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PHYSICAL REVIEW A 68, 041802 ~2003!

How to measure the duration of subfemtosecond xuv laser pulses using asymmetric photoionizatio

Andre D. Bandrauk,* Szczepan Chelkowski, and Nguyen Hong Shon†

Laboratoire de Chimie The´orique, Faculte´ des Sciences, Universite´ de Sherbrooke, Que´bec, Canada J1K 2R1~Received 7 January 2003; published 21 October 2003!

A technique for directly measuring the duration of subfemtosecond extreme ultraviolet~xuv! laser pulses isproposed. Using the numerical solutions of the time-dependent Schro¨dinger equation for a H atom in thecombined field of a near-infrared femtosecond and an xuv attosecond~asec! laser pulse~both linearly polar-ized!, we show that the photoelectrons are emitted preferentially in a forward~or backward! direction, alongthe laser polarization vector, depending on the delay between two pulses. Our calculations show that thenormalized maximal asymmetryP3(tasec) ~i.e., the signal difference measured by a forward and a backwarddetector divided by the total signal! exhibits a simple linear dependence on the durationtasecof the attosecondpulse thus allowing one to easily compute the attosecond-pulse duration from measured asymmetry.

DOI: 10.1103/PhysRevA.68.041802 PACS number~s!: 42.65.Re, 42.65.Ky, 32.80.Rm

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Shortly after the first experimental success on the gention of an isolated, soft extreme ultraviolet~xuv!, 650 asec~attasecond! pulse@1,2#, the first genuine application of thipulse for time-resolved attosecond spectroscopy wasported @3#. Attosecond physics has become a new reawhich promises unprecedented applications in different aof science and technology~see, for example Refs.@4–7#!.One of the most challenging problems in attosecond phyis the measurement of the asec-pulse durationtasec, forwhich the traditional autocorrelation technique~AT! fails.The first proposal for this measurement was based on theand consists in the following: a Ti:sapphire~800 nm! femto-second~fs! laser pulse is split into two parts~with control-lable delay!. Both are focused on Ar gas forsimultaneousgeneration and measurement of the created asec pulsefirst derivative of the harmonic energy~asec-pulse energy!with respect to the delay time between two fs pulses seas a measure oftasec@8#. The physics of this measurementstill controversial@9# and the method is not widely usedAnother interesting method for measuringtasec @10# relieson the measurement of thetotal ~integrated over all direc-tions! ionization yield as a function of delay timetdel be-tween the intense 800-nm fs and the xuv asec pulse.depth of the modulation of this function serves as a measfor tasec. This method is applicable when the photon eneof the asec pulse,\vasec, is smaller than the ionization potential I p of an atom; therefore for measuring xuv pulses oshould use atoms with very highI p such as He1 . The thirdmethod@1,2,11,12# is also based on the cross correlation btween visible fs and xuv asec pulses. It exploits the measment of the full width at half maximumDW of the photo-electron energy spectrum as a function oftdel . By fittingexperimental data with the calculated functionDW(tdel)which uses the fit parametertasec, one can determine thvalue oftasec with an accuracy;625% (6506150 asec).The method is applicable when\vasec@I p .

In this paper we propose another variant of the abcross-correlation techniques for directly measuring the dution of xuv asec pulse. We suggest to exploit the strong

*Electronic address: andre.bandrauk@USherbrooke.ca†Deceased.

1050-2947/2003/68~4!/041802~4!/$20.00 68 0418

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rectional asymmetry of photoionization that occurs in cobined fields of 800-nm fs and xuv~40–70 nm! asec pulses aswas described in our previous paper@7#. The shape of bothpulses is shown in Fig. 1~a!. The xuv photon energyvasec ischosen to be slightly higher than the ionization potentialI p

FIG. 1. ~a! Electric field of IR fs~dashed line! and of xuv asec~solid line! pulses (tdel52T/4). ~b! Function P2(tdel)5P22P1

for different asec-pulse durations: solid line with squares,tasec

50.7 fs; solid line with circles,tasec50.5 fs; dashed line, normalized vector potentialAl(t); and solid line with triangles,P1(tdel)5P11P2 for tasec50.7 fs. In all caseslasec570 nm andP6 areevaluated at the end of the laser pulse.

