Optical attosecond pulses and tracking the nonlinear response of bound electrons

Journal name:
Nature
Volume:
530,
Pages:
66–70
Date published:
DOI:
doi:10.1038/nature16528
Received
Accepted
Published online

The time it takes a bound electron to respond to the electromagnetic force of light sets a fundamental speed limit on the dynamic control of matter and electromagnetic signal processing. Time-integrated measurements of the nonlinear refractive index1 of matter indicate that the nonlinear response of bound electrons to optical fields is not instantaneous; however, a complete spectral characterization of the nonlinear susceptibility tensors2—which is essential to deduce the temporal response of a medium to arbitrary driving forces using spectral measurements—has not yet been achieved. With the establishment of attosecond chronoscopy3, 4, 5, the impulsive response of positive-energy electrons to electromagnetic fields has been explored through ionization of atoms6 and solids7 by an extreme-ultraviolet attosecond pulse8 or by strong near-infrared fields9, 10, 11. However, none of the attosecond studies carried out so far have provided direct access to the nonlinear response of bound electrons. Here we demonstrate that intense optical attosecond pulses synthesized in the visible and nearby spectral ranges allow sub-femtosecond control and metrology of bound-electron dynamics. Vacuum ultraviolet spectra emanating from krypton atoms, exposed to intense waveform-controlled optical attosecond pulses, reveal a finite nonlinear response time of bound electrons of up to 115 attoseconds, which is sensitive to and controllable by the super-octave optical field. Our study could enable new spectroscopies of bound electrons in atomic, molecular or lattice potentials of solids12, as well as light-based electronics operating on sub-femtosecond timescales and at petahertz rates13, 14, 15.

At a glance

Figures

  1. Nonlinear response of bound electrons of Kr to an optical attosecond pulse.
    Figure 1: Nonlinear response of bound electrons of Kr to an optical attosecond pulse.

    a, Nonlinear dipole moment ρ(t) of Kr atoms subjected to an intense half-cycle optical field (dashed red line) calculated using the adiabatic (ρad(t), black line) and the TDSE (ρTDSE(t), blue line) model. The dashed grey line represents the bound-electron response in the energy range (10–15 eV) that is calculated using the TDSE model. The inset shows a close-up of the peak at time t = 0, highlighting the delay τ between the predictions of the two models. b, c, Time–frequency analysis of the nonlinear dipoles ρad(t) (b) and ρTDSE(t) (c) (left panels); simulated nonlinear dipole spectra (5.7–8.2 eV) as a function of the global phase ϕG of the optical attosecond pulse predicted by the adiabatic (b) and TDSE (c) Kr models (right panels). The red line represents the instantaneous intensity of the driving field. The dashed white lines represent the centre of mass along the x axis of each plot. The colour scale represents spectral intensity in arbitrary units (a.u.). d, e, Low-pass-filtered (0–8 eV) normalized nonlinear dipoles for three peak intensities of the driving field: 4 × 1013 W cm−2 (blue lines), 6 × 1013 W cm−2 (orange lines) and 8 × 1013 W cm−2 (green lines) for global phases ϕG ≈ 0 (d) and ϕG ≈ π/2 rad (e).

  2. Synthesis of an optical attosecond pulse.
    Figure 2: Synthesis of an optical attosecond pulse.

    a, Photograph of an attosecond light-field synthesizer (top). The beams (artificially visualized) of ultra-wide-band pulses (about 1.1–4.6 eV) are divided by dichroic beam-splitters into four, almost equally wide, spectral bands. The pulses in each band are compressed so that they have durations of a few femtoseconds (the temporal intensity profiles are shown in the bottom panels), before they are spatiotemporally superimposed to yield a single beam/pulse at the exit of the apparatus. The insets in the bottom panels show representative beam profiles in the far field. b, Attosecond-streaking spectrogram of an optical attosecond pulse. The colour bar represents the yield of photoelectron counts in arbitrary units. c, d, Evaluated electric field (c) and instantaneous intensity profile (d; blue line) of the pulse. The intensity profile has a FWHM duration of approximately 380 as. The inset of d shows a close-up of this instantaneous intensity profile (blue line, ϕG ≈ 0) along with that of the same pulse, but with ϕG ≈ π/2 rad (dashed red line), and their common intensity envelope (dashed black line), which has a FWHM duration of approximately 975 as.

  3. Attosecond control of bound electrons in Kr.
    Figure 3: Attosecond control of bound electrons in Kr.

    a, e, VUV spectra generated in neutral Kr atoms driven by an optical attosecond pulse (a) and a single-cycle pulse (e) of the same peak intensity (about 5 × 1013 W cm−2), at global phase settings of ϕG ≈ 0 (red line) and ϕG ≈ π/2 rad (black line). b, f, Spectrograms composed of 25 VUV spectra (about 5.5–15 eV) recorded as a function of ϕG for an optical attosecond pulse (b) and a single-cycle pulse (f). c, g, Sections (about 5.5–8 eV) of the global-phase spectrograms in b and f, respectively. d, h, Representative driving waveforms at ϕG ≈ 0 (d) and ϕG ≈ π/2 rad (h). The colour scales in b, c, f and g indicate intensity (in arbitrary units).

  4. Sub-femtosecond, delayed nonlinear response of bound electrons in Kr.
    Figure 4: Sub-femtosecond, delayed nonlinear response of bound electrons in Kr.

    ac, Global-phase spectrograms of the optical attosecond pulse recorded at peak-intensity settings of about 5 × 1013 W cm−2 (a), 7 × 1013 W cm−2 (b) and 9 × 1013 W cm−2 (c) are shown in the left panels. Corresponding reconstructed spectrograms based on equation (1) are shown in the right panels. The colour bar represents spectral intensity in arbitrary units. d, e, Low-pass-filtered (0–8 eV) nonlinear dipoles obtained using the intensity settings in ac (dark blue, orange, green lines, respectively) are shown along with the instantaneous response (black line) simulated for global-phase settings of the optical attosecond pulse of ϕG ≈ 0 (d) and ϕG ≈ π/2 rad (e). Standard errors of the mean for the delays (τ) indicated are evaluated from the reconstruction of three data sets recorded under identical experimental conditions. The dashed red lines are the normalized instantaneous intensities of the input electric fields.

  5. Supercontinuum spectra and experimental set-up.
    Extended Data Fig. 1: Supercontinuum spectra and experimental set-up.

    a, The supercontinuum spectra spans more than two octaves (1.1–4.6 eV), and is shown by the black line; the spectra of the individual channels ChNIR, Chvis, Chvis-UV and ChDUV are shown by the red, orange, blue and violet lines, respectively. b, Schematic illustration of the experiment set-up.

  6. Spectral equalization by a metal–dielectric–metal (MDM) coating.
    Extended Data Fig. 2: Spectral equalization by a metal–dielectric–metal (MDM) coating.

    a, The designed reflectivity (red line) and spectral phase (black) of the MDM coating applied on the outer mirror of the double mirror module shown in Extended Data Fig. 1. b, Multi-octave spectrum at the exit of the synthesizer (red line) and upon reflection off the MDM mirror (black line).

  7. Global phase and CEP difference in a sub-cycle regime.
    Extended Data Fig. 3: Global phase and CEP difference in a sub-cycle regime.

    a, Half-cycle, cosine-like (blue line) and sine-like (red line) waveforms, compared with the carrier wave (dotted blue and red lines) and envelope (dashed blue lines). b, Limits of CEP decomposition in the time domain. Retrieved CEP (solid lines) versus global-phase offset (dashed lines) as a function of the ratio between the FWHM of the intensity envelope and the period of the carrier wave for ϕG = π/4 rad (blue line), ϕG = π/2 rad (green line) and ϕG = 3π/4 rad (red).

  8. Simulated global-phase spectrograms.
    Extended Data Fig. 4: Simulated global-phase spectrograms.

    a, b, Calculated global-phase (ϕG) spectrograms using the adiabatic model (a) and the TDSE model (b) for experimentally sampled waveforms (optical attosecond pulse) and an intensity of about 4 × 1013 W cm−2. c, Simulated global-phase spectrograms for a sub-cycle pulse (about 2 fs; shown in inset). The polarization response is insensitive to global-phase variation of the waveform; see also experiments in Fig. 3. The colour scale represents the spectral intensity in arbitrary units.

  9. Nonlinearity study of the interaction of an optical attosecond pulse with a Kr atom.
    Extended Data Fig. 5: Nonlinearity study of the interaction of an optical attosecond pulse with a Kr atom.

    ad, Data points show the spectrally integrated yield of the nonlinear polarization (0–8 eV) as a function of the intensity of the optical attosecond pulse for two global phases ϕG of the driving waveform, ϕG ≈ 0 (a, c) and ϕG ≈ π/2 rad (b, d), evaluated from TDSE simulations (a, b) and our experiments in Fig. 4 (c, d ). The error bars represent the standard error of the mean of the input field intensity and the calculated integrated spectral yield of the nonlinear polarization. Δ is the slope of the linear fitting of the log–log plots; the errors shown indicate the standard error of mean, with n = 3. Insets show temporal profiles of the driving electric fields in arbitrary units.

  10. Representation of the possible multiphoton transitions on a Kr atom energy level diagram assuming third- and fifth-order nonlinear processes.
    Extended Data Fig. 6: Representation of the possible multiphoton transitions on a Kr atom energy level diagram assuming third- and fifth-order nonlinear processes.
  11. Synthetic global-phase spectrograms and nonlinear dipole fittings.
    Extended Data Fig. 7: Synthetic global-phase spectrograms and nonlinear dipole fittings.

    ac, Synthetic global-phase (ϕG) spectrograms determined using the simple model (equation (1) with the same set of parameters (a, b, c) and delays dt = 0 as (a), dt = 20 as (b) and dt = 30 as (c). The colour scale represents spectral intensity in arbitrary units. d, e, Fits (red lines) of the spectrally filtered (d, 5.5–8 eV; e, 0–8 eV) nonlinear dipoles (ρ(t), black lines) with ϕG ≈ 0 (left column) and ϕG ≈ π/2 rad (right column) using equation (1) and 6 × 1013 W cm−2.

  12. Reconstruction of the global-phase spectrogram calculated by the TDSE model.
    Extended Data Fig. 8: Reconstruction of the global-phase spectrogram calculated by the TDSE model.

    a, Calculated spectrogram at different global-phase settings (ϕG = −1.2π−1.2π rad) of the optical attosecond pulse using the TDSE model. b, Reconstructed spectrogram using the model of equation (1). The colour bar represents spectral intensity in arbitrary units. c, d, Calculated nonlinear dipoles using the TDSE model at ϕG ≈ 0 (c) and ϕG ≈ π/2 rad (d) are plotted in black; the reconstructed nonlinear dipoles and the adiabatic dipoles are shown in red and blue, respectively.

  13. Intensity dependence of the nonlinear dipole delay.
    Extended Data Fig. 9: Intensity dependence of the nonlinear dipole delay.

    a, b, The delay between the adiabatic dipole and the calculated dipole using the TDSE model as a function of the peak intensity according to TDSE simulations (a) and evaluated from the experimental data (b; see Fig. 4).

Tables

  1. Ionization and excitation probabilities calculated using TDSE simulations of Kr for a range of peak intensities of the optical attosecond pulse
    Extended Data Table 1: Ionization and excitation probabilities calculated using TDSE simulations of Kr for a range of peak intensities of the optical attosecond pulse

References

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Author information

  1. These authors contributed equally to this work.

    • M. Th. Hassan &
    • T. T. Luu

Affiliations

  1. Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany

    • M. Th. Hassan,
    • T. T. Luu,
    • A. Moulet,
    • M. Garg,
    • N. Karpowicz,
    • F. Krausz &
    • E. Goulielmakis
  2. Department für Physik, Ludwig-Maximilians-Universität, Am Coulombwall 1, D-85748 Garching, Germany

    • O. Raskazovskaya,
    • V. Pervak &
    • F. Krausz
  3. Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA

    • P. Zhokhov &
    • A. M. Zheltikov
  4. Physics Department, International Laser Center, M.V. Lomonosov Moscow State University, 119992 Moscow, Russia

    • P. Zhokhov &
    • A. M. Zheltikov

Contributions

M.Th.H., T.T.L., A.M. and M.G. conducted the experiments; O.R. and V.P. developed the dispersive optics for the experiments; E.G. planned the experiments and supervised the project; M.Th.H., T.T.L., P.Z., A.M. and N.K. conducted the simulations; and M.Th.H., T.T.L., A.M., A.M.Z., F.K. and E.G. interpreted the data and contributed to the preparation of the manuscript.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to:

Author details

Extended data figures and tables

Extended Data Figures

  1. Extended Data Figure 1: Supercontinuum spectra and experimental set-up. (182 KB)

    a, The supercontinuum spectra spans more than two octaves (1.1–4.6 eV), and is shown by the black line; the spectra of the individual channels ChNIR, Chvis, Chvis-UV and ChDUV are shown by the red, orange, blue and violet lines, respectively. b, Schematic illustration of the experiment set-up.

