- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Phase-of-the-phase spectroscopy in the few-cycle regime

D. Habibović1 and D. B. Milošević1,2

PDF

Phys. Rev. A 113, 033116 – Published 20 March, 2026

DOI: https://doi.org/10.1103/jdsz-p838

Abstract

We present a theoretical study of the phase-of-the-phase spectroscopy in the few-cycle regime and analyze its explicit dependence on the carrier-envelope phase (CEP) and the pulse length. The process of above-threshold detachment of negative ions driven by ultrashort two-color laser pulses is investigated within the strong-field approximation formulated for few-cycle pulses with two carrier frequencies. While the CEP effects are known to be difficult to extract from the photoelectron yield of direct electrons, we demonstrate that the CEP leaves a clear and robust imprint on the phase of the phase, even when only direct electrons are considered. We analyze two-color configurations with parallel and orthogonal polarizations of the driving and perturbation pulses and identify characteristic CEP-dependent structures in both momentum-resolved and energy-averaged phase-of-the-phase spectra. The role of symmetry and its reduction in orthogonally polarized configurations is discussed in detail. Importantly, we show that the CEP dependence of the phase of the phase survives focal averaging and becomes particularly smooth under realistic experimental conditions. In addition, we demonstrate that the phase of the phase exhibits a pronounced sensitivity to the pulse length in the few-cycle regime, while converging for longer pulses, indicating its potential for characterizing ultrashort pulse durations. Our results establish phase-of-the-phase spectroscopy as a sensitive and experimentally viable tool for extracting information about strong-field processes driven by few-cycle laser pulses.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (55)