©2003 The American Physical Society02-1

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BANDRAUK, CHELKOWSKI, AND SHON PHYSICAL REVIEW A68, 041802 ~2003!

of an atom. Under the combined action of these two pulthe ionization signal significantly increases as comparedthe one produced by a single fs or asec pulse@7#. The pho-toelectron signals in forward (P1) and backward (P2) di-rections~along the polarization vector of laser pulses! exhibitstrong asymmetry~up to 90% electrons ionizing in one drection!. The difference P25P22P1 as a function of thedelay time between two pulses reproduces very well the vtor potentialAl(t) of the electric field in the IR pulse whichcan exploited for the measurement of the electric field offemtosecond pulse@7#. We suggest to use the time delaywhich the asymmetryP2(tdel) is maximum, e.g.,tdel52T/4 as shown in Fig. 1~a!, for the measurement of thdurationtasec of the attosecond pulse. Our numerical calclations show that thenormalized asymmetry P35P2 /P1,calculated at this maximum, falls rapidly and nearly lineaas function oftasec. We also derive this simple, strong, linedependence from a simple semiclassical model in whichelectron is instantaneously ionized via an absorption ofattosecond photon and next it moves in the electric fieldthe femtosecond laser field as a classical particle. This leus to propose an experimental method based on the meament of the normalized asymmetryP3 at time delay corre-sponding the maximal asymmetry, from which the attosond pulse durationtasec can be computed either fromnumerical simulations based on time-dependent Schro¨dingerequation~TDSE! or from simple semiclassical model. Thexperimental setup for our measurement technique shoulvery similar to the one described in experiments@1–3# andour method permits evaluation oftasec with high accuracy.

Our investigation is based on the numerical solutionthe exact three-dimensional TDSE for a H atom@13# in thefield of linearly polarized laser pulses~along thez axis!

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where me and e are the electron mass and charge,El andEasec are electric fields of IR fs and xuv attosecond pulsThe electric fields are given byEj (t)52]Aj /]t, where

Aj~ t !52A2I j

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f j~ t2t j !

v jsin@v j~ t2t j !1f j #, j 5 l , asec.

v j andI j are frequencies and intensities, andf j (t) andt j arefield envelopes and the peak-positions of the pulses, restively, in SI units. The ionization signals measured insidecone within a small solid angleu,u0 are defined as theelectron fluxes@P6(t)# passing the surfaces perpendicularthe z axis at6z0 @13#, where6z0 is the observation pointwhich has been chosen near the absorbing boundary.@P6 are

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dimensionless probabilities of electrons ionized into forwabackward directions. The real number of emitted electronproportional to the product ofP6 and atomic density in theinteraction volume.# In our calculations we use two lasepulses, both have the sin-square pulse envelopes. Thenm, 5-fs driver pulse has the peak intensityI l5431013 W/cm2 and the xuv asec pulse has the peak intsity I asec5131013 W/cm2. Unless specified, the angleu0515°. In Fig. 1~a! we plot the electric fields of IR fs and xuasec pulses and define the delay timetdel5tasec2tdel be-tween them. In Fig. 1~b! we plot the asymmetry of ionizationsignalsP25P22P1 as a function of the delay timetdel forthe case whenlasec570 nm (\vasec517.7 eV) for differ-ent asec-pulse durations. For comparison, we also plotnormalized vector potentialAl(t) of IR fs pulse and the totaionizationP15P11P2 . We note that the functionP2(tdel)oscillates at the optical periodT with large modulation am-plitude, and reproduces well the vector potentialAl(t),whereas the functionP1 oscillates atT/2 with much smallermodulation depth and large background. We also notethe modulation ofP1, as shown in Fig. 1~b! (\vasec.I p), ismuch weaker and badly distinguishable as compared tocase when\vasec,I p @7,10#. The modulations ofP2 aremuch closer to the electric-field oscillations than the modlations of the total probabilityP1, since by calculating thesignal differenceP25P22P1 one eliminates the background contained in the total signalP1. Thus by usingP2instead ofP1 we improve the time resolution of the fielmeasurements by a factor of 2 and we believe thatchoice also improves in the same way the measurementhe attosecond-pulse durationtasec proposed in this work.We have done a series calculations of functionsP2(tdel) forvarious values of\vasec.I p and for several values oattosecond-pulse durations 0.5 fs,tasec,1.2 fs, and wefound thatP2(tdel) has always a maximum at the sametdelasAl(t l1tdel), e.g., attdel52T/4, independent of the asecpulse duration. This interesting feature can also be obtaianalytically from the classical equation of motion, see beloWe have done similar calculations for larger angleu0530°and found that, though the absolute value ofP2 is reduced,this notable feature still remains. This leads to proposingmethod for the measurement of the attosecond pulse dtion, i.e., we propose to find experimentally the maximumthe asymmetry measureP2(tdel) and to normalize it by com-puting P35P2 /P1 at the maximum ofP2(tdel). Next, onecan obtain the asec-pulse duration from theory, e.g., frsimulations based on TDSE as presented here or from osimpler models. We believe that normalized asymmetryP3 ismore suitable for this purpose thanP2, since by dividing byP1 we eliminate the linear dependence of ionization proability on pulse durationtasec and intensityI asec, as ex-pected from one-photon absorption perturbative ionizatrates. Thus the obvious benefit of using the normalizasymmetryP3 ~instead of the usingP2) is that it maximallyreduces uncertainties caused by uncontrollable intensityasec pulse, see Fig. 2 which shows that forl570 nmP3 isnearly independent ofI asec, whereas for lower wavelengthl540 nm, P3 slightly rises as a function ofI asec. Besides,the variation ofP3 as a function oftasec is expected to be

