Abstract
The interaction of intense near and midinfrared laser pulses with rare gases has produced bursts of radiation with spectral content extending into the extreme ultraviolet and soft xray region of electromagnetic spectrum. On the other end of the spectrum, laserdriven gas plasmas has been shown to produce coherent subharmonic optical waveforms, covering from terahertz (THz) to mid and nearinfrared frequency spectral band. Both processes can be enhanced via a combination of a driving field and its second harmonic. Despite this striking similarity, only limited experimental and theoretical attempts have been made to address these two regimes simultaneously. Here we present systematic experiments and a unifying picture of these processes, based on our extension of the semiclassical threestep model. Further understanding of the generation and coherent control of timesynchronized transients with photon energies from meV to 1 keV can lead to numerous technological advances and to an intriguing possibilities of ultrabroadband investigations into complex condensed matter systems.
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Introduction
Generation and manipulation of coherent radiation at extreme wavelengths has advanced immensely in the recent years. Significant progress has been made in the field of high harmonic generation (HHG) where researchers routinely produce bursts of extreme ultraviolet (XUV)^{1,2} and soft xray^{3,4} radiation by focusing intense nearinfrared laser pulses into rare gas targets. Along with the advent of novel diagnostic schemes^{5,6,7}, these advancements have led to the generation of isolated sub100 attosecond pulses^{8,9} and can potentially support zeptosecond pulses in the near future^{4}. At the same time, great progress has been made in the generation and detection of intense optical waveforms covering from sub −1 THz to beyond 100 THz frequency range^{10,11,12,13,14}. Broadband electromagnetic pulses in both of these spectral extremes can be efficiently generated and manipulated using a twocolor excitation scheme where the fundamental driving pulse is cofocused with its second harmonic (SH) onto a rare gas plasma^{15,16,17}.
The generation of high harmonics from rare gases follows a nonperturbative mechanism, with its salient features captured by a classical threestep model^{18,19}. In this picture, electrons are liberated in a tunneling ionization process and accelerated every half cycle of the driving laser pulse. Depending on their birth time, a fraction of these electrons can acquire sufficient kinetic energy to trigger pulsed emission of highfrequency photons upon recombination with the parent ion. Taking into account polar symmetry of the multicycle laser field and the centrosymmetry of the gaseous media, the resulting XUV pulse train contains only odd harmonics of the carrier frequency ω_{0}. Injection of a small fraction of second harmonic (SH) field facilitates breaking of the polar symmetry enabling emission of odd as well as even harmonics^{15,20,21}. Information on the time of birth of the high frequency emission^{20,22,23} is encoded in the spectral structure of the high harmonics, thus providing a useful route for gating single attosecond pulses^{9}.
The presence of second harmonic drastically enhances emission in the THz band. Similar to the HHG, the THz generation mechanism is described semiclassically including tunneling ionization of electrons followed by their classical response to an oscillating optical field^{24}, mimicking the first two steps of Corkum’s threestep model. Those electrons that do not return to the parent ion may instead create a transient net current which can support bandwidth in excess of 100 THz when excited with fewoptical cycle pulses^{25}. Therefore, it is interesting to note that coherent electromagnetic transients in the extreme frequency range spanning from THz to XUV can be produced by essentially the same method of optical twocolor excitation while sharing similar origin in the physical description (Fig. 1). The intriguing similarities in the physics of HHG and THz generation together with the practical implications of temporal synchronization provide motivation for understanding their correlations and coherent control.
In recent years, various groups have investigated the spectral features of high harmonic generation under weak polar asymmetry in twocolor excitation^{20,22,26}. Additionally, there has been an investigation of simultaneous THz and XUV emission under similar conditions. Most of the above data have been reconciled qualitatively with the prediction of the semiclassical strongfieldapproximation (SFA). In this paper, we extend the experimental scope to include the regime of strong polar asymmetry and investigate HHG correlation from short as well as long electron trajectories with the synchronous THz emission. Most importantly, we present a simple and intuitive formalism that extends the Corkum’s threestep model. By analyzing the classical “closed” electron trajectories, we are able to explain all the salient features observed not only in our data but all the previously published experiments on high harmonic generation in gasses by twocolor fields. Instead, for “open” trajectories the XUV yield with strong asymmetry is predicted to be minimized and instead being replaced by THz radiation, thus recovering the results of the plasma current model^{24}. More generally, by treating the two processes on an equal footing we are able to establish the correlations between the XUV and THz emission mechanisms and use this information to shed further light onto the generation of XUV in the cases of weakpolar asymmetric pulses.
