Nonsequential double ionization by co-rotating two-color circularly polarized laser fields

Cheng Huang; Mingmin Zhong; Zhengmao Wu

doi:10.1364/OE.27.007616

1. Introduction

An electron can tunnel through the suppressed potential barrier of atoms and molecules by a strong laser field. Then the ionized electron may be driven back by the subsequent laser field and collide with the parent ion [1, 2], which can trigger a broad range of nonlinear phenomena, such as high harmonic generation (HHG), high-order above threshold ionization (ATI) and nonsequential double ionization (NSDI) [3–6]. In NSDI, the two electrons induced by recollision are highly correlated, which provides a simple model for the study of correlated electron dynamics and thus attracted many experimental [7–12] and theoretical attentions [13–21] in the past decades. Direct recollision-impact ionization (RII) [7, 22] and recollision-induced excitation with subsequent ionization (RESI) [22–26] have been well known as the two ionization channels of NSDI.

Recollision, the heart of NSDI, is strongly dependent on the time evolution of the electric field vector. Single few-cycle pulse and parallel two-color pulses are used to steer the recollision trajectory and control the phase of laser field and the return energy at the recollision instant [27–30]. In recent years, much effort has been devoted to the exquisite two dimensional (2D) combined electric fields [31–33], in which the ionized electron can return back to the parent ion from different angles. And the return angle can be well controlled by changing the parameters of those components constituting the combined electric fields. One class of such combined electric fields is the orthogonal two-color (OTC) pulses consisting of a fundamental and a second harmonic linearly polarized pulses, which has been used to control the photoelectron interference in ATI [34] and correlation behaviors of the electron pairs in NSDI [35, 36]. Another class is two-color circularly polarized (TCCP) laser fields [37–41]. The TCCP laser field can generate high-brightness circularly polarized harmonics in the extreme ultraviolet and soft x-ray regions [42], enabling new capabilities for probing magnetic materials and chiral molecules. Recently, the above-threshold ionization in counter-rotating TCCP fields has been studied [43, 44] and the rescattering process is optimized when the ponderomotive energies for two fields are equal [45]. A subcycle interference is experimentally observed in the electron momentum component along the light propagation direction for ionization of He by counter-rotating TCCP laser fields [46]. Three types of photoelectron holographic interferences in counter-rotating TCCP fields are well resolved [47]. Co-rotating TCCP laser fields have been used to attoclock photoelectron interferometry for probing the phase and the amplitude of emitting wave packets [48]. Co-rotating and counter-rotating TCCP laser fields have been compared to show that the nonadiabatic offset of the initial electron momentum distribution in strong field tunneling ionization of argon [49].

Recently, the study of NSDI is extended to He atom in counter-rotating TCCP laser fields by Chaloupka $e t a l$ [50]. Four types of recollision trajectories are identified and their contributions depend on field amplitude ratios. Subsequently,the enhancement of double ionization yield and its dependence on field ratios are demonstrated by experiments for Ar in counter-rotating TCCP laser fields [51, 52]. Double-recollision trajectory is demonstrated in NSDI of Ar in counter-rotating TCCP laser fields [53]. Intensity dependence of proportion of double-recollision trajectory and momentum distributions of the two electrons from NSDI of He in counter-rotating TCCP laser fields are also studied [54]. Very recently, the angular distributions of the correlated electrons in NSDI of Mg in counter-rotating TCCP laser fields are discussed [55]. For molecules, electron recollision assisted enhanced ionization is experimentally observed for N $_{2}$ molecule in counter-rotating TCCP laser fields, but it is almost absent for O $_{2}$ molecule [56] owing to the antisymmetric profile of the outermost orbital.

All of these studies on NSDI above use counter-rotating TCCP laser fields. It is because that probability of recollision in co-rotating TCCP fields is extremely low [39, 43, 45]. But HHG induced by recollision in co-rotating TCCP fields have been experimentally observed [39]. In this paper, we focus on NSDI of Ar induced by co-rotating TCCP fields. Our numerical results indicate that co-rotating TCCP fields can induce NSDI by recollision process. But the yield is an order of magnitude lower than counter-rotating case. Moreover, NSDI yield in co-rotating TCCP fields strongly depends on field ratio of the two colors and achieves its maximum at a ratio of 2.4. The short recollision trajectory with traveling time smaller than one cycle is dominant. The electron momentum distributions show a shift in time evolution direction of the negative vector potential with the field ratio increasing. It is attributed to the shift of recollision time with the field ratio.

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 decades numerous studies have resorted to classical models [57, 58] 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 [59] proposed by Eberly $e t a l$ . to study the electron dynamics in NSDI by 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

_{t h}

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=1.5 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_{1600} f (t) [c o s (ω t) \hat{x} - s i n (ω t) \hat{y}] + E_{800} f (t) [c o s (2 ω t) \hat{x} \pm 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. E

_{1600}

and E

_{800}

are the electric field amplitudes for the 1600-nm and 800-nm pulses, respectively.

f (t)

=sin

^{2}

(

π t / N T

) is the envelope of the laser pulse with N being the number of the optical cycle and T being the period of the 1600-nm laser field. N is chosen to be 10 in the simulation. In this work, our results are reported in terms of the intensity of linearly polarized light corresponding to the same peak field amplitude (E

_{1600}

+E

_{800}

).

To obtain the initial conditions for Eq. (1), the ensemble is populated starting from a classically allowed position for the energy of -1.59 a.u., corresponding to the sum of the first and second ionization potentials of Ar. 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 endof 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.

Fig. 1 Double ionization probabilities of Ar as a function of the laser intensity by co- (a) and counter-rotating (b) TCCP laser fields with different field amplitude ratios.

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3. Results and discussions

Figure 1 shows double ionization probabilities of Ar as a function of the laser intensity by co- (a) and counter-rotating (b) TCCP laser fields with field amplitude ratios 0.8, 1, 2, 3, 4 and 5. For counter-rotating TCCP laser fields as shown in Fig. 1(b), the knee structure is present for field amplitude ratios from 0.8 to 4. It means that recollision and NSDI is dominant for field amplitude ratios between 0.8 and 4. More interestingly, the clear knee structure is also observed for co-rotating TCCP laser fields as shown in Fig. 1(a). It is present only for field amplitude ratios 2 and 3. Although the prominent enhanced ionization in co-rotating TCCP laser fields occurs in a much smaller range of field amplitude ratio and the ionization probability is an order of magnitude lower than that from counter-rotating case, these results demonstrate that recollision and NSDI can occur in co-rotating TCCP laser fields.

Fig. 2 (a) Double ionization probability versus field amplitude ratio for co- (blue curve) and counter-rotating (red curve) TCCP laser fields at 2 × 10¹⁴ W/cm². For visual convenience, those values from co-rotating TCCP fields have been multiplied by a factor of 45. Panels (b) and (c) show the combined laser electric field E(t) (dotted curve) and the corresponding negative vector potential -A(t) (solid curve) for co- and counter-rotating TCCP laser fields with field ratio 2.0 and intensity 2 × 10¹⁴ W/cm². The arrows indicate the time evolution direction. The black dots mark a field maximum and its negative vector potential.

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Figure 2(a) shows double ionization probability versus field amplitude ratio for co- (blue curve) and counter-rotating (red curve) TCCP laser fields at 2×10 $^{14}$ W/cm $^{2}$ . The ionization probability from co-rotating TCCP fields is much smaller than that from counter-rotating fields. For visual convenience, those values from co-rotating TCCP fields have been multiplied by a factor of 45. The double ionization probability in counter-rotating fields is strongly dependent on the field ratio, which has been reported in previous studies [50–52]. For 1600+800nm and 2×10 $^{14}$ W/cm $^{2}$ , NSDI in counter-rotating fields achieves the highest value at a ratio of 1.2. There exists a tiny deviation from previous experimental studies on NSDI of Ar in 800+400nm fields, where NSDI yield peaks at the ratio of 1.4 [51, 52]. The tiny deviation may originate from the use of longer wavelength (1600+800nm) in our work. The probability tends to become zero for very small or very large field ratio because the combined fields will tend to become circularly polarized laser fields, where recollision is significantly suppressed. Furthermore, our result in Fig. 2(a) indicates that in co-rotating fields double ionization probability also strongly depends on the field ratio. It reaches the maximum at a ratio of 2.4.

Figures 2(b) and 2(c) show the combined laser electric field E(t) (dotted curve) and the corresponding negative vector potential -A(t) (solid curve) for co- and counter-rotating TCCP laser fields with field ratio 2.0. The arrows indicate the time evolution direction. The black dots mark a field maximum and its negative vector potential. For counter-rotating TCCP fields, the electric field and negative vector potential both have three maxima in one cycle and threefold symmetry. The maximal electric field corresponds to the minimal vector potential. For co-rotating TCCP fields, there is only a maximal electric field value and its corresponding vector potential is also a maximum.

