High-order harmonic generation by chirped and self-guided femtosecond laser pulses.II. Time-frequency analysis

V. Tosa,1,2 H. T. Kim,1 I. J. Kim,1 and C. H. Nam11Department of Physics, KAIST, Yuseong-gu, Daejeon, 305-701 Korea

2National Institute for R&D of Isotopic and Molecular Technologies, Cluj-Napoca, RomaniasReceived 7 December 2004; published 16 June 2005d

We present a time-dependent analysis of high-order harmonics generated by a self-guided femtosecond laserpulse propagating through a long gas jet. A three-dimensional model is used to calculate the harmonic fieldsgenerated by laser pulses, which only differ by the sign of their initial chirp. The time-frequency distributionsof the single-atom dipole and harmonic field reveal the dynamics of harmonic generation in the cutoff. Atime-dependent phase-matching calculation was performed, taking into account the self-phase modulation ofthe laser field. Good phase matching holds for only few optical cycles, being dependent on the electrontrajectory. When the cutoff trajectory is phase matched, emitted harmonics are locked in phase and the emis-sion intensity is maximized.

DOI: 10.1103/PhysRevA.71.063808 PACS numberssd: 42.65.Ky, 32.80.Rm, 52.38.Hb

I. INTRODUCTION

High-order harmonicssHHd generation is one of the mostpromising ways of producing subfemtosecond coherent ra-diation in the soft-x-ray rangef1,2g. HH generation is alwaysa complex interplay between the single-atom response to alaser field often distorted by the propagation effects and theHH field propagation, which is affected by absorption andphase matching. One has to account for these factors in orderto find optimized conditions for generation. When the at-tosecond pulse is obtained by superposition of HH, the aimof the optimization is the phase lockingssynchronizationd ofthe emitted harmonics and the maximization of the HH flux.In this case, the emitted x-ray field will achieve high inten-sity and temporal coherence. As shown inf2g, the HH syn-chronization is the key to the optimization of attosecondpulse generation by superposition of HH. As a rule, the lowharmonics in the plateau are temporarily shifted in emissionwith respect to the following ones, while harmonics in thecutoff are fully phase locked. However, in the latter case theauthors off2g found a rapid decrease of the harmonic flux. Inthis paper we investigate the attosecond pulse formationwhen the laser pulse propagates in a self-guided configura-tion induced in a long gas jet. We show that in these condi-tions, the HH that survive in propagation are specificallythose in the cutoff. Unlike the HH generated in the cutoff ina short medium, in the long jet the generation optimizationcan be done and was experimentally achievedf3g in bothspace and frequency domains. We demonstrate here thatthese harmonics are locked in phase and that the emittedx-ray field is a train of few pulses, each having 180–220 asduration.

Systematic investigations have been conducted recently inour group in order to achieve generation optimizationf3–6g.The work recently reported inf3g maximized the harmonicgeneration volume by inducing a profile-flattened, self-guided regime of beam propagation in a long gas jet, andachieved the generation of spectrally sharp harmonics byadding an appropriate chirp to the laser frequency. The emis-

sion volume was maximized by placing the jet about oneRayleigh lengthszRd before the focus, while the brightnessoptimum was obtained when using negatively chirpedsNCdpulses. On the contrary, with positively chirpedsPCd pulses,the spectra were diffuse, although the energy contained inone harmonic did not decrease too much.

Intensive numerical modeling under these particular con-ditions was performed in order to understand the experimen-tal findings. The results are reported in the preceding articlef7g, hereafter referred to as I. We used a three-dimensionals3Dd model f8g for the propagation of both the driving fieldand the harmonic field. Our results confirmed the formationof the self-guided configuration for the laser beam, andqualitatively reproduced the spatial and spectral distributionof the generated HH. In particular, when using NC pulses,the distribution of the harmonic field was found to be spa-tially and spectrally confined, being generated compactlyaround the propagation axis with a sharp spectral profile. ThePC pulse generates a broad spectral distribution on axis, andsharper off axis, but in this case with higher spatial diver-gence. We found that the spatial and temporal modificationsproduced to the laser pulse during propagation are reflectedin the single-atom response and ultimately in the HH char-acteristics.