  1. W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G. G. Paulus, and H. Walther, Above-threshold ionization: From classical features to quantum effects, Adv. At. Mol. Opt. Phys. 48, 35 (2002).
  2. D. B. Milošević and F. Ehlotzky, Scattering and reaction processes in powerful laser fields, Adv. At. Mol. Opt. Phys. 49, 373 (2003).
  3. A. Becker and F. H. M. Faisal, Intense field many-body S-matrix theory, J. Phys. B 38, R1 (2005).
  4. D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, Above-threshold ionization by few-cycle pulses, J. Phys. B 39, R203 (2006).
  5. S. V. Popruzhenko, Keldysh theory of strong field ionization: History, applications, difficulties and perspectives, J. Phys. B 47, 204001 (2014).
  6. D. B. Milošević, in Ultrafast Dynamics Driven by Intense Light Pulses: From Atoms to Solids, from Lasers to Intense X-Rays, edited by M. Kitzler and S. Gräfe, Springer Series on Atomic, Optical, and Plasma Physics, Vol. 86 (Springer, Cham, 2016), pp. 27–47.
  7. C. Figueira de Morisson Faria and A. S. Maxwell, It is all about phases: Ultrafast holographic photoelectron imaging, Rep. Prog. Phys. 83, 034401 (2020).
  8. Y. Huismans, A. Rouzée, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, et al., Time-resolved holography with photoelectrons, Science 331, 61 (2011).
  9. Y. Huismans, A. Gijsbertsen, A. S. Smolkowska, J. H. Jungmann, A. Rouzée, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, and T. Marchenko, et al., Scaling laws for photoelectron holography in the midinfrared wavelength regime, Phys. Rev. Lett. 109, 013002 (2012).
  10. D. B. Milošević, G. G. Paulus, and W. Becker, Ionization by few-cycle pulses: Tracing the electron orbits, Phys. Rev. A 71, 061404(R) (2005).
  11. D. G. Arbó, K. L. Ishikawa, K. Schiessl, E. Persson, and J. Burgdörfer, Intracycle and intercycle interferences in above-threshold ionization: The time grating, Phys. Rev. A 81, 021403(R) (2010).
  12. R. Reichle, H. Helm, and I. Y. Kiyan, Photodetachment of H− in a strong infrared laser field, Phys. Rev. Lett. 87, 243001 (2001).
  13. I. Yu Kiyan and H. Helm, Production of energetic electrons in the process of photodetachment of F−, Phys. Rev. Lett. 90, 183001 (2003).
  14. A. Gazibegović-Busuladžić, D. B. Milošević, and W. Becker, Gauge dependence of the strong-field approximation: Theory vs experiment for photodetachment of F−, Opt. Commun. 275, 116 (2007).
  15. S. Bivona, G. Bonanno, R. Burlon, and C. Leone, Photodetachment of F− by short laser pulses. Comparison between experiments and numerical results, Laser Phys. 19, 805 (2009).
  16. A. Gazibegović-Busuladžić, D. B. Milošević, W. Becker, B. Bergues, H. Hultgren, and I. Y. Kiyan, Electron rescattering in above-threshold photodetachment of negative ions, Phys. Rev. Lett. 104, 103004 (2010).
  17. Y. Albeck, G. Lerner, D. M. Kandhasamy, V. Chandrasekaran, and D. Strasser, Intense field double detachment of atomic versus molecular anions, Phys. Rev. A 92, 061401(R) (2015).
  18. T. Sun, H. Lv, J. Chen, H. Shi, Y. Liang, D. Gao, H. Liang, J. Guo, and H. Xu, Photodetachment of the negative iodine ions in strong femtosecond laser fields, New J. Phys. 27, 093202 (2025).
  19. G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, Absolute-phase phenomena in photoionization with few-cycle laser pulses, Nature (London) 414, 182 (2001).
  20. D. B. Milošević, G. G. Paulus, and W. Becker, Phase-dependent effects of a few-cycle laser pulse, Phys. Rev. Lett. 89, 153001 (2002).
  21. D. B. Milošević, G. G. Paulus, and W. Becker, High-order above-threshold ionization with few-cycle pulse: A meter of the absolute phase, Opt. Express 11, 1418 (2003).
  22. F. Grasbon, G. G. Paulus, H. Walther, P. Villoresi, G. Sansone, S. Stagira, M. Nisoli, and S. De Silvestri, Above-threshold ionization at the few-cycle limit, Phys. Rev. Lett. 91, 173003 (2003).
  23. J. Réhault, M. Maiuri, A. Oriana, and G. Cerullo, Two-dimensional electronic spectroscopy with birefringent wedges, Rev. Sci. Instrum. 85, 123107 (2014).
  24. F. Schlaepfer, M. Volkov, N. Hartmann, A. Niedermayr, Z. Schumacher, L. Gallmann, and U. Keller, Phase stabilization of an attosecond beamline combining two IR colors, Opt. Express 27, 22385 (2019).
  25. A.-K. Raab, M. Schmoll, E. R. Simpson, M. Redon, Y. Fang, C. Guo, A.-L. Viotti, C. L. Arnold, A. L'Huillier, and J. Mauritsson, Highly versatile, two-color setup for high-order harmonic generation using spatial light modulators, Rev. Sci. Instrum. 95, 073002 (2024).
  26. J. Mathijssen, E. J. Salumbides, K. S. E. Eikema, and S. Witte, Sub-cycle dynamics in two-color high-harmonic generation from laser-produced plasmas, Opt. Express 32, 30824 (2024).
  27. S. Skruszewicz, J. Tiggesbäumker, K.-H. Meiwes-Broer, M. Arbeiter, T. Fennel, and D. Bauer, Two-color strong-field photoelectron spectroscopy and the phase of the phase, Phys. Rev. Lett. 115, 043001 (2015).
  28. M. A. Almajid, M. Zabel, S. Skruszewicz, J. Tiggesbäumker, and D. Bauer, Two-color phase-of-the-phase spectroscopy in the multiphoton regime, J. Phys. B 50, 194001 (2017).
  29. J. Tan, Y. Li, Y. Zhou, M. He, Y. Chen, M. Li, and P. Lu, Identifying the contributions of multiple-returning recollision orbits in strong-field above-threshold ionization, Opt. Quantum Electron. 50, 57 (2018).
  30. D. Würzler, S. Skruszewicz, A. M. Sayler, D. Zille, M. Möller, P. Wustelt, Y. Zhang, J. Tiggesbäumker, and G. G. Paulus, Accurate retrieval of ionization times by means of the phase-of-the-phase spectroscopy, and its limits, Phys. Rev. A 101, 033416 (2020).
  31. J. Braß, R. Milbradt, S. Villalba-Chávez, G. G. Paulus, and C. Müller, Two-color phase-of-the-phase spectroscopy applied to nonperturbative electron-positron pair production in strong oscillating electric fields, Phys. Rev. A 101, 043401 (2020).
  32. J. Tan, S. Xu, X. Han, Y. Zhou, M. Li, W. Cao, Q. Zhang, and P. Lu, Resolving and weighing the quantum orbits in strong-field tunneling ionization, Adv. Photonics 3, 035001 (2021).
  33. J. Tan, Y. Zhou, S. Xu, Q. Ke, J. Liang, X. Ma, W. Cao, M. Li, Q. Zhang, and P. Lu, Analyzing the electron trajectories in strong-field tunneling ionization with the phase-of-the-phase spectroscopy, Opt. Express 29, 37927 (2021).
  34. B. S. Krebs, V. Tulsky, L. Kazak, M. Zabel, D. Bauer, and J. Tiggesbäumker, Phase-of-the-phase electron momentum spectroscopy on single metal atoms in helium nanodroplets, J. Phys. Chem. Lett. 13, 1526 (2022).
  35. M. Yu, Y. Zhou, M. Li, and P. Lu, Probing the effect of orbital deformation on the atomic tunneling-ionization-time distribution by phase-of-the-phase spectroscopy, Phys. Rev. A 105, 063103 (2022).
  36. X. Huo, W. Yang, F. Jin, S. Ben, and X. Song, Application of the generalized Bessel function to two-color phase-of-the-phase spectroscopy, Mathematics 10, 4642 (2022).
  37. D. Würzler, N. Eicke, M. Möller, D. Seipt, A. M. Sayler, S. Fritzsche, M. Lein, and G. G. Paulus, Velocity map imaging of scattering dynamics in orthogonal two-color fields, J. Phys. B 51, 015001 (2018).
  38. V. A. Tulsky, M. A. Almajid, and D. Bauer, Two-color phase-of-the-phase spectroscopy with circularly polarized laser pulses, Phys. Rev. A 98, 053433 (2018).
  39. V. A. Tulsky, B. Krebs, J. Tiggesbäumker, and D. Bauer, Revealing laser-coherent electron features using phase-of-the-phase spectroscopy, J. Phys. B 53, 074001 (2020).
  40. M. Han, P. Ge, J. Wang, Z. Guo, Y. Fang, X. Ma, X. Yu, Y. Deng, H. J. Wörner, Q. Gong, and Y. Liu, Complete characterization of sub-Coulomb-barrier with phase-of-phase attoclock, Nat. Photonics 15, 765 (2021).
  41. M. Yu, K. Liu, M. Li, J. Yan, C. Cao, J. Tan, J. Liang, K. Guo, W. Cao, P. Lan, Q. Zhang, Y. Zhou, and P. Lu, Full experimental determination of tunneling time with attosecond-scale streaking method, Light: Sci. Appl. 11, 215 (2022).
  42. W. Jin, T. Jiang, J. Liu, S. Luo, D. Ren, X. Li, C. Wang, Y. Lang, X. Wang, J. Zhao, Z. Zhao, and D. Ding, Strong field ionization dynamics resolved by two-color elliptical phase-of-phase spectroscopy, Ultrafast Sci. 4, 0066 (2024).
  43. T. Rathje, N. G. Johnson, M. Möller, F. Sußmann, D. Adolph, M. Kübel, R. Kienberger, M. F. Kling, G. G. Paulus, and A. M. Sayler, Review of attosecond resolved measurement and control via carrier-envelope phase tagging with above-threshold ionization, J. Phys. B 45, 074003 (2012).
  44. M. F. Kling, J. Rauschenberger, A. J. Verhoef, E. Hasović, T. Uphues, D. B. Milošević, H. G. Muller, and M. J. J. Vrakking, Imaging of carrier-envelope phase effects in above-threshold ionization with intense few-cycle laser fields, New J. Phys. 10, 025024 (2008).
  45. K. Amini, J. Biegert, F. Calegari, A. Chacón, M. F. Ciappina, A. Dauphin, D. K. Efimov, C. Figueira de Morisson Faria, K. Giergiel, P. Gniewek, et al., Symphony on strong field approximation, Rep. Prog. Phys. 82, 116001 (2019).
  46. D. Habibović and D. B. Milošević, Phase-dependent effects in strong-field ionization induced by a two-component ultrashort laser pulse, Eur. Phys. J. D 78, 8 (2024).
  47. D. Habibović, A. S. Jašarević, M. Busuladžić, and D. B. Milošević, High-order above-threshold ionisation of diatomic molecules by few-cycle bicircular and orthogonally polarised two-colour pulses, Phys. Chem. Chem. Phys. 26, 19008 (2024).
  48. D. Habibović and D. B. Milošević, Strong-field processes induced by an ultrashort linearly polarized pulse with two carrier frequencies, Phys. Rev. A 109, 043110 (2024).
  49. B. Fetić, M. Šišić Čaluk, D. Habibović, and D. B. Milošević, High-order above-threshold ionization: Improved strong-field approximation vs numerical solution of the time-dependent Schrödinger equation, J. Phys. B 58, 195601 (2025).
  50. The term "modified" is used to stress that neither S(p̃,t) nor S(p̃,t)−Ipt is the integral of the classical Lagrangian of the system [see, for example, Eq. (6.75) in [51] and the text below it].
  51. C. J. Joachain, N. J. Kylstra, and R. M. Potvliege, Atoms in Intense Laser Fields (Cambridge University Press, Cambridge, 2011).
  52. A. A. Radzig and B. M. Smirnov, Reference Data on Atoms, Molecules, and Ions (Springer, Berlin, 1985).
  53. G. N. Gibson, R. R. Freeman, T. J. McIlrath, and H. G. Muller, Excitation and ionization dynamics in short-pulse multiphoton ionization, Phys. Rev. A 49, 3870 (1994).
  54. M. A. Walker, P. Hansch, and L. D. Van Woerkom, Intensity-resolved multiphoton ionization: Circumventing spatial averaging, Phys. Rev. A 57, R701(R) (1998).
  55. R. Kopold, W. Becker, M. Kleber, and G. G. Paulus, Channel-closing effects in high-order above-threshold ionization and high-order harmonic generation, J. Phys. B 35, 217 (2002).

Outline

Information

Phys. Rev. A 113, 033116– Published 20 March, 2026
  • Received 2 January 2026
  • Accepted 3 March 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation