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Abstract
We theoretically investigate the frustrated double ionization (FDI) of Ar atoms with counter-rotating two-color circular (CRTC) laser fields using the three-dimensional (3D) classical ensemble method. Our results show that the FDI probability depends upon the intensity ratio of the CRTC laser fields. The FDI event accompanied with the recollision excitation with subsequent ionization is prevalent and three pathways exist in FDI processes driven by CRTC laser fields. The momentum distribution of a recaptured electron at the ionization time after recollision indicates that the momentum being close to the vector potential is a necessary condition for FDI events to occur. In addition, the recaptured electron most probably transitions to a Rydberg state of which the quantum number is ten in the CRTC fields.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The ionization of gas-phase atoms and molecules and the following motion of the freed electrons driven by strong laser fields bring about a large variety of fascinating phenomena, such as high-order above threshold ionization (HATI) [1,2], high harmonic generation (HHG) [3–5], nonsequential double ionization (NSDI) [6,7], and electron self-diffraction (ESD) [8–10]. In the interaction of atoms or molecules with strong laser fields, a substantial fraction of liberated electron with near-zero kinetic energies can be trapped in high-lying Rydberg states at the end of laser fields, a process dubbed as frustrated ionization [11–15]. The trapped electron of high-lying Rydberg states has gained extensive attention in recent ten years, such as explanation for the mechanisms of the certain harmonic order disappearing below the ionization threshold [16], the suppression of the probability of near-zero-momentum electron [17], the coherent extreme-ultraviolet emission induced by frustrated tunneling ionization [18], and the dissociative frustrate ionization of molecules [19].
In the frustrated double ionization (FDI) process, one of the two released electrons occupies a Rydberg state at the end of the strong laser pulse, which has been theoretically and experimentally explored for atoms and molecules in the past years [20–25]. For linear laser pulses, the frustrated NSDI is sensitive to the conditions of the ionization [20], energy sharing of molecular fragmentation depends on the laser pulse duration [26], and two-electron effects are key to the formation of the highly excited H* [22]. Recently, Larimian et al. have observed the variation of the momentum distribution in the FDI events from the double-hump structure to single-hump structure with the increase of the laser intensity [24]. Furthermore, using a 3D semiclassical method, Chen et al. have reproduced the observed spectrum of the double-hump structure and explained ionization mechanism leading to FDI [27]. For elliptically polarized fields, the excited neutral fragments come from the Coulomb explosion in the molecular FDI [28]. For orthogonally polarized two-color laser pulses, the FDI of the triatomic molecular D$_3^+$ can be sub-cycle attosecond controlled [29]. Yet, the electron dynamics of the FDI event for atoms in the counter-rotating two-color circular (CRTC) laser fields remains unclear.
In recent years, the CRTC laser fields have been widely applied to the HHG [30,31], investigating properties of magnetic structures [32,33], and controlling the NSDI [34–38]. In the NSDI process, an electron is first ionized via over-the-barrier or tunneling mechanism, when the Coulomb field is distorted by the laser electric fields. Then, the first ionized electron is accelerated and driven back to the parent ion core by the laser field, and collides inelastically with it, leading to the ionization of the second electron. Shomsky et al. have found that one of the two liberated electrons may be recaptured by the parent ion core after recollision at the end of the laser field, which constitutes a pathway of FDI and is responsible for the electronic correlation [20]. Katsoulis et al. have explained how the ratio of the two laser field strengths for each pair of wavelengths significantly enhances the pathway FFIS [a pathway of FDI for two-electron diatomic and triatomic molecules, one electron tunnel ionizes early (first step), while the bound electron does so later in time (second step). If the first ionization step is “frustrated”, it is the FFIS pathway] with electronic correlation being roughly absent. In addition, they show that the main features of FDI for a range of pairs of wavelengths is predicted by the simple model [39]. Despite the great progresses on the double-ionization dynamics in the CRTC laser fields, we note that no systematic theoretical study has been presented to illustrate the FDI accompanied with NSDI of atoms in the CRTC laser fields.
