Abstract

We investigate the strong-field ionization from p± orbitals driven by circularly polarized laser fields by solving the two-dimensional time-dependent Schrödinger equation in polar coordinates with the Lagrange mesh technique. Enhancement of ionization is found in the deep multiphoton ionization regime when the helicity of the laser field is opposite to that of the p electron, while this enhancement is suppressed when the helicities are the same. It is found that the enhancement of ionization is attributed to the multiphoton resonant excitation. The helicity sensitivity of the resonant enhancement is related to the different excitation-ionization channels in left and right circularly polarized laser fields.

© 2016 Optical Society of America

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    [Crossref]
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    [Crossref]
  3. D. B. Milošević and F. Ehlotzky, “Scattering and reaction processes in powerful laser fields,” Adv. At. Mol. Opt. Phys. 49, 373–532 (2003).
    [Crossref]
  4. A. Becker and F. H. M. Faisal, “Intense-field many-body S-matrix theory,” J. Phys. B: At. Mol. Opt. Phys. 38, R1–R56 (2005).
    [Crossref]
  5. B. Walker, B. Sheehy, L. F. DiMauro, P. Agostini, K. J. Schafer, and K. C. Kulander, “Precision measurement of strong field double ionization of helium,” Phys. Rev. Lett. 73, 1227 (1994).
    [Crossref] [PubMed]
  6. R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
    [Crossref] [PubMed]
  7. C. F. M. Faria and X. Liu, “Electron-electron correlation in strong laser fields,” J. Mod. Opt. 58, 1076–1131 (2011).
    [Crossref]
  8. 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] [PubMed]
  9. Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
    [Crossref] [PubMed]
  10. E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
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    [Crossref] [PubMed]
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  13. L. He, P. Lan, Q. Zhang, C. Zhai, F. Wang, W. Shi, and P. Lu, “Spectrally resolved spatiotemporal features of quantum paths in high-order-harmonic generation,” Phys. Rev. A 92, 043403 (2015).
    [Crossref]
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    [Crossref] [PubMed]
  17. A. N. Pfeiffer, C. Cirelli, M. Smorlarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nat. Phys. 8, 76–80 (2012).
    [Crossref]
  18. J. Wu, M. Meckel, L. P. Schmidt, M. Kunitski, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Probing the tunnelling site of electrons in strong field enhanced ionization of molecules,” Nat. Commun. 3, 1113 (2012).
    [Crossref] [PubMed]
  19. J. Wu, L. Schmidt, M. Kunitski, M. Meckel, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Multiorbital tunneling ionization of the CO molecule,” Phys. Rev. Lett. 108, 183001 (2012).
    [Crossref] [PubMed]
  20. L. Holmegaard, J. L. Hansen, L. Kalhøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
    [Crossref]
  21. J. Hansen, H. Stapelfeldt, D. Dimitrovski, M. Abu-samha, C. Martiny, and L. Madsen, “Time-resolved photo-electron angular distributions from strong-field ionization of rotating naphthalene molecules,” Phys. Rev. Lett. 106, 073001 (2011).
    [Crossref]
  22. X. Zhu, Q. Zhang, W. Hong, P. Lu, and Z. Xu, “Molecular orbital imaging via above-threshold ionization with circularly polarized pulses,” Opt. Express 19, 13722–13731 (2011).
    [Crossref] [PubMed]
  23. I. Petersen, J. Henkel, and M. Lein, “Signatures of molecular orbital structure in lateral electron momentum distributions from strong-field ionization,” Phys. Rev. Lett. 114, 103004 (2015).
    [Crossref] [PubMed]
  24. G. Yudin and M. Ivanov, “Nonadiabatic tunnel ionization: looking inside a laser cycle,” Phys. Rev. A 64, 013409 (2001).
    [Crossref]
  25. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electrom,” Sov. Phys. JETP 64, 1191 (1986).
  26. X. M. Tong, Z. X. Zhao, and C. D. Lin, “Theory of molecular tunneling ionization,” Phys. Rev. A 66, 033402 (2002).
    [Crossref]
  27. I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields: physical picture and calculations,” Phys. Rev. A 84, 063415 (2011);
    [Crossref]
  28. I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields. II. Derivation of formulas,” Phys. Rev. A 87, 013433 (2013).
    [Crossref]
  29. I. Barth and M. Lein, “Numerical verification of the theory of nonadiabatic tunnel ionization in strong circularly polarized laser fields,” J. Phys. B: At. Mol. Opt. Phys. 47, 204016 (2014).
    [Crossref]
  30. T. Herath, L. Yan, S. K. Lee, and W. Li, “Strong-field ionization rate depends on the sign of the magnetic quantum number,” Phys. Rev. Lett. 109, 043004 (2012).
    [Crossref] [PubMed]
  31. Y. Li, P. Lan, H. Xie, M. He, X. Zhu, Q. Zhang, and P. Lu, “Nonadiabatic tunnel ionization in strong circularly polarized laser fields: counterintuitive angular shifts in the photoelectron momentum distribution,” Opt. Express 23, 28801–28807 (2015).
    [Crossref] [PubMed]
  32. J. Kaushal, F. Morales, and O. Smirnova, “Opportunities for detecting ring currents using an attoclock setup,” Phys. Rev. A 92, 063405 (2015).
    [Crossref]
  33. I. Barth and O. Smirnova, “Spin-polarized electrons produced by strong-field ionization,” Phys. Rev. A 88, 013401 (2013).
    [Crossref]
  34. A. D. Bandrauk and Huizhong Lu, High Performance Computing Systems and Applications (Springer, 2010), Chapter 8.
  35. C. Lefebvre, Huizhong Lu, and A. D. Bandrauk, “Numerical method for dynamic imaging of nonlinear polyatomic molecules,” J. Phys. Coference Series 341, 012004 (2012).
    [Crossref]
  36. M. Vincke, L. Malegat, and D. Baye, “Regularization of singularities in Lagrange-mesh calculations,” J. Phys. B: At. Mol. Opt. Phys. 26, 811–826 (1993).
    [Crossref]
  37. D. Baye, L. Filippin, and M. Godefroid, “Accurate solution of the Dirac equation on Lagrange meshes,” Phys. Rev. E 89, 043305(2014).
    [Crossref]
  38. D. Dundas, “Accurate and efficient non-adiabatic quantum molecular dynamics approach for laser-matter interactions,” J. Phys. B: At. Mol. Opt. Phys. 37, 2883–2901 (2004).
    [Crossref]
  39. L. Y. Peng, J. F. McCann, D. Dundas, K. T. Taylor, and I. D. Williams, “A discrete time-dependent method for metastable atoms and molecules in intense fields,” J. Chem. Phys. 120, 10046 (2004).
    [Crossref] [PubMed]
  40. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1965).
  41. M. D. Perry, A. Szoke, and K. C. Kulander, “Resonantly enhanced above-threshold ionization of helium,” Phys. Rev. Lett. 63, 1058–1061 (1989).
    [Crossref] [PubMed]
  42. M. Li, P. Zhang, S. Luo, Y. Zhou, Q. Zhang, P. Lan, and P. Lu, “Selective enhancement of resonant multiphoton ionization with strong laser fields,” Phys. Rev. A 92, 063404 (2015).
    [Crossref]
  43. K. Liu, Q. Li, P. Lan, and P. Lu, “Enhanced dissociation of H2+ into highly excited states by UV pulses,” Molecular Physics 21, 3247–3252 (2015).
    [Crossref]

