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Proposal for complete characterization of attosecond pulses from relativistic plasmas

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Abstract

In this study, we propose two full-optical-setup and single-shot measurable approaches for complete characterization of attosecond pulses from surface high harmonic generation (SHHG): SHHG-SPIDER (spectral phase interferometry for direct electric field reconstruction) and SHHG-SEA-SPIDER (spatially encoded arrangement for SPIDER). 1D- and 2D-EPOCH PIC (particle-in-cell) simulations were performed to generate the attosecond pulses from relativistic plasmas under different conditions. Pulse trains dominated by single isolated peak as well as complex pulse train structures are extensively discussed for both methods, which showed excellent accuracy in the complete reconstruction of the attosecond field with respect to the direct Fourier transformed result. Kirchhoff integral theorem has been used for the near-to-far-field transformation. This far-field propagation method allows us to relate these results to potential experimental implementations of the scheme. The impact of comprehensive experimental parameters for both apparatus, such as spectral shear, spatial shear, cross-angle, time delay, and intensity ratio between the two replicas has been investigated thoroughly. These methods are applicable to complete characterization for SHHG attosecond pulses driven by a few to hundreds of terawatts femtosecond laser systems.

© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Recent progresses on ultrafast dynamics in atoms [1,2], molecules [3,4], liquid [5] and solids [6,7] have witnessed the remarkable benefits of attosecond temporal resolution. To date, the available tabletop attosecond sources in the extreme ultraviolet (XUV) regime are based on high harmonic generation (HHG) from gases [8] and solid density plasma-surfaces [9,10]. Presently, gas-based HHG is the preferred approach to produce attosecond pulses in XUV spectral region for applications. Nevertheless, the saturation intensity [11] and the cut-off scaling law [12] limit the photon flux and maximum frequency. It is extremely difficult to use mid-infrared [13] or apply the loose-focusing geometry [14] to achieve bright attosecond sources at water window or keV regime – especially when high single pulse energy is required. An alternative approach for delivering high-energy attosecond pulses up to the X-ray regime is solid-surface HHG (SHHG) from relativistic plasmas [15,16]. In principle, there is no driving intensity restriction to SHHG. At very high intensities the relativistic oscillating mirror (ROM) scheme is dominant, giving rise to the power law $I(\omega )\propto {\omega ^{ - 8/3}}$ ($\omega $ is the high harmonic frequency) [17], which was experimentally verified for photon energies up to 3 keV [16]. However, so far, only few spatial [18] and temporal [9,19] characterizations have been applied to this source of high-brilliance XUV.

Complete characterization of attosecond pulses is essential for source development and applications in studies of ultrafast dynamics. In the XUV regime, temporal characterization is challenging due to the lack of efficient dispersion-free optics and spectrally flat with instantaneous temporal response materials. One cannot direct translate the visible or infrared characterization techniques into the XUV regime. For the attosecond pulses from gas-based HHG, the cross-correlation (CC) and auto-correlation (AC) approaches have been widely used for the temporal structure study [20]. CC includes the second-order intensity volume autocorrelation (2-IVAC) [21] and frequency resolved optical gating (FROG) [22] techniques. AC contains reconstruction of attosecond beating by interference of two-photon transitions (RABBIT) [8], attosecond streak camera [23], FROG for complete reconstruction of attosecond field (FROG-CRAB) [24] and the phase retrieval by omega oscillation filtering (PROOF) [25]. Both CC and AC techniques have different applicability, either on the XUV peak intensity, photon energy, or spectral bandwidth [20]. Moreover, all the above techniques are based on photoelectron spectroscopy and require repetitive measurements as well as significant averaging to reach adequate signal-to-noise ratio. Although the SHHG signal has been demonstrated as a robust attosecond light source [10,26,27], the temporal characterization has been only implemented in the XUV range of 12-20 eV via 2-IVAC technique [9], for experimental parameters where the coherent wake emission (CWE) scheme is dominant. Due to the experimental difficulties and the presence of even-order harmonics in the XUV spectrum, the previously mentioned techniques have quite limited applicability to the SHHG signal. Therefore, it is worth considering a direct diagnostic approach to reveal the full spatial-temporal distribution of the attosecond pulses from SHHG.

In current letter, we propose a direct, rapid, and robust with respect to noise and full-optical method for the complete characterization of SHHG attosecond field. The proposal includes two full-optical set-ups: spectral phase interferometry for direct electric field reconstruction of SHHG sources (SHHG-SPIDER) and the SHHG-SPIDER with spatially encoded arrangement (SHHG-SEA-SPIDER). These two optical methods, with the advantages of insensitivity to spectral response and non-iterative reconstruction algorithm, provide a promising solution to the complete spatial-temporal characterization of the SHHG source even in the presence of shot-to-shot fluctuations caused by the stringent requirements on focus properties and target technologies. The SPIDER technique [28] has been successfully used in the characterization of gas-based HHG sources [29] and x-ray free electron lasers [30,31]. The SEA-SPIDER technique [32] has also been proposed for temporal characterization of XUV pulses from gas-HHG [33], however, until now, no experiment has been implemented. The current work numerically analyzes the practicability of the SHHG-SPIDER and SHHG-SEA-SPIDER techniques where attosecond pulses were simulated via EPOCH particle-in-cell (PIC) code [34,35]. In association with a novel far-field propagation arithmetic, the current work gives effective technical support to the implementation of the two Fourier transformation-based approaches for complete attosecond field characterization of SHHG signals.

2. Methods

Due to the lack of proven optical elements for XUV manipulation, modifications are required for the implementation of the Takeda inversion algorithm [36] on the characterization of attosecond pulses. Two identical but spectrally sheared attosecond replicas with a proper time delay are required in this interferometry-based method. In the current work, a controllable spectral shift $\mathrm{\Omega }$ and time delay $\mathrm{\tau }$ were introduced to the driving pulses. We choose driving pulses with central wavelength ${\lambda _L} \approx 400\; nm$, using a previously used spectral shear creating technique via frequency doubling [30]. The spectral shear of the ${n^{th}}$ harmonics between the attosecond replicas was then inherited as $n\mathrm{\Omega }$. The spectral phase and intensity profile can be extracted from the interferogram between the two attosecond replicas using the Takeda inversion algorithm. The complete temporal profile is finally reconstructed via Fourier transformation.

Numerical simulations on SHHG were carried out with the EPOCH PIC code. The PIC simulation provides the opportunity to verify spectral phase reconstructions via the SHHG-SPIDER and SHHG-SEA-SPIDER methods. Figure 1(a) shows the schematic layout of the PIC simulation. A Gaussian beam was focused on a glass target from $y ={-} 5\; \mu m$ with incident angle at ${\theta _i} = {45^ \circ }$. Inset image is showing the specular reflected attosecond pulse train (APT). Analysis of the reflected pulses in the near-field is considered at $x = 5\; \mu m.$ Few-cycle pulses ($2\; fs\; at\; FWHM$ of intensity) with ${a_0} = 5$ were focused on a target with density scale length $0.12{\lambda _L}$ and a maximum electron density of 100 times the critical density ${n_c}$. . The electron density distribution at the vacuum-plasma interface with attosecond emission (purple lines) is illustrated in Fig. 1(b). Only harmonic orders higher than 10 were included in the attosecond pulses. The lineout of the spectrum for the specular reflected field at$\; y = 0\; \mu m$ is plotted in Fig. 1(c). As shown in Fig. 1(d), an attosecond pulse train dominated by a single peak was observed in the direct Fourier transformation (DFT) of the filtered 12th to 16th harmonics within the gray shaded region in Fig. 1(c).

 figure: Fig. 1.

Fig. 1. SHHG simulation using the EPOCH code. (a) Schematic of the simulation model. (b) Electron density distribution at vacuum-plasma interface with attosecond emission (purple lines). (c) Lineout of spectrum for specular reflected field at$\; y = 0\; \mu m$. (d) Temporal envelope of the filtered harmonics (the 12th – 16th orders) via DFT.

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To aid the experimental design based on the two-dimensional (2D) EPOCH simulation, a near-to-far-field transformation code was developed. This 2D numerical code is based on the Kirchhoff integration theorem. The far-field spectrum in polar coordinate is given by

$$f({\omega ,P} )={-} \frac{i}{4}\mathop {\oint }\limits_C [k\textrm{cos}(\theta )f({\omega ,{\mathbf r}} )H_1^1({k\textrm{s}} )+ H_0^1({k\textrm{s}} )\frac{{\partial f({\omega ,{\mathbf r}} )}}{{\partial x}}]d{\mathbf r},\; $$
where $P({s,\; \theta } )$ is the posion of far-field spectrometer, $\theta $ is the angle between line from near-field point to P and the normal vector of output planes, s is the distance from the source point, $H_\alpha ^{(n )}(x )$ is the Hankel function. The integral is performed on the only non-zero output plane placed at $x = 5\; \mu m$.The finite-difference based far-field algorithm is discussed in the Supplement 1.

3. Results and discussions

3.1 SHHG-SPIDER

Any incarnation of the SPIDER method is an embodiment of the spectral-shearing interferometry, in which two temporally identical pulsed replicas with different central frequency are spectrally resolved. For the SHHG-SPIDER apparatus, the spectral shear in the XUV regime is inherited from the driving pulses. Reconstruction of a near-isolated attosecond pulse (AP) by an inversion algorithm is described in the flowchart shown in Fig. 2. The high harmonics driven by $400\; \textrm{nm}$ (solid line, ${\; }4.7\; \textrm{rad/fs}$) and $398{\; }\textrm{nm}$ (dashed line) fundamental pulses are illustrated in Fig. 2(a) with a Butterworth spectral filter. Time delay $\tau $ between the two replicas is $25\; fs$. The interferogram is then given by

$$\begin{array}{*{20}{c}} {S(\omega )= {{\left|{\tilde{E}(\omega )+ \tilde{E}\left( {\mathrm{\omega } + \frac{\mathrm{\Omega }}{{{\mathrm{\omega }_\textrm{F}}}}\mathrm{\omega }} \right){\textrm{e}^{\mathrm{i\omega \tau }}}} \right|}^2}}\\ { \approx 2I\left( {\omega + \frac{{\mathrm{\Omega \omega }}}{{2{\mathrm{\omega }_\textrm{F}}}}} \right) + 2E(\omega )E\left( {\omega + \frac{{\mathrm{\Omega }\omega }}{{{\omega_F}}}} \right)\cos ({{\phi_\mathrm{\Delta }}} ),} \end{array}$$
where $\tilde{E}(\omega )= E(\omega ){e^{i\phi (\omega )}}$, $\phi (\omega )$ is the spectral phase in the frequency domain, and ${\omega _F}$ is the frequency of driving laser.

 figure: Fig. 2.

Fig. 2. SHHG-SPIDER inversion routine. (a) Spectral shifted identical replicas, (b) interferogram, (c) Fourier pattern in pseudo-time domain, (d) reconstructed ($Recon.$) phase gradient$\; {d_\omega }\phi $, (e) reconstructed spectral intensity and phase, (f) final retrieved temporal profile.

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The phase difference given by$\; {\phi _\mathrm{\Delta }} = \omega \left( {\tau + \frac{\mathrm{\Omega }}{{{\omega_F}}}{d_\omega }\phi } \right)$ is encoded in the fringe pattern shown in Fig. 2(b). Following the Takeda algorithm, the interferogram was Fourier transformed into the pseudo-time domain. Three distinct terms are visible in Fig. 2(c): one direct current (DC) and two alternating current (AC+ and AC-) terms. The DC term only carries the spectral amplitude information, while the AC terms encrypt both amplitude and phase information of the attosecond pulses. A super-Gaussian filter was used to select the AC+ term $C({t - \tau } )= \mathrm{\mathfrak{F}}\left\{ {2E(\omega )E\left( {\omega + \frac{{\mathrm{\Omega }\omega }}{{{\omega_F}}}} \right){e^{i{\phi_\mathrm{\Delta }}}}} \right\}$ for extracting the spectral phase difference … In order to remove the linear phase $\omega \tau $, the time delay normally needs to be calibrated in a separated step. The phase gradient ${d_\omega }\phi $ (solid line) was then obtained in this two-step scheme, see Fig. 2(d), which is in good agreement with the DFT result (dashed line). The spectral phase (dashed line) was subsequently retrieved with a numerical integration algorithm, as shown in Fig. 2(e). The spectral amplitude profile was obtained from DC term as plotted in solid line in Fig. 2(e). Eventually, the temporal profile of attosecond pulses was reconstructed using an inverse Fourier transform. The envelope of the reconstructed attosecond pulses (solid line) is shown in Fig. 2(f), which is in excellent agreement with the DFT profile (dashed line). A temporal width of the main attosecond pulse (AP-1) of $404\; as$ (FWHM) was retrieved. To evaluate the reconstruction error in the inversion routine, the envelope of two individual attosecond pulses (AP-1 and AP-2) have been compared with the DFT results. With the definition of $({{I_{rec}} - {I_{DFT}}} )/{I_{DFT}}$, the reconstruction error for AP-1 and AP-2 are 0.3% and 8.3%, respectively.

The scheme discussed above follows the standard SPIDER implementation. Furthermore, we describe a new one-step scheme for the broadband XUV signal reconstruction without additional calibration in SPIDER routine, as shown in Fig. S3 in Supplement 1. Principally, the SPIDER phase ${\phi _\mathrm{\Delta }}$ consists of the approximation of $\frac{{\omega \mathrm{\Omega }}}{{{\omega _F}}}{d_\omega }\phi $ and the linear phase $\omega \tau $ induced by the time delay. In the SPIDER algorithm, only the former part is related to the spectral phase reconstruction. In the case of ultrafast XUV sources with low dispersion effect, such as SHHG ROM signals, the linear phase is dominant in the SPIDER phase, while $\frac{{\omega \mathrm{\Omega }}}{{{\omega _F}}}{d_\omega }\phi $ can be considered as a perturbation term. One can numerically remove the linear phase without a dedicated calibration process to extract the perturbation part from the SPIDER phase. By associating the final temporal profile with the correlation of spectral phase, the suppression of temporal leakage referring to the intensity of retrieved field envelope can be regarded as a decrease of the phase slope on the spectral phase gradient. Thus, the linear term can be eliminated properly when the maximum temporal intensity is reached by scanning the linear phase in the one-step scheme. Finally, the reconstructed temporal profile is in excellent agreement with DFT result (see Fig. S3). The one-step scheme is more robust than the complete two-step scheme [28] to characterize the temporal structure of attosecond pulses from SHHG. Further details are discussed in Supplement 1. The one-step SHHG-SPIDER method allows single-shot characterization for XUV sources with large shot-to-shot fluctuations. Reconstruction error of the one-step (black lines) and two-step (blue lines) schemes for different central HH orders are illustrated in Fig. 3. Due to the intensity advantage, reconstruction error of AP-1 (within 1%) is lower than that of AP-2 by one order of magnitude.

 figure: Fig. 3.

Fig. 3. Reconstruction errors for APT with different central HH orders. Solid lines and dashed lines are representing the errors of AP-1 and AP-2, respectively. Black dot and blue triangle correspond to one-step and two-step scheme.

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Also, this one-step scheme is applicable for recovering APTs with more complex temporal structures. Cases using SHHG pulses with driving pulse durations of $4\; fs$ and $8\; fs$ at FWHM were studied. The interferogram of the two replicas driven by central wavelengths at ${\lambda _1} = 400\; nm$ and ${\lambda _2} = 398\; nm$ with time delay of $\tau = 30\; fs$ are displayed in Fig. 4(a) and 4(c). The reconstructed multi-cycle APTs using the one-step scheme are illustrated in Fig. 4(b) and 4(d), respectively. The reconstruction error of the main attosecond pulse is within $5\%$. It is noted that the special temporal structure inherited from the laser-plasma interaction is well retrieved. In conclusion, the one-step SHHG-SPIDER routine shows high robustness for different attosecond pulse structures.

 figure: Fig. 4.

Fig. 4. The interferogram (left) and reconstructed APTs (right) in the inversion routine using driving pulse with durations of 4 fs (upper) and 8 fs (lower). HH signals from ${12^{th}}$ to ${16^{th}}$ orders were selected.

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3.2 Experimental implementation of the SHHG-SPIDER

The SHHG-SPIDER approach originates from the spectral interferometry technique. The major difference is that the two interference replicas are encoded with a dedicated frequency shear in the SHHG-SPIDER. Although the spectral shearing technique at XUV regime is challenging, frequency shear generating approaches for the driving pulses at optical wavelengths are more flexible, such as by means of Dazzler [29], double-frequency procedure [30], or optical parametric amplification process. The resulted spectral shift to the XUV replicas is hence magnified by n times for the ${n^{th}}$ harmonic order. To perform spectrally resolved spectral shearing interferometry, the ideal scenario would be to have the two replicas co-propagating with a proper delay between them. There are two configurations to fulfil this requirement: 1) co-axial propagation of the driving pulses; 2) beam-combining of the XUV replicas. As shown in Fig. 5(a) and 5(b), the former configuration requires a ‘sustained target’ with the same shooting position for both pulses, while the later scheme allows two spatially separated shooting spots by means of an XUV beam splitter.

 figure: Fig. 5.

Fig. 5. Experimental configurations for SHHG-SPIDER (a, b) and SHHG-SEA-SPIDER (c, d) apparatus.

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To verify the feasibility of the ‘sustained target’ scheme, a simulation with two spectrally sheared driving pulses shooting at the same target point has been performed, as shown in Fig. 6. Two few-cycle driving pulses ($4\; fs$ at FWHM) with equivalent focusing intensity and CEP are arriving at the target with a time delay of $15\; fs$. Central wavelengths of the two driving pulses (Drive-1 and Drive-2) are at $400\; nm$ and $398\; nm$, respectively. Due to the nature of Fourier transformation and vector assumption, requirements for remaining the similarity of spectra between the reflected APTs are twofold. First, the intensity of temporal peaks should be consistent (the temporal-spectral relationship of SHHG pulse has been discussed thoroughly in Supplement 1). Second, the time intervals between the individual attosecond pulses of the APTs are controllable or observable. According to the low-dispersion nature of signals via the ROM SHHG, the two APTs are considered as identical when meeting above criteria. However, in Fig. 6(b), obvious modulation on the temporal structure was observed in the specular reflection from the target. Moreover, a spectral red shift of the APT-2 with respect to APT-1 was detected, as it is shown in Fig. 6(a).

 figure: Fig. 6.

Fig. 6. (a) comb-like spectra of the reflected pulses (RP), the zoom in figure represents for the filtered spectral intensity from the 12th to the 16th HH orders, (b) the envelope of DFT APTs.

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To give an illustration for these temporal and spectral modulations, a series of simulations have been performed by varying the time delay $\mathrm{\Delta }t$ between the two replicas. Figure 7(a) exhibits the variation of the intensity ratio between the individual APs driven by the second pulse in the $2\; fs$ simulations. Referring to the intensity ratio in the APT-1 (blue dashed line), the spectral phase of APT-2 is supposed to evolve substantially over$\; \mathrm{\Delta }t$. Additionally, Fig. 7(b) shows that the time interval between two individual pulses in the APTs also changes with$\; \mathrm{\Delta }t$, which represents a $\mathrm{\Delta }$t-dependent spectral shift. These temporal evolutions result from the electron motion driven by the charge separation field, which further depend on the driving laser intensity. Therefore, the stochastic frequency shift and the non-identical features of APTs bring up difficulty to implement SHHG-SPIDER experiment with the sustained target scheme. Alternatively, using the scheme described in Fig. 5(b), two independent foci are generated in the same focusing system, and the two APTs are combined into one beam via an XUV beam splitter. The influence of the first driving-pulse on the second one can be avoided in this configuration. The XUV beam splitter normally requires a thin film to have reasonable transmissivity and a multilayer coating to reach tolerable reflectivity. The silicon nitride membranes with Mo/Si multilayer coating have been successfully demonstrated with efficiency of ∼15% for both reflection and transmission at $13.9\; nm$ [37]. An XUV Mach-Zehnder interferometry setup can be realized by use of the beam splitter. Subsequently, the interferogram can be captured by a high-resolution spectrometer, typically with $\frac{{\Delta \lambda }}{\lambda } < 6.67 \times {10^{ - 18}}/\tau \; @\; 30\; nm$ (see Supplement 1).

 figure: Fig. 7.

Fig. 7. (a) Variation of the intensity ratio between peaks of APT-2 with the time delay$\; \mathrm{\Delta }t$. (b) The time interval between the peaks of APT-2 as a function of$\; \mathrm{\Delta }t$. Blue dashed lines indicate the intensity ratio and time interval of the peaks in APT-1, respectively.

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3.3 SHHG-SEA-SPIDER

Although the co-axial propagation requirement of the SHHG-SPIDER configuration can be fulfilled by employing a dedicated beam splitter, the dispersion introduced by the transmitted arm would affect the similarity of the two APTs. Another difficulty is that the spectrometer needs high resolution for a broadband APT at lower wavelength, which is hard to achieve. These concerns can be addressed by an alternative interferometry geometry, which resolves the spatial interferogram. The spatially encoded SPIDER scheme is shown schematically in Fig. 5(c). By focusing the driving pulses onto the solid surface independently, two spectrally sheared attosecond replicas are generated in spatially separated plasmas simultaneously. Considering the source-to-detector distance, the far-field XUV spots can be fully overlapped when a proper cross angle ${\theta _c}$ is selected. Eventually, the interferogram is spectrally filtered and imaged onto a spectrometer. Assuming that the APTs propagate symmetrically about the normal vector of the imaging interface, the detected interferogram from this optimum alignment has a form expressed by

$$\begin{array}{c} {S({\omega ,{\; }\textrm{y}} )= {{\left|{\tilde{E}({\omega ,y} )+ \tilde{E}\left( {\mathrm{\omega } + \frac{\mathrm{\Omega }}{{{\mathrm{\omega }_\textrm{F}}}}\mathrm{\omega },\; \textrm{y}} \right){\textrm{e}^{\mathrm{i\Delta ky}}}} \right|}^2}}\\ { \approx 2I\left( {\omega + \frac{{\mathrm{\Omega \omega }}}{{2{\mathrm{\omega }_\textrm{F}}}}} \right) + 2E(\omega )E\left( {\omega + \frac{{\mathrm{\Omega }\omega }}{{{\omega_F}}}} \right)\cos ({{\phi_\mathrm{\Delta }}} ),} \end{array}$$
where $\Delta k(\omega )= ({{{\mathbf k}_1} - {{\mathbf k}_2}} )\cdot \widehat {{j_y}} = 2\frac{\omega }{c}\textrm{sin}({{\theta_c}/2} )$ is the difference between the spatial projections of two wave vectors. Thus, the spectral phase difference ${\phi _\mathrm{\Delta }} \approx \omega \frac{\mathrm{\Omega }}{{{\omega _F}}}{d_\omega }\phi + \mathrm{\Delta }ky$ is encoded in the spatial fringes on the interferogram. Figure 8 shows the inversion algorithm applied to the far-field interferogram obtained with cross angle of$\; {\theta _c} = 2.4\; mrad$ and source-to-detector distance of $d = 500\; mm$. In the 2D PIC simulation, the near-field information of the specular reflected APTs is acquired individually with identical driving pulses (${a_0} = 5$, focal spot of $1.4\; \mu m$ and central wavelengths of ${\lambda _L} = 400\; nm$ and $398\; nm$). Utilizing the near-to-far-field transformation code based on the Kirchhoff integration theorem, the wave front curvature induced by the propagating phase is displayed in Fig. 8(a), and the corresponding angular spectrum of ${12^{th}}\sim {16^{th}}$ harmonics is shown in Fig. 8(b). The spatial interferogram at $d = 500\; mm$ (Fig. 8(c)) is hence generated by two spectrally sheared APTs with fundamental wavelengths at $400\; nm$ and $398\; nm$, where the lineout of the spatial fringes marked by the white dashed line is shown in the right inset. A 2D Fourier transformation was performed, isolated sidebands can be observed in the Fourier pseudo-time domain (Fig. 8(d)). Same as the properties in the SHHG-SPIDER method, the AC terms carries both spectral phase gradient and amplitude information, while the DC term only contains information about the spectral amplitude. Two filters have been used to select one of the AC terms for retrieving the spectral phase via integration algorithm and to select the DC term for recovering the spectral amplitude. Consequently, the complete attosecond field reconstruction was achieved by an inversion Fourier transformation (see Fig. 8(e)), which is consistent with the DFT results (Fig. 8(f)). Three typical temporal lineouts marked in the above two images are plotted in Fig. 8(g) to 8(i).

 figure: Fig. 8.

Fig. 8. SHHG-SEA-SPIDER inversion routine. (a) Temporal and (b) spectral profiles of one replica in the far-field. (c) Spatial interferogram of the two replicas. (d) 2D Fourier transformation, (e) reconstructed and (f) DFT temporal profiles. (g) - (i) lineouts of the temporal profiles.

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Furthermore, the temporal characterization of attosecond pulses with more complex spatial-temporal structures has been investigated with the SHHG-SEA-SPIDER method. Figure 9 illustrates the temporal characterization of XUV pulses driven by a multi-cycle laser pulse. The upper and lower panel shows the reconstruction results of ${12^{th}}$ to ${16^{th}}\; $harmonics driven by lasers of $\tau = 4\; \textrm{fs}$ and $8\; \textrm{fs}$ (FWHM) pulse duration, respectively. The far-field profiles of selected harmonics are displayed in Fig. 9(a) and 9(d) after removing the propagating phase. Spatial modulations of the temporal profiles have been observed in both cases. The reconstructed profiles are illustrated in Fig. 9(b) and 9(e), which shows consistent spatial distribution with respect to the DFT results. Lineouts of the reconstructed and DFT temporal profiles at $y = 0\; mm$ are displayed in Fig. 9(c) and 9(f). This indicates that the SHHG-SEA-SPIDER method is well suited to characterize the signal with spatial temporal coupling, which is significant for the single-shot measurement of laser driving plasma properties. Note that the mild distortion of the intensity envelope is due to the accumulated spectral phase noise at far-field. The phase noise is attributed to the integration algorithm used for full field signal from EPOCH simulation. A numerical filter was applied to the field extracted from EPOCH PIC simulations for noise suppressing, which is critical for the successful reconstruction with the simulated data with low signal-to-noise ratio in the XUV region (see Supplement 1).

 figure: Fig. 9.

Fig. 9. Spatial-temporal characterization of complex field profile of APT driven by $\tau = 4\; fs\; $(upper panel) and$\; 8\; fs$ (lower panel) pulse. (a) and (d) are the DFT temporal profiles, (b) and (e) are the reconstructed attosecond field, (c) and (f) are the lineouts at$\; y = 0\; mm$.

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3.4 Experimental implementation of the SHHG-SEA-SPIDER

Two basic configurations of the SHHG-SEA-SPIDER have been schematically demonstrated in Fig. 5(c) and 5(d) indicating the crossed and parallel propagation scheme respectively. Each utilizes a different focusing setup. In the crossed propagation scheme, a proper cross angle ${\theta _c}$ and foci separation s between the two driving pulses would guarantee a full overlap at spectrometer detection plane. The interferogram is composed of spatial fringes with spacing${\; }{\mathrm{\Delta }_{fringe}} = \frac{{2\pi }}{{\mathrm{\Delta }k}} \approx \frac{{{\lambda _c}}}{{2\; \textrm{sin}({{\theta_c}/2} )}}$, where ${\lambda _c}$ is the central wavelength of the attosecond pulse, and $\mathrm{\Delta }k$ is the wave number difference between the two spectrally sheared replicas. For a spectrometer with spatial resolution of ${\mathrm{\Delta }_s}\; (m )$, at least five sample points per fringe are required for extracting phase from the interferogram, giving rise to an upper limit of the cross angle ${\theta _c} < 2{\sin ^{ - 1}}\left( {\frac{{{\lambda_c}}}{{2 \times 5{\mathrm{\Delta }_s}}}} \right) \approx \frac{{{\lambda _c}}}{{5{\mathrm{\Delta }_s}}}\; rad$. Meanwhile, the source-to-detector distance d is determined by the imaging configuration of the detection system, for instance, the flat focal field spectrometer. In order to achieve high resolution spectrograph at a flat-focal field, mechanically ruled aberration-corrected concave gratings are preferred [38]. In the soft x-ray regime, two typical source-to-detector distances $d = 472.3\; mm$ and $819.0\; mm\; $are used for attosecond pulse wavelengths below and above$\; 20\; nm$, respectively. According to the source-to-detector distance, lower limit of the cross angle is due to the two-focus separation on the target. To avoid the influence between the two replicas, a minimum foci separation s of five times of the focal spot size (radius$\; r$) is recommended, leading to$\; {\theta _c} > 2ta{n^{ - 1}}({3r/2d\; } )$. Another important factor to the cross angle lower limit is the AC-DC separation in the Fourier domain (see Fig. 8(d)). In principle, the minimum gap between the AC and DC terms is the double width of the DC term at k-domain, i.e., $2\pi \sin \left( {\frac{{{\theta_c}}}{2}} \right)/{\lambda _c} = \mathrm{\Delta }k > 2\pi /{D_f}$ (${D_f}\; $is the beam diameter at detection plane). For example, to resolve spatial interferograms with central wavelength at $10\; nm$ and$\; 30\; nm$, the maximum cross angles are $4\; mrad$ and $12\; mrad,$ respectively, while the corresponding foci separations are ${\sim} 1.9\; mm$ and ${\sim} 9.8\; mm$. These focusing requirements of the driving laser can be satisfied by employing either a customized off-axis-parabola (OAP) system with two independent parabolas, or a single OAP with two deviated incident angles. The latter technique will need to consider aberrations caused by imperfect alignment with the OAP’s principal axis that this entails [39]. Figure 8 and Fig. 9 show the temporal structure reconstruction via the SHHG-SEA-SPIDER approach at different driving pulse durations under the cross-angle propagation scheme. It is evident that the spatially distributed profile of the temporal structure for the attosecond pulses can be accurately retrieved by the cross-angle scheme of the SHHG-SEA-SPIDER method.

Another scheme is to allow parallel propagation of the two attosecond replicas, as shown in Fig. 5(d). With benefit of the large divergence of SHHG signals [16], the two parallel-propagated attosecond pulses would mostly overlap at the far-field detection plane. As the two driving pulses are focused on the target in independent regions, a spatial shear between the centers of the APs beams is therefore inherited. As is illustrated in Fig. 8(e), the simulated SHHG APT has an angular divergence of ${\sim} 70\; mrad$ at FWHM and the corresponding beam size at the detection plane is ${r_f} \approx 57\; mm$ at FWHM for wavelengths above $20\; nm$. With the assumption of a uniform angular phase distribution, the APT temporal profiles can be recovered successfully in the overlapped region in the simulation performed by introducing a spatial shear to the replicas (see Fig. 10). The spatial shear as defined by the distance between the far-field spot center $s = \; 10\; mm$ (Fig. 10(a) and 10(d)) and $20\; mm$ (Fig. 10(b) and 10(e)) for both cases with driving pulse durations of $2\; fs$ and $4\; fs$ were tested with the SHHG-SEA-SPIDER method. In Fig. 10(c) and 10(f), lineouts of the temporal profile on the white dashed lines marked in the 2D reconstructed profiles are compared with the DFT result. The reconstructed temporal profiles in the overlapped region are in good agreement with the DFT results. In general, a spatial shear of $s < {r_f}/2$ is recommended for a SHHG APT with uniform angular phase distribution driven by a Gaussian pulse. It is worth noting that the spatial intensity variation has limited effect on the temporal shape with a consistent phase profile. Due to the stringent requirement of driving pulse polarization for effective SHHG process, the often-used two foci generation technique by means of calcite crystals [40] is not suitable for the current parallel propagation scheme. To implement this configuration, a focusing system with two compact parallel-installed OAPs is recommended.

 figure: Fig. 10.

Fig. 10. Reconstruction results with spatial shear. Upper and lower panels are the reconstruction of APT driven by $\tau = 2\; fs\; ({FWHM} )$ and $\tau = 4\; fs({FWHM} )$ pulses, respectively. The first and second columns are showing the reconstructed APT temporal distribution with $s = 10\; mm$ and $20\; mm$ separately. (c) and (f) are the lineouts of reconstructed temporal profiles marked in dashed lines with respect to the DFT profiles.

Download Full Size | PPT Slide | PDF

4. Conclusion

In conclusion, we have proposed two interferometry methods for complete temporal characterization of attosecond pulses from relativistic plasmas. By means of spectral interferometry with spectrally sheared replicas, namely SHHG-SPIDER, temporal profiles as a function of spatial positions can be accurately retrieved for APs from near-isolated to multi-pulse shapes. A one-step SHHG-SPIDER approach without delay calibration has been introduced with the benefit of multi-shear attosecond replicas, giving rise to a single-shot temporal characterization method for HH sources. To relax the stringent requirement of co-axial propagation of the SHHG-SPIDER method, the SHHG-SEA-SPIDER is considered as an alternative approach to achieve complete characterization of attosecond pulses with high precision. The spatial interferometry-based technique has several practical advantages, making it a promising method for spatial-temporal characterization of attosecond field from relativistic plasmas. Spatially resolved interferograms in SHHG-SEA-SPIDER relax the spectrometer resolution by one order of magnitude with respect to the SHHG-SPIDER method. Various experimental parameters, such as spectral shear, time delay, intensity ratio, foci separation of the two attosecond replicas, have been discussed, giving effective suggestions to the experimental implementations. The current proposal is suitable for attosecond pulses from surface high harmonic generation driven by few to hundreds of TW laser pulses. These methods, in association with a high numerical aperture wave front detection technique [41], provide a route to unveiling the complete spatial, temporal and spectral distribution of the attosecond field generated from relativistic plasmas.

Funding

National Key Research and Development Program of China (Grant No. 2016YFA0401100); National Natural Science Foundation of China (Grant No. 11875092); Natural Science Foundation of Top Talent of SZTU (Grant No. 2019010801001, No. 2020107).

Acknowledgments

We gratefully acknowledge Dr. Philippe Zeitoun and Dr. Sergey G. Rykovanov for fruitful discussion.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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References

  • View by:

  1. E. Goulielmakis, Z. H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, and F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010).
    [Crossref]
  2. U. S. Sainadh, H. Xu, X. Wang, A. Atia-Tul-Noor, W. C. Wallace, N. Douguet, A. Bray, I. Ivanov, K. Bartschat, A. Kheifets, R. T. Sang, and I. V. Litvinyuk, “Attosecond angular streaking and tunnelling time in atomic hydrogen,” Nature 568(7750), 75–77 (2019).
    [Crossref]
  3. F. Calegari, D. Ayuso, A. Trabattoni, L. Belshaw, S. De Camillis, S. Anumula, F. Frassetto, L. Poletto, A. Palacios, P. Decleva, J. B. Greenwood, F. Martín, and M. Nisoli, “Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses,” Science 346(6207), 336–339 (2014).
    [Crossref]
  4. S. Beaulieu, A. Comby, A. Clergerie, J. Caillat, D. Descamps, N. Dudovich, B. Fabre, R. Géneaux, F. Légaré, S. Petit, B. Pons, G. Porat, T. Ruchon, R. Täieb, V. Blanchet, and Y. Mairesse, “Attosecond-resolved photoionization of chiral molecules,” Opt. InfoBase Conf. Pap.Part F86-HILAS 2018(December), 1288–1294 (2018).
  5. I. Jordan, M. Huppert, D. Rattenbacher, M. Peper, D. Jelovina, C. Perry, A. Von Conta, A. Schild, H. J. Wörner, and M. Carlo, “Attosecond spectroscopy of liquid water,” Science 369(6506), 974–979 (2020).
    [Crossref]
  6. A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449(7165), 1029–1032 (2007).
    [Crossref]
  7. F. Siegrist, J. A. Gessner, M. Ossiander, C. Denker, Y. P. Chang, M. C. Schröder, A. Guggenmos, Y. Cui, J. Walowski, U. Martens, J. K. Dewhurst, U. Kleineberg, M. Münzenberg, S. Sharma, and M. Schultze, “Light-wave dynamic control of magnetism,” Nature 571(7764), 240–244 (2019).
    [Crossref]
  8. P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
    [Crossref]
  9. Y. Nomura, R. Hörlein, P. Tzallas, B. Dromey, S. Rykovanov, Z. Major, J. Osterhoff, S. Karsch, L. Veisz, M. Zepf, D. Charalambidis, F. Krausz, and G. D. Tsakiris, “Attosecond phase locking of harmonics emitted from laser-produced plasmas,” Nat. Phys. 5(2), 124–128 (2009).
    [Crossref]
  10. J. A. Wheeler, A. Borot, S. Monchocé, H. Vincenti, A. Ricci, A. Malvache, R. Lopez-Martens, and F. Quéré, “Attosecond lighthouses from plasma mirrors,” Nat. Photonics 6(12), 829–833 (2012).
    [Crossref]
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    [Crossref]
  12. A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H. C. Bandulet, D. Comtois, F. Légaré, M. Giguère, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Wavelength scaling of high harmonic generation efficiency,” Phys. Rev. Lett. 103(7), 073902 (2009).
    [Crossref]
  13. T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the kev x-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
    [Crossref]
  14. A. Nayak, I. Orfanos, I. Makos, M. Dumergue, S. Kühn, E. Skantzakis, B. Bodi, K. Varju, C. Kalpouzos, H. I. B. Banks, A. Emmanouilidou, D. Charalambidis, and P. Tzallas, “Multiple ionization of argon via multi-XUV-photon absorption induced by 20-GW high-order harmonic laser pulses,” Phys. Rev. A 98(2), 023426 (2018).
    [Crossref]
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    [Crossref]
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    [Crossref]
  17. S. Gordienko, A. Pukhov, O. Shorokhov, and T. Baeva, “Coherent focusing of high harmonics: A new way towards the extreme intensities,” Phys. Rev. Lett. 94(10), 103903 (2005).
    [Crossref]
  18. A. Leblanc, S. Monchocé, H. Vincenti, S. Kahaly, J. L. Vay, and F. Quéré, “Spatial Properties of High-Order Harmonic Beams from Plasma Mirrors: A Ptychographic Study,” Phys. Rev. Lett. 119(15), 155001 (2017).
    [Crossref]
  19. L. Chopineau, A. Denoeud, A. Leblanc, E. Porat, P. Martin, H. Vincenti, and F. Quéré, “Spatio-temporal characterization of attosecond pulses from plasma mirrors,” Nat. Phys. 17(8), 968–973 (2021).
    [Crossref]
  20. I. Orfanos, I. Makos, I. Liontos, E. Skantzakis, B. Förg, D. Charalambidis, and P. Tzallas, “Attosecond pulse metrology,” APL Photonics 4(8), 080901 (2019).
    [Crossref]
  21. P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosecond light bunching,” Nature 426(6964), 267–271 (2003).
    [Crossref]
  22. T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, “Nonlinear optics in the extreme ultraviolet,” Nature 432(7017), 605–608 (2004).
    [Crossref]
  23. E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct Measurement of Light Waves,” Science 305(5688), 1267–1269 (2004).
    [Crossref]
  24. Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A 71(1), 011401 (2005).
    [Crossref]
  25. M. Chini, S. Gilbertson, S. D. Khan, and Z. Chang, “Characterizing ultrabroadband attosecond lasers,” Opt. Express 18(12), 13006 (2010).
    [Crossref]
  26. M. Yeung, S. Rykovanov, J. Bierbach, L. Li, E. Eckner, S. Kuschel, A. Woldegeorgis, C. Rödel, A. Sävert, G. G. Paulus, M. Coughlan, B. Dromey, and M. Zepf, “Experimental observation of attosecond control over relativistic electron bunches with two-colour fields,” Nat. Photonics 11(1), 32–35 (2017).
    [Crossref]
  27. O. Jahn, V. E. Leshchenko, P. Tzallas, A. Kessel, M. Krüger, A. Münzer, S. A. Trushin, G. D. Tsakiris, S. Kahaly, D. Kormin, L. Veisz, V. Pervak, F. Krausz, Z. Major, and S. Karsch, “Towards intense isolated attosecond pulses from relativistic surface high harmonics,” Optica 6(3), 280 (2019).
    [Crossref]
  28. C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric field reconstruction of ultrashort optical pulses,” Opt. Lett. 23(10), 792 (1998).
    [Crossref]
  29. Y. Mairesse, O. Gobert, P. Breger, H. Merdji, P. Meynadier, P. Monchicourt, M. Perdrix, P. Salières, and B. Carré, “High harmonic XUV spectral phase interferometry for direct electric-field reconstruction,” Phys. Rev. Lett. 94(17), 173903 (2005).
    [Crossref]
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2021 (1)

L. Chopineau, A. Denoeud, A. Leblanc, E. Porat, P. Martin, H. Vincenti, and F. Quéré, “Spatio-temporal characterization of attosecond pulses from plasma mirrors,” Nat. Phys. 17(8), 968–973 (2021).
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2020 (2)

2019 (5)

J. Lu, E. F. Cunningham, Y. S. You, D. A. Reis, and S. Ghimire, “Interferometry of dipole phase in high harmonics from solids,” Nat. Photonics 13(2), 96–100 (2019).
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U. S. Sainadh, H. Xu, X. Wang, A. Atia-Tul-Noor, W. C. Wallace, N. Douguet, A. Bray, I. Ivanov, K. Bartschat, A. Kheifets, R. T. Sang, and I. V. Litvinyuk, “Attosecond angular streaking and tunnelling time in atomic hydrogen,” Nature 568(7750), 75–77 (2019).
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F. Siegrist, J. A. Gessner, M. Ossiander, C. Denker, Y. P. Chang, M. C. Schröder, A. Guggenmos, Y. Cui, J. Walowski, U. Martens, J. K. Dewhurst, U. Kleineberg, M. Münzenberg, S. Sharma, and M. Schultze, “Light-wave dynamic control of magnetism,” Nature 571(7764), 240–244 (2019).
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I. Orfanos, I. Makos, I. Liontos, E. Skantzakis, B. Förg, D. Charalambidis, and P. Tzallas, “Attosecond pulse metrology,” APL Photonics 4(8), 080901 (2019).
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O. Jahn, V. E. Leshchenko, P. Tzallas, A. Kessel, M. Krüger, A. Münzer, S. A. Trushin, G. D. Tsakiris, S. Kahaly, D. Kormin, L. Veisz, V. Pervak, F. Krausz, Z. Major, and S. Karsch, “Towards intense isolated attosecond pulses from relativistic surface high harmonics,” Optica 6(3), 280 (2019).
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2018 (1)

A. Nayak, I. Orfanos, I. Makos, M. Dumergue, S. Kühn, E. Skantzakis, B. Bodi, K. Varju, C. Kalpouzos, H. I. B. Banks, A. Emmanouilidou, D. Charalambidis, and P. Tzallas, “Multiple ionization of argon via multi-XUV-photon absorption induced by 20-GW high-order harmonic laser pulses,” Phys. Rev. A 98(2), 023426 (2018).
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2017 (3)

A. Leblanc, S. Monchocé, H. Vincenti, S. Kahaly, J. L. Vay, and F. Quéré, “Spatial Properties of High-Order Harmonic Beams from Plasma Mirrors: A Ptychographic Study,” Phys. Rev. Lett. 119(15), 155001 (2017).
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M. Yeung, S. Rykovanov, J. Bierbach, L. Li, E. Eckner, S. Kuschel, A. Woldegeorgis, C. Rödel, A. Sävert, G. G. Paulus, M. Coughlan, B. Dromey, and M. Zepf, “Experimental observation of attosecond control over relativistic electron bunches with two-colour fields,” Nat. Photonics 11(1), 32–35 (2017).
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P. Finetti, H. Höppner, E. Allaria, C. Callegari, F. Capotondi, P. Cinquegrana, M. Coreno, R. Cucini, M. B. Danailov, A. Demidovich, G. De Ninno, M. Di Fraia, R. Feifel, E. Ferrari, L. Fröhlich, D. Gauthier, T. Golz, C. Grazioli, Y. Kai, G. Kurdi, N. Mahne, M. Manfredda, N. Medvedev, I. P. Nikolov, E. Pedersoli, G. Penco, O. Plekan, M. J. Prandolini, K. C. Prince, L. Raimondi, P. Rebernik, R. Riedel, E. Roussel, P. Sigalotti, R. Squibb, N. Stojanovic, S. Stranges, C. Svetina, T. Tanikawa, U. Teubner, V. Tkachenko, S. Toleikis, M. Zangrando, B. Ziaja, F. Tavella, and L. Giannessi, “Pulse duration of seeded free-electron lasers,” Phys. Rev. X 7(2), 021043 (2017).
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2015 (1)

2014 (1)

F. Calegari, D. Ayuso, A. Trabattoni, L. Belshaw, S. De Camillis, S. Anumula, F. Frassetto, L. Poletto, A. Palacios, P. Decleva, J. B. Greenwood, F. Martín, and M. Nisoli, “Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses,” Science 346(6207), 336–339 (2014).
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2012 (2)

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the kev x-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
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J. A. Wheeler, A. Borot, S. Monchocé, H. Vincenti, A. Ricci, A. Malvache, R. Lopez-Martens, and F. Quéré, “Attosecond lighthouses from plasma mirrors,” Nat. Photonics 6(12), 829–833 (2012).
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2011 (1)

G. Sansone, L. Poletto, and M. Nisoli, “High-energy attosecond light sources,” Nat. Photonics 5(11), 655–663 (2011).
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2010 (3)

E. Goulielmakis, Z. H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, and F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010).
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C. Thaury and F. Quéré, “High-order harmonic and attosecond pulse generation on plasma mirrors: Basic mechanisms,” J. Phys. B: At., Mol. Opt. Phys. 43(21), 213001 (2010).
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M. Chini, S. Gilbertson, S. D. Khan, and Z. Chang, “Characterizing ultrabroadband attosecond lasers,” Opt. Express 18(12), 13006 (2010).
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2009 (2)

Y. Nomura, R. Hörlein, P. Tzallas, B. Dromey, S. Rykovanov, Z. Major, J. Osterhoff, S. Karsch, L. Veisz, M. Zepf, D. Charalambidis, F. Krausz, and G. D. Tsakiris, “Attosecond phase locking of harmonics emitted from laser-produced plasmas,” Nat. Phys. 5(2), 124–128 (2009).
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A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H. C. Bandulet, D. Comtois, F. Légaré, M. Giguère, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Wavelength scaling of high harmonic generation efficiency,” Phys. Rev. Lett. 103(7), 073902 (2009).
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2008 (1)

S. G. Rykovanov, M. Geissler, J. Meyer-ter-Vehn, and G. D. Tsakiris, “Intense single attosecond pulses from surface harmonics using the polarization gating technique,” New J. Phys. 10(2), 025025 (2008).
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2007 (1)

A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449(7165), 1029–1032 (2007).
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2006 (3)

F. Quéré, C. Thaury, P. Monot, S. Dobosz, P. Martin, J. P. Geindre, and P. Audebert, “Coherent wake emission of high-order harmonics from overdense plasmas,” Phys. Rev. Lett. 96(12), 125004 (2006).
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A. S. Wyatt, I. A. Walmsley, G. Stibenz, and G. Steinmeyer, “Sub-10 fs pulse characterization using spatially encoded arrangement for spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 31(12), 1914 (2006).
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2005 (4)

E. Cormier, I. A. Walmsley, E. M. Kosik, A. S. Wyatt, L. Corner, and L. F. Dimauro, “Self-referencing, spectrally, or spatially encoded spectral interferometry for the complete characterization of attosecond electromagnetic pulses,” Phys. Rev. Lett. 94(3), 033905 (2005).
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Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A 71(1), 011401 (2005).
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S. Gordienko, A. Pukhov, O. Shorokhov, and T. Baeva, “Coherent focusing of high harmonics: A new way towards the extreme intensities,” Phys. Rev. Lett. 94(10), 103903 (2005).
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2004 (2)

T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, “Nonlinear optics in the extreme ultraviolet,” Nature 432(7017), 605–608 (2004).
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E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct Measurement of Light Waves,” Science 305(5688), 1267–1269 (2004).
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2003 (2)

P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosecond light bunching,” Nature 426(6964), 267–271 (2003).
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P. Arguijo and M. S. Scholl, “Exact ray-trace beam for an off-axis paraboloid surface,” Appl. Opt. 42(16), 3284 (2003).
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2002 (1)

2001 (1)

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
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1998 (1)

1983 (1)

1982 (1)

Agostini, P.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
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Ališauskas, S.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the kev x-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
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Alj, D.

Allaria, E.

P. Finetti, H. Höppner, E. Allaria, C. Callegari, F. Capotondi, P. Cinquegrana, M. Coreno, R. Cucini, M. B. Danailov, A. Demidovich, G. De Ninno, M. Di Fraia, R. Feifel, E. Ferrari, L. Fröhlich, D. Gauthier, T. Golz, C. Grazioli, Y. Kai, G. Kurdi, N. Mahne, M. Manfredda, N. Medvedev, I. P. Nikolov, E. Pedersoli, G. Penco, O. Plekan, M. J. Prandolini, K. C. Prince, L. Raimondi, P. Rebernik, R. Riedel, E. Roussel, P. Sigalotti, R. Squibb, N. Stojanovic, S. Stranges, C. Svetina, T. Tanikawa, U. Teubner, V. Tkachenko, S. Toleikis, M. Zangrando, B. Ziaja, F. Tavella, and L. Giannessi, “Pulse duration of seeded free-electron lasers,” Phys. Rev. X 7(2), 021043 (2017).
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Andriukaitis, G.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the kev x-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
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Anumula, S.

F. Calegari, D. Ayuso, A. Trabattoni, L. Belshaw, S. De Camillis, S. Anumula, F. Frassetto, L. Poletto, A. Palacios, P. Decleva, J. B. Greenwood, F. Martín, and M. Nisoli, “Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses,” Science 346(6207), 336–339 (2014).
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Arguijo, P.

Arpin, P.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the kev x-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
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Atia-Tul-Noor, A.

U. S. Sainadh, H. Xu, X. Wang, A. Atia-Tul-Noor, W. C. Wallace, N. Douguet, A. Bray, I. Ivanov, K. Bartschat, A. Kheifets, R. T. Sang, and I. V. Litvinyuk, “Attosecond angular streaking and tunnelling time in atomic hydrogen,” Nature 568(7750), 75–77 (2019).
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Audebert, P.

F. Quéré, C. Thaury, P. Monot, S. Dobosz, P. Martin, J. P. Geindre, and P. Audebert, “Coherent wake emission of high-order harmonics from overdense plasmas,” Phys. Rev. Lett. 96(12), 125004 (2006).
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Augé, F.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
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Ayuso, D.

F. Calegari, D. Ayuso, A. Trabattoni, L. Belshaw, S. De Camillis, S. Anumula, F. Frassetto, L. Poletto, A. Palacios, P. Decleva, J. B. Greenwood, F. Martín, and M. Nisoli, “Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses,” Science 346(6207), 336–339 (2014).
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Azzeer, A. M.

E. Goulielmakis, Z. H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, and F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010).
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Baeva, T.

S. Gordienko, A. Pukhov, O. Shorokhov, and T. Baeva, “Coherent focusing of high harmonics: A new way towards the extreme intensities,” Phys. Rev. Lett. 94(10), 103903 (2005).
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Balciunas, T.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the kev x-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
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Balcou, P.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
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Baltuska, A.

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct Measurement of Light Waves,” Science 305(5688), 1267–1269 (2004).
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Baltuška, A.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the kev x-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
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A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449(7165), 1029–1032 (2007).
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Bandulet, H. C.

A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H. C. Bandulet, D. Comtois, F. Légaré, M. Giguère, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Wavelength scaling of high harmonic generation efficiency,” Phys. Rev. Lett. 103(7), 073902 (2009).
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Banks, H. I. B.

A. Nayak, I. Orfanos, I. Makos, M. Dumergue, S. Kühn, E. Skantzakis, B. Bodi, K. Varju, C. Kalpouzos, H. I. B. Banks, A. Emmanouilidou, D. Charalambidis, and P. Tzallas, “Multiple ionization of argon via multi-XUV-photon absorption induced by 20-GW high-order harmonic laser pulses,” Phys. Rev. A 98(2), 023426 (2018).
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Bartschat, K.

U. S. Sainadh, H. Xu, X. Wang, A. Atia-Tul-Noor, W. C. Wallace, N. Douguet, A. Bray, I. Ivanov, K. Bartschat, A. Kheifets, R. T. Sang, and I. V. Litvinyuk, “Attosecond angular streaking and tunnelling time in atomic hydrogen,” Nature 568(7750), 75–77 (2019).
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Beaulieu, S.

S. Beaulieu, A. Comby, A. Clergerie, J. Caillat, D. Descamps, N. Dudovich, B. Fabre, R. Géneaux, F. Légaré, S. Petit, B. Pons, G. Porat, T. Ruchon, R. Täieb, V. Blanchet, and Y. Mairesse, “Attosecond-resolved photoionization of chiral molecules,” Opt. InfoBase Conf. Pap.Part F86-HILAS 2018(December), 1288–1294 (2018).

Becker, A.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the kev x-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
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Belshaw, L.

F. Calegari, D. Ayuso, A. Trabattoni, L. Belshaw, S. De Camillis, S. Anumula, F. Frassetto, L. Poletto, A. Palacios, P. Decleva, J. B. Greenwood, F. Martín, and M. Nisoli, “Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses,” Science 346(6207), 336–339 (2014).
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Bierbach, J.

M. Yeung, S. Rykovanov, J. Bierbach, L. Li, E. Eckner, S. Kuschel, A. Woldegeorgis, C. Rödel, A. Sävert, G. G. Paulus, M. Coughlan, B. Dromey, and M. Zepf, “Experimental observation of attosecond control over relativistic electron bunches with two-colour fields,” Nat. Photonics 11(1), 32–35 (2017).
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Blanchet, V.

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O. Jahn, V. E. Leshchenko, P. Tzallas, A. Kessel, M. Krüger, A. Münzer, S. A. Trushin, G. D. Tsakiris, S. Kahaly, D. Kormin, L. Veisz, V. Pervak, F. Krausz, Z. Major, and S. Karsch, “Towards intense isolated attosecond pulses from relativistic surface high harmonics,” Optica 6(3), 280 (2019).
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Y. Nomura, R. Hörlein, P. Tzallas, B. Dromey, S. Rykovanov, Z. Major, J. Osterhoff, S. Karsch, L. Veisz, M. Zepf, D. Charalambidis, F. Krausz, and G. D. Tsakiris, “Attosecond phase locking of harmonics emitted from laser-produced plasmas,” Nat. Phys. 5(2), 124–128 (2009).
[Crossref]

S. G. Rykovanov, M. Geissler, J. Meyer-ter-Vehn, and G. D. Tsakiris, “Intense single attosecond pulses from surface harmonics using the polarization gating technique,” New J. Phys. 10(2), 025025 (2008).
[Crossref]

P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosecond light bunching,” Nature 426(6964), 267–271 (2003).
[Crossref]

Tzallas, P.

I. Orfanos, I. Makos, I. Liontos, E. Skantzakis, B. Förg, D. Charalambidis, and P. Tzallas, “Attosecond pulse metrology,” APL Photonics 4(8), 080901 (2019).
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O. Jahn, V. E. Leshchenko, P. Tzallas, A. Kessel, M. Krüger, A. Münzer, S. A. Trushin, G. D. Tsakiris, S. Kahaly, D. Kormin, L. Veisz, V. Pervak, F. Krausz, Z. Major, and S. Karsch, “Towards intense isolated attosecond pulses from relativistic surface high harmonics,” Optica 6(3), 280 (2019).
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A. Nayak, I. Orfanos, I. Makos, M. Dumergue, S. Kühn, E. Skantzakis, B. Bodi, K. Varju, C. Kalpouzos, H. I. B. Banks, A. Emmanouilidou, D. Charalambidis, and P. Tzallas, “Multiple ionization of argon via multi-XUV-photon absorption induced by 20-GW high-order harmonic laser pulses,” Phys. Rev. A 98(2), 023426 (2018).
[Crossref]

Y. Nomura, R. Hörlein, P. Tzallas, B. Dromey, S. Rykovanov, Z. Major, J. Osterhoff, S. Karsch, L. Veisz, M. Zepf, D. Charalambidis, F. Krausz, and G. D. Tsakiris, “Attosecond phase locking of harmonics emitted from laser-produced plasmas,” Nat. Phys. 5(2), 124–128 (2009).
[Crossref]

P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosecond light bunching,” Nature 426(6964), 267–271 (2003).
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Uiberacker, M.

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct Measurement of Light Waves,” Science 305(5688), 1267–1269 (2004).
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A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449(7165), 1029–1032 (2007).
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Vakakis, N.

B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. S. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, “High harmonic generation in the relativistic limit,” Nat. Phys. 2(7), 456–459 (2006).
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Vanbostal, L.

Varju, K.

A. Nayak, I. Orfanos, I. Makos, M. Dumergue, S. Kühn, E. Skantzakis, B. Bodi, K. Varju, C. Kalpouzos, H. I. B. Banks, A. Emmanouilidou, D. Charalambidis, and P. Tzallas, “Multiple ionization of argon via multi-XUV-photon absorption induced by 20-GW high-order harmonic laser pulses,” Phys. Rev. A 98(2), 023426 (2018).
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A. Leblanc, S. Monchocé, H. Vincenti, S. Kahaly, J. L. Vay, and F. Quéré, “Spatial Properties of High-Order Harmonic Beams from Plasma Mirrors: A Ptychographic Study,” Phys. Rev. Lett. 119(15), 155001 (2017).
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[Crossref]

Y. Nomura, R. Hörlein, P. Tzallas, B. Dromey, S. Rykovanov, Z. Major, J. Osterhoff, S. Karsch, L. Veisz, M. Zepf, D. Charalambidis, F. Krausz, and G. D. Tsakiris, “Attosecond phase locking of harmonics emitted from laser-produced plasmas,” Nat. Phys. 5(2), 124–128 (2009).
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A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H. C. Bandulet, D. Comtois, F. Légaré, M. Giguère, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Wavelength scaling of high harmonic generation efficiency,” Phys. Rev. Lett. 103(7), 073902 (2009).
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L. Chopineau, A. Denoeud, A. Leblanc, E. Porat, P. Martin, H. Vincenti, and F. Quéré, “Spatio-temporal characterization of attosecond pulses from plasma mirrors,” Nat. Phys. 17(8), 968–973 (2021).
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A. Leblanc, S. Monchocé, H. Vincenti, S. Kahaly, J. L. Vay, and F. Quéré, “Spatial Properties of High-Order Harmonic Beams from Plasma Mirrors: A Ptychographic Study,” Phys. Rev. Lett. 119(15), 155001 (2017).
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J. A. Wheeler, A. Borot, S. Monchocé, H. Vincenti, A. Ricci, A. Malvache, R. Lopez-Martens, and F. Quéré, “Attosecond lighthouses from plasma mirrors,” Nat. Photonics 6(12), 829–833 (2012).
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I. Jordan, M. Huppert, D. Rattenbacher, M. Peper, D. Jelovina, C. Perry, A. Von Conta, A. Schild, H. J. Wörner, and M. Carlo, “Attosecond spectroscopy of liquid water,” Science 369(6506), 974–979 (2020).
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U. S. Sainadh, H. Xu, X. Wang, A. Atia-Tul-Noor, W. C. Wallace, N. Douguet, A. Bray, I. Ivanov, K. Bartschat, A. Kheifets, R. T. Sang, and I. V. Litvinyuk, “Attosecond angular streaking and tunnelling time in atomic hydrogen,” Nature 568(7750), 75–77 (2019).
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Walowski, J.

F. Siegrist, J. A. Gessner, M. Ossiander, C. Denker, Y. P. Chang, M. C. Schröder, A. Guggenmos, Y. Cui, J. Walowski, U. Martens, J. K. Dewhurst, U. Kleineberg, M. Münzenberg, S. Sharma, and M. Schultze, “Light-wave dynamic control of magnetism,” Nature 571(7764), 240–244 (2019).
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U. S. Sainadh, H. Xu, X. Wang, A. Atia-Tul-Noor, W. C. Wallace, N. Douguet, A. Bray, I. Ivanov, K. Bartschat, A. Kheifets, R. T. Sang, and I. V. Litvinyuk, “Attosecond angular streaking and tunnelling time in atomic hydrogen,” Nature 568(7750), 75–77 (2019).
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T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, “Nonlinear optics in the extreme ultraviolet,” Nature 432(7017), 605–608 (2004).
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B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. S. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, “High harmonic generation in the relativistic limit,” Nat. Phys. 2(7), 456–459 (2006).
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E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct Measurement of Light Waves,” Science 305(5688), 1267–1269 (2004).
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J. A. Wheeler, A. Borot, S. Monchocé, H. Vincenti, A. Ricci, A. Malvache, R. Lopez-Martens, and F. Quéré, “Attosecond lighthouses from plasma mirrors,” Nat. Photonics 6(12), 829–833 (2012).
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E. Goulielmakis, Z. H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, and F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010).
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P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosecond light bunching,” Nature 426(6964), 267–271 (2003).
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M. Yeung, S. Rykovanov, J. Bierbach, L. Li, E. Eckner, S. Kuschel, A. Woldegeorgis, C. Rödel, A. Sävert, G. G. Paulus, M. Coughlan, B. Dromey, and M. Zepf, “Experimental observation of attosecond control over relativistic electron bunches with two-colour fields,” Nat. Photonics 11(1), 32–35 (2017).
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I. Jordan, M. Huppert, D. Rattenbacher, M. Peper, D. Jelovina, C. Perry, A. Von Conta, A. Schild, H. J. Wörner, and M. Carlo, “Attosecond spectroscopy of liquid water,” Science 369(6506), 974–979 (2020).
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E. Cormier, I. A. Walmsley, E. M. Kosik, A. S. Wyatt, L. Corner, and L. F. Dimauro, “Self-referencing, spectrally, or spatially encoded spectral interferometry for the complete characterization of attosecond electromagnetic pulses,” Phys. Rev. Lett. 94(3), 033905 (2005).
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U. S. Sainadh, H. Xu, X. Wang, A. Atia-Tul-Noor, W. C. Wallace, N. Douguet, A. Bray, I. Ivanov, K. Bartschat, A. Kheifets, R. T. Sang, and I. V. Litvinyuk, “Attosecond angular streaking and tunnelling time in atomic hydrogen,” Nature 568(7750), 75–77 (2019).
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E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct Measurement of Light Waves,” Science 305(5688), 1267–1269 (2004).
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E. Goulielmakis, Z. H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, and F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010).
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M. Yeung, S. Rykovanov, J. Bierbach, L. Li, E. Eckner, S. Kuschel, A. Woldegeorgis, C. Rödel, A. Sävert, G. G. Paulus, M. Coughlan, B. Dromey, and M. Zepf, “Experimental observation of attosecond control over relativistic electron bunches with two-colour fields,” Nat. Photonics 11(1), 32–35 (2017).
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Zepf, M.

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M. Yeung, S. Rykovanov, J. Bierbach, L. Li, E. Eckner, S. Kuschel, A. Woldegeorgis, C. Rödel, A. Sävert, G. G. Paulus, M. Coughlan, B. Dromey, and M. Zepf, “Experimental observation of attosecond control over relativistic electron bunches with two-colour fields,” Nat. Photonics 11(1), 32–35 (2017).
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Y. Nomura, R. Hörlein, P. Tzallas, B. Dromey, S. Rykovanov, Z. Major, J. Osterhoff, S. Karsch, L. Veisz, M. Zepf, D. Charalambidis, F. Krausz, and G. D. Tsakiris, “Attosecond phase locking of harmonics emitted from laser-produced plasmas,” Nat. Phys. 5(2), 124–128 (2009).
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B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. S. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, “High harmonic generation in the relativistic limit,” Nat. Phys. 2(7), 456–459 (2006).
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E. Goulielmakis, Z. H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, and F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010).
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Ziaja, B.

P. Finetti, H. Höppner, E. Allaria, C. Callegari, F. Capotondi, P. Cinquegrana, M. Coreno, R. Cucini, M. B. Danailov, A. Demidovich, G. De Ninno, M. Di Fraia, R. Feifel, E. Ferrari, L. Fröhlich, D. Gauthier, T. Golz, C. Grazioli, Y. Kai, G. Kurdi, N. Mahne, M. Manfredda, N. Medvedev, I. P. Nikolov, E. Pedersoli, G. Penco, O. Plekan, M. J. Prandolini, K. C. Prince, L. Raimondi, P. Rebernik, R. Riedel, E. Roussel, P. Sigalotti, R. Squibb, N. Stojanovic, S. Stranges, C. Svetina, T. Tanikawa, U. Teubner, V. Tkachenko, S. Toleikis, M. Zangrando, B. Ziaja, F. Tavella, and L. Giannessi, “Pulse duration of seeded free-electron lasers,” Phys. Rev. X 7(2), 021043 (2017).
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APL Photonics (1)

I. Orfanos, I. Makos, I. Liontos, E. Skantzakis, B. Förg, D. Charalambidis, and P. Tzallas, “Attosecond pulse metrology,” APL Photonics 4(8), 080901 (2019).
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Appl. Opt. (3)

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J. Phys. B: At., Mol. Opt. Phys. (1)

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Nat. Photonics (4)

J. Lu, E. F. Cunningham, Y. S. You, D. A. Reis, and S. Ghimire, “Interferometry of dipole phase in high harmonics from solids,” Nat. Photonics 13(2), 96–100 (2019).
[Crossref]

M. Yeung, S. Rykovanov, J. Bierbach, L. Li, E. Eckner, S. Kuschel, A. Woldegeorgis, C. Rödel, A. Sävert, G. G. Paulus, M. Coughlan, B. Dromey, and M. Zepf, “Experimental observation of attosecond control over relativistic electron bunches with two-colour fields,” Nat. Photonics 11(1), 32–35 (2017).
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J. A. Wheeler, A. Borot, S. Monchocé, H. Vincenti, A. Ricci, A. Malvache, R. Lopez-Martens, and F. Quéré, “Attosecond lighthouses from plasma mirrors,” Nat. Photonics 6(12), 829–833 (2012).
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B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. S. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, “High harmonic generation in the relativistic limit,” Nat. Phys. 2(7), 456–459 (2006).
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Y. Nomura, R. Hörlein, P. Tzallas, B. Dromey, S. Rykovanov, Z. Major, J. Osterhoff, S. Karsch, L. Veisz, M. Zepf, D. Charalambidis, F. Krausz, and G. D. Tsakiris, “Attosecond phase locking of harmonics emitted from laser-produced plasmas,” Nat. Phys. 5(2), 124–128 (2009).
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L. Chopineau, A. Denoeud, A. Leblanc, E. Porat, P. Martin, H. Vincenti, and F. Quéré, “Spatio-temporal characterization of attosecond pulses from plasma mirrors,” Nat. Phys. 17(8), 968–973 (2021).
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Nature (6)

P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosecond light bunching,” Nature 426(6964), 267–271 (2003).
[Crossref]

T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, “Nonlinear optics in the extreme ultraviolet,” Nature 432(7017), 605–608 (2004).
[Crossref]

A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449(7165), 1029–1032 (2007).
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F. Siegrist, J. A. Gessner, M. Ossiander, C. Denker, Y. P. Chang, M. C. Schröder, A. Guggenmos, Y. Cui, J. Walowski, U. Martens, J. K. Dewhurst, U. Kleineberg, M. Münzenberg, S. Sharma, and M. Schultze, “Light-wave dynamic control of magnetism,” Nature 571(7764), 240–244 (2019).
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E. Goulielmakis, Z. H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, and F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010).
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U. S. Sainadh, H. Xu, X. Wang, A. Atia-Tul-Noor, W. C. Wallace, N. Douguet, A. Bray, I. Ivanov, K. Bartschat, A. Kheifets, R. T. Sang, and I. V. Litvinyuk, “Attosecond angular streaking and tunnelling time in atomic hydrogen,” Nature 568(7750), 75–77 (2019).
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New J. Phys. (1)

S. G. Rykovanov, M. Geissler, J. Meyer-ter-Vehn, and G. D. Tsakiris, “Intense single attosecond pulses from surface harmonics using the polarization gating technique,” New J. Phys. 10(2), 025025 (2008).
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A. Leblanc, S. Monchocé, H. Vincenti, S. Kahaly, J. L. Vay, and F. Quéré, “Spatial Properties of High-Order Harmonic Beams from Plasma Mirrors: A Ptychographic Study,” Phys. Rev. Lett. 119(15), 155001 (2017).
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A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H. C. Bandulet, D. Comtois, F. Légaré, M. Giguère, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Wavelength scaling of high harmonic generation efficiency,” Phys. Rev. Lett. 103(7), 073902 (2009).
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E. Cormier, I. A. Walmsley, E. M. Kosik, A. S. Wyatt, L. Corner, and L. F. Dimauro, “Self-referencing, spectrally, or spatially encoded spectral interferometry for the complete characterization of attosecond electromagnetic pulses,” Phys. Rev. Lett. 94(3), 033905 (2005).
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Y. Mairesse, O. Gobert, P. Breger, H. Merdji, P. Meynadier, P. Monchicourt, M. Perdrix, P. Salières, and B. Carré, “High harmonic XUV spectral phase interferometry for direct electric-field reconstruction,” Phys. Rev. Lett. 94(17), 173903 (2005).
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Supplementary Material (1)

NameDescription
Supplement 1       Supplemental document: Numerical methods and error analysis

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (10)

Fig. 1.
Fig. 1. SHHG simulation using the EPOCH code. (a) Schematic of the simulation model. (b) Electron density distribution at vacuum-plasma interface with attosecond emission (purple lines). (c) Lineout of spectrum for specular reflected field at$\; y = 0\; \mu m$. (d) Temporal envelope of the filtered harmonics (the 12th – 16th orders) via DFT.
Fig. 2.
Fig. 2. SHHG-SPIDER inversion routine. (a) Spectral shifted identical replicas, (b) interferogram, (c) Fourier pattern in pseudo-time domain, (d) reconstructed ($Recon.$) phase gradient$\; {d_\omega }\phi $, (e) reconstructed spectral intensity and phase, (f) final retrieved temporal profile.
Fig. 3.
Fig. 3. Reconstruction errors for APT with different central HH orders. Solid lines and dashed lines are representing the errors of AP-1 and AP-2, respectively. Black dot and blue triangle correspond to one-step and two-step scheme.
Fig. 4.
Fig. 4. The interferogram (left) and reconstructed APTs (right) in the inversion routine using driving pulse with durations of 4 fs (upper) and 8 fs (lower). HH signals from ${12^{th}}$ to ${16^{th}}$ orders were selected.
Fig. 5.
Fig. 5. Experimental configurations for SHHG-SPIDER (a, b) and SHHG-SEA-SPIDER (c, d) apparatus.
Fig. 6.
Fig. 6. (a) comb-like spectra of the reflected pulses (RP), the zoom in figure represents for the filtered spectral intensity from the 12th to the 16th HH orders, (b) the envelope of DFT APTs.
Fig. 7.
Fig. 7. (a) Variation of the intensity ratio between peaks of APT-2 with the time delay$\; \mathrm{\Delta }t$. (b) The time interval between the peaks of APT-2 as a function of$\; \mathrm{\Delta }t$. Blue dashed lines indicate the intensity ratio and time interval of the peaks in APT-1, respectively.
Fig. 8.
Fig. 8. SHHG-SEA-SPIDER inversion routine. (a) Temporal and (b) spectral profiles of one replica in the far-field. (c) Spatial interferogram of the two replicas. (d) 2D Fourier transformation, (e) reconstructed and (f) DFT temporal profiles. (g) - (i) lineouts of the temporal profiles.
Fig. 9.
Fig. 9. Spatial-temporal characterization of complex field profile of APT driven by $\tau = 4\; fs\; $(upper panel) and$\; 8\; fs$ (lower panel) pulse. (a) and (d) are the DFT temporal profiles, (b) and (e) are the reconstructed attosecond field, (c) and (f) are the lineouts at$\; y = 0\; mm$.
Fig. 10.
Fig. 10. Reconstruction results with spatial shear. Upper and lower panels are the reconstruction of APT driven by $\tau = 2\; fs\; ({FWHM} )$ and $\tau = 4\; fs({FWHM} )$ pulses, respectively. The first and second columns are showing the reconstructed APT temporal distribution with $s = 10\; mm$ and $20\; mm$ separately. (c) and (f) are the lineouts of reconstructed temporal profiles marked in dashed lines with respect to the DFT profiles.

Equations (3)

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f ( ω , P ) = i 4 C [ k cos ( θ ) f ( ω , r ) H 1 1 ( k s ) + H 0 1 ( k s ) f ( ω , r ) x ] d r ,
S ( ω ) = | E ~ ( ω ) + E ~ ( ω + Ω ω F ω ) e i ω τ | 2 2 I ( ω + Ω ω 2 ω F ) + 2 E ( ω ) E ( ω + Ω ω ω F ) cos ( ϕ Δ ) ,
S ( ω , y ) = | E ~ ( ω , y ) + E ~ ( ω + Ω ω F ω , y ) e i Δ k y | 2 2 I ( ω + Ω ω 2 ω F ) + 2 E ( ω ) E ( ω + Ω ω ω F ) cos ( ϕ Δ ) ,

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