Metrology of time-domain soft X-ray attosecond pulses and re-evaluation of pulse durations of three recent experiments
MMetrology of time-domain soft X-ray attosecond pulses and re-evaluation of pulsedurations of three recent experiments
Xi Zhao , Su-Ju Wang , Wei-Wei Yu , , Hui Wei , and C. D. Lin Department of Physics, Kansas State University, Manhattan, KS 66506, USA School of Physics and Electronic Technology, Liaoning Normal University, Dalian 116029, People’s Republic of China
Attosecond pulses in the soft-X-ray (SXR) to water-window energy region offer the tools forcreating and studying target specific localized inner-shell electrons or holes in materials, enablemonitoring or controlling charge and energy flows in a dynamic system on attosecond timescales.Recently, a number of laboratories have reported generation of continuum harmonics in the hundred-electron-volt to kilovolt region with few-cycle long-wavelength mid-infrared lasers. These harmonicshave the bandwidth to support pulses with duration of few- to few-ten attoseconds. But harmonicsgenerated in a gas medium have attochirps that cannot be fully compensated by materials over abroad spectral range; thus, realistically what are the typical shortest attosecond pulses that one cangenerate? To answer this question, it is essential that the temporal attosecond pulses be accuratelycharacterized. By re-analyzing the soft X-ray harmonics reported in three recent experiments [1–3] using a newly developed broadband phase retrieval algorithm, we show that their generatedattosecond pulses are all longer than about 60 as. Since broadband pulses tend to have high-orderchirps away from the spectral center of the pulse, the algorithm has to be able to retrieve accuratelythe phase over the whole bandwidth. Our re-evaluated pulse durations are found to be longer thanthose previously reported. We also introduce the autocorrelation (AC) of the streaking spectrogram.By comparing the ACs from the experiments and from the retrieved SXR pulses, the accuracy ofthe retrieved results can be directly visualized to ensure that correct phases have been obtained.Our retrieval method is fast and accurate, and it shall provide a powerful tool for the metrology offew-ten-attosecond pulses in the future.
PACS numbers: 32.80.Rm, 42.50.Hz, 42.65.Ky
I. MAIN
Since the first report of isolated attosecond pulses(IAP) in 2001 [4], the majority of IAPs are generatedonly in EUV (or XUV) region, with photon energy be-low about 120 eV [5–7]. To study temporal electron dy-namics, bond breaking, and energy flow in a bio-chemicalreaction, extension of IAPs to soft-X-ray region (SXR) ishighly desirable since SXR would excite inner-shell elec-trons to create a localized initial hole, which would finger-print the flow of charges (electrons or holes) and energy,giving access to chemical processes at the most funda-mental level with best spatial and temporal resolutions.Within the last decade, the development of high-energylong-wavelength driving lasers together with pulse com-pression to few cycles has allowed experimentalists togenerate broadband continuum high-energy photons upto 1.6 keV [8–20]. These pulses have spectral bandwidthto support transform-limited pulses of a few attosecondsor even zeptoseconds. However, to claim such short dura-tions would require accurate characterization of the spec-tral phase over the whole spectral bandwidth.Recently, using two-cycle 1.80 to 1.85 µ m lasers, threegroups have reported broadband SXR harmonics [1–3].In all three experiments, streaking spectra were also mea-sured. Two groups reported pulse durations of 53 ± ± E SXR (Ω) = U (Ω) e i Φ(Ω) is available. The spec-tral amplitude U (Ω), where Ω is the photon energy, canbe obtained from photoelectron spectra of some rare-gasatoms ionized by