Simultaneous generation of XUV and THz radiation is investigated by varying polar asymmetry of a copolarized twocolor excitation by adjusting the ratio (η) of the second harmonic to the fundamental field from weak η = 0.005 (0.5%) to strong η = 0.1 (10%) (see Methods section). For a fixed ratio of ω_{0} and 2ω_{0} fields, asymmetry can also be continuously modulated by changing the relative phase δϕ between these phaselocked constituents. We demonstrate that the underlying complexity of the response can only be observed when both η and δϕ are varied systematically. In comparison, earlier attempts to draw conclusions on the underpinning relationship between THz and XUV generation mechanisms were considered only in the regime of weak SH injection, resulting in casespecific observations, which depended on the particular values of the observed order of the high harmonics^{27,28}.
To elucidate the intricate nature of the highly nonlinear interaction of matter with such designer waveforms, we conducted measurements of simultaneous THz and XUV emission from argon gas (Fig. 1) as a function of δϕ for strong (η = 0.1, Fig. 2(a,b)) and weak (η = 0.005, Fig. 2(c–f)) polar asymmetry in the driving field. As expected, in both of these cases the emission shows a characteristic oscillatory dependence on the subcycle polar asymmetry of the driving field, controlled by δϕ. First we concentrate on the details of simultaneous emission for η = 0.1. Fig. 2(a,b) shows the dependence of the XUV and THz emission on the relative phase δϕ, respectively. Independent of the axial position of the gas nozzle with respect to the focus, modulations of emitted even and odd harmonics versus δϕ appear predominantly in phase (dotted line in Fig. 2(a)). Comparison of the two emissions (XUV and THz) reveals opposite δϕ dependence: while the XUV is maximized for δϕ ≈ mπ, the simultaneously emitted THz is maximized at δϕ ≈ (2m + 1)π/2. This anticorrelated dependence is observed here for the first time and can be qualitatively understood by considering classical electron trajectories. According to the 3step model, those electrons responsible for high harmonic emission follow closed trajectories that terminate in a collision with a parent ion within a halfcycle of the driving pulse. Instead, in accord with the plasma current model (PCM)^{24}, THz radiation is generated from a timevarying net charge displacement of tunnelionized electrons in open trajectories, accumulating with each halfcycle of the driving field. This accumulation is possible only if the instantaneous drift velocity v is polar asymmetric and since , therefore it is the polar symmetric field (with δϕ ≈ (2m + 1)π/2) which maximizes THz yield. The emitted power is predicted to follow sin^{2}(δϕ) which is in close agreement with our observation (see fits in Fig. 2(b,d,f)), earlier results in ref. 24 and Eq S1 of the Supplementary Information). Good agreement of the THz yield with the fits predicted by the PCM indicate that the effects of Coulomb attraction and subsequent soft recollisions^{29} may not play a significant role in the overall THz emission process, within the demonstrated level of our experimental uncertainty. Realizing that the observed THz power follows ≈sin^{2}(δϕ), independent of η (Fig. 2(b,d,f)), provides us with a convenient and an independent measure of the relative phase δϕ for cases of weak polar asymmetry.
In contrast to strong asymmetry, lowering η (=0.005) while the gas injector is positioned away from the focal plane produces negative slope in harmonics vs δϕ (dotted line in Fig. 2(c)). However, placement of the nozzle closer to the focus results in appearance of an additional positive slope for lower harmonic orders q < 28 (ω_{q} = qω_{0}, see dotted lines in Fig. 2(e) and e.g.^{23,27}). Here, it is instructive to note that emission from the short electron trajectories are known to be more efficiently phase matched further away from focus due to dipole phase contribution^{30}, therefore suggesting that the two observed slopes in Fig. 2(e) potentially originate from different electron trajectories. The combined results in Fig. 2(a,c,e) clearly demonstrate that the spectra of high harmonics intricately depend on precise conditions of the nonperturbative emission process, in turn controlled by the details of the synthesized driving field (η and δϕ parameters) as well as propagation considerations. Deciphering such complexity requires experiments with a good degree of systematic control of the key parameters, thus allowing us to glean a more complete physical picture than previously attempted.
To gain further insight into the twocolordriven XUV and THz emission processes, we develop a simple analytical model based on classical electron trajectories used in the plasma current model. To this end, we broaden Corkum’s Simple Man’s model^{18} to include an intuitive and compact expression for the HHG spectrum, which is valid for arbitrary linearly polarized twocolor excitation with both weak and strong polar asymmetry. As a result, our extended threestep (ETS) model captures all essential experimental details for various symmetrybroken fields and successfully explains the observed correlations of XUV and THz radiation, both processes arising out of superposed emissions from different classical electron trajectories. Furthermore, the model yields a quantitative prediction on the amplitude of the slope lines versus δϕ and their dependence on the intensity of the driving pulse. As is shown below, these predictions are in good agreement with our observations as well as with a broad range of independent experiments.
We consider twocolor excitation pulses having a combined electric field at a fixed position along the propagation axis E(t, δϕ) = A_{1}(t)cos(ω_{0}t) + A_{2}(t)cos(2ω_{0}t + δϕ), where A_{1}(t) and are the pulse envelope at the fundamental ω_{0}, and secondharmonic 2ω_{0} frequencies with η denoting the fraction of power in the secondharmonic field. The local HHG electric field E_{H}(ω, δϕ) is obtained by summing the contributions from all classical electron trajectories that arise during the excitation pulse:
The sum is performed over all the birth times t_{i} terminating at time t_{r} in a recollision which in turn is taken to be deterministically emitting an XUV photon with energy ħω = U(t_{i}, t_{r}, δϕ) + I_{p} with U and I_{p}being the return energy of the electrons and ionization potential of the gas, respectively. is the ionization rate, sgn(E) ensures the centrosymmetry of the medium (see Supplementary Information for additional details). Equation (1) constitutes the central result of the ETS model, providing a direct route for calculation of the spectrum of the emitted harmonics directly from the classical electron trajectories. In deriving this simple equation, therefore, we do not attempt to predict the absolute magnitude of the HHG conversion yield as we exclude the propagation effects (phasematching) and the quantummechanical probabilities in the recombination process; the latter is a consequence of the classical nature of the model. The dependence of the electron transit time Δt = t_{r} − t_{i} and return energy U together with the ionization rate on the subcycle time structure of the driving field is shown in Fig. 3 for the cases of weak (Fig. 3(a,b)) and strong (Fig. 3(c,d)) polar asymmetries. Before highlighting the comparison of these results to the experimental findings, we would like to emphasize the motivation for resorting to a simple classical model leading to equation (1). While realizing that more rigorous approaches using quantum mechanical processes may be exploited, our simple and intuitive model using classical trajectories enables us to clearly identify the various underlying physical mechanisms manifested in the experimental observations. First, we note that within a single cycle of the excitation, the harmonic field E_{H}(ω) is given as a coherent superposition of four emission contributions arising from recollision events (Fig. 3). These result from two long (L_{1}, L_{2}) and two short (S_{1}, S_{2}) electron trajectories, each with their respective amplitudes and phases qω_{0}t_{r}(t_{r}, δϕ), determined from the tunneling rate and the classical equation of motion. The pairwise interference of the subsequent long and short trajectories is governed by their respective differential phase terms. With no SH injection (η = 0), Δφ^{S} = φ^{L} = qπ due to an exact polar symmetry of the driving field. Deviation of ∆φ^{L,S} from qπ due to SH injection is thus a direct consequence of the broken polar symmetry which can be controlled by the phase δϕ and relative amplitude of the SH field. Numerical evaluations of ∆φ^{L,S} indicate that, for a given return energy, the corresponding temporal separation between recollision events in the two halfcycles shrinks and expands for the long and short trajectories, respectively (Fig. 3).
Starting with the case of high polar asymmetry (η = 0.1) for δϕ = mπ (Fig. 3(c)), the extreme nonlinearity of the tunnel ionization suppresses the amplitude () in the adjacent half cycle, thus leading to a nearly fullcycle periodicity of the HHG signal in the time domain. Fourier analysis translates this periodicity into the formation of both odd and even harmonics. At phase delays δϕ = (2m + 1)π/2, the timedomain signal returns to halfcycle periodicity but with much lower amplitude (Fig. 3(d)). Thus, for strong polar asymmetry the XUV emission is predicted to peak at δϕ ≈ mπ for all harmonic orders, as can also be directly seen from the calculated spectral dependence on δϕ (Fig. 4(a)). This result is in good qualitative agreement with the measurements (Fig. 2(a)). Furthermore, the XUV dependence on the relative phase is precisely anticorrelated with the THz emission process (Fig. 2(b)), thus recovering the intuitive picture of essentially mutually exclusive nature of XUV and THz generation for the case of strong SH injection.
We now turn to the analysis of weak second harmonic injection. Using equation (1) with input of the calculated dynamic parameters from Fig. 3(a,b), one obtains the dependence of the XUV spectra on the relative phase (Fig. 4(b)). In general, the simulated spectral map reproduces the qualitative features observed in our experiment (Fig. 2(e)) as well as by other groups^{20,21,23,27,28}. In particular, we note that for harmonic orders q ≤ 28 the calculation and experiments both show the shift in the XUV extrema with increasing q as a function of phase delay δϕ, i.e. positive values of the slopes . In addition to the correct dependence, the ETS model prediction of the slope of = +16 harmonics per radian is also in a remarkable agreement with the experiment (dotted lines in Figs 4(b) and 2(e) for q ≤ 28, respectively).
While the observed agreement validates the model for lower harmonics, the question remains as to why the model seemingly struggles to reproduce the experimental features seen for higher harmonic orders. As was already mentioned, the XUV spectra strongly depend on the phase matching conditions, with preference for short or long trajectories depending on the position of the gas injector. Since the ETS model does not take propagation effects into account, it is therefore unrealistic to expect a general agreement of the calculations based on conditions which might be beyond the applicability. Nonetheless, since equation (1) includes contributions from both trajectories (together with results in Fig. 2(c,e)) one anticipates that the electrons following short(long) trajectories are responsible for observed spectral features at higher(lower) harmonics and the two dominant regions are separated by a spectral band where both contributions are comparable (e.g. 28 < q < 32 in Fig. 2(e) for particular phasematching conditions). To elaborate on this we turn our attention to the amplitudes of these emission events, the temporal structures of which already played a decisive role in breaking halfcycle periodicity for strong polar asymmetry (Fig. 3(c,d)). Instead, for weak polar asymmetry the electrons which undergo long trajectory recollision events are generated at times before the peak of the electric field (Fig. 3(a,b)). This fact leads us to conclude that the highly nonlinear tunneling ionization rate favors these events over the short trajectories that are in contrast born at later times. To check the validity of these observations, we calculate the spectral dependence of the amplitudes of short and long trajectories for δϕ = 0 and π/2, depicted in Fig. 3(e,f) with blue and red lines, respectively. At low harmonics, the total amplitude is dominated by the contribution from the long trajectories, while long and short trajectories assume comparable amplitudes as harmonics approach the cutoff region. This explains why all the experiments (including ours) show the positive slope at lower harmonics, corresponding to long trajectories. In contrast, at energies approaching the cutoff harmonics, one expects a more complex behavior resulting from an interference of all 4 trajectories. This is indeed the case in the experiments where there is no additional selectivity that distinguishes short and long trajectories. In practice, such selectivity can be implemented by exploiting the farfield divergence of the XUV emission leading to a spatial separation of contributions from long and short trajectories^{26} or preferential phasematching^{31}.
