Intensity-dependent two-electron emission dynamics in nonsequential double ionization by counter-rotating two-color circularly polarized laser fields

Cheng Huang; Mingmin Zhong; Zhengmao Wu

doi:10.1364/OE.26.026045

1. Introduction

When an atom or a molecule is exposed in a strong laser field, an electron can be liberated via tunnel ionization. Subsequently the electron is driven by the laser field. Depending on the phase of the laser field at the tunnel instant, the electron may be driven back to the parent ion once or multiple times [1, 2]. The return electron recollides with the parent ion inelastically, resulting in nonsequential double ionization (NSDI) [3–6]. Because of the recollision process, the two electrons involved in NSDI are highly correlated. In the past decades, a great number of experimental [7–12] as well as theoretical [13–24] studies have been performed to explore the correlated dynamics of the electron pairs in NSDI. By recollision the system either immediately releases the second electron or forms a transition excited state (ionic excited state or doubly excited state) with subsequent ionization by the laser electric field. The two pathways are referred to as direct recollision-impact ionization (RII) [7, 25] and recollision-induced excitation with subsequent ionization (RESI) [25–29].

In the continuum, the electron is governed by the laser field. One can steer the electron motion and further control the recollision dynamics by tailoring the time evolution of the electric field vector of the laser pulse. Single few-cycle pulse and parallel two-color pulses only can steer the electron motion in one spatial dimension [30–33]. In recent years, much effort has been devoted to the exquisite two dimensional (2D) combined electric fields [34–37], which can control the electron motion in a 2D plane. One class of such combined electric fields is two-color circularly polarized (TCCP) laser fields [38–42]. The TCCP laser field has been used to generate high-brightness circularly polarized harmonics in the extreme ultraviolet and soft x-ray regions [43], enabling new capabilities for probing magnetic materials and chiral molecules. Recently, the TCCP laser field has also been used to study strong-field above-threshold ionization. It has been shown that counter-rotating TCCP fields can enable electron-ion recollision, but co-rotating TCCP fields can not [44]. Moreover, the counter-rotating TCCP fields allow the tunneling and recollision processes to occur at different angles [45]. The rescattering process is optimized when the ponderomotive energies for two fields are equal [46]. A subcycle interference is experimentally observed in the electron momentum component along the light propagation direction for ionization of He by TCCP laser fields [47]. Three types of photoelectron holographic interferences between the forward scattered and nonscattered trajectories in counter-rotating TCCP fields are well resolved [48]. Recently, Chaloupka et al. studied theoretically double ionization of helium in counter-rotating TCCP laser fields [49]. They explored the dependence of NSDI yields on field amplitude ratios of the two colors and identified four types of recollision trajectories whose contributions depend on field amplitude ratios. The enhancement of double ionization yield and its dependence on field ratios are demonstrated by subsequent experiments for Ar in counter-rotating TCCP laser fields [50, 51]. Multiple-recollision trajectory is demonstrated in NSDI of Ar in counter-rotating TCCP laser fields [52]. The enhanced ionization via the recollision is also observed for N₂ molecule in counter-rotating TCCP laser fields, but it is absent for O₂ molecule [53]. Very recently, co-rotating TCCP laser fields are used to attoclock photoelectron interferometry for probing the phase and the amplitude of emitting wave packets [54].

In this paper, we investigate the intensity dependence of the ultrafast dynamics of the two electrons in NSDI by counter-rotating TCCP laser fields for the field ratio of 1. The laser intensity spans over a range from 6.5×10¹⁴ W/cm² to 1 × 10¹⁶ W/cm². Numerical results show that at high intensity the momentum distribution of the two electrons in the field plane exhibits a double-triangle structure, and at moderate and low intensities the two electrons distribute on the triangle of negative vector potential. Moreover, at the low intensity the population shows a shift to the time evolution direction of negative vector potential relative to the moderate intensity. Back analysis indicates that RESI is always the dominant mechanism and the contribution from double-recollision events on NSDI increases gradually with decreasing laser intensity.