Solving the wave equation for the harmonic field propa-gation does not reveal the mechanism of HH formation dur-ing propagation. To understand the mechanism of harmonicfield buildup, a time-frequency analysis and a time-dependent phase-matching calculation have been performed.The time-frequency characteristics of the single-atom dipole,when compared to the corresponding distribution for the har-monic field, reveals the dynamics of harmonic generationand propagation. In particular, we clearly demonstrate thatonly specific trajectories, mainly those close to the cutofftrajectory, are enhanced in the propagation. An additionalphase-matching calculation helps us to evidence the roleplayed by different electron trajectories in the emission ofHH and to reveal the dynamics of the harmonic generationnear the cutoff. The time-frequency analysis combined with

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the time-dependent phase-matching investigation shows thatwhen the cutoff trajectory is phase matched, HH are emittedlocked in phase and emission intensity is maximized. Thecalculations show that in this case the harmonic emission is atrain of attosecond pulses, each having,200 as in full widthat half-maximum.

The paper is organized as follows. In Sec. II we brieflydescribe the numerical model used for calculating the funda-mental and harmonic field propagation. The details of thetime-dependent phase-matching calculation are also pre-sented. The results of the modeling are shown in Sec. III.Time-frequency distributions of the harmonic field are ana-lyzed in connection with the corresponding distributions ofthe single-atom response. The phase-matching distributionsfor the 61st harmonicsH61d are calculated at different timesduring harmonic emission. By corroborating these results weare able to coherently describe the harmonic generation withself-guided and chirped laser pulses. Conclusions are pre-sented in Sec. IV.

II. MODELING

HH generation requires the application of intense femto-second laser pulse to a high-density gas medium that is par-tially ionized during interaction. A certain degree of mediumionization is always present and is indeed necessary for HHgeneration. However, besides being a fundamental ingredientfor HH generation, ionization also poses serious limits byaltering the applied laser field. Theoretically, there are threedistinct aspects of the generation process:sid propagation ofthe driving field through the ionized medium,sii d single-atom radiation in which HH are generated and,siii d propa-gation of the HH through the medium. Any realistic model-ing of HH generation should give equal consideration tothese three aspects. Propagation through the medium inducesspatial, temporal, and spectral modifications to the originallaser field, and the single-atom response will be directly af-fected by these modifications. Finally, harmonic field propa-gation must be considered because of absorption and phase-matching effects.

The 3D model used by us was proposed inf9g, reviewedin I, and described in detail inf10g, so that we recall hereonly the main features. The evolution of the laser pulse in theionizing medium is calculated by numerically solving thewave equation Eq.s1d of I, for E1sr ,z,td, the axially sym-metric transverse electric field of frequencyv0. The effectiverefractive indexh of the medium includes dispersion, ab-sorption, electron plasma, and nonlinear Kerr contribution.The free electron concentration was calculated using theAmosov-Delone-Krainov formulaf11g. We recall that, to setthe initial conditions for the differential equation we used themeasuredenergy of the pulse, assuming a Gaussian beam inspace and time withmeasuredbeam waist and pulse dura-tion.

For the single-atom response, we used the strong-fieldapproximationsSFAd developed analytically by Lewensteinand co-workersf13g, as it proved to be able to model ourexperimental data. Solving the Schrödinger equationf12g ateach grid point of the interaction region could improve the

quality of the modeling but limits the number of runs as it ismuch more time consuming.

Having obtained the single-atom response, we numeri-cally solved the wave equation for the harmonic fieldEhsr ,z,td fsee Eq.s2d in Ig. In a typical calculation the fun-damental and harmonic field is defined in 256 points peroptical cycle, the number of optical cycles depending on itsdurationsfor example, we used 64 cycles for a 27 fs pulsed.The spatial grid built over the interaction region has a 150-point uniform mesh in the axial direction and a 200-pointnonuniform mesh in the radial direction. The nonuniformmesh solves the problem of rapid variations of the solution atthe periphery of the interaction region and allows consider-ing wide integration volumes in order to avoid reflectionsfrom its boundaries.

