## Revealing the multi-electron effects in sequential double ionization using classical simulations |

Optics Express, Vol. 20, Issue 18, pp. 20201-20209 (2012)

http://dx.doi.org/10.1364/OE.20.020201

Acrobat PDF (1294 KB)

### Abstract

We theoretically investigated sequential double ionization (SDI) of Ar by the nearly circularly polarized laser pulses with a fully correlated classical ensemble model. The ion momentum distributions of our numerical results at various laser intensities and pulse durations agree well with the experimental results. The experimentally observed multi-electron effects embodied in the joint momentum spectrum of the two electrons is also reproduced by our correlated classical calculations. Interestingly, our calculations show that the angular distribution of the first photoelectron from the trajectories which eventually suffer SDI differs from the distribution of the photoelectrons from above-threshold ionization trajectories. This observation provides additional evidence of multi-electron effects in strong field SDI.

© 2012 OSA

## 1. Introduction

21. S. L. Haan, L. Breen, A. Karim, and J. H. Eberly, “Recollision dynamics and time delay in strong-field double ionization,” Opt. Express **15**, 767–778 (2007). [CrossRef] [PubMed]

25. 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]

26. Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical simulations including electron correlations for sequential double ionization,” Phys. Rev. Lett. **109**, 053004 (2012). [CrossRef]

20. A. N. Pfeiffer, C. Cirelli, M. Smolarski, X. Wang, J. H. Eberly, R. Döner, and U. Keller, “Breakdown of the independent electron approximation in sequential double ionization,” New. J. Phys. **13**, 093008 (2011). [CrossRef]

27. A. N. Pfeiffer, C. Cirelli, M. Smolarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nature Phys. **8**, 76–80 (2011). [CrossRef]

28. N. I. Shvetsov-Shilovski, D. Dimitrovski, and L. B. Madsen, “Ionization in elliptically polarized pulses: Multi-electron polarization effects and asymmetry of photoelectron momentum distributions,” Phys. Rev. A **85**, 023428 (2012). [CrossRef]

## 2. The classical ensemble model

26. Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical simulations including electron correlations for sequential double ionization,” Phys. Rev. Lett. **109**, 053004 (2012). [CrossRef]

**r**

*and*

_{i}**p**

*are the position and canonical momentum of the*

_{i}*i*th electron, respectively.

**E**(

*t*) is the electric field, which is given as

*ω*,

*ε*and

*φ*are the laser frequency, the ellipticity and carrier-envelope phase (CEP), respectively.

*V*(

_{H}*r*,

_{i}*p*) is the Heisenberg-core potential, which is expressed as [29

_{i}29. C. L. Kirschbaum and L. Wilets, “Classical many-body model for atomic collisions incorporating the Heisenberg and Pauli principles,” Phys. Rev. A **21**, 834–841 (1980). [CrossRef]

*α*indicates the rigidity of the Heisenberg core and is chosen to be 2 in this paper. For a given

*α*, the parameter

*ξ*is chosen to match the second ionization potential of the target. Here we set

*ξ*=1.225 for target Ar. This Heisenberg-core potential was introduced by Kirschbaum

*et al*[29

29. C. L. Kirschbaum and L. Wilets, “Classical many-body model for atomic collisions incorporating the Heisenberg and Pauli principles,” Phys. Rev. A **21**, 834–841 (1980). [CrossRef]

30. D. Zajfman and D. Maor, “Heisenberg core in classical-trajectory monte carlo calculations of ionization and charge exchange,” Phys. Rev. Lett. **56**, 320–323 (1986). [CrossRef] [PubMed]

32. W. A. Beck and L. Wilets, “Semiclassical description of proton stopping by atomic and molecular targets,” Phys. Rev. A **55**, 2821–2829 (1997). [CrossRef]

26. Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical simulations including electron correlations for sequential double ionization,” Phys. Rev. Lett. **109**, 053004 (2012). [CrossRef]

10. S. L. Haan, L. Breen, A. Karim, and J. H. Eberly, “Variable time lag and backward ejection in full-dimension analysis of strong field double ionization,” Phys. Rev. Lett. **97**, 103008 (2006). [CrossRef] [PubMed]

21. S. L. Haan, L. Breen, A. Karim, and J. H. Eberly, “Recollision dynamics and time delay in strong-field double ionization,” Opt. Express **15**, 767–778 (2007). [CrossRef] [PubMed]

24. Phay J. Ho, R. Panfili, S. L. Haan, and J. H. Eberly, “Nonsequential double ionization as a completely classical photoelectric effect,” Phys. Rev. Lett. **94**, 093002 (2005). [CrossRef] [PubMed]

**109**, 053004 (2012). [CrossRef]

**109**, 053004 (2012). [CrossRef]

25. 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]

25. 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]

## 3. Results and discussions

19. A. N. Pfeiffer, C. Cirelli, M. Smolarski, R. Döner, and U. Keller, “Timing the release in sequential double ionization,” Nature Phys. **7**, 428–433 (2011). [CrossRef]

20. A. N. Pfeiffer, C. Cirelli, M. Smolarski, X. Wang, J. H. Eberly, R. Döner, and U. Keller, “Breakdown of the independent electron approximation in sequential double ionization,” New. J. Phys. **13**, 093008 (2011). [CrossRef]

19. A. N. Pfeiffer, C. Cirelli, M. Smolarski, R. Döner, and U. Keller, “Timing the release in sequential double ionization,” Nature Phys. **7**, 428–433 (2011). [CrossRef]

33. C. M. Maharjan, A. S. Alnaser, X. M. Tong, B. Ulrich, P. Ranitovic, S. Ghimire, Z. Chang, I. V. Litvinyuk, and C. L. Cocke, “Momentum imaging of doubly charged ions of Ne and Ar in the sequential ionization region,” Phys. Rev. A **72**, 041403R(2005). [CrossRef]

34. 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]

19. A. N. Pfeiffer, C. Cirelli, M. Smolarski, R. Döner, and U. Keller, “Timing the release in sequential double ionization,” Nature Phys. **7**, 428–433 (2011). [CrossRef]

^{2}with the volume effect taken into account. Figure 1(c) show that for the 7-fs case the distribution exhibits a three-band shape even at the high laser intensity. This is in good agreement with the experimental observation [20

20. A. N. Pfeiffer, C. Cirelli, M. Smolarski, X. Wang, J. H. Eberly, R. Döner, and U. Keller, “Breakdown of the independent electron approximation in sequential double ionization,” New. J. Phys. **13**, 093008 (2011). [CrossRef]

33. C. M. Maharjan, A. S. Alnaser, X. M. Tong, B. Ulrich, P. Ranitovic, S. Ghimire, Z. Chang, I. V. Litvinyuk, and C. L. Cocke, “Momentum imaging of doubly charged ions of Ne and Ar in the sequential ionization region,” Phys. Rev. A **72**, 041403R(2005). [CrossRef]

**7**, 428–433 (2011). [CrossRef]

**7**, 428–433 (2011). [CrossRef]

**13**, 093008 (2011). [CrossRef]

**13**, 093008 (2011). [CrossRef]

**109**, 053004 (2012). [CrossRef]

**13**, 093008 (2011). [CrossRef]

**13**, 093008 (2011). [CrossRef]

**7**, 428–433 (2011). [CrossRef]

**13**, 093008 (2011). [CrossRef]

35. S. V. Popruzhenko, G. G. Paulus, and D. Bauer, “Coulomb-corrected quantum trajectories in strong-field ionization,” Phys. Rev. A **77**, 053409 (2008). [CrossRef]

36. S. P. Goreslavski, G. G. Paulus, S. V. Popruzhenko, and N. I. Shvetsov-Shilovski, “Coulomb asymmetry in above-threshold ionization,” Phys. Rev. Lett. **93**, 233002 (2004). [CrossRef] [PubMed]

37. A. Jaroń, J. Z. Kamińskia, and F. Ehlotzky, “Asymmetries in the angular distributions of above threshold ionization in an elliptically polarized laser field,” Opt. Commun. **163**, 115–121 (1999). [CrossRef]

27. A. N. Pfeiffer, C. Cirelli, M. Smolarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nature Phys. **8**, 76–80 (2011). [CrossRef]

27. A. N. Pfeiffer, C. Cirelli, M. Smolarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nature Phys. **8**, 76–80 (2011). [CrossRef]

^{2}, respectively. Because of the ellipticity, electrons are almost ionized along the major axis of the polarization (x axis). Without further interaction with the remaining ion, the most probable electron momentum distribution is along the minor polarization axis (y axis). In Fig. 3(a), a small but visible offset angle with respect to the minor polarization axis exists, indicating the interaction between the remaining ion and the escaping electron [34

34. 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]

36. S. P. Goreslavski, G. G. Paulus, S. V. Popruzhenko, and N. I. Shvetsov-Shilovski, “Coulomb asymmetry in above-threshold ionization,” Phys. Rev. Lett. **93**, 233002 (2004). [CrossRef] [PubMed]

*α*= 2 and

*ξ*= 0.7225 to fit the first ionization potential of Ar. At the end of the laser pulses, we collect the singly ionized trajectories and analysis the angular distributions of the photoelectrons. Figure 4(a) displays the offset angle of the photoelectrons as a function of laser intensity.

**8**, 76–80 (2011). [CrossRef]

## 4. Conclusion

**8**, 76–80 (2011). [CrossRef]

## Acknowledgment

## References and links

1. | P. B. Corkum, “Plasma perspective on strong-field multiphoton ionization,” Phys. Rev. Lett. |

2. | K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett. |

3. | Th. Weber, H. Giessen, M. Weckenbrock, G. Urbasch, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, M. Vollmer, and R. Dörner, “Correlated electron emmision in multiphoton double ionization,” Nature |

4. | B. Walker, B. Sheehy, L. F. Dimauro, P. Agostini, K. J. Schafer, and K. C. Kulander, “Precision measurement of strong field double ionizaiton of helium,” Phys. Rev. Lett. |

5. | B. Feuerstein, R. Moshammer, D. Fischer, A. Dorn, C. D. Schröter, J. Deipenwisch, J. R. Crespo Lopez-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 ionizaion of Ar: the role of excitation tunneling,” Phys. Rev. Lett. |

6. | M. Weckenbrock, D. Zeidler, A. Staudte, Th. Weber, M. Schöffler, M. Meckel, S. Kammer, M. Smolarski, O. Jagutzki, V. R. Bhardwaj, D.M. Rayner, D.M. Villeneuve, P. B. Corkum, and R. Dörner, “Fully differential rates for femtosecond multiphoton double ionization of neon,” Phys. Rev. Lett. |

7. | A. Rudenko, K. Zrost, B. Feuerstein, V. L. B. de Jesus, C. D. Schröter, R. Moshammer, and J. Ullrich, “Correlated multielectron dynamics in ultrafast laser pulse interactions with atoms,” Phys. Rev. Lett. |

8. | M. Lein, E. K. U. Gross, and V. Engel, “Intense-field double ionization of helium: identifying the mechanism,” Phys. Rev. Lett. |

9. | A. Staudte, C. Ruiz, M. Schöffler, S. Schössler, D. Zeidler, Th. Weber, M. Meckel, D. M. Villeneuve, P. B. Corkum, A. Becker, and R. Dörner, “Binary and recoil collisions in strong field double ionization of helium,” Phys. Rev. Lett. |

10. | S. L. Haan, L. Breen, A. Karim, and J. H. Eberly, “Variable time lag and backward ejection in full-dimension analysis of strong field double ionization,” Phys. Rev. Lett. |

11. | Y. Zhou, C. Huang, A. Tong, Q. Liao, and P. Lu, “Correlated electron dynamics in nonsequential double ionization by orthogonal two-color laser pulses,” Opt. Express |

12. | Y. Zhou, C. Huang, Q. Liao, W. Hong, and P. Lu, “Control the revisit time of the electron wave packet,” Opt. Lett. |

13. | A. Becker, R. Dörner, and R. Moshammer,“Multiple fragmentation of atoms in femtosecond laser pulses,” J. Phys. B |

14. | Q. Liao, Y. Zhou, C. Huang, and P. Lu, “Multiphoton Rabi oscillations of correlated electrons in strong-field nonsequential double ionization,” New J. Phys. |

15. | B. Chang, P. R. Bolton, and D. N. Fittinghoff, “Closed-form solutions for the production of ions in the collisionless ionization of gases by intense lasers,” Phys. Rev. A |

16. | K. I. Dimitriou, S. Yoshida, J. Burgdöfer, H. Shimada, H. Oyama, and Y. Yamazaki, “Momentum distribution of multiply charged ions produced by intense (50–70-PW/cm |

