871

Laser Physics, Vol. 15, No. 6, 2005, pp. 871–879.

Original Text Copyright © 2005 by Astro, Ltd.Copyright © 2005 by

MAIK “Nauka /Interperiodica”

(Russia).

1. INTRODUCTION

Subfemtosecond pulse generation via high-har-monic generation (HHG) is of great practical and fun-damental interest as a way of obtaining the shortestelectromagnetic pulses currently possible. It was pre-dicted in [1, 2] and shown experimentally in [3, 4] thata train of VUV pulses of attosecond duration can resultfrom the appropriate phase locking of different har-monics. The generation of a single attopulse is muchmore promising for applications than the generation ofthe train. There are currently two methods that could beused to obtain a single attopulse. The first one involvesusing a few-cycle (~5-fs) linearly polarized fundamen-tal pulse such that the emission of the cutoff harmonicsis naturally confined to a fraction of the laser oscillationperiod at the peak of the pulse [5]. The second way is togenerate harmonics with a pulse whose polarizationchanges with time [6]. HHG is extremely sensitive tothe ellipticity of the fundamental field, and, typically, a10–15% ellipticity reduces the generation efficiency bya factor of two. Thus, the temporal modulation of thefundamental ellipticity confines the harmonic emissioninside the temporal “gate,” where the polarization isclose to linear, as was shown in recent experiments [7].If this gate is short enough, one could achieve emissionof a single attosecond pulse [8–11]. An advantage ofthis method is that one could achieve single attosecondpulse generation using a longer fundamental pulse thanthat required by the first method. Besides, using a fun-damental with a time-modulated ellipticity provides fortemporal confinement of all high harmonics, i.e., bothplateau and cutoff. Practically, this implies the possibil-ity of obtaining a single attosecond pulse with a broadspectral width. Then, using a spectral filter, one canselect the desired spectral range for optimizing thepulse duration or for a specific application.

In this paper, we study both theoretically and exper-imentally the latter way of temporally confining HHG.The paper is organized as follows. First, we present amethod for temporally modulating the ellipticity, andwe discuss the properties of the resulting fundamentalfield. Then, we present briefly our approach for the cal-culation of VUV generation with an arbitrarily polar-ized laser field. We compare calculated and experimen-tally observed harmonic spectra obtained with rela-tively long (35-fs) fundamental pulses and observe agood agreement. Finally, we present the calculatedresults for shorter fundamental pulses and show thatusing 10-fs incident pulses results in the production ofan isolated attosecond pulse.

2. FUNDAMENTAL FIELD WITH A POLARIZATION GATING

The fundamental field with time-varying ellipticityis obtained in our experiments by using two replicas ofa fundamental pulse with crossed polarization andintroducing a delay between them [7, 9–13]. This trans-formation of the field is achieved by using two quartzquarter-wave plates; tuning the angle between therespective neutral axes of the plates allows us to controlthe ellipticity evolution as further described in [13]. Inthis section, we present the results of an analytical anal-ysis of the fundamental field.

The notations concerning the orientation of thewave plates and the polarization of the fundamentalfield are presented in Fig. 1. We assume an incidentvertically polarized laser field. First, the laser fieldgoes through a multiple-order quarter-wave plate suchthat its axis

o

1

is at 45

°

with respect to the initial direc-tion of laser polarization. Then, the laser field goesthrough a zero-order quarter-wave plate; its axis

o

2

is

Attosecond Pulses Generation with an Ellipticity-Modulated Laser Pulse

V. Strelkov

1, 2,

*, A. Zaïr

1

, O. Tcherbakoff

1

, R. López-Martens

3

, E. Cormier

1

, E. Mével

1

, and E. Constant

1

1

CELIA, Université Bordeaux 1, 351 Cours de la Liberation, Talence, 33405 France

2

General Physics Institute, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia

3

Department of Physics, Lund Institute of Technology, P.O. Box 118, S-22100, Lund, Sweden

*e-mail: v-strelkov@yandex.ruReceived January 31, 2005

Abstract

—Attosecond pulse emission using a laser field with time-dependent ellipticity is studied experimen-tally and theoretically. Our theoretical approach is validated by comparison with experimental VUV spectra.We show in calculations that the VUV emission can be confined into a narrow temporal window in which thefundamental polarization is quasi-linear. We find an analytical equation describing the duration of this windowand compare it with numerical results. The requirements for the fundamental field that are necessary to confinethis emission to a single attosecond pulse generation are formulated.

ATTOSECOND SCIENCEAND TECHNOLOGY

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LASER PHYSICS

Vol. 15

No. 6

2005

STRELKOV

et al

.

at an angle

β

with respect to the direction of initial laserpolarization.

Let the initial fundamental field be

(1)

where

E

0

(

t

) is a slowly varying envelope. Let us define

(2)

where

δ

is the delay introduced by the multiple-orderwave plate between the orthogonal field components

(

t

) and (

t

).

For the analysis of the field behind the two waveplates, it is convenient to project it onto the axes (

x

,

y

),which are turned by 45

°

with respect to the axes of thesecond wave plate (

o

2

,

e

2

):

(3)

(4)

One can see that, in the center of the pulse (i.e., around

t

= 0), the

x

component of the field is zero; i.e., the fieldis almost linearly polarized along the

y

axis, indepen-dently of the angle

β

. That is why we have chosen thisorientation of the (

x

,

y

) axes.

E 2E0 t( ) ω0t φ π4---–+⎝ ⎠

⎛ ⎞ ,cos=

E0±

t( ) E0 tδ2---±⎝ ⎠

⎛ ⎞ ,=

E0+

E0–

Ex t( ) E0+

t( ) E0–

t( )–[ ] ω0t φ β–+( ),sin=

Ey t( ) E0+

t( ) E0–

t( )+[ ] ω0t φ β+ +( )cos .=

The ellipticity of the field described in Eqs. (3), (4) is

(5)

Thus, the ellipticity is zero in the center of the pulse forevery

β

. It remains zero during the whole pulse for

β

=45

°

+

k

×

90

°

,

k

= 0,

±

1,

±

2, …. We call this field con-figuration a large gate configuration, because there is alarge temporal gate for harmonic emission [13]. On thecontrary, for

β

≠

45

°

+

k

×

90

°

, the ellipticity varies intime. For

β

= 0

°

+

k

×

90

°

, the variation rate is maxi-mum and the field remains linearly polarized for a veryshort time, so we call the field configuration a narrowgate configuration. However, from Eqs. (3), (4), onecan see that the intensity of the field does not depend on

β

. Thus, our scheme allows us to continuously controlthe rate of ellipticity variation (changing

β

) withoutaffecting the intensity profile of the pulse.

The direction of the main axis of the polarizationellipse (denoted as

y

* in Fig. 1) also varies in time. Theangle

ψ

between this direction and

y

axis is

(6)

We are mainly interested in the part of the pulse wherethe polarization is close to linear and the intensity isclose to maximum, i.e., near

t

= 0. In the vicinity of thistime,

(7)

(8)

where

For a Gaussian fundamental pulse with duration

τ

(FWHM), we have

(9)

Below, we study in more detail the field componentalong the main axis of the polarization ellipse. Let uspresent it in the form

where (

t

) and

χ

(

t

) are the slowly varying amplitudeand phase, respectively. For this phase we obtain

(10)

ε t( ) 12---

E0+2

t( ) E0 2–

t( )–

E0+2

t( ) E0 2–

t( )+-------------------------------------- 2β( )cos

⎝ ⎠⎜ ⎟⎛ ⎞

arcsin .tan=

ψ t( ) 12---

E0+2

t( ) E02–

t( )–

2E0+

t( )E0–

t( )------------------------------------- 2β( )sin .arctan=

ψ αt 2β( ),sin–≈

ε αt 2β( ),cos–≈

α 1E0 δ/2( )------------------- ∂E0

∂t---------

δ/2

.=

α 2 2δτ2----.ln=

Ey* E0 t( ) ω0t χ t( )+( ),cos=

E0

χ t( ) φ β α2t

2

2---------- 4β( ).sin–+≈

x

45°

β

ψ

y

y*

e2

o2

Inital polarization direction

Fig. 1. Orientation of the fundamental field polarizationdirection before the wave plates, axes of the zero-orderquarter wave plate (o2, e2), and polarization ellipse of thefield behind the wave plates.

LASER PHYSICS Vol. 15 No. 6 2005

ATTOSECOND PULSES GENERATION 873

Thus, the frequency of the field component varies intime:

(11)

Note that this variation is present only in the intermedi-ate gate (i.e., for β ≠ k × 45°), not in the narrow andlarge gate configurations; it depends periodically on βwith a period of 90°.

We would like to stress that (11) is a variation of fre-quency of a field projection on the rotating axis y*.Thus, one hardly could call this variation a chirp,because the total field (Eqs. (3), (4)) is nonchirped.Nevertheless, we will see below that this “effectivechirp” may influence some features of the harmonicspectrum.

3. SINGLE-ATOM RESPONSE FOR HHG

The theoretical approach that we use here to calcu-late the single-atom response is based on a HHG theorydeveloped in [14]. We generalize this theory to the caseof an arbitrarily polarized fundamental field. Ourapproach will be published in detail elsewhere [15].Here, we briefly present the principles of this theory. Itis based on the three-step model for high-order-har-monic generation [16]. The three steps of the process(namely, ionization, electron motion, and short-wave-length radiation generation by the electron comingback to the parent ion) are described quantum mechan-ically. We take into account the electron wavepacketspreading after ionization using the results of [17].Moreover, we estimate the wavepacket modificationdue to the Coulomb attraction by the parent ion. Formiddle-plateau harmonics, there are several signifi-cantly different electron trajectories [18, 19] that lead tothe generation of radiation with the same frequency(albeit with different phase properties [19–21]). Theimportant feature of our approach is the possibility ofseparating the contributions of these trajectories withinthe frame of the single-atom response. Below, discuss-ing properties of the two main contributions, we labelthe harmonic fields generated with electrons travelingalong the shorter and the longer quantum paths as theSP component and LP component, respectively.

The theory utilizes explicitly the ionization rate thatis calculated separately. To calculate the ionization rate,one can use analytical formulas (such as the ADK for-mula [22] or the “tunneling” formula [23]). Still, theaccuracy of these formulas is low (see, for example,[24]). In this paper, we use an other approach based onthe interpolation of a numerically computed ionizationrate. Although it is more cumbersome than using theabove-mentioned analytical formulas, it provides real-istic ionization rates. We solve the Schrödinger equa-tion for a single-electron atom in a linearly polarizedfield with the B-spline method described in [25] with amodel atomic potential [26]. Then, we reconstruct the

∂ω∂t------- α2

4β( ).sin–≈

approximate instantaneous field-dependent ionizationrate for this atom and use it to calculate the ionizationprobability in the arbitrarily polarized laser field.

4. SOME RESULTS ON SINGLE-ATOM RESPONSE

In this section, we present some results of the single-atom response calculation and compare them withresults of other theories and experiments. For the sakeof clarity, in this section we consider long fundamentalpulses leading to well-resolved harmonic lines in thespectrum and artificially suppress the ground-statedepletion.

For the phase of high-harmonic generated in Ar witha Ti:sapphire laser (800 nm), we obtain analytically thefollowing approximate expressions:if the harmonic is in the plateau region,

(12)

(13)

if the harmonic is in the cutoff region,

(14)

where is a term that depends on the har-monic order q but does not depend on either the funda-mental field intensity or the ellipticity; the factors

depend on the harmonic order and the fun-damental intensity but do not depend on the ellipticity.

The expressions (12)–(14) are approximate ones. InFig. 2 we present the phases and the intensities of theSP and LP components as functions of the fundamentalintensity for the 19th and 29th harmonics calculatedusing our theory without any approximations. The fun-damental field is linearly polarized. One can see fromFig. 2a that the phase dependence in the plateau regionis very close to linear. The slope is significantly differ-ent for the SP and LP components but depends justslightly on the harmonic order. In the cutoff region, tra-jectories leading to the generation of SP and LP com-ponents get very close to each other (and exactly coin-cide for a harmonic in the cutoff). When the harmonicis close to the cutoff, the phase dependence on intensitybecomes close for the SP and LP components (seeFig. 2a).

From Fig. 2b one can see that the intensities of SPand LP components are comparable in the wide rangeof the fundamental intensities. This agrees with thenumerical results of [14, 27].

In Fig. 3 we present the dependence of the 21st har-monic generated in Ar on fundamental laser ellipticity.The fundamental wavelength is 800 nm, and the peakintensity is 2 × 1014 W/cm2. In the same figure we

ϕqLP

24.7 1 μqLPε2

+( )I 1014

W/cm2[ ]– δq

LP,+≈

ϕqSP

24.7 0 μqSPε2

+( )I 1014

W/cm2[ ]– δq

SP;+≈

ϕqcut-off

–13 1 μqcut-offε2

+( )I 1014

W/cm2[ ]≈

+ δqcut-off( ),

δqLP SP cut-off, ,

μqLP SP cut-off, ,

874

LASER PHYSICS Vol. 15 No. 6 2005

STRELKOV et al.

present experimental results [29, 30] of the ellipticitydependence for this harmonic (circles and triangles cor-respond to results with left- and right-polarized funda-mental fields, which were presented in [29]). There is avery good agreement between our calculations and theexperimental data. Note that the SP and LP componentsdepend on the fundamental laser ellipticity in a similarway. The explanation is that a higher drift of the elec-tron traveling along a longer trajectory is approxi-mately compensated for by the greater spreading of thewavepacket.

5. PHASE MATCHING

In this section we consider the propagation and dif-fraction of the harmonics. Our approach is close to theone suggested in [31]. In order to simulate the macro-scopic emission, we calculate the single-atom responseat a number of locations throughout the gas target usingthe above-mentioned theory. Then, we calculate thepropagated harmonic field and its angular distributionin the far field. In this procedure, we take into accountthe (on- and off-axis) phase matching of HHG. We con-sider both geometrical and material (gas) dispersion.Also, we take into account absorption of harmonics bythe gas, which is quite important [32] for the relativelylow-order harmonics that we consider in this paper.Within the framework of our approach, refraction andabsorption of the medium at the harmonic frequencyare considered to be constant; thus, the model is validonly if the ionization probability is significantly lessthan unity. In contrast, the change in the fundamentalfield due to ionization of the medium could be takeninto account by introducing the additional temporal andspatial dependence of the phase of the fundamentalfield. However, in this paper we deal with low ioniza-tion probability and, thus, consider this phase to be aconstant.

6. COMPARISON OF THEORY AND EXPERIMENT

In this section we present a comparison between themeasured and calculated harmonic properties in thespectral domain. Our experimental conditions are closeto the ones described in [10, 13]. The laser pulse dura-tion (before the wave plates used to obtain ellipticitymodulation) is τ0 = 35 fs; the delay introduced with themultiple-order wave plate is τ = 31.4 fs.

In Fig. 4 we present the experimental and calculatedspectra of harmonics generated in the narrow and largegate configurations of the fundamental field. One cansee a general agreement of the envelope of the spectra(note that the experimental harmonic signal is pre-sented without calibration for different spectral ranges).In the insets of Fig. 4a, one can see in detail some fea-tures of the experimental spectrum: splitting of the har-monic lines in the large gate configuration and spectralbroadening and red shift of the lines in the narrow gate.

–50

0.5 1.0

Harmonic phase, rad

Fundamental intensity, 1014 W/cm21.5 2.0 2.5 3.0

–40

–30

–20

–10

0

H19, SP (short path)H19, LP (long path)H29, SP H29, LP

(a)

00.5 1.0

Harmonic intensity, arb. units

Fundamental intensity, 1014 W/cm21.5 2.0 2.5 3.0

50

100

150H19, SPH19, LPH29, SP H29, LP

(b)

Fig. 2. Phases (a) and intensities (b) of the SP (short path)and LP (long path) components of the 19th and 29th har-monics generated in Ar as a function of the fundamentalintensity. The fundamental polarization is linear.

0.01

10–3

0

Harmonic intensity, arb. units

Fundamental ellipticity0.1 0.2 0.3 0.4

0.1

1

ExperimentTheoryTheory, SP-componentTheory, LP-component

, ,

Fig. 3. Intensity of the 21st harmonic generated in Ar underthe fundamental intensity 2 × 1014 W/cm2 as a function ofthe fundamental ellipticity. Lines—calculation, symbols—experiment (circles and triangles—results from [29],squares—results from [30]).

LASER PHYSICS Vol. 15 No. 6 2005

ATTOSECOND PULSES GENERATION 875

In Fig. 5 we present experimental and calculatedspectra for the 15th harmonic generated in the largegate. One can see a red-shifted satellite (and no blue-shifted one) in the experimental spectrum. In the calcu-lated spectrum, the red-shifted satellite is more pro-nounced than the blue-shifted one.

An important advantage of our method of obtaininga fundamental field with a time-varying ellipticity is thepossibility of changing the configuration of the funda-mental field continuously from a large to narrow gate(through the intermediate gate) just by rotating the sec-

ond wave plate (i.e., by changing the angle β; seeFig. 1). In Fig. 6 we present the width of the 15th har-monic line (namely, the width of the Lorentzian fit tothe line) as a function of β. Note that the dependence isasymmetric with respect to the large gate. We interpretthis as the influence of effective chirp (11) in the funda-mental field component along the main axis of thepolarization ellipse: indeed, the properties of the single-atom response for a given harmonic (for example,instants of ionization and recombination) dependmainly on the time evolution of the fundamental field

0

2

4

6

8

10

Har

monic

sig

nal

, ar

b. unit

sLarge gate

Narrow gate

23.0 23.5

H15H27

41.5 42.0 42.5

25 30 35 40 45Photon energy, eV

0

2

4

6

8

10

Fig. 4. Experimentally measured (a) and calculated (b) spectra generated in the large and narrow gate configurations. Fundamentalwavelength is 800 nm; peak intensity in the focus is 1.4 × 1014 W/cm2; initial pulse duration is τ0 = 35 fs; delay is τ = 31.4 fs.Harmonics are generated in 15-mm-long capillary placed in the focus and filled with Ar of a backing pressure of 30 mbar. Spectraare measured and calculated in the far field and integrated over the angle.

(a)

(b)

876

LASER PHYSICS Vol. 15 No. 6 2005

STRELKOV et al.

component along the main axis of the polarizationellipse. Thus, its effective chirp is somehow transferredin the harmonic chirp and, thus, influences the har-monic linewidth.

The general conclusion that can be drawn from thecomparison between experiment and theory is thatquite fine features of the experimental spectra arereproduced in our calculations (see also the comparisonbetween experiment and theory presented in [33]). Thisvalidates our theoretical approach for VUV generationwith time-dependent fundamental polarization. Below,we use the theory to obtain results directly in the tem-poral domain.

7. CALCULATION RESULTSIN THE TIME DOMAIN

In Fig. 7 we present the calculated intensity tempo-ral profile of the VUV field (for harmonic orders higherthan q = 19) generated in the narrow and large gate con-figurations under our experimental conditions. One cansee a significant temporal confinement of the VUV gen-eration in the narrow gate configuration; specifically,changing β allows us to shorten the pulse train from36 fs in the large gate to 8 fs in the narrow gate.

Using a 10-fs incident fundamental pulse, one canobtain (under a certain absolute fundamental phase) asingle attosecond pulse, as shown in Fig. 8. The dura-tion of this pulse is about 200 as; note that this is farfrom the Fourier-limited duration, which is about120 as for this pulse. This difference is due to the chirpthat is inherent in attosecond pulses obtained via HHG[34–36] and that, in principle, could be compensatedfor [35, 37]. Although we deal with a quite modest fun-damental intensity and relatively low harmonic orders,the obtained pulse duration of 200 as is significantlyshorter than that achieved by the spectral selection of

cutoff harmonics generated with a linearly polarizedpulse [5].

Averaging the VUV intensity over the fundamentalphase, we obtain the “gate envelope” that is also pre-sented in Fig. 8. Now, we can also accurately define thegate duration as the FWHM of the envelope. To stressthe effect of the delay δ on the gate duration, wepresent, in Fig. 8, the results obtained under equal inci-dent pulse durations but different delays.

If the gate duration is several optical cycles or more,the ellipticity varies relatively slowly in time, and, thus,the VUV generation efficiency changes adiabaticallywith the ellipticity. The dependence of the VUV gener-ation efficiency on the fundamental ellipticity is closeto a Gaussian (see Fig. 3). In the center of the gate, theellipticity variation in time is close to linear (seeEq. (8)); thus, the temporal envelope of the gate in thiscase is also almost a Gaussian.

Taking into account that the VUV generation effi-ciency decreases twofold when the ellipticity reachescertain threshold εtr, we obtain, from Eqs. (8) and (9),that the gate duration is

(15)

This equation is valid if the gate duration is limited withellipticity variation but not with the intensity variation;thus, under approximately Δ2 > τ2 + δ2 (in particular,this equation is certainly invalid in the large gate con-figuration, i.e., for β = 45°). Note that, for the caseβ = 0° (narrow gate), this equation coincides with oneobtained in [11] for another method of temporally mod-ulating the fundamental ellipticity.

For a gate duration much longer than an opticalperiod, Eq. (15) with εtr = 0.15 agrees very well with thecalculated duration. However, if the fundamental polar-ization varies very quickly (i.e., if the gate duration is

Δεtrτ

2

δ 2βcos 2ln------------------------------.=

1

22.5 23.0

Harmonic signal, arb. units

Photon energy, eV

2

3

4

023.5 24.0

Experiment

Theory

Fig. 5. Measured and calculated spectrum of the 15th har-monic generated in the large gate configuration under thesame conditions as in Fig. 4.

0.10

0

H15 linewidth, eV

Angle, deg

0.05

0.15

0.20

0.25

20 40 60 80 100

ExperimentTheory

Nar

row

Lar

ge

Nar

row

Fig. 6. Measured and calculated line width of the 15th har-monic as a function of the angle between the axis of thezero-order wave plate and the initial polarization direction(angle β).

H15, large date

LASER PHYSICS Vol. 15 No. 6 2005

ATTOSECOND PULSES GENERATION 877

comparable with the optical cycle duration), the VUVproduction efficiency does not vary adiabatically withthe ellipticity; in fact, the ellipticity itself cannot bedefined accurately. In Fig. 8 one can see that, in this

case, the gate envelope is not a Gaussian any more:now, it has a flat top. Strictly speaking, Eq. (15) is notapplicable in this case. To check if Eq. (15) gives atleast a correct estimate, we carry out the calculation of

0.2

–20

Ell

ipti

city

Time, fs

0.4

0.6

0.8

1.0

00 20

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0

I(t)

ε(t)

Fundam

enta

l in

tensi

ty, 10

14 W

/cm

2

Fig. 7. The fundamental field intensity, its ellipticity in the narrow gate configuration, and VUV (photon energy >27 eV) intensitycalculated in the large (thin line) and in the narrow (thick line) gate for the conditions of Fig. 4. VUV intensity is calculated in thefar field on the axis.

–1.0 –0.5 0 0.5 1.0 1.5 2.0Time, fs

0

0.5

1.0VUV intensity, arb. units

200 as 1.7 fs

1.2 fs

Fig. 8. Calculated VUV intensity (photon energy from 25 to 45 eV) generated in the narrow gate configuration for the incident laserpulse of duration τ = 10 fs and the delay δ = 13 fs and for two absolute phases of the fundamental laser field differing from oneanother by π/2 (solid and dot), and temporal gate envelope (dash). The fundamental field intensity in the center of the pulse is 2.2 ×1014 W/cm2, and the VUV intensity is calculated on the axis, resulting in the selection of the short path component. The inset showsthe VUV emission with a 10-fs input pulse and an 18-fs delay leading, after spectral selection, to the emission of an isolated attosec-ond pulse with an efficiency that fluctuates with the absolute phase.

878

LASER PHYSICS Vol. 15 No. 6 2005

STRELKOV et al.

the gate duration under different τ, δ, β for fundamentalintensities of 2.2 × 1014 W/cm2 and 3 × 1014 W/cm2 (seeFig. 9). The agreement is surprisingly good if the gateduration is comparable to or even shorter than an opti-cal cycle (down to 1 fs). The calculated duration isslightly shorter than that predicted by (15), but this dif-ference is typically less than 10–15%. The calculatedduration depends just slightly on the fundamentalintensity: for a higher intensity, it is shorter.

Taking into account that the attosecond pulses in thetrain are separated with an interval of approximatelyhalf an optical cycle, we can now formulate the require-ments for the fundamental field to achieve singleattosecond pulse generation via ellipticity gating. Ifduration (15) is shorter than an optical cycle (forinstance, in the narrow gate (β = 0) under τ = 15 fs, δ =20 fs; τ = 12 fs, δ = 12 fs; or τ = 10 fs, δ = 13 fs), thena single attosecond pulse is generated under a certainrange of fundamental phases (see, for instance, Fig. 8);if duration (15) is shorter than half an optical cycle (forinstance, in the narrow gate under τ = 10 fs, δ = 18 fs orτ = 6 fs, δ = 6 fs), an isolated attosecond pulse is gener-ated under every fundamental phase, and changes in theabsolute phase result only in fluctuations in the isolatedattosecond pulse emission efficiency, as shown in Fig. 8(inset). Strictly speaking, the single attopulse is gener-ated under every phase except one, under which twoattopulses with equal intensity are generated. However,this intensity is lower than the intensity of a singleattopulse generated under all other phases, so this caseis practically unimportant.

Dealing with a short train of attosecond pulses (orwith a single attosecond pulse), one should take into

account the contribution of secondary recollisions tothe harmonic generation process. These contributionsnaturally increase the duration of the train. However,we find in our calculations that these contributions aresignificant in the off-axis harmonic field and are muchless significant on the axis. This can be understood bytaking into account that the phase of the VUV gener-ated due to secondary recollisions is very sensitive tothe fundamental intensity. Thus, the wave front of thisradiation is very curved [27, 28], the beam is very diver-gent, and the impact of the secondary recollisions in thefar field on the axis is small. If necessary, this impactcan be suppressed by selecting the central part of thebeam with a diaphragm.

8. CONCLUSIONS

We have developed a theory of attosecond pulsegeneration by a fundamental laser field with arbitraryellipticity. In our experiments we measured spectra ofHH generated in narrow, large, and intermediate gatesusing 35-fs fundamental pulses, and we find agreementbetween features of the measured and calculated spec-tra. Calculations show that, under our experimentalconditions (τ = 35 fs, δ = 31.4 fs), we can go from a36-fs-long train of attosecond pulses generated in thelarge gate to an 8-fs train in the narrow gate. We applyour theory to study attopulse generation with shorterfundamental pulses, find Eq. (15) to describe the gateduration obtained via temporal variation of the funda-mental polarization, and formulate the requirements forthe fundamental field that are necessary to achieve sin-gle attosecond pulse generation. The gate durationdepends on the incident pulse duration, the delaybetween pulses, and the orientation of the wave plates.In particular, in the narrow gate configuration using adelay equal to the incident pulse duration, one canobtain the generation of a single attopulse under a cer-tain absolute phase of the fundamental using an inci-dent pulse as short as 12 fs, and, with the 6-fs pulse, thesingle attopulse is generated under every absolutephase (the attopulse intensity still depends on the abso-lute phase). Furthermore, we find that the contributionof secondary recollisions does not significantly affectthe attosecond pulses on the axis in the far field.

ACKNOWLEDGMENTS

We gratefully acknowledge discussions with B. Carré,A. L’Huillier, and V.T. Platonenko. This study was sup-ported by the region Aquitaine, the French Ministère dela Recherche, and the European network “ATTO”no. HPRN-2000-00133.

REFERENCES

1. G. Farkas and C. Toth, Phys. Lett. A 168, 447 (1992).2. S. E. Harris, J. J. Macklin, and T. W. Hëansch, Opt. Com-

mun. 100, 487 (1993).

10

10

Gate duration, fs

0.22 τ2/|δcos2β|, fs1

1

EstimationCalculation, 2 × 1014 W/cm2

Calculation, 3 × 1014 W/cm2

Fig. 9. Comparison of the gate duration predicted byEq. (15) with εtr = 0.15 and the gate duration calculatednumerically under different incident pulse durations τ,delays δ, and angles β. In calculations the fundamentalintensity in the center of the gate is 2 × 1014 W/cm2 (circles)and 3 × 1014 W/cm2 (squares).

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