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arxiv: 1906.11325 · v2 · pith:TR7N4HF5new · submitted 2019-06-26 · ⚛️ physics.optics · physics.atm-clus

Controlling Sub-Cycle Optical Chirality in the Photoionization of Chiral Molecules

Pith reviewed 2026-05-25 14:52 UTC · model grok-4.3

classification ⚛️ physics.optics physics.atm-clus
keywords photoionizationchiral moleculesoptical chiralitysub-cycle dynamicstwo-color laser fieldphotoelectron circular dichroismattosecond processes
0
0 comments X

The pith

A laser field with zero net ellipticity but controllable instantaneous chirality produces opposite forward-backward electron asymmetries from consecutive half-cycles in chiral molecule photoionization.

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

The paper demonstrates a method to engineer an electric field whose instantaneous chirality switches sign within each optical cycle by combining phase-locked, orthogonally polarized fundamental and second-harmonic fields. This field ionizes chiral molecules such that electrons show a forward or backward preference relative to the propagation direction that reverses between one half-cycle and the next. The resulting chiral signal isolates the dynamical effect of instantaneous optical chirality on photoionization, beyond what cycle-averaged polarization schemes can access. A sympathetic reader would care because it resolves sub-cycle chiroptical processes in attosecond detail.

Core claim

Electrons ionized from two consecutive half cycles of the laser show opposite forward/backward asymmetries because the instantaneous chirality switches sign from one half cycle to the next. This chiral signal provides a unique insight into the influence of instantaneous chirality in the dynamical photoionization process of chiral molecules, using a composite field that has zero net ellipticity.

What carries the argument

The two-color composite electric field whose instantaneous chirality is set by the delay between the fundamental and second-harmonic components while net ellipticity remains zero over the full cycle.

If this is right

  • Electrons ejected in one half-cycle exhibit a forward-backward preference determined by molecular handedness that reverses in the next half-cycle.
  • The chiral signal originates specifically from the sub-cycle variation of polarization rather than any cycle-averaged ellipticity.
  • Sub-cycle polarization shaping constitutes a new route to study and manipulate chiroptical processes in photoionization.
  • Standard photoelectron circular dichroism can be resolved into contributions from individual half-cycles of the driving field.

Where Pith is reading between the lines

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

  • The technique could be extended to map how molecular response to chirality evolves on attosecond time scales within a single pulse.
  • Analogous field shaping might isolate sub-cycle effects in other symmetry-sensitive light-matter interactions such as high-harmonic generation from chiral targets.
  • Experimental confirmation would require verifying that the asymmetry reversal tracks the two-color delay with attosecond precision.

Load-bearing premise

The composite field has exactly zero net ellipticity over each optical cycle while its instantaneous chirality remains independently controllable by the two-color delay.

What would settle it

A direct measurement showing non-zero net ellipticity in the composite field, or the absence of asymmetry reversal when the two-color delay is shifted by half a fundamental period, would falsify the claim that the observed signal arises from sub-cycle chirality control.

Figures

Figures reproduced from arXiv: 1906.11325 by Antoine Comby, Baptiste Fabre, Bernard Pons, Dominique Descamps, Etienne Bloch, Nirit Dudovich, Sandra Beauvarlet, Shaked Rozen, Stephane Petit, Valerie Blanchet, Yann Mairesse.

Figure 1
Figure 1. Figure 1: Principle of the experiment: the shaped electric [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic description of the sub-cycle instanta [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: TDSE calculations of photoionization of a toy model chiral molecule by a composite bichromatic field. (a,g,m) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Experimental data. (a,g,m) Normalized projections in the y-z plane of the the photoelectron angular distribution [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Sub-cycle control over the instantaneous chirality [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Skeleton and isocontour representation of the fun [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Influence of the carrier-envelop phase φCEP in the TDSE calculations at I = 5 ×1012 W.cm−2 and for a relative phase between the two components of the laser field ϕ = 3π/4. (a-c) Projections of the ESCARGOT signal, defined as the antisymmetric part of the photoelectron distribution with re￾spect to the propagation axis z. (d-f) Corresponding projec￾tions of the ESCARGOT signal, defined as the antisymmet￾ric… view at source ↗
Figure 9
Figure 9. Figure 9: Convergence check with respect to the maximum [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Calibration of the two-color phase using 4 [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Experimental data without up-down antisymmetrization. (a) Normalized projection in the y-z plane of the the [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the ESCARGOT signal obtained in (+)-Fenchone (first and third rows) and a racemic mixture [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
read the original abstract

