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arxiv: 2604.27386 · v1 · pith:5H3BYDTWnew · submitted 2026-04-30 · ⚛️ physics.atom-ph · physics.optics

Intermediate-state Coulomb-corrected strong-field approximation for rescattering processes

Chunli Miao , Jiarui Qin , Chan Li , Xiaolei Hao , Weidong Li , Jing Chen This is my paper

Pith reviewed 2026-05-07 09:58 UTC · model grok-4.3

classification ⚛️ physics.atom-ph physics.optics
keywords strong-field approximationCoulomb correctionabove-threshold ionizationrescatteringCoulomb focusingS-matrixatomic hydrogenlaser-atom interaction
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The pith

Incorporating Coulomb interactions during the electron's intermediate propagation improves the strong-field approximation for rescattering in above-threshold ionization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives an all-order strong-field S-matrix series that adds Coulomb-Volkov corrections in the intermediate states rather than only the final state. For the second-order rescattering term applied to atomic hydrogen in linear laser fields, the resulting ICSFA spectra match time-dependent Schrödinger equation solutions more closely than either the plain strong-field approximation or the final-state Coulomb-corrected version. The improvement occurs because the intermediate corrections increase the contribution of third- and fourth-return rescattering trajectories and change the interference structure in the energy spectrum, effects the authors link to Coulomb focusing through modified ionization yields and scattering cross sections.

Core claim

The authors analytically derive the all-order strong-field S-matrix series incorporating intermediate-state Coulomb-Volkov corrections. Focusing on the second-order rescattering term, they show that this ICSFA provides superior agreement with TDSE results for ATI spectra of hydrogen compared to standard SFA and FCSFA. Intermediate-state Coulomb effects enhance the yield of multi-return-recollision trajectories, equivalent to a Coulomb focusing effect, by modifying ionization yield and scattering cross-section.

What carries the argument

The intermediate-state Coulomb-Volkov corrected all-order S-matrix series, truncated at the second-order rescattering term, which accounts for Coulomb interactions while the freed electron propagates before rescattering.

If this is right

  • The yield of third- and fourth-return rescattering trajectories increases relative to lower-order paths.
  • Interference patterns in the photoelectron energy spectrum are altered by the intermediate corrections.
  • The net effect is equivalent to Coulomb focusing acting on both ionization and scattering stages.
  • Analytical predictions of rescattering processes become more reliable without requiring full numerical solution of the Schrödinger equation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same intermediate-state correction could be tested on molecular targets where orientation-dependent rescattering matters.
  • Extending the truncation to third-order terms might reveal whether further gains appear at higher photoelectron energies.
  • The approach supplies a practical route to model Coulomb effects in high-harmonic generation or non-sequential double ionization without abandoning the S-matrix framework.

Load-bearing premise

Truncating the all-order S-matrix series to the second-order rescattering term captures the dominant physics in the ATI spectra of hydrogen without significant omissions from higher-order terms.

What would settle it

Direct numerical comparison of ICSFA spectra against TDSE results for a different atom such as helium or for circularly polarized driving fields at comparable intensities would show whether the reported agreement holds or breaks.

Figures

Figures reproduced from arXiv: 2604.27386 by Chan Li, Chunli Miao, Jiarui Qin, Jing Chen, Weidong Li, Xiaolei Hao.

Figure 1
Figure 1. Figure 1: FIG. 1: (color online) The photoelectron spectra of hydroge view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (color online) The logarithm of the ionization proba view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (color online) Photoelectron energy spectra (PES) view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (color online) (a) shows the contributions from the view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: (color online) Photoelectron energy spectra at 800 n view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: (color online) Dependence of the proportion of differ view at source ↗
read the original abstract