Fig. 3 Traveling times (left column) and delay times between double ionization and recollision (right column) for field ratios 1.8 (a,b), 2.4 (c,d) and 3.0 (e,f) at 2 × 10¹⁴ W/cm².

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In order to obtain a deep understanding for ultrafast dynamics of NSDI in co-rotating TCCP fields and its dependence on the field ratio, we trace the classical NSDI trajectories and find that NSDI still proceeds by recollision. All NSDI trajectories only experience one recollision. It is different from the counter-rotating case, where double-recollision trajectory has a non-negligible contribution for NSDI. To perform statistical analysis, we find out the single ionization time t $_{S I}$ , the recollision time t $_{R}$ and the double ionization time t $_{D I}$ . Here, the single ionization time is defined as the instant when one electron achieves positive energy or is outside the nuclear well. The double ionization time is defined as the instant when both electrons achieve positive energies. The recollision time is defined as the instant of the closest approach after the first departure of one electron from the parent ion. Based on the final ionization order after recollision the two electrons are defined as the first and the second electron.

Figure 3 shows traveling times (left column) and delay times between double ionization and recollision (right column) for field ratios 1.8 (a,b), 2.4 (c,d) and 3.0 (e,f) at 2×10 $^{14}$ W/cm $^{2}$ . From Figs. 3(a), 3(c) and 3(e) one can see that for most of NSDI, after ionization the free electron experiences an excursion with a traveling time smaller than one cycle before recollision, we can refer them as short recollision trajectory (SRT). In addition, a close examination can find that there are some events with traveling time longer than one cycle, which is called long recollision trajectory (LRT). The contribution from LRT is much smaller than that from SRT. According to our statistics, the proportions of LRT in NSDI are respectively 32%, 12% and 18% for field ratio 1.8, 2.4 and 3.0. After recollision the first electron is ionized quickly. The second electron may be ionized quickly or be promoted to an excited state and then be released by the laser field at a later time. The two pathways are referred to as direct recollision-impact ionization (RII) and recollision-induced excitation with subsequent ionization (RESI). The distributions of the delay times between double ionization and recollision as shown in right column of Fig. 3 indicate RII and RESI mechanisms are comparable for field ratios 1.8, 2.4 and 3.0. Here we define RESI (RII) as the ionization mechanism when the delay time between double ionization and recollision is more than 0.25T (less than 0.25T). Statistics shows that the proportions of RESI in NSDI are respectively 41%, 53% and 58% for field ratios 1.8, 2.4 and 3.0. For the three field ratios, the proportions of RESI in LRT events are respectively 77%, 23% and 32%, and the proportions of RESI in SRT events are respectively 54%, 26% and 44%.

Fig. 4 Two sample NSDI trajectories show electron energies versus time (a, d), electron distances from the parent ion versus time (b, e) and the electrons’ path in the field plane (c, f) for field ratio 2.4. The arrows indicate the time evolution direction.

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Two sample NSDI trajectories in Fig. 4 show electron energies versus time (a, d), electron distances from the parent ion versus time (b, e) and the electrons’ path in the field plane (c, f) for field ratio 2.4. The two trajectories intuitively show motionpaths of the two electrons and the recollision process in co-corotating TCCP fields is unambiguously demonstrated. The traveling time of the trajectory shown in the first row is about 0.57T. It is a SRT. After recollision the returning electron keeps free and the second electron is excited and ionized with a delay time of 2T [see Figs. 4(a) and 4(b)]. As shown in Fig. 4(c) the recollision trajectory is nearly elliptical. The second row of Fig. 4 exhibits a LRT. After ionization the free electron travels about 2.75T before recollision. The second electron is directly knocked out by recollision.

Fig. 5 Electron momentum distributions in the field plane at intensity of 2 × 10¹⁴ W/cm² with field ratio 1.8 (a), 2.4 (b) and 3.0 (c). Both electrons are included.

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Fig. 6 Momentum distributions of the first and second electrons in the field plane at intensity of 2 × 10¹⁴ W/cm² with field ratio 1.8 (left column), 2.4 (middle column) and 3.0 (right column).