Controlling the polarization state of electromagnetic radiation enables the investigation of fundamental symmetry properties of matter through chiroptical processes. Many strategies have been developed to reveal structural or dynamical information about chiral molecules, from the microwave to the extreme ultraviolet range. Most schemes employ circularly or elliptically polarized radiation, and more sophisticated configurations involve, for instance, light pulses with time-varying polarization states. In all these schemes, the polarization state of light is always considered as constant over one optical cycle. In this study, we zoom into the optical cycle in order to resolve and control a subcyle attosecond chiroptical process. We engineer an electric field whose instantaneous chirality can be controlled within the optical cycle, by combining two phase-locked orthogonally polarized fundamental and second harmonic fields. While the composite field has zero net ellipticity, it shows an instantaneous optical chirality which can be controlled via the two-color delay. We theoretically and experimentally investigate the photoionization of chiral molecules with this controlled chiral field. We find that electrons are preferentially ejected forward or backward relative to the laser propagation direction depending on the molecular handedness, similarly to the well-established photoelectron circular dichroism process. However, since the instantaneous chirality switches sign from one half cycle to the next, electrons ionized from two consecutive half cycles of the laser show opposite forward/backward asymmetries. This chiral signal provides a unique insight into the influence of instantaneous chirality in the dynamical photoionization process. Our results demonstrate the important role of sub-cycle polarization shaping of electric fields, as a new route to study and manipulate chiroptical processes.

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 manuscript claims that a composite two-color laser field (orthogonally polarized fundamental and second-harmonic components) can be engineered to possess zero net (cycle-averaged) ellipticity while allowing independent control of its instantaneous chirality via the two-color delay. Theory and experiment on photoionization of chiral molecules then show that the forward/backward photoelectron asymmetry reverses sign between consecutive half-cycles, which the authors attribute directly to the sign switch of the sub-cycle chirality.

Significance. If the central attribution holds, the work supplies a concrete experimental route to isolate and control sub-cycle chiroptical effects, extending conventional photoelectron circular dichroism into the attosecond domain. The combination of a parameter-free field-construction principle with both theoretical modeling and measured asymmetry reversal constitutes a clear advance over cycle-averaged polarization schemes.

major comments (2)
  1. [Abstract and experimental methods] The load-bearing assumption that the composite field maintains exactly zero net ellipticity for all two-color delays (while instantaneous chirality remains independently tunable) is stated in the abstract but is not accompanied by quantitative verification. No Stokes-parameter measurements versus delay, no explicit calculation of the time-dependent ellipticity, and no error analysis confirming that residual ellipticity lies below the detection threshold of the asymmetry are shown; without these data the reversal cannot be unambiguously assigned to sub-cycle rather than cycle-averaged polarization.
  2. [Results section (asymmetry data)] The reported photoelectron asymmetry reversal lacks quantitative error bars, a description of data-exclusion criteria, and an explicit check that the measured net ellipticity remains zero within experimental precision across the scanned delays. These omissions directly affect the strength of the claim that the observed sign change originates from instantaneous chirality switching.
minor comments (2)
  1. [Theory section] Notation for the two-color delay and the definition of instantaneous versus cycle-averaged ellipticity should be introduced once with an equation and then used consistently.
  2. [Figures] Figure captions should explicitly state whether the plotted fields are calculated or measured and whether error bands are included.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The two points raised concern the quantitative verification of zero net ellipticity and the presentation of the asymmetry data. We agree that explicit supporting material will strengthen the central claim and outline the revisions below.

read point-by-point responses
  1. Referee: [Abstract and experimental methods] The load-bearing assumption that the composite field maintains exactly zero net ellipticity for all two-color delays (while instantaneous chirality remains independently tunable) is stated in the abstract but is not accompanied by quantitative verification. No Stokes-parameter measurements versus delay, no explicit calculation of the time-dependent ellipticity, and no error analysis confirming that residual ellipticity lies below the detection threshold of the asymmetry are shown; without these data the reversal cannot be unambiguously assigned to sub-cycle rather than cycle-averaged polarization.

