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Optical attosecond pulses and tracking the nonlinear response of bound electrons


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.

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Figure 1: Nonlinear response of bound electrons of Kr to an optical attosecond pulse.
Figure 2: Synthesis of an optical attosecond pulse.
Figure 3: Attosecond control of bound electrons in Kr.
Figure 4: Sub-femtosecond, delayed nonlinear response of bound electrons in Kr.

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This work was supported by European Research Council grant (Attoelectronics-258501), the Deutsche Forschungsgemeinschaft Cluster of Excellence: Munich Centre for Advanced Photonics (, the Max Planck Society and the European Research Training Networks ATTOFEL and MEDEA, the Russian Foundation for Basic Research (projects numbers 13-02-01465, 13-02-92115 and 13-04-40335) and the Welch Foundation (grant number A-1801).

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Authors and Affiliations



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.

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Correspondence to E. Goulielmakis.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 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.

Extended Data Figure 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).

Extended Data Figure 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).

Extended Data Figure 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.

Extended Data Figure 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.

Extended Data Figure 6 Representation of the possible multiphoton transitions on a Kr atom energy level diagram assuming third- and fifth-order nonlinear processes.

Extended Data Figure 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.

Extended Data Figure 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.

Extended Data Figure 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).

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

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Hassan, M., Luu, T., Moulet, A. et al. Optical attosecond pulses and tracking the nonlinear response of bound electrons. Nature 530, 66–70 (2016).

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