2. The 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 classical models [55, 56] which have been widely recognized as reliable and useful approaches in exploring electron dynamics in NSDI. In this paper, we employ the 3D fully classical ensemble model [57] proposed by Eberly et al. to study the electron dynamics in NSDI by counter-rotating TCCP laser fields. In this model the evolution of the three-particle system is described by the Newton’s equations of motion (atomic units are used throughout until stated otherwise):

\frac{d^{2} r_{i}}{d t^{2}} = - \nabla [V_{n e} (r_{i}) + V_{e e} (r_{1}, r_{2})] - E (t),

where the subscript i=1, 2 is the label of the two electrons and r_i is the coordinate of the i_th electron. The potentials

V_{n e} (r_{i}) = - 2 / \sqrt{r_{i}^{2} + a^{2}}

and

V_{e e} (r_{1}, r_{2}) = 1 / \sqrt{{(r_{1} - r_{2})}^{2} + b^{2}}

represent the ion-electron and electron-electron Coulomb interactions. The softening parameter a=0.825 is introduced here to avoid autoionization. The electron-electron softening parameter b is included primarily for numerical stability and here is set to be 0.05. The electric field of the laser pulse is given by

E (t) = E_{800} f (t) [c o s (ω t) \hat{x} - s i n (ω t) \hat{y}] + E_{400} f (t) [c o s (2 ω t) \hat{x} + s i n (2 ω t) \hat{y}],

where

\hat{x}

and

\hat{y}

are the unit vectors along the x and y directions, respectively. E₈₀₀ and E₄₀₀ are the electric field amplitudes for the 800-nm and 400-nm pulses, respectively. f(t) =sin²(πt/NT) is the envelope of the laser pulse with N being the number of the optical cycle and T being the period of the 800-nm laser field. N is chosen to be 10 in the simulation. The electric field amplitude ratio is chosen to be 1, i.e., the intensities of the two pulses are the same. In this work, our results are reported in terms of the intensity of linearly polarized light corresponding to the same peak field amplitude (E₈₀₀+E₄₀₀).

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 the two 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 double ionization event is determined if both electrons achieve positive energies, where the energy of each electron contains the kinetic energy, potential energy of the electron-ion interaction, and half electron-electron repulsion energy.

3. Results and discussions

Figure 1 shows the combined laser electric field (thin) and the corresponding negative vector potential (thick) at the intensity of 1×10¹⁵ W/cm². The electric field and negative vector potential trace out a trefoil pattern and a triangle, respectively. A lobe of the electric field corresponds to a side of the triangle of the negative vector potential. Each lobe is marked by different colors. The negative vector potentials corresponding to the electric field maxima are located in the middle of each side of the triangle of the negative vector potential, which are indicated by the dots in Fig. 1. According to simple-man model, where the initial momentum at the ionization instant and the effect of the Coulomb potential to the ionized electron are ignored, the final momentum of the ionized electron is equal to the negative vector potential -A(t) at the ionization instant. So the electrons mainly distribute on the the triangle of the negative vector potential.

Fig. 1 The combined laser electric field E(t) (thin) and the corresponding negative vector potential -A(t) (thick) at the intensity of 1×10¹⁵ W/cm². The arrows indicate the time evolution direction. The electric field and negative vector potential trace out a trefoil pattern and a triangle, respectively. A lobe of the electric field corresponds to a side of the triangle of the negative vector potential. The dots mark the field maxima and their negative vector potential.

