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We have investigated the correlated electron dynamics in nonsequential double ionization (NSDI) of helium by the orthogonally polarized two-color pulses that consisted of an 800-nm and a 400-nm laser fields using the classical ensemble model. Depending on the relative phase of the two-color field, the electron momentum distributions along the polarization direction of the 800-nm field exhibit a surprisingly strong anticorrelated or correlated behavior. Back analysis reveals that recollisions eventually leading to NSDI are concentrated in a time window as short as several hundreds attoseconds with this scheme. By changing the relative phase of the two-color field, the revisit time of recolliding electron wave packet has been controlled with attosecond precision, which is responsible for the various correlated behaviors of the two electrons. Our results reveal that the orthogonally polarized two-color field can serve as a powerful tool to control the correlated electron dynamics in NSDI.
© 2011 Optical Society of America
1. Introduction
Atoms or molecules exposed to an intense femtosecond laser pulse suffer tunneling ionization. The ionized electron is accelerated in the laser field and can be pulled back by the laser force to collide with the parent ion. This recolliding process plays the center role in the attosecond physics [1]. The recolliding electron wave packet can be exploited to produce attosecond (as) extreme ultraviolet pulses [2], to obtain an image of the electron orbital [3], or to probe molecular dynamics with attosecond resolution [4]. The improvements in these areas are based on the enhanced controlling of the recolliding electron wave packet, which is realized by the rapid advance of optical technology. For instance, with the waveform-controlled sub-1.5-cycle near-infrared laser, an isolated sub-100-as extreme ultraviolet pulse is obtained [5]. The control of the recolliding electron wave packet is a hot topic of current research.
Nonsequential double ionization (NSDI), in which the second electron is ionized through the recollision of the first tunneled electron that driven back by the laser field [6], has been widely investigated because it reveals a highly correlated electron behavior [7–10]. As an important strong-field recollision process, the control on the recolliding electron wave packet can unambiguously manipulate the electron correlations in NSDI. It has been reported that with the few-cycle pulses the momentum distributions of the doubly ionized ions can be directed by the carrier envelope phase [11]. Recently, we have demonstrated that in the linearly polarized two-color few-cycle pulses the correlated electron momentum reveals a novel arc-like structure [12].
The orthogonally polarized two-color field, as an efficient tool to control the electron dynamics in strong-field laser-matter interaction, has been widely investigated. For instance, it has been demonstrated that with the orthogonally polarized two-color laser field, the birth and recolliding angles, the temporal structure of the recolliding electron wave packet can be precisely controlled [13]. Recently, via manipulating the recollision angle of the recolliding electron wave packet with the orthogonally polarized two-color field, the atomic orbital symmetry [14] as well as the molecular orbital symmetry [15] is clearly probed. In this paper, we employ the orthogonally polarized two-color pulse that consisted of an 800-nm and a 400-nm laser fields to control the recolliding electron wave packet in NSDI. Compared to the linear single-color case, only a selective part of electron wave packet can be driven back to collide with the parent ion in the orthogonal two-color field. Thus, the recollisions time window is significantly compressed. Furthermore, by changing the relative phase of the two fields, the revisit time of recolliding electron wave packet can be controlled with attosecond precision. Based on this precise control of the revisit time of recolliding electron wave packet, the two-electron momentum distributions along the polarization of the 800-nm field are manipulated to exhibit a surprisingly strong anticorrelated or correlated behavior.
2. The classical ensemble model
We employ the classical model that proposed by Haan and Eberly et al and it has been detailedly described in Refs. [16, 17]. In this study we restrict the electrons to motion in the x–y polarization plane for simplicity because the out-of-plane effects are negligible [18]. The evolution of the two-electron system is determined by the Newton’s classical motion equations (atomic units are used throughout this paper unless otherwise stated):
where the subscript i = 1, 2 is the electron label. Ex(t) and Ey(t) are the x and y components of the electric field, respectively. To obtain the initial values, the ensemble is populated starting from a classically allowed position for the helium ground-state energy of −2.9035 a.u. The available kinetic energy is distributed between the two electrons randomly in momentum space. Then the electrons are allowed to evolve a sufficient long time (100 a.u.) in the absence of the laser field to obtain stable position and momentum distributions [19, 20]. To avoid autoionization, we set the screening parameter a2 to be 0.75. b2 is set to be 0.01. In our calculations, the orthogonally polarized two-color field is composed by an 800-nm field which is linearly polarized along x axis and a 400-nm field which is linearly polarized along y axis. The synthetical electric field is written as: E(t) = f(t)[Ex(t)x̂ + Ey(t)ŷ]. x̂ and ŷ are the polarization vectors. f(t) is the laser envelope, which turns on and turns off linearly during two optical cycles of the 800-nm field and keeps full strength for four cycles of the 800-nm field. Ex(t) = Ex0cos(ωxt) and Ey(t) = Ey0cos[κ(ωxt + ϕ)], where κ = 2, Ex0, ωx are the amplitude and frequency of the 800-nm field, respectively. Ey0 is the amplitude of the 400-nm field. ϕ is the relative phase between the 800-nm and 400-nm fields. In this paper, both intensities of the 800-nm and 400-nm fields are set to be 4.0 ×1014 W/cm2.3. Results and discussions
