigated the recollision dynamics and the electron correlation behavior in strong-field NSDI for several long laser wavelengths (1200–3000 nm). The results show that for these long wavelengths NSDI events are mainly from multiple-return trajectory and the contribution from multiple-return trajectory increases gradually with increasing laser wavelength. We have shown the intuitive classical trajectories of the multiple-return event in NSDI. Moreover, with the laser wavelength increasing the NSDI events move from the diagonal to the two axes in the correlated electron momentum distributions, and finally form an experimentally observed prominent V-shaped structure [Phys. Rev. X 5, 021034 (2015)] in the first and third quadrants. Back analysis indicates that the asymmetric energy sharing between the two electrons during recollision results in the final momentum difference and thus off-diagonal distribution. For 3000 nm, the more energetic recollisions lead to more asymmetric energy sharing, and thus the correlated electron longitudinal momentum distribution shows a near-axes V-shaped structure.