VOLUME 89, NUMBER 28 P H Y S I C A L R E V I E W L E T T E R S 31 DECEMBER 2002

Measuring the Electric Field of Few-Cycle Laser Pulses by Attosecond Cross Correlation

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

Laboratoire de Chimie Theorique, Faculte des Sciences, Universite de Sherbrooke, Quebec, J1K 2R1, Canada(Received 5 August 2002; published 31 December 2002)

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A new technique for directly measuring the electric field of linearly polarized few-cycle laser pulsesis proposed. Based on the solution of the time-dependent Schrodinger equation (TDSE) for an H atomin the combined field of infrared (IR) femtosecond (fs) and ultraviolet (UV) attosecond (as) laser pulseswe show that, as a function of the time delay between two pulses, the difference (or equivalently,asymmetry) of photoelectron signals in opposite directions (along the polarization vector of laserpulses) reproduces very well the profile of the electric field (or vector potential) in the IR pulse. Suchionization asymmetry can be used for directly measuring the carrier-envelope phase difference (i.e.,the relative phase of the carrier frequency with respect to the pulse envelope) of the IR fs laser pulse.

DOI: 10.1103/PhysRevLett.89.283903 PACS numbers: 42.65.Re, 32.80.Rm, 42.65.Ky

[16]. Thus using two laser pulses: an IR fs and an UV as E�t� � El�t� � Eas�t�; (1)

Recently, the first experimental results on the produc-tion and measurement of a single, soft x-ray, 650 as pulse,were reported [1–4] using high-order harmonic genera-tion (HHG). Such as pulses can be focused to achieve theintensities that are comparable to the conventionalTi:sapphire lasers, i.e., Ias � 1014 W=cm2 [5,6]. Thisopens up new possibilities for studying and controllingelectron dynamics in atoms and molecules on as timescales [7–10]. Few-cycle intense laser pulses in the 800-nm region are now routinely available [11]. The carrier-envelope phase difference of such pulses �l (defined asthe relative phase of the carrier frequency with respect tothe pulse envelope) becomes a physical important quan-tity as it affects significantly most nonlinear phenomenain laser-atom interactions, e.g., HHG [11], angular distri-bution of photoelectrons [12–14], etc. For measuring andstabilizing the carrier-envelope phase difference in afew-cycle laser pulse different schemes have been pro-posed [11–15]. In particular, the possibility of measuringthe carrier-envelope phase difference of circularly polar-ized laser pulses, based on asymmetric angular distribu-tions of photoelectrons, was theoretically investigated in[12] and experimentally demonstrated in [14]. In thistechnique atoms are ionized by a short laser pulse, andthe photoelectrons are recorded with two oppos-ing detectors in a plane perpendicular to the laser beam.The anticorrelation in the analysis of the electron yieldserves as a measure for determination of the carrier-envelope phase difference. However, these techniquescannot measure the evolution of the electric field.Furthermore, the photoelectron yield produced by asingle optical laser pulse is small, which will affect theaccuracy of measurements.

In this Letter, we propose an efficient technique fordirectly measuring the electric field, and consequently,the carrier-envelope phase difference �l of a linearlypolarized IR pulse. The physics of the technique is basedon the directional asymmetry of photoionization as itoccurs in two-color (!� 2!) IR ionization schemes

0031-9007=02=89(28)=283903(4)$20.00

pulses one ionizes an atomic (or molecular) gas. The UVphoton energy h!as is chosen to be close to the ionizationpotential Wp. It will be shown that under the action ofcombined IR fs and UVas pulses the ionization probabil-ity significantly increases (by 1� 3 orders as comparedto the one produced by a single fs or as pulse). Plottingthe difference of photoelectron numbers in opposite di-rections (along the polarization vector of the laser field)allows for measuring the asymmetry as a function ofa time delay of the as pulse with respect to the IR fspulse. For optimum as pulse duration as � T=4 [17](T � 2�=!l) this asymmetry is modulated at the periodT of the IR laser pulse and reproduces very well theprofile of its electric field. By simple fitting the calcu-lated data with free parameters Il and �l (Il is the maxi-mum intensity of the laser pulse), we can determine thecarrier-envelope phase difference of IR fs pulse �l withan accuracy �99%. We note that the attosecond crosscorrelation technique was, first, proposed for measuringthe as pulse duration in Ref. [18] and was developed invarious contexts [1,2,8,9]. The main differences betweenour technique and previous ones are as follows: (i) inRefs. [1,2,8,9] h!as � Wp while in our technique h!as �Wp; (ii) we measure the difference of ionization proba-bilities in forward and backward directions (within asmall solid angle �) while in [18] the integrated (overall directions) ionization yield is used. As a function ofthe time delay between the two pulses, the integratedprobability modulates at T=2 with a noticeable back-ground, while in our case the difference oscillates at anoptical period T [it also automatically eliminates a back-ground produced by the above-threshold ionization (ATI)along the polarization vector of the pulses]. Therefore, formeasuring as, our technique has better time resolution(by a factor 2).

Our work is based on the numerical solution of theexact 3D TDSE for an H atom [13] in the field of linearlypolarized laser pulses

2002 The American Physical Society 283903-1

-6 -4 -2 0 2 4 61E-6

1E-5

1E-4

1E-3

0.01

0.1

1

-2 -1 0 1 2-0.04-0.03-0.02-0.010.000.010.020.030.04 τ

del

Ele

ctric

field

[a.u

.]

Time [laser cycle]

fs pulse

asec pulse

70 nm

115 nm

(P++

P −)

Time [laser cycle]

FIG. 1. Total ionization P1 � P� � P� in the combined fieldof two laser pulses (solid line: �as � 115 nm, dashed line:�as � 70 nm). The dotted lines show P1�t� produced by single800 nm 5 fs, and 115 nm 600 as pulses alone. The inset showselectric fields of the two pulses (del � �T=2).

VOLUME 89, NUMBER 28 P H Y S I C A L R E V I E W L E T T E R S 31 DECEMBER 2002

where El and Eas are the electric fields of IR fs and UVaspulses (we assume both laser pulses are polarized alongthe z axis). The electric fields are given by Ej�t� ���@Aj=@t�, where

Aj�t� � �1

!j

��������2Ij"0c

sfj�t� tj� sin�!j�t� tj� ��j; (2)

j � l, as; "0 is the permittivity in vacuum; c is the speedof light, !j, �j, and Ij are frequencies, carrier-envelopephases, and intensities, respectively, and fj�t� and tj arefield envelopes and the peak positions of pulses, respec-tively. In this Letter we are interested in �l, and forsimplicity, we put �as � 0. Note that, in the ultrashortpulse limit, the definition of electric fields via vectorpotentials as in Eq. (2) is accurate as it ensures thecondition

R1�1 E�t�dt � 0 (the transferred momentum

from laser pulses to the electron is zero). In the notation(2) the delay time of an as pulse with respect to the IR fspulse is del � tas � tl. The ionization probabilities mea-sured inside the cone within a small solid angle � aredefined as the electron fluxes P� passing the surfacesperpendicular to the z axis at �z0 [13], where �z0 isthe observation points, which has been chosen near theabsorbing boundary. (P� are dimensionless probability ofelectrons ionized into forward, backward directions. Thereal number of emitted electrons is equal to the product ofP� and the number of atoms in the interaction volume). Inour calculations we use two laser pulses; both have thesin-square pulse envelopes [20]. The 800 nm, 5 fs driverpulse has the peak intensity Il � 4� 1013 W=cm2 and theUV as pulse has the peak intensity Ias � 1�1013 W=cm2, respectively. Unless specified, the durationof the UV pulse is 600 as and the angle � � 15 .