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HOW TO MEASURE THE DURATION OF . . . PHYSICAL REVIEW A 68, 041802 ~2003!

larger, between zero~long pulses! and unity~shortest pulses!,than the variation ofP2 so that this will also improve theaccuracy of measurements. In Fig. 3 we plot the calculafunction P3(tasec) for different values of asec-pulse wavlengths, computed attdel52T/4 @i.e., at the maximum ofP2(tdel)]. The upper panel~a! is for the case\vasec.I p andthe lower panel~b! is for the case\vasec<I p and for\vasec

only slightly exceedingI p (l580 nm). We note that thenormalized asymmetryP3 is the highest for frequencies verclose toI p : for lasec570, 80, and 115 nm the asymmetcan be as high as;90% for tasec;400–500 asec, then idecreases monotonically as a function oftasec. We observethat for all displayed frequencies and attasec>T/4 this de-crease is almost linear. Most surprisingly, the slope of thcurves is scaled~with high accuracy! to the wavelength ofthe asec pulse~i.e., 70:60:50:40!. For lasec>80 nm(\vasec<15.5 eV), the functionP3(tasec) exhibits maxi-mum. We believe that this maximum is related to the huspectral width (\Dv53.6 eV) of the the 500-asec pulsand thus lower-energy photons contained in the pulse caionize via one-photon absorption; for 80 nm,\vasec

FIG. 2. Intensity dependence ofP3 for lasec570 and 40 nm(tasec50.6 fs), respectively

FIG. 3. The calculated functionP3(tasec) for different values ofasec-pulse wavelengths~a! \vasec.I p ; ~b! \vasec,I p ~or slightlylarger thanI p). lasec5115 nm, tdel520.375T, and for all othercasestdel52T/4. The vertical dashed line is located atT/4.

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515.5 eV, which only slightly exceedsI p513.6 eV.Clearly, asymmetriesP3 as functions oftasec show veryregular, nearly universal behavior suggesting a reliamethod for determining of the attosecond pulse durattasec from the measurement of the normalized asymmeP3.

For a better understanding of the results of our quantcalculations showing simple regular behavior, we constrand exploit a simple two-step semiclassical model, similathat used in Refs.@1,2,11,12,14#. In the first step we assumthat by an instantaneous absorption of an xuv photon, attime t0 , t0.tasec, and at the time delaytdel52T/4, forwhich P2 reaches a maximum, the electron is moved direcinto the continuum and starts to move as a classical parin the laser electric fieldEl ~we neglect the Coulomb forcewhich is justifiable for sufficiently high frequencyvasec andfor high laser intensityI l) according to Newton’s equation omotion

me

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whereeW z is a unitary vector along the laser polarization suposed to be parallel to thez axis (ueW zu51). We have ne-glected above the Coulomb force which is justifiable for sficiently high vasec and for high laser intensityI l . Equation~2! can be integrated analytically yielding

vW f5vW ~ t f !5vW 01vW d , vW d5e

meAl~ t0!eW z , ~3!

wherevW f is the final velocity at the final timet f at which thelaser is turned off, i.e.,Al(t f)50, andvW 05vW (t0) is the initialvelocity such thatuvW 0u5A2me(\v2I p) with v restricted tothe interval defined byuv2vasecu,3/2Dvasec, where weassumed the spectral width of a Gaussian attosec pDvasec54 ln(2)/tasec. For fixedt0 , v, and a fixed angleu8

betweenvW 0 and the laser polarizationeW z , the final electronvelocity vW f is determined via Eq.~3!, thus allowing us tomake some simple predictions for asymmetries in photoetron angular distributions. First, when there is no femtosond laser~i.e., whenAl50) the photoelectrons are emittesymmetrically alongeW z with angular distribution propor-tional to cos2(u8), as expected from perturbative calculatioof the photoeffect~see, e.g., Ref.@15#!. Similarly, for a timedelay between two pulsestdel such thatA(tasec)50, noasymmetry is expected from Eq.~3! whereas maximal asymmetry should occur whenuA(tasec)u is maximum, e.g. fortdel52T/4 shown in Fig. 1~a!. Clearly, a simple classicamodel based on Eq.~3! explains well the main features oasymmetries displayed in Fig. 1~b!. By assuming some statistical distribution of parameterst0 , v, and cos(u8) and bysimple counting of how many vectors fall within the solangle within 0,u,u0 or within p,u,p2u0, one can getsignalsP1 or P2 , respectively, and next computeP3 fortime delaytdel52T/4. We assumed thatt0 values from theinterval defined byut02tasecu,3/2tasec are simply distrib-uted according to the attosecond-pulse intensity time pro

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Similarly, we use intensity spectral profile for a distributioof v8 ~both assumed to be Gaussian!, and we weigh theangular distribution of the vectorvW 0 by a factor cos2(u8) @15#.The resultingP3 of such counting, performed for each valuof tp , using 200 points for each of the above three intervare plotted in Fig. 4 together with asymmetries obtainform simulations based on TDSE. The agreement is surp

FIG. 4. AsymmetriesP3 calculated from a classical modebased on Eq.~3!.