Detailed time-frequency dependence of the harmonic fieldon the atomic dipole were analyzed by using the short-timeFourier transformsSTFTd f14g, which yields the frequencycontent of a signal locally in time. It can be written as

Sst,vd =1

2pE

−`

+`

sst8dgst8 − tde−ivt8dt8, s1d

wheresstd is the time signal under inspection. The windowfunctiongstd is sliding along the signalsstd and for each shiftgst− t8d we compute the usual Fourier transform of the prod-uct functionsst8dgst− t8d. The square of the above distribu-tion senergy density spectrumd is called a spectrogramf14gand was calculated for both atomic dipole and harmonicfield. In the phase matching calculations, STFT distributionwas used to obtain the instantaneous frequency of the drivingfield as

vstd =E Sst,vdvdvE Sst,vddv . s2d

As the self-phase modulationsSPMd is nonuniform overthe interaction region, the phase-matching analysis must ac-count for time-dependent phase of theE1std driving field.The time-independent analysis, as originally proposed byBalcouet al. f15g, does not account for variations in time ofthe field phase as it assumes that this variation is uniformeverywhere in the interaction region. We extended theirmethod f15g to include the time-dependent phase, whicharises due to the initial chirp and SPM. For theqth harmonicwe calculated the polarization wave vector as

kpol = q ¹ ff1sr,z,tdg + ¹ faĪsr,z,tdg, s3d

where f1 is the fundamental field phase, anda the phasecoefficient f16g for the qth harmonic. This coefficient wascalculated for the long and short trajectories and also for thecutoff trajectories, as will be described in Sec. III. To obtainthe phasef1sr ,z,td of the fundamental, we assumed that thespatial and temporal parts can be estimated separately as

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f1sr,z,td = w8sr,zd + w9std. s4d

The spatial termw8sr ,zd is obtained from the solutionE1sr ,z,vd of the propagation equation, by using the methoddescribed inf10g. The time-dependent term, due to the chirpand SPM, can be calculated from the instantaneous fre-quency, as

w9std =E vst8ddt8. s5dIn this calculation, we first estimatedw8sr ,zd, which can beregarded as the phase at the end of the laser pulse. The time-dependent partw9std was then calculated and added over thespatial part. These two parts are not completely independent,however, they cannot be easily disentangled. Consideringthem additively seems not to be a bad approximation becausethe results of the time-dependent phase-matching analysisare in very good agreement with other results in the paper.

In Eq. s3d, Īsr ,z,td was calculated from the propagatedsolution E1sr ,z,td by averaging over one optical cyclearound t. Following Balcou f15g, we define a coherencelength Lcoh=p /dk where dk=2pq/l− ukpolu, which will benow a function of space and time coordinates. We want to

stress again thatĪsr ,zd and f1sr ,zd are here obtained fromthe calculated solution of the propagation equation,not fromthe unperturbed fundamental field. The advantage of thisprocedure is that, as all the nongeometrical effectssdisper-sion, absorption, ionizationd are already included inLcoh, oneneeds not considering them separately, as is done in general.

III. RESULTS AND DISCUSSION

A. Time-frequency analysis

The propagation of the laser pulse was modeled usingparameters that mimic our optimized experimental condi-tions as closely as possible. As described inf3g, the optimumharmonic emission was obtained by placing the long gas jetat a distancez0

tion on generation dynamics, especially for the NC case.In contrast to the results plotted in Fig. 2, the time-

frequency distribution of the dipole generated after 9 mm ofpropagation are very different. In Fig. 3 we have plotted, forboth the PCsad and NCsbd pulse, the time-frequency distri-bution of the atomic dipole as it develops atz=9 mm andr =0. One can see the shift towards earlier times and short-ening in time, following the driving field plotted in Fig. 1. Inboth cases the dipole radiates mainly in the range fromt=−10 to t=−2 optical cycles, that is, in the rising ionizationfront, when both pulses have a positive chirp. The value ofthis chirp will determine the time-frequency distribution ofthe dipole. Indeed, in this time range the chirp is 2–4 timeshigher for the PC pulse compared to the NC pulse. As aresult, the single-atom response is weaker in intensity andbroader in frequency for the PC pulse, while for the NCpulse the distribution is more confined in frequency, and hasa stronger intensity. Despite the distortion of the time-

frequency distribution, we can still identify the short andlong trajectories in Fig. 3, if we take into account the factthat the short trajectory has a smaller intrinsic chirp, thus, itstime variation almost follows the time dependence of theinstantaneous laser frequency. We confirmed this was correctby recalculating the dipole with the contributions only fromshort trajectories, excluding the contributions from the longtrajectories.

We solved the wave equation for the harmonic field, andobtainedEhsr ,zd, defined in the grid points over the interac-tion region. Shown in Fig. 3 are the time-frequency distribu-tions ofEh at z=9 mm andr =0, for PCsad and NCsbd cases,respectively. The harmonic field and dipole distributions donot completely overlap, which shows beautifully the selec-tion of specific trajectories as a result of harmonic fieldpropagation. Some trajectories are enhanced even at low-intensity dipole radiation; other trajectories, althoughstrongly present in the single-atom response, do not survivein the propagation. Both fields have a maximum intensitydistributed mainly at the confluence between the long and theshort trajectories, aroundt=−4 for NC and t=−5 for PCpulse. We note that the maximum H61 field is born at low-intensity dipole radiation, at the minimum necessary laserintensity to generate the single dipole for H61; this clearlyindicates the generation in the cutoff regime.

It is well known that for a harmonic in the plateau, thereare swithin one optical cycled two trajectories that yield thesame kinetic energy, and which therefore contribute to thesame harmonicfsee the inset of Fig. 7sadg. With increasingharmonic order, thus approaching the cutoff, the two trajec-tories become closer and, for the maximum achievable ki-netic energy, they coincide. As shown inf20g, the two trajec-tories do not contribute equally to the total emission;moreover; there are more trajectoriesswith larger excursiontimesd that become important, especially for the harmonics inthe plateau. These trajectories are not included in our mod-eling, as the integration time for the dipole moment was only1.2 optical cycles. This is reasonable because the harmonicsin study are very close to the cutoff.

Bearing in mind the above consideration, we can nowanalyze the data in Fig. 3. One can see that the final har-monic field is built up from the contributions of both shortand long trajectories. However, in the NC case, the harmonicfield builds up mainly on the short trajectory pattern of thesingle-atom response, despite the fact that SFA tends to over-estimatef20g the long trajectory contribution. The result is a

FIG. 2. sColor onlined Spectro-grams of the atomic dipolesloga-rithmic scale blue: red=1:10d, asgenerated by the unperturbed PCsad and NC sbd pulse atz=0 mmand r =0. The instantaneous fre-quency of the corresponding laserfields is also shown by the dashedline.

FIG. 3. sColor onlined Spectrograms of the atomic dipolesloga-rithmic scale blue: red=1:10d and harmonic fieldscontour plot,logarithmic scale 1:10d at r =0 andz=9 mm, as generated by the PCsad and NCsbd pulse. Black/white dots help tracing of the long/shorttrajectories.

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frequency profile even narrower than that of the correspond-ing single dipole. The emission spans in time over 10 opticalcycles, but in a narrow frequency range, which generates ahigh spectral brightness. In the case of PC pulse, harmonicsare also built from both short and long trajectories; however,the final field spans over two harmonic orders in frequency,which practically means overlapping between two adjacentorders, in full agreement with the experimental findingsf3g.

Using the instantaneous frequency data we can estimatethe average blueshift of the fundamental field for the timeinterval corresponding to the atomic dipole emission;namely, from aboutt=−10 tot=−2 optical cycles. We obtainon average 1.5%v0 and 3.1%v0 for NC and PC pulses,respectively. Comparing these values with the spectral spreadof the atomic dipoles0.9v0 and 1.9v0, respectivelyd, we con-clude that the chirp of the driving laser pulse is transferred tothe qth component of the dipole asq times the fundamentalchirp. However, the driving field chirp isnot directly trans-ferred to theqth harmonic, contrary to what was reportedrecentlyf21,22g. As we see from Fig. 3, this is because in along target the phase matching plays a very selective role inthe HH generation process, by eliminating some trajectoriesand enhancing others. Inf21g, the observation of sharp har-monic peaks for relatively low-intensity PC pulses supportthe assumption of dominant single-atom effects, in agree-ment with the transfer ofq times the chirp from the drivingfield to theqth harmonic, which was also observed.

We can conclude that the time-frequency analysis showsthe tight causal connections between the driving field and thesingle-atom dipole, and between the latter and the harmonicfield characteristics. The dipole radiates in a very restrictedtime range during laser pulse; namely, during the rising ion-ization front. This was shown to be specific for the cutoff HHf23g. We also demonstrated that the harmonic field is formedby selecting specific trajectories at specific time momentsfrom the dipole response. For the understanding of this tra-jectory selection, a phase-matching analysis was proved tobe necessary.