In this paper, we systematically investigate the FDI of Ar atoms accompanied with NSDI in the CRTC laser fields using a three dimensional classical model. Our results show that the FDI yield strongly depends on the ratio $\gamma _{E}$. There are three pathways in FDI events: FDI accompanied with recollision impact ionization (FDI-RII), FDI accompanied with recollision excitation with subsequent ionization involving doubly excited state (FDI-RESI-DES), and FDI accompanied with recollision excitation with subsequent ionization involving singly excited states mechanism (FDI-RESI-SES). They are all related to the electronic correlation. Trajectory analysis of the FDI events shows that the FDI event accompanied with the recollision excitation with subsequent ionization mechanism is prevalent in the CRTC laser fields. Besides, one condition for the FDI event to occur is that the momentum distribution of recaptured electron at the ionization time after recollision is close to the negative vector potential. It is most probable that the recaptured electron transitions to a Rydberg state of which the quantum number is ten in the CRTC laser fields.
2. Method
Precise investigation of multi-electron system in a strong laser field demands numerically solving the time-dependent Schrödinger equation [40,41]. This requires a number of computational load. In order to explain the various kinds of experimental phenomena and gain insight into mechanisms of the strong-field double ionization, classical and semiclassical methods are employed [42–48]. Moreover, the trajectory can be traced during the whole laser pulse using the two methods, thus a visual picture on the dynamics process of the two electrons can be provided [49,50]. With the 3D classical ensemble method, we investigate the electron dynamics of the FDI by the CRTC laser fields [20]. The total ensemble size is more than $10^{7}$ and the time step is 0.05 a.u..
In this model, the frustrated electron and the ionized electron are described by Newtonian equations of motion (atomic units are used unless stated otherwise)
In Eq. (1), $\mathbf {E}(t)= \mathbf {E}_{r}(t)+ \mathbf {E}_{b}(t)$ is the form of the CRTC laser pulse. $\mathbf {E}_{r}(t)$ and $\mathbf {E}_{b}(t)$ are the fundamental and second harmonic laser pulse, given by
The initial momenta and positions of the electrons are randomly assigned, satisfying the classically allowed position for the total energy of −1.59 a.u.. It approximately equals the sum of the first and second ionization energy to match Ar atom. Once the initial conditions are ready, the laser pulse will be turned on and the trajectories of all electrons can be recorded. Firstly, the energies of the two electrons are more than zero at some time during the laser duration, and then one of the ionized electrons is recaptured with the negative energy at the end of the laser filed. This process is defined as the FDI events. In this paper, we just pay attention to the FDI events accompanied with NSDI processes, where contains the multiple return events but recollision occurs only once (except for Fig. 1, which includes multiple return and collision events).
Fig. 1. Probabilities of DI (blue squares) and FDI (red circles) versus the combined laser intensity for (a) $\gamma _{E}$=1.4, and (b) $\gamma _{E}$=3.3. (c) The ratios of yields between FDI and DI events versus the combined laser intensity at the three field ratios $\gamma _{E}$. (d) The probabilities of FDI as a function of field ratio at the four combined laser intensities.
3. Results and discussions
Figures 1(a) and 1(b) show the FDI (red circles) and the double ionization (blue squares) probabilities as a function of combined laser intensity at different ratios $\gamma _{E}=1.4$ and 3.3, respectively. The probability curves of FDI events increase from 0.01 PW/cm$^2$ to 2 PW/cm$^2$ and their trends are similar to the corresponding curves of double ionization (DI) events. The probability curves between FDI and DI exhibit a crossing at about 0.07 PW/cm$^2$ in Figs. 1(a) and 1(b), and the probability of FDI events is higher (lower) than that of DI events below (above) 0.07 PW/cm$^2$. The phenomena demonstrated in Figs. 1(a) and 1(b) are similar to the previous theory on FDI of Ar by 400-nm linearly polarized laser fields where it is shown that the crossing is around 9$\times 10^{13}$ W/cm$^2$ [25]. Figure 1(c) displays the ratios of probabilities between FDI and DI versus combined laser intensity for the case of $\gamma _{E}$=0.8, 1.4, and 3.3, respectively. The ratios of probabilities between FDI and DI decrease rapidly with the increase of the laser intensity, and the probability of the FDI event is greater (the ratios greater than 1.0) than that for the DI event below 0.07 PW/cm$^2$. On the contrary, the probability of the FDI event is less than that for the DI event above 0.07 PW/cm$^2$. When a pair of electrons are emitted during the laser pulse, only one of the two events can happen, which constitutes a “competition” for emitted electron pairs. Whether it is an FDI event or a DI event depends on whether one of two electrons can be recaptured by the parent ion core at the end of pulse.
Figure 1(d) shows the FDI probability as a function of the amplitude ratio $\gamma _{E}$ at four different combined laser intensities. Generally, the NSDI events occur when the laser is within the range of “knee” structure [35,51]. The frustrated NSDI is studied here. Therefore, we choose the cases of 0.4 PW/cm$^2$, 0.6 PW/cm$^2$, 0.8 PW/cm$^2$, and 1.0 PW/cm$^2$ at the plateau area for DI and FDI events, as shown in Figs. 1(a) and 1(b), respectively. The FDI probabilities strongly depend on $\gamma _{E}$ and exhibit different peaks around $\gamma _E\approx {1-3}$. The peak shifts to the right slightly as the increase of the laser intensity. According to the simple-man model [52], where the initial momentum at the ionization instant and the effect of the Coulomb potential to the ionized electron are ignored. The recollision could happen for the first ionized electron with zero initial velocity in the CRTC laser fields, when $\gamma _{E}$ equals to the wavelength ratio $\lambda _{b}/\lambda _{r}$ [53]. In fact, Coulomb interaction between the ionized electron and the parent ion should be taken into account. The intensity ratio for the most probable FDI should shift to the predicted ($\gamma _{E}=2$) by the simple-man model. As the laser intensity increases, ionized electron is more far away from the parent ion and the Coulomb interaction becomes less important. Consequently, the FDI yield peaks are more close to the value predicted by the simple-man model. Besides, the recollision has the maximum probability and NSDI yield is most probable. We focus on the FDI after NSDI occurrence and $\lambda _{r}$=790 nm and $\lambda _{b}$=395 nm are the wavelength of fundamental and second harmonic laser pulse. Therefore, the peaks of the FDI probability exhibit around $\gamma _{E}\approx {2.0}$, as shown in Fig. 1(d). The Coulomb interaction influences the probability of the FDI events [24,25]. For the relatively weaker laser intensity, the Coulomb interaction plays a key role in the recaptured electron processes. As the laser intensity increases, the Coulomb interaction becomes less important. Accordingly, the FDI probability decreases as the laser intensity increases. It indicates that the FDI event prefers the relative lower laser intensity at around $\gamma _E\approx {1-3}$ in the CRTC laser fields.
To obtain more details about microscopic electron dynamics of FDI events, we trace the electron trajectories and perform statistical analysis of the recollision time (t$_r$) and the delay-time (t$_{td}$). The recollision time is the instant when the distance is closest between both electrons after single ionization time, and the delay-time is the interval between the recollision time and the double ionization time before the FDI occurrence. Figure 2 shows the distributions of the delay-time versus the recollision time. The ratios $\gamma _{E}$ are 1.4 and 3.3, and the combined laser intensity are 0.4 PW/cm$^2$ (top row) and 0.6 PW/cm$^2$ (bottom row), respectively. The different mechanisms of frustrated nonsequential double ionization can be distinguished by the delay-time (t$_{dt}$). The delay-time t$_{dt}$=0.25 optical cycles (o.c.) is marked by the white dashed line in Fig. 2. The area below the white dashed line (t$_{dt}$ < 0.25 o.c.) corresponds to the FDI events accompanied with the recollision impact ionization (FDI-RII) mechanism. The area above the white dashed line (t$_{dt}$ > 0.25 o.c.) corresponds to the FDI events accompanied with recollision excitation with subsequent ionization (FDI-RESI). The delay-time (t$_{dt}$) distributes about 0.5 o.c., 0.83 o.c. and 1.1 o.c. etc., as shown in Fig. 2. This means that the returning electron collides with the bound electron not strongly and the bound electron is ionized until the next field maximum by the laser filed. The probabilities of the FDI-RESI event are about 93.3$\%$, 90.7$\%$, 93.1$\%$, and 89.9$\%$ for Figs. 2(a)–2(d), respectively. It indicates that the FDI-RESI event is prevalent. This is consistent with the previous theoretical result of Shomsky et al. [20], and it shows that the “ionized” electron though the RESI mechanism can be recaptured easily at the end of the laser fields.
