## Scaling law for energy-momentum spectra of atomic photoelectrons |

Optics Express, Vol. 19, Issue 21, pp. 20849-20856 (2011)

http://dx.doi.org/10.1364/OE.19.020849

Acrobat PDF (925 KB)

### Abstract

A scaling law which was used to classify photoelectron angular distributions (PADs) is now extended to photoelectron kinetic energy spectra. Both a theoretical proof and an independent verification are presented. Considering PADs are of photoelectron momentum spectra, this extension really extends the scaling law to the entire energy-momentum spectra. The scaling law for photoelectron energy-momentum spectra applies to both directly ionized and rescattered photoelectrons. Re-scaling experimental input parameters without loosing the physical essence with this scaling law may ease the experimental conditions and reduce the material and the energy consumptions in the experiments.

© 2011 OSA

## 1. Introduction

1. D.-S. Guo, J. Zhang, Z. Xu, X. Li, P. Fu, and R. R. Freeman, “Practical scaling law for photoelectron angular distributions,” Phys. Rev. A **68**, 043404 (2003). [CrossRef]

2. A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H.-C. Bandulet, D. Comtois, F. Legare, M. Giguere, J-C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Wavelength scaling of high harmonic generation efficiency,” Phys. Rev. Lett. **103**, 073902 (2009). [CrossRef] [PubMed]

3. K. L. Ishikawa, E. J. Takahashi, and K. Midorikawa, “Wavelength dependence of high-order harmonic generation with independently controlled ionization and ponderomotive energy,” Phys. Rev. A **80**, 011807 (2009). [CrossRef]

4. J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. **98**, 013901(2007). [CrossRef] [PubMed]

6. J. Chen, B. Zeng, X. Liu, Y. Cheng, and Z. Xu, “Wavelength scaling of high-order harmonic yield from an optically prepared excited state atom,” N. J. Phys. **11**, 113021 (2009). [CrossRef]

11. S. Micheau, Z. Chen, A. T. Le, J. Rauschenberger, M. F. Kling, and C. D. Lin, “Accurate retrieval of target structures and laser parameters of few-cycle pulses from photoelectron momentum spectra,” Phys. Rev. Lett. **102**, 073001 (2009). [CrossRef] [PubMed]

14. R. Torres, N. Kajumba, J. G. Underwood, J. S. Robinson, S. TriTriBaker, J. W. G. Tisch, R. de Nalda, W. A. Bryan, R. Velotta, C. Altucci, I. C. E. Turcu, and J. P. Marangos, “Probing orbital structure of polyatomic molecules by high-order harmonic generation,” Phys. Rev. Lett. **98**, 203007 (2007). [CrossRef] [PubMed]

15. J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P.B. Corkum, and D.M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature (London) **432**, 867–871 (2004). [CrossRef]

16. M. Okunishi, T. Morishita, G. Prumper, K. Shimada, C. D. Lin, S. Watanabe, and K. Ueda, “Experimental retrieval of target structure information from laser-induced rescattered photoelectron momentum distributions,” Phys. Rev. Lett. **100**, 143001 (2008). [CrossRef] [PubMed]

17. Y. Guo, P. Fu, Z.-C. Yan, J. Gong, and B. Wang, “Imaging the geometrical structure of the **80**, 063408 (2009). [CrossRef]

18. B. Wang, Y. Guo, B. Zhang, Z. Zhao, Z.-C. Yan, and P. Fu, “Charge-distribution effect of imaging molecular structure by high-order above-threshold ionization,” Phys. Rev. A **82**, 043402 (2010). [CrossRef]

## 2. The scaling law of photoelectrons

*KE*, shined by a laser beam of frequency

_{b}*Kω*and intensity

*K*

^{3}

*I*is

*K*times of that for an atom with binding energy

*E*, shined by a laser beam of frequency

_{b}*ω*and intensity

*I*. According to this scaling law, different PADs linked by scaling transformations will have the same shape, subjected to the scaling ratio

*K*; while their corresponding kinetic energy spectra will have the similar curve up or down, also subjected to the scaling ratio

*K*. We can also say: Under a scaling transformation which maps (

*E*,

_{b}*ω*,

*I*) to (

*KE*,

_{b}*Kω*,

*K*

^{3}

*I*), the photoelectron energy-momentum spectra in a logarithmic scale are invariant but up to a constant position difference log

*K*.

19. D.-S. Guo, T. Åberg, and B. Crasemann, “Scattering theory of multiphoton ionization in strong fields,” Phys. Rev. A **40**, 4997–5005 (1989). [CrossRef] [PubMed]

*h̄*=

*c*= 1 is used throughout this paper) [20] where

*m*is the rest mass of electron and

_{e}*q*is the number of photons absorbed during the overall ionization process and denotes the ATI order. The first transition matrix element is for the directly emitted photoelectrons in which

*j*and

_{i}*j*are the numbers of absorbed photons in excitation and exit processes, respectively;

_{f}**P**

*is the final momentum of photoelectrons; The second transition matrix element in Eq. (1) comes from the second term in Eq. (33) of Ref. [19*

_{f}19. D.-S. Guo, T. Åberg, and B. Crasemann, “Scattering theory of multiphoton ionization in strong fields,” Phys. Rev. A **40**, 4997–5005 (1989). [CrossRef] [PubMed]

*n*and

_{i}*n*are the numbers of photons before and after interaction, respectively, Φ

_{f}*denotes the initial wave function of the bound electron, and*

_{i}*ϕ*is the final plane wave of the photoelectron.

_{f}*V*is the interaction operator between the electron and the laser field,

*U*denotes the attraction of the ionic core to the electron; The quantity ℰ

_{P,n}is the eigenenergy of the quantized-field Volkov state |Ψ

_{P,n}〉, in which

**P**is the momentum of the electron and

*n*is the number of background photons [19

19. D.-S. Guo, T. Åberg, and B. Crasemann, “Scattering theory of multiphoton ionization in strong fields,” Phys. Rev. A **40**, 4997–5005 (1989). [CrossRef] [PubMed]

**P**| = (2

*m*)

_{e}ω^{1/2}(

*j*

_{i}*− u*

_{p}*−*

*ɛ*)

_{b}^{1/2}, and

*U*(

**P**) is the Fourier transform of the binding potential. In our calculations, we set

*u*equal to

_{p}*j*[19

_{f}**40**, 4997–5005 (1989). [CrossRef] [PubMed]

*z*and

*z*′ are complex variables with the arguments in Eq. (2) and (4) given by where

*ɛ*is the polarization vector of the laser beam. For directly emitted photoelectrons, the order and the arguments of the GPB functions in

*(|*

_{i}**P**

*|) does not affect the PADs in the long-wavelength limit [22*

_{f}22. J. Zhang, W. Zhang, Z. Xu, X. Li, P. Fu, D.-S. Guo, and R. R. Freeman, “The calculation of photoelectron angular distributions with jet-like structure from scattering theory,” J. Phys. B **35**, 4809–4818 (2002). [CrossRef]

_{−jf}(

*ζ*,

_{f}*η*) is still the same, while the values of other two Bessel functions are not. The arguments and the order of the rest two Bessel functions vary with the number of photons absorbed in the ionization process, say

*j*. Since this is a dummy index of the summation, it does not result in any change in the summation of Bessel functions, thus the dynamic part of the formula keeps the same under the scaling change. The form of

_{i}*U*(

**P**

*–*

_{f}**P**) depends on the Coulomb potential. For the widely used screened Coulomb potential we obtain Since the momentum

**P**is also a dummy variable, we expect the value of

*U*(

**P**

*–*

_{f}**P**) does not depend on the direction of

**P**

*. Thus, the PADs of rescattered photoelectrons obey the scaling law.*

_{f}## 3. Numerical verification by the analytical formula

*K*times, and (3) changing the laser intensity as

*K*

^{2},

*K*

^{3}and

*K*

^{4}times of the original value respectively, then (4) comparing the main features of the calculated PADs with the original one. One comparison is depicted in Fig. 1, in which (a) is the PAD served as the reference. In these PADs, we see clearly the main lobes along the laser polarization and the jets striking from the waist of the main lobes. Figure 1(a) depicts the PAD of the 23rd ATI peak of the hydrogen atom irradiated by the laser field of wavelength 800nm and intensity 1.04 × 10

^{14}W/cm

^{2}, while the others depict the PADs of the 23rd ATI peak of a model atom with binding energy 27.20 eV irradiated by the laser field of wavelength 400nm and intensity 4.16 (b), 8.32 (c) and 16.64 (d) × 10

^{14}W/cm

^{2}, respectively. We see the PAD in (c) agrees well with the referenced PAD in (a), while the other two PADs show more or less differences from the referenced PAD, in the size and the number of jets and the breadth of main lobes. The fact that the PADs formed by rescattered photoelectrons are invariant under the scaling transformation confirms the scaling law of PADs of rescattered photoelectrons.

*u*. Since keeping

_{p}*u*unchanged is the most important features of the scaling law, one may anticipate that the energy spectra obey the same scaling law. On the other hand, since the PADs disclose only the relative variation of the ionization rate with the azimuthal angle, the difference in the absolute value of PADs may change the energy spectra of photoelectrons. The question is to what extent this difference changes the photoelectron energy spectra.

_{p}*K*. For directly emitted photoelectrons, as we discussed above, the GPB functions keep unchanged under the scaling transformation, then any change in the energy spectra is caused by the initial wave function Φ

*(|*

_{i}**P**

*|). For hydrogen-like 1s wave function, Φ*

_{f}*(|*

_{i}**P**

*|) is given by where*

_{f}*β*

^{2}and

*K*. The wave function, as well as the value of

*K*

^{−3/4}under the scaling transformation. For rescattered photoelectrons, the changes in the energy spectra depend also on the value of

*U*(

**P**

*–*

_{f}**P**) which varies with the charge number

*Z*and the screen parameter

*λ*. For analytical simplicity, we set

*λ*= 0. According to the relation

*E*∝ –

_{b}*Z*

^{2}/

*n*

^{*2}where

*n*

^{*}is the principle quantum defect number, the value of

*Z*changes as

*K*times. Under the scaling transformation, the value of |

**P**

*–*

_{f}**P**|

^{2}is enlarged by

*K*times, thus the value of

*U*(

**P**

*–*

_{f}**P**) is enlarged by

*K*

^{−1/2}times. Considering the momentum |

**P**| in front of the integral, we find the value of

*(|*

_{i}**P**

*|) and is also enlarged by*

_{f}*K*

^{−3/4}times in the rescattering case. Considering the factor

*ω*

^{5/2}in front of the summation, we find the ionization yield is enlarged by

*K*times under the scaling transformation. This means the energy spectra obey the scaling law, because the overall ionization rate scales as

*K*times under the scaling transformation.

*K*for both directly emitted photoelectrons and re-scattered photoelectrons. Thus we conclude that the kinetic energy spectra obey the same scaling law as the PADs do. In a logarithmic scale, the energy spectra of the re-scaled case are overall shifted by a constant log

*K*.

## 4. Verification by the numerical method

23. D. Bauer and P. Koval, “QPROP: A Schrodinger-solver for intense laser-atom interaction,” Comp. Phys. Comm. **174**, 396–421 (2006) [CrossRef]

23. D. Bauer and P. Koval, “QPROP: A Schrodinger-solver for intense laser-atom interaction,” Comp. Phys. Comm. **174**, 396–421 (2006) [CrossRef]

25. K. J. Schafer and K.C. Kulander, “Energy analysis of time-dependent wave functions: Application to above-threshold ionization,” Phys. Rev. A **42**, 5794–5797 (1990). [CrossRef] [PubMed]

*γ*is the width of the energy bin centered at energy ℰ. In our calculations, we set

*γ*= 5.44 × 10

^{−2}eV and

*n*= 3.