  2. Extended Data Figure 2: Spectral equalization by a metal–dielectric–metal (MDM) coating. (266 KB)

    a, The designed reflectivity (red line) and spectral phase (black) of the MDM coating applied on the outer mirror of the double mirror module shown in Extended Data Fig. 1. b, Multi-octave spectrum at the exit of the synthesizer (red line) and upon reflection off the MDM mirror (black line).

  3. Extended Data Figure 3: Global phase and CEP difference in a sub-cycle regime. (134 KB)

    a, Half-cycle, cosine-like (blue line) and sine-like (red line) waveforms, compared with the carrier wave (dotted blue and red lines) and envelope (dashed blue lines). b, Limits of CEP decomposition in the time domain. Retrieved CEP (solid lines) versus global-phase offset (dashed lines) as a function of the ratio between the FWHM of the intensity envelope and the period of the carrier wave for ϕG = π/4 rad (blue line), ϕG = π/2 rad (green line) and ϕG = 3π/4 rad (red).

  4. Extended Data Figure 4: Simulated global-phase spectrograms. (158 KB)

    a, b, Calculated global-phase (ϕG) spectrograms using the adiabatic model (a) and the TDSE model (b) for experimentally sampled waveforms (optical attosecond pulse) and an intensity of about 4 × 1013 W cm−2. c, Simulated global-phase spectrograms for a sub-cycle pulse (about 2 fs; shown in inset). The polarization response is insensitive to global-phase variation of the waveform; see also experiments in Fig. 3. The colour scale represents the spectral intensity in arbitrary units.

  5. Extended Data Figure 5: Nonlinearity study of the interaction of an optical attosecond pulse with a Kr atom. (205 KB)

    ad, Data points show the spectrally integrated yield of the nonlinear polarization (0–8 eV) as a function of the intensity of the optical attosecond pulse for two global phases ϕG of the driving waveform, ϕG ≈ 0 (a, c) and ϕG ≈ π/2 rad (b, d), evaluated from TDSE simulations (a, b) and our experiments in Fig. 4 (c, d ). The error bars represent the standard error of the mean of the input field intensity and the calculated integrated spectral yield of the nonlinear polarization. Δ is the slope of the linear fitting of the log–log plots; the errors shown indicate the standard error of mean, with n = 3. Insets show temporal profiles of the driving electric fields in arbitrary units.

  6. Extended Data Figure 6: Representation of the possible multiphoton transitions on a Kr atom energy level diagram assuming third- and fifth-order nonlinear processes. (196 KB)
  7. Extended Data Figure 7: Synthetic global-phase spectrograms and nonlinear dipole fittings. (427 KB)

    ac, Synthetic global-phase (ϕG) spectrograms determined using the simple model (equation (1) with the same set of parameters (a, b, c) and delays dt = 0 as (a), dt = 20 as (b) and dt = 30 as (c). The colour scale represents spectral intensity in arbitrary units. d, e, Fits (red lines) of the spectrally filtered (d, 5.5–8 eV; e, 0–8 eV) nonlinear dipoles (ρ(t), black lines) with ϕG ≈ 0 (left column) and ϕG ≈ π/2 rad (right column) using equation (1) and 6 × 1013 W cm−2.

  8. Extended Data Figure 8: Reconstruction of the global-phase spectrogram calculated by the TDSE model. (338 KB)

    a, Calculated spectrogram at different global-phase settings (ϕG = −1.2π−1.2π rad) of the optical attosecond pulse using the TDSE model. b, Reconstructed spectrogram using the model of equation (1). The colour bar represents spectral intensity in arbitrary units. c, d, Calculated nonlinear dipoles using the TDSE model at ϕG ≈ 0 (c) and ϕG ≈ π/2 rad (d) are plotted in black; the reconstructed nonlinear dipoles and the adiabatic dipoles are shown in red and blue, respectively.

  9. Extended Data Figure 9: Intensity dependence of the nonlinear dipole delay. (54 KB)

    a, b, The delay between the adiabatic dipole and the calculated dipole using the TDSE model as a function of the peak intensity according to TDSE simulations (a) and evaluated from the experimental data (b; see Fig. 4).

Extended Data Tables

  1. Extended Data Table 1: Ionization and excitation probabilities calculated using TDSE simulations of Kr for a range of peak intensities of the optical attosecond pulse (41 KB)

Additional data