## Abstract

Subfemtosecond light pulses can be obtained by superposing several high harmonics of an intense laser pulse. Provided that the harmonics are emitted simultaneously, increasing their number should result in shorter pulses. However, we found that the high harmonics were not synchronized on an attosecond time scale, thus setting a lower limit to the achievable x-ray pulse duration. We showed that the synchronization could be improved considerably by controlling the underlying ultrafast electron dynamics, to provide pulses of 130 attoseconds in duration. We discuss the possibility of achieving even shorter pulses, which would allow us to track fast electron processes in matter.

The generation of soft x-ray, subfemtosecond light pulses is providing new opportunities for time-resolved spectroscopy (*1*). With the recent production of pulses as short as 650 attoseconds (as) (*2*) and trains of 250-as pulses (*3*), the time-resolved dynamics of particles in the attosecond (10^{–18} s) regime is becoming a reality. The key method of synthesizing attosecond pulses (*4*, *5*) is the generation, in rare gases, of high harmonics of an infrared femtosecond laser pulse (*6*, *7*). Such coherent harmonics can be generated to high orders (300 or more) (*8*, *9*), covering a spectral bandwidth of hundreds of electron volts. The physical origin of the harmonic emission can be described by a semi-classical three-step model (*10*, *11*). First, at some initial time *t*_{i}, close to the peak of the laser electric field, an electron wavepacket tunnels through the potential barrier formed by the combined Coulomb and laser fields; next, it oscillates almost freely in the laser field, gaining kinetic energy; finally, this energy is converted into a high-energy photon through recombination with the parent ion at time *t*_{r}. An infrared pulse lasting a few cycles allows the production of a single x-ray burst with a continuous spectrum (*12*, *13*). Using a multicycle (50-fs) laser pulse, this sequence is repeated every half cycle of the laser optical field, leading to periodic emission of light bursts with a discrete spectrum (*14*, *15*) containing only odd multiples of the laser frequency: ω* _{q}* =

*q*ω

_{0}(where ω

_{0}is the fundamental laser angular frequency, and

*q*is the harmonic order). The temporal profile

*I*(

*t*) of the intensity of the emitted pulses can be determined from knowledge of the spectral amplitudes (

*A*) and phases [φ(ω

_{q}*)] of the*

_{q}*N*contributing harmonic fields through the expression

The spectral phase φ(ω* _{q}*), which is sampled only at the harmonic frequencies, reflects their synchronization during the emission process. A linear relationship, φ(ω

*) =*

_{q}*q*ω

_{0}

*t*

_{e}, with

*t*

_{e}(harmonic emission time) independent of

*q*, results in so-called Fourier-limited pulses of duration τ

_{FL}∝ 1/

*N*, which is the shortest duration allowed by the bandwidth. In that situation, all harmonics are emitted at the same time, (Δφ is the spectral phase difference between two consecutive harmonics). Because of the uncertainty principle, one cannot define an instant associated with a single energy: The emission time of harmonic

*q*applies to a group of harmonics centred on ω

*; that is, the group delay. The first spectral phase measurements of five consecutive harmonics generated in argon revealed a near-linear phase relationship, corresponding to a train of 250- as pulses (*

_{q}*3*). The synthesis of much shorter bursts demands that this linearity hold for a much wider spectral range.

The generation of attosecond pulses is best understood in terms of the quantum-mechanical theory developed in the Feynman's path integral formalism (*16*, *17*), which recovers the semiclassical picture: The emission of a given harmonic is associated with at least two quantum orbits that are complex electron trajectories labeled in terms of their short or long return times, *t*_{r} – *t*_{i}. We find that the real part of their recombination times determines the values for *t*_{e} (*18*). The semiclassical calculations reveal that over a large spectral range, *t*_{e} varies quasilinearly with ω* _{q}* (linear chirp), with a positive slope for the branch corresponding to the short trajectories (that is, the higher harmonics are emitted after the lower ones) and a negative slope for the long trajectory branch (Fig. 1A). This lack of synchronization is a direct consequence of the temporal broadening of the electron wavepacket, and can be quantified by the time shift between emissions of two consecutive harmonics,

*q*and

*q*+ 2: Δ

*t*

_{e}(ω

_{q}_{+1}) =

*t*

_{e}(ω

_{q}_{+2}) –

*t*

_{e}(ω

*). This temporal drift in harmonic emission has dramatic effects on the duration of the emitted attosecond pulses: Selecting the entire available spectral range no longer provides the shortest possible pulses. The value of this time shift determines the minimum pulse duration τ*

_{q}_{opt}. Assuming Gaussian pulses with linear chirp, it can be shown analytically (and numerical calculations with a discrete harmonic spectrum confirm this) that at the optimum number of harmonics . Our calculations also show that Δ

*t*

_{e}∝ ω

_{0}/

*U*

_{p}, where is the ponderomotive energy in atomic units (

*I*

_{IR}is laser intensity) and could thus be minimized through the generating conditions. One would expect the generation of (at least) two bursts per half laser period because of the short and long trajectories (

*19*). However, in some harmonic generating conditions, the macroscopic response is predicted to contain only the shorttrajectory contributions, resulting in regular harmonic phases (

*19*,

*20*).

The measurement of the relative harmonic phases can be performed through two-photon two-color ionization (*3*, *21*). Photoelectron spectra obtained by superposing the harmonic pulse and a weak infrared field consist of peaks corresponding to the harmonic energies and of sidebands located in between. When the delay between harmonic and infrared fields varies, the sideband amplitudes oscillate. The phase of this oscillation directly provides the phase difference between two consecutive harmonics and thus the harmonic emission time: ϕ_{SB}(ω_{q}_{+1}) = φ(ω_{q}_{+2}) – φ(ω* _{q}*) – Δφ

_{q}_{+1}

^{at}≈ 2ω

_{0}

*t*

_{e}(ω

_{q}_{+1}), because Δφ

_{q}_{+1}

^{at}(the atomic phase term) is only a small correction (

*18*).

Our experimental setup allows a broadband harmonic analysis and fine-tuning of the measurement parameters (Fig. 2). The photoelectron spectra obtained by using argon as a generating and detecting gas are displayed as a function of delay for *I*_{IR} = 1.2 × 10^{14} W/cm^{2} (Fig. 3A). Macroscopic selection of the short trajectory contributions (*19*, *20*) was achieved through phase matching by focusing the laser beam ahead of the gas jet (*22*, *23*). The dephasing of the sideband oscillations as a function of order is a direct indication of the lack of synchronization in the harmonic emission, which is confirmed by the plot of the emission time (Fig. 3B). A linear dependence is obtained, corresponding to a time shift Δ*t*_{e}^{exp} = 106 ± 8 as.

We carried out a thorough study of the time shift as a function of laser intensity and generating gas (Fig. 1B). In all cases, the synchronization improves when the laser intensity is increased up to the ionization intensity of the generating medium. Agreement with the predictions of our theoretical analysis is good over a large range of intensity, despite a slight upshift of the experimental data. In good phase-matching conditions, the macroscopic response is indeed predicted to be very close to the single-atom one restricted to the selected trajectories, and we checked that our time shift calculations agree quantitatively with the linear chirp resulting from a full macroscopic study (*20*). The single-atom response provides a limit that can be experimentally approached by optimizing the generating conditions, but never reached. The experimental curves deviate from the 1/*I*_{IR} dependence close to the saturation intensity of each gas: When the medium becomes ionized, we observe strongly perturbed phase relationships, because propagation effects (in particular phase mismatch) become dominant.

The minimum time shift is obtained by using neon as the generating gas. Its spectrum extends to much higher orders, allowing a broadband study of the harmonic synchronization with helium as the detecting gas (its ionization cross-section is flatter). The emission times measured at *I*_{IR} = 3.8 × 10^{14} W/cm^{2} indicate a good linearity from H_{25} to H_{55}, with Δ*t*_{e}^{exp} = 33 ± 3 as, close to the single-atom prediction Δ*t*_{e}^{th} = 26.1 ± 0.2 as (Fig. 4A). The temporal profile of harmonic emission, as well as its timing with respect to the generating laser field, can be reconstructed from the measured spectral amplitudes and phases (Fig. 4B) (*3*, *24*). When the temporal profile of groups of five consecutive harmonics is plotted, their different emission times are clearly evidenced: The time separating two groups is 5Δ*t*_{e} ≈ 150 as. Low harmonics are emitted close to the peak of the laser field, with the highest harmonics being emitted close to zero field. This temporal drift results in a strong distortion of the pulse corresponding to the full spectral bandwidth (H_{25} to H_{69}), and in a temporal broadening to 150 as, which is three times the Fourier limit. The minimum pulse duration imposed by Δ*t*_{e} is obtained by selecting *N*_{opt} = 11 harmonics, resulting in close–to–Fourier-limited pulses of duration τ_{opt} = 130 as.

This measurement constitutes a direct observation of the electron return times with a temporal resolution of 50 as, set by the available spectral width. In that respect, it can be considered as an unambiguous validation of the three-step model. Moreover, deconvolution from the recombination probability provides a characterization of the temporal shape of the recombining high-energy electron wavepacket, which is crucial for its direct use as an attosecond probe of molecular dynamics (*25*, *26*). It opens up new avenues for generating even shorter pulses. Increasing the saturation intensity by using ultrashort laser pulses and helium as the generating gas should help to improve the harmonic synchronization. Another possibility is to increase the laser wavelength using currently developed mid-infrared lasers (*27*), because the time shift varies as ω_{0}^{3}. Finally, ultrashort pulse durations could be obtained by optical compression techniques: The temporal drift can be compensated by propagation in a medium or optical system where the high frequencies travel faster than the low ones. From that perspective, our accurate knowledge of the phase relationship is essential. One could, for instance, use a “plasma compressor,” because free electrons induce a negative group velocity dispersion. In the case presented in Fig. 4A, propagation through 5 mm of fully ionized helium gas with an electron density *n*_{e} of 9.7 × 10^{19} cm^{–3} would lead to pulse compression down to 75 as; that is, about 3 atomic units, using H_{25} to H_{55}, without loss in transmission (Fig. 4C). Alternatively, thin filters with linear negative group velocity dispersion could be used to compensate more accurately for the temporal drift. Our analysis holds for single attosecond bursts produced by few-femtosecond laser pulses or by temporal confinement of harmonic emission (*28*, *29*). We look forward to eventually reducing pulse duration below one atomic unit of time and thus tracking the fastest electron processes in matter with few-cycle soft x-ray pulses.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/302/5650/1540/DC1

Materials and Methods

Fig. S1

References