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Abstract
With the three-dimensional (3D) classical ensemble method, we theoretically investigate the recollision dynamics in strong-field nonsequential double ionization (NSDI) of Ar by counter-rotating two-color circularly polarized laser fields. With the analysis of the NSDI trajectories, we find that not only multiple-recollision but also single-recollision processes occur in the double ionization events. Furthermore, the multiple-recollision and single-recollision processes both undergo the recollision-induced excitation with subsequent ionization (RESI) and recollision-induced ionization (RII). The angle between the momentum and the force of the laser field at the recollision moment can affect the times of the recollision.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
When intense laser fields interact with atoms or molecules, many new physical phenomena can happen, such as nonsequential double ionization (NSDI) [1–3], above-threshold ionization (ATI) [4–6] and high harmonic generation (HHG) [7], etc. The recollision process [8, 9] is known as the responsible mechanism for NSDI. In this picture, an electron through tunneling or over-the-barrier ionization mechanism is accelerated by the oscillating laser field, and can be driven back to its parent ion core when the electric field changes its direction, leading to the second electron ionized by an inelastic recollison. This process is most effective with linear polarization but it is significantly suppressed with circular polarization [10,11].
In 1995, bichromatic circularly polarized pulses was theoretically proposed to drive HHG [12, 13] and was realized in the experiment by Eichmann et al. [14]. The electron trajectories responsible for the emission of particular harmonics are identified [15]. This field has been successfully used to drive high-order above-threshold ionization (HATI) [16,17], and to investigate molecular photoelectron momentum distribution [18]. Recently, the double ionization of molecules and atoms on the comparison study is demonstrated [19].
In recent year, NSDI in counter-rotating two-color circularly polarized (TCCP) laser field is illustrated and the analysis of NSDI in this laser field is made [20–24]. With the development of the experimental technic [25–27] and the theoretical methods [28–35], the details of correlated electron dynamics in NSDI is demonstrated during the past two decades. For example, at high laser intensities, the correlated electron momentum spectra showed the “fingerlike” (V-shaped) structure [36] and cross-shaped structure [37] in the experimental observation. The theoretical analysis has shown that these structures result from asymmetric energy sharing between the two electrons during recollision processes [38,39]. At lower laser intensities, there are doubly excited states and the decay dynamics of the doubly excited state in the NSDI processes, which has been explored detailedly [40,41].
During the study of recollision picture, most of NSDI experiments have been carried out using linearly polarized laser fields, where the first ionized electron can return several times to the parent ion and the recollision mainly occurs at the first return. However, the recollision-induced phenomenon resulting from the recollision at later returns is also demonstrated in recent studies [42,43]. Moreover, multiple recollisions have been shown in recent experiments, which results from the low-energy peaks in the photoelectron energy spectrum [44–47]. Theoretical studies have shown that multiple recollisions are more prevalent at lower laser intensities and shorter wavelengths [48].
In our early paper [24], we explain how the relative intensity ratio of the two colors controls the correlated electron dynamics and optimizes the ionization yields by tracing the history of the recollision trajectories. In this paper, however, we use the three-dimensional classical ensemble method to investigate multiple recollisions of NSDI in a counter-rotating TCCP laser field. We investigate the recollision dynamics in NSDI by counter-rotating TCCP laser fields. We find that the multiple-recollision and single-recollision processes both undergo RESI and RII. The analysis of the NSDI trajectories shows that the single-recollision (SR) NSDI events occur more quickly than the double-recollisioin (DR) NSDI events. The times of recollision are affected by the angle between the momentum and the force of the laser field at the recollision moment.
2. Methods
In order to accurately study NSDI, we need to use full quantum theory. However, solving the time-dependent Schrödinger equation of the multielectron systems in the strong laser fields demands a huge computational condition [29, 49, 50]. Here, we use the classical ensemble method [51, 52] to explore the ionization dynamics of Ar in a counter-rotating TCCP laser field. This method has widespread application in investigating strong-field double ionization processes [31,53,54]. Moreover, we can intuitively present the underlying processes in NSDI by tracing the classical trajectories.
