ABOVE-THRESHOLDIONIZATION: …people.physics.tamu.edu/ggp/Publications/027-advances02.pdf38 W.Beckeretal. [I A.Experimental Methods ATIisobservedintheintensityregime1012W/cm2 to1016W/cm2.Atsuch

ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 48

ABOVE-THRESHOLD IONIZATION:FROM CLASSICAL FEATURES TOQUANTUM EFFECTSW. BECKER 1, F. GRASBON 2, R. KOPOLD 1, D.B. MILOSEVIC 3,G.G. PAULUS 2 and H. WALTHER 2,41Max-Born-Institut, Max-Born-Str. 2A, 12489 Berlin, Germany; 2Max-Planck-Institut furQuantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany; 3Faculty of Science,University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Hercegovina;4Ludwig-Maximilians-Universitat Munchen, Germany

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35A. Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38B. Theoretical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

II. Direct Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40A. The Classical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40B. Quantum-mechanical Description of Direct Electrons . . . . . . . . . . . . . . . . . 43C. Interferences of Direct Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

III. Rescattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50A. The Classical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50B. Quantum-mechanical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

IV. ATI in the Relativistic Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73A. Basic Relativistic Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73B. Rescattering in the Relativistic Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

V. Quantum Orbits in High-order Harmonic Generation . . . . . . . . . . . . . . . . . . . . 76A. The Lewenstein Model of High-order Harmonic Generation . . . . . . . . . . . . . 77B. Elliptically Polarized Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78C. HHG by a Two-color Bicircular Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78D. HHG in the Relativistic Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

VI. Applications of ATI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86A. Characterization of High Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86B. The “Absolute Phase” of Few-cycle Laser Pulses . . . . . . . . . . . . . . . . . . . . 90

VII. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91VIII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

I. Introduction

With the discovery of above-threshold ionization (ATI) by Agostini et al.(1979) intense-laser atom physics entered the nonperturbative regime. Theseexperiments recorded the photoelectron kinetic-energy spectra generated by laserirradiation of atoms. Earlier experiments had measured total ionization rates

35 Copyright © 2002 Elsevier Science (USA)All rights reserved

ISBN 0-12-003848-X/ISSN 1049-250X/01 $35.00

36 W. Becker et al. [I

Fig. 1. Photoelectron spectrum in the above-threshold-ionization (ATI) intensity regime. The seriesof peaks corresponds to the absorption of photons in excess of the minimum required for ionization.The figure shows the result of a numerical solution of the Schrodinger equation (Paulus, 1996).

by way of counting ions, and the data were well described by lowest-orderperturbation theory (LOPT) with respect to the electron–field interaction. ThisLOPT regime was already highly nonlinear (see, e.g., Mainfray and Manus,1991), the lowest order being the minimal number N of photons necessary forionization. An ATI spectrum consists of a series of peaks separated by the photonenergy, see Fig. 1. They reveal that an atom may absorb many more photons thanthe minimum number N , which corresponds to LOPT.In the 1980s, the photon spectra emitted by laser-irradiated gaseous media

were investigated at comparable laser intensities and were found to exhibit peaksat odd harmonics of the laser frequency (McPherson et al., 1987; Wildenauer,1987). The spectra of this high-order harmonic generation (HHG) display aplateau (Ferray et al., 1988), i.e., the initial decrease of the harmonic yieldwith increasing harmonic order is followed by a flat region where the harmonicintensity is more or less independent of its order. This plateau region terminatesat some well-defined order, the so-called cutoff.A simple semiclassical model of HHG was furnished by Kulander et al. (1993)

and by Corkum (1993): At some time, an electron enters the continuum byionization. Thereafter, the laser’s linearly polarized electric field accelerates theelectron away from the atom. However, when the field changes direction, then,depending on the initial time of ionization, it may drive the electron back toits parent ion, where it may recombine into the ground state, emitting its entireenergy – the sum of the kinetic energy that it acquired along its orbit plus the