Fig. 2. The map of the recollision time (t$_{r}$) versus the delay-time (t$_{dt}$) in the CRTC laser fields at 0.4 PW/cm$^2$ (top row) and 0.6 PW/cm$^2$ (bottom row). The field ratios are 1.4 (a, c), and 3.3 (b, d), respectively.
The probability of the FDI-RESI events decreases a little as the combined laser intensity increases for the same ratio $\gamma _{E}$. The reason is that the recollision energy of the returning electrons increases with the increase of the laser intensity and the hard collision can lead to the bound electron to be ionized immediately. Thus, the probability of FDI-RESI event decreases. As $\gamma _{E}$ increases, the recollision energy of the returning electron increases [35,51] but the FDI-RESI event decreases for the same laser intensity. It not only indicates that recollision energy play an important role in the FDI processes, but also that the momentum of the frustrated electron at the ionization time is an important factor, as shown in Fig. 5.
The above analysis of the distribution of recollision time versus the delay-time shows that recollision can induce the two pathways leading to the FDI. The studies have shown that the hard or soft collision between ionized electron and parent ion core is responsible for the RII or RESI mechanisms in NSDI processes [54,55]. Thus, recollision energies $E_{r}(t_{r}+\Delta {t})$ of the recaptured electrons and $E_{i}(t_{r}+\Delta {t})$ energy of ionized electrons at the moment $\Delta {t} = 0.06$ o.c. [20] after the recollision are shown in Fig. 3. By tracing these FDI trajectories, we separately show the energy distributions of the FDI-RESI events and the FDI-RII events in Figs. 3(a) and 3(b), respectively.
Fig. 3. Energy of the recaptured electron vs the ionized electron at the moment $\Delta {t} = 0.06$ o.c. after recollision. (a) The FDI event accompanied with the RESI mechanism. (b) The FDI event accompanied with the RII mechanism.
It can be clearly seen that, for the FDI-RESI events [Fig. 3(a)], the largest part of the clusters are on both sides of the main diagonal (away from the main diagonal), which means the two electrons often achieve different energies during recollision. However, for the FDI-RII events [Fig. 3(b)], a part of the cluster move to the main diagonal, meaning that two electrons achieve similar energy during recollision processes. In addition, there are also FDI accompanied with the doubly excited states with subsequent ionization involving doubly excited state (FDI-RESI-DES) by tracing the energy trajectories, as shown in Figs. 4(d) and 4(e). The energy of the recaptured electron and the ionized electron are both negative as shown in the third quadrant of Fig. 3(a). We also note that the asymmetry of the energy sharing is obvious during the recollision processed and six kinds of trajectories exist in FDI events. When the energies of recaptured electron and ionized electron are both positive after recollision, the FDI event does not occur.
Fig. 4. Time evolution of the electron energies for illustrative the FDI event accompanied with the RII (left column) and the RESI (right column) mechanism, respectively. The red and the blue lines mark the recaptured electrons and ionized electrons at the end of the laser field, respectively. The arrows indicate the recollisions (the cyan), the moment $\Delta {t}=0.06$ o.c. after recollision (the orange), and the double ionization time (the violet). The double-headed arrows denote the delay time.
