## Abstract

We theoretically investigated sequential double ionization (SDI) of Ar by the nearly circularly polarized laser pulses with a fully correlated classical ensemble model. The ion momentum distributions of our numerical results at various laser intensities and pulse durations agree well with the experimental results. The experimentally observed multi-electron effects embodied in the joint momentum spectrum of the two electrons is also reproduced by our correlated classical calculations. Interestingly, our calculations show that the angular distribution of the first photoelectron from the trajectories which eventually suffer SDI differs from the distribution of the photoelectrons from above-threshold ionization trajectories. This observation provides additional evidence of multi-electron effects in strong field SDI.

© 2012 OSA

## 1. Introduction

Strong-field double ionization (DI) can proceed through two different processes, sequential double ionization (SDI) and nonsequential double ionization (NSDI). In the linear polarized and moderate intensity laser field, DI is dominated by the nonsequential process, where the second electron is released by the recollision of the first tunneled electron [1, 2]. Because of the recollision, the two electrons reveal a highly correlated behavior [3]. During the past decades, a great number of studies have been performed in this area and the rich dynamics of recollision in NSDI has been well explored [4–14]. When the laser field is circularly polarized or its intensity is high enough, sequential process dominates DI, where the two electrons are ionized sequentially by the laser field [15–17]. In SDI, it was usually assumed that there is no correlation between the two electrons and they can be treated independently. However, a recent experiment shows that the two electrons from SDI exhibit a strong angular correlation [18], implying that SDI can not be treated as two independent steps. Especially, previous work [19, 20] has given strong evidences of the breakdown of the independent electron assumption in SDI. The first feature is the measured release time of the second electron, which is much earlier than the prediction of the independent electron model [19]. The second feature is the ratio of the parallel to antiparallel emissions of the SDI events, which shows an oscillating behavior as a function of the laser intensity [20]. The third feature is displayed in the joint momentum spectrum of the two electrons in the direction of the minor polarization axis of the laser field. The experimental results show that the distributions exhibit different shapes for the parallel and antiparallel emission events, in contrast to the prediction of the independent electron SDI model which predicts the same shape for the parallel and antiparallel events [20]. These features indicate that the electron correlations should be carefully treated in the theoretical description of SDI.

Because of the enormous computational demand of the nonperturbative quantum approaches in the two- and multi-electron system, present theoretical studies have resorted to classical methods [21–25]. In our previous work, we investigated the SDI of Ar with a Heisenberg-potential classical model (HPCM) [26]. In that paper, several experimental results, for instance, the ion momentum distribution, the ratio of the parallel to antiparallel emissions as a function of laser intensity and the ionization times of both electrons, are well reproduced by our classical calculations. In this paper, with the HPCM, we perform further investigations on SDI of Ar by the nearly circular laser pulses. We show that the third evidence of multi-electron effects in SDI, i.e., the difference in the shapes of the joint electron momentum distributions of the parallel and antiparallel emissions [20], is also reproduced by our classical calculation.

It has been demonstrated that the angular distribution of the photoelectrons is sensitive to the details of the remaining ion and thus the multi-electron effects have an obvious manifestation on the angular distribution [27, 28]. Here, we investigate the angular distributions of the photo-electrons from SDI, aiming to check the multi-electron effects in SDI. We analyze the angular distributions of the photoelectrons of the singly and doubly ionized trajectories separately. Our results reveal that the angular distributions of the first electron of the SDI trajectories are very different from those of the above-threshold ionization (ATI) trajectories. This difference indicates the different ion-electron interactions during the escaping of the outmost electron for the ATI and SDI trajectories. Thus, it provides additional evidence of the breakdown of the independent-step assumption in SDI.

## 2. The classical ensemble model

In the HPCM [26], the Hamiltonian of the two-electron system in the presence of the laser field is written as (atomic units are used throughout this paper unless stated otherwise):

**r**

*and*

_{i}**p**

*are the position and canonical momentum of the*

_{i}*i*th electron, respectively.

**E**(

*t*) is the electric field, which is given as $\mathbf{E}(t)=f(t)\left[\frac{1}{\sqrt{{\epsilon}^{2}+1}}\mathit{cos}\left(\omega t+\phi \right)\widehat{\mathbf{x}}+\frac{\epsilon}{\sqrt{{\epsilon}^{2}+1}}\mathit{sin}\left(\omega t+\phi \right)\widehat{\mathbf{y}}\right]$, where $\text{f}(\text{t})={E}_{0}\mathit{exp}\left[-\frac{1}{2}{\left(\frac{t}{\tau}\right)}^{2}\right]$ is the field envelope.

*ω*,

*ε*and

*φ*are the laser frequency, the ellipticity and carrier-envelope phase (CEP), respectively. $2\sqrt{\mathit{ln}2}\tau $ denotes the pulse duration (FWHM).