^{13}W/cm

^{2}is chosen as the reference. Same as the aforementioned procedure, we enlarge the atomic binding energy and the laser frequency as 2 times, and vary the laser intensity to 4, 8 and 16 times the original intensity, respectively. Figure 3(a) is the kinetic energy spectrum served as the reference, which exhibits a falloff in the low-energy region and a plateau followed by a cutoff as the kinetic energy increases. Owing to resonant enhancement caused by the level shift in strong laser field [26

26. H. G. Muller and F. C. Kooiman, “Bunching and focusing of tunneling wave packets in enhancement of high-order above-threshold ionization,” Phys. Rev. Lett. **81**, 1207–1210 (1998). [CrossRef]

^{14}W/cm

^{2}. The spectrum exhibits a rapid decrease as the ATI order increases, but the plateau is not remarkable which indicates that the rescattering effect is not notable. Figure 3(d) depicts the calculated spectrum for the model atom for the laser intensity 1.09 × 10

^{15}W/cm

^{2}(2

^{4}times the original). The spectrum shows a striking plateau that is longer than that in Fig. 3(a), and the resonance structure becomes significantly shifted to high-energy part. Figure 3(c) is the spectrum for the model atom for the laser intensity enlarged by 2

^{3}times the original one. The spectrum resembles the referenced one in many aspects. The shape of the two spectra is quite similar to each other and both the spectra begin to ramp down at the same order. A detailed similarity is that the resonance structure starts after the fourth peak. We thus conclude that the spectrum for laser intensity changed by

*K*

^{3}times shows the best agreement with the original one, which verifies the scaling law.

## 5. Conclusions and discussions

## Acknowledgments

## References and links

1. | D.-S. Guo, J. Zhang, Z. Xu, X. Li, P. Fu, and R. R. Freeman, “Practical scaling law for photoelectron angular distributions,” Phys. Rev. A |

2. | A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H.-C. Bandulet, D. Comtois, F. Legare, M. Giguere, J-C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Wavelength scaling of high harmonic generation efficiency,” Phys. Rev. Lett. |

3. | K. L. Ishikawa, E. J. Takahashi, and K. Midorikawa, “Wavelength dependence of high-order harmonic generation with independently controlled ionization and ponderomotive energy,” Phys. Rev. A |

4. | J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. |

5. | J. A. Pérez-Hernández, J. Ramos, L. Roso, and L. Plaja, “Harmonic generation beyond the strong-field approximation: phase and temporal description,” Laser Phys. |

6. | J. Chen, B. Zeng, X. Liu, Y. Cheng, and Z. Xu, “Wavelength scaling of high-order harmonic yield from an optically prepared excited state atom,” N. J. Phys. |

7. | H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A |

8. | L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP |

9. | X. Zhang, J. Zhang, L. Bai, Q. Gong, and Z. Xu, “Verification of a scaling law in few-cycle laser pulses,” Opt. Express |

10. | J. Zhang, L. Bai, S. Gong, and Z. Xu, “A scaling law of photoionization in few-cycle regime,” Opt. Express |

11. | S. Micheau, Z. Chen, A. T. Le, J. Rauschenberger, M. F. Kling, and C. D. Lin, “Accurate retrieval of target structures and laser parameters of few-cycle pulses from photoelectron momentum spectra,” Phys. Rev. Lett. |

12. | K. Yoshii, G. Miyaji, and K. Miyazaki, “Retrieving Angular Distributions of high-order harmonic generation from a single molecule,” Phys. Rev. Lett. |

13. | A. D. Shiner, B. E. Schmidt, C. Trallero-Herrero, H. J. Worner, S. Patchkovskii, P. B. Corkum, J-C. Kieffer, F. Legare, and D. M. Villeneuve, “Probing collective multi-electron dynamics in xenon with high-harmonic spectroscopy,” Nat. Phys. |

14. | R. Torres, N. Kajumba, J. G. Underwood, J. S. Robinson, S. TriTriBaker, J. W. G. Tisch, R. de Nalda, W. A. Bryan, R. Velotta, C. Altucci, I. C. E. Turcu, and J. P. Marangos, “Probing orbital structure of polyatomic molecules by high-order harmonic generation,” Phys. Rev. Lett. |

15. | J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P.B. Corkum, and D.M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature (London) |