Figure 1 shows the correlated electron momentum distributions along x axis, i.e., the polarization direction of the 800-nm field. Figure 1(a) displays the correlated electron momentum from NSDI by the linearly polarized 800-nm field. While figs. 1(b)–1(f) represent the spectra for the orthogonally polarized two-color fields, where the relative phases of the two fields are 0.25π, 0.3π, 0.35π, 0.4π and 0.45π, respectively. For the 800-nm field, in accord with previous results [7, 8], the correlated momenta are clustered in the two circular regions in the first and third quadrants, meaning the parallel emission of the two electrons. While for the two-color field with ϕ = 0.25π, the correlated electron momenta reveal a surprising anticorrelated behavior, i.e., the two electrons escape with opposite momenta. Though the antiparallel emission has been reported in NSDI of Ar by the linearly polarized field at intensity below recollision threshold in Ref. [21], the origin of anticorrelation in the orthogonal two-color field is completely different from that in Ref. [21], as will be shown below. When ϕ changes to 0.3π, the electron momenta also exhibits overall maxima in the second and fourth quadrants. However, the distribution extends to the first and third quadrants and the anticorrelation is not as strong as that for ϕ = 0.25π. When ϕ = 0.35π, 0.4π and 0.45π, the electron momentum spectra are dominated by correlated emission, i.e., the two electrons escape into the same hemisphere.
Fig. 1 Correlated electron momentum distributions along the polarization direction of the 800-nm field for NSDI of helium (a) by the 800-nm field, and (b)–(f) by the orthogonally polarized two-color fields with relative phase ϕ = 0.25π, 0.3π, 0.35π, 0.4π, 0.45π, respectively. The ensemble sizes are 2 millions.
In Fig2 we display the correlated electron momentum distributions along the y axis. For the 800-nm filed, the distribution reveals obvious repulsion behavior, which is consistent with previous study [8]. While for the orthogonal two-color field with ϕ = 0.25π, the momenta are distributed in a circular region in the first quadrant, meaning that both electrons emit to the positive direction with similar momenta. When ϕ changes to 0.3π, the events extend to the second and fourth quadrants. When ϕ = 0.35π, the distribution shows two maxima in the second and fourth quadrants. When ϕ increases further, the two maxima move toward the third quadrant.
Fig. 2 The same as Fig. 1 but for the electron momentum along the polarization direction of the 400-nm field.
The results above mean that the electron correlations are strongly depend on the relative phase of the orthogonally polarized two-color pulses. In order to understand how the two-color field manipulates the correlated dynamics in NSDI, we trace back the history of the two-electron trajectories. We find out the recollision time of the double ionization (DI) trajectories, where the recollision time is defined as the instant of the closest approach after the first departure of one electron from the core [16, 19]. In Fig. 3, we present the counts of DI events versus laser phase at recollision. For the 800-nm field, recollisions that eventually leading to NSDI occur over a wide range of laser phase. However, for the two-color fields, the recollisions are clustered in a much narrower time window, which is as short as 300 attoseconds. The pronounced suppression of the recollision time window results from the fact that only selective parts of trajectories can fulfill the recollision conditions in the two dimensions. This characteristic implies that the recollision dynamics in NSDI can be explored with a much high time resolution by the orthogonally polarized two-color field.
Fig. 3 Counts of DI events versus laser phase at recollision. Plots (a)–(f) correspond to the events in figs. 1(a)–1(f), respectively. The solid green curve and dashed black curve represent the electric fields of the 800-nm and 400-nm pulses, respectively. T1 is the laser cycle of the 800-nm field.
Further investigation of Fig. 3 reveals that the revisit time of the recolliding electron wave packet changes with variation of the relative phase. For ϕ = 0.25π, recollisions take place just before the maximum of the 800-nm field. When ϕ changes from 0.25π to 0.45π, the revisit time of the recolliding electron wave packet moves gradually toward the zero crossing of the 800-nm field. This variation is more clearly seen in Fig. 4(a), where only the bunch of electron current that returns to the core around 3.5T1 (T1 is the laser period of the 800-nm field) is counted. Obviously, the revisit time of the recolliding electron wave packet has been controlled with attosecond resolution by adjusting the relative phase of the two-color field.
Fig. 4 (a) Counts of DI events versus laser phase at recollision. Only the bunch of electron current that revisits the core around 3.5T1 is counted. The solid green curves donate the electric field of the 800-nm laser. The dashed lines are added to guide the eye. (b) Energy structure of the recolliding electron wave packets in the 800-nm field. (c) The same as (b) but for the orthogonal two-color field with ϕ = 0.45π.