In this method, the evolution of the two-electron system is followed by Newton’s equation of motion (atomic units are used throughout until stated otherwise):
In the above equation, the subscript i is the label of two electrons and ri is the position of the ith electron, and the E(t) = Er(t) + Eb(t) is the counter-rotating TCCP laser field, where the Er(t) is the fundamental (red) laser pulse and the Eb(t) is the second harmonic (blue) laser pulse. The laser pulses are written as
where E0 is the maximum combined electric field amplitude, γE is the electric field amplitude ratio between the second harmonic laser pulse and the fundamental laser pulse, ωr = 0.0576 a.u. (790 nm in wavelength) is the fundamental frequency and ωb = 0.115 a.u. (395 nm in wavelength) is the second harmonic frequency. f(t) is a trapezoidal pulse with a two-cycle turn on, five cycles at full strength, and two-cycle turn off and the φ0 is the random carrier-envelope phase. The potential of the ion-electron interaction and the electron-electron interaction can be given by with the soft core parameters a = 1.5 and b = 0.05 to avoid the autoionization and remove the numerical singularity, respectively [39].The initial conditions for Eq. (1) in our calculation is that evolution ensemble starts from a classical allowance position for the energy of −1.59 a.u.. It approximately equals to the sum of the first and second ionization energy of Ar. The available kinetic energies of two electrons and the directions of the momentum vectors of two electrons are assigned randomly. Then the system is allowed to evolve in the field-free case to obtain stable position and momentum distributions. Once the initial ensemble is ready, the laser field is turned on and all trajectories can be traced in the laser fields. When the energies of both electrons are greater than zero at the end of the laser field, we define it as the double ionization event.
3. Results and discussion
Figure 1(a) shows the total ion momentum distribution by a larger combined electron field amplitude ratio γE = 2.0 (the yield of high-energy rescattered electron is optimized for this ratio) [21] with the combined laser intensity of 0.25 PW/cm2. The electric fields of the counter-rotating TCCP case are shown in the insets of Figs. 1(a)–1(c). We can clear see that the counter-rotating TCCP laser fields have specific dynamical symmetries of the total net electric field. Thus, the distribution exhibits triangle structure symmetrically. Furthermore, it can demonstrate the correlation between two electrons in this laser field. The net momentum of the ion is smaller when the two electrons are emitted in opposite directions than that when the two electrons are emitted in the same direction [55]. The distribution shows a prominent anticorrelated behavior in Fig. 1(a), because the ion momenta are more populated near the origin. This phenomenon has been presented in the experimental observations and theoretical studies [23,24].
Fig. 1 (a) Total ion momentum distribution for NSDI of Ar by the field amplitude ratio γE = 2.0 with the combined laser intensity of 0.25 PW/cm2. (b) Ion momentum distribution for the NSDI but only the SR events occurred. (c) Ion momentum distribution for the NSDI but only the DR events occurred. The insets in (a)–(c) show the electric fields of the counter-rotating TCCP laser field.
In order to demonstrate the correlation of the SR and DR events, we show the ion momentum distribution when only the SR occurs and when only the DR occurs in Figs. 1(b) and 1(c), respectively. Because the inner part of the triangle structure is more populated than the outer part, both of them exhibit prominent anticorrelated behavior. This phenomenon is different from that of Ar by the linearly polarized laser field illustrated [48]. It was demonstrated in [48], Ma et al. that the correlated electron momentum distribution exhibits a prominent anticorrelated phenomenon for the SR events, while it is evenly distributed in the four quadrants for the DR events. The multiple recollisions induced NSDI in linearly polarized laser pulses has been explored, which occurs through double excited states [40,48,56] at low laser intensities.
In order to illustrate the phenomenon of the ion momentum distribution shown in Fig. 1, we trace the energy and distance trajectories of the recollision process in NSDI events. The left column of Fig. 2 shows the time evolution of the two electron energies and the right column shows the time evolution of the distance between the nucleus and each electron. Figures 2(a) and 2(c) show the SR events in the NSDI process. The ionized electron returns to the parent ion and collides with another electron at the first return as shown in Fig. 2(a) and it collides with another electron at the second return as shown in Fig. 2(c). Figures 2(e) and 2(g) show the DR events in the NSDI process, where the first recollision happens at the first return and the second recollision happens at the second [Fig. 2(e)] or other [Fig. 2(g)] return. Moreover, we find out that RESI [as shown in Fig 2(a) and 2(e)] and RII [as shown in Figs. 2(c) and 2(g)] [39,57] can, respectively, occur in SR and DR processes by analyzing the trajectories. In the DR events, however, we define the RESI as the NSDI mechanism that second recollision induces excitation with subsequent ionization mechanism. The time evolution of the distance as shown in the right column of Fig. 2 is in good agreement with that of the electron energy trajectories shown in the left column of Fig. 2.