I] ATI: CLASSICAL TO QUANTUM 37

binding energy – in the form of one single photon. This simple model beautifullyexplains the cutoff energy of the plateau, as well as the fact that the yield of HHGstrongly decreases when the laser field is elliptically polarized. In this event, theelectron misses the ion. This model is often referred to as the simple-man model.The model suggests (Corkum, 1993) that the electron, when it recollides with

the ion, may very well scatter off it, either elastically or inelastically. Elasticscattering should contribute to ATI. Indeed, the corresponding characteristicfeatures in the angular distributions were observed by Yang et al. (1993), andan extended plateau in the energy spectra due to this mechanism, much likethe plateau of HHG, was identified by Paulus et al. (1994c). Under the sameconditions, a surprisingly large yield of doubly charged ions was recorded(l’Huillier et al., 1983; Fittinghoff et al., 1992) that was incompatible with asequential ionization process. A potential mechanism causing this nonsequentialionization (NSDI) is inelastic scattering. It was only recently, however, thatthis inelastic-scattering scenario emerged as the dominant mechanism of NSDI,through analysis of measurements of the momentum distribution of the doublycharged ions (Weber et al., 2000a,b; Moshammer et al., 2000).The semiclassical rescattering model sketched above has proved invaluable

in providing intuitive understanding and predictive power. It was embedded infully quantum-mechanical descriptions of HHG (Lewenstein et al., 1994; Beckeret al., 1994b) and ATI (Becker et al. 1994a; Lewenstein et al., 1995a). Thiswork has led to the concept of “quantum orbits,” a fully quantum-mechanicalgeneralization of the classical orbits of the simple-man model that retains theintuitive appeal of the former, but allows for interference and incorporatesquantum-mechanical tunneling. The quantum orbits arise naturally in the contextof Feynman’s path integral (Salieres et al., 2001).This review will concentrate on ATI and the various formulations of the

rescattering model, from the simplest classical model to the quantum orbits forelliptical polarization. Alongside with theory, we will provide a review of theexperimental status of ATI. We also give a brief survey of recent applications ofATI. High-order harmonic generation is considered only insofar as it providesfurther illustrations of the concept and application of quantum orbits. We do notdeal with the important collective aspects of HHG, and no attempt is made torepresent the vast literature on HHG. For this purpose, we refer to the recentreviews by Salieres et al. (1999) and Brabec and Krausz (2000). Earlier reviewspertinent to ATI have been given by Mainfray and Manus (1991), DiMauro andAgostini (1995), and Protopapas et al. (1997). The entire field of laser–atomphysics has been compactly surveyed by Kulander and Lewenstein (1996) and,recently, by Joachain et al. (2000). Both of these reviews concentrate on thetheory. Nonsequential double ionization is well covered in a recent focus issueof Optics Express, Vol. 8.

38 W. Becker et al. [I

A. Experimental Methods

ATI is observed in the intensity regime 1012 W/cm2 to 1016 W/cm2. At suchintensities, atoms may ionize so quickly that complete ionization has takenplace before the laser pulse has reached its maximum. This calls, on theone hand, for atoms with high ionization potential (i.e. the rare gases) and,on the other, for ultrashort laser pulses. Owing to the rapid progress infemtosecond laser technology, in particular since the invention of titanium–sapphire (Ti:Sa) femtosecond lasers (Spence et al., 1991), generation of laserfields with strengths comparable to inner atomic fields has become routine. Theprerequisite of detailed investigations of ATI, however, has been the developmentof femtosecond laser systems with high repetition rate. Owing to the latter,the detection of faint but qualitatively important features of ATI spectra withlow statistical noise has become possible. This holds, in particular, if multiplydifferential ATI spectra are to be studied, such as angle-resolved energy spectra,or spectra that are very weak, such as for elliptical polarization or outside theclassically allowed regions. State-of-the-art pulses are as short as 5 fs (Nisoliet al., 1997) and repetition rates reach 100 kHz (Lindner et al., 2001).The most widespread method of analyzing ATI electrons is time-of-flight

spectroscopy. When the laser pulse creates a photoelectron, it simultaneouslytriggers a high-resolution clock. The electrons drift in a field-free flight tubeof known length towards an electron detector, which then gives the respectivestop pulses to the clock. Now, their kinetic energy can easily be calculated fromtheir time of flight. This approach has by far the highest energy resolution andis comparatively simple. However, the higher the laser repetition rate, the moredemanding becomes the data aquisition.Other approaches include photoelectron imaging spectroscopy (Bordas et al.,

1996), which is able to record angle-resolved ATI spectra, and so-called cold-target recoil-ion-momentum spectroscopy (COLTRIMS) technology (Dorneret al., 2000), which is capable of providing complete kinematic determination ofthe fragments of photoionization, i.e. the electrons and ions. It requires, however,conditions such that no more than one atom is ionized per laser shot. Therefore,it can take particular advantage of high laser repetition rates. The disadvantageof COLTRIMS is the poor energy resolution for the electrons and the exactingtechnology.