17. | N. I. Shvetsov-Shilovski, A. M. Sayler, T. Rathje, and G. G. Paulus, “Momentum distributions of sequential ionization,” Phys. Rev. A |

18. | A. Fleischer, H. J. Wörner, L. Arissian, L. R. Liu, M. Meckel, A. Rippert, R. Dörner, D. M. Villeneuve, P. B. Corkum, and A. Staudte, “Probing angular correlations in sequential double ionization,” Phys. Rev. Lett. |

19. | A. N. Pfeiffer, C. Cirelli, M. Smolarski, R. Döner, and U. Keller, “Timing the release in sequential double ionization,” Nature Phys. |

20. | A. N. Pfeiffer, C. Cirelli, M. Smolarski, X. Wang, J. H. Eberly, R. Döner, and U. Keller, “Breakdown of the independent electron approximation in sequential double ionization,” New. J. Phys. |

21. | S. L. Haan, L. Breen, A. Karim, and J. H. Eberly, “Recollision dynamics and time delay in strong-field double ionization,” Opt. Express |

22. | Y. Zhou, Q. Liao, and P. Lu, “Mechanism for high-energy electrons in nonsequential double ionization below the recollision-excitation threshold,” Phys. Rev. A |

23. | Y. Zhou, Q. Liao, and P. Lu, “Complex sub-laser-cycle electron dynamics in strong-field nonsequential triple ionizaion,” Opt. Express |

24. | Phay J. Ho, R. Panfili, S. L. Haan, and J. H. Eberly, “Nonsequential double ionization as a completely classical photoelectric effect,” Phys. Rev. Lett. |

25. | Y. Zhou, Q. Liao, and P. Lu, “Asymmetric electron energy sharing in strong-field double ionization of helium,” Phys. Rev. A |

26. | Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical simulations including electron correlations for sequential double ionization,” Phys. Rev. Lett. |

27. | A. N. Pfeiffer, C. Cirelli, M. Smolarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nature Phys. |

28. | N. I. Shvetsov-Shilovski, D. Dimitrovski, and L. B. Madsen, “Ionization in elliptically polarized pulses: Multi-electron polarization effects and asymmetry of photoelectron momentum distributions,” Phys. Rev. A |

29. | C. L. Kirschbaum and L. Wilets, “Classical many-body model for atomic collisions incorporating the Heisenberg and Pauli principles,” Phys. Rev. A |

30. | D. Zajfman and D. Maor, “Heisenberg core in classical-trajectory monte carlo calculations of ionization and charge exchange,” Phys. Rev. Lett. |

31. | J. S. Cohen, “Quasiclassical-trajectory Monte Carlo methods for collisions with two-electron atoms,” Phys. Rev. A |

32. | W. A. Beck and L. Wilets, “Semiclassical description of proton stopping by atomic and molecular targets,” Phys. Rev. A |

33. | C. M. Maharjan, A. S. Alnaser, X. M. Tong, B. Ulrich, P. Ranitovic, S. Ghimire, Z. Chang, I. V. Litvinyuk, and C. L. Cocke, “Momentum imaging of doubly charged ions of Ne and Ar in the sequential ionization region,” Phys. Rev. A |

34. | X. Wang and J. H. Eberly, “Effects of elliptical polarization on strong-field short-pulse double ionization,” Phys. Rev. Lett. |

35. | S. V. Popruzhenko, G. G. Paulus, and D. Bauer, “Coulomb-corrected quantum trajectories in strong-field ionization,” Phys. Rev. A |

36. | S. P. Goreslavski, G. G. Paulus, S. V. Popruzhenko, and N. I. Shvetsov-Shilovski, “Coulomb asymmetry in above-threshold ionization,” Phys. Rev. Lett. |

37. | A. Jaroń, J. Z. Kamińskia, and F. Ehlotzky, “Asymmetries in the angular distributions of above threshold ionization in an elliptically polarized laser field,” Opt. Commun. |

**OCIS Codes**

(020.4180) Atomic and molecular physics : Multiphoton processes

(260.3230) Physical optics : Ionization

(270.6620) Quantum optics : Strong-field processes

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: July 5, 2012

Manuscript Accepted: August 8, 2012

Published: August 20, 2012

**Citation**

Yueming Zhou, Cheng Huang, and Peixiang Lu, "Revealing the multi-electron effects in sequential double ionization using classical simulations," Opt. Express **20**, 20201-20209 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-20201

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### References

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- K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett.70, 1599–1602 (1993). [CrossRef] [PubMed]
- Th. Weber, H. Giessen, M. Weckenbrock, G. Urbasch, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, M. Vollmer, and R. Dörner, “Correlated electron emmision in multiphoton double ionization,” Nature405, 658–661 (2000). [CrossRef] [PubMed]
- B. Walker, B. Sheehy, L. F. Dimauro, P. Agostini, K. J. Schafer, and K. C. Kulander, “Precision measurement of strong field double ionizaiton of helium,” Phys. Rev. Lett.73, 1227–1230 (1994). [CrossRef] [PubMed]
- B. Feuerstein, R. Moshammer, D. Fischer, A. Dorn, C. D. Schröter, J. Deipenwisch, J. R. Crespo Lopez-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 ionizaion of Ar: the role of excitation tunneling,” Phys. Rev. Lett.87, 043003 (2001). [CrossRef] [PubMed]
- M. Weckenbrock, D. Zeidler, A. Staudte, Th. Weber, M. Schöffler, M. Meckel, S. Kammer, M. Smolarski, O. Jagutzki, V. R. Bhardwaj, D.M. Rayner, D.M. Villeneuve, P. B. Corkum, and R. Dörner, “Fully differential rates for femtosecond multiphoton double ionization of neon,” Phys. Rev. Lett.92, 213002 (2004). [CrossRef] [PubMed]
- A. Rudenko, K. Zrost, B. Feuerstein, V. L. B. de Jesus, C. D. Schröter, R. Moshammer, and J. Ullrich, “Correlated multielectron dynamics in ultrafast laser pulse interactions with atoms,” Phys. Rev. Lett.93, 253001 (2004). [CrossRef]
- M. Lein, E. K. U. Gross, and V. Engel, “Intense-field double ionization of helium: identifying the mechanism,” Phys. Rev. Lett.85, 4707–4710 (2000). [CrossRef] [PubMed]
- A. Staudte, C. Ruiz, M. Schöffler, S. Schössler, D. Zeidler, Th. Weber, M. Meckel, D. M. Villeneuve, P. B. Corkum, A. Becker, and R. Dörner, “Binary and recoil collisions in strong field double ionization of helium,” Phys. Rev. Lett.99, 263002 (2007). [CrossRef]
- S. L. Haan, L. Breen, A. Karim, and J. H. Eberly, “Variable time lag and backward ejection in full-dimension analysis of strong field double ionization,” Phys. Rev. Lett.97, 103008 (2006). [CrossRef] [PubMed]
- Y. Zhou, C. Huang, A. Tong, Q. Liao, and P. Lu, “Correlated electron dynamics in nonsequential double ionization by orthogonal two-color laser pulses,” Opt. Express19, 2301–2308 (2011). [CrossRef] [PubMed]
- Y. Zhou, C. Huang, Q. Liao, W. Hong, and P. Lu, “Control the revisit time of the electron wave packet,” Opt. Lett.36, 2758–2760 (2011). [CrossRef] [PubMed]
- A. Becker, R. Dörner, and R. Moshammer,“Multiple fragmentation of atoms in femtosecond laser pulses,” J. Phys. B38, S753–S772 (2005). [CrossRef]
- Q. Liao, Y. Zhou, C. Huang, and P. Lu, “Multiphoton Rabi oscillations of correlated electrons in strong-field nonsequential double ionization,” New J. Phys.14, 013001 (2012). [CrossRef]
- B. Chang, P. R. Bolton, and D. N. Fittinghoff, “Closed-form solutions for the production of ions in the collisionless ionization of gases by intense lasers,” Phys. Rev. A47, 4193–4203 (1993). [CrossRef] [PubMed]
- K. I. Dimitriou, S. Yoshida, J. Burgdöfer, H. Shimada, H. Oyama, and Y. Yamazaki, “Momentum distribution of multiply charged ions produced by intense (50–70-PW/cm2) lasers,” Phys. Rev. A75, 013418 (2007). [CrossRef]
- N. I. Shvetsov-Shilovski, A. M. Sayler, T. Rathje, and G. G. Paulus, “Momentum distributions of sequential ionization,” Phys. Rev. A83, 033401 (2011). [CrossRef]
- A. Fleischer, H. J. Wörner, L. Arissian, L. R. Liu, M. Meckel, A. Rippert, R. Dörner, D. M. Villeneuve, P. B. Corkum, and A. Staudte, “Probing angular correlations in sequential double ionization,” Phys. Rev. Lett.107, 113003 (2011). [CrossRef] [PubMed]
- A. N. Pfeiffer, C. Cirelli, M. Smolarski, R. Döner, and U. Keller, “Timing the release in sequential double ionization,” Nature Phys.7, 428–433 (2011). [CrossRef]
- A. N. Pfeiffer, C. Cirelli, M. Smolarski, X. Wang, J. H. Eberly, R. Döner, and U. Keller, “Breakdown of the independent electron approximation in sequential double ionization,” New. J. Phys.13, 093008 (2011). [CrossRef]
- S. L. Haan, L. Breen, A. Karim, and J. H. Eberly, “Recollision dynamics and time delay in strong-field double ionization,” Opt. Express15, 767–778 (2007). [CrossRef] [PubMed]
- Y. Zhou, Q. Liao, and P. Lu, “Mechanism for high-energy electrons in nonsequential double ionization below the recollision-excitation threshold,” Phys. Rev. A80, 023412 (2009). [CrossRef]
- Y. Zhou, Q. Liao, and P. Lu, “Complex sub-laser-cycle electron dynamics in strong-field nonsequential triple ionizaion,” Opt. Express18, 16025–16034 (2010). [CrossRef] [PubMed]
- Phay J. Ho, R. Panfili, S. L. Haan, and J. H. Eberly, “Nonsequential double ionization as a completely classical photoelectric effect,” Phys. Rev. Lett.94, 093002 (2005). [CrossRef] [PubMed]
- Y. Zhou, Q. Liao, and P. Lu, “Asymmetric electron energy sharing in strong-field double ionization of helium,” Phys. Rev. A82, 053402 (2010). [CrossRef]
- Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical simulations including electron correlations for sequential double ionization,” Phys. Rev. Lett.109, 053004 (2012). [CrossRef]
- A. N. Pfeiffer, C. Cirelli, M. Smolarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nature Phys.8, 76–80 (2011). [CrossRef]
- N. I. Shvetsov-Shilovski, D. Dimitrovski, and L. B. Madsen, “Ionization in elliptically polarized pulses: Multi-electron polarization effects and asymmetry of photoelectron momentum distributions,” Phys. Rev. A85, 023428 (2012). [CrossRef]
- C. L. Kirschbaum and L. Wilets, “Classical many-body model for atomic collisions incorporating the Heisenberg and Pauli principles,” Phys. Rev. A21, 834–841 (1980). [CrossRef]
- D. Zajfman and D. Maor, “Heisenberg core in classical-trajectory monte carlo calculations of ionization and charge exchange,” Phys. Rev. Lett.56, 320–323 (1986). [CrossRef] [PubMed]
- J. S. Cohen, “Quasiclassical-trajectory Monte Carlo methods for collisions with two-electron atoms,” Phys. Rev. A54, 573–586 (1996). [CrossRef] [PubMed]
- W. A. Beck and L. Wilets, “Semiclassical description of proton stopping by atomic and molecular targets,” Phys. Rev. A55, 2821–2829 (1997). [CrossRef]
- C. M. Maharjan, A. S. Alnaser, X. M. Tong, B. Ulrich, P. Ranitovic, S. Ghimire, Z. Chang, I. V. Litvinyuk, and C. L. Cocke, “Momentum imaging of doubly charged ions of Ne and Ar in the sequential ionization region,” Phys. Rev. A72, 041403R(2005). [CrossRef]
- 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]
- S. V. Popruzhenko, G. G. Paulus, and D. Bauer, “Coulomb-corrected quantum trajectories in strong-field ionization,” Phys. Rev. A77, 053409 (2008). [CrossRef]
- S. P. Goreslavski, G. G. Paulus, S. V. Popruzhenko, and N. I. Shvetsov-Shilovski, “Coulomb asymmetry in above-threshold ionization,” Phys. Rev. Lett.93, 233002 (2004). [CrossRef] [PubMed]
- A. Jaroń, J. Z. Kamińskia, and F. Ehlotzky, “Asymmetries in the angular distributions of above threshold ionization in an elliptically polarized laser field,” Opt. Commun.163, 115–121 (1999). [CrossRef]

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