Fig. 2 Left column: The sample energy trajectories of the two electrons for four different NSDI processes. (a) The RESI mechanism for the SR event; (c) The RII mechanism for the SR event; (e) The RESI mechanism for DR event; (g) The RII mechanism for DR event. Right column: The sample distance trajectories between the nucleus and each electron, which corresponds to the electron energy trajectories in the left column. The pink arrows indicate the recollision occurred.
Figure 3 shows the ion momentum distributions for four different NSDI processes, which is in accordance with the trajectories shown in Fig. 2. Here, we define the recollision time (tr) as the instant when the returning electron goes back to the core area, so that the distance between two electrons is smaller than 2.0 a.u.. The delay time (tD) is defined as the time interval between the recollision time (for the single-recollision) or the second recollision time (for the double-recollision) and final double ionization moment. We define RESI (RII) as the ionization mechanism that the delay time is more than 0.25 o.c. (less than 0.25 o.c.) [58].
Fig. 3 Ion momentum distributions for the NSDI processes with different mechanisms. (a) The RESI mechanism for the SR events; (b) The RII mechanism for the SR events. (c) The RESI mechanism for the DR events; (d) The RII mechanism for the DR events.
For the ion momentum distribution of single-recollision RESI channel [as shown in Fig. 3(a)], the inner part of the triangle structure is more populated than the outer part because both electrons still populate bound states with strong electron-electron repulsion and one electron passes over Coulom barrier at the coming maximum field, while leaving the other one, and releases in the opposite direction until the next maximum field. A similar phenomenon was also analyzed in [56], Hann et al., and in [59], Ye et al..
For the ion momentum distribution of single-recollision RII channel [as shown in Fig. 3(b)], the outer part of the triangle structure is more populated than the inner part because the two electrons are released almost simultaneously in the same instant laser field resulting in the same released direction.
A similar phenomenon is shown in Figs. 3(c) and 3(d) for the ion momentum distributions of double-recollision RESI and RII mechanisms, respectively. For the double-recollision RESI channel, the ion momentum distribution exhibits a prominent anticorrelated phenomenon, while for the double-recollision RII channel, it exhibits a prominent correlated phenomenon.
We perform the statistical analysis by tracing the SR and DR classical trajectories, which is shown in Fig. 4. The single ionization time (tSI) is defined as the instant of one electron achieving positive energy or passing through the nuclear well. The double ionization time (tDI) is defined as the instant of both electrons achieving positive energy after recollision. The travelling time (tr − tSI) is the lag between the recollision and the single ionization time.
Fig. 4 (a) The probability of SR as a function of the traveling time. (b) The probability of the SR (red squares) and DR (blue dots) as a function of the delay time. (c) The probability of the first recollision (black squares) and second recollision (pink dots) in the double-recollision NSDI processes as a function of the interval between the first recollision time and the tunneling ionization time and the second recollision time and the tunneling ionization time. (d) The probability of double-recollision NSDI events as a function of the lag between the second and the first recollision time.
Figure 4(a) shows the probability of only one recollision (single-recollision) as a function of the traveling time (tr − tSI), which displays the highest peak located around 0.37 o.c. (optical cycle) and several lower peaks located around 0.62 o.c. and 1.05 o.c., 1.39 o.c. etc. It means that the recollision occurs at the first, third, fifth, seventh, etc, returns, respectively. According to the recollision trajectory, the recollision electron has the highest energy at the first return [48], but the energy of the third, fifth, seventh, etc, returns are lower than that of the first return. The recollision energies of the second, fourth, etc, returns are too low to induce the double ionization so that the corresponding peaks are suppressed as shown in Fig. 4(a).
We show the probability of the SR and DR as a function of the delay time (tDI − tr) in Fig. 4(b). We found that the probability peak is located around 0.25 o.c. for the SR events and is located around 0.32 o.c. and 0.50 o.c. for the double-recollision events. It is obvious that the delay time of the SR is shorter than that of the DR, meaning that the double ionization of the SR events occur more quickly.