16. | M. Okunishi, T. Morishita, G. Prumper, K. Shimada, C. D. Lin, S. Watanabe, and K. Ueda, “Experimental retrieval of target structure information from laser-induced rescattered photoelectron momentum distributions,” Phys. Rev. Lett. |

17. | Y. Guo, P. Fu, Z.-C. Yan, J. Gong, and B. Wang, “Imaging the geometrical structure of the |

18. | B. Wang, Y. Guo, B. Zhang, Z. Zhao, Z.-C. Yan, and P. Fu, “Charge-distribution effect of imaging molecular structure by high-order above-threshold ionization,” Phys. Rev. A |

19. | D.-S. Guo, T. Åberg, and B. Crasemann, “Scattering theory of multiphoton ionization in strong fields,” Phys. Rev. A |

20. | Yan Wu, Huiliang Ye, Jingtao Zhang, and D.-S. Guo are preparing a manuscript to be called “Nonperturbative quantum electrodynamics theory of rescattering effect of photoelectrons in intense laser fields.” |

21. | X. Hu, H. X. Wang, and D.-S. Guo, “Phased bessel functions,” Can. J. Phys. |

22. | J. Zhang, W. Zhang, Z. Xu, X. Li, P. Fu, D.-S. Guo, and R. R. Freeman, “The calculation of photoelectron angular distributions with jet-like structure from scattering theory,” J. Phys. B |

23. | D. Bauer and P. Koval, “QPROP: A Schrodinger-solver for intense laser-atom interaction,” Comp. Phys. Comm. |

24. | H. G. Muller, “An efficient propagation scheme for the time-dependent Schrödinger equation in the velocity gauge,” Laser Phys. |

25. | K. J. Schafer and K.C. Kulander, “Energy analysis of time-dependent wave functions: Application to above-threshold ionization,” Phys. Rev. A |

26. | H. G. Muller and F. C. Kooiman, “Bunching and focusing of tunneling wave packets in enhancement of high-order above-threshold ionization,” Phys. Rev. Lett. |

**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: August 9, 2011

Revised Manuscript: September 19, 2011

Manuscript Accepted: September 20, 2011

Published: October 5, 2011

**Citation**

Huiliang Ye, Yan Wu, Jingtao Zhang, and D.-S. Guo, "Scaling law for energy-momentum spectra of atomic photoelectrons," Opt. Express **19**, 20849-20856 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20849