*V*(

_{H}*r*,

_{i}*p*) is the Heisenberg-core potential, which is expressed as [29]:

_{i}*α*indicates the rigidity of the Heisenberg core and is chosen to be 2 in this paper. For a given

*α*, the parameter

*ξ*is chosen to match the second ionization potential of the target. Here we set

*ξ*=1.225 for target Ar. This Heisenberg-core potential was introduced by Kirschbaum

*et al*[29] and it has been widely used in the the classical investigations of atomic and molecular collisions [30–32]. In our previous paper [26], we employed it to investigate the ionization process of the two-electron system exposed to strong laser fields.

From the classical point of view, the two-electrons system suffers autoionization. In the past decade, the soften-core potential has been widely used to avoid autoionization [10, 21, 24]. However, the soften-core potential can not reproduce the ionization potentials of the electrons correctly. In tunneling and over-the-barrier ionization, the ionization rate is very sensitive to the ionization energy. Thus, the soften-core potential may be deficient in describing DI that occurs through tunneling or over-the-barrier escape. In our previous paper [26], we employed the Heienberg-core potential in the classical two-electron model, where the Heisenberg-core potential plays two roles. One role is to avoid autoionization and the other role is to make the first and the second ionization potentials of the model atom match with those of the realistic target. With this HPCM, we reproduced the experimental observations at the quantitative level [26].

The evolution of the two-electron system is determined by the following equations:

## 3. Results and discussions

First, we show the momentum distributions of the doubly ionized ion in the laser polarization plane. The upper line of Fig. 1 displays the results of the 7-fs pulses and the bottom line corresponds to the results of the 33-fs pulses. We mention that in all of the calculations of this paper, the laser parameters are chosen to be the same as those in the experimental reports [19, 20], i.e, the wavelength and ellipticity are 740 nm (788 nm) and 0.78 (0.77) for the 7-fs (33-fs) pulses, respectively. It is clearly shown that for both pulse durations, the distributions exhibit a three-band structure at the relatively low laser intensity [see Figs. 1(a) and 1(d)] while a four-band structure at the relatively high laser intensity [see Figs. 1(b) and 1(e)]. The origin of the four-band structure is interpreted as due to the different values of the electric field at which the two electrons are released [19, 33, 34]. The outer bands correspond to the events where the two electrons emit into the parallel directions whereas the inner bands result from the events where the two electrons release into the antiparallel directions. At the relatively low intensity, both electrons release around the center of the pulses. Thus the momentum amplitudes of the two electrons are almost the same, leading to the two inner bands (corresponding to the antiparallel emissions) evolving into a one-band structure located at zero momentum [19]. In Figs. 1(c) and 1(f), we show the distribution of the ion at 3.0 PW/cm^{2} with the volume effect taken into account. Figure 1(c) show that for the 7-fs case the distribution exhibits a three-band shape even at the high laser intensity. This is in good agreement with the experimental observation [20, 33]. For the 33-fs case, the four-band structure is still discernable when the focal volume effect is considered [Fig. 1(f)], which is also consistent with the experimental results [19].

Definitely, the results above are well understood within the independent-electron assumption. Nevertheless, previous experiments [19, 20] have displayed evidence of multi-electron effects in SDI. For example, the ratio of the parallel to antiparallel emissions along the minor polarization axis shows an oscillating behavior as a function of laser intensity [20]. This behavior has been well reproduced with our HPCM where the electron correlations are taken into account during the entire ionization process [26]. Further evidence of multi-electron effects in SDI is revealed by the joint electron momentum distribution along the minor polarization axis [20]. It is shown that the shapes of the distributions for the parallel and antiparallel emission events are very different [20]. In Fig. 2 we display our numerical result. The difference in the shapes of the distributions for the parallel and antiparallel emissions is discernable though it is fainter than the experimental data. This result further confirms that the multi-electron effects in SDI have been expressed in our fully classical calculations.

For the laser parameters used in our calculations and the recent experiments [19, 20], recollision is forbidden because of the large ellipticity. Thus the electron correlations in SDI most possibly arise during the process of the first ionization. In this process, the electron correlations may leave footprints on the remaining ion. It has been shown that the interaction between the remaining ion and the escaping electron significantly corrects the trajectory of the escaping electron [35]. The final angular distribution of the electron can reveal the details of the remaining ion [36, 37]. For instance, it has been shown that the offset angles of the electrons from ATI of Ar and He exhibit different behaviors as a function of the laser intensity [27]. For Ar, the offset angle decreases with the increase of the laser intensity, while it keeps unchanged for He. This difference is ascribed to the multi-electron effects (laser-induced polarization) [27]. For Ar, it is much easier to polarize than He, leading to a strong dependence of offset angle on laser intensity. Thus, by examining the offset angle of the ionized electron, it is possible to extract information of the remaining ion which may hint some multi-electron effects. In the following, we investigate the angular distributions of the electrons from different processes, intending to check the multi-electron effects in SDI.