Figure 4 displays six sample trajectories to provide an intuitive picture of the FDI processes. For the lower laser intensity, the lower energy transfer cannot directly cause the bound electron to be released after recollision, but it is excited by recollision and then ionized later by the subsequent electric field. For the doubly exited states mechanism, the two electrons could not be freed directly, instead they pass through a doubly excited states after recollision. For RII mechanism, both electrons could be freed directly after recollision. The energy evolution of the FDI events are shown in Fig. 4. There are three pathways in FDI events: FDI accompanied with recollision impact ionization (FDI-RII) [as shown in Figs. 4(a)-(c)], FDI accompanied with recollision excitation with subsequent ionization involving doubly excited state (FDI-RESI-DES) [as shown in Figs. 4(d) and 4(e)], and FDI accompanied with recollision excitation with subsequent ionization involving singly excited states mechanism (FDI-RESI-SES) [as shown in Fig. 4(f)]. The arrows indicate the recollisions (the cyan), the moment $\Delta {t}=0.06$ o.c. after recollision (the orange), and the double ionization time (the violet). The double-headed arrows denote the delay time.
From Figs. 4(a) and 4(d), we can see that, after recollision in the FDI processes, the recaptured electron and the ionized electron are with negative and positive energy, respectively. It is more likely that the electron with negative energy will be the stuck electron as shown by the distribution of the second quadrant in Fig. 3. When the recaptured and ionized electrons are both with negative energies after recollision for the trajectories in Figs. 4(d) and 4(e), it is the doubly excited states mechanism for producing the FDI, corresponding to the distribution of the third quadrant in Fig. 3(a). Either electron may be recaptured with equal probability at the end of the laser field, when the two electrons are with the negative energies after recollision. The trajectories in Figs. 4(c) and 4(f) are the cases of recaptured electron with positive energy and the ionized electron with negative energy after recollision, which indicates that the electron with positive energy can also be captured at the end of the laser field. The trajectory corresponds to the distribution of the fourth quadrant in Fig. 3. The analysis above illustrates that there is an other reason to produce the FDI events, which will be explained below. Moreover, either the electron that is ionized before the recollision [the red line of Figs. 4(a), 4(c), 4(d) and 4(f)] or the electron that is bound before the recollision [the red line of Figs. 4(b) and 4(e)] can be recaptured by the parent ion core at the end of laser fields, which is related to the electron-electron correlation in the CRTC laser fields, as reported in Ref. [39]
To further explore which condition is responsible for the FDI events, we show the momentum distribution of recaptured electron at the ionization time after recollision by the field amplitude ratios $\gamma _{E}$=1.4 [Figs. 5(a)–5(c)], and $\gamma _{E}$=3.3 [Figs. 5(d)–5(f)], respectively. The ionization time after recollision is defined as the instant that the energy of the recaptured electron become positive after recollision. For the cases of $\gamma _{E}$=1.4 and $\gamma _{E}$=3.3, the momentum distributions of all FDI events are shown in Figs. 5(a) and 5(d), respectively. The triangles of the negative vector potential (the white curve) of the laser field are also shown in Figs. 5(a) and 5(d). The negative vector potential marked by the pink curve is close to the momentum distribution of recaptured electron. For the double ionization without the recaptured electron at the end of the laser field, the momenta of the ionized electron distribute on the triangles of the negative vector potential [56]. However, one can see that the momentum distribution of the recaptured electron shows three parts, which are close to the negative vector potential marked by the pink curve. It indicates that the FDI event occurrence is due to the special initial momentum of recaptured electron. According to the simple-man model, the negative vector potential equals the final momentum of the ionized electron. When the momentum of the recaptured electron at the ionization time after recollision is equal to the negative vector potential, the “ionized” electron will be recaptured by the Coulomb attraction. This is similar to the theoretical study on FDI of Ar by the linearly polarized laser fields where it is shown that velocity of the recaptured electron must be equal to the vector potential of laser field at the ionization time [25]. In Fig. 5 we separately display the momentum distribution of the FDI-RESI events (b) and (e), the FDI-RII events (c) and (f), respectively. The momentum distribution exhibits a “fat” trefoil-like shape as $\gamma _{E}$ increases, which is near the negative vector potential. The momentum distribution of the FDI-RESI (FDI-RII) events is similar to that of all FDI events, which indicates that the FDI events accompanied with the two mechanisms have the same initial momentum (close to the negative vector potential), and the FDI-RESI event is prevalent.