We analytically derive the all-order strong-field S-matrix series incorporating intermediate-state Coulomb-Volkov corrections (ICSFA). Focusing on rescattering processes described by the second-order term, we systematically investigate the impact of intermediate-state Coulomb interactions on above-threshold ionization (ATI) spectra of atomic hydrogen in linearly polarized laser fields. Crucially, ICSFA spectra demonstrate superior agreement with the results obtained by numerically solving the time-dependent Schr\"{o}dinger equation compared to the standard strong-field approximation (SFA) and final-state Coulomb-corrected SFA (FCSFA). Our analysis reveals that intermediate-state Coulomb corrections enhance the yield of the third- and fourth-return-recollision trajectories while modifying interference patterns in the energy spectrum. The observed enhancement of the multi-return-recollision trajectories can be attributed to modifications of the ionization yield and scattering cross-section, which are induced by intermediate-state Coulomb effects. These effects are equivalent to the so-called Coulomb focusing effect.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper analytically derives the all-order strong-field S-matrix series incorporating intermediate-state Coulomb-Volkov corrections (ICSFA). It focuses on the second-order rescattering term and compares above-threshold ionization (ATI) spectra of atomic hydrogen in linearly polarized laser fields against the standard SFA and final-state Coulomb-corrected SFA (FCSFA). The central claim is that the ICSFA truncation yields superior agreement with TDSE solutions, with intermediate-state Coulomb effects enhancing third- and fourth-return recollision trajectories via modifications to ionization yield and scattering cross-section, equivalent to Coulomb focusing.

Significance. If the truncation is justified and the TDSE agreement is quantitatively robust across parameter ranges, the work would provide a useful analytic framework for incorporating intermediate Coulomb corrections into rescattering calculations without full numerical TDSE solution. The link between intermediate-state effects and Coulomb focusing offers interpretive value for ATI spectra. The all-order derivation itself is a technical contribution, though its practical impact hinges on validation of the second-order approximation.

major comments (2)
  1. [Theory section on S-matrix series and results on ATI spectra] The central claim of superior TDSE agreement rests on the second-order truncation of the all-order ICSFA series. No convergence test, estimate of third- or higher-order contributions, or comparison of spectra with and without higher terms is provided for the hydrogen ATI cases at the intensities and wavelengths considered. This is load-bearing because the reported enhancement of third- and fourth-return trajectories could be altered if higher orders affect ionization or scattering comparably.
  2. [Abstract and comparison with TDSE results] The abstract and results assert superior agreement with TDSE over SFA and FCSFA, but no quantitative metrics (e.g., integrated error, pointwise deviations, or R² values for spectral features) or error analysis are referenced. Without these, the improvement cannot be assessed as systematic rather than visual or parameter-specific.
minor comments (2)
  1. [Figures showing spectra] Laser parameters (intensity, wavelength, pulse duration) should be stated explicitly in the figure captions or a dedicated table for reproducibility of the ATI spectra.
  2. [Theory section] Notation for the intermediate-state Coulomb-Volkov wave functions could be clarified with a brief reminder of the Volkov phase and Coulomb correction terms when first introduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the opportunity to clarify aspects of our work on the ICSFA. We address each major comment below.

read point-by-point responses

  1. Referee: [Theory section on S-matrix series and results on ATI spectra] The central claim of superior TDSE agreement rests on the second-order truncation of the all-order ICSFA series. No convergence test, estimate of third- or higher-order contributions, or comparison of spectra with and without higher terms is provided for the hydrogen ATI cases at the intensities and wavelengths considered. This is load-bearing because the reported enhancement of third- and fourth-return trajectories could be altered if higher orders affect ionization or scattering comparably.

    Authors: We acknowledge the importance of justifying the second-order truncation. The all-order ICSFA series is derived analytically in the manuscript, with higher-order terms corresponding to additional rescatterings whose amplitudes scale with the product of ionization and scattering matrix elements. For the laser parameters considered (linearly polarized fields at intensities where single rescattering dominates), these contributions are suppressed by more than an order of magnitude relative to the second-order term, as can be estimated from the known decay of rescattering probabilities in SFA. We will add a brief scaling argument and order-of-magnitude estimate in the revised Theory section to support the truncation without performing full higher-order numerics. revision: partial

  2. Referee: [Abstract and comparison with TDSE results] The abstract and results assert superior agreement with TDSE over SFA and FCSFA, but no quantitative metrics (e.g., integrated error, pointwise deviations, or R² values for spectral features) or error analysis are referenced. Without these, the improvement cannot be assessed as systematic rather than visual or parameter-specific.

    Authors: The manuscript demonstrates the improvement through direct overlay of spectra in the figures, where ICSFA reproduces TDSE peak positions, heights, and the enhancement of third- and fourth-return features more accurately than SFA or FCSFA. While no explicit error metrics were computed, the agreement is systematic across the displayed energy range and multiple intensities. We will incorporate quantitative measures such as integrated absolute deviation between each approximation and the TDSE reference in the revised Results section to allow objective assessment. revision: yes

Circularity Check

0 steps flagged

Analytic S-matrix derivation is self-contained with external TDSE benchmark

full rationale

The paper derives the all-order ICSFA S-matrix series analytically from the time-dependent Schrödinger equation in the strong-field limit, then restricts attention to the second-order rescattering term without fitting parameters or invoking self-citations for the core result. Superior agreement with TDSE spectra is reported as an independent numerical check rather than a tautological outcome. No load-bearing step reduces to a prior fitted quantity, renamed ansatz, or self-referential uniqueness theorem; the truncation is explicitly an approximation whose accuracy is tested externally.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields limited visibility into parameters or assumptions; the derivation rests on the standard strong-field S-matrix framework and Volkov states.

axioms (1)
  • domain assumption Strong-field approximation and S-matrix expansion for laser-driven electron dynamics
    Invoked as the base for adding intermediate-state corrections; standard in the field but not re-derived here.

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Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [1]

    Krausz and M

    F. Krausz and M. Ivanov, Attosecond physics, Rev. Mod. Phys. 81, 163 (2009)

    work page 2009

  2. [2]

    Ferray, A

    M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompre, G. Mainfray, and C. Manus, Multiple-harmonic conversion of 1064 nm radiation in rare gases, J. Phys. B: At. Mol. Opt. Phys. 21, L31 (1988)

    work page 1988

  3. [3]

    D. S. Guo, C. Yu, J. T. Zhang, J. Gao, Z. -W. Sun, and Z. R. Sun, On the cutoff law of laser induced high harmonic spectra, Front. Phys. 10(2), 103201 (2015)

    work page 2015

  4. [4]

    C. Yu, J. T. Zhang, Z. -W. Sun, Z. R. Sun, and D. S. Guo, A nonperturbative quantum electrodynamic approach to the theory of laser induced high harmonic generation, Front. Phys. 10(4), 103202 (2015)

    work page 2015

  5. [5]

    Gohle, T

    C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, T. W. H¨ ansch, A frequency comb in the extreme ultraviolet, Nature 436(7048), 234 (2005)

    work page 2005

  6. [6]

    Y. Bai, F. Fei, S. Wang, N. Li, X. Li, F. Song, R. Li, Z. Xu, and P. Liu, High-harmonic generation from topolog- ical surface states, Nat. Phys. 17(3), 311 (2021)

    work page 2021

  7. [7]

    Heide, Y

    C. Heide, Y. Kobayashi, A. C. Johnson, F. Liu, T. F. Heinz, D. A. Reis, and S. Ghimire, Probing electron-hole coherence in strongly driven 2D materials using high- harmonic generation, Optica 9(5), 512 (2022)

    work page 2022

  8. [8]

    Y. Y. Lv, J. Xu, S. Han, C. Zhang, Y. Han, J. Zhou, S. H. Yao, X. P. Liu, M. H. Lu, H. Weng, Z. Xie, Y. B. Chen, J. Hu, Y. F. Chen, and S. Zhu, High-harmonic generation in Weyl semimetal β -WP2 crystals, Nat. Commun. 12, 6437 (2021)

    work page 2021

  9. [9]

    S. Cui, X. Huang, C. Sun, H. Yang, and X. Li, Efficient generation of polarization multiplexed OAM using levi- tated metasurfaces, Front. Phys. (Beijing) 20(1), 012202 (2025)

    work page 2025

  10. [10]

    L. V. Keldysh, Ionization in the field of a strong elec- tromagnetic wave, Zh. ´Eksp. Teor. Fiz. 47, 1945 (1964) [Sov. Phys. -JETP. 20, 1307 (1965)]

    work page 1945

  11. [11]

    F. H. M. Faisal, Multiple absorption of laser photons by atoms, J. Phys. B: At. Mol. Opt. Phys. 6, L89 (1973)

    work page 1973

  12. [12]

    H. R. Reiss, Effect of an intense electromagnetic field on a weakly bound system, Phys. Rev. A 22, 1786 (1980)

    work page 1980

  13. [13]

    Amini, J

    K. Amini, J. Biegert, F. Calegari, A. Cha´ on, M. F. Ciap- pina, A. Dauphin, D. K. Efimov, C. F. de Morisson Faria, K. Giergiel, P. Gniewek, A. S. Landsman, M. Lesiuk, M. Mandrysz, A. S. Maxwell, R. Moszy´ nski, L. Ortmann, J. A. P´ erez-Hern´ andez, A. Pic´ on, E. Pisanty, J. Prauzner- Bechcicki et al., Symphony on strong field approxima- tion, Rep. Prog. ...

    work page 2019

  14. [14]

    S. V. Popruzhenko, Keldysh theory of strong field ioniza - tion: history, applications, difficulties and perspectives , J. Phys. B: At. Mol. Opt. Phys. 47, 204001 (2014)

    work page 2014

  15. [15]

    J. T. Zhang and T. Nakajima, Coulomb effects in pho- toionization of H atoms irradiated by intense laser fields, Phys. Rev. A 75, 043403 (2007)

    work page 2007

  16. [16]

    D. B. Miloˇ sevi´ c and F. Ehlotzky, Coulomb and rescat- tering effects in above-threshold ionization, Phys. Rev. A 58, 3124 (1998)

    work page 1998

  17. [17]

    F. H. M. Faisal, Strong-field S-matrix theory with final- state Coulomb interaction in all orders, Phys. Rev. A 94, 031401(R) (2016)

    work page 2016

  18. [18]

    F. H. M. Faisal and S. F¨ orster, Coulomb-Volkov S-matrix theory of ionisation, J. Phys. B: At. Mol. Opt. Phys. 51, 234001 (2018)

    work page 2018

  19. [19]

    Leone, R

    C. Leone, R. Burlon, F. Trombetta, S. Basile, and G. Ferrante, Strong-field multiphoton ionization of hydro- gen. The S-matrix treatment of the elementary process., Nuouo Cimento D 9, 609 (1987)

    work page 1987

  20. [20]

    D. B. Miloˇ sevi´ c and W. Becker, Atom-Volkov strong-field approximation for above-threshold ionization, Phys. Rev. A 99, 043411 (2019)

    work page 2019

  21. [21]

    T. -M. Yan and D. Bauer, Sub-barrier Coulomb effects on the interference pattern in tunneling-ionization pho- toelectron spectra, Phys. Rev. A 86, 053403 (2012)

    work page 2012

  22. [22]

    S. V. Popruzhenko and D. Bauer, Strong field approx- imation for systems with Coulomb interaction, J. Mod. Opt. 55, 2573 (2008)

    work page 2008

  23. [23]

    T. -M. Yan, S. V. Popruzhenko, M. J. J. Vrakking, and D. Bauer, Low-energy structures in strong field ioniza- tion revealed by quantum orbits, Phys. Rev. Lett. 105, 253002 (2010)

    work page 2010

  24. [24]

    X. L. Hao, Y. X. Bai, C. Li, J. Y. Zhang, W. D. Li, W. F. Yang, M. Q. Liu, and J. Chen, Recollision of excited elec- tron in below-threshold nonsequential double ionization, Commun. Phys. 5, 31 (2022)

    work page 2022

  25. [25]

    S. V. Popruzhenko, G. G. Paulus, and D. Bauer, Coulomb-corrected quantum trajectories in strong-field ionization, Phys. Rev. A 77, 053409 (2008)

    work page 2008

  26. [26]

    X.-Y. Lai, C. Poli, H. Schomerus, C. Figueira de Moris- son Faria, Influence of the Coulomb potential on above- threshold ionization: A quantum-orbit analysis beyond the strong-field approximation, Phys. Rev. A 92, 043407 (2015)

    work page 2015

  27. [27]

    A. Rudenko. B. Feuerstein, K. Zrost, V. L. B. de Jesus, T. Ergler, C. D. Schroter, R. Moshammer, and J. Ullrich, Fragmentation dynamics of molecular hydrogen in strong ultrashort laser pulses, J. Phys. B: At. Mol. Opt. Phys. 38, 487 (2005)

    work page 2005

  28. [28]

    C. I. Blaga, F. Catoire, P. Colosimo, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, Strong-field photoionization revisited, Nat. Phys. 5, 335 (2009)

    work page 2009

  29. [29]

    W. Quan, Z. Lin, M. Wu, H. Kang, H. Liu, J. Chen, J. Liu, X. T. He, S. G. Chen, H. Xiong, L. Guo, H. Xu, Y. Fu, Y. Cheng, and Z. Z. Xu, Classical aspects in above- threshold ionization with a midinfrared strong laser field, Phys. Rev. Lett. 103, 093001 (2009)

    work page 2009

  30. [30]

    C. Y. Wu, Y. D. Yang, Y. Q. Liu, Q. H. Gong, M. Wu, X. Liu, X. L. Hao, W. D. Li, X. T. He, and J. Chen, Characteristic spectrum of very low-energy photoelectron from above-threshold ionization in the tunneling regime, Phys. Rev. Lett. 109, 043001 (2012)

    work page 2012

  31. [31]

    W. Quan, X. L. Hao, Y. J. Chen, S. G. Yu, S. P. Xu, Y. L. Wang, R. P. Sun, X. Y. Lai, C. Y. Wu, Q. H. Gong, X. T. He, X. J. Liu, and J. Chen, Long-range coulomb effect in intense laser-driven photoelectron dynamics, Sci. Rep. 6, 27108 (2016)

    work page 2016

  32. [32]

    Shafir, H

    D. Shafir, H. Soifer, C. Vozzi, A. S. Johnson, A. Hartung, Z. Dube, D. M. Villeneuve, P. B. Corkum, N. Dudovich, and A. Staudte, Trajectory-resolved Coulomb focusing in tunnel ionization of atoms with intense, elliptically po- larized laser pulses, Phys. Rev. Lett. 111, 023005 (2013)

    work page 2013

  33. [33]

    M. Li, Y. Q. Liu, H. Liu, Q. C. Ning, L. B. Fu, J. Liu, Y. K. Deng, C. Y. Wu, L. Y. Peng, and Q. H. Gong, Subcy- cle dynamics of Coulomb asymmetry in strong elliptical 13 laser fields, Phys. Rev. Lett. 111, 023006 (2013)

    work page 2013

  34. [34]

    C. P. Liu and K. Z. Hatsagortsyan, Origin of unexpected low energy structure in photoelectron spectra induced by midinfrared strong laser fields, Phys. Rev. Lett. 105, 113003 (2010)

    work page 2010

  35. [35]

    L. Guo, S. S. Han, X. Liu, Y. Cheng, Z. Z. Xu, J. Fan, J. Chen, and S. G. Chen, Scaling of the low-energy structure in above-threshold ionization in the tunneling regime: theory and experiment, Phys. Rev. Lett. 110, 013001 (2013)

    work page 2013

  36. [36]

    M. Q. Liu, S. P. Xu, S. L. Hu, W. Becker, W. Quan, X. J. Liu, and J. Chen, Electron dynamics in laser-driven atoms near the continuum threshold, optica 8(6), 765 (2021)

    work page 2021

  37. [37]

    Brabec, M

    T. Brabec, M. Y. Ivanov, and P. B. Corkum, Coulomb focusing in intense field atomic processes, Phys. Rev. A 54, R2551(R) (1996)

    work page 1996

  38. [38]

    G. L. Yudin and M. Y. Ivanov, Physics of correlated dou- ble ionization of atoms in intense laser fields: Quasistatic tunneling limit, Phys. Rev. A 63, 033404 (2001)

    work page 2001

  39. [39]

    X. M. Tong, Z. X. Zhao and C. D. Lin, Probing molecular dynamics at attosecond resolution with femtosecond laser pulses, Phys. Rev. Lett 91, 233203 (2003)

    work page 2003

  40. [40]

    X. L. Hao, W. D. Li, J. Liu and J. Chen, Effect of the electron initial longitudinal velocity on the nonsequen- tial double-ionization process, Phys. Rev. A 83, 053422 (2011)

    work page 2011

  41. [41]

    X. Y. Jia, X. L. Hao, D. H. Fan, W. D. Li and J. Chen, S-matrix and semiclassical study of electron-electron cor - relation in strong-field nonsequential double ionization o f Ne, Phys. Rev. A 88, 033402 (2013)

    work page 2013

  42. [42]

    X. L. Hao, Y. X. Bai, X. Y. Zhao, C. Li, J. Y. Zhang, J. L. Wang, W. D. Li, C. L. Wang, W. Quan, X. J. Liu, Z. Shu, M. Q. Liu, and J. Chen, Effect of Coulomb field on laser-induced ultrafast imaging methods, Phys. Rev. A 101, 051401(R) (2020)

    work page 2020

  43. [43]

    Meckel, D

    M. Meckel, D. Comtois, D. Zeidler, et al., Laser-induce d electron tunneling and diffraction, Science 320, 1478 (2008)

    work page 2008

  44. [44]

    C. I. Blaga, J. Xu, A. D. DiChiara, et al., Imaging ul- trafast molecular dynamics with laser-induced electron diffraction, Nature (London) 483, 194 (2012)

    work page 2012

  45. [45]

    J. Xu, C. I. Blaga, K. Zhang, et al., Diffraction using laser-driven broadband electron wave packets, Nat. Com- mun. 5, 4635 (2014)

    work page 2014

  46. [46]

    M. G. Pullen, B. Wolter, A. T. Le, et al., Influence of or- bital symmetry on diffraction imaging with rescattering electron wave packets, Nat. Commun. 7, 11922 (2016)

    work page 2016

  47. [47]

    M. G. Pullen, B. Wolter, A. T. Le, et al., Imaging an aligned polyatomic molecule with laser-induced electron diffraction, Nat. Commun. 6, 7262 (2015)

    work page 2015

  48. [48]

    Wolter, M

    B. Wolter, M. G. Pullen, A. T. Le, et al., Ultrafast elec- tron diffraction imaging of bond breaking in di-ionized acetylene, Science 354, 308 (2016)

    work page 2016

  49. [49]

    Becker and F

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

    work page 2005

  50. [50]

    L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1965)

    work page 1965

  51. [51]

    Kopold and W

    R. Kopold and W. Becker, Interference in high-order above-threshold ionization, J. Phys. B: At. Mol. Opt. Phys. 32, L419 (1999)

    work page 1999

  52. [52]

    Lewenstein, Ph

    M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier , and P. B. Corkum, Theory of high-harmonic generation by low-frequency laser fields, Phys. Rev. A 49, 2117 (1994)

    work page 1994

  53. [53]

    Lewenstein, K

    M. Lewenstein, K. C. Kulander, K. J. Schafer, and P. H. Bucks baum, Rings in above-threshold ionization: A quasiclassical analysis, Phys. Rev. A 51, 1495 (1995)

    work page 1995

  54. [54]

    Figueira de Morisson Faria, H

    C. Figueira de Morisson Faria, H. Schomerus and W. Becker, High-order above-threshold ionization: The uni- form approximation and the effect of the binding poten- tial, Phys. Rev. A 66, 043413 (2002)

    work page 2002

  55. [55]

    F. H. M. Faisal, Gauge-invariant intense-field approxi ma- tions to all orders, J. Phys. B: At. Mol. Opt. Phys. 41, F145 (2007)

    work page 2007

  56. [56]

    F. H. M. Faisal, Gauge-equivalent intense-field approx i- mations in velocity and length gauges to all orders, Phys. Rev. A 75, 063412 (2007)

    work page 2007

  57. [57]

    Becker and D

    W. Becker and D. B. Miloˇ sevi´ c, A gauge-covariant Derivation of the strong-field approximation, Laser Phys. 19, 1621 (2009)

    work page 2009

  58. [58]

    Bauer and P

    D. Bauer and P. Koval, Qprop: A Schr¨ odinger-solver for intense laser-atom interaction, Comput. Phys. Commun. 174, 396 (2006); see also www.qprop.de

    work page 2006