Skip to main content Accessibility help

Analysis of above-threshold ionization by “Wigner-distribution-like function” method

  • Li Guo (a1), Mingqing Liu (a2) (a3), Ronghua Lu (a1), Shensheng Han (a1) and Jing Chen (a2) (a4)...


Above-threshold ionization (ATI) is one of the most fundamental processess when atoms or molecules are subjected to intense laser fields. Analysis of ATI process in intense laser fields by a Wigner-distribution-like (WDL) function is reviewed in this paper. The WDL function is used to obtain various time-related distributions, such as time-energy distribution, ionization time distribution, and time-emission angle distribution and so on, of atoms in laser field pulses with different laser parameters. For the linearly polarized laser pulses, the time-energy distribution intuitively shows from a quantum point of view the relationship between the ionization moment and the final energy and clearly reveals the origin of interference structures in the photoelectron spectrum. In particular, for linearly polarized few-cycle laser pulses, all calculated distributions show the dependence of electron behavior on the ionization time, emission direction, and carrier-envelope phase (CEP). For elliptically polarized few-cycle pulses, we calculate the angular distribution, ionization time distribution, and time-emission distribution, which are compared with the semiclassical calculations. Analysis shows that the offset angle (difference between positions of the peaks in the angular distributions obtained by two methods) in the angular distributions does not correspond to the offset time (difference between positions of the peaks in the ionization time distributions obtained by two methods) in the ionization time distributions, which implies that the attosecond angular streaking technique based on this correspondence between the offset angle and time is in principle inaccurate. Furthermore, the offset time cannot be interpreted as tunneling time.


Corresponding author

Author for correspondence: Jing Chen, Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing100088, China and HEDPS, Center for Applied Physics and Technology, Collaborative Innovation Center of IFSA, Peking University, Beijing100084, China, E-mail:


Hide All
Agostini, P, Fabre, F, Mainfray, G and Petite, G (1979) Free-free transitions following six-photon ionization of xenon atoms. Physical Review Letters 42, 1127.
Arbó, DG, Ishikawa, KL, Schiessl, K, Persson, E and Burgdörfer, J (2010) Intracycle and intercycle interferences in above-threshold ionization: The time grating. Physical Review A 81, 021403(R).
Arbó, DG, Ishikawa, KL, Persson, E and Burgdörfer, J (2012) Doubly differential diffraction at a time grating in above-threshold ionization: Intracycle and intercycle interferences. Nuclear Instruments and Methods in Physics Research B 279, 24.
Barth, I and Smirnova, O (2011) Nonadiabatic tunneling in circularly polarized laser fields: Physical picture and calculations. Physical Review A 84, 063415.
Becker, W, Grasbon, F, Kopold, R, Milošević, DB and Paulus, GG (2002) Above-threshold ionization: From classical features to quantum effects. Advances In Atomic, Molecular, and Optical Physics 48, 3598.
Boge, R, Cirelli, C, Landsman, AS, Heuser, S, Ludwig, A, Maurer, J, Weger, M, Gallmann, L and Keller, U (2013) Probing nonadiabatic effects in strong-field tunnel ionization. Physical Review Letters 111, 103003.
Brabec, T, Ivanov, MY and Corkum, PB (1996) Coulomb focusing in intense field atomic processes. Physical Review A 54, R2551R2554.
Camus, N, Yakaboylu, E, Fechner, L, Klaiber, M, Laux, M, Mi, Y, Hatsagortsyan, KZ, Pfeifer, T, Keitel, CH and Moshammer, R (2017) Experimental evidence for quantum tunneling time. Physcal Review Letters 119, 023201.
Chen, J, Liu, J and Chen, SG (2000) Rescattering effect on phase-dependent ionization of atoms in two-color intense fields. Physcial Review A 61, 033402.
Chen, J, Chu, SI and Liu, J (2006) Time-frequency analysis of molecular high-harmonic generation spectrum by means of wavelet transform and Wigner distribution techniques. Journal of Physics B Atomic Molecular and Optical Physics 39, 4747.
Cohen, L (1989) Time-Frequency Distributions a Review. Proceedings of the IEEE 77, 941981.
Corkum, PB (1993) Plasma perspective on strong-field multiphoton ionization. Physical Review Letters 71, 19941997.
Czirják, A, Kopold, R, Becker, W, Kleber, M and Schleich, WP (2000) The Wigner function for tunneling in a uniform static electric field. Optics Communications 179, 29.
Della Picca, R, Gramajo, AA, Garibotti, CR, López, SD and Arbó, DG (2016) Nonconstant ponderomotive energy in above-threshold ionization by intense short laser pulses. Physical Review A 93, 023419.
Eckle, P, Smolarski, M, Schlup, P, Biegert, J, Staudte, A and Schöffler, M (2008 a) Attosecond angular streaking. Nature Physics 4, 565570.
Eckle, P, Pfeiffer, AN, Cirelli, C, Staudte, A and Dörner, R (2008 b) Attosecond ionization and tunneling delay time measurements in helium. Science 322, 15251529.
Eslami, E and Basereh, K (2013) Effects of plasma and ultrashort laser pulse on residual electron energy in optical-field-ionized oxygen plasma. Laser And Particle Beams 31, 187193.
Faisal, FMH (1973) Multiple absorption of laser photons by atoms. Journal of Physics B Atomic Molecular and Optical Physics 6, L89.
Garg, JB, Rainwater, J and Havens, WW (1965) Neutron resonance spectroscopy. V. Nb Ag I and Cs. Physical Review 137, B547.
Guo, L, Han, SS and Chen, J (2010) Time-energy analysis of above-threshold ionization. Optics Express 18, 12401248.
Guo, L, Han, SS and Chen, J (2012) Time-energy analysis of above-threshold ionization in few-cycle laser pulses. Physical Review A 86, 053409.
Guo, L, Han, SS and Chen, J (2016) Study of above-threshold ionization by “Wigner-distribution-like function”. Acta Physica Sinica 65, 223203.
Guo, L, Han, SS, Hu, SL and Chen, J (2017) Time-energy analysis of above-threshold ionizaiton in the transverse direction of the lineary polarized laser pulses. Journal of Physics B Atomic Molecular and Optical Physics 50, 125006.
Guo, L, Hu, SL, Liu, MQ, Shu, Z, Liu, XW, Li, J, Yang, WF, Lu, RH, Han, SS and Chen, J (2019) Accuracy of the semiclassical picture of photoionization in intense laser fields. Available at
Hao, XL, Li, WD, Liu, J and Chen, J (2011) Effect of the electron initial longitudinal velocity on the nonsequential double-ionization process. Physical Review A 83, 053422.
Heuvell, HB, van Linden van den, and Muller, HG (1988) Limiting cases of excess-photon ionization. In Smith, SJ and Knight, PL (eds), Multiphoton Processes. Cambridge: Cambridge University Press, pp. 2534.
Hofmann, C, Landsman, AS, Cirelli, C, Pfeiffer, AN and Keller, U (2013) Comparison of different approaches to the longitudinal momentum spread after tunnel ionization. Journal of Physics B Atomic Molecular and Optical Physics 46, 125601.
Hofmann, C, Zimmermann, T, Zielinski, A and Landsman, AS (2016) Non-adiabatic imprints on the electron wave packet in strong field ionization with circular polarization. New Journal of Physics 18, 043011.
Hu, B, Liu, J and Chen, SG (1997) Plateau in above-threshold-ionization spectra and chaotic behavior in rescattering processes. Physical Letter A 236, 533542.
Ivanov, IA and Kheifets, AS (2014) Strong-field ionization of He by elliptically polarized light in attoclock configuration. Physical Review A 89, 021402.
Keldysh, LV (1965) Ionization in the field of a strong electromagnetic wave. Journal of Experimental and Theoretical Physics 20, 1307.
Kim, JH, Lee, DG, Shin, HJ and Nam, CH (2001) Wigner time-frequency distribution of high-order harmonics. Physical Review A 63, 063403.
Klaiber, M, Hatsagortsyan, Z and Keitel, CH (2015) Tunneling dynamics in multiphoton ionization and attoclock calibration. Physical Review Letters 114, 83001.
Kopold, R, Becker, W, Kleber, M and Paulus, GG (2002) Channel-closing effects in high-order above-threshold ionization and high-order harmonic generation. Journal of Physics B Atomic Molecular and Optical Physics 35, 217.
Korneev, PA, Popruzhenko, SV, Goreslavski, SP, Yan, TM and Bauer, D (2012) Interference Carpets in Above-threshold ionization: from the Coulomb-free to the Coulomb-dominated regime. Physical Review Letters 108, 223601.
Krainov, VP (1997) Ionization rates and energy and angular distributions at the barrier-suppression ionizaiton of complex atoms and atomic ions. Journal of the Optical Society of America B 14, 425.
Krausz, F and Ivanov, M (2009) Attosecond physics. Reviews of Modern Physics 81, 163234.
Kruit, P and Read, FH (1983) Magnetic-field parallelizer for 2-pi electronspectrometer and electron-image magnifier. Journal of Physics E: Scientific Instruments 16, 313324.
Landsman, AS, Weger, M, Maurer, J, Boge, R, Ludwig, A, Heuser, S, Cirelli, C, Gallmann, L and Keller, U (2014) Ultrafast resolution of tunneling delay time. Optica 1, 343349.
Larochelle, SFJ, Talebpoury, A and Chin, SL (1998) Coulomb effect in multiphoton ionization of rare-gas atoms. Journal of Physics B Atomic Molecular and Optical Physics 31, 1215.
Lindner, F, Schätzel, MG, Walther, H, Baltuška, A, Goulielmakis, E, Krausz, F, Milošević, DB, Bauer, D, Becker, W and Paulus, GG (2005) Attosecond double-slit experiment. Physical Review Letters 95, 040401.
Lompré, LA, L'Huillier, A, Mainfpay, G and Manus, C (1985) Laser-intensity effects in the energy-distributions of electrons produced in multiphoton ionization of rare-gases. Journal of the Optical Society of America B xtbf2, 1906.
Milošević, DB, Paulus, GG, Bauer, D and Becker, W (2006) Above-threshold ionization by few-cycle pulses. Journal of Physics B Atomic Molecular and Optical Physics 39, R203R262.
Offenberger, AA, Blyth, W, Preston, SG, Wark, JS, Key, MH, Dangor, AE, Modena, A, Najmudin, Z, Djaoui, A and Key, MH (1995) Optical ionization and heating of gases by intense picosecond KrF laser radiation. Laser And Particle Beams 13, 1931.
Pfeiffer, AN, Cirelli, C, Smolariski, M, Dörner, R and Keller, U (2011) Timing the release in sequential double ionization. Nature Physics 7, 428433.
Pfeiffer, AN, Cirelli, C, Smolarski, M, Dimitrovski, D, Abu-samhaet, M, Madsen, LB and Keller, U (2012 a) Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms. Nature Physics 8, 7680.
Pfeiffer, AN, Cirelli, C, Landsman, AS, Smolarski, M, Dimitrovski, D, Madsen, LB and Keller, U (2012 b) Probing the longitudinal momentum spread of the electron wave packet at the tunnel exit. Physical Review Letters 109, 083002.
Reed, VC and Burnett, K (1991) Role of resonances and quantum-mechanical interference in the generation of above-threshold-ionization spectra. Physical Review A 43, 62176226.
Reiss, HR (1980) Effect of an intense electromagnetic field on a weakly bound system. Physical Review A 22, 1786.
Teeny, N, Yakaboylu, E, Bauke, H and Keitel, CH (2016) Ionization time and exit momentum in strong-field tunnel ionization. Physical Review Letters 116, 063003.
Torlina, L, Morales, F, Kaushal, J, Ivanov, I, Kheifets, A, Zielinski, A, Scrinzi, A, GeertMuller, H, Sukiasyan, S, Ivanov, M and SmirnovaL, O (2015) Interpreting attoclock measurements of tunnelling times. Nature Physics 11, 503508.
Wang, CL, Lai, XY, Hu, ZL, Chen, YJ, Quan, W, Kang, HP, Gong, C and Liu, XJ (2014) Strong-field atomic ionization in elliptically polarized laser fields. Physical Review A 90, 013422.
Wickenhauser, M, Tong, XM and Lin, CD (2006) Laser-induced substructures in above-threshold-ionization spectra from intense few-cycle laser pulses. Physical Review A 73, 011401(R): 1–4.
Wu, J, Schmidt, LPH, Kunitski, M, Meckel, M, Voss, S, Sann, H, Kim, H, Jahnke, T, Czasch, A and Dörner, R (2012) Multiorbital tunneling ionization of the CO molecule. Physical Review Letters 108, 183001.
Yudin, GL and Ivanov, MY (2001) Nonadiabatic tunnel ionization: Looking inside a laser cycle. Physical Review A 64, 013409.


Analysis of above-threshold ionization by “Wigner-distribution-like function” method

  • Li Guo (a1), Mingqing Liu (a2) (a3), Ronghua Lu (a1), Shensheng Han (a1) and Jing Chen (a2) (a4)...


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed