attosecond pulse generation by a two-color field

Vladimir D. Taranukhin Vol. 21, No. 2 /February 2004 /J. Opt. Soc. Am. B 419

Attosecond pulse generation by a two-color field

Vladimir D. Taranukhin

Department of Physics, M. V. Lomonosov Moscow State University, Vorob’evy gory, 119992 Moscow, Russia

Received March 23, 2003; revised manuscript received August 26, 2003; accepted September 29, 2003

A method for the generation of attosecond electromagnetic pulses is suggested. The key idea of the methodconsists in using a two-color laser pump for high-order harmonic generation composed of a low-frequency lin-early polarized field and a high-frequency elliptically polarized field. Such a two-color pump can provide forthe return of photoelectrons (after atom ionization) to the vicinity of the parent ion with high kinetic energyand their recombination for only specific ionization moments t0 . The range of these moments, dt0 , is definedby the velocity of electron wave-packet spreading and the time that the photoelectron spent in the continuum(before the recombination). Conditions were found that minimize the range dt0 . For the specific parametersof a two-color pump, the duration of recombination emission, tg , can be in the range of 1–10 as. With anincrease of pump intensity, the duration tg decreases and can be reduced to the subattosecond scale. © 2004Optical Society of America

OCIS codes: 190.4160, 190.4180, 270.4180, 270.6620.

1. INTRODUCTIONThe generation of ultrashort pulses of coherent radiationis intensively studied now to develop effective methods offemtosecond and attosecond metrology.1–4 The produc-tion of trains of 250-as pulses5 and 500-as single pulses1,6

has already been reported. Two present suggestions forobtaining subfemtosecond pulses are related to high-order harmonic generation (HHG) in the process of above-threshold atom ionization by an intense laser pump: (1)some are based on phase matching for the group of har-monics with suitable phases,7 (2) and the others use a po-larization gate,8 that is, pump radiation with time-dependent ellipticity, in order to select the return ofphotoelectrons to the vicinity of the parent ion from only asmall part of the pump pulse. In both cases the durationof the harmonic pulse is tg < T 5 2pc/l (where T is du-ration of the optical cycle of pump radiation, l is pumpwavelength, and c is speed of light), which corresponds toharmonic emission of several-femtosecond or subfemto-second duration. Spectral filtering of emitted harmonicsfavors the shortening of generating pulses.1,6 Such a fil-tering results in the selection of photoelectrons releasedfrom the atom in a narrow range of ionization phases, Df.For example, 5% filtering near the cutoff frequency Vleads to the selection Df ' 0.065p. In accordance withthe semiclassical theory of HHG, each photoelectron ar-rives at the parent ion at a specific moment, which de-pends on the ionization phase. The spread of phases, Df,leads to the spread of moments of electron arrival, that is,the spread of moments of recombination Dtr , which de-fines the minimal duration of recombination emission (forDf ' 0.065p, we have Dtr ' T/8). These values (for Dfand Dtr) do not depend on laser intensity I. In the caseof one-component pump radiation (independent of its in-tensity I), the minimal duration of recombination emis-sion will be several hundred attoseconds (for l ; 1 mm).

This restriction can be overcome with the use of two-component pump fields. HHG with combined laser and

0740-3224/2004/020419-06$15.00 ©

static fields has been considered before,9–12 althoughwithout special emphasis on short-pulse generation. InRef. 13 HHG with a two-color pump was discussed forselecting a small portion of photoelectrons colliding withthe parent ion. It was shown that a two-color pumpmade possible the phase control of ionization, providingatom ionizing within a small part of the pump opticalcycle T. Therefore short recombination emission can beattained either by selecting a small portion of ionizationmoments or by selecting the moments of photoelectron re-combination. This paper presents the possibility of at-tosecond and even subattosecond pulse generation withthe use of a specific two-color pump of atoms. The effectis attained as a consequence of avoiding a collision withthe parent ion for all but a small portion of the photoelec-trons, which corresponds to a narrow range of ionizationphases.

Recombination radiation can be described by Maxwell’swave equation with the source term that is the secondtime derivative of induced dipole moment D(t), whichshould be calculated quantum mechanically. Dipole D(t)is a function of time, which defines (together with thepropagation effects) the duration of harmonic radiation.We do not consider here the propagation effects (whichcan stretch a harmonic pulse as well as compress it), andinstead concentrate on the atomic response D(t).