    Authors: We agree that an explicit verification is necessary to unambiguously attribute the observed reversal to sub-cycle chirality. In the revised manuscript we will add (i) analytic and numerical calculations of the instantaneous and cycle-averaged Stokes parameters as a function of two-color delay, confirming that the net ellipticity remains zero to within 10^{-4} for all delays, and (ii) a discussion of the experimental tolerance on residual ellipticity relative to the measured asymmetry amplitude. These additions will be placed in a new subsection of the methods and referenced from the abstract and results. revision: yes

  2. Referee: [Results section (asymmetry data)] The reported photoelectron asymmetry reversal lacks quantitative error bars, a description of data-exclusion criteria, and an explicit check that the measured net ellipticity remains zero within experimental precision across the scanned delays. These omissions directly affect the strength of the claim that the observed sign change originates from instantaneous chirality switching.

    Authors: We will revise the results section to include (i) statistical error bars on all asymmetry values derived from the full dataset, (ii) a concise description of the data-selection criteria and background subtraction procedure, and (iii) a supplementary figure or table showing the measured (or upper-bound) net ellipticity for each scanned delay, obtained from auxiliary polarization diagnostics. These changes will make the quantitative support for the sub-cycle interpretation explicit. revision: yes

Circularity Check

0 steps flagged

No circularity; experimental observation is independent of fitted inputs or self-citations

full rationale

The paper reports an experimental measurement of forward/backward asymmetry reversal in photoelectron circular dichroism using a two-color field engineered for zero net ellipticity and tunable instantaneous chirality. No derivation chain reduces a claimed prediction to a fitted parameter by construction, nor does any load-bearing premise rest on a self-citation whose content is unverified or equivalent to the target result. The central attribution follows from direct observation and field characterization rather than from any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard assumptions of strong-field ionization and the definition of instantaneous chirality from the two-color field; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The electric field formed by orthogonally polarized fundamental and second-harmonic components can be treated as having well-defined instantaneous chirality that flips sign each half-cycle while net ellipticity remains zero.
    Invoked when the abstract states the composite field has zero net ellipticity yet controllable instantaneous chirality.

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Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · 1 internal anchor

  1. [1]

    C- shaped field

    On the other hand, when ϕ = π, the instantaneous chirality is maximal when the instantaneous ellipticity is close to zero. The color-coding of the shape of the elec- tric field presented in Figure 2(a) reveals the origin of the complex behavior of the instantaneous chirality: it maps the rotational velocity of the electric field [15]. III. PHOTOELECTRON CIR...

  2. [2]

    Y sin,cos are (sine and cosine) real spheri- cal harmonics, and jl(klr) are spherical Bessel functions

    Details on the theoretical framework The bound and continuum eigenstates φEi of the toy model molecule described in the main text are obtained by diagonalizing the field-free Hamiltonian H0 in a basis of functions jl(klr)Y sin,cos lm (Ωr) with 0 ≤ l≤ lmax and −l≤ m≤ l. Y sin,cos are (sine and cosine) real spheri- cal harmonics, and jl(klr) are spherical Be...

  3. [3]

    Influence of the carrier-envelop phase φCEP in the TDSE calculations at I = 5 × 1012 W.cm−2 and for a relative phase between the two components of the laser fieldϕ = 3π/4

    Influence of short pulse duration The limited number of optical cycles of the laser pulses used in the calculations can lead to artifacts, related to Figure 7. Influence of the carrier-envelop phase φCEP in the TDSE calculations at I = 5 × 1012 W.cm−2 and for a relative phase between the two components of the laser fieldϕ = 3π/4. (a-c) Projections of the ESC...

  4. [4]

    (6) using numerical quadratures on countable orientations with Euler angle spacing ∆α = ∆β = ∆γ =π/4

    Convergence checks It is stated in the main text that averaging over molec- ular orientations is performed in eq. (6) using numerical quadratures on countable orientations with Euler angle spacing ∆α = ∆β = ∆γ =π/4. We illustrate in Figure 8 how increasing the number of orientations according to ∆α = ∆β = ∆γ = π/5 left the symmetric and normal- ized antis...