Finally, to further elucidate the roles of amplitude and phase of the trajectories and to further clarify the complexity of the spectrum for harmonics, we cast the expression in equation (1) into an even simpler form. For a square driving pulse containing N cycles at fundamental frequency, an approximation of scaling as E^{Q} and consideration of the dominant birth times to be near the peaks of each cycle, produces an HHG spectrum:
where C_{N}(q) = sin^{2}(Nπq)/sin^{2}(πq) is a comb function, are the differential phase terms defined earlier and Q ≈ 7–9 for typical laser intensities involved. A plot of this simple expression for long and short trajectories during weak polar asymmetry (Fig. 4(c,d)) reproduces all the essential features of the more exact calculation (Fig. 4(b)) and experimental observations (Fig. 2(c,e)).The calculation also faithfully reproduces XUV spectra for strong polar asymmetry (not shown). Furthermore, the differential phase terms in equation (2) are determined to follow the following empirical functions: , and , where with q_{0} = 2U_{p}/ħω_{0}, U_{P} being the ponderomotive energy and denoting the Keldysh parameter (see also Supplementary Information). The peaks of harmonics occur at corresponding to a stationary phase (=mπ, m integer) of the sine functions. Evaluated at these stationary points, the two expressions yield opposite signs of slopes and for long and short trajectories, respectively (also can be directly seen in Fig. 4(c,d)). Furthermore, these slopes are found to depend on the peak intensity and the center wavelength of the fundamental field: (harmonics per radian) (see Fig. S2 in Supplementary Information). Evaluating this result for λ = 0.8 μm, we find which correspond to slopes of ≈ ±16 harmonics/radian for long and short trajectories, based on our estimated focal intensity of I_{0} = 4 × 10^{14} W/cm^{2}. Both in sign and magnitude, these slopes are in an excellent agreement with our experimental findings (Fig. 2(e)). Notably, in addition to the agreement with our results, the prediction of the ETS model of the amplitudes of the slopes is also in a very good agreement with a rather general set of reported experimental observations^{20,21,23,27,28}. Finally, we estimate reduction in the peak intensity of our fundamental pulse by a factor 3–4 when translating the gas injector away from the focal plane to a new position (conditions in Fig. 2(c)). We observe a commensurate reduction in the slope from 16 ± 0.3 harmonics/rad (Fig. 2(e)) to a value of 5 ± 0.3 harmonics/rad (Fig. 2(c)). The overall drop in the slope value by a factor of 3.2 ± 0.6 during an estimated 3–4 times drop in the intensity shows very good agreement with the predictions of the ETS model (see previous paragraph and Supplementary Fig. S2).
In conclusion, we report detailed investigation of the XUV emission spectra from argon gas when driven by intense femtosecond pulses while being subjected to a transient optical bias induced by the phaselocked second harmonic pulse replica. By varying the intensity ratio (η) and the relative phase (δϕ) of the copolarized second harmonic pulses we are able to systematically study the resulting spectra of the XUV emission and the yields of the simultaneouslyemitted THz radiation. For strong injection of the second harmonic (10% of the fundamental intensity or strong polar asymmetry) we find that XUV and THz radiation are anticorrelated in their dependence on the relative phase δϕ. Instead, for weak polar asymmetry we find that while THz dependence on δϕ remains unchanged, the XUV spectra show remarkably complex structure, which, while also dependent on the phasematching considerations, reveals two distinct slopes, defined by a phase slip between the adjacent harmonic orders q. To explain these features we devised an extended threestep model (ETS) based on the classical electron trajectories. With the gained physical insight on the origin of XUV emission to arise from interference of XUV bursts from multiple recollision events of short and long trajectories within a single cycle of the driving pulse, we are able to provide qualitative explanations for all the experimentally observed features. Very good qualitative agreement between experimental results and the model points to the robustness of the semiclassical approach of the original 3step model of high harmonic generation. Furthermore, quantitative prediction of the ETS model for the sign and magnitude of the slopes has been verified in our experiments and found to also reproduce a wide array of previously published results by other groups. Curiously, dependence of these slopes on the intensity of the fundamental pulse might suggest future routes for XUV pulse, as verified by our experiments, compression schemes with control of the spectral phase of harmonics, known as attochirp^{21}.
Finally, gained insight into the contributions of short and long trajectories in cases of weak and strong polar asymmetry in the driving twocolor field allows us to draw further conclusions about the relation of the XUV and THz emission processes. It is clear from the nature of the nontrivial phase shifts with increasing harmonic orders that in case of weak polar asymmetry no absolute phase relationship between the two processes can be established a priori. In turn, XUV emission resulting from strong polar asymmetry has shown a clear phase dependence which is anticorrelated in the relative phase parameter with the production of the THz. It is only from a careful comparison of these two regimes that precise conclusions on the phasedependence of all the processes can be drawn. Thus our experiments and the ETS model provide a unified picture of XUV and THz emission mechanisms within the framework of classical trajectories and arbitrary degree of polar asymmetry in the driving field. Furthermore, the agreement of the observed correlation features in the THz and XUV emission with our theory suggests that the two processes have the same physical origin, only with different electron trajectories. Together with the high degree of control, perhaps it is sources like these that in the future could provide ultrabroad emission in the entire THzXUV frequency window with potential technological and scientific applications.
Methods
Experimental Details
The experimental setup for synchronous generation and detection of XUV and THz radiation is shown schematically in Fig. 1. A train of 1 kHz, 40 fs pulses with a center wavelength of 800 nm produces high harmonics when focused onto the output of an argon gas injection nozzle with peak intensity of (4–5) × 10^{14} W/cm^{2}. Our experiments are done at high gas pressures of up to 1000 Torr and the laser beam is loosely focused under the gas injection nozzle with a diameter of 0.3 mm. Under these conditions, both long and short trajectories can be phasematched. Preferential generation of short trajectories can be selected by locating the nozzle after the focus^{31}.
The SH of fundamental field at angular frequency ω_{0} is generated in a 150 μm typeI BBO crystal, the orientation and position of which control the conversion efficiency of SH in the range of 0.5% to approximately 10%. Parallel polarization of the two fields is ensured by an ultrathin, true zeroorder halfwave plate for the fundamental wavelength. As expected from symmetry arguments and confirmed by measurements, this results in linearly polarized THz and XUV emission. A pair of glass wedges controls the relative phase δϕ and hence the degree of asymmetry between two pulses with an overall accuracy ≈π/5 that is independently calibrated in a separate experiment^{32} (also see Supplementary Information). All of these optical components are placed in a low background pressure chamber to ensure minimal phaseslip between the fundamental and SH pulses. An offaxis parabolic mirror is located after the gas nozzle and serves to collimate the THz field out of the vacuum chamber through a silicon viewport. This window also acts as an optical filter to remove the excitation light. A 500 μm diameter central hole in the parabolic mirror transmits the lowdivergence XUV beam, which then passes through a 200nm thick aluminum filter and is routed to a grazingincidence XUV spectrometer (McPherson Inc.), as shown in Fig. 1. Terahertz emission is fieldresolved with electrooptic sampling (500μm thick (110) ZnTe detector) and direct detected by a pyroelectric sensor. Comparison of measured signals by both of these techniques verifies that THz yield is frequency independent, resulting in the overall scaling of the magnitude of the emission process with the variation of the relative phase δϕ. Polarization states of the XUV and THz pulses are ensured using polarizationdependent anisotropy of the grating reflectivity and a broadband polarizer, respectively, both with polarization contrast better than 10 to 1.
Additional Information
How to cite this article: Gragossian, A. et al. Classical trajectories in polarasymmetric laser fields: Synchronous THz and XUV emission. Sci. Rep. 6, 34973; doi: 10.1038/srep34973 (2016).
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Acknowledgements
The work at UNM was partially supported by the NSF under Award DMR1207489, and by the Los Alamos National Laboratory (LDRD) program 20120246ER. DVS acknowledges support by the NSF (Grant No. 1160764) and partial support by the EU FP7 Marie Curie Zukunftskolleg Incoming Fellowship Program, University of Konstanz (Grant No. 291784). The authors also thank Dr. Vladislav S. Yakovlev for useful discussions.
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D.V.S. and M.S.B. conceived the idea and designed the experiment. A.G. and D.V.S. performed the experiments and M.S.B. supervised the effort and developed the ETS model. All authors contributed to the analysis and reviewed the manuscript.
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Gragossian, A., Seletskiy, D. & SheikBahae, M. Classical trajectories in polarasymmetric laser fields: Synchronous THz and XUV emission. Sci Rep 6, 34973 (2016). https://doi.org/10.1038/srep34973
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DOI: https://doi.org/10.1038/srep34973
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Mansoor SheikBahae (1956–2023)
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