Fig. 5. Momentum distributions of the recaptured electrons for the all FDI events at the ionization time after recollision by the field amplitude ratios (a) $\gamma _{E}$=1.4 and (d) $\gamma _{E}$=3.3 with the combined laser intensity of 0.4 PW/cm$^2$. The momentum distributions of the FDI-RESI events for the case of (b) $\gamma _{E}$=1.4 and (e) $\gamma _{E}$=3.3, respectively. The momentum distributions of the FDI-RII events for the case of (c) $\gamma _{E}$=1.4 and (f) $\gamma _{E}$=3.3, respectively.
Next, we will identify the energy variation of the recaptured electron with the increase of combined laser intensity and the relative ratio $\gamma _{E}$. In FDI events, one electron is ionized and the other is recaptured in the Rydberg state at the end of laser fields. Figure 6(a) shows the energy distributions of the recaptured electrons for the case of $\gamma _{E}$=1.4 and $\gamma _{E}$=3.3, and the combined intensities of 0.4 PW/cm$^2$ and 0.6 PW/cm$^2$, respectively. For the combined laser intensity of 0.4 PW/cm$^2$ (0.6 PW/cm$^2$), the energy decreases slightly as the relative ratio increases. The similar result has been reported that the recaptured electron is most probable at the Rydberg state with $n=\sqrt {-2/E_{r}}\approx {10}$ to produce FDI of triatomic molecules with CRTC laser fields [39]. For the ratio $\gamma _{E}$=1.4 ($\gamma _{E}$=3.3), the energy increases slightly as the combined laser intensity increases. The reason is that the higher the combined laser intensity, the greater the electron oscillating distance (the distance between electron and ion core), and the less the negative potential energy [25]. Figure 6(b) displays the distribution of the effective principal quantum number with the same laser field as Fig. 6(a). The peaks in the distribution are at n$\approx {10}$. It indicates that forming the Rydberg state with n$\approx {10}$ in the FDI events of Ar atoms is the most probable in the CRTC laser fields.
Fig. 6. (a) The energy distributions of the recaptured electrons at the end of laser fields for the case of $\gamma _{E}$=1.4 and $\gamma _{E}$=3.3, and the combined laser intensities of 0.4 PW/cm$^2$ and 0.6 PW/cm$^2$, respectively. (b) The effective principal quantum number distributions of the recapture electrons with the same combined laser field as (a).
4. Conclusions
In summary, we have theoretically investigated the frustrated nonsequential double ionization of Ar atoms in the CRTC laser fields. Our result shows that the FDI events prefer relative lower intensity at around ratio ($\gamma _{E}\approx {1-3}$) of the two field strengths and the yield of the FDI events can be controlled by the electric field amplitude ratio of the CRTC laser fields. By analyzing the FDI trajectories, we find that there are six kinds of trajectories and the FDI-RESI event is more probable in the CRTC laser fields. A certain condition for the FDI events to occur is given, that the momentum of the recaptured electron at the ionization time after recollision is close to the negative vector potential. Moreover, the stuck electron most probably transitions to high-lying Rydberg states with n$\approx {10}$ in the CRTC laser fields. This work illustrates that the recollision excitation with subsequent ionization involving doubly excited state, recollision excitation with subsequent ionization involving singly excited states, and recollision impact ionization are the three pathways for producing FDI events in the CRTC laser fields. Though the FDI processes are more complex than that in the linearly polarized laser fields, the processes provide multiple mechanisms to gain insights into the electron dynamics of FDI in CRTC laser fields. We hope our work will be helpful for the study of further related experiments.
Funding
China Postdoctoral Science Foundation (2019M661108); Fundamental Research Funds for the Central Universities (N2005023); Postdoctoral Science Foundation of Northeastern University (20200324).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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