  5. [5]

    N. W. R. Berova N, Polavarapu P L,Comprehensive Chi- roptical Spectroscopy vol 1 (New York: Wiley, 2012)

  6. [6]

    Goulielmakis, M

    E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Bal- tuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, 14 U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, Science 305, 1267 (2004)

  7. [7]

    K. T. Kim, C. Zhang, A. D. Shiner, S. E. Kirk- wood, E. Frumker, G. Gariepy, A. Naumov, D. M. Vil- leneuve, and P. B. Corkum, Nature Physics (2013), 10.1038/nphys2525

  8. [8]

    R. Boge, S. Heuser, M. Sabbar, M. Lucchini, L. Gall- mann, C. Cirelli, and U. Keller, Optics express 22, 2696726975 (2014)

  9. [9]

    Carpeggiani, M

    P. Carpeggiani, M. Reduzzi, A. Comby, H. Ahmadi, S. Khn, F. Calegari, M. Nisoli, F. Frassetto, L. Po- letto, D. Hoff, J. Ullrich, C. Schr¨ oter, R. Moshammer, G. Paulus, and G. Sansone, Nature Photonics 11, 383 (2017)

  10. [10]

    Kitzler and M

    M. Kitzler and M. Lezius, Phys. Rev. Lett. 95, 253001 (2005)

  11. [11]

    Shafir, Y

    D. Shafir, Y. Mairesse, D. M. Villeneuve, P. B. Corkum, and N. Dudovich, Nature Physics 5, 412 (2009)

  12. [12]

    Shafir, H

    D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Y. Smirnova, and N. Dudovich, Nature 485, 343 (2012)

  13. [13]

    Zhang, X

    L. Zhang, X. Xie, S. Roither, D. Kartashov, Y. Wang, C. Wang, M. Sch¨ offler, D. Shafir, P. B. Corkum, A. Baltuˇ ska, I. Ivanov, A. Kheifets, X. Liu, A. Staudte, and M. Kitzler, Phys. Rev. A 90, 061401 (2014)

  14. [14]

    O. Kfir, P. Grychtol, E. Turgut, R. Knut, D. Zusin, D. Popmintchev, T. Popmintchev, H. Nembach, J. M. Shaw, A. Fleischer, H. Kapteyn, M. Murnane, and O. Cohen, Nature Photonics 9, 99 (2015)

  15. [15]

    Richter, M

    M. Richter, M. Kunitski, M. Sch¨ offler, T. Jahnke, L. P. H. Schmidt, M. Li, Y. Liu, and R. D¨ orner, Phys. Rev. Lett. 114, 143001 (2015)

  16. [16]

    Smirnova, Y

    O. Smirnova, Y. Mairesse, and S. Patchkovskii, Journal of Physics B: Atomic, Molecular and Optical Physics 48, 234005 (2015)

  17. [17]

    Ferr´ e, H

    A. Ferr´ e, H. Soifer, O. Pedatzur, C. Bourassin-Bouchet, B. Bruner, R. Canonge, F. Catoire, D. Descamps, B. Fabre, E. M´ evel, S. Petit, N. Dudovich, and Y. Mairesse, Physical Review Letters116, 053002 (2016)

  18. [18]

    W¨ urzler, N

    D. W¨ urzler, N. Eicke, M. M¨ oller, D. Seipt, A. M. Sayler, S. Fritzsche, M. Lein, and G. G. Paulus, Journal of Physics B: Atomic, Molecular and Optical Physics 51, 015001 (2017)

  19. [19]

    Neufeld and O

    O. Neufeld and O. Cohen, Phys. Rev. Lett. 120, 133206 (2018)

  20. [20]

    Baykusheva and H

    D. Baykusheva and H. J. W¨ orner, Phys. Rev. X8, 031060 (2018)

  21. [21]

    M. Li, H. Xie, W. Cao, S. Luo, J. Tan, Y. Feng, B. Du, W. Zhang, Y. Li, Q. Zhang, P. Lan, Y. Zhou, and P. Lu, Phys. Rev. Lett. 122, 183202 (2019)

  22. [22]

    Strelkov, A

    V. Strelkov, A. Zair, O. Tcherbakoff, R. L´ opez-Martens, E. Cormier, E. M´ evel, and E. Constant, Applied Physics B 78, 879 (2004)

  23. [23]

    Misawa, Advances in Physics: X 1, 544 (2016), https://doi.org/10.1080/23746149.2016.1221327

    K. Misawa, Advances in Physics: X 1, 544 (2016), https://doi.org/10.1080/23746149.2016.1221327

  24. [24]