2015 (7)

L. He, P. Lan, Q. Zhang, C. Zhai, F. Wang, W. Shi, and P. Lu, “Spectrally resolved spatiotemporal features of quantum paths in high-order-harmonic generation,” Phys. Rev. A 92, 043403 (2015).
[Crossref]

F. Wang, L. He, C. Zhai, W. Shi, Q. Zhang, P. Lan, and P. Lu, “Time-dependent phase matching of high-order-harmonic generation,” Phys. Rev. A 92, 063839 (2015).
[Crossref]

I. Petersen, J. Henkel, and M. Lein, “Signatures of molecular orbital structure in lateral electron momentum distributions from strong-field ionization,” Phys. Rev. Lett. 114, 103004 (2015).
[Crossref] [PubMed]

J. Kaushal, F. Morales, and O. Smirnova, “Opportunities for detecting ring currents using an attoclock setup,” Phys. Rev. A 92, 063405 (2015).
[Crossref]

M. Li, P. Zhang, S. Luo, Y. Zhou, Q. Zhang, P. Lan, and P. Lu, “Selective enhancement of resonant multiphoton ionization with strong laser fields,” Phys. Rev. A 92, 063404 (2015).
[Crossref]

K. Liu, Q. Li, P. Lan, and P. Lu, “Enhanced dissociation of H2+ into highly excited states by UV pulses,” Molecular Physics 21, 3247–3252 (2015).
[Crossref]

Y. Li, P. Lan, H. Xie, M. He, X. Zhu, Q. Zhang, and P. Lu, “Nonadiabatic tunnel ionization in strong circularly polarized laser fields: counterintuitive angular shifts in the photoelectron momentum distribution,” Opt. Express 23, 28801–28807 (2015).
[Crossref] [PubMed]

2014 (3)

D. Baye, L. Filippin, and M. Godefroid, “Accurate solution of the Dirac equation on Lagrange meshes,” Phys. Rev. E 89, 043305(2014).
[Crossref]

I. Barth and M. Lein, “Numerical verification of the theory of nonadiabatic tunnel ionization in strong circularly polarized laser fields,” J. Phys. B: At. Mol. Opt. Phys. 47, 204016 (2014).
[Crossref]

X.-B. Bian and A. D. Bandrauk, “Probing nuclear motion by frequency modulation of molecular high-order harmonic generation,” Phys. Rev. Lett. 113, 193901 (2014).
[Crossref] [PubMed]

2013 (2)

I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields. II. Derivation of formulas,” Phys. Rev. A 87, 013433 (2013).
[Crossref]

I. Barth and O. Smirnova, “Spin-polarized electrons produced by strong-field ionization,” Phys. Rev. A 88, 013401 (2013).
[Crossref]

2012 (6)

C. Lefebvre, Huizhong Lu, and A. D. Bandrauk, “Numerical method for dynamic imaging of nonlinear polyatomic molecules,” J. Phys. Coference Series 341, 012004 (2012).
[Crossref]

T. Herath, L. Yan, S. K. Lee, and W. Li, “Strong-field ionization rate depends on the sign of the magnetic quantum number,” Phys. Rev. Lett. 109, 043004 (2012).
[Crossref] [PubMed]

A. N. Pfeiffer, C. Cirelli, M. Smorlarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nat. Phys. 8, 76–80 (2012).
[Crossref]

J. Wu, M. Meckel, L. P. Schmidt, M. Kunitski, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Probing the tunnelling site of electrons in strong field enhanced ionization of molecules,” Nat. Commun. 3, 1113 (2012).
[Crossref] [PubMed]

J. Wu, L. Schmidt, M. Kunitski, M. Meckel, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Multiorbital tunneling ionization of the CO molecule,” Phys. Rev. Lett. 108, 183001 (2012).
[Crossref] [PubMed]

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] [PubMed]

2011 (4)

C. F. M. Faria and X. Liu, “Electron-electron correlation in strong laser fields,” J. Mod. Opt. 58, 1076–1131 (2011).
[Crossref]

J. Hansen, H. Stapelfeldt, D. Dimitrovski, M. Abu-samha, C. Martiny, and L. Madsen, “Time-resolved photo-electron angular distributions from strong-field ionization of rotating naphthalene molecules,” Phys. Rev. Lett. 106, 073001 (2011).
[Crossref]

I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields: physical picture and calculations,” Phys. Rev. A 84, 063415 (2011);
[Crossref]

X. Zhu, Q. Zhang, W. Hong, P. Lu, and Z. Xu, “Molecular orbital imaging via above-threshold ionization with circularly polarized pulses,” Opt. Express 19, 13722–13731 (2011).
[Crossref] [PubMed]

2010 (1)

L. Holmegaard, J. L. Hansen, L. Kalhøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
[Crossref]

2009 (1)

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
[Crossref] [PubMed]

2008 (3)

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
[Crossref] [PubMed]

P. Eckle, M. Smorlarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. 4, 565–570 (2008).
[Crossref]

P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, H. G. Muller, M. Bütticker, and U. Keller, “Attosecond ionization and tunneling delay time measurements in helium,” Science 322, 1525–1529 (2008).
[Crossref] [PubMed]

2006 (1)

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B: At. Mol. Opt. Phys. 39, R203–R262 (2006).
[Crossref]

2005 (1)

A. Becker and F. H. M. Faisal, “Intense-field many-body S-matrix theory,” J. Phys. B: At. Mol. Opt. Phys. 38, R1–R56 (2005).
[Crossref]

2004 (2)

D. Dundas, “Accurate and efficient non-adiabatic quantum molecular dynamics approach for laser-matter interactions,” J. Phys. B: At. Mol. Opt. Phys. 37, 2883–2901 (2004).
[Crossref]

L. Y. Peng, J. F. McCann, D. Dundas, K. T. Taylor, and I. D. Williams, “A discrete time-dependent method for metastable atoms and molecules in intense fields,” J. Chem. Phys. 120, 10046 (2004).
[Crossref] [PubMed]

2003 (2)

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
[Crossref] [PubMed]

D. B. Milošević and F. Ehlotzky, “Scattering and reaction processes in powerful laser fields,” Adv. At. Mol. Opt. Phys. 49, 373–532 (2003).
[Crossref]

2002 (2)

W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G.G. Paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effects,” Adv. At. Mol. Opt. Phys. 48, 35–98 (2002).
[Crossref]

X. M. Tong, Z. X. Zhao, and C. D. Lin, “Theory of molecular tunneling ionization,” Phys. Rev. A 66, 033402 (2002).
[Crossref]

2001 (1)

G. Yudin and M. Ivanov, “Nonadiabatic tunnel ionization: looking inside a laser cycle,” Phys. Rev. A 64, 013409 (2001).
[Crossref]

2000 (1)

R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
[Crossref] [PubMed]

1994 (1)

B. Walker, B. Sheehy, L. F. DiMauro, P. Agostini, K. J. Schafer, and K. C. Kulander, “Precision measurement of strong field double ionization of helium,” Phys. Rev. Lett. 73, 1227 (1994).
[Crossref] [PubMed]

1993 (1)

M. Vincke, L. Malegat, and D. Baye, “Regularization of singularities in Lagrange-mesh calculations,” J. Phys. B: At. Mol. Opt. Phys. 26, 811–826 (1993).
[Crossref]

1989 (1)

M. D. Perry, A. Szoke, and K. C. Kulander, “Resonantly enhanced above-threshold ionization of helium,” Phys. Rev. Lett. 63, 1058–1061 (1989).
[Crossref] [PubMed]

1986 (1)

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electrom,” Sov. Phys. JETP 64, 1191 (1986).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1965).

Abu-samha, M.

A. N. Pfeiffer, C. Cirelli, M. Smorlarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nat. Phys. 8, 76–80 (2012).
[Crossref]

J. Hansen, H. Stapelfeldt, D. Dimitrovski, M. Abu-samha, C. Martiny, and L. Madsen, “Time-resolved photo-electron angular distributions from strong-field ionization of rotating naphthalene molecules,” Phys. Rev. Lett. 106, 073001 (2011).
[Crossref]

L. Holmegaard, J. L. Hansen, L. Kalhøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
[Crossref]

Agostini, P.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
[Crossref] [PubMed]

B. Walker, B. Sheehy, L. F. DiMauro, P. Agostini, K. J. Schafer, and K. C. Kulander, “Precision measurement of strong field double ionization of helium,” Phys. Rev. Lett. 73, 1227 (1994).
[Crossref] [PubMed]

Ammosov, M. V.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electrom,” Sov. Phys. JETP 64, 1191 (1986).

Aquila, A. L.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
[Crossref] [PubMed]

Attwood, D. T.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
[Crossref] [PubMed]

Bandrauk, A. D.

X.-B. Bian and A. D. Bandrauk, “Probing nuclear motion by frequency modulation of molecular high-order harmonic generation,” Phys. Rev. Lett. 113, 193901 (2014).
[Crossref] [PubMed]

C. Lefebvre, Huizhong Lu, and A. D. Bandrauk, “Numerical method for dynamic imaging of nonlinear polyatomic molecules,” J. Phys. Coference Series 341, 012004 (2012).
[Crossref]

A. D. Bandrauk and Huizhong Lu, High Performance Computing Systems and Applications (Springer, 2010), Chapter 8.

Barth, I.

I. Barth and M. Lein, “Numerical verification of the theory of nonadiabatic tunnel ionization in strong circularly polarized laser fields,” J. Phys. B: At. Mol. Opt. Phys. 47, 204016 (2014).
[Crossref]

I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields. II. Derivation of formulas,” Phys. Rev. A 87, 013433 (2013).
[Crossref]

I. Barth and O. Smirnova, “Spin-polarized electrons produced by strong-field ionization,” Phys. Rev. A 88, 013401 (2013).
[Crossref]

I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields: physical picture and calculations,” Phys. Rev. A 84, 063415 (2011);
[Crossref]

Bauer, D.

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B: At. Mol. Opt. Phys. 39, R203–R262 (2006).
[Crossref]

Baye, D.

D. Baye, L. Filippin, and M. Godefroid, “Accurate solution of the Dirac equation on Lagrange meshes,” Phys. Rev. E 89, 043305(2014).
[Crossref]

M. Vincke, L. Malegat, and D. Baye, “Regularization of singularities in Lagrange-mesh calculations,” J. Phys. B: At. Mol. Opt. Phys. 26, 811–826 (1993).
[Crossref]

Becker, A.

A. Becker and F. H. M. Faisal, “Intense-field many-body S-matrix theory,” J. Phys. B: At. Mol. Opt. Phys. 38, R1–R56 (2005).
[Crossref]

Becker, W.

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B: At. Mol. Opt. Phys. 39, R203–R262 (2006).
[Crossref]

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X.-B. Bian and A. D. Bandrauk, “Probing nuclear motion by frequency modulation of molecular high-order harmonic generation,” Phys. Rev. Lett. 113, 193901 (2014).
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P. Eckle, M. Smorlarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. 4, 565–570 (2008).
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Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
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P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, H. G. Muller, M. Bütticker, and U. Keller, “Attosecond ionization and tunneling delay time measurements in helium,” Science 322, 1525–1529 (2008).
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Carré, B.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
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Cirelli, C.

A. N. Pfeiffer, C. Cirelli, M. Smorlarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nat. Phys. 8, 76–80 (2012).
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P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, H. G. Muller, M. Bütticker, and U. Keller, “Attosecond ionization and tunneling delay time measurements in helium,” Science 322, 1525–1529 (2008).
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Corkum, P.

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
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Czasch, A.

J. Wu, M. Meckel, L. P. Schmidt, M. Kunitski, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Probing the tunnelling site of electrons in strong field enhanced ionization of molecules,” Nat. Commun. 3, 1113 (2012).
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J. Wu, L. Schmidt, M. Kunitski, M. Meckel, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Multiorbital tunneling ionization of the CO molecule,” Phys. Rev. Lett. 108, 183001 (2012).
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Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
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M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electrom,” Sov. Phys. JETP 64, 1191 (1986).

DiMauro, L. F.

B. Walker, B. Sheehy, L. F. DiMauro, P. Agostini, K. J. Schafer, and K. C. Kulander, “Precision measurement of strong field double ionization of helium,” Phys. Rev. Lett. 73, 1227 (1994).
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Dimitrovski, D.

A. N. Pfeiffer, C. Cirelli, M. Smorlarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nat. Phys. 8, 76–80 (2012).
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J. Hansen, H. Stapelfeldt, D. Dimitrovski, M. Abu-samha, C. Martiny, and L. Madsen, “Time-resolved photo-electron angular distributions from strong-field ionization of rotating naphthalene molecules,” Phys. Rev. Lett. 106, 073001 (2011).
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L. Holmegaard, J. L. Hansen, L. Kalhøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
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Dinu, L. C.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
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Dorn, A.

R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
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Dörner, R.

J. Wu, M. Meckel, L. P. Schmidt, M. Kunitski, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Probing the tunnelling site of electrons in strong field enhanced ionization of molecules,” Nat. Commun. 3, 1113 (2012).
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J. Wu, L. Schmidt, M. Kunitski, M. Meckel, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Multiorbital tunneling ionization of the CO molecule,” Phys. Rev. Lett. 108, 183001 (2012).
[Crossref] [PubMed]

P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, H. G. Muller, M. Bütticker, and U. Keller, “Attosecond ionization and tunneling delay time measurements in helium,” Science 322, 1525–1529 (2008).
[Crossref] [PubMed]

P. Eckle, M. Smorlarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. 4, 565–570 (2008).
[Crossref]

Dudovich, N.

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
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D. Dundas, “Accurate and efficient non-adiabatic quantum molecular dynamics approach for laser-matter interactions,” J. Phys. B: At. Mol. Opt. Phys. 37, 2883–2901 (2004).
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L. Y. Peng, J. F. McCann, D. Dundas, K. T. Taylor, and I. D. Williams, “A discrete time-dependent method for metastable atoms and molecules in intense fields,” J. Chem. Phys. 120, 10046 (2004).
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P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, H. G. Muller, M. Bütticker, and U. Keller, “Attosecond ionization and tunneling delay time measurements in helium,” Science 322, 1525–1529 (2008).
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P. Eckle, M. Smorlarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. 4, 565–570 (2008).
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D. B. Milošević and F. Ehlotzky, “Scattering and reaction processes in powerful laser fields,” Adv. At. Mol. Opt. Phys. 49, 373–532 (2003).
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A. Becker and F. H. M. Faisal, “Intense-field many-body S-matrix theory,” J. Phys. B: At. Mol. Opt. Phys. 38, R1–R56 (2005).
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C. F. M. Faria and X. Liu, “Electron-electron correlation in strong laser fields,” J. Mod. Opt. 58, 1076–1131 (2011).
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R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
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Filippin, L.

D. Baye, L. Filippin, and M. Godefroid, “Accurate solution of the Dirac equation on Lagrange meshes,” Phys. Rev. E 89, 043305(2014).
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Filsinger, F.

L. Holmegaard, J. L. Hansen, L. Kalhøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
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Frasinski, L. J.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
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Gagnon, J.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
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Godefroid, M.

D. Baye, L. Filippin, and M. Godefroid, “Accurate solution of the Dirac equation on Lagrange meshes,” Phys. Rev. E 89, 043305(2014).
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Goulielmakis, E.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
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Grasbon, F.

W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G.G. Paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effects,” Adv. At. Mol. Opt. Phys. 48, 35–98 (2002).
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Gullikson, E. M.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
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Hansen, J.

J. Hansen, H. Stapelfeldt, D. Dimitrovski, M. Abu-samha, C. Martiny, and L. Madsen, “Time-resolved photo-electron angular distributions from strong-field ionization of rotating naphthalene molecules,” Phys. Rev. Lett. 106, 073001 (2011).
[Crossref]

Hansen, J. L.

L. Holmegaard, J. L. Hansen, L. Kalhøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
[Crossref]

He, L.

F. Wang, L. He, C. Zhai, W. Shi, Q. Zhang, P. Lan, and P. Lu, “Time-dependent phase matching of high-order-harmonic generation,” Phys. Rev. A 92, 063839 (2015).
[Crossref]

L. He, P. Lan, Q. Zhang, C. Zhai, F. Wang, W. Shi, and P. Lu, “Spectrally resolved spatiotemporal features of quantum paths in high-order-harmonic generation,” Phys. Rev. A 92, 043403 (2015).
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He, M.

Henkel, J.

I. Petersen, J. Henkel, and M. Lein, “Signatures of molecular orbital structure in lateral electron momentum distributions from strong-field ionization,” Phys. Rev. Lett. 114, 103004 (2015).
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Herath, T.

T. Herath, L. Yan, S. K. Lee, and W. Li, “Strong-field ionization rate depends on the sign of the magnetic quantum number,” Phys. Rev. Lett. 109, 043004 (2012).
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Hoffmann, K.

R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
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Hofstetter, M.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
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Holmegaard, L.

L. Holmegaard, J. L. Hansen, L. Kalhøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
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Hong, W.

Huang, C.

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

G. Yudin and M. Ivanov, “Nonadiabatic tunnel ionization: looking inside a laser cycle,” Phys. Rev. A 64, 013409 (2001).
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Ivanov, M. Y.

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
[Crossref] [PubMed]

Jahnke, T.

J. Wu, L. Schmidt, M. Kunitski, M. Meckel, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Multiorbital tunneling ionization of the CO molecule,” Phys. Rev. Lett. 108, 183001 (2012).
[Crossref] [PubMed]

J. Wu, M. Meckel, L. P. Schmidt, M. Kunitski, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Probing the tunnelling site of electrons in strong field enhanced ionization of molecules,” Nat. Commun. 3, 1113 (2012).
[Crossref] [PubMed]

Kalhøj, L.

L. Holmegaard, J. L. Hansen, L. Kalhøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
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Kaushal, J.

J. Kaushal, F. Morales, and O. Smirnova, “Opportunities for detecting ring currents using an attoclock setup,” Phys. Rev. A 92, 063405 (2015).
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Keller, U.

A. N. Pfeiffer, C. Cirelli, M. Smorlarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nat. Phys. 8, 76–80 (2012).
[Crossref]

P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, H. G. Muller, M. Bütticker, and U. Keller, “Attosecond ionization and tunneling delay time measurements in helium,” Science 322, 1525–1529 (2008).
[Crossref] [PubMed]

P. Eckle, M. Smorlarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. 4, 565–570 (2008).
[Crossref]

Kienberger, R.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
[Crossref] [PubMed]

Kim, H.

J. Wu, M. Meckel, L. P. Schmidt, M. Kunitski, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Probing the tunnelling site of electrons in strong field enhanced ionization of molecules,” Nat. Commun. 3, 1113 (2012).
[Crossref] [PubMed]

J. Wu, L. Schmidt, M. Kunitski, M. Meckel, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Multiorbital tunneling ionization of the CO molecule,” Phys. Rev. Lett. 108, 183001 (2012).
[Crossref] [PubMed]

Kleineberg, U.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
[Crossref] [PubMed]

Kopold, R.

W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G.G. Paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effects,” Adv. At. Mol. Opt. Phys. 48, 35–98 (2002).
[Crossref]

Korn, G.

R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
[Crossref] [PubMed]

Kovacev, M.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
[Crossref] [PubMed]

Kragh, S. L.

L. Holmegaard, J. L. Hansen, L. Kalhøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
[Crossref]

Krainov, V. P.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electrom,” Sov. Phys. JETP 64, 1191 (1986).

Krausz, F.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
[Crossref] [PubMed]

Kulander, K. C.

B. Walker, B. Sheehy, L. F. DiMauro, P. Agostini, K. J. Schafer, and K. C. Kulander, “Precision measurement of strong field double ionization of helium,” Phys. Rev. Lett. 73, 1227 (1994).
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M. D. Perry, A. Szoke, and K. C. Kulander, “Resonantly enhanced above-threshold ionization of helium,” Phys. Rev. Lett. 63, 1058–1061 (1989).
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Kunitski, M.

J. Wu, L. Schmidt, M. Kunitski, M. Meckel, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Multiorbital tunneling ionization of the CO molecule,” Phys. Rev. Lett. 108, 183001 (2012).
[Crossref] [PubMed]

J. Wu, M. Meckel, L. P. Schmidt, M. Kunitski, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Probing the tunnelling site of electrons in strong field enhanced ionization of molecules,” Nat. Commun. 3, 1113 (2012).
[Crossref] [PubMed]

Küpper, J.

L. Holmegaard, J. L. Hansen, L. Kalhøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
[Crossref]

Lan, P.

L. He, P. Lan, Q. Zhang, C. Zhai, F. Wang, W. Shi, and P. Lu, “Spectrally resolved spatiotemporal features of quantum paths in high-order-harmonic generation,” Phys. Rev. A 92, 043403 (2015).
[Crossref]

F. Wang, L. He, C. Zhai, W. Shi, Q. Zhang, P. Lan, and P. Lu, “Time-dependent phase matching of high-order-harmonic generation,” Phys. Rev. A 92, 063839 (2015).
[Crossref]

K. Liu, Q. Li, P. Lan, and P. Lu, “Enhanced dissociation of H2+ into highly excited states by UV pulses,” Molecular Physics 21, 3247–3252 (2015).
[Crossref]

M. Li, P. Zhang, S. Luo, Y. Zhou, Q. Zhang, P. Lan, and P. Lu, “Selective enhancement of resonant multiphoton ionization with strong laser fields,” Phys. Rev. A 92, 063404 (2015).
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Y. Li, P. Lan, H. Xie, M. He, X. Zhu, Q. Zhang, and P. Lu, “Nonadiabatic tunnel ionization in strong circularly polarized laser fields: counterintuitive angular shifts in the photoelectron momentum distribution,” Opt. Express 23, 28801–28807 (2015).
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Lee, S. K.

T. Herath, L. Yan, S. K. Lee, and W. Li, “Strong-field ionization rate depends on the sign of the magnetic quantum number,” Phys. Rev. Lett. 109, 043004 (2012).
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Lefebvre, C.

C. Lefebvre, Huizhong Lu, and A. D. Bandrauk, “Numerical method for dynamic imaging of nonlinear polyatomic molecules,” J. Phys. Coference Series 341, 012004 (2012).
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Lein, M.

I. Petersen, J. Henkel, and M. Lein, “Signatures of molecular orbital structure in lateral electron momentum distributions from strong-field ionization,” Phys. Rev. Lett. 114, 103004 (2015).
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I. Barth and M. Lein, “Numerical verification of the theory of nonadiabatic tunnel ionization in strong circularly polarized laser fields,” J. Phys. B: At. Mol. Opt. Phys. 47, 204016 (2014).
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Li, M.

M. Li, P. Zhang, S. Luo, Y. Zhou, Q. Zhang, P. Lan, and P. Lu, “Selective enhancement of resonant multiphoton ionization with strong laser fields,” Phys. Rev. A 92, 063404 (2015).
[Crossref]

Li, Q.

K. Liu, Q. Li, P. Lan, and P. Lu, “Enhanced dissociation of H2+ into highly excited states by UV pulses,” Molecular Physics 21, 3247–3252 (2015).
[Crossref]

Li, W.

T. Herath, L. Yan, S. K. Lee, and W. Li, “Strong-field ionization rate depends on the sign of the magnetic quantum number,” Phys. Rev. Lett. 109, 043004 (2012).
[Crossref] [PubMed]

Li, Y.

Liao, Q.

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

X. M. Tong, Z. X. Zhao, and C. D. Lin, “Theory of molecular tunneling ionization,” Phys. Rev. A 66, 033402 (2002).
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M. Li, P. Zhang, S. Luo, Y. Zhou, Q. Zhang, P. Lan, and P. Lu, “Selective enhancement of resonant multiphoton ionization with strong laser fields,” Phys. Rev. A 92, 063404 (2015).
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K. Liu, Q. Li, P. Lan, and P. Lu, “Enhanced dissociation of H2+ into highly excited states by UV pulses,” Molecular Physics 21, 3247–3252 (2015).
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Y. Li, P. Lan, H. Xie, M. He, X. Zhu, Q. Zhang, and P. Lu, “Nonadiabatic tunnel ionization in strong circularly polarized laser fields: counterintuitive angular shifts in the photoelectron momentum distribution,” Opt. Express 23, 28801–28807 (2015).
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L. He, P. Lan, Q. Zhang, C. Zhai, F. Wang, W. Shi, and P. Lu, “Spectrally resolved spatiotemporal features of quantum paths in high-order-harmonic generation,” Phys. Rev. A 92, 043403 (2015).
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M. Li, P. Zhang, S. Luo, Y. Zhou, Q. Zhang, P. Lan, and P. Lu, “Selective enhancement of resonant multiphoton ionization with strong laser fields,” Phys. Rev. A 92, 063404 (2015).
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A. N. Pfeiffer, C. Cirelli, M. Smorlarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nat. Phys. 8, 76–80 (2012).
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M. Vincke, L. Malegat, and D. Baye, “Regularization of singularities in Lagrange-mesh calculations,” J. Phys. B: At. Mol. Opt. Phys. 26, 811–826 (1993).
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J. Hansen, H. Stapelfeldt, D. Dimitrovski, M. Abu-samha, C. Martiny, and L. Madsen, “Time-resolved photo-electron angular distributions from strong-field ionization of rotating naphthalene molecules,” Phys. Rev. Lett. 106, 073001 (2011).
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L. Holmegaard, J. L. Hansen, L. Kalhøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
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L. Y. Peng, J. F. McCann, D. Dundas, K. T. Taylor, and I. D. Williams, “A discrete time-dependent method for metastable atoms and molecules in intense fields,” J. Chem. Phys. 120, 10046 (2004).
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J. Wu, M. Meckel, L. P. Schmidt, M. Kunitski, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Probing the tunnelling site of electrons in strong field enhanced ionization of molecules,” Nat. Commun. 3, 1113 (2012).
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Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
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W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G.G. Paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effects,” Adv. At. Mol. Opt. Phys. 48, 35–98 (2002).
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Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
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J. Kaushal, F. Morales, and O. Smirnova, “Opportunities for detecting ring currents using an attoclock setup,” Phys. Rev. A 92, 063405 (2015).
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R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
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P. Eckle, M. Smorlarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. 4, 565–570 (2008).
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P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, H. G. Muller, M. Bütticker, and U. Keller, “Attosecond ionization and tunneling delay time measurements in helium,” Science 322, 1525–1529 (2008).
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Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
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O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
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D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B: At. Mol. Opt. Phys. 39, R203–R262 (2006).
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Paulus, G.G.

W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G.G. Paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effects,” Adv. At. Mol. Opt. Phys. 48, 35–98 (2002).
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L. Y. Peng, J. F. McCann, D. Dundas, K. T. Taylor, and I. D. Williams, “A discrete time-dependent method for metastable atoms and molecules in intense fields,” J. Chem. Phys. 120, 10046 (2004).
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M. D. Perry, A. Szoke, and K. C. Kulander, “Resonantly enhanced above-threshold ionization of helium,” Phys. Rev. Lett. 63, 1058–1061 (1989).
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A. N. Pfeiffer, C. Cirelli, M. Smorlarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nat. Phys. 8, 76–80 (2012).
[Crossref]

P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, H. G. Muller, M. Bütticker, and U. Keller, “Attosecond ionization and tunneling delay time measurements in helium,” Science 322, 1525–1529 (2008).
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R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
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Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
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R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
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J. Wu, L. Schmidt, M. Kunitski, M. Meckel, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Multiorbital tunneling ionization of the CO molecule,” Phys. Rev. Lett. 108, 183001 (2012).
[Crossref] [PubMed]

J. Wu, M. Meckel, L. P. Schmidt, M. Kunitski, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Probing the tunnelling site of electrons in strong field enhanced ionization of molecules,” Nat. Commun. 3, 1113 (2012).
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B. Walker, B. Sheehy, L. F. DiMauro, P. Agostini, K. J. Schafer, and K. C. Kulander, “Precision measurement of strong field double ionization of helium,” Phys. Rev. Lett. 73, 1227 (1994).
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J. Wu, L. Schmidt, M. Kunitski, M. Meckel, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Multiorbital tunneling ionization of the CO molecule,” Phys. Rev. Lett. 108, 183001 (2012).
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J. Wu, M. Meckel, L. P. Schmidt, M. Kunitski, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Probing the tunnelling site of electrons in strong field enhanced ionization of molecules,” Nat. Commun. 3, 1113 (2012).
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R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
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Schöffler, M.

P. Eckle, M. Smorlarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. 4, 565–570 (2008).
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R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
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Schultze, M.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
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B. Walker, B. Sheehy, L. F. DiMauro, P. Agostini, K. J. Schafer, and K. C. Kulander, “Precision measurement of strong field double ionization of helium,” Phys. Rev. Lett. 73, 1227 (1994).
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L. He, P. Lan, Q. Zhang, C. Zhai, F. Wang, W. Shi, and P. Lu, “Spectrally resolved spatiotemporal features of quantum paths in high-order-harmonic generation,” Phys. Rev. A 92, 043403 (2015).
[Crossref]

F. Wang, L. He, C. Zhai, W. Shi, Q. Zhang, P. Lan, and P. Lu, “Time-dependent phase matching of high-order-harmonic generation,” Phys. Rev. A 92, 063839 (2015).
[Crossref]

Smirnova, O.

J. Kaushal, F. Morales, and O. Smirnova, “Opportunities for detecting ring currents using an attoclock setup,” Phys. Rev. A 92, 063405 (2015).
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I. Barth and O. Smirnova, “Spin-polarized electrons produced by strong-field ionization,” Phys. Rev. A 88, 013401 (2013).
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I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields. II. Derivation of formulas,” Phys. Rev. A 87, 013433 (2013).
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I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields: physical picture and calculations,” Phys. Rev. A 84, 063415 (2011);
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O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
[Crossref] [PubMed]

Smorlarski, M.

A. N. Pfeiffer, C. Cirelli, M. Smorlarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nat. Phys. 8, 76–80 (2012).
[Crossref]

P. Eckle, M. Smorlarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. 4, 565–570 (2008).
[Crossref]

Stapelfeldt, H.

J. Hansen, H. Stapelfeldt, D. Dimitrovski, M. Abu-samha, C. Martiny, and L. Madsen, “Time-resolved photo-electron angular distributions from strong-field ionization of rotating naphthalene molecules,” Phys. Rev. Lett. 106, 073001 (2011).
[Crossref]

L. Holmegaard, J. L. Hansen, L. Kalhøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
[Crossref]

Staudte, A.

P. Eckle, M. Smorlarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. 4, 565–570 (2008).
[Crossref]

P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, H. G. Muller, M. Bütticker, and U. Keller, “Attosecond ionization and tunneling delay time measurements in helium,” Science 322, 1525–1529 (2008).
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M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1965).

Szoke, A.

M. D. Perry, A. Szoke, and K. C. Kulander, “Resonantly enhanced above-threshold ionization of helium,” Phys. Rev. Lett. 63, 1058–1061 (1989).
[Crossref] [PubMed]

Taïzeb, R.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft X-rays,” Science 302, 1540–1543 (2003).
[Crossref] [PubMed]

Taylor, K. T.

L. Y. Peng, J. F. McCann, D. Dundas, K. T. Taylor, and I. D. Williams, “A discrete time-dependent method for metastable atoms and molecules in intense fields,” J. Chem. Phys. 120, 10046 (2004).
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R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
[Crossref] [PubMed]

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E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
[Crossref] [PubMed]

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R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
[Crossref] [PubMed]

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O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
[Crossref] [PubMed]

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M. Vincke, L. Malegat, and D. Baye, “Regularization of singularities in Lagrange-mesh calculations,” J. Phys. B: At. Mol. Opt. Phys. 26, 811–826 (1993).
[Crossref]

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J. Wu, L. Schmidt, M. Kunitski, M. Meckel, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Multiorbital tunneling ionization of the CO molecule,” Phys. Rev. Lett. 108, 183001 (2012).
[Crossref] [PubMed]

J. Wu, M. Meckel, L. P. Schmidt, M. Kunitski, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Probing the tunnelling site of electrons in strong field enhanced ionization of molecules,” Nat. Commun. 3, 1113 (2012).
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B. Walker, B. Sheehy, L. F. DiMauro, P. Agostini, K. J. Schafer, and K. C. Kulander, “Precision measurement of strong field double ionization of helium,” Phys. Rev. Lett. 73, 1227 (1994).
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W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G.G. Paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effects,” Adv. At. Mol. Opt. Phys. 48, 35–98 (2002).
[Crossref]

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F. Wang, L. He, C. Zhai, W. Shi, Q. Zhang, P. Lan, and P. Lu, “Time-dependent phase matching of high-order-harmonic generation,” Phys. Rev. A 92, 063839 (2015).
[Crossref]

L. He, P. Lan, Q. Zhang, C. Zhai, F. Wang, W. Shi, and P. Lu, “Spectrally resolved spatiotemporal features of quantum paths in high-order-harmonic generation,” Phys. Rev. A 92, 043403 (2015).
[Crossref]

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L. Y. Peng, J. F. McCann, D. Dundas, K. T. Taylor, and I. D. Williams, “A discrete time-dependent method for metastable atoms and molecules in intense fields,” J. Chem. Phys. 120, 10046 (2004).
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R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84, 447–450 (2000).
[Crossref] [PubMed]

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J. Wu, L. Schmidt, M. Kunitski, M. Meckel, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Multiorbital tunneling ionization of the CO molecule,” Phys. Rev. Lett. 108, 183001 (2012).
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J. Wu, M. Meckel, L. P. Schmidt, M. Kunitski, S. Voss, H. Sann, H. Kim, T. Jahnke, A. Czasch, and R. Dörner, “Probing the tunnelling site of electrons in strong field enhanced ionization of molecules,” Nat. Commun. 3, 1113 (2012).
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Xu, Z.

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E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
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Figures (6)

Fig. 1
Fig. 1 Time-dependent depletions of the initial wavefunctions P 1(t) and time-dependent populations in the continuum P 2(t) for (a) 2p and (b) 2p+ orbitals driven by a right circularly polarized 3-cycle laser field with 800 nm and E 0 = 0.09 a.u.
Fig. 2
Fig. 2 Ionization probabilities for the 2p orbital and for laser intensities I = 5 × 1014 W/cm2 (blue curve) and I = 1 × 1015 W/cm2 (red curve) and laser wavelengths varying from 110 nm to 6400 nm. Results for right circularly polarized laser fields are presented as solid curves and results for left circularly polarized laser fields are presented as dashed curves.
Fig. 3
Fig. 3 Ionization probabilities for (a) 2p and (b) 2p+ orbitals and for laser intensities from I = 1.1 × 1015 W/cm2 to 1.5 × 1015 W/cm2 and laser wavelengths varying from 110 nm to 300 nm. Ionization probabilities for the 2p orbital and for much higher intensities I = 2.0 × 1015 W/cm2 and I = 3.0 × 1015 W/cm2 of the left circularly polarized laser field are also shown as additional dotted curves in panel (a).
Fig. 4
Fig. 4 Ionization probabilities for (a) 2p and (b) 2p+ orbitals driven by linearly polarized laser fields with laser intensities from I = 1.1×1015 W/cm2 to 1.5 × 1015 W/cm2 and laser wavelength varying from 110 nm to 400 nm. With linearly polarized laser fields, the curves of the ionization probabilities for 2p and 2p+ orbitals are the same for each laser intensity.
Fig. 5
Fig. 5 Time evolution of the occupations of the bound states with different magnetic quantum numbers m starting from the 2p orbital driven by a (a) right or (b) left circularly polarized laser field with laser intensity I = 1.1 × 1015 W/cm2 and laser wavelength 150 nm. The color denotes the value of log10[C(m,t)], where C(m,t) is the sum of modulus square of the projections of the wave function ψ(t) onto field-free bound eigenstates χ k ( m ) for particular m.
Fig. 6
Fig. 6 Analysis of the origin of the ionization peaks. The blue curves present the ionization probabilities for the 2p orbital and for the laser intensity I = 1.1 × 1015 W/cm2 of right (solid) and left (dashed) circularly polarized laser fields. The horizontally arranged red triangles denote the possible resonant wavelengths for the channels m = −1 → −1 + ξ (driven by right circularly polarized laser fields) and the horizontally arranged black rectangles denote the possible resonant wavelengths for the channels m = −1 → −1 −ξ (driven by left circularly polarized laser fields). Red arrows indicate the relationship between the ionization peaks and the excitation channels.

Tables (1)

Tables Icon

Table 1 Energies of the first 10 lowest-lying eigenstates for m = 0, ±1, ±2, ±3, ±4, ±5.

Equations (24)

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i Ψ ( r , t ) t = [ 2 2 + V c ( r ) + V e ( r , t ) ] Ψ ( r , t ) = [ 1 2 ( 1 ρ ρ ρ ρ + 1 ρ 2 2 θ 2 ) + V c ( ρ ) + r E ± ( t ) ] Ψ ( r , t ) ,
E ± ( t ) = E 0 sin 2 ( t π τ ) [ cos ( ω 0 t ) e x ± sin ( ω 0 t ) e y ]
V c ( ρ ) = Z c ρ 2 + a = 1 + 9 exp ( ρ 2 ) ρ 2 + a
ψ ( ρ , θ , t ) = 1 ρ Ψ ( ρ , θ , t ) .
i ψ ( ρ , θ , t ) t = [ 1 2 ( 2 ρ 2 + 1 4 ρ 2 + 1 ρ 2 2 θ 2 ) + V c ( ρ ) + r E ± ( t ) ] ψ ( ρ , θ , t ) .
ψ ( ρ , θ , t ) = 1 M m = M / 2 M / 2 1 ϕ m ( ρ , t ) exp ( i m θ ) ,
i ϕ m ( ρ , t ) t = [ T ρ + T θ + V c ( ρ ) + r E ± ( t ) ] ϕ m ( ρ , t ) = [ 1 2 ( 2 ρ 2 + 1 4 ρ 2 ) + m 2 2 ρ 2 + V c ( ρ ) + r E ± ( t ) ] ϕ m ( ρ , t ) .
g n ( x ) = ( Γ ( n + 1 ) Γ ( α + n + 1 ) ) 1 / 2 x α / 2 e x / 2 L n α ( x ) ,
0 g l ( x ) g n ( x ) = δ ln .
f i ( x ) = λ i 1 / 2 ( 1 g N ( x i ) ) g N ( x ) x x i ,
λ i = 1 x i g N ( x i ) 2 .
( T ρ ) i j = 1 2 h 2 × { ( α + 1 ) 2 4 x i 2 + S i i ( i = j ) ( 1 ) i j [ α + 1 2 x i x j ( 1 x i + 1 x j ) + S i j ] ( i j ) ,
S i j = x i x j p i , j 1 x p ( x p x j ) ( x p x i ) .
( T θ ) i j = m 2 2 h 2 x i 2 i j
( V c ) i j = Z c h 2 x i 2 + a δ i j .
ψ ( ρ , θ , t + δ t ) = exp ( i H 0 δ t / 2 ) exp ( i V e δ t ) exp ( i H 0 δ t / 2 ) ψ ( ρ , θ , t ) + O ( δ t 3 ) ,
ψ ( 1 ) ( ρ , θ , t ) = exp ( i H 0 δ t / 2 ) ψ ( ρ , θ , t ) = 1 M m = M / 2 M / 2 1 exp ( i m θ ) exp [ i H 0 ( m ) δ t / 2 ] ϕ m ( ρ , t ) ,
exp [ i H 0 ( m ) δ t / 2 ] = k = 1 N exp ( i η k ( m ) δ t / 2 ) × | χ k ( m ) χ k ( m ) | .
ψ ( 2 ) ( ρ , θ , t ) = exp ( i V e δ t ) × ψ ( 1 ) ( ρ , θ , t ) .
ψ ( 3 ) ( ρ , θ , t ) = exp ( i H 0 δ t / 2 ) ψ ( 2 ) ( ρ , θ , t ) = 1 M m = M / 2 M / 2 1 exp ( i m θ ) exp [ i H 0 ( m ) δ t / 2 ] ϕ m ( 2 ) ( ρ , t ) .
W ( ρ ) = { 1 , ( ρ ρ a b s ) sin 2 [ ( ρ max ρ ) π 2 ( ρ max ρ a b s ) ] ( ρ > ρ a b s ) .
P 1 ( t ) = 1 | ψ ( ρ , θ , t ) | 2 .
P 2 ( t ) = 1 m = 1 M k = 1 η k ( m ) < 0 N | χ k ( m ) | ψ ( ρ , θ , t ) | 2 .
C ( m , t ) = k = 1 η k ( m ) < 0 N | χ k ( m ) | ψ ( t ) | 2

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