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ingly close between two calculations, in particular, for highvasec values where a classical model is expected to be mcorrect. In particular, classical model gives correct slope vues in all four cases fortp.T/4 . Thus Eq.~3! becomes veryuseful for estimating suitable values ofI p , frequencies, andlaser intensitiesI l for the measurement of the attoseconpulse duration, which should be chosen in a such wayuvW 0u and uvW du in Eq. ~3! are comparable.

Comparison of the normalized asymmetryP3, calculatedfrom the measured forward backward ionization signaltdel52T/4, with theoretical asymmetries can give the valof tasec with very high accuracy fortasec>T/5 ~50.51 fs,for a Ti: sapphire laser!. Thus the laser cycleT determinesthe lowest limit for the attosecond-pulse measurement. Tleads us to suggest one to use shorter wavelength thannm, for the measurement of pulses shorter than 0.5 fs.example, by using 200-nm lasers instead of 800-nm ocould extend the above lower limit down to 0.1 fs. So lopulse durations were not yet, to our knowledge, measuThus judging from Eq.~3! such experiment should use atomwith higher I p , e.g., helium atoms and wavelength 10–nm for the attosecond pulse and four times more inte200-nm laser thanI l5431013 W/cm2.

In summary, we have suggested a technique for diremeasuring the duration of subfemtosecond xuv pulse.have shown that the cross-correlation signal of IR fs andasec pulses exhibits a strong directional asymmetry alongpolarization vector for\vasec.I p , which can be exploitedfor the measurement of the attosecond-pulse duration.method requires a measurement of the total number of ptoelectrons, which is experimentally simpler than the meods based on measurement of electron kinetic-energy sp@1,2#. Besides, the method relies on a subfemtosecondpulse synchronized to the optical pulse. This tool is navailable @3#. We believe that this technique would begreat interest in the new attosecond science.

We thank F. Krausz and A. Sokolov for valuable discusions

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@1# M. Hentschelet al., Nature~London! 414, 509 ~2001!.@2# M. Drescher, M. Hentschel, R. Kienberger, G. Tempea,

Spielmann, G.A. Reider, P.B. Corkum, and F. Krausz, Scie291, 1923~2001!.

@3# R. Kienbergeret al., Science297, 1144~2002!.@4# F. Krausz, Opt. Photonics News13, 62 ~2002!; Phys. World

14, 41 ~2001!.@5# H. Niikura et al., Nature~London! 417, 917 ~2002!.@6# A.D. Bandrauk and Nguyen Hong Shon, Phys. Rev. A66,

031401~R! ~2002!.@7# A.D. Bandrauk, S. Chelkowski, and Nguyen Hong Shon, Ph

Rev. Lett.89, 283903~2002!.@8# N.A. Papadogiannis, B. Witzel, C. Kalpouzos, and

Charalambdis, Phys. Rev. Lett.83, 4289~1999!.@9# G. Tempea, A. Scrinzi, F. Krausz, and T. Brabec, Phys. R

Lett. 87, 109402~2002!; N.A. Papadogiannis, B. Witzel, C

.e

.

v.

Kalpouzos, and D. Charalambdis,ibid. 87, 109402 ~2001!;Phys. Rev. A66, 025803~2002!; P. Corkum, Nature~London!403, 845 ~2000!.

@10# A. Scrinzi, M. Geissler, and T. Brabec, Phys. Rev. Lett.86, 412~2001!.

@11# M. Kitzler, N. Milosevic, A. Scrinzi, F. Krausz, and T. BrabecPhys. Rev. Lett.88, 173904~2002!.

@12# J. Itatani, F. Quee´re, G.L. Yudin, M.Y. Ivanov, F. Krausz, andP.B. Corkum, Phys. Rev. Lett.88, 173903~2002!.

@13# S. Chelkowski and A.D. Bandrauk, Phys. Rev. A65, 061802~2002!.

@14# S. Chelkowski, M. Zamojski, and A.D. Bandrauk, Phys. Rev63, 023409~2001!.

@15# H.A. Bethe and E.S. Salpeter,Quantum Mechanics of One-anTwo-Electron Atoms~Springer-Verlag, Berlin, 1957!, Sec. 70.

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