B. Theoretical Methods

The single-active-electron approximation (SAE) replaces the atom in the laserfield by a single electron that interacts with the laser field and is bound by aneffective potential so optimized as to reproduce the ground state and singlyexcited states. Up to now, in single ionization no qualitative effect has beenidentified that would reveal electron–electron correlation. The SAE has found

I] ATI: CLASSICAL TO QUANTUM 39

Fig. 2. (a) Measured and (b) calculated photoelectron spectrum in argon for 800 nm, 120 fs pulsesat the intensities given in TW/cm2 in the figure (10UP = 39 eV). From Nandor et al. (1999).

its most impressive support in the comparison of experimental ATI spectra inargon with spectra calculated by numerical solution of the three-dimensionaltime-dependent Schrodinger equation (TDSE) (Nandor et al., 1999); see Fig. 2.The agreement between theory and experiment is as remarkable as it has beenachieved for low-order ATI in hydrogen; cf. Dorr et al. (1990) for the Sturmian–Floquet calculation and Rottke et al. (1990) for the experiment. For helium, acomparison of total ionization rates with and without the SAE in the above-barrier regime has lent further support to the SAE (Scrinzi et al., 1999).Numerical solution of the one-particle TDSE in one dimension was instru-

mental for the understanding of ATI in its early days; for a review, see Eberlyet al. (1992). For the various methods of solving the TDSE in more than onedimension we refer to Joachain et al. (2000). Comparatively few papers havedealt with high-order ATI in three (that is, in effect, two) dimensions. This isparticularly challenging since the emission of plateau electrons is caused byvery small changes in the wave function, and the large excursion amplitudesof free-electron motion in high-intensity low-frequency fields necessitate a largespatial grid. This is exacerbated for energies above the cutoff and for ellipticalpolarization. Expansion of the radial wave function in terms of a set of B-splinefunctions was used by Paulus (1996), by Cormier and Lambropoulos (1997),and by Lambropoulos et al. (1998). Matrix-iterative methods were employed byNurhuda and Faisal (1999). The most detailed calculations have been carriedout by Nandor et al. (1999) and by Muller (1999a,b, 2001a,b). The techniquesare detailed by Muller (1999c). To our knowledge, no results for high-order ATIfor elliptical polarization based on numerical solution of the TDSE have beenpublished to this day.Recently, numerical solution of the TDSE for a two-dimensional model

atom by means of the split-operator method has been widely used in order to

40 W. Becker et al. [II

investigate various problems such as elliptical polarization (Protopapas et al.,1997), stabilization (Patel et al., 1998; Kylstra et al., 2000), magnetic-drift effects(Vazquez de Aldana and Roso, 1999; Vazquez de Aldana et al., 2001) andvarious low-order relativistic effects (Hu and Keitel, 2001).Efforts to deal with the two-electron TDSE and, in particular, to compute

double-electron ATI spectra are under way (Smyth et al., 1998; Parker et al.,2001; Muller, 2001c). In one dimension for each electron, such spectra havebeen obtained by Lein et al. (2001).An approach that is almost complementary to the solution of the TDSE starts

from the analytic solution for a free electron in a plane-wave laser field, the so-called Volkov solution (Volkov, 1935), which is available for the Schrodingerequation as well as for relativistic wave equations, and considers the bindingpotential as a perturbation. The stronger the laser field, the lower its frequency,and the longer the pulse becomes, the more demanding is the solution of theTDSE, and the more the Volkov-based methods play out their strengths.This review concentrates on methods of the latter variety.

II. Direct Ionization

A. The Classical Model

The classical model of strong-field effects divides the ionization process intoseveral steps (van Linden van den Heuvell and Muller, 1988; Kulander et al.,1993; Corkum, 1993; Paulus et al., 1994a, 1995). In a first step, an electronenters the continuum at some time t0. If this is caused by tunneling (Chin et al.,1985; Yergeau et al., 1987; Walsh et al., 1994), the corresponding rate is a highlynonlinear function of the laser electric field E(t0). For example, the quasistaticAmmosov–Delone–Krainov (ADK) tunneling rate (Perelomov et al., 1966a,b;Ammosov et al., 1986) is given by (in atomic units)

G(t) = AEIP

(4√2E3IP|E(t)|

)2n∗ − |m| − 1exp

(−4√2E3IP

3|E(t)|

), (1)

where E(t) is the instantaneous electric field, EIP > 0 is the ionizationpotential of the atom, n∗ = Z/

√2EIP is the effective principal quantum number,

Z is the charge of the nucleus, and m is the projection of the angular momentumon the direction of the laser polarization. The constant A depends on the actualand the effective quantum numbers. The rate G(t) was derived on the assumptionthat the laser frequency is low, excited states play no role, and the Keldyshparameter g =

√EIP/2UP is small compared with unity [UP is the ponderomotive

potential of the laser field; see Eq. (3) below]. Instantaneous rates that holdfor arbitrary values of the Keldysh parameter have been presented by Yudin

II] ATI: CLASSICAL TO QUANTUM 41

and Ivanov (2001b). For the discussion below, the important feature of theinstantaneous ionization rate G(t) is that it develops a sharp maximum at timeswhen the field E(t) reaches a maximum.The classical model considers the orbits of electrons that are released into the

laser-field environment at some time t0. The contribution of such an orbit willbe weighted according to the value of the rate G(t0). Classically, an electron bornby tunneling will start its orbit with a velocity of zero at the classical “exit of thetunnel” at r ~ EIP/ |eE|, which, for strong fields, is a few atomic units away fromthe position of the ion. We will, usually, ignore this small offset and have theelectronic orbit start at x(t = t0) = 0 (the position of the ion) with x(t = t0) = 0.If, after the ionization process, the interaction of the electron with the ion

is negligible, we speak of a “direct” electron, in contrast to the case, to beconsidered below in Sect. III, where the electron is driven back to the ion andrescatters. An unambiguous distinction between direct and rescattered electrons,in particular for low energy, is possible only in theoretical models.

A.1. Basic kinematics

The second step of the classical model is the evolution of the electron trajectoryin the strong laser field. During this step, the influence of the atomic potentialis neglected. For an intense laser field, the electron’s oscillation amplitude ismuch larger than the atomic diameter, and so this is well justified. For a vectorpotential A(t) that is chosen so that its cycle average 〈A(t)〉T is zero, theelectron’s velocity is

mv(t) = e(A(t0) − A(t)) ≡ p − eA(t), (2)

where e = −|e| is the electron’s charge. The velocity consists of a constant termp ≡ eA(t0), which is the drift momentum measured at the detector, and a termthat oscillates in phase with the vector potential A(t). The kinetic energy of thiselectron, averaged over a cycle T of the laser field, is

E =m

2〈v(t)2〉T = p2

2m+e2

2m〈A(t)2〉T ≡ Edrift + UP. (3)

The ponderomotive energy

UP =e2

2m〈A(t)2〉T , (4)

viz. the cycle-averaged kinetic energy of the electron’s wiggling motion, isfrequently employed to characterize the laser intensity. A useful formula is

Up [eV] = 0.09337I [W/cm2] l2[m]

for a laser with intensity I and wavelength l. If the electron is to have a nonzerovelocity v0 at time t0, one has to replace eA(t0) by eA(t0) + mv0 ≡ p in thevelocity (2).


Most of the time, we will be concerned with the monochromatic ellipticallypolarized laser field (−1 � x � 1)

E(t) = wA√1 + x2

(x sinwt − xy coswt) (5)

with vector potential

A(t) =A√1 + x2

(x coswt + xy sinwt) (6)

and ponderomotive energy UP = (eA)2/4m. The drift energy Edrift = (eA(t0))2/2mis restricted to the interval

2x2

1 + x2UP � Edrift �

21 + x2

UP. (7)

For linear polarization, it can acquire any value between 0 and 2UP , whilefor circular polarization it is restricted to the value UP . Quantum mechanicsconsiderably softens these classical bounds. However, these bounds are useful asbenchmarks in the analysis of experimental spectra (Bucksbaum et al., 1986),in particular for high intensity (Mohideen et al., 1993).In general, it is important to recall that the ionization probability depends

on the electric field, while the drift momentum p = eA(t0) is proportionalto the vector potential, both at the time t0 of ionization. The probability ofa certain drift momentum is weighted with the ionization rate at time t0. Theelectron is preferably ionized when the absolute value of the electric field isnear its maximum. Then, for linear polarization, the vector potential and, hence,the drift momentum are near zero. In order to reach the maximal drift energyof 2UP , the electron must be ionized when the electric field is zero and, hence,the ionization rate is very low. This explains the pronounced drop of the ATIelectron spectrum for increasing energy, see Fig. 1. Sometimes, this interplaybetween the instantaneous ionization rate and the drift momentum has surprisingconsequences, notably for fields where the connection between the two is lessstraightforward than for a linearly polarized sinusoidal field, e.g. for a two-colorfield (Paulus et al., 1995; Chelkowski and Bandrauk, 2000; Ehlotzky, 2001).Another illustration is the dodging phenomenon for the direct ATI electronsin an elliptically polarized laser field (Paulus et al., 1998; Goreslavskii andPopruzhenko, 1996; Mur et al., 2001), see Fig. 3.We have tacitly assumed that pulses are short enough to pass over the electron

before it has a chance to experience the transverse spatial gradient of thefocused pulse. In this event, the spatial dependence of the vector potential A(t)can truly be neglected. Hence the drift momentum p is conserved and is


Fig. 3. Dependence of the photoelectron yield as a function of the ellipticity x of the ellipticallypolarized laser field (5) for electrons with an energy of 16.1 eV. Only electrons emitted parallel tothe major axis of the polarization ellipse are recorded. The ATI spectrum corresponding to linearpolarization (x = 0) is shown in the inset. The laser intensity was 0.8×1014 W/cm2 at a wavelengthof 630 nm. The figure illustrates the dodging effect mentioned in Sect. II.A.1: ionization primarilytakes place when the electric field is near an extremum. For elliptical polarization, the electric fieldthen points in the direction of the major axis of the polarization ellipse, and the vector potential inthe direction of the minor axis. Hence, the electron’s drift momentum p = eA(t0) is in the direction ofthe minor axis. It is the larger, the larger the ellipticity x is. Consequently, emission in the directionof the large component of the field decreases with increasing ellipticity: the electron dodges thestrong component of the field. The effect vanishes when circular polarization is approached and the

distinction between the major and minor axes disappears. From Paulus et al. (1998).

indeed the momentum recorded at the detector outside the field (Kibble, 1966;Becker et al., 1987). The wiggling energy UP is lost or, in a self-consistentdescription, returned to the field when the electron is left behind by the trailingedge of the pulse. In the opposite case, where the electron escapes from thepulse perpendicularly to its direction of propagation, the wiggling energy isconverted into drift energy (Muller et al., 1983). The effects of a space-dependentponderomotive potential UP were observed in the “surfing” experiments ofBucksbaum et al. (1987).

B. Quantum-mechanical Description of Direct Electrons

There is an enormous body of work on the quantum-mechanical description oflaser-induced ionization. For reviews, we refer to Delone and Krainov (1994,1998). Here we want to concentrate on the analytical approach dating back toKeldysh (1964) and Perelomov, Popov and Terent’ev (Perelomov et al. 1966a,b;


Perelomov and Popov, 1967). The goal is to find a suitable approximation to theprobability amplitude for detecting an ATI electron with drift momentum p thatoriginates from laser irradiation of an atom that was in its ground state |y0〉before the laser pulse arrived:

Mp = limt→ ∞, t′ → −∞

〈yp(t) |U (t, t′)|y0(t′)〉 . (8)

Here, U (t, t′) is the time-evolution operator of the Hamiltonian (à = 1)

H (t) = −12m∇2 − er · E(t) + V (r), (9)

which includes the atomic binding potential V (r) and the interaction −er · E(t)with the laser field. Furthermore, we introduce the Hamiltonians for the atomwithout the field and for a free electron in the laser field without the atom,

Ha ≡ Hatom = − 12m∇2 + V (r), (10)

Hf (t) ≡ Hfield(t) = − 12m∇2 − er · E(t). (11)

The corresponding time-evolution operators are denoted by Ua and Uf , re-spectively. In Eq. (8), |yp〉 and |y0〉 are a scattering state with asymptoticmomentum p and the ground state, respectively, of the atomic Hamiltonian Ha.The eigenstates of the time-dependent Schrodinger equation with the Hamilto-nian Hf (t) are known as the Volkov states and are of compact analytical form.In the length gauge, one has

|y (Vv)p (t)〉 = |p − eA(t)〉e−iSp(t), (12)

with |p − eA(t)〉 a plane-wave state [〈r|p − eA(t)〉 = (2p )−3/2 exp i[(p − eA(t)) · r]and

Sp(t) =12m

∫ t

dt [p − eA(t)]2. (13)

The lower limit of the integral is immaterial. It introduces a phase that does notcontribute to any observable.The time-evolution operator U (t, t′) satisfies integral equations (Dyson equa-

tions) that are convenient if one wants to generate perturbation expansions withrespect to either the interaction HI (t) = −er · E(t) with the laser field,

U (t, t′) = Ua(t, t′) − i

t∫t′

dt U (t, t)HI (t)Ua(t , t′), (14)

or the binding potential V (r),

U (t, t′) = Uf (t, t′) − i

t∫t′

dt Uf (t, t)V U (t , t′). (15)

Equation (14) also holds if U and Ua in the second term on the right-hand sideare interchanged. The equivalent is true of Eq. (15). With the help of the integral


equation (14), using the orthogonality of the eigenstates of Ha, we rewrite Eq. (8)in the form

Mp = −i limt→ ∞

∫ t

−∞dt 〈yp(t) |U (t, t)HI (t)|y0(t)〉 , (16)

which is still exact. A crucial simplification occurs if we now introduce thestrong-field approximation. That is, we make the substitutions |yp〉 → |y (Vv)p 〉and U → Uf , with the result

Mp = −i∫ ∞

−∞dt0⟨y (Vv)p (t0) |HI (t0)|y0(t0)

⟩. (17)

The physical content of this substitution is that, after the electron has beenpromoted into the continuum at time t0 due to the interaction HI (t0) = −er · E(t0)with the laser field, it no longer feels the atomic potential. This satisfies the abovedefinition of a “direct electron.” Amplitudes of the type (17) are called Keldysh–Faisal–Reiss (KFR) amplitudes (Keldysh, 1964; Perelomov et al., 1966a,b;Faisal, 1973; Reiss, 1980); for a comparison of the various forms that exist seeReiss (1992).In the amplitude (17), one may write

− er · E(t0) = Hf (t0) − Ha + V (r) = −i←−−ð

ðt0− i−−→ð

ðt0+ V (r). (18)

Via integration by parts, the amplitude (17) can then be rewritten as

Mp = −i∫ ∞

−∞dt0⟨y (Vv)p (t0) |V (r)|y0(t0)

⟩. (19)

This form is particularly useful for a short-range or zero-range potential, sincethese restrict the range of the spatial integration in the matrix element.Further evaluation of the amplitudes (17) or (19) leads to expansions

in terms of Bessel functions. For sufficiently high intensity (small Keldyshparameter g), the saddle-point method (method of steepest descent) can beinvoked (Dykhne, 1960). This consists in expanding the phase of the integrandabout the points where the phase is stationary. Given the form of the Volkovwave functions (12) and the time dependence of the ground-state wave function,|y0(t)〉 = exp(iEIPt)|y0〉, this amounts to determining the solutions of

d

dt[EIPt + Sp(t)] = EIP + 1

2 [p − eA(t)]2 = 0. (20)


Let us consider a periodic (not necessarily monochromatic) vector potential withperiod T = 2p /w. In terms of the solutions ts of Eq. (20), the amplitude (19)can then be written as

Mp ∝∑n

d(

p2

2m+ EIP + UP − nw

)

×∑s

(2p iS ′′

p (ts)

)1/2ei[EIPts + Sp(ts)]〈p − eA(ts)|V |y0〉,

(21)

where S ′′p denotes the second derivative of the action (13) with respect to time.

The sum over s extends over those solutions of Eq. (20) within one period ofthe field (e.g. such that 0 � Re ts < T ) that have a positive imaginary part.Obviously, the saddle points are complex unless EIP = 0. For EIP = 0, we retrievