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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 21 — Oct. 10, 2011
  • pp: 20849–20856
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Scaling law for energy-momentum spectra of atomic photoelectrons

Huiliang Ye, Yan Wu, Jingtao Zhang, and D.-S. Guo  »View Author Affiliations


Optics Express, Vol. 19, Issue 21, pp. 20849-20856 (2011)
http://dx.doi.org/10.1364/OE.19.020849


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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

Scaling and re-scaling are powerful tools for analyzing various physical phenomena. Searching for new scaling laws and applying these laws are interesting research topics and possess both fundamental importance and practical applications. For example, in designing a strong-laser experiment, if one can re-choose the sample atom with one-half binding energy and the laser beam with one-half frequency, also re-scale the laser beam intensity as one eighth, according to our previously published scaling law [1

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]

], one will observe the same physical phenomenon. If experiments can be performed in a re-scaled way, not only the experimental conditions will be greatly eased, but also the energy and the material consumptions will be greatly reduced. Many discussions have been devoted to scaling laws in treating atoms and molecules interacting with intense laser fields. Most studies of scaling relations of high-order harmonic generation in intense laser fields employed numerical methods [2

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

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]

]. Many scaling relations on the wavelength of the driving laser field were disclosed [4

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

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]

] by numerical methods while the deeper physical understandings still remain open. Thus, the theoretical study of scaling laws from basic physics relations is still needed with immediate applications.

Recently, the high-energy photoelectrons released from atoms and molecules were used to extract the structure information of targets [11

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

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]

]. When these laser-induced photoelectrons fall back to the vicinity of the target core, the Coulomb force dominates their subsequent motion which brings out the structure information of the target core. The harmonics generated during the recombination of photoelectron with its parent core were used to reconstruct the tomography of the outmost shell of target molecules [15

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]

]. The high-energy photoelectrons rescattered by the ionic parent core were used to retrieve the structure information of target atoms [16

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]

] and molecules [17

17. 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. A 80, 063408 (2009). [CrossRef]

, 18

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]

]. To reach this purpose, one needs to use the scaling law to classify all information from the irradiated targets. However, a question is, does the scaling relation still hold for rescattered photoelectrons, since the scaling law mentioned above is obtained for photoelectrons directly emitted?

2. The scaling law of photoelectrons

The extended scaling law states: the photoelectron production rate in one direction, as well as the total rate, for an atom with binding energy KEb, shined by a laser beam of frequency and intensity K3I is K times of that for an atom with binding energy Eb, shined by a laser beam of frequency 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 (Eb, ω, I) to (KEb, , K3I), the photoelectron energy-momentum spectra in a logarithmic scale are invariant but up to a constant position difference logK.

The scaling relation between the laser frequency and the atomic binding energy is trivial, while the cubic relation pertinent to the laser intensity is the key of the scaling law and highly nontrivial.

3. Numerical verification by the analytical formula

Then, we numerically verify the scaling law using the derived formula through the following steps: (1) choosing the PAD of one ATI peak as a reference, (2) enlarging the atomic binding energy and the laser frequency as K times, and (3) changing the laser intensity as K2, K3 and K4 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 × 1014 W/cm2, 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) × 1014 W/cm2, 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.

Fig. 1 Polar plots of the PADs of the 23rd ATI peaks. (a) serves as a reference which is for hydrogen atoms irradiated by laser field of intensity 1.04 × 1014 W/cm2 and of wavelength 800 nm. (b), (c) and (d) are from a model atoms with binding energy 27.20 eV, irradiated by laser field of wavelength 400 nm and of intensity 4.16, 8.32, and 16.64 × 1014 W/cm2, respectively. The similarity between the PADs in (a) and (c) verifies the scaling law. We set λ = 0.1 and change it as K times in the scaling transformation.

Next, we prove that the scaling law which applies well for the PADs also applies for the energy spectra of photoelectrons. The ionization rate of a given ATI peak is the angle-integrated PAD. Thus the energy spectra of photoelectrons are expected to obey the same scaling law. Furthermore, in the unit of laser photon energy, the plateau in the energy spectra cuts off at 10up. Since keeping up 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.

We first analysis the change of energy spectra under the scaling transformation with a ratio constant 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 Φi(|Pf|). For hydrogen-like 1s wave function, Φi(|Pf|) is given by
Φi|Pf|=8(πβ5)1/2(β2+|Pf|2)2,
(9)
where β=2meEb. In the denominator both β2 and Pf2=2me(qωEbupω) scale as constant K. The wave function, as well as the value of Tfid, scales as K−3/4 under the scaling transformation. For rescattered photoelectrons, the changes in the energy spectra depend also on the value of U(PfP) which varies with the charge number Z and the screen parameter λ. For analytical simplicity, we set λ = 0. According to the relation Eb ∝ – Z2/n*2 where n* is the principle quantum defect number, the value of Z changes as K times when the binding energy is enlarged by K times. Under the scaling transformation, the value of |PfP|2 is enlarged by K times, thus the value of U(PfP) is enlarged by K−1/2 times. Considering the momentum |P| in front of the integral, we find the value of Tfir changes as that of Φi(|Pf|) and is also enlarged by K−3/4 times in the rescattering case. Considering the factor ω5/2 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.

To show numerically, we calculate the energy spectra changed according to the scaling law, then compare them with the original one. The comparison is shown in Fig. 2 for two different scaling ratios. The re-scaled photoelectron energy spectra are almost the same as the original one, and the only difference is that the rescaled spectra are overall shifted by a ratio of 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 logK.

Fig. 2 (Color online) The calculated energy spectra for different scaling ratios. The original one is for the hydrogen atoms irradiated by laser field of wavelength 800 nm and intensity 1.04 × 1014 W/cm2. The rest two spectra are these calculated according to the scaling law with different scaling ratios. The value of λ is chosen as 0.1.

4. Verification by the numerical method

Verification by independent methods provides a strong check to the scaling law. In the following we verify the scaling law of photoelectron energy spectra by the time-dependent Schrödinger equation (TDSE) method, which is widely used in the study of the interaction process of atoms and molecules with intense laser fields. Under the single active electron approximation, the TDSE is solved by the QPROP code [23

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

] in the velocity gauge, in which the spherical coordinate system is employed and the wave function is expanded on the radial-position angular-moment grid. The detailed procedure is presented in Refs. [23

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

,24

24. H. G. Muller, “An efficient propagation scheme for the time-dependent Schrödinger equation in the velocity gauge,” Laser Phys. 9, 138–148 (1999).

]. The simulation is done with 60 angular momenta involved. The laser field is chosen as that a 16-cycle flat-top laser pulse with a half-cycle in both the raising and the tailing edges. The target is chosen as the argon atom in its ground state with binding energy 15.76 eV. The kinetic energy spectrum of photoelectrons is obtained by employing a window operator on the time-resolved wave function. The window operator is taken as [25

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]

]
Wγ()=γ2n(H0)2n+γ2n,
(10)
where 2γ 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.

The photoelectron spectrum of the argon atom in the laser pulse of wavelength 800 nm and intensity 6.80×1013 W/cm2 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]

], the 5th to 8th ATI peaks become higher than their neighboring peaks. Figure 3(b) depicts the spectrum for a model atom with binding energy 31.52 eV irradiated by laser field of wavelength 400 nm and intensity 2.72 × 1014 W/cm2. 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 × 1015 W/cm2 (24 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 23 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 K3 times shows the best agreement with the original one, which verifies the scaling law.

Fig. 3 The kinetic energy spectra of photoelectrons calculated by the TDSE method: (a) is for argon atoms in laser field of wavelength 800nm and intensity 6.80 × 1013 W/cm2; (b)–(d) are calculated spectra for a model atom with binding energy 31.52 eV irradiated by laser field of wavelength 400 nm and intensity 2.72 × 1014 W/cm2 (b); 5.44 × 1014 W/cm2 (c); 1.09 × 1015 W/cm2 (d). For convenience of comparison, the abscissa is plotted in the unit of ω.

5. Conclusions and discussions

In this paper we establish the scaling law of photoelectron energy spectra including the rescattered photoelectrons. We present the detailed verifications of the scaling law by analytical and numerical methods. We show that the scaling law working for photoelectron momentum spectra can be well extended to the photoelectron energy spectra. The well-extended scaling relation for photoelectron four-momentum spectra applies for both directly ionized and rescattered photoelectrons.

The scaling law can be used to obtain the necessary reference to extract detailed structure information of the targets. This has potential applications in the study of molecules. Since the scaling law states the same dynamics behavior while the wave function determines the initial electronic state before interaction, the scaling relation depends less on the wave function. The numerical calculations also support this statement. This result implies that the scaling law holds for the molecular case, although the single-active-electron wave function of molecules is quite different from that of atoms.

Acknowledgments

This work was supported by the Chinese National Natural Science Foundation under Grant Nos. 10774153, 61078080, and 11174304, and by the 973 Program of China under Grant Nos. 2010CB923203 and 2011CB808103.

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 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]

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. 20, 1044–1050 (2010). [CrossRef]

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]

7.

H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A 22, 1786–1813 (1980). [CrossRef]

8.

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).

9.

X. Zhang, J. Zhang, L. Bai, Q. Gong, and Z. Xu, “Verification of a scaling law in few-cycle laser pulses,” Opt. Express 13, 8708–8716 (2005). [CrossRef] [PubMed]

10.

J. Zhang, L. Bai, S. Gong, and Z. Xu, “A scaling law of photoionization in few-cycle regime,” Opt. Express 15, 7261–7268 (2007). [CrossRef] [PubMed]

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]

12.

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]

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. 7, 464–467 (2011). [CrossRef]

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 H2+ molecular ion by high-order above-threshold ionization in an intense laser field,” Phys. Rev. A 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]

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]

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. 96, 863–870 (2008).

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]

23.

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

24.

H. G. Muller, “An efficient propagation scheme for the time-dependent Schrödinger equation in the velocity gauge,” Laser Phys. 9, 138–148 (1999).

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]

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]

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

  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. A68, 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. A80, 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]
  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.20, 1044–1050 (2010). [CrossRef]
  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]
  7. H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A22, 1786–1813 (1980). [CrossRef]
  8. L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP20, 1307–1314 (1965).
  9. 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]
  10. 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]
  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]
  12. 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]
  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.7, 464–467 (2011). [CrossRef]
  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 H2+ molecular ion by high-order above-threshold ionization in an intense laser field,” Phys. Rev. A80, 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. A82, 043402 (2010). [CrossRef]
  19. D.-S. Guo, T. Åberg, and B. Crasemann, “Scattering theory of multiphoton ionization in strong fields,” Phys. Rev. A40, 4997–5005 (1989). [CrossRef] [PubMed]
  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.96, 863–870 (2008).
  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. B35, 4809–4818 (2002). [CrossRef]
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