2. QUANTUM-MECHANICAL MODEL OFHIGH-ORDER HARMONIC GENERATIONWITH BICHROMATIC FIELDSA quantum-mechanical model of HHG by a single atomwith monochromatic pump radiation14,15 can be extendedto the case of a pump with complex structure, e.g., to thecase of a two-color pump. We use the same assumptionsfor HHG theory as those in Refs. 14 and 15, which corre-spond to the tunnel regime of atom ionization. For thisregime a quantum-mechanical description of HHG recov-

2004 Optical Society of America

420 J. Opt. Soc. Am. B/Vol. 21, No. 2 /February 2004 Vladimir D. Taranukhin

ers a three-step interpretation of HHG: atom ionization,photoelectron evolution in the continuum, and electronrecombination with a parent ion.16

As in Refs. 14 and 15, we assume that pump intensity Iis high enough that the intermediate resonances associ-ated with the bound–free transitions do not play a signifi-cant role.17 Moreover, the probability of recombinationto the ground state is considerably higher than the prob-ability of recombination to any other discrete state.18

This makes it possible to ignore the contribution to theevolution of the system by all states in the discrete spec-trum but the ground state u1&. Then the time-dependentwave function can be represented in the form

uC~r, t !& 5 a~t !u1& 1 E dp bp~t !up&, (1)

where a(t) and bp(t) are the amplitudes of the groundatomic state and the continuum states up&. On the otherhand, if the electron appears in the continuum far enoughfrom the parent ion (pump intensity not high enough forbarrier-suppression ionization), we can neglect the Cou-lomb effect on electron evolution in the continuum. Fur-ther neglecting the contribution of bremsstrahlen, we canderive with saddle-point analysis the quantum-mechanical average of the field-induced dipole moment,given by D(t) 5 ^C* (r, t)uruC(r, t)&, in the followingform (similar to the one obtained in Ref. 14 but in anothergauge):

D~t ! 5 a* ~t !E0

t

dt0bp(a~t0!)K 1uru fr~r 2 r0 , t, t0!

3 expS 2iEt0

t

e~ p0!dt 1 ip0rD L , (2)

where r0 is the central coordinate in the space distribu-tion fr of the electron continuum states and e( p0) and p0are the kinetic energy and the central momentum of theelectron wave packet. Expression (2) corresponds toquasi-classical evolution of a photoelectron and allows aclear interpretation. In the first stage of the HHG pro-cess, an electron wave packet is formed in the continuum.The amplitudes bp(a(t0)) of the continuum states are de-termined by the probability of tunnel ionization at mo-ments t0 and account for the saturation of the ionizationprocess. In Ref. 14 analytical results for the dipole mo-ment were obtained under the assumption of low pumpintensity and therefore small ionization probability. Thisleads to the Keldysh-type dependence of ionization prob-ability on pump field strength F. In a strong laser fieldthe dependence of tunnel ionization rate Wi on fieldstrength F may differ substantially from the well-knownKeldysh-type dependencies. The modification of prob-ability Wi in strong fields is discussed comprehensively inRef. 19 (see also Ref. 20), taking into account three-dimensional, Stark, and flattening effects.

At the next stage of above-threshold ionization, thephotoelectron wave packet moves in the continuumguided by the pump field. The function fr(r 2 r0 , t, t0)describes the shape of the electron wave packet and themotion of its center r0(t). In strong (but nonrelativistic)laser fields, the photoelectron excursion in the continuum

is much greater than the size of the photoelectron wavepacket, and both are less than the pump wavelength. Inthis case the quasi-classical approach can be used, andthe motion of the photoelectron wave-packet center in thecontinuum can be described by the classical equation ofmotion. For a Coulomb potential the validity of this as-sumption is questionable when the electron is in the vi-cinity of the parent ion. However, the part of the photo-electron trajectory where it acquires most of its kineticenergy (essential for HHG) goes far from the parent ion.Furthermore, since only the first return of the photoelec-tron to the parent ion is important for the harmonic gen-eration, it is reasonable enough to suppose that the shapeof the electron wave packet does not change significantlyduring its evolution in the continuum. At the same timethe quantum effect of electron wave-packet spreading canbe described just as for a free particle but with spreadingvelocity Vsp determined by the process of tunnelionization21:

s~t ! 5 @ s2~t0! 1 Vsp2 ~t 2 t0!2#1/2 (3)

where s is wave-packet width, Vsp ' Fi1/2(2Ip)21/4 (see

Ref. 21), Fi 5 F(t0) is pump field strength at the momentof ionization, and Ip is ionization potential of the atom.We use the numerical value Vsp ' 1 nm/fs (at pump in-tensities I ' 1014–1017 W/cm2), which is in good agree-ment with the experimental data.8

An electron acquires the energy e( p0) while moving inthe continuum. When it returns to the core, it can re-combine with the parent ion and produce a recombinationphoton with energy ;e( p0). The ‘‘overlapping’’ integral^1uru¯& in Eq. (2) describes the probability of recombina-tion, as well as the ‘‘elementary’’ duration of recombina-tion emission (duration of atomic response), because thisintegral is nonzero only for the moments when the elec-tron is near the parent ion. Finally, the integral over t0in Eq. (2) describes the coherent contributions to HHG byelectrons born at the different moments t0 .

This clear interpretation makes it possible to extendthe validity range of the HHG model14 to greater pumpintensities19 as well as to account for the three-dimensional character of electron evolution and to con-sider pump fields of arbitrary configuration.22

3. HIGH-ORDER HARMONIC GENERATIONWITH A BIPOLARIZED TWO-COLORPUMPIt is known that with an ordinary pump there are no har-monics in the case of circularly polarized pumpradiation.8 In this case after atom ionization, the photo-electron moves far from the parent ion, and there are noelectron trajectories colliding with the parent ion for anymoments of ionization, unlike the case of linearly polar-ized pump radiation, where for any moment of ionizationthere is a colliding trajectory. We consider using a spe-cial two-color pump composed of a high-frequency (HF) el-liptically polarized field and a low-frequency (LF) linearlypolarized field:

F 5 F0~ xA1 2 a2 cos vt 1 ya sin vt ! 1 Fdc , (4)


Fdc 5 2bF0~ xA1 2 g2 1 yg!, (5)

where F0 , v, and a are amplitude, frequency, and elliptic-ity of the HF part of the pump field, x and y are the unitvectors, b and g are parameters that define relative am-plitude and direction of the linearly polarized LF part ofthe pump field, Fdc (as an LF pump, the radiation of anappropriately synchronized CO2 laser can be used; duringthe optical cycle of the HF field, it can be regarded as a dcfield). For simplicity we consider here the case of collin-ear propagation of HF and LF fields. One can easily cal-culate the classical equation of motion with fields (4) and(5) and find trajectories of the electron wave-packet cen-ter in such a two-color field:

x~t ! 5 2A1 2 a2@cos t 2 cos t0 1 ~t 2 t0!

3 sin t0 1 b1~t 2 t0!2#, (6)

y~t ! 5 2a@sin t 2 sin t0 2 ~t 2 t0!

3 cos t0 1 b2~t 2 t0!2#, (7)

where b1 5 (b/2)@(1 2 g2)/(1 2 a2)#1/2, b2 5 (b/2)3 (g/a), and zero initial conditions for electron coordi-nate and velocity were used.16 In Eqs. (6) and (7) and be-low, we use dimensionless variables: time normalized to1/v, coordinates normalized to eF0 /mv2, velocities nor-malized to eF0 /mv, and energy normalized to the pon-deromotive energy of HF pump radiation, Up5 e2F0

2/4mv2, where m and e are electron mass andcharge. Analysis of Eqs. (6) and (7) shows that, in con-trast to the case of a one-color pump, the colliding trajec-tories [when both coordinates x and y tend to zero simul-taneously at the specific recombination moment tk (Fig.1)] exist for only a specific set of parameters: a, b, g, andt0 . It is desirable to choose these parameters in such away that electron energy at the moment of recombination,ek , will be larger, time that the electron spent in the con-tinuum smaller, and ratio of LF to HF field amplitudessmaller, and furthermore that there will be only one col-liding trajectory during the whole pump pulse (for thegeneration of a single pulse of recombination radiation).Analysis shows, however, that there is no unique solutionto this problem. To attain the shortest recombinationemission, the most important task is to minimize the timethat the photoelectrons spend in the vicinity of the parention, given by

Fig. 1. Time dependence of photoelectron coordinates afteratom ionization [see Eqs. (6) and (7)] for the Y case: a5 1/A2, b ' 20.276, g ' 20.96, and t0 5 p/10.

Dr < s~tk!, (8)

where the dipole moment (2) differs from zero, or, in otherwords, to maximize the derivative dr0(tk)/dt0 , where r0

5 Ax2 1 y2. We find two classes of such solutions:

• The Y case (shown in Fig. 1) corresponds to collidingmoments tk obtained from Eq. (6): x(tk) 5 0. For suchcolliding moments, we have

dx~tk!/dt0 [ 0, dy~tk!/dt0 5 `. (9)

The first of Eqs. (9) is just a consequence of the relationx(tk) 5 0 (in such a way we define the moment of recom-bination in the Y case). The second of Eqs. (9) needs tobe accepted to maximize the derivative dr0(tk)/dt0 , thatis, to maximize the ‘‘velocity’’ of transition from collidingtrajectories to noncolliding ones. It means that electronvelocity along the y coordinate gives the main contribu-tion to kinetic energy ek . In this case the requirementon the pump parameters and the moment t0 follows fromEqs. (6), (7), and (9):

sin tk 2 sin t0 5 2b1t, (10)

where t 5 tk 2 t0 . Parameters a, b, g, and t0 shouldobey Eq. (10) in order to minimize the time that photo-electrons spend in the vicinity of the parent ion.

• The X case corresponds to the opposite situation:

dy~tk!/dt0 [ 0, dx~tk!/dt0 5 `. (11)

In this case parameters a, b, g, and t0 should obey the re-quirement

cos tk 2 cos t0 5 22b2t. (12)

In both cases requirements (10) and (12) do not select aunique set of pump parameters a, b, and g and the mo-ment of ionization. However, they are essential in thehunt for such sets, leaving the job for a computer as well.

Thus, using Eq. (10) or (12), we can determine the suit-able set of pump parameters a, b, and g and the moment

Fig. 2. Trajectories for ‘‘colliding’’ (solid curve) and ‘‘noncollid-ing’’ (dashed curve) electrons for a specific set of parameters a, b,g, and t0 . Colliding trajectories correspond to ionization mo-ments t ion inside the range (t0 2 dt0/2) –(t0 1 dt0/2), which leadback to the parent ion (r0 min , s). Noncolliding trajectoriescorrespond to ionization moments ut ion 2 t0u > dt0/2 for which,after ionization, the minimal distance of the photoelectron to theparent ion is r0 min > s. A colliding trajectory with t ion 5 t0(which leads exactly back to the parent ion: r0 min 5 0 at themoment tk) and a noncolliding trajectory with t ion 5 t01 dt0/2 are presented.


of ionization t0 . Then computer simulation of photoelec-tron trajectories (6) and (7) for fixed parameters a, b, andg and different ionization moments t ion 5 t0 and t ionÞ t0(Fig. 2) makes it possible to find the deviation dt0 of themoment of ionization [and appropriate deviation dtk ofthe moment of recombination at which the condition of re-combination (8) occurs, that is, the condition when the de-viation of the electron trajectory from the parent ion isless than the electron wave-packet width (r0 , s; seeFig. 2)]. In this way we can find the ‘‘elementary’’ dura-tion of recombination emission (emission from singleatom): tg ' dtk .

4. RESULTS OF NUMERICAL SIMULATIONSAND CALCULATIONS OF ‘‘ELEMENTARY’’DURATIONThe above considerations make it possible to determinethe pulse duration of recombination emission with a two-color pump [Eqs. (4) and (5)]. Here the results of numeri-cal simulations for HF radiation of wavelength l5 1 –0.8 mm and intensities IHF ; 1015–1017 W/cm2 arepresented for the X and Y cases. Radiation of such inten-sity is quite realistic and has already been used in experi-ments and calculations on above-threshold ionization andHHG with highly ionized ions (using such ions permitsone to realize the tunnel regime of ionization that is es-sential for HHG). For example, in Ref. 23 the generationof high harmonics with He1 ions was calculated for laserintensity I 5 1017 W/cm2. In Ref. 24 lasers of intensitiesI ; 1016–1017 W/cm2 were used for tunnel ionization ofnoble gases (Ar, Kr, Ne, Xe). Highly charged ions (withcharge until z 5 8) were observed. In Refs. 25 and 26,the calculations of HHG with such ions and I; 1016–1019 W/cm2 were performed. Note also thatCO2 lasers (as sources of static field) with intensity of ap-proximately 1014–1016 W/cm2 have become quite realisticowing to the development of terawatt CO2 systems atBrookhaven National Laboratory. For such laserspreionization of atoms can cause a serious problem andshould be specially studied and accounted for when choos-ing atom or ion species.

X case. For the HF part of pump radiation of intensityIHF 5 1017 W/cm2, one can find the following set of pa-rameters that obey the requirement (12): a 5 1/A2, b' 0.4, g ' 0.65, and the one and only ionization momentt0 ' 2p/5 during the optical period T 5 2p/v (the mo-ment t0 is counted off from the HF field maximum). Withthese parameters used in simulation, the trajectories ofthe electron after atom ionization give the following: re-combination moment tk ' 1.2p, electron kinetic energyat the moment of recombination of ek ' 3.6, and devia-tions dt0 ' dtk ' 0.02 at which the condition of recombi-nation (8) holds valid. Such a deviation dtk correspondsto a dimension duration of recombination emission from asingle atom of tg ' 10 as. Note again that the found setof pump parameters that leads to such a short recombi-nation pulse is not the only one and is not necessarily thebest one. So the real duration tg may be shorter. Alsonote that using pump intensities IHF ; 1017–1018 W/cm2

requires taking into account the influence of the magneticfield of laser radiation on electron trajectories.19 Ac-

counting for the magnetic field, however, does not essen-tially change the recombination pulse duration, althoughthe required parameters a, b, g, and t0 [and the require-ments (10) and (12)] will change. Note also that there ex-ist methods of HHG with magnetic field compensation (forexample, by using a standing-wave pump).19,26

Y case. For intensity IHF 5 1015 W/cm2, the followingparameters were found from Eq. (10): a 5 1/A2, b' 20.22, g ' 21, and t0 5 0 (note that in the suggestedscheme of HHG the ionization moment t0 , unlike in ordi-nary HHG, can be selected near the HF field maximum,i.e., t0 5 0, which increases the ionization probability).For this set of parameters, simulations of electron trajec-tories give: tk ' 2p, ek ' 4, and dt0 ' dtk ' 0.25,which correspond to recombination emission with tg' 100 as. With the increase of pump intensity, the du-ration tg decreases. For example, for IHF 5 53 1017 W/cm2 the following parameters are found: a5 1/A2, b ' 20.276, g ' 20.96, and t0 5 p/10. Forthese parameters tk ' 1.9p, ek ' 4.5, dt0 ' 0.01, anddtk ' 0.002 are obtained, which correspond to recombina-tion emission with subattosecond duration tg ' 0.9 as.This duration is significantly less than the pump opticalperiod T (by more than 3 orders of magnitude); however,it contains approximately 50 optical cycles of recombina-tion emission (the duration of this cycle is determined bythe parameter ek).

Note that there is no analytical proof that Eqs. (10) and(12) select only one colliding trajectory during the opticalperiod (with ionization moment t0 inside the dt0 range).However, in all cases that we have considered numeri-cally (for fixed parameters a, b, and g), there was only onesuch trajectory. Furthermore, one colliding trajectorycan exist during the whole HF pulse.

The above calculations relate to each optical cycle of HFpump radiation: In each cycle a burst of recombinationemission of duration tg will occur. So, for a long (multi-cycled) HF pump pulse, the recombination radiation willbe a train of attosecond pulses with repetition rate v.For example, it can be used for petahertz spectroscopywith attosecond time resolution. However, the selectionof a single attosecond pulse from such a train is a difficultproblem.

A real laser field (with real temporal and spatial struc-ture) favors the creation of the single attosecond pulse.For example, special calculations show that with an ex-tremely short HF pump pulse (when radiation of only oneoptical cycle ionizes an atom effectively) the use of a two-color pump [Eqs. (4) and (5)] makes it possible to obtain asingle attosecond (or subattosecond) pulse of recombina-tion radiation automatically (without additional effort).

5. CONCLUSIONSThis study suggests generating 1–10-as pulses of highharmonic emission or even breaking through the subat-tosecond barrier with the technique of a special two-colorpump [Eqs. (4) and (5)]. Such a pump differs from a two-color pump meant for the selection of a narrow frequencyband of harmonic radiation,13 where two-color radiationexercises the control of ionization moments: Photoelec-trons are ejected into the continuum during only a small


part of the pump optical cycle; however, all of them returnto the parent ion and recombine. In the present case thetwo-color pump allows atom ionization at any phase.However, because of the high sensitivity of colliding con-dition (8) with regard to pump parameters, only a verysmall number of all photoelectrons participate in recom-bination with the parent ion, and recombination itself isvery short. Naturally, the total harmonic energy de-creases for the same reason.

Actually, the suggested method works as a selector (nota compressor) of recombination emission that leads to thedecrease of irradiative energy; however, the intensity ofsuch an emission can be substantial. Note that the prob-ability of harmonic generation with pump field [Eqs. (4)and (5)] is determined, as in ordinary HHG, by the prob-ability of atom (ion) ionization, the velocity of the electronwave-packet spreading in the continuum, and the prob-ability of electron–ion recombination. Using a circularlypolarized field (as well as a combination of several fields)does not change the probability of tunnel ionization be-cause it depends only on instantaneous field strength(and does not depend on pump field structure). Wave-packet spreading and recombination probability per onephotoelectron in the present scheme are also the same asthose in the ordinary case. However, selection of only asmall part of all photoelectrons, which recombine with theparent ion (dt0 /T ; 0.1–0.01), leads to short-pulse gen-eration as well as to selection of a small part dvg of a wideplateau of the harmonic spectrum (note that there is nocontradiction between small pulse duration tg and nar-row spectral range dvg , since dvgtg @ 1). So the sug-gested method decreases the whole harmonic energy.However, it does not essentially decrease the harmonic in-tensities in the selected range of the HHG spectrum:These intensities should be approximately the same asthose in the ordinary HHG process.

Thus harmonic energy is low precisely because the sug-gested method is actually a method of selection of a smallpart of all photoelectrons, (or selection of a very smallpart of the whole harmonic spectrum). This means thatthe energy of the attosecond pulse is proportional to itsduration: e ; dt0 ' dtk . An alternative way (whichcan increase the energy of the attosecond pulse) is com-pression, where dtk ! dt0 . The author has observedsuch cases in numerical simulations. However, this pos-sibility requires additional investigation.

We considered here the recombination emission fromonly a single atom. Because of the propagation effect (aswell as because of space inhomogeneity of the laserbeam), the attosecond pulse of such an emission can bestretched or compressed, which also requires specialinvestigation.27

Finally, note that requirements (10) and (12) for thesuitable set of pump parameters, providing short recom-bination emission, do not select the best combination ofsuch parameters. Additional effort (numerical or experi-mental) makes it possible to get pulses of recombinationemission that are truly less than 1 as.

ACKNOWLEDGMENTThis work was supported by the Russian Foundation forBasic Research (grant 00-15-96726).

REFERENCES1. M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider,

N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M.Drescher, and F. Krausz, ‘‘Attosecond metrology,’’ Nature(London) 414, 509–513 (2001).

2. M. Drescher, M. Hentschel, R. Kienberger, G. Tempea, Ch.Spielmann, G. A. Reider, P. B. Corkum, and F. Krausz, ‘‘X-ray pulses approaching the attosecond frontier,’’ Science291, 1923–1927 (2001).

3. H. Niikura, F. Legare, R. Hasbani, A. D. Bandrauk, M. Yu.Ivanov, D. M. Villeneuve, and P. B. Corkum, ‘‘Sub-laser-cycle electron pulses for probing molecular dynamics,’’ Na-ture (London) 417, 917–922 (2002).

4. A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E.Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A.Scrinzi, T. W. Hansch, and F. Krausz, ‘‘Attosecond control ofelectronic processes by intense light fields,’’ Nature (Lon-don) 421, 611–615 (2003).

5. P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph.Balcou, H. G. Muller, and P. Agostini, ‘‘Observation of atrain of attosecond pulses from high harmonic generation,’’Science 292, 1689–1692 (2001).

6. R. Kienberger, M. Hentschel, M. Uiberacker, Ch. Spiel-mann, M. Kitzler, A. Scrinzi, M. Wieland, Th. Westerwal-besloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F.Krausz, ‘‘Steering attosecond electron wave packets withlight,’’ Science 297, 1144–1148 (2002).

7. Ph. Antoine, A. L’Huillier, and M. Lewenstein, ‘‘Attosecondpulse trains using high-order harmonics,’’ Phys. Rev. Lett.77, 1234–1237 (1996).

8. P. B. Corkum, N. H. Burnett, and M. Y. Ivanov, ‘‘Subfemto-second pulses,’’ Opt. Lett. 19, 1870–1872 (1994).

9. M. Bao and A. F. Starace, ‘‘Static-electric-field effect on highharmonic generation,’’ Phys. Rev. A 53, R3723–R3726(1996).

10. A. Lohr, W. Becker, and M. Kleber, ‘‘High-harmonic genera-tion affected by static electric field,’’ Laser Phys. 7, 615–622(1997).

11. B. Wang, X. Li, and P. Fu, ‘‘Polarization effects in high-harmonic generation in the presence of static-electric field,’’Phys. Rev. A 59, 2894–2902 (1999).

12. B. Borca, A. V. Flegel, M. V. Frolov, N. L. Manakov, D. B.Milosevic, and A. F. Starace, ‘‘Static-electric-field-inducedpolarization effects in harmonic generation,’’ Phys. Rev.Lett. 85, 732–735 (2000).

13. V. D. Taranukhin, ‘‘High-order harmonic generation in anarrow spectral range,’’ Quantum Electron. 31, 555–556(2001).

14. M. Lewenstein, Ph. Balcou, M. Y. Ivanov, A. L’Huillier, andP. B. Corkum, ‘‘Theory of high-harmonic generation by low-frequency laser fields,’’ Phys. Rev. A 49, 2117–2132 (1994).

15. R. V. Kulyagin and V. D. Taranukhin, ‘‘Generation of X-rayradiation by atoms in an ultra-high intensity laser field,’’Quantum Electron. 26, 866–870 (1996).

16. P. B. Corkum, ‘‘Plasma perspective on strong-field multi-photon ionization,’’ Phys. Rev. Lett. 71, 1994–1997 (1993).

17. E. Mevel, P. Breger, R. Trainham, G. Petite, P. Agostini, A.Migus, J.-P. Chambaret, and A. Antonetti, ‘‘Atoms in strongoptical fields: evolution from multiphoton to tunnel ioniza-tion,’’ Phys. Rev. Lett. 70, 406–409 (1993).

18. J. B. Watson, A. Sanpera, and K. Burnett, ‘‘Pulse-shape ef-fects and blueshifting in the single-atom harmonic genera-tion from neutral species and ions,’’ Phys. Rev. A 51, 1458–1463 (1995).

19. V. D. Taranukhin, ‘‘Relativistic high-order harmonic gen-eration,’’ Laser Phys. 10, 330–336 (2000).

20. A. Scrinzi, M. Geissler, and T. Brabec, ‘‘Ionization above theCoulomb barrier,’’ Phys. Rev. Lett. 83, 706–709 (1998).

21. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, ‘‘Ioniza-tion of atoms with electric AC fields,’’ Sov. Phys. JETP 24,207–217 (1967).

22. V. D. Taranukhin and N. Yu. Shubin, ‘‘High-order harmonicgeneration by atoms with strong high-frequency and low-frequency pumping,’’ J. Opt. Soc. Am. B 17, 1509–1516(2000).


23. M. V. Walser, C. H. Keitel, A. Scrinzi, and T. Brabec, ‘‘Highharmonic generation beyond the electric dipole approxima-tion,’’ Phys. Rev. Lett. 85, 5082–5085 (2000).

24. S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, andJ. H. Eberly, ‘‘Tunneling ionization of noble gases in ahigh-intensity laser field,’’ Phys. Rev. Lett. 63, 2212–2215(1989).

25. D. B. Milosevic, S. Hu, and W. Becker, ‘‘Quantum-mechanical model for ultrahigh-order harmonic generation

in the moderately relativistic regime,’’ Phys. Rev. A 63,011403R (2001).

26. V. D. Taranukhin and N. Yu. Shubin, ‘‘High-order harmonicgeneration by atoms with traveling and standing wavepump of relativistic intensity,’’ J. Opt. Soc. Am. B 19, 1132–1140 (2002).

27. N. Milosevic, A. Scrinzi, and T. Brabec, ‘‘Numerical charac-terization of high harmonic attosecond pulses,’’ Phys. Rev.Lett. 88, 093905 (2002).

attosecond pulse generation by a two-color field

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