In Fig. 1 we plot the time evolution of the total ioniza-tion probability P1 � P� � P� in the combined field oftwo laser pulses. Solid and dashed lines correspond to thecases �as � 115 nm and 70 nm, respectively. For compar-ison we also plot P1�t� produced by IR fs and by 115 nmUV as pulse alone (dotted lines). At the end of the pulse,these single pulse probabilities (1:2� 10�5 and 6� 10�5,respectively) still are more than 2 and 1 orders of magni-tude smaller than the combined pulse results. The mostsurprising and unexpected results shown in Fig. 1 are thatthe ionization under the action of combined IR and UVfields is drastically enhanced as compared to the onesproduced by single pulses. We have calculated P1 andP2 � P� � P� (at the end of the laser pulse) as a functionof UV-pulse wavelength and have found that the enhance-ment exhibits clear resonant behavior at �as � 115 nm,which is close to the 1s-2p H atom transition.

Next, we consider the asymmetry in the regime whenthe UV photon energy is close to the 1s-2p transition andis smaller than Wp (�as � 115 nm, h!as � 10:78 eV). InFig. 2(a) we plot the difference of ionization probabilitiesP2 � P� � P� as a function of time delay del (solid linewith circles, P� are evaluated at the end of the laser

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pulse). For comparison, we also plot the normalizedelectric field of IR fs pulse (�l � 0, dashed lines) andtotal ionization function P1�del� (solid line with squares).As can be seen, the difference oscillates at the opticalperiod T with large modulation amplitude, and reprodu-ces well the profile of the electric field El, whereas thetotal ionization oscillates at T=2 with much smallermodulation depth and large background. The differentresults for P1 and P2 can be explained once the differenceP2 eliminates backgrounds and singles out the asymmetryof ionization while the total probability P1 superposes thebackground and asymmetry. In general, the functionP2�del� is shifted [as compared to the input field El�t�].We have made calculations with different carrier-envelope phase differences �l [see Fig. 2(b)] and havefound that, for fixed parameters of UV pulses, the shiftof the function P2�del� with respect to El�t� remainsnearly unchanged ( ’ 0:388T with an error �1%). Forexample, the curve corresponding to �l � �=2 is shiftedexactly by T=4 with respect to the curve corresponding to�l � 0. We have also considered the asymmetry in theregime when the UV photon energy is slightly larger thanWp (�as � 70 nm, h!as � 17:7 eV). In Fig. 3 we plot thefunction P2�del� (solid line with circles). For comparisonwe also plot the electric field El�t� (dash-dotted line) andvector potential Al�t� (dashed line). The most surprisingand important observation in this regime is that theasymmetry reproduces, with very high accuracy, thevector potential Al�t� rather than the electric field El�t�.The shift between function P2�del� and El�t� and Al�t� are’ 0:277T and �0:02T, respectively. We will show belowthat in the limiting case, when h!as � Wp, the asymme-try is indeed proportional to Al�t�, so the shift as com-pared to the El is 0.25 T. Thus when UV photon energy ishigh the asymmetry directly measures the vector poten-tial Al�t�, and by simple derivation one can get the electricfield El�t�.

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-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

P −−P

+

τdel

[laser cyc.]

FIG. 3. �as � 70 nm. P2�del� (solid line with circles). Dashedand dash-dotted lines show the normalized vector potentialAl�t� and electric field El�t� (�l � 0), respectively.

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-0.010

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0.010

0.015(b)

P −−P

+

τdel

[laser cyc.]

φl=0

φl=π/4

φl=π/2

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-0.010

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φl=0

P1,P

2

FIG. 2. �as � 115 nm. (a) P2�del� (solid line with circles)and P1�del� (solid line with squares). Dashed lines show thenormalized electric field of the IR fs pulse (�l � 0).(b) P2�del� with different �l.

VOLUME 89, NUMBER 28 P H Y S I C A L R E V I E W L E T T E R S 31 DECEMBER 2002

For a better understanding of our quantum calculationresults we construct and exploit a simple two-step semi-classical model, similar to that used in [19] or in[1,2,8,9]. In the first step we assume that by an instanta-neous absorption of a XUV photon, at the time when anas pulse reaches its peak value, the electron is movedinto a superposition of high Rydberg or continuum states.An important assumption of our model is that at t � tasthe classical electron is still very close to the core(jzj< 3 bohr) but it has a considerable initial velocity(positive or negative) and before it is completely ionizedthe electric field may change considerably. We checkedthis assumption with the help of our quantum code andhave verified that the electron wave packet, in both casesof �as � 115 nm and �as � 70 nm, is indeed centerednear z � �2 bohr with zero probability at z � 0 whenthe as pulse reaches its peak value t � tas. In the secondstep, we consider the electron as a classical particlemoving in the combined laser field El�t� and Coulombpotential VC�z� � �e2=�4�"0

��������������z2 � 1

p�. The electron tra-

jectory can be described by the 1D Newton’s equation

d2z

dt2� eEl�t� �

@@z

VC�z�: (3)

We emphasize the importance of including the Coulomb

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potential into the Eq. (3) when h!as �Wp. In the limitingcase, when h!as � Wp, the Coulomb potential in Eq. (3)can be neglected. Then Eq. (3) can easily be integrated,yielding the final electron velocity vf � vd � v0, where

vd � eAl�t0�=me, v0 � ���������������������������������2me� h!�Wp�

q(t0 � tas)

[2,13]. Thus when jvdj > jv0j, the asymmetry P2 / �vdwill reproduce exactly the vector potential Al�t0� andthe shift between asymmetry (Al) and El�t� is T=4 asshown in Fig. 3. We solved Eq. (3) for a series of 100values of the photon energies h! around h!as to accountfor a frequency spread of a 600 as pulse. For each photonenergy, 100 initial z0 values [z0 � z�t0�] were chosenfrom the intervals 1:5< jz0j< 2:5 bohr. Each z0 valuecorresponds to two initial velocities v0 � �f2me� h!�Wp � VC�z0� � eE�t0�z0g1=2. P� and P� were defined asa fraction of trajectories ionizing toward z > 0 and z < 0,respectively. In Fig. 4(a) we plot the function P2�del�obtained by this simple model. By comparing Fig. 4(a)with Fig. 2(a) we conclude that our model reproducesquite well the asymmetries with a similar delay betweenEl�t� and P2�del�. We have chosen a particular value oftime delay del � �0:25T [the dot in Fig. 4(a)] and dis-played in Fig. 4(b) four trajectories initialized at twopoints z0 � �2 bohr. Each of them corresponds to twoinitial velocities. As seen, it took 0.3 cycle for trajectoriesthree and four to get to the right turning points and startits movement in the direction of a classical force �El�t�,which has risen considerably during this motion. Thusthese trajectories suggest that the final direction of tra-jectories is determined not by the field at t � tas, but bythe field at time roughly 0:3T later. Therefore the functionP2 should be shifted by this amount to the right in orderto represent directly the electric field.

Our technique also suggests a potential way for mea-suring the duration of UV as pulses. We have calculatedthe asymmetry P2 (at del � 0) as a function of UV aspulse duration as. We found that the asymmetry increaseslinearly with as within the range 300–900 as and reaches

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-1.0

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E(t)

(a)

P −−P+

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-10

-8

-6

-4

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0

2

4

6

E(t)

(b)43

21

z(t)

[a.u

.]

Time [laser cyc.]

FIG. 4. (a) P2�del� calculated by solving Newton’s equation(3) for 10 000 trajectories. (b) Four selected trajectories(1, 2, 3, 4) for time delay del � �0:25T [the dot in (a)] ini-tialized at z0 � �2 bohr (2, 3) and z0 � 2 bohr (1, 4). Thetrajectories (1, 2) and (3, 4) have negative and positive veloci-ties, respectively.