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Figure 2(a) shows double ionization probabilities of He as a function of the laser intensity by counter-rotating TCCP laser fields. The knee structure, the signature of strong-field NSDI, is clearly shown. NSDI is dominant below 1×10¹⁶ W/cm². In this work we study two-electron emission dynamics in NSDI regime. Figure 2(b)–2(d) show electron momentum distributions in the field plane at intensities of 6.5×10¹⁴ W/cm², 1×10¹⁵ W/cm² and 8×10¹⁵ W/cm². Both the first and the second electron are included. The negative vector potentials -A(t) for the peak intensity of each pulse are shown. For 6.5×10¹⁴ W/cm² and 1×10¹⁵ W/cm², the electrons mainly distribute on the triangle of the negative vector potential. A closer look reveals that at 1×10¹⁵ W/cm² these electrons cluster around the middle of each side of the triangle of the negative vector potential, whereas the population at the low intensity 6.5×10¹⁴ W/cm² deviates from the center and shows a shift to the time evolution direction of negative vector potential. Different from the two lower intensities, at 8×10¹⁵ W/cm², besides some electrons distribute on the triangle of the negative vector potential there are a large number of electrons located inside the triangle of the negative vector potential, which also approximately shows a triangle shape. Overall, the distribution shows a double-triangle structure. It is noteworthy that these momentum distributions from our calculations for He are remarkably different from those distributions from the experimental data for Ar in [51]. The momentum distributions in our calculations are mainly concentrated on the parametric plot of the vector potential, whereas the experimental data shows a three-pronged distribution with a maximum around zero. It may be because that in our calculation the use of higher intensities for He results in a larger acceleration from the laser field [Fig. 1] relative to the experiment in [51]. In this situation the ionized electron is quickly pulled away from the parent ion and thus the effect of Coulomb potential becomes also negligible. So the momentum distributions in our calculations are concentrated on the parametric plot of the vector potential.

Fig. 2 (a) Double ionization probabilities of He as a function of the laser intensity by counter-rotating TCCP laser fields. (b)–(d) Electron momentum distributions in the field plane at intensities of 6.5×10¹⁴ W/cm², 1×10¹⁵ W/cm² and 8×10¹⁵ W/cm². Both the first and the second electron are included. The negative vector potentials -A(t) at the peak intensities of the pulses are shown.

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In order to obtain a deep understanding for two-electron emission dynamics in NSDI by counter-rotating TCCP laser fields and its dependence on the laser intensity, we trace the classical NSDI trajectories and perform statistical analysis. We find that besides single-recollision trajectories there are many double-recollision trajectories [23, 52, 58] contributing to NSDI, where the free electron returns to the parent ion twice and significant energy exchange occurs during the two returns. A sample double-recollision NSDI trajectory for 1×10¹⁵ W/cm² is shown in Fig. 3. Figure 3(a) presents the time evolution of electron distances from the parent ion. Two recollisions are clearly shown, which happen at t=3.02T and t=3.35T respectively. The recollision directions of the two returns are also marked in the 2D electron trajectory shown in Fig. 3(b). Statistical analysis indicates that the proportion of the double-recollision trajectory in NSDI increases sharply with the laser intensity decreasing, as shown in Fig. 4. The tendency is the same as the single linearly polarized pulse [23, 58]. As the intensity drops to 6.5×10¹⁴ W/cm², the contribution of double-recollision trajectory is up to 18%. It is because that at the lower intensity the return electron more likely has smaller energy which is not enough to ionize the second electron, so more than one recollision are required to transfer enough energy to the second electron.

Fig. 3 A sample double-recollision NSDI trajectory shows electron distances from the parent ion versus time (a) and the electrons’ path in the field plane (b) for 1×10¹⁵ W/cm².

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Fig. 4 Proportion of double-recollision trajectory as a function of the laser intensity.

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After the last recollision, the two electrons may quickly ionize (i.e., achieve positive energy) or be delayed some time and then released near subsequent field maxima. Based on the final ionization order after recollision the two electrons are defined as the first and the second electron. Their final ionization times are denoted by t_i₁ and t_i₂ respectively. The delay time is defined as the time interval between the final ionization of each electron and the recollision (the second recollision for the double-recollision trajectory). The distributions of the delay time of the two electrons are shown in Fig. 5 for 6.5×10¹⁴ W/cm² (a), 1×10¹⁵ W/cm² (b) and 8×10¹⁵ W/cm² (c). At the high intensity 8×10¹⁵ W/cm² and the moderate intensity 1×10¹⁵ W/cm², the first electrons are mainly ionized within 0.1T after recollision. For the second electrons, except a small part quickly released within 0.1T after recollision, most of them are delayed longer time to ionize. It indicates that RESI is dominant at these intensities. One can see that there are three ionization bursts for the second electrons every cycle which corresponds to the fact that there exist three field maxima every cycle. By comparing Fig. 5(b) and 5(c), it is easy to see that more electrons are delayed longer time for 8×10¹⁵ W/cm² than for 1×10¹⁵ W/cm². At the low intensity 6.5×10¹⁴ W/cm², the distribution is remarkably different from the high and moderate intensities. The distribution exhibits two vertical bands, which represent two final ionization times of the first electrons. It means that the first electrons either ionize quickly after recollision or with a delay of 0.28T. Almost all the second electrons have a considerable time delay with respect to the recollision. So RESI is the dominant mechanism at the low intensity. The left band corresponds to an ion excited state and the right band corresponds to a doubly excited state. Furthermore, we present the intensity dependence of the proportions of RII and RESI mechanisms in Fig. 6. Here we define RESI (RII) as the ionization mechanism when the delay time of the second electron t_i₂-t_r is more than 0.2T (less than 0.2T). It is obvious that in the intensity range considered in this work, RESI is always the dominant mechanism. The contribution of RII channel firstly increases and then decreases. It reaches the largest value at 1×10¹⁵ W/cm².

