Abstract
Attosecond extremeultraviolet pulses^{1} have a complex space–time structure^{2}. However, at present, there is no method to observe this intricate detail; all measurements of the duration of attosecond pulses are, to some extent, spatially averaged^{1,3,4,5}. A technique for determining the full space–time structure would enable a detailed study of the highly nonlinear processes that generate these pulses as a function of intensity without averaging^{6,7}. Here, we introduce and demonstrate an alloptical method to measure the space–time characteristics of an isolated attosecond pulse. Our measurements show that intensitydependent phase and quantumpath interference both play a key role in determining the pulse structure. In the generating medium, the attosecond pulse is strongly modulated in space and time. Propagation modifies but does not erase this modulation. Quantumpath interference of the singleatom response, previously obscured by spatial and temporal averaging, may enable measuring the laserfielddriven ion dynamics with subcycle resolution.
Main
Fully defining an attosecond pulse requires knowledge of its phase variation both temporally and spatially. Until now, temporal^{1,3,4,5,8,9} and spatial^{10,11,12} measurements are achieved only separately. As the temporal characterization methods (known as RABBIT; ref. 1 and CRAB; refs 3, 4, 5, 9, 13) rely on the photoelectric effect, they average over the spatial profile of a pulse, mixing the contribution from the different emitters of the extremeultraviolet (XUV) source at a secondary target. On the other hand, spatial measurements of XUV emission have been achieved using small apertures^{10,11} or two foci^{12}. Although temporal information remains available in principle, neither method seems compatible with RABBIT or CRAB. Therefore, space–time measurements of attosecond pulses have never been made.
To solve the space–time problem, we turn to an in situ technique. The in situ method is a unique method of measurement that is feasible only for highly nonlinear processes^{14,15,16}. It relies on the fact that adding a single photon to an already highly nonlinear process only weakly perturbs the process^{17,18}. Yet, it can modify the spatial and spectral pattern of a beam. The in situ method has been considered in attosecond pulse metrology to determine only the temporal profile of the average attosecond pulse within attosecond pulse trains^{14}. For that measurement, a weak secondharmonic beam copropagates with the fundamental beam to break the symmetry between adjacent attosecond pulses, thereby allowing an evenorder harmonic signal. Temporal information was encoded in the evenorder harmonic signal as a function of the phase delay between the fundamental and secondharmonic laser pulses^{14}.
For our spatially encoded in situ measurement, we produce XUV radiation using the fundamental laser pulse with a timedependent polarization^{3} as illustrated in Fig. 1a and described in the Methods. We bring the weak perturbing beam into the medium at a small angle θ_{p} to gently modify the process of attosecond pulse generation in space as well as in time. This seemingly small change has a large impact. It perturbs the wavefront of the XUV radiation in the nearfield (in the medium), which, in turn, modifies the farfield (at the detector) pattern as illustrated in Fig. 1a. As we delay the perturbing beam relative to the fundamental, we modulate the spatial dependence of the phase and therefore, the near and farfield patterns. Our perturbing beam serves as a spatial gate. Measuring the farfield pattern as a function of delay τ shown in Fig. 1b allows us to reconstruct the amplitude and phase of the attosecond pulse in space and time in the region where it is produced and, therefore, everywhere in space.
Figure 2a,b shows experimental maps constructed from these twodimensional (2D) images for the energy range of 46±0.8 eV and 79±0.8 eV. In these figure panels, each XUV component is isolated and plotted individually. The vertical axis is the spatial image of the XUV whereas the horizontal axis consists of 48 images placed sidebyside, each taken at a different time delay between the fundamental and the secondharmonic perturbing beam shown on the horizontal axis. Measuring the farfield distribution as a function of delay allows us to determine the amplitude and phase of XUV emission in the nearfield.
To analyse the data and reconstruct the pulse, it is convenient to use the strong field approximation (SFA). This is the most intuitive way to treat highharmonic and attosecond pulse generation^{19,20}. In the SFA, the electron moves along quantum trajectories under the force of the laser field alone. The phase of the XUV emission is determined by the classical action and the XUV emission time. The perturbation to the quantum trajectories induced by the perturbing beam can be derived from the SFA (ref. 17). The phase modification is insensitive to the approximations in the SFA to first order^{14}. In Supplementary Section S1, we arrive at the farfield intensity pattern for the XUV photon energy of ɛ under a 1D approximation as shown in equation (1):
In equation (1), E_{ɛ}^{far}(θ,τ)^{2} is the farfield pattern measured as a function of the propagation angle θ and the time delay τ. k_{ɛ} is the wavenumber of the XUV emission, c is the speed of the light and θ_{p} is the angle between the fundamental and the perturbation beam. E_{ɛ}^{near} is the complex spectral component of the unperturbed XUV as a function of the vertical position y in the nearfield. The superscripts, s and l, are used to denote short and long quantum trajectories. The gate function, G_{ɛ}(y) = (1+{\alpha}_{\varepsilon}){\text{e}}^{i{\sigma}_{\varepsilon}}, represents the firstorder approximation of the amplitude (α_{ɛ}) and the phase (σ_{ɛ}) modification due to the perturbing pulse.
For a given photon energy, the relation between the near and farfield shown in equation (1) is expressed as a superposition of two 2D spectrograms for short and long trajectories. Reconstructing the spatial structure of an attosecond pulse is equivalent to solving the phase retrieval problem of the unperturbed harmonics (E_{ɛ}^{near}) and the modulation (G_{ɛ}) for both quantum trajectories. If the XUV generation is dominated by a single quantum trajectory, equation (1) becomes a simple spectrogram, and it is the spatial equivalent to the spectrogram used in the frequencyresolved optical gating to reconstruct the spectral amplitude and phase of femtosecond laser pulses^{21,22,23}. One can use the principal component generalized projection algorithm^{22} for the reconstruction as described in Supplementary Section S2.
To completely reconstruct the XUV radiation generated from both quantum trajectories, we used a parametric fitting procedure as described in Supplementary Section S3. Reconstruction results are shown for 46±0.8 eV and 79±0.8 eV in Fig. 2c,d. The procedure determines the functions E_{ɛ}^{near(s,l)} and G_{ɛ}^{(s,l)} by searching for optimum parameters that produce the observed farfield intensity distribution E_{ɛ}^{far}(θ,τ)^{2}. We perform this reconstruction for all energies. Thus, the amplitude and the phase as a function of position at each XUV frequency are directly obtained for both quantum trajectories.
To obtain the temporal information, we recall that the perturbing field imposes a phase modulation^{14}, σ_{ɛ}^{(s)} = A_{ɛ}^{(s)}sin[k_{p}θ_{p}(y−c τ/θ_{p})+Φ_{ɛ}^{(s)}], on the XUV wavefront with a frequencydependent amplitude A_{ɛ}^{(s)} and phase Φ_{ɛ}^{(s)}, where k_{p} is the wavenumber of the perturbation. The phase of the modulation, Φ_{ɛ}^{(s)}, depends linearly on the XUV emission time (or group delay)^{14} that defines the relative spectral phase of the shorttrajectory components at the centre of the beam. This connects the spectral and spatial phase. The difference in the phase Φ_{ɛ}^{(s)} can be directly seen from the modulation as shown in Fig. 2c,d. The phase difference of 1.1 radindicates that the emission times for 46±0.8 eV and 79±0.8 eV radiation differ by 360 as. In this manner, the amplitude and phase of the XUV emission are fully determined for both quantum trajectories in space and time.
Figure 2f clearly shows that the XUV emission is strongly modulated in the nearfield (for reference, the unperturbed farfield pattern is shown in Fig. 2e). This modulation occurs because two major quantum paths—short and long trajectories—contribute to the atomic dipole. This quantumpath interference depends on the local intensity, producing the radial dependence of the modulation. In Fig. 2g, we compare the experimental result (data points) with SFA calculations (solid curves). The first destructive interference is observed at 77 eV for the XUV emission generated at the centre of the medium (I_{0}(y = 0) = 3.5×10^{14} W cm^{−2}). At 7 μm offcentre, we observe that the interference has the opposite phase (I_{0}(y = 7 μm) = 3.2×10^{14} W cm^{−2}). If all of this radiation is captured by a collection mirror and reimaged for a streak camera measurement, it is the average of this structure that is measured by RABBIT or CRAB.
Figure 3a,b shows a spatiotemporal snapshot of an isolated attosecond pulse in the nonlinear medium. Figure 3a shows the electric field (colour) of the attosecond pulse as a function of time and position. Figure 3b highlights how the temporal profile changes by moving from the pulse centre (red) to y = 12 μm offaxis (blue). The temporal profile of the attosecond pulse is delayed as we move offaxis. This delay arises because of the intensitydependent emission time τ_{ɛ} of the shorttrajectory components of the pulse. The emission time contours for the short trajectories at 46 (τ_{46 eV}^{(s)}) and 79 eV (τ_{79 eV}^{(s)}) are shown with dotted lines in Fig. 3a. In contrast, the longtrajectory components (τ_{79 eV}^{(l)}) shows the opposite direction of the delay.
The emission time of different frequency components of the attosecond pulses can be seen in the time–frequency plots in Fig. 3b, insets. The slope of the emission times measured in the experiment agrees with the slope of the emission times calculated by the saddle point approximation^{20} shown as circles in Fig. 3b, insets. The slope of the emission time (dτ_{ɛ}^{(s)}/dɛ), called the attochirp, is 8.5 as eV^{−1} and 11.3 as eV^{−1} for I_{0}(y = 0) = 3.5×10^{14} W cm^{−2} and I_{0}(y = 12 μm) = 2.6×10^{14} W cm^{−2}, showing the intensity dependence of the attochirp (dτ_{ɛ}/dɛ 1/I_{0}λ; refs 4, 15, 24).
The XUV pulse observed at the farfield (Fig. 3c) is the coherent superposition of all emissions in the nearfield. Time–frequency analysis (Fig. 3d, top inset) shows that the attosecond pulse onaxis in Fig. 3d is mainly composed of lowerenergy XUV photons owing to their flat wavefront in the nearfield (Supplementary Fig. S3). However, the attosecond pulse at 2.5 mrad offaxis in Fig. 3d is mainly composed of highenergy XUV photons of both the short and longtrajectory components as shown in Fig. 3d, bottom inset.
All of our measurements have been performed in a thin, lowdensity jet where limits imposed by the destruction of phase matching are unimportant. However, our approach can be applied more generally. As we reconstruct the XUV radiation from the farfield, the reconstructed XUV radiation in the nearfield is the coherent superposition of the radiation in the medium along the propagation direction. For a longer medium, the XUV radiation may vary along the propagation direction. However, the phase matching condition strictly limits the variation of the XUV radiation along the propagation direction. The phase of the XUV radiation cannot slip by more than a half wavelength with respect to the fundamental laser pulse (λ_{XUV}(50 eV)/2 = 12 nm) before phase matching is destroyed. In practice, this slip occurs on the fundamental. A similar phase velocity mismatch between the fundamental and second harmonic would have very little influence on our measurement. That means that our in situ measurement can be applied to any condition where phase matching is possible. Although we have concentrated on the case where the jet is at the centre of the focus, our spatially encoded in situ measurement is applicable to other jet positions. In addition, we have assumed a cylindrically symmetric XUV beam because the farfield pattern is sampled by a vertical slit. The reconstruction of asymmetric XUV radiation can also be achieved by measuring the 2D spatial pattern of the XUV beam. For aligned molecules, the XUV radiation may have elliptical polarization^{25}. As the weak perturbing beam mainly affects the propagation of the electron wave packet, the same amount of phase shift will be imposed for orthogonal polarization directions. Therefore, the XUV radiation can be independently reconstructed if the XUV polarization components can be separately measured. Finally, our in situ measurement can be generalized to attosecond pulse trains. For trains, the only modification is that the perturbing beam must have the same wavelength as the fundamental owing to the inversion symmetry of adjacent pulses in the train.
Although we have emphasized space–time characterization of an isolated attosecond pulse, the technique can also be used to measure the timedependent waveform of the perturbing laser pulse because the deflection angle of the XUV emission is proportional to the field strength of the perturbing beam at the time that the perturbation occurs. Thus, we have constructed an alloptical transient recorder^{26} that allows us to measure the complete waveform of a laser pulse, including its carrierenvelope phase, with high temporal resolution.
There are three important implications. First, quantumpath interference is prominently revealed in our space–time measurement. This interference is very sensitive to (and therefore will resolve) changes that occur in the ion between the short and longtrajectory emission times^{27}. It will allow us an unprecedented subcycle view of electricfielddriven electronic^{28} dynamics initiated by tunnelling.
Second, the spatial dimension has never been used for probing attosecond dynamics. Yet, any intensitydependent phenomenon—such as attosecond hole dynamics^{7}—will be imprinted on the spatial profile of an attosecond or highharmonic pulse. Our spatially encoded in situ measurement makes the spatial dimension experimentally accessible.
Finally, our results are important for any attosecond pump–probe experiments because space–time coupling limits the temporal resolution that can be achieved. Our alloptical approach characterizes the attosecond pulse in space and time for both quantum trajectories, which can be applied even for weak XUV sources^{29}. The only uncharacterized parameter of an attosecond pulse that remains to be measured is its carrierenvelope phase.
Methods
We used a carrierenvelopephasestabilized 5fs 760nm Ti:sapphire laser pulse. The stability of the carrierenvelope phase was measured by a singleshot f– 2f interferometer to have a rootmeansquare value of 400 mrad. After the secondharmonic generation (200μmthick barium borate crystal), the laser beam was divided by a longwavelengthpass beam splitter. The transmitted beam passed through a 120μmthick quartz plate and a broadband quarterwave plate to create a beam suitable for polarization gating^{3}. The reflected second harmonic was used as a perturbing beam. The two beams were combined, with the perturbing beam parallel but 7 mm below the fundamental, then focused by a 300 mm concave mirror. The angle of the perturbation beam (θ_{p} = 23 mrad) was chosen to maximize the deflection angle of the XUV emission without leading to separate diffraction orders^{17}. For a longer medium, the angle of the perturbing beam θ_{p} should be chosen to minimize the phase slip between the fundamental and the secondharmonic pulses along the propagation direction (that is, 2k_{0}L(1−cosθ_{p})<π). Here, k_{0} is the wavenumber of the fundamental and L is the medium length. The time delay between the two pulses was controlled by a piezo stage. Neon gas from a supersonic jet (nozzle diameter of 250 μm with a backing pressure of 4 bar) at the focus served as the nonlinear medium. The laser beam is focused roughly 200 μm below the nozzle with an intensity of I_{0} = 3.5×10^{14} W cm^{−2}. The peak intensity of the fundamental beam was estimated from the cutoff energy of the XUV radiation. The offaxis intensity was inferred from the beam profile (40 μm at fullwidth at halfmaximum) measured by a CCD (chargecoupled device) camera. The ratio of intensities between the fundamental (I_{0}) and the perturbing beam (I_{2ω}) was I_{2ω}/I_{0} = 3×10^{−4}. XUV radiation passed through a ∼250μmwide spectrometer slit. The frequencyresolved farfield pattern was recorded with an imaging microchannel plate and CCD camera. The time–frequency analysis shown in Fig. 3 was made by a shorttime Fourier transform with a 200 as Gaussian time window. The details of the reconstruction procedure are explained in the Supplementary Information.
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Acknowledgements
We gratefully acknowledge the technical assistance of D. Crane and B. Avery. We also acknowledge financial support from NSERC, AFOSR and MURI Grant No. W911NF0710475. In addition, E.F. acknowledges support from the Marie Curie International Outgoing fellowship.
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K.T.K. and P.B.C. conceived the idea and designed the experiment. K.T.K., C.Z., A.D.S., S.E.K. and G.G. performed the experiment and collected the data. K.T.K. provided the theoretical analysis and analysed the experimental data. K.T.K., D.M.V. and P.B.C. prepared the initial manuscript. All authors contributed in writing the manuscript.
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Kim, K., Zhang, C., Shiner, A. et al. Manipulation of quantum paths for space–time characterization of attosecond pulses. Nature Phys 9, 159–163 (2013). https://doi.org/10.1038/nphys2525
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DOI: https://doi.org/10.1038/nphys2525
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