B. Phase-matching analysis

The initial chirp of the driving field combined with theplasma-induced SPM plays an important role in determiningthe single-atom response; thus, one must also account forthese effects in the phase-matching analysis. A first benefit ofsuch an analysis would be to determine the way they influ-ence the final harmonic field: whether the consideration ofonly the single-atom response is enough to explain the har-monic output or whether the phase-matching process shouldbe considered as well. Second, from the analysis we wouldbe able to determine the relative importance of these twofactors and their specific action in building the harmonicfield.

The dipole emission for a given harmonic, as shown inFig. 3, takes place in a very limited time interval during thepulse evolution. The condition of the laser field during thistime interval, such as its amplitude and phase, directly influ-ences atomic dipole amplitude and phase. However, we haveseen in the preceding section that only specific dipole phase

components are enhanced by the propagation and give thefinal harmonic field. Therefore, one must analyze the phase-matching condition over the time range where dipole emis-sion is significant. We should point out that the time-dependent phase matching calculation proved to be essentialfor the analysis of harmonic field generation.

The phase-matching status of HH generation can be wellrepresented by the coherence lengthLcoh. The calculationof Lcoh for different trajectories is as follows. We calculatethe time-frequency distribution of the dipole generated bya chirpfree 42 fs pulse, and identify the short and longtrajectories. From these calculations we derivedas=4 andal =22310

14 cm2/W for the short and long trajectories, re-spectively. These values are well known and agree well withsimilar calculationsf19g. In the cutoff region we assumedthat the dipole phaseF can still be approximated by thelinear relationF=−aI and, usingdv=]F /]t=−a]I /]t, weestimated a valueac=13310

14 cm2/W, for the trajectoriesclose to the cutoff one. These values were used in phase-matching calculations, as will be shown further on. However,we should keep in mind that this value ofac is not a con-stant, but depends on the laser intensity and harmonic order,and hence it can only be used in a limited range of laserintensitiess,3.531014 W/cm2 in our cased.

We calculatedLcoh distributions for different time instantsin the range fromt=−10 to t= +2 optical cycles. This is thetime measured in the moving frame and referenced to thecenter of the unperturbed laser pulse. Plotted in Figs. 4–6are the values ofLcoh for the short, long, and cutoff trajecto-ries, respectively, and fromt=−6 to t=−1 optical cycles. Fort,−6 we did not observe significant changes in theLcohmap, while forz=9 mm andt.−1 the laser intensity is be-low the cutoff intensity for H61. In general, the obtainedLcohmaps are similar for the PC and NC pulses, in all three cases.Nevertheless, the differences can be singled out and assignedto specific characteristics of the harmonic field. The calcula-tion in Figs. 4–6 was performed for a 15 mm long jet; how-ever, we have to keep in mind that we are interested in onlythe first 9 mm of propagation.

The short trajectory in Fig. 4 develops a good on-axisphase matching fort ranging from −5 to −1, with maximumof Lcoh at t=−1. For t.−1, Lcoh is increasing on axis;however, the case is of no further interest because of lowlaser intensity. TheLcoh maximum is betweenz=6 mm andz=9 mm, so that this good phase matching will give anon-axis contribution to the harmonic field. The coherencelength is clearly higher for NC pulses, which indicates thatshort trajectories are better phase matched with NC pulses.The off-axis region of good phase matching, developing atr

However, the analysis of the phase matching created bythe two trajectories, either short or long, is only indicativefor the characterization of the generation process in our case.As we can see from Fig. 3, the main harmonic field developsat the confluence between the short and long trajectories. Infact, all trajectories surviving in the propagation are closesinterms of birth and recollision moment of the electrond to thecutoff one. The calculatedLcoh for cutoff trajectories is plot-ted in Fig. 6. First, one must note that in this case the differ-ences between the maps generated with PC and NC pulsesare still observable, but are smaller than the differences ob-tained for long and short trajectories. If we start our analysisfrom the PC case in Fig. 6, we find that there is a good

on-axis phase matching starting att=−5 aroundz=9 mm. Att=−4 the maximum ofLcoh is clearly localized between 6and 9 mm for both PC and NC case. Further on, att=−3, HHare generated under the phase-matching condition, becauseLcoh is higher than the propagation length. This takes placefor both PC and NC pulse, between 5 and 9 mm and for aregion extending radially up to 40mm. For t=−2 andt=−1the good phase matching is maintained, but only off axisaround 30mm, with a slightly largerLcoh for the PC pulse.

At this point we should recall that, solving the propaga-tion equation for the harmonic field, we obtained in I theradial and spectral distribution of the harmonic field afterz=9 mm of propagation. In particular, when using NC pulses,

FIG. 4. Coherence length distributions for H61, corresponding to the shortsSd trajectory, for positivelysPd and negativelysNd chirpedpulses. Time in optical cycles is referred to the unperturbed laser pulse peak.

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each harmonic around H61 was generated as spectrally sharpand spatially compact around the propagation axis. An initialpositive chirp in the fundamental pulse generates a broadspectral distribution on axis, and sharper off axis. We arenow in a position to assume that the good phase-matchingevent, taking place fromt=−5 to t=−3, is responsible for thegood emission close to the propagation axis, which was ob-served in experimentf3g and reproduced in modeling re-ported in I for both PC and NC pulses. Experimentally, wefound that the energy contained in one harmonic was com-parable for both pulses. The chirp of the PC and NC pulses,within this range of time, produces the difference in spectraldistributions. We can see from Fig. 1 that for the NC pulse

the chirp is positive but small, vanishing aroundt=−3, whilethe PC pulse still has a large positive chirp. The PC pulseyields a narrow spectral width field at off axis, as seen in I.This narrow spectral width is the combined effect of twocauses: the off-axis generationswhere the SPM is smallerthan on axisd and the later generation in timesat t.−2 wherethe positive chirp is decreasingd.

We also see from Fig. 3 that the harmonic field extends atlater times, notwithstanding that the corresponding dipole isextremely weak. This is due to the very good phase matchingof the short trajectoryssee Fig. 4d for t.−1, which addsconstructively thez,9 mm contributions of the single di-pole. The time-frequency distribution of these contributions

FIG. 5. Coherence length distributions for H61, corresponding to the longsLd trajectory, for positivelysPd and negativelysNd chirpedpulses.

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shows that, with decreasingz, they are more extended andless shifted in time, giving a sizable contribution fort.−1.

C. Harmonic synchronization in the cutoff

The time-frequency distribution in Fig. 3 reveals the fre-quency details of the harmonic field, and for this purpose weused a large temporal windows4 optical cyclesd. To see thedetails in time domain we recalculated the distribution with anarrow s0.125 optical cyclesd time window. The frequencyresolution is lost, but the temporal resolution relevant for theharmonic emission is recovered. Doing this for every har-monic q, i.e., calculating the STFT distribution of the field

Ehsr =0,z=9,vd for sq−1dv0,v, sq+1dv0, we obtainedthe necessary time resolution so as to estimate the moment ofmaximal emission for every harmonic, at each half opticalcycle. The results are given in Fig. 7 for the NC pulse, andfor the HH between H55 and H65, which are the most in-tense in both the calculated and the experimental spectraf3g.We obtained similar results for the PC pulse. The locking intime is slightly better and the total harmonic pulse is nar-rower in time, in agreement with the spectrally broad HHobserved in this casef3g.

The reading of these results can be easily done if we havein mind the electron classical dynamics during the HH gen-eration processfsee the inset of Fig. 7sadg. During each op-

FIG. 6. Coherence length distributions for H61 corresponding to the cutoffsCd trajectory, for positivelysPd and negativelysNd chirpedpulses.

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tical half-cycle, a harmonic can be emitted two times,namely, when the short or long trajectory closes and electronrecombines with the parent atom. The two recombinationmoments become closer to each other as the harmonic orderincreases and approaches the cutoff order. In other words,along the short trajectory, the later the recombination mo-ment, the higher the kinetic energy gained, thus, higher har-monics are created later. Long trajectories generate higherharmonics when electron recombines at earlier time, thus,higher harmonics are created sooner. This enables us to pre-cisely describe the generation dynamics during pulse evolu-tion. Analyzing the earlier interval of emissionslaser cycleswith t,−5.5d, we observed that higher harmonics are gener-ated before lower harmonics, thus they are generated fromlong paths. This is in agreement with the phase-matchingmap in Fig. 5, which indicates a good phase matching forlong trajectories at earlier times. The harmonics from H55 toH65 are almost fully locked in time for the time rangearound −5.5, which is also close to the maximum emissionintensity. For later times, the emission order for the HH isreversed: lower harmonics are generated first, which meansthat the harmonics are generated along the short path, butwith lower intensity. This is again in agreement with thephase-matching maps, as for the cutoff paths the good phasematching is evidenced in Fig. 6 for −5, t,−2, while for theshort pathsLcoh in Fig. 4 is increasing for later times.