The probability of the first recollision and second recollision in the double-recollision NSDI processes as a function of the travelling time is shown in Fig. 4(c). For the first recollision, the probability peak mainly locates around 0.37 o.c., which means that the recollision occurs at the first return, while for the second recollision it mainly locates around 0.62 o.c., which means that the recollision occurs at the second return.
We show the probability of double-recollision NSDI events as a function of the time between the second and the first recollision in Fig. 4(d). The probability peak locates around 0.25 o.c., which means that the second recollision mainly occurs at next the return after the first recollision.
In order to illustrate the single- and double-recollision in NSDI processes, we demonstrate the probability distribution of the recollision energy by tracing the energy transfer during the recollision processes. The recollision energy (Er (tr − Δt)) is defined as the energy of the recollision electron at the moment Δt = 3a.u. before the recollision tr [48,56]. For the single-recollision NSDI events, the transfer energy is Etrans = Er (tr − Δt). For the double-recollision NSDI events, there are two times energy transfers () and , and the total transfer energy is [48,56].
In Fig. 5 (a), we show the probability distribution of the recollision energy for single-recollision and double-recollision NSDI events, where the recollision energy of the double-recollision NSDI events is the sum of the first and the second recollision energy transfer during the two recollisions processes. We can find that the probability distribution peak for the SR locates around 0.3 a.u. while for the DR it locates around 0.2 a.u.. Thus, energy transfer during the recollision for the SR is slight larger than that for the DR, indicating that the double-recollision NSDI event occurs a little slower than the single-recollision NSDI event, as shown in Fig. 4(b).
Fig. 5 (a) The probability distribution of the recollision energy for the SR (red squares) NSDI events and the probability distribution of the sum of the two recollision energy for the DR (blue dots) NSDI events. (b) The probability distribution of recollision energy of the first (black squares) and the second (pink dots) recollision for the double-recollision NSDI events.
For the double-recollision NSDI events, we show the probability distributions of the first and the second recollision energies in Fig. 5(b). It is obvious that the first energy transfer for the DR trajectories is lower than that for the SR trajectories [the red squares in Fig. 5(a)]. Thus, it indicates that only part of the recollision energy is transferred to the bond system in the first recollision process of double-recollision NSDI events.
Figure 6(a) displays the angle distribution between momentum and the force of the laser field at the recollision time for the single-recollision NSDI events and the double-recollision NSDI events. The angle is mainly distributed at 30° ∼ 60° and 300° ∼ 330°. The angle distribution of the SR (black square line) is smaller than the angle distribution of the first recollision in the DR processes (red circle line), thus the force of the laser field projecting to the momentum direction for the SR events is larger than that for the first recollision of the DR events.
Fig. 6 (a) The angle distribution between momentum and the force of the laser field at the recollision time for the single-recollision NSDI events (black line) and double-recollison NSDI events (red short-dashed line and blue dash-dotted line). (b) The probability distribution of the momentum at the recollision time for the single-recollison NSDI events (black square line) and the double-recollision NSDI events (red circle line and blue up triangle line).
Figure 6(b) displays the momentum distribution at the recollision time for the above-mentioned SR and DR events in NSDI processes. The momentum at the recollision time for the SR event is slight larger than that for the first recollision of DR events. Thus, only one recollision occurrence can result in the single-recollision NSDI event with the smaller angle distribution and the larger momentum.
From Figs. 6(a) and 6(b), we can see that the angle distribution of the second recollison for the DR event is the smallest and the momentum is the largest, thus the second recollision can lead to the NSDI events. We deduce that the angle and the momentum can affect the times of multiple recollisions.
4. Conclusion
In summary, we use the 3D classical ensemble method to investigate the NSDI processes of Ar in the counter-rotating TCCP laser field. By analyzing the NSDI trajectories, we find that not only SR but also DR processes occur in the double ionization events. We also perform statistical analysis of the classical trajectories, which indicates that single-recollision NSDI events occur more quickly than the double-recollisioin NSDI events. Furthermore, the different delay time of the single-recollision (double-recollision) NSDI events result in the different ion momentum distributions. From the statistical analysis of the angle between the momentum and the force of the laser field and the momentum at the recollision moment, we see that this angle can affect the times of the multiple recollisions.
Funding
National Natural Science Foundation of China (Grant No. 61575077).
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