    D. M. Lipkin, Journal of Mathematical Physics 5, 696 (1964)

  25. [25]

    Tang and A

    Y. Tang and A. E. Cohen, Physical Review Letters 104, 163901 (2010)

  26. [26]

    Tang and A

    Y. Tang and A. E. Cohen, Science 332, 333 (2011), http://science.sciencemag.org/content/332/6027/333.full.pdf

  27. [27]

    P. V. Demekhin, A. N. Artemyev, A. Kastner, and T. Baumert, arXiv:1807.01097v1 (2018)

  28. [28]

    Ritchie, Phys

    B. Ritchie, Phys. Rev. A 13, 1411 (1976)

  29. [29]

    Powis, The Journal of Chemical Physics 112, 301 (2000)

    I. Powis, The Journal of Chemical Physics 112, 301 (2000)

  30. [30]

    B¨ owering, T

    N. B¨ owering, T. Lischke, B. Schmidtke, N. M¨ uller, T. Khalil, and U. Heinzmann, Physical Review Letters 86, 1187 (2001)

  31. [31]

    Nahon, G

    L. Nahon, G. A. Garcia, and I. Powis, Journal of Elec- tron Spectroscopy and Related Phenomena Gas phase spectroscopic and dynamical studies at Free-Electron Lasers and other short wavelength sources, 204, Part B, 322 (2015)

  32. [32]

    Patterson, M

    D. Patterson, M. Schnell, and J. M. Doyle, Nature 497, 475 (2013)

  33. [33]

    Beaulieu, A

    S. Beaulieu, A. Comby, D. Descamps, B. Fabre, G. Gar- cia, R. G´ eneaux, A. Harvey, F. L´ egar´ e, Z. Maˇ s´ ın, a. A. O. Laurent Nahon, S. Petit, B. Pons, Y. Mairesse, O. Smirnova, and V. Blanchet, Nature Physics 14, 484 (2018)

  34. [34]

    B¨ owering, T

    N. B¨ owering, T. Lischke, B. Schmidtke, N. M¨ uller, T. Khalil, and U. Heinzmann, Phys. Rev. Lett. 86, 1187 (2001)

  35. [35]

    C. Lux, M. Wollenhaupt, T. Bolze, Q. Liang, J. K¨ ohler, C. Sarpe, and T. Baumert, Angewandte Chemie Inter- national Edition 51, 5001 (2012)

  36. [36]

    C. S. Lehmann, N. B. Ram, I. Powis, and M. H. M. Janssen, The Journal of Chemical Physics 139, 234307 (2013)

  37. [37]

    Beaulieu, A

    S. Beaulieu, A. Ferr´ e, R. G´ eneaux, R. Canonge, D. Descamps, B. Fabre, N. Fedorov, F. L´ egar´ e, S. Pe- tit, T. Ruchon, V. Blanchet, Y. Mairesse, and B. Pons, New Journal of Physics 18, 102002 (2016)

  38. [38]

    C. Lux, M. Wollenhaupt, C. Sarpe, and T. Baumert, ChemPhysChem 16, 115 (2015)

  39. [39]

    Comby, E

    A. Comby, E. Bloch, C. Bond, D. Descamps, J. Miles, S. Petit, S. Rozen, J. Greenwood, V. Blanchet, and Y. Mairesse, Nature Communications 9 (2018), 10.1038/s41467-018-07609-9

  40. [40]

    Kl¨ under, J

    K. Kl¨ under, J. M. Dahlstr¨ om, M. Gisselbrecht, T. Fordell, M. Swoboda, D. Gu´ enot, P. Johnsson, J. Caillat, J. Mauritsson, A. Maquet, R. Ta¨ ıeb, and A. L’Huillier, Phys. Rev. Lett. 106, 143002 (2011)

  41. [41]

    Gu´ enot, D

    D. Gu´ enot, D. Kroon, E. Balogh, E. W. Larsen, M. Ko- tur, M. Miranda, T. Fordell, P. Johnsson, J. Maurits- son, M. Gisselbrecht, K. Varj, C. L. Arnold, T. Carette, A. S. Kheifets, E. Lindroth, A. LHuillier, and J. M. Dahlstr¨ om, Journal of Physics B: Atomic, Molecular and Optical Physics 47, 245602 (2014)

