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The carrier-envelop-phase (CEP) dependence of nonsequential double ionization (NSDI) of atomic Ar with few-cycle elliptically polarized laser pulse is investigated using 2D classical ensemble method. We distinguish two particular recollision channels in NSDI, which are recollision-impact ionization (RII) and recollision-induced excitation with subsequent ionization (RESI). We separate the RII and RESI channels according to the delay time between recollision and final double ionization. By tracing the history of the trajectories, we find the electron correlation spectra as well as the competition between the two channels are sensitively dependent on the laser field CEP. Finally, control can be achieved between the two channels by varying the CEP.
© 2016 Optical Society of America
1. Introduction
Nonsequential double ionization (NSDI) [1, 2] is one of the most interesting phenomena in strong-field ionization and contains extensive information about electron-electron correlation. Theoretical and experimental works have explored nonsequential ionization (NSI) mechanisms in detail [3–12]. Researchers widely hold the idea that NSI is described by a three-step model [13]. In this model, one electron tunnel ionizes when the laser field is strong enough and then is driven back to the parent ion as the oscillating electric field reverses its direction. It recollides with the parent ion and transfers a part of energy to it, which enables the second NSI event. Further studies [14–16] reveal that if the returning electron energy is not enough to kick out the other directly but induces it to an excited state, then NSI could occur in subsequent laser field. This process implies longer delay time between recollision and NSI, and is defined as recollision-induced excitation with subsequent ionization (RESI). Accordingly, the direct ionization channel with prompt NSI is defined as recollision-impact ionization (RII).
Ultrafast lasers with pulses as short as a few optical cycles are now available [17–21]. Few-cycle laser pulse can simplify the NSDI process because one can achieve a single recollision event. Few-cycle laser pulse also presents carrier-envelope-phase (CEP) effects, enabling the control of transitions between different mechanisms or phenomena.
Elliptically polarized or circularly polarized laser fields are also employed in research on double ionization [22–25]. It is generally known that increasing ellipticity can depress recollision and then decrease probability of NSDI because of additional freedom in transverse direction. However, elliptical or circular polarization NSDI have been observed experimently and theoretically in Mg and other molecules [26–30]. In [24], Wang and Eberly undertook calculations that exhibited clear NSDI picture under elliptical polarization and electron recollision in NSDI through an elliptical trajectory.
In this paper, we apply the 2D classical ensemble method to investigate the CEP dependence in NSDI with the few-cycle elliptically polarized laser pulse. We can clearly distinguish two particular channels (RII and RESI) in NSDI by varying the CEP. The results show that few-cycle elliptically polarized laser pulse could simplify understanding of NSDI, as this external field eliminates some recollision trajectories and favors selecting a few distinct pathways with more information about the two channels.
2. Methods
We carry out simulated calculation by classical ensemble method which is proved to be useful in strong field ionization and especially in electron-electron correlation [10,30–34]. The general idea of this method used in strong field ionization is to mimic the predictions of a quantum wavefunction by using distributions of position, momenta, etc., generated by following the dynamics of members of an initially randomized ensemble of classically modeled atoms. In addition, the lack of cylindrical symmetry puts the case of elliptically polarized fields beyond the range of essentially all quantum calculation approaches which are relevant to strong-field double ionization; see [22] and references therein.
Details of ensemble generation process could be seen in [31]. The symplectic method is used in numerically calculation, which can preserve the symplectic structure of the system and be suitable for many-step calculations. Here, we use a 2D model of Ar with initial ensemble of 10 million scale generated by following equation in the absence of external laser fields
where Etot is the total energy of two electrons whose negative value is equal to the sum of first and second ionization potential. The subscript i = 1,2 labels the two electrons. is the electron momentum, and is the electron coordinate. In equation (1), two terms in bracket represent kinetic energy and soft-core Coulomb potential between the ion core and the ith electron, final term is the potential between the two electrons. For Ar, Etot = –1.59 (Using atomic units throughout the paper if not stated otherwise). Softening parameters a and b are set to be 1.5 and 0.1, respectively, which could avoid autoionization. To obtain the stable initial ensemble, system is allowed to evolve a sufficient long time in the absence of the external laser field by classical equations of motion. With initial value available, laser field is turned on The electric field can be written as here f (t) = sin2 (πt/T0), ω = 0.0584 a.u. that corresponds to a wavelength of 780 nm, the full duration of laser pulse is four cycles. φ is the carrier-envelope-phase. Advantage of transverse influence with elliptical polarized field requires the value of ellipticity ε could be as large as possible. However, recollision sharply decreases if ε is more than 0.35, which means the NSDI is not the main mechanism anymore. If ε is more than 0.4, the NSDI disappears. Therefore ε is suitably set to be 0.3.3. Results and discussion
To specify an appropriate laser field intensity, we sketch the double ionization yield curves versus laser intensity for four typical CEPs as shown in Fig. 1. It is indicated that all of the curves twist together, which illustrates the CEP effects of NSDI. Overall, each of the CEP phases exhibits the knee structure in the yield representing NSDI. Meanwhile, the curves overlapping over high field range show sequential double ionization (SDI). Along the SDIs route and beyond the area, we designate the working laser intensity as I0 = 1.0PW/cm2, which ensures the maximum recollision.
Fig. 1 The CEP-dependent double ionization yield versus laser intensity. To guide the eye, pink dashed curve marks out the SDI area, next to which a green vertical line marks the laser field intensity I0 = 1.0PW/cm2, which is used in the calculation.
Figure 2 shows the correlation spectra of double electron momenta in major polarization direction (x direction) for different CEPs with the designated laser intensity I0 = 1.0PW/cm2. A near cross-like structure can be observed. Similar structures have been paid attention to both experimentally and theoretically [17, 34]. In those works, the results and analysis indicated that the RESI mainly contributes to the structure, and it is crucial that the delay time between recollision and final ionization leads to the two ionized electrons probably exposed to different external field environment, especially in few-cycle laser pulse. So the two electrons may have very different final momenta, which forms the spectra close to axis and exhibits a cross-like shape. Figure 2 also shows the increasing distribution in first quadrant with the increasing of φ value, which leads us to infer that the result could comprise different underlying mechanisms. We will analyze the details latter and separate the different channels in NSDI according to the delay time.
Fig. 2 Correlation spectra of double electron momenta in major polarization direction for different CEPs. They show a cross-like structure or assemble in the first quadrant (actually, the underlying different double ionization channels just are RESI and RII, which will be talked about in details. see details below).
The recolliding process between two electrons will make the Coulomb repulsion energy increased instantaneously. We define the recollision time (tr) as the instant when the repulsion energy reaches its maximum after the first electron ionized. The final double ionization time (tDI) is defined as the instant when the electron finally become free. The delay time between recollision and final double ionization can be defined as tDI – tr. The relationship between tr and tDI as well as the statistical counts of delay time is shown in Fig. 3 for two typical CEPs (φ = 0.25π and φ = 0.625π). The similar analysis can be found in [34]. We mainly address two regions marked by B1 and B2 in Figs. 3(a) and 3(b), in which the diagonals mark out the area where the final double ionization time is equal to the recollision time. The cluster B1 which is far from the diagonal just shows a longer delay time for final double ionization after recolliding. In contrast, the cluster B2 near the diagonal indicates that the final double ionization occurring follows the recollision with shorter delay time. In addition, we find from Figs. 3(a) and 3(b) that the relative density of the clusters B1 and B2 is different for the two typical CEPs. The underlying mechanisms of the two clusters actually show RESI and RII processes. The counts of delay time shown in Figs. 3(c) and 3(d) give a comprehensive statistic analysis of the two channels for the two typical CEPs. The counts reach minimum around 0.2–0.6 cycles, which separate shorter and longer delay ionization clearly. Therefore we defined the RII channel as the ionization processes with delay time shorter than 0.4 cycles. Accordingly, the others with longer delay time are classified as the RESI channel. This standard of classification will also be used in latter discussion.
Fig. 3 (a) (b) The final double ionization time (tDI) versus the recollision time (tr) for two typical CEPs (φ = 0.25π and φ = 0.625π). The cluster B1 and B2 represent the two channels (RESI and RII). (c) (d) Counts of delay time for the two typical CEPs. The statistic results show the different channels with some peaks of counts. In this way, RESI and RII channels with longer and shorter delay time can be recorded, and the boundary of the two channels is set to be 0.4 cycle time delay.
Figure 4 shows two typical classical trajectories. A shorter trajectory shown in Fig. 4(a) corresponds to the cluster B1 for φ = 0.25π, where the lower energy of outer returning electron (red line) cannot kick out the inner one (black line) directly but induces it to an excited state. Then the excited electron is free when subsequent laser field is strong enough. This process describes the RESI channel. However, the longer elliptical trajectory of returning electron (red line) shown in Fig. 4(b) (corresponds to the cluster B2 for φ = 0.625π) could transfer enough energy to the inner electron (black line), which make the inner one ionized quickly after recolliding. This process with few delay time just describes the RII channel. Furthermore, the different density of cluster B1 or B2 for φ = 0.25π and φ = 0.625π shows that the RESI or RII plays a dominant role for the two specific CEP conditions. Thus we suppose that the two channels would compete in other CEP circumstances such as φ = 0.375π or φ = 0.5π and that controlling of the two channels is available by CEPs.
Fig. 4 Two typical classical trajectories of double ionized electrons. (a) shows the trajectories of two electrons (corresponds to the cluster B1 for φ = 0.25π) in RESI channel where a shorter travel of the outer electron (red line) colliding the inner one (black line), lower energy transfer enable the inner electron to be excited for a moment and then ionized with subsequent laser field. (b) shows the RII channel (corresponds to the cluster B2 for φ = 0.625π), where the outer electron (red line) in a longer elliptical recollision orbit with higher energy could impact the inner one (black line) and make it ionized directly.
The correlation spectra of the double electron momenta shown in Fig. 2 can be further separated into two parts (RII channel and RESI channel), which is shown in Fig. 5. The separation standard is the same as that used in defining the two channels. Correlation spectra shown in the upper panel of Fig. 5 (RII channel) tend to distribute in the first quadrant which shows that the two electrons tend to fly into the same direction because the shorter delay time makes them escaped from the core almost during the same time, and then they are both driven by the external field. However, lower panel of Fig. 5 (RESI channel) tends to form a cross-like structure, which has been mentioned above. A similar analytical method also was used in [34].
Fig. 5 Correlation spectra of double-electron momenta for different CEPs, which shows the different NSDI channels (the upper panel shows the RII channel and the lower panel shows the RESI channel). This is just the separating image of the blended one, see in Fig. 2. RIIs spectra mainly assemble in the first quadrant while RESIs shows a cross-like structure. The boundary of the two channels is set to be 0.4 cycle time delay as well.
Finally, we define the ratio of the two channels as NRESI/(NRESI + NRII) and NRII/(NRESI + NRII), where NRESI and NRII are the counts of RESI and RII in a definite CEP, respectively. Therefore, an overview of the channels control mechanism could be obtained as shown in Fig. 6. The schema shows that RESI and RII can be controlled by changing CEP. For φ = 0.25π and φ = 0.625π, the channel of RII (red line) could be appropriately turned off or turned on, respectively, and it is contrary to RESI. In addition, the competition behavior of two channels changes following a π cycle on account of circular symmetry for Ar atom.
Fig. 6 Ratio between definite ionization mechanism and total ionization versus CEPs. Two channels can be appropriately turned off or turned on respectively. In addition, their behaviors change following a π cycle on account of circular symmetry for Ar atom. The boundary of the two channels is set to be 0.4 cycle time delay as well.
In addition, the strongest control occurs at φ = 0.25π and φ = 0.625π, which shows an asymmetry phenomenon (not 0.25π and 0.25π + 0.5π), which is interesting and could be investigated by comparing with other system (such as He, Ne, Xe, Mg or even some molecule) in further work. Circumstance of different cycles of laser pulse or changing wavelength could also be explored.
4. Conclusion
In conclusion, with the 2D classical ensemble model, we investigate the two mechanisms in NSDI of Ar by few-cycle elliptical laser pulses. The results show that the CEP effects in elliptically polarized few-cycle laser pulses are evident. The CEP-dependent mechanisms of NSDI display two particular channels, and the regular controllable transition between RESI and RII is available. The transverse influence in elliptical laser field helps one to select few obvious trajectories and simplifie the researches on the two channels.
In addition, this transverse influence has a pervasive significance in controlling some kind of trajectories or other phenomena. Some researchers have used orthogonal two-color fields to achieve resolution of a quarter optical cycle in the photoelectron or attosecond-resolved electron emission [9,35–38]. In our work, elliptical laser field can also be regarded as orthogonal fields (with same frequency naturally) with certain relative phase, and the controlling mechanism is achieved by adjusting absolute phase (CEPs of the elliptical laser pulse), the recollision events occur at high resolution as well. We anticipate that if we change the relative phase, which actually vary the ellipticity, the corresponding phenomena could be explored in deep work.
Acknowledgments
We wish to thank Dr. Yingpeng Wang for grammar check of the paper. This work was supported by the National Natural Science Foundation of China. (Grants No. 61575077, 11271158 and 11574117).
References and links
1. Th. Weber, H. Giessen, M. Weckenbrock, G. Urbasch, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, M. Vollmer, and R. Dörner, “Correlated electron emission in multiphoton double ionization,” Nature (London) 405,658–661 (2000). [CrossRef]
2. D. N. Fittinghoff, P. R. Bolton, B. Chang, and K. C. Kulander, “Observation of nonsequential double ionization of helium with optical tunneling,” Phys. Rev. Lett. 69, 2642–2645 (1992). [CrossRef] [PubMed]
3. B. Walker, B. Sheehy, L. F. DiMauro, P. Agostine, K. J. Schafer, and K. C. Kulander, “Precision measurement of strong field double ionization of helium,” Phys. Rev. Lett. 73, 1227–1230 (1994). [CrossRef] [PubMed]
4. Y. Zhou, Q. Liao, and P. Lu, “Asymmetric electron energy sharing in strong-field double ionization of helium,” Phys. Rev. A 82, 053402 (2010). [CrossRef]
5. B. Walker, X. Liu, P. J. Ho, and J. H. Eberly, “Theories of photoelectron correlation in laser-driven multiple atomic ionization,” Rev. Mod. Phys. 84, 1011 (2012). [CrossRef]
6. J. Guo and X. Liu, “Exploration of nonsequential-double-ionization dynamics of Mg atoms in linearly and circularly polarized laser fields with different potentials,” Phys. Rev. A 88, 023405 (2013). [CrossRef]
7. A. Tong, Y. Zhou, C. Huang, and P. Lu, “Electron dynamics of molecular double ionization by circularly polarized laser pulses,” J. Chem. Phys. 139, 074308(2013). [CrossRef] [PubMed]
8. T. Wang, X. Ge, J. Guo, and X. Liu, “Sensitivity of strong-field double ionization to the initial ensembles in circularly polarized laser fields,” Phys. Rev. A 90, 033420 (2014). [CrossRef]
9. Z. Yuan, D. Ye, Z. Xia, and L. Fu, “Intensity-dependent two-electron emission dynamics with orthogonally polarized two-color laser fields,” Phys. Rev. A 91, 063417 (2015). [CrossRef]
10. S. Dong, L. Zhang, H. Bai, and T. Zhang, “Scaling law of nonsequential double ionization,” Phys. Rev. A 92, 033409 (2015). [CrossRef]
11. H. Xu, F. He, T. Sang, and I. V. Litvinyuk, “Experimental observation of the elusive double-peak structure in R-dependent strong-field ionization rate of H2+,” Sci. Rep. 5, 13527 (2015). [CrossRef]
12. Z. Xia, F. Ye, Y. Han, and J. Liu, “Momentum distribution of near zero-energy photoelectrons in the strong-field tunneling ionization in the long wavelength limit,” Sci. Rep. 5, 11473 (2015). [CrossRef]
13. P. B. Corkum, “Plasma perspective on strong-field multiphoton ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993). [CrossRef] [PubMed]
14. B. Feuerstein, R. Moshammer, D. Fischer, A. Dorn, C. D. Schöter, J. Deipenwisch, J. R. CrespoLopez-Urrutia, C. Höhr, P. Neumayer, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, and W. Sandner, “Separation of recollision mechanisms in nonsequential strong field double ionization of Ar: the role of excitation tunneling,” Phys. Rev. Lett. 87, 043003 (2001). [CrossRef] [PubMed]
15. L. Zhang, H. Bai, and T. Zhang, “Double ionization of Ar below the recollision threshold Intensity,” Phys. Rev. Lett. 90, 023410 (2014).
16. F. Mauger, A. Kamor, C. Chandre, and T. Uzer, “Mechanism of delayed double ionization in a strong laser field,” Phys. Rev. Lett. 108, 063001 (2012). [CrossRef] [PubMed]
17. B. Bergues, M. Kübel, N. G. Johnson, B. Fischer, N. Camus, K. J. Betsch, O. Herrwerth, A. Senftleben, A. M. Sayler, T. Rathje, T. Pfeifer, I. Ben-Itzhak, R. R. Jones, G. G. Paulus, F. Krausz, R. Moshammer, J. Ullrich, and M. F. Kling, “Attosecond tracing of correlated electron-emission in non-sequential double ionization,” Nature Commun. 3, 813 (2012). [CrossRef]
18. X. Liu and C. Figueira de Morisson Faria, “Nonsequential double ionization with few-cycle laser pulses,” Phys. Rev. Lett. 92, 133006 (2004). [CrossRef] [PubMed]
19. X. Liu, H. Rottke, E. Eremina, W. Sandner, E. Goulielmakis, K. O. Keeffe, M. Lezius, F. Krausz, F. Lindner, M. G. Schätzel, G. G. Paulus, and H. Walther, “Nonsequential ionization at the single-optical-cycle limit,” Phys. Rev. Lett. 93, 263001 (2004). [CrossRef]
20. C. Figueira de Morisson Faria, T Shaaran, and M. T. Nygren, “Time-delayed nonsequential double ionization with few-cycle laser pulses: Importance of the carrier-envelope phase,” Phys. Rev. A 86, 053405 (2012). [CrossRef]
21. M. Kübel, Nora G. Kling, K. J. Betsch, N. Camus, A. Kaldun, U. Kleineberg, I. Ben-Itzhak, R. R. Jones, G. G. Paulus, T. Pfeifer, J. Ullrich, R. Moshammer, M. F. Kling, and B. Bergues, “Nonsequential double ionization of N2 in a near-single-cycle laser pulse,” Phys. Rev. A 88, 023418 (2013). [CrossRef]
22. X. Wang and J. H. Eberly, “Effects of elliptical polarization on strong-field short-pulse double ionization,” Phys. Rev. Lett. 103, 103007 (2009). [CrossRef] [PubMed]
23. X. Wang and J. H. Eberly, “Elliptical polarization and probability of double ionization,” Phys. Rev. Lett. 105, 083001 (2010). [CrossRef] [PubMed]
24. X. Wang and J. H. Eberly, “Elliptical trajectories in nonsequential double ionization,” New J. Phys. 12, 093047 (2010). [CrossRef]
25. M. Wu, Y. Wang, X. Liu, W. Li, X. Hao, and J. Chen, “Coulomb-potential effects in nonsequential double ionization under elliptical polarization,” Phys. Rev. A 87, 013431 (2013). [CrossRef]
26. C. Guo, M. Li, J. P. Nibarger, and G. N. Gibson, “Single and double ionization of diatomic molecules in strong laser fields,” Phys. Rev. A 58, R4271–R4274 (1998). [CrossRef]
27. G. D. Gillen, M. A. Walker, and L. D. Van Woerkom, “Enhanced double ionization with circularly polarized light,” Phys. Rev. A 64, 043413 (2001). [CrossRef]
28. C. Guo and G. N. Gibson, “Ellipticity effects on single and double ionization of diatomic molecules in strong laser fields,” Phys. Rev. A 63, 040701 (2001). [CrossRef]
29. F. Mauger, C. Chandre, and T. Uzer, “Recollision and correlated double ionization with circularly polarized light,” Phys. Rev. Lett. 105, 083002 (2010). [CrossRef]
30. T. Xu, S. Ben, T. Wang, J. Zhang, J. Guo, and X. Liu, “Exploration of the nonsequential double-ionization process of a Mg atom with different delay time in few-cycle circularly polarized laser fields,” Phys. Rev. A 92, 033405 (2015). [CrossRef]
31. R. Panfili, J. H. Eberly, and S. L. Haan, “Comparing classical and quantum dynamics of strong-field double ionization,” Opt. Express 8, 431 (2001). [CrossRef] [PubMed]
32. X. Wang, J. Tian, and J. H. Eberly, “Angular correlation in strong-field double ionization under circular polarization,” Phys. Rev. Lett. 110, 073001 (2013). [CrossRef] [PubMed]
33. F. Mauger, C. Chandre, and T. Uzer, “Strong field double ionization: the phase space perspective,” Phys. Rev. Lett. 102, 073002 (2009). [CrossRef]
34. C. Huang, Y. Zhou, Q. Zhang, and P. Lu, “Contribution of recollision ionization to the cross-shaped structure in nonsequential double ionization,” Opt. Express 21, 11382 (2013). [CrossRef] [PubMed]
35. M. Kitzler and M. Lezius, “Spatial control of recollision wave packet with attosecond precision,” Phys. Rev. Lett. 95, 253001 (2005). [CrossRef]
36. L. Chen, Y. Zhou, C. Huang, Q. Zhang, and P. Lu, “Attosecond-resolved electron emission in nonsequential double ionization,” Phys. Rev. A 88, 043425 (2013). [CrossRef]
37. L. Zhang, X. Xie, S. Roither, D. Kartashov, Y. Wang, C. Wang, M. Schöffler, D. Shafir, P. B. Corcum, A. Baltuška, I. Ivanov, A. Kheifets, X. Liu, A. Staudte, and M. Kitzler, “Laser-sub-cycle two-dimensional electron-momentum mapping using orthogonal two-color fields,” Phys. Rev. A 90, 061401 (2014). [CrossRef]
38. L. Zhang, X. Xie, S. Roither, Y. Zhou, P. Lu, D. Kartashov, M. Schöffler, D. Shafir, P. B. Corcum, A. Baltuška, A. Staudte, and M. Kitzler, “Subcycle control of electron-electron correlation in double ionization,” Phys. Rev. Lett. 112, 193002 (2014). [CrossRef] [PubMed]
