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Coherent kiloelectronvolt x-rays generated by subcycle optical drivers: a feasibility study A. Moulet,1 V. Tosa,2,* and E. Goulielmakis1 1

2

Max-Planck-Institut fur Quantenoptik, Hans-Kopfermann-Str. 1, Garching D-85748, Germany National Institute for R&D of Isotopic and Molecular Technologies, Cluj-Napoca 400293, Romania *Corresponding author: tosa@itim‑cj.ro Received August 1, 2014; revised September 26, 2014; accepted September 30, 2014; posted September 30, 2014 (Doc. ID 220146); published October 22, 2014

We theoretically explored the feasibility of high-harmonic generation in the kiloelectronvolt spectral range by optical waveforms of durations progressively shortened from multicycle to subcycle. Our study revealed that subcycle optical pulses offer a clear advantage in generating isolated x-ray attosecond pulses. In combination with their sub-fs optical drivers these pulses will open the route for x-ray attosecond pump-optical attosecond probe experiments, advancing attosecond streaking and attosecond absorption techniques to new realms of investigation of electronic processes. © 2014 Optical Society of America OCIS codes: (320.7110) Ultrafast nonlinear optics; (340.7480) X-rays, soft x-rays, extreme ultraviolet (EUV); (140.7090) Ultrafast lasers. http://dx.doi.org/10.1364/OL.39.006189

The generation of extreme ultraviolet (EUV) attosecond pulses by few-cycle laser fields spectrally centered at near infrared and optical frequencies and their synergy in pump-probe experiments [1–6] have enabled the establishment of a versatile toolbox [7] for the exploration of electronic processes on the attosecond time scale [8]. Extension of the capabilities of this toolbox would benefit from (i) compression of the optical fields to attosecond time intervals and (ii) generation of attosecond pulses spectrally extending up to the kiloelectronvolt (keV) regime and beyond, precisely synchronized to the intense optical fields. Progress in multioctave light-field synthesis [9,10] has recently allowed reaching the first of these goals by extending synthesis toward the deep ultraviolet range [11]. However, the generation of high-harmonic radiation nearing or exceeding keV energies has so far been only possible by infrared and midinfrared pulses [12,13] whereby attosecond measurements such as streaking [14] or transient absorption [6] are still lacking. Moreover, a long carrier period prevents pulse compression to attosecond half-cycle fields, affecting the resolution of potential x-ray pump optical-probe schemes. Here, in a numerical study that explores systematically high-harmonic generation (HHG) by multicycle, fewcycle, and subcycle optical driving fields we demonstrate that the latter lend themselves as a promising tool for attosecond pulse generation in the keV range and offer potential for extension of existing attosecond transient absorption or streaking techniques to the x-ray regime. To explore the potential of keV HHG by optical laser drivers we have conducted numerical simulations by extending a nonadiabatic three-dimensional model describing HHG by a two-color field [15] to the case of four fields of arbitrary spectral amplitudes, carrier envelope phases (CEP), and relative delays. Our numerical method allows simulation of HHG by field waveforms, the generation of which is enabled by a modern light-field synthesizer [9–11] and thus offers the impetus for future experimental studies. We solve the wave equations for the four fields propagating in a space-time-dependent refractive 0146-9592/14/216189-04$15.00/0

index that is build up for each field by accounting for dispersion from neutral atoms and from electron plasma as well as for optical Kerr effect [15]. The electric field of the synthesized waveform produces a plasma density, which is calculated using the exact static ionization rates reported for He [16]—the atomic system used in our study at a constant pressure of 50 Torr. Next, by using the strong field approximation [17,18], we estimate the single-dipole response to the propagated field along the interaction region. Finally, the gas polarizability is used as a source term to solve the propagation equation for the harmonic field, again taking into account the frequencydependent absorption and dispersion. The result of the calculation is the high-harmonic field H
r; z; t or its frequency counterpart H
r; z; ω, where r and z are the cylindrical coordinates in a grid set over the interaction region, t the time, and ω the angular frequency. Even though our model accounts for nearly arbitrary waveforms, for the sake of clarity we have limited this first study to a set of four waveforms (thereafter WFs) having the same peak intensity (1016 W∕cm2 ) and central wavelength (540 nm) but increasing pulse duration, as shown in the inset of Fig. 1(a). Studies based on more complex periodic waveforms can be found elsewhere [19–21]. For convenience they are labeled WF1, WF2, WF4, and WF8, according to their intensity envelope full width at half-maximum (FWHM) duration of 1, 2, 4, and 8 fs, respectively. As we intend to explore the limits of this approach, the peak intensity was chosen such as to yield full ionization of the medium for WF1, which also warrants full ionization for the other WFs. The spatial beam waist is taken to be 40 μm, constant for all spectral bands (in experiments this condition is met by imaging the output of a hollow-core fiber [9]), and the 250 μm medium starts at the common focus of the beams. Figure 1(a) shows single-dipole frequency response at the center of the medium entrance for the four WFs, the instantaneous intensity of which can be seen in the inset (same CEP of π∕2 was assumed for all). The advantage of the shorter driver field (WF1) is evident already at this stage of our study. The emission by WF1 forms a single © 2014 Optical Society of America

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cutoff extended to 1 keV, while the emission of the other waveforms is fragmented to multiple cutoffs of lower photon energies, each corresponding to a half-cycle. In these pulses, the dipoles indeed develop in early half-cycles having intensities lower than the peak, thus leading to lower photon energies. Moreover, ionization completely depletes the ground state before the peak intensity time, hindering generation of the highest possible photon energies. The dipole intensity in Fig. 1(a) is lowest for WF1, which reflects the intracycle depletion [22] of the ground state. Indeed, dipole amplitude is proportional to the ground-state amplitude at both ionization and recombination instances. For WF1, our calculations indicate that the ground-state amplitude at ionization is close to unity, but that the next half-cycle nearly depletes the ground 1E -20 2

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state, hence the smaller dipole amplitude. Moreover, they show that for multicycle pulses, in the cycles preceding the peak, the two ground-state amplitudes are far from 0, thus having a smaller influence on dipole amplitude. After 250 μm of propagation the single-dipole response [see Fig. 1(b)] has markedly changed. This is manifested by a reduction of the cutoff energy of the radiation emitted by all waveforms. This is more pronounced for WF1 (from 1200 to 600 eV), and it is gradually less pronounced for WF2, WF4, and WF8. In fact, the dipole induced by WF8 is nearly unaffected for this propagation distance. This cutoff energy reduction is directly connected with the spatiotemporal distortions each driving WF suffers during propagation. These distortions are highlighted in the inset of Fig. 1(b) for WF1. The most essential of these distortions is the decrease of the peak intensity of the first ionizing half-cycle due to defocusing and self-phase modulation at optical-cycle level [23,24], which is associated with the sharp variation of the refractive index on that time scale. It is worth noting that both these effects contribute to the reduction of the cutoff energy during propagation. The total harmonic emission, radially integrated at the medium exit, is shown in Fig. 1(c) for the case-study waveforms. WF4 and WF8 extend up to 800 eV in reasonable agreement with the reported experiments [25], but fail to extend beyond 1 keV. WF1 is the only waveform whose generated harmonic signal builds up at energies up to 1.2 keV. By contrast, the harmonic emission at lower energies is most efficient with WF8 for energies up to 350 eV and WF2 from 350 to 850 eV. For optical-driver fields, such a short duration could be optimum for the generation of photons in the water-window range. It is worth mentioning that the keV beam has a typical calculated divergence of 0.6 mrad full angle, and is more collimated than the lower-energy photons (2.8 mrad at 150 eV). An important matter is to clarify whether phase matching plays an essential role in the observed spectra. To elaborate on this we plot in Fig. 2 the buildup of the photon flux at representative energies as a function of the propagation length. The quadratic increase of the signal for photon energies lower than 400 eV [Fig. 2(a)], suggest that phase-matching conditions are reached for the radiation emitted in this spectral regime [26]. However, for higher photon energies [Figs. 2(b) and 2(c)], an initial increase of the harmonic signal is followed by a plateau, indicating that the corresponding single dipole is eliminated due to the drop of driving-field intensity below the threshold intensity required for the emission of this photon energy. Clearly, the higher the photon energy the shorter the distance over which the harmonic field builds up, as seen in Figs. 2(b) and 2(c). Besides the possibility of generating coherent x-ray radiation by these fields, the potential for isolated attosecond pulses (IAP) is yet another aspect to expanding the attosecond toolbox. To explore this, we calculated the time structure of the total harmonic field emitted at the exit plane of the medium highpass filtered above 0.8  E cutoff (i.e., 530 eV for WF8 and WF4, 720 eV for WF2, and 880 eV for WF1). The corresponding temporal envelopes are displayed on Fig. 3. Including all spatial effects and without any chirp compensation attosecond pulses are formed for WF1 (28 as), WF2 (67 as), and WF4 (195 as).

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WF1:WF2:WF4:WF8 ≈ 1∶5∶160∶40, while the ratios of the areas under the peaks are 1∶1.4∶2250∶4500. It is also worth mentioning that by filtering the harmonic fields in a common spectral range (e.g., from 400 eV to the cutoff), we obtain a different picture: WF1 and WF2 still produce IAP (84 as and 105 as, respectively) but only when their CEP is set around zero. For WF4, we observe IAP with satellites regardless of CEP, while WF8 produces a train of bursts. The peak-intensity ratio of the IAPs is in this case is WF1:WF2  1∶36. In conclusion we showed that intense optical subcycle fields readily available by modern field synthesizers offer the potential of extending the generation of soft-x-ray attosecond pulses in the keV range. Such pulses, in combination with their optical attosecond counterparts [11], hold promise for expanding the scope of techniques of attosecond spectroscopy to encompass phenomena that are initiated by deep-shell excitation of gases and solids and their real-time tracing. V. T. acknowledges financial support from projects PNII-ID-PCE-2012-4-0342, PNII-PT-PCCA-2011-3.1-0886, and from DAAD fellowship. A. M. and E. G. acknowledge support by European Research Council grant (Attoelectronics-258501), the Deutsche Forschungsgemeinschaft Cluster of Excellence: Munich Centre for Advanced Photonics (www.munich‑photonics.de), the Max Planck Society, and the European Initial Training Network ATTOFEL.

Fig. 2. Buildup of harmonic field at different photon energies for the 1 fs waveform.

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