the SXR alone since accurate photoion-ization cross sections of rare-gas atoms are available fromexperiments or theoretical calculations. To get informa-tion on the spectral phase Φ(Ω), photoelectron spectraare generated by the same SXR in the presence of a time-delayed driving laser that is used to generate harmonics.The resulting electron spectra vs the time delay is calledthe spectrogram (or the streaking trace) and is then an-alyzed to retrieve the spectral phase. In this method, itis assumed that the spectrogram can be calculated usingthe so-called strong-field approximation (SFA)[21, 22]: a r X i v : . [ phy s i c s . a t m - c l u s ] M a y S ( E, τ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞−∞ E SXR ( t − τ ) d ( p + A ( t )) × e − iφ ( p,t ) e i (cid:16) p + I P (cid:17) t dt (cid:12)(cid:12)(cid:12)(cid:12) , (1)where E = p / τ is thetime delay between the two pulses, A ( t ) is the vectorpotential of the laser, d is the dipole transition matrixelement, I p is the ionization potential of the atom, and φ ( p, t ) = (cid:82) ∞ t [ pA ( t (cid:48) ) + A ( t (cid:48) ) / dt (cid:48) is the action of theelectron in the laser field.Atomic units are used throughout the paper unlessotherwise stated. The vector potential A should not betoo large so it cannot contribute to the ionization of thetarget atom. In this model, the electron is removed bythe SXR and then streaked by the vector potential ofthe laser until the pulse is over. To obtain the phase of E SXR , previously the so-called FROG-CRAB [21, 22] wasused. It is based on the FROG method for retrieving thespectral phase of a femtosecond laser. To use the algo-rithm written for FROG, an additional approximation,called central momentum approximation (CMA), has tobe made by replacing the p in the action φ above by themomentum of the electron at the center of U (Ω). Thisapproximation is not severe if the bandwidth is narrow,say about 10 to 20 eV. Thus, earlier attosecond pulseswere retrieved using the FROG-CRAB method [21, 23].It is generally believed that the method works for thesenarrowband pulses.To retrieve broadband pulses, the CMA has tobe removed. Three methods have been proposed,PROOF [24], VTGPA [25], and PROBP [26]. ThePROOF is based on approximating the action φ in Eq. (1)by taking the limit when A is small, and that the streak-ing IR field is a monochromatic wave. In PROOF, thestreaking shift should be small, and thus it is more sensi-tive to errors in the streaking spectra. PROOF was usedin [1] and [23] even when the IR is a short few-cycle pulse.In the experiment of [2], the spectral phase was retrievedusing the ML-VTGPA method, which is a modificationof VTGPA [25] to account for photoelectrons generatedfrom multiple shells of the atom. In [3], the bandwidthof the SXR is about 200 eV. Using FROG-CRAB, thepulse duration obtained was about 24 as. Based on theattosecond lighthouse model, it was estimated that thepulse duration should be less than 322 as.Among the phase retrieval algorithms, includingFROG-CRAB, all are based on iterative methods. Thecalculation is terminated after tens of thousands of iter-ations when the merit is not changing. To “prove” thatthe retrieved SXR pulse (also for XUV pulse) is correct,the retrieved pulse is then used in Eq. (1) to calculatethe spectrogram. By comparing the experimental spec-trogram with the one from the retrieved pulse visually, itis often deemed that the agreement is good. This proce-dure is probably acceptable for narrowband pulses, but not for broadband pulses.To illustrate this point, we first compare how the spec-trogram calculated using Eq. (1) depends on the spectralphase. For simplicity, consider a SXR pulse in the energydomain, E SXR (Ω) = U (Ω) e i Φ(Ω) , with a Gaussian spec-tral amplitude U (Ω) = U e − (Ω − Ω0)2(∆Ω)2 and a quadraticspectral phase Φ(Ω) = a − Ω ) (∆Ω / . Here we use Ω = 164eV as the central photon energy and ∆Ω = 94 eV as thefull width at half maximum (FWHM) bandwidth, whichcan support a transform-limited (TL) pulse (correspond-ing to a = 0) of 20 as. The coefficient a is a measure ofthe attochirp, or equivalently we can use the parameter γ = ∆ τ / ∆ τ TL , which is the ratio of the duration of thechirped pulse to the TL duration. Figures 1(a)-(c) com-pare the spectrograms simulated according to Eq. (1).The mid-IR (MIR) used is 1800 nm in wavelength, 5.7 fsin duration, and 2 . × W/cm in peak intensity.From Figs. 1(a)-(c), visually the three spectrogramsshow little difference, even though their pulse durationsare 20, 90, and 177 as, respectively. To magnify theircontrast, we calculate the auto-correlation (AC or Q ) ofthe spectrogram, defined by Q ( τ , τ ) = (cid:90) ∞ S ( E, τ ) S ( E, τ ) dE. (2)Figures 1(d)-(f) show the corresponding AC patterns overone optical period. For the TL pulse, the AC shape isclose to a square in the center. As the linear chirp isincreased, the square gradually deforms and skews alongthe diagonal axis. Clearly, the more the deformation is,the larger is the linear chirp (and pulse duration). Todelineate the relative deformation of the AC with respectto the TL pulse, we define a normalized volume V norm foreach AC pattern: V norm = (cid:82) (cid:82) Q ( τ , τ ) dτ dτ (cid:82) (cid:82) Q T L ( τ , τ ) dτ dτ , (3)in which we integrate the AC pattern (transform-limitedAC pattern) over an area of half an optical cycle T alongeach axis centered at the coordinate with the maximumvalue of the AC pattern (transformed-limited AC pat-tern). Figure 1(g) shows that V norm drops very quicklywith γ if the TL pulse is 70 as, but very slowly if the TLpulse is 20 as. This speaks that attosecond pulses witha broader bandwidth are much more difficult to retrieve.The MIR used for Fig. 1(g) has the wavelength of 1800nm.To elaborate the advantage of using the AC insteadof the spectrogram for spectral phase retrieval, in Figs.2(a,e), we show the spectrograms from experiment withthe one obtained from the retrieved pulse reported in [2].We calculate the ACs from these two spectrograms, andthe results are shown in Figs. 2(b,f). The AC patternsare expected to repeat reasonably well for each optical - 6 - 3 0 3 68 01 8 02 8 0 Electron energy (eV)
T L 2 0 a s ( a ) - 6 - 3 0 3 6( b ) - 6 - 4 - 2 0 2 4 6( c ) - 4 - 2 0- 4- 20 ( d )
Time delay(fs)
T i m e d e l a y ( f s ) - 4 - 2 0( e ) B - 4 - 2 0( f ) B ( g ) Volume(normalized) g ( D t / D t
T L ) T L 2 0 a s
T L 5 0 a s
T L 7 0 a s
FIG. 1: (Color online) Theoretically calculated spectrograms (a,b,c) and the corresponding AC patterns (d,e,f). The centralenergy of the three XUV pulses is 160 eV, with a FWHM bandwidth of 94 eV. The duration of the TL pulse is 20 as, whilethe other two are 90 and 177 as, respectively. The MIR used in the simulation is 1800 nm in wavelength, 5.7 fs in FWHMduration and 2 . × W/cm in peak intensity. (g) Normalized volume of the AC pattern vs pulse duration in units of theduration of the transform-limited pulse. The normalized volume drops more slowly versus the scaled duration for broadbandpulses than for narrowband pulses, making it harder to retrieve broadband pulses. The examples assume that the XUV puleshave quadratic phases only; see text. cycle of the MIR laser. We label several blocks (blocks1-7), and one can see that the ACs from the experimentand from the simulated pulse do not agree well. Similar,in Figs. 2(c,g), the spectrograms from the experimentand from the simulated pulse of [1] are compared, andtheir corresponding ACs are compared in Figs. 2(d,h).Clearly, the latter shows that the experimental one andthe retrieved one do not match well. Such