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### References

- D.-S. Guo, J. Zhang, Z. Xu, X. Li, P. Fu, and R. R. Freeman, “Practical scaling law for photoelectron angular distributions,” Phys. Rev. A68, 043404 (2003). [CrossRef]
- A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H.-C. Bandulet, D. Comtois, F. Legare, M. Giguere, J-C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Wavelength scaling of high harmonic generation efficiency,” Phys. Rev. Lett.103, 073902 (2009). [CrossRef] [PubMed]
- K. L. Ishikawa, E. J. Takahashi, and K. Midorikawa, “Wavelength dependence of high-order harmonic generation with independently controlled ionization and ponderomotive energy,” Phys. Rev. A80, 011807 (2009). [CrossRef]
- J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett.98, 013901(2007). [CrossRef] [PubMed]
- J. A. Pérez-Hernández, J. Ramos, L. Roso, and L. Plaja, “Harmonic generation beyond the strong-field approximation: phase and temporal description,” Laser Phys.20, 1044–1050 (2010). [CrossRef]
- J. Chen, B. Zeng, X. Liu, Y. Cheng, and Z. Xu, “Wavelength scaling of high-order harmonic yield from an optically prepared excited state atom,” N. J. Phys.11, 113021 (2009). [CrossRef]
- H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A22, 1786–1813 (1980). [CrossRef]
- L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP20, 1307–1314 (1965).
- X. Zhang, J. Zhang, L. Bai, Q. Gong, and Z. Xu, “Verification of a scaling law in few-cycle laser pulses,” Opt. Express13, 8708–8716 (2005). [CrossRef] [PubMed]
- J. Zhang, L. Bai, S. Gong, and Z. Xu, “A scaling law of photoionization in few-cycle regime,” Opt. Express15, 7261–7268 (2007). [CrossRef] [PubMed]
- S. Micheau, Z. Chen, A. T. Le, J. Rauschenberger, M. F. Kling, and C. D. Lin, “Accurate retrieval of target structures and laser parameters of few-cycle pulses from photoelectron momentum spectra,” Phys. Rev. Lett.102, 073001 (2009). [CrossRef] [PubMed]
- K. Yoshii, G. Miyaji, and K. Miyazaki, “Retrieving Angular Distributions of high-order harmonic generation from a single molecule,” Phys. Rev. Lett.106, 013904 (2011). [CrossRef] [PubMed]
- A. D. Shiner, B. E. Schmidt, C. Trallero-Herrero, H. J. Worner, S. Patchkovskii, P. B. Corkum, J-C. Kieffer, F. Legare, and D. M. Villeneuve, “Probing collective multi-electron dynamics in xenon with high-harmonic spectroscopy,” Nat. Phys.7, 464–467 (2011). [CrossRef]
- R. Torres, N. Kajumba, J. G. Underwood, J. S. Robinson, S. TriTriBaker, J. W. G. Tisch, R. de Nalda, W. A. Bryan, R. Velotta, C. Altucci, I. C. E. Turcu, and J. P. Marangos, “Probing orbital structure of polyatomic molecules by high-order harmonic generation,” Phys. Rev. Lett.98, 203007 (2007). [CrossRef] [PubMed]
- J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P.B. Corkum, and D.M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature (London)432, 867–871 (2004). [CrossRef]
- M. Okunishi, T. Morishita, G. Prumper, K. Shimada, C. D. Lin, S. Watanabe, and K. Ueda, “Experimental retrieval of target structure information from laser-induced rescattered photoelectron momentum distributions,” Phys. Rev. Lett.100, 143001 (2008). [CrossRef] [PubMed]
- Y. Guo, P. Fu, Z.-C. Yan, J. Gong, and B. Wang, “Imaging the geometrical structure of the H2+ molecular ion by high-order above-threshold ionization in an intense laser field,” Phys. Rev. A80, 063408 (2009). [CrossRef]
- B. Wang, Y. Guo, B. Zhang, Z. Zhao, Z.-C. Yan, and P. Fu, “Charge-distribution effect of imaging molecular structure by high-order above-threshold ionization,” Phys. Rev. A82, 043402 (2010). [CrossRef]
- D.-S. Guo, T. Åberg, and B. Crasemann, “Scattering theory of multiphoton ionization in strong fields,” Phys. Rev. A40, 4997–5005 (1989). [CrossRef] [PubMed]
- Yan Wu, Huiliang Ye, Jingtao Zhang, and D.-S. Guo are preparing a manuscript to be called “Nonperturbative quantum electrodynamics theory of rescattering effect of photoelectrons in intense laser fields.”
- X. Hu, H. X. Wang, and D.-S. Guo, “Phased bessel functions,” Can. J. Phys.96, 863–870 (2008).
- J. Zhang, W. Zhang, Z. Xu, X. Li, P. Fu, D.-S. Guo, and R. R. Freeman, “The calculation of photoelectron angular distributions with jet-like structure from scattering theory,” J. Phys. B35, 4809–4818 (2002). [CrossRef]
- D. Bauer and P. Koval, “QPROP: A Schrodinger-solver for intense laser-atom interaction,” Comp. Phys. Comm.174, 396–421 (2006) [CrossRef]
- H. G. Muller, “An efficient propagation scheme for the time-dependent Schrödinger equation in the velocity gauge,” Laser Phys.9, 138–148 (1999).
- K. J. Schafer and K.C. Kulander, “Energy analysis of time-dependent wave functions: Application to above-threshold ionization,” Phys. Rev. A42, 5794–5797 (1990). [CrossRef] [PubMed]
- H. G. Muller and F. C. Kooiman, “Bunching and focusing of tunneling wave packets in enhancement of high-order above-threshold ionization,” Phys. Rev. Lett.81, 1207–1210 (1998). [CrossRef]

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