Figure 3(a) depicts the momentum distribution in the laser polarization plane for the electrons from ATI of Ar [modeled by the two-active-electron atom, see formula (1)]. The ellipticity, wavelength, pulse duration and laser intensity are 0.78, 740 nm, 7 fs and 1.0 PW/cm^{2}, respectively. Because of the ellipticity, electrons are almost ionized along the major axis of the polarization (x axis). Without further interaction with the remaining ion, the most probable electron momentum distribution is along the minor polarization axis (y axis). In Fig. 3(a), a small but visible offset angle with respect to the minor polarization axis exists, indicating the interaction between the remaining ion and the escaping electron [34–36]. In order to read the value of the offset angle, we perform a double gaussian fitting on the angular distribution of the photoelectrons, as shown in Fig. 3(b). In the following, we extract the offset angles with this way.

In order to test the validity of our HPCM in describing the subtlety of the interaction of the remaining ion and the escaping electron, we performed two calculations on ATI of Ar. In the first calculation, we employ the single-active-electron assumption, Thus the multi-electron effects (laser-induced polarization) are excluded. In this calculation, the hamiltonian of Ar is written as:

*α*= 2 and

*ξ*= 0.7225 to fit the first ionization potential of Ar. At the end of the laser pulses, we collect the singly ionized trajectories and analysis the angular distributions of the photoelectrons. Figure 4(a) displays the offset angle of the photoelectrons as a function of laser intensity.

In the second calculation, the multi-electron effects are included by considered two active electrons, i.e., the hamiltonian of the Ar is expressed as formula (1) and the evolution of the system in the laser field is determined by Eq. (3). At the end of the pulse we analyzed the angular distributions of the photoelectrons from the singly ionized trajectories. The offset angle as a function of the laser intensity for these electrons is shown in Fig. 4(b). It is clearly seen that for the single-active-electron calculation where the multi-electron effects are excluded, no significant intensity dependence of offset angle is observed over the investigated intensity range [Fig. 4(a)]. However, for the second calculation, where the multi-electron effects are included, a monotonous decrease of the offset angle with increasing laser intensity is obvious. This behavior is in good agreement with recent experimental observations [27]. These results indicate that the multi-electron behavior revealed in the angular distributions of photoelectrons can be well described by our classical model, confirming the validity of our HPCM in analyzing the angular distributions of the photoelectrons below.

Now, we examine the angular distributions of the photoelectrons from SDI calculated by the HPCM. At the end of the pulses, we select out the singly and the doubly ionized trajectories. It is interesting to compare the angular distribution of the photoelectrons from the singly ionized trajectories (ATI photoelectron) with that of the first electron from SDI trajectories because in both cases the electrons start from the same state (the ground state of Ar). In Fig. 5 we display the offset angles of the photoelectrons from the ATI trajectories (green squares) and those of the first photoelectrons from the SDI trajectories (magenta triangles). The top and the bottom plots correspond to the results from the 7-fs and 33-fs pulses, respectively. It is clearly seen that for both pulse durations the offset angles for the ATI photoelectrons are much larger than those of the first photoelectron from SDI trajectories. With the assumption that the two ionization events in SDI are independent, the offset-angle distribution of the first electrons from SDI trajectories will be the same as that of the photoelectrons from ATI trajectories. The results shown in Fig. 5 are in contrast to the prediction of this independent-electron assumption of SDI. Thus it definitely confirms the multi-electron effects in SDI though the details of this faint effect in SDI is currently not clear.

## 4. Conclusion

In summary, we have investigated the strong-field SDI of Ar by the elliptical laser pulses with the HPCM. The experimentally measured ion momentum distributions at various laser intensities and durations are obtained with our classical calculations. The experimentally observed multi-electron effects revealed by the joint electron momentum spectrum is also catched with our calculations. Furthermore, we have investigated the angular distribution of the photoelectrons from SDI. In order to confirm the validity of the classical model in studying the multi-electron effects revealed by the angular distribution, we first performed calculations on ATI of Ar with two different treatments, one including and the other excluding the multi-electron effects. The offset angles as a function of the laser intensity for these two treatments exhibit very different behaviors, which agrees well with recent experimental results [27], and thus confirms the validity of our classical model. Based on these results, we examined the offset angle of the electrons from SDI. We find that the offset angle for the first electron from SDI is very different from that for photoelectrons from the ATI trajectories (the trajectories that eventually suffer single ionization), in contrast to the prediction of the independent-electron assumption of SDI. Thus, our calculations predict an additional evidence of multi-electron effects in SDI. A full understanding of the subtle multi-electron effects in SDI calls for more sophisticated studies.

## Acknowledgment

We thank Dr. A. N. Pfeiffer for his useful suggestions. This work was supported by the National Science Fund (No. 60925021) and the 973 Program of China (No. 2011CB808103).

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