discrepancypoints that the SXR pulses reported in these experimentswere not accurately retrieved. These results also demon-strate that the ACs serve as a good metric for evaluatingthe quality of the broadband SXR pulses retrieved, inde-pendent of whatever retrieval method is used.The method we used to retrieve the spectral phase ofthe SXR pulse is called PROBP-AC [27]. It is a revisionfrom the earlier proposed PROBP method [26]. Here,we first report the re-analysis of the data from [2], wherethe ML-VTGPA method was used. From Fig. 2(a), thespectrogram from [2] appears to be well-behaved from τ = -9 fs to -1 fs. Thus we choose to analyze the AC inblock number 5.[For more details, see sections I and III ofthe Supplementary Information (SI)]. The ACs for blocknumber 5 from the experimental and the retrieved pulsesfrom [2] are shown in Figs. 3(a) and (b), respectively.We can see that they show little resemblance. Using ourPROBP-AC method, the resulting AC is shown in Fig.3(c). It is much closer to the experimental data givenin Fig. 3(a). In our PROBP-AC method, the spectral amplitude of the SXR is obtained from the experiment.The vector potential of the 1800-nm MIR laser is alsoretrieved. From our results, we reconstruct the intensityof the SXR in the time domain, see Fig. 3(d). Our resultdoes not agree very well with the one reported in [2]. TheFWHM pulse duration from our new evaluation is 62 as,as compared to 43 as reported in [2]. In Fig. 3(e), wecompare the spectral phases. The phase obtained in [2]is very small over the whole spectral range, thus theyretrieved a near-TL pulse. Our result from PROBP-ACshows large chirps away from the central energy. Notethat the linear term in the spectral phase has been re-moved. Thus, the spectral phases presented in Fig. 3(e)show the phases that contribute to the pulse durationbeyond the TL pulse. The reason for the discrepancy be-tween the present result from [2] is further discussed inSec.II and III of SI. On the other hand, the better agree-ment of Fig. 3(c) than Fig. 3(b) with the experimentaldata Fig. 3(a) is a good indication that the present re-trieved result is more accurate. We can also calculatethe merit of the retrieved pulses (see Sec. IV of SI) us-ing the AC or the spectrogram. From the AC, our ( [2])merit is 0.03 (0.09), and from the spectrogram is 0.010(0.019), both showing our method obtains better merits.Fig. 3(f) compares the retrieved vector potential of thetwo methods. On femtosecond timescales the agreementbetween the two retrieved vector potentials is quite good.Next we re-examine the SXR pulse generated by a two- FIG. 2: Experimental spectrograms and corresponding AC patterns. The two left columns are from [2] and the two rightcolumns are from [1]. The upper row data are from the experiments and the lower row data are from the reconstructedtheoretical data using the retrieved SXR and MIR pulses. Each pair along the column should match well if the retrieval isaccurate. cycle pulse near 1.8 µ m reported in [1]. Their spectrumextends from about 100 to 300 eV; the bandwidth is 94eV and the TL pulse is 20 as. This pulse was retrievedin [1] based on the PROOF method which approximatesthe SFA, Eq. (1), under the condition that the streak-ing MIR is monochromatic and that the vector potentialof the MIR is very weak. In this experiment, the targetatom for the streaking spectra is helium. Fig. 4 summa-rizes the retrieved results of [1] using PROOF and fromour PROBP-AC method. Fig. 4(c) shows the AC fromour retrieved result. It agrees better with the AC fromthe experimental data in [1] shown in Fig. 4(a) than theAC obtained from the retrieved SXR using PROOF (Fig.4(b)). The pulse duration retrieved from our method is61 as, while from PROOF it is 53 ± .
036 vs 0 .
061 basedon the ACs, or 0 .
012 vs 0 .
032 using the spectrograms (seesec IV of SI).We also retrieved the streaking data from [3]. The cen-tral photon energy of the SXR is at 250 eV, with a band-width of about 200 eV, corresponding to a TL pulse ofabout 10 as. Ionization of the SXR on Kr atoms wouldgenerate most of the photoelectrons from the 3 d . Fig.5(a) shows the experimental spectrogram. Due to theweak signal, the spectrogram does not show clean oscil-lation with respect to the optical period of the Mid-IRpulse. Since the spectrogram shows good oscillatory be-havior from 15 to 35 fs in Fig. 5(a), we calculate the ACfor this range, and the result is shown in Fig. 5(b). Us-ing this AC pattern, we retrieve the spectral phase. Theretrieved time-domain intensity profile is shown in Fig.5(c). It has a FWHM pulse duration of 165 as, abouthalf of the upper limit of 322 as reported in [3] based on FIG. 3: (Color online) (a)Experimental AC pattern and, (b) the retrieved AC pattern from [2]. (c) The retrieved AC patternfrom the present method. (d) The retrieved temporal intensity envelope, (e) the spectral phase of the SXR, and (f) the vectorpotential of the MIR. The retrieval is obtained from the AC pattern of block 5 in Fig. 2(c). In (d-f), the solid red lines arefrom the present retrieval method and the blue dashed lines are from [2]. the attosecond lighthouse model. The spectral phase ob-tained from our retrieval method is given in the inset ofFig. 5(c). Comparing the AC using the retrieved pulses,as shown in Fig. 5(d), it does show an overall globalagreement with Fig. 5(b), but clearly the details are dif-ferent. Due to the broader bandwidth and higher photonenergies, the weaker signals and longer time (about 10hrs) in collecting the spectrogram, it is not clear thatthe 165-as duration retrieved is the “shortest” pulse thatcan be obtained for this broadband pulse. Since broaderbandwidth would incur larger chirps in the spectral phaseaway from the center, a shorter TL pulse may not neces-sarily be a better way for generating a shorter attosecondpulse below 50 or 60 as.
II. DISCUSSION
One of the grand goals of ultrafast and attosecondphysics is to generate even shorter light pulses in the softX-ray region for probing inner-shell electron dynamicsof materials. Often it was assumed that one would justhave to keep generating continuum harmonics over everincreasing bandwidth. This would work if the spectralphase can be compensated over the whole broad energyregion. As addressed in [3], there is still no practicalmethod available to do that. In the meanwhile, accu-rate phase retrieval of a broadband pulse using the spec-trogram has been shown to be very slowly converging.In this work, we demonstrated that phase retrieved di-rectly from the autocorrelation (AC) of the spectrogram is more efficient and more accurate. We also demon-strated that correct phase is retrieved when the AC fromthe experimental data agrees with the AC from the re-trieved pulse. The PROBP-AC method is expected toprovide the metrology of broadband pulses in the future.
III. METHODS
The phase retrieval of broadband pulses (PROBP) wasfirst introduced in [26], where the unknown laser and/orSXR are to be retrieved from the spectrogram. it doesnot impose the central momentum approximation. InPROBP, the spectral amplitude U (Ω) of the SXR isknown from the experiment. The vector potential of thestreaking MIR field is also expressed in the energy do-main A (Ω) = f (Ω) e i Ψ(Ω) . (4)Each of the unknown functions Φ(Ω), f (Ω), and Ψ (Ω),respectively, is expanded in terms of B-spline basis func-tions f ( x ) = n (cid:88) i =1 g i B ki ( x ) . (5)With some guessed parameters of these unknown func-tions, the constructed SXR and MIR are used in Eq. (1) - 1 0 0 0 1 0 0 2 0 0 - 2 0 2 4- 2024 C ( d ) Time delay (fs)
T i m e d e l a y ( f s ) ( a ) - 2 0 2 4- 2024 ( e )( b ) - 2 0 2 4- 2024 C ( c ) Phase(radian)
R e f [ 1 ] 5 3 a s
P R O B P A C 6 1 a s
Intensity(arb.unit)
T i m e ( a s )
P h o t o n e n e r g y ( e V ) ( f )
A(t) (atomic unit)
T i m e ( f s )
FIG. 4: (Color online) (a) and (b). AC patterns from the experimental and the retrieved spectrograms reported in Ref. [1],respectively. (c) Reconstructed AC pattern from the present method. (d)Temporal intensity envelopes of the SXR. (e) Spectralphase of the SXR retrieved and (f) vector potential of the MIR retrieved from the present method. In (d) and (e), the red solidlines are from the present method, and the blue-dashed lines are from the PROOF results of Ref. [1]. to obtain the spectrogram. By comparing the resultingspectrogram with the experiment, a genetic algorithmwas used to select the new guesses for the next iteration.The iterative process is terminated after tens of thou-sands steps or after some preselected merit is reached.The PROBP method has been shown to work well forpulses with bandwidth up to about 100 eV [26]. Theconvergence becomes much slower for pulses with largerchirps or broader bandwidths. Since the AC appears tobe a more sensitive marker of the spectral phase than thespectrogram, in the PROBP-AC method [27], we retrievethe phase directly from the experimental AC patterns.In the iterative method, we apply the genetic algo-rithm. The fitness function is defined as the sum of E = (cid:88) i,j min ( Q ( i, j ) , Q ( i, j )) , (6)where Q and Q are the normalized input AC from theexperiment and the reconstructed AC, respectively. Themin ( x, y ) is defined as the smaller of x and y . If the inputand reconstructed ACs are exactly the same, the fitnessfunction is equal to 1. In the numerical computation wediscretize the spectrogram S ( E, τ ) and the AC pattern Q ( τ , τ ) on grid points. We use the genetic algorithm(GA) to find the optimal parameters that would minimizeEq. (6).For narrower bandwidth pulses, previous studies [27] show that the PROBP-AC method converges much fasterand more accurately. For the broadband pulses dis-cussed here, we use the PROBP-AC method only sincethe PROBP method does not converge or take a longtime to reach convergence. The amplitude and phase ofthe vector potential A of the MIR field are also obtained.The experimental spectrogram is first normalized withinthe same time delay domain. IV. ACKNOWLEDGEMENT
We thank Professors Hans Jakob Woerner, ZenghuChang and Jens Biegert for providing the digital spec-trograms from their experiments for us to analyze us-ing our phase retrieval algorithm and for input to ourinitial draft of the manuscript. We also wish to thankDr. P. D. Keathley for communicating some comparisonwith the result from the VTGPA method. This researchwas supported in part by the Chemical Sciences, Geo-sciences, and Biosciences Division, Office of Basic EnergySciences, Office of Science, US Department of Energy,under Grant No. DE-FG02-86ER13491. W.Y. wouldalso like to acknowledge partial support by the ChineseScholarship Council (CSC), and by the National NaturalScience Foundation of China under Grant No. 11604131. - 4 0 0 - 2 0 0 0 2 0 0 4 0 0 6 0 0 8 0 00 . 00 . 20 . 40 . 60 . 81 . 0 0 5 1 0 1 5 2 0 2 5 3 0 3 55 01 0 01 5 02 0 02 5 03 0 0
Energy(eV)
T i m e d e l a y ( f s ) ( a ) 1 5 2 0 2 5 3 0 3 51 52 02 53 03 5 ( b )
Time delay (fs) ( c )
Intensity(arb.unit)
T i m e ( a s )1 6 5 a s
Phase(radian)
P h o t o n e n e r g y ( e V )
FIG. 5: (Color online) (a) and (b). Experimental trace and the derived AC pattern from [3], respectively. (d) The AC obtainedfrom the present retrieved pulse. It is only in marginal agreement with the one in Fig. 5(b). (c) Envelope of the temporalintensity of the SXR; the inset is the spectral phase. Both are from the present retrieval method.[1] Li, J. et al.
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