Now we explain the responsible process for the correlated electron momentum distributions in Figs. 1 and 2. As is well known, an electron ionizes to an oscillatory field E0cos(ωt) at time ti with initial velocity υ0 will achieve final momentum υf = υ0 − (E0/ω)sin(ωti). In the linear 800-nm field, most of recollisions occur around the zero crossing of the electric field (see Fig. 3(a)). If the two electrons emit simultaneously after recollision, the final momenta of both electrons are mainly determined by the latter term [(E0/ω)sin(ωti)] due to the fact that the latter term is often larger than the former term υ0. Consequently, the two electrons emit into the same hemisphere, resulting in the correlated behavior in Fig. 1(a).
In the two-color field with ϕ = 0.25π, recollisions occur just before the maximum of the 800-nm field (Fig. 3(b)). As a result, the latter term is negligible and the final momenta of the two electrons along the polarization of the 800-nm field are determined by υ0. Thus the final momentum distribution represents the correlated behavior of υ0. The antiparallel emission in Fig. 1(b) implies that the two electrons often obtain opposite momenta during the recollision, which might be due to the electron repulsion. When ϕ increases, the revisit time of the recolliding wave packet moves gradually from the maximum toward the zero crossing of the electric field (see Fig. 4(a)). Thus the latter part [(E0/ω)sin(ωti)] becomes more important and the final momenta are determined by the competition of the two terms: the former term υ0 drives the two electron to emit into the opposite hemispheres while the latter term pushes them into the same hemisphere. When ϕ = 0.45π, most of the recollisions occur around the field zero (see Fig. 3(f)), thus the latter term dominates the final electron momenta. Consequently, the correlated electron momentum distribution is clustered along the main diagonal, while the off-diagonal detail mainly results from the different initial velocities υ0 of the two electrons [22]. In the single-color 800-nm field, recollisions occur over a wide range of laser phase. Thus various correlated patterns coincidently contribute to the electron momentum distribution, which complicates the correlation behaviors. While in the orthogonal two-color field, recollisions are concentrated in a very short time window, which makes the correlated behavior much stronger.
The origin of the correlation behaviors along polarization direction of the 400-nm filed is similar as that in the polarization direction of the 800-nm field. For ϕ = 0.25π, the laser phase of the 400-nm field at recollision is zero, and thus the two electrons achieve the same final momentum which is determined by the electric field. For ϕ = 0.35π, recollision occurs at the maximum of the 400-nm field, thus the final momenta are determined by υ0, which are opposite for the two electrons. One feature should be mentioned that in figs. 2(b)–2(f) the distributions are asymmetric with respect to the diagonal P1y + P2y = 0. It is well known that the momentum distribution will be symmetric with respect to the diagonal P1y + P2y = 0 if the recollision currents repeat every half optical cycle. However, in the orthogonal two-color field the repetition period is one optical cycle of the 400-nm field (see figs. 3(b)–3(f)), not half optical cycle, Thus, the momentum distributions along the polarization of the 400-nm field are asymmetric with respective to the diagonal P1y + P2y = 0.
The above analysis illustrates that the manipulation on the electron correlation of NSDI is based on the precise control of the recollision time. The orthogonal two-color field can significantly suppress the recollision time window. Moreover, the revisit time of the recolliding electron wave packet can be controlled with attosecond resolution, which results in the surprisingly strong anticorrelation and correlation in the electron momentum distributions. Another interesting characteristic of the recolliding electron wave packet caused by the orthogonal two-color field is its energy structure, as shown in figs. 4(b) and 4(c). In the 800-nm single-color field, the recollision energy exhibits a very wide distribution (Fig. 4(b)). While in the orthogonal two-color field, the distribution shows a sharp peak. As is well known, the electron dynamics in NSDI strongly depends on the recollision energy [22, 23]. In the single-color field, it is difficult to perform a clean investigation to separate the various electron dynamics at different recollision energies because of the wide energy band of the recolliding electron wave packet. However, the narrow energy band of the recolliding electron wave packet in Fig. 4(c) implies that the orthogonally polarized two-color field can provide a potential tool to investigate the complex electron dynamics at various recollision energies in a clean way.
4. Conclusion
In summary, we have investigated the correlated electron dynamics in NSDI of helium by the orthogonally polarized two-color pulses. In this field, the correlated momentum patterns of the two electrons along the polarization of the 800-nm field exhibit a surprisingly strong anticor-related or correlated behavior, which can be manipulated by changing the relative phase of the two-color field. Back analysis shows that the recollision time window has been significantly suppressed by the orthogonal two-color field and the revisit time of the recolliding electron wave packet can be controlled with attosecond precision by changing the relative phase of the two-color field. The precise control of the temporal characteristic of the recolliding electron wave packet is responsible for the surprisingly strong anticorrelated and correlated behaviors. Additionally, the narrow energy band of the recolliding electron wave packet implies that the orthogonal two-color field can serve as a potential tool to explore the complex sub-laser-cycle dynamics at various recollision energies in a clean way.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant No. 10774054, National Science Fund for Distinguished Young Scholars under Grant No. 60925021, and the 973 Program of China under Grant No. 2011CB808103. This work was partially supported by the State Key Laboratory of Precision Spectroscopy of Huadong Normal university.
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