The sum of time-frequency distributions of the harmonicsfrom H55 to H65, plotted in Fig. 7sbd, show that the total

emission field is a train of attosecond pulses. These data,correlated to that in Fig. 6, demonstrate that the HH aroundH61 are generated mainly from cutoff trajectories, whichprovide the maximum contribution to the emission intensi-ties. Long trajectories have a sizable contribution at earliertimes, while short trajectories have a smaller contribution.The long trajectories have the phase matching correlated intime with the maximum of the laser intensity, both takingplace att.−5, which explains their larger contribution to thetotal emission. At later times, when the short trajectorieshave their best phase matching, the laser intensity is de-creased to values which are below the cutoff value for H61,which means a smaller contribution to the emission.

The total intensity ofEhsr ,z=9,td for the NC case, ob-tained through an inverse Fourier transform of the solutionEhsr ,z=9,vd, is shown in Fig. 8 forr =0 sad and integratedfrom r =0 to r =25 mm sbd. The r =0 data are similar to thatin Fig. 7, and the emitted attosecond pulses atr =0 are wellsynchronized to the corresponding pulses of the integratedfield. Good synchronization in the radial direction is a con-sequence of homogeneous generation conditions that exist inthe self-guided region for the NC pulse. For the same reason,in this r range, H61 is emitted with almost the same intensity,as is shown in Fig. 7 of I. The maximum harmonic field atr =0 is emitted att=−5, but for r .0, this maximum shiftstowards later times, due to the corresponding shift of thedriving field. This shift changes the intensity distribution of

FIG. 7. sad Emission time of the harmonics from H55 to H65 forthe NC pulse. The inset shows the cutoff electron trajectory in aframe oscillating at the field frequencysKramers Hennebergerframed in which the electron moves in straight lines. The two ar-rows show the direction of increasing harmonic order along theshortsSd and longsLd trajectories.sbd The sum of the STFT modu-lus of the same harmonics.

FIG. 8. Time dependence of the harmonic field, for the NCpulse, at the exit from the interaction region, forr =0 sad and inte-grated up tor =25 mm sbd. It was calculated as the inverse Fouriertransform ofEhsvd, where 54v0,v,66v0.

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the pulses in Fig. 8sbd, enhancing the pulse intensities at latertimes. From the data in Fig. 8sbd, we can estimate the dura-tion of the emitted attosecond pulses, as ranging from180 to 225 as. The duration of the total signal, includingcontributions from largerr will be probably longer, however,the good synchronization will keep the attosecond pulse du-ration within the same range.

IV. CONCLUSIONS

We presented an analysis of HH generation when chirpedpulses are propagated in a self-guided configuration througha long gas jet. The time-frequency distributions of the single-atom response and of the harmonic field demonstrate thatonly the cutoff trajectories survive to build up the final har-monic field. A time-dependent phase-matching calculation,taking into account the chirp and self-phase modulation ofthe driving field, was performed. Phase-matching distribu-tions for different time instants during dipole emission wereobtained. For the cutoff trajectories, phase-matched HH are

generated by only a few optical cycles during single-atomemission.

In a self-guided configuration the harmonic generation isaccompanied by an appreciable degree of ionization, whichaffects the process in several ways. However, the decrease ofthe initial laser intensity can be controlled to some extent byselecting the atom, or by adjusting the pulse duration and/orpressure. The generation volume can be controlled and maxi-mized by changing the jet position with respect to the focus,as was experimentally demonstratedf3g. We found that asmall number of harmonics, those which have the singleatom response in the plateau but close to the cutoff, prove tohave favorable phase-matching conditions, and are finallygenerated from cutoff trajectories, with high spectral bright-nesssby adjusting the initial chirpd and good temporal phaselocking.

ACKNOWLEDGMENT

The research was supported by the Korea Science andEngineering FoundationsKOSEFd through the Creative Re-search Initiative Program.

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