VOLUME 89, NUMBER 28 P H Y S I C A L R E V I E W L E T T E R S 31 DECEMBER 2002

a maximum around the value as � 1 fs. To evaluate theUV-pulse duration one may proceed as indicated inRef. [1], i.e., to simulate the cross-correlation data witha single fit parameter as. We also have done similarcalculations for H�

2 ion using a frozen 1D model. Cal-culations show that our technique for measuring theelectric field or vector potential also works well in mo-lecular system. Details of our investigations will bepresented elsewhere [21]. We also have done calculationswith the as UV peak intensity twice smaller (5�1012 W=cm2). The behavior of functions P1 and P2 re-main unchanged, except their absolute value is reduced bya factor 2. Such values are experimentally detectable.

In summary, we have suggested a new and simpletechnique for directly measuring the electric field andconsequently, the carrier-envelope phase difference oflinearly polarized ultrashort optical laser pulses. Wehave shown that the cross-correlation signal of IR fsand UVas pulses exhibits a strong directional asymmetryalong the polarization vector. The asymmetry of thephotoelectron signals (the difference of the number ofphotoelectrons in opposite directions) as a function ofthe time delay between two pulses reproduces well theprofile of the electric field in IR laser pulses. [In theparticular case, when h!as > Wp, the asymmetry exactlyreproduces the vector potential Al]. The calculations inthis Letter were done for the H atom (Wp � 13:6 eV). Inpractice, we can choose atoms or ions with different

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ionization potentials to achieve the best measurementwith specific �as. Our technique relies on a subfemto-second UV or XUV pulse synchronized to the opticalpulse to be characterized, and this tool is now available[1,4]. The investigation in this paper demonstrates one ofthe new potential applications of as XUV pulses in laser-atom physics.

We thank F. Krausz, A. Sokolov, and F. Legare forvaluable discussions.

*Electronic address: andre.bandrauk@usherbrooke.ca†Permanent address: Institute for Nuclear Sciences andTechnique, Hanoi, Vietnam.

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[3] P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph.Balcou, H. G. Muller, and P. Agostini, Science 292, 1689(2001).

[4] R. Kienberger et al., Science 297, 1144 (2002).[5] F. Krausz (private communication).[6] E. Takahashi, Y. Nabekawa, T. Otsuka, M. Obara, and

K. Modorikawa, Phys. Rev. A 66, 021802 (R) (2002).[7] H. Niikura et al., Nature (London) 417, 917 (2002).[8] M. Kitzler, N. Milosevic, A. Scrinzi, F. Krausz, and

T. Brabec, Phys. Rev. Lett. 88, 173904 (2002).[9] J. Itatani, F. Quere, G. L. Yudin, M.Y. Ivanov, F. Krausz,

and P. B. Corkum, Phys. Rev. Lett. 88, 173903 (2002).[10] A. D. Bandrauk and Nguyen Hong Shon, Phys. Rev. A 66,

031401(R) (2002).[11] T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000).[12] P. Dietrich, F. Krausz, and P. B. Corkum, Opt. Lett. 25, 16

(2000).[13] S. Chelkowski and A. D. Bandrauk, Phys. Rev. A 65,

061802 (2002).[14] G. G. Paulus et al., Nature (London) 414, 182 (2001).[15] A. Baltuska, T. Fuji, and T. Kobayashi, Phys. Rev. Lett.

88, 133901 (2002).[16] A. D. Bandrauk and S. Chelkowski, Phys. Rev. Lett. 84,

3562 (2000).[17] This condition ensures that the as is short enough for

‘‘sharp’’ timing of the as pulse within an optical periodT, and long enough for sufficient as pulse energy forproducing high ionization signals.

[18] A. Scrinzi, M. Geissler, and T. Brabec, Phys. Rev. Lett.86, 412 (2001).

[19] S. Chelkowski, M. Zamojski, and A. D. Bandrauk, Phys.Rev. A 63, 023409 (2001).

[20] We use sine-square pulse envelope because it allowssignificant reduction in computation time compared toother envelope shapes which usually have long tails. Webelieve that the physics and conclusions of our work arenot affected by this choice.

[21] A. D. Bandrauk, S. Chelkowski, and Nguyen Hong Shon(unpublished).

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