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According to simple-man model, where the initial momentum at the ionization instant and the effect of the Coulomb potential on 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 are expected to mainly distribute along the negative vector potential curve shown in Fig. 2(b) and have a maximum near y axis. However, as shown in Fig. 5, these momentum distributions including the two electrons for three different field ratios do not always exhibit a maximum near y axis. For field ratio 1.8 most electrons are located in the second quadrant. For field ratio 2.4 the most probable momentum distributes near y axis. When the field ratio increases to 3.0, most electrons cluster in the first quadrant. Overall, the electron momentum distribution shows a shift in time evolution direction of the negative vector potential with the field ratio increasing.

Fig. 7 Distributions of recollision energy for field ratios 1.8 (blue), 2.4 (red) and 3.0 (green).

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Fig. 8 Distributions of recollision time for field ratios 1.8 (a), 2.4 (b) and 3.0 (c). The solid gray lines show the laser electric field in arbitrary units. In order to more clearly show the laser phase at recollision, the recollision time is transferred into one optical cycle [kT-0.5T, kT+0.5T].

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In order to clarify the origin of the shift of the total momentum distribution of two electrons with the field ratio, we separately present the momentum distributions of the first electrons (the first row) and the second electrons (the second row) in Fig. 6. By comparing the momentum distributions of the first and second electrons, it is easy to see that the shift of the momentum distribution of the first electrons with the field ratio is accordance with the shift of the total momentum distribution of two electrons. If the initial momentum at the ionization instant and the effect of the Coulomb potential on the ionized electron are ignored, the final emission direction of the ionized electron is mostly determined by its ionization time. For the laser parameters considered in this work, the recollision energy of the returning electron mainly distributes in 0.5-0.8 a.u., as shown in Fig. 7. After recollision the first electron is freed immediately with a near-zero initial momentum. This establishes a connection between the final momentum of the first electron and the recollision time. Finally the first electron is most likely emitted to the direction of -A(t $_{R}$ ). Further, we examine the recollision time for three field ratios as shown in Fig. 8. It is obvious that the recollision time strongly depends on the field ratio. For field ratio 2.4, one can see that recollision mainly occurs at field maximum. After recollision the first electron is ionized immediately. So subsequent electric field drives the first electron to +y direction finally. For field ratio 1.8, recollision occurs before field maximum. After the acceleration of the subsequent electric field the first electron is emitted to the second quadrant. For field ratio 3.0, recollision mainly occurs after field maximum. Finally the first electron is released to the first quadrant. So the shift of the recollision time with the field ratio results in the shift of the momentum distribution. Conversely, the electron momentum distribution can be regarded as the map of the recollision time, which allows one to access the subcycle dynamics of the recollision process. The information about recollision time can be obtained from the electron momentum distribution. In addition, our results also demonstrate that the recollision can be steered by varying the field ratio of the two colors in co-rotating TCCP laser fields.

In addition, a closer examination for Figs. 6 and 8 indicates that the final emission direction of the first electrons does not exactly point to the negative vector potential at recollision instant -A(t $_{R}$ ). It is because the Coulomb potential slightly deflects the emission angle of the electron [60]. In RESI channel the second electron is excited by recollision and then is emitted by tunneling ionization. In this case besides the Coulomb potential the non-zero initial momentum originating from the nonadiabaticity of tunneling ionization also results in the deflection of the emission directrion [61]. Consequently, the centers of momentum distributions of the second electrons show a very small deviation from y axis [see the second row of Fig. 6]. However, they can not qualitatively affect the shift of the recollision time and the electron momentum distribution with the field ratio.

4. Conclusion

In conclusion, we have investigated the ultrafast dynamics of NSDI in co-rotating TCCP laser fields. Numerical results indicate that co-rotating TCCP fields can induce NSDI by recollision process. But the yield is an order of magnitude lower than counter-rotating case. Moreover, NSDI yield in co-rotating TCCP fields strongly depends on field ratio of the two colors and achieves its maximum at a ratio of 2.4. Two types of recollision trajectories are demonstrated. The short recollision trajectory with traveling time smaller than one cycle is dominant. The recollision time in co-rotating TCCP laser fields strongly depends on the field ratio. The recollision can be steered by changing the amplitude ratio of the two field in co-rotating TCCP laser fields. Therecollision time leaves a footprint in the electron momentum distribution. This provides an avenue to obtain information about recollision time and access the subcycle dynamics of the recollision process.

Funding

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

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Nonsequential double ionization by co-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 (8)

Equations (2)

Optics Express