  42. [42]

    Gallmann, I

    L. Gallmann, I. Jordan, H. J. W¨ orner, L. Cas- tiglioni, M. Hengsberger, J. Osterwalder, C. A. Ar- rell, M. Chergui, E. Liberatore, U. Rothlisberger, and U. Keller, Structural Dynamics 4, 061502 (2017), https://doi.org/10.1063/1.4997175

  43. [43]

    Haessler, B

    S. Haessler, B. Fabre, J. Higuet, J. Caillat, T. Ruchon, P. Breger, B. Carr´ e, E. Constant, A. Maquet, E. M´ evel, P. Sali´ eres, R. Taeb, and Y. Mairesse, Phys. Rev. A80, 011404 (2009)

  44. [44]

    J. Vos, L. Cattaneo, S. Patchkovskii, T. Zimmermann, C. Cirelli, M. Lucchini, A. Kheifets, A. S. Landsman, and U. Keller, Science 360, 1326 (2018)

  45. [45]

    Beaulieu, A

    S. Beaulieu, A. Comby, A. Clergerie, J. Caillat, D. Descamps, N. Dudovich, B. Fabre, R. G´ eneaux, F. L´ egar´ e, S. Petit, B. Pons, G. Porat, T. Ruchon, R. Ta¨ ıeb, V. Blanchet, and Y. Mairesse, Science 358, 1288 (2017). 15

  46. [46]

    Cormier and P

    E. Cormier and P. Lambropoulos, Journal of Physics B: Atomic, Molecular and Optical Physics 29, 1667 (1996)

  47. [47]

    Goldstein, The Euler Angles (Addison-Wesley, 1980)

    H. Goldstein, The Euler Angles (Addison-Wesley, 1980)

  48. [48]

    Agostini, F

    P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, Phys. Rev. Lett. 42, 1127 (1979)

  49. [49]

    Lewenstein, P

    M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Physical review. A, 49, 2117 (1994)

  50. [50]

    Multielectron high har- monic generation: Simple man on a complex plane,

    O. Smirnova and M. Ivanov, “Multielectron high har- monic generation: Simple man on a complex plane,” in Attosecond and XUV Physics (John Wiley & Sons, Ltd,

  51. [51]

    Chap. 7, pp. 201–256

  52. [52]

    Cireasa, A

    R. Cireasa, A. E. Boguslavskiy, B. Pons, M. C. H. Wong, D. Descamps, S. Petit, H. Ruf, N. Thir, A. Ferr, J. Suarez, J. Higuet, B. E. Schmidt, A. F. Al- harbi, F. Lgar, V. Blanchet, B. Fabre, S. Patchkovskii, O. Smirnova, Y. Mairesse, and V. R. Bhardwaj, Nature Physics 11, 654 (2015)

  53. [53]

    Calegari, G

    F. Calegari, G. Sansone, S. Stagira, C. Vozzi, and M. Nisoli, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 062001 (2016)

  54. [54]

    Pedatzur, G

    O. Pedatzur, G. Orenstein, V. Serbinenko, H. Soifer, B. Bruner, A. Uzan, D. Brambila, A. Harvey, L. Tor- lina, F. Morales, O. Smirnova, and N. Dudovich, Nature Physics 11, 815 (2015)

  55. [55]

    Ferr´ e, H

    A. Ferr´ e, H. Soifer, O. Pedatzur, C. Bourassin-Bouchet, B. D. Bruner, R. Canonge, F. Catoire, D. Descamps, B. Fabre, E. M´ evel, S. Petit, N. Dudovich, and Y. Mairesse, Phys. Rev. Lett. 116, 053002 (2016)

  56. [56]

    Pons, Phys

    B. Pons, Phys. Rev. Lett. 84, 4569 (2000)

  57. [57]

    Pons, Phys

    B. Pons, Phys. Rev. A 67, 040702 (2003)

  58. [58]

    S. A. T. W. H. Press, B. P. Flannery and W. T. Vet- terling, Numerical Recipes (Cambridge University Press, 1992)

  59. [59]

    B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules (Pearson Education, 2003)

  60. [60]

    A. N. Artemyev, A. D. M¨ uller, D. Hochstuhl, and P. V. Demekhin, The Journal of Chemical Physics142, 244105 (2015), https://doi.org/10.1063/1.4922690