Fig. 5 Distributions of delay time between final ionizations of the first and second electrons and the recollision for 6.5×10¹⁴ W/cm² (a), 1×10¹⁵ W/cm² (b) and 8×10¹⁵ W/cm² (c). The two electrons are numbered based on the final ionization order after recollision.

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Fig. 6 Proportions of RII and RESI mechanisms as a function the laser intensity.

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Figure 7 shows the distributions of single ionization time, recollision time, final ionization times of the first and second electrons after recollision from up to down for 8×10¹⁵ W/cm² (left column) and 1×10¹⁵ W/cm² (right column). For 8×10¹⁵ W/cm², the single ionization occurs around t=1.67T, 2.0T and 2.33T [Fig. 7(a)], i.e., the single ionization reaches saturation at the turn-on stage of the laser pulse. After a short traveling time of roughly 0.2T [Fig. 7(b)], the free electron is driven back to collide with the parent ion. This type of short excursion trajectory has been illustrated in Fig. 4(a) and 4(b) in [49]. After recollision, the return electron still keeps free [Fig. 7(c)]. A small part of the second electrons are ionized immediately, but most of the second electrons are delayed to ionize near subsequent field maxima, even being delayed to t=6T. The maximal ionization probability of the second electrons is at 4.3T which is very close to the peak of the pulse. For the relatively low laser intensity 1×10¹⁵ W/cm², one can see that the maximal ionization probability of the first electrons is at t=4.4T. It is 4.7T for the second electrons. The above discussion indicates that for the relatively low laser intensity after recollision both electrons are released around the peak of the pulse. Ignoring the initial momentum and the effect of Coulomb potential on the ionized electron, the electron final momentum is equal to the acceleration of the electric field after ionization. So for the relatively low laser intensity both electrons follow the triangle of the negative vector potential at the peak intensity of the pulse. However, for the high intensity, the laser intensities at which the two electrons are released are considerable different. So the two electrons are not possible to distribute on the same the triangle of the negative vector potential. In Fig. 8, we separately present the momentum distributions of the first electrons (a) and the second electrons (b). The triangles of the negative vector potential for the intensity at t=2.3T and the peak intensity of the pulse are also shown in Fig. 8(a) and 8(b) respectively. It is obvious that the first electrons well distributed on the triangle of the negative vector potential for the intensity at t=2.3T, which is the ionization time of the first electrons. The population of the second electrons is well on the triangle of the negative vector potential for the peak intensity of the pulse. It means that the origin of the double-triangle structure is due to the different values of the laser intensity at which the two electrons are released after recollision. After a closer examination, one can see that the first electrons do not distribute in the middle of the side of the triangle of the negative vector potential and there is a shift opposite to the time evolution direction [Fig. 8(a)]. It means that after recollision the first electron is liberated before the maximum of the field. It is because that at high intensity the single ionization occurs in the early rising edge of the pulse, where the electric field which pulls the free electron away from the parent ion is much smaller than the electric field which drives it back [Fig. 7(a)]. So the free electron returns and collides with the parent ion before the maximum of the field [Fig. 7(b)]. As a result, after recollision the first electron is liberated before the maximum of the field [Fig. 7(c)].

Fig. 7 Single ionization time, recollision time, final ionization times of the first and second electrons after recollision are shown from up to down for 8×10¹⁵ W/cm² (left column) and 1×10¹⁵ W/cm² (right column).

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Fig. 8 Momentum distributions of the first electrons (a) and the second electrons (b) for 8×10¹⁵ W/cm².

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Now we turn to the electron momentum distribution at the low intensity (6.5×10¹⁴ W/cm²). The distribution in Fig. 2(b) shows a shift deviating from the middle of the side of the triangle of the negative vector potential. In Fig. 9 we separately present the momentum distributions of the first electrons (a) and the second electrons (b) for the low intensity 6.5×10¹⁴ W/cm². One can see that the momentum distribution of the first electrons shows three parts with a shift deviating from the middle of the side of the triangle of the negative vector potential, which is similar to the total distribution of the two electrons. The distribution of the second electrons almost is uniform following the triangle of the negative vector potential. Figure 10 shows recollision time (a), final ionization times of the first (b) and second (c) electrons after recollision for 6.5×10¹⁴ W/cm². We can see that the recollision mainly occurs at the maxima of the electric field [Fig. 10(a)]. The first electrons are ionized with a short time delay after the recollision. Because the first electrons are ionized within a narrow time window after the field maxima [Fig. 10(b)], they obtain an acceleration deviating from the middle of the side of the triangle of the negative vector potential, showing a shift to the time evolution direction. However, the second electrons are released within a broad time range for each field maximum, resulting in a approximately uniform distribution on the triangle of the negative vector potential. For 1×10¹⁵ W/cm² the situation is different. The first electrons are ionized at the field maxima [Fig. 7(g)] and the second electrons are released in a broad time window [Fig. 7(h)], resulting in the momentum distribution of the two electrons shows a maximum at the middle of each side of the triangle of the negative vector potential.

Fig. 9 Momentum distributions of the first electrons (a) and the second electrons (b) for 6.5×10¹⁴ W/cm².

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Fig. 10 Recollision time (a), final ionization times of the first (b) and second (c) electrons after recollision for 6.5×10¹⁴ W/cm².

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In addition, in Fig. 10 the main peaks of the ionization times of the first and the second electrons are accompanied by some secondary peaks. At the low intensity, RESI mechanism is dominant and there also exist some NSDI events from RII mechanism. The distribution of ionization time of the second electrons from RESI mechanism shows a broad main peak. Those RII events result in a secondary peak after the maximum of the field. Furthermore, in RESI mechanism, there are two ionization pathways. One is that after recollision the first electron is liberated immediately, leaving the second electron in an ion excited state. By this pathway, the first electron is ionized after the maximum of the field [the main peak in Fig. 10(b)]. The other pathway is that after recollision the return electron is captured and a doubly excited state is formed. In this pathway, the first electron is delayed to ionize before the next field maximum [the secondary peak in Fig. 10(b)].

4. Conclusion

In conclusion, we have investigated the emission dynamics of the two electrons in NSDI by counter-rotating TCCP laser fields at different laser intensities. At moderate intensity, the momentum distribution of the two electrons shows a maximum at the middle of each side of the triangle of the negative vector potential. At high intensity, the momentum distribution shows a double-triangle structure, the inner and outer triangles correspond to the former ionization and later ionization electrons respectively. At low intensity, the momentum distribution shows a shift deviating from the middle of the side of the triangle of the negative vector potential. It is because that the first electrons are ionized within a narrow time window after the field maximum. In addition, at low intensity, double-recollision trajectories and NSDI events from doubly excited state induced by recollision are also demonstrated.

Funding

National Natural Science Foundation of China (NSFC) (11504302, 61178011, 61475127, 61475132).

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Intensity-dependent two-electron emission dynamics in nonsequential double ionization by counter-rotating two-color circularly polarized laser fields

Abstract

1. Introduction

2. The classical ensemble model

3. Results and discussions

4. Conclusion

Funding

References

Cited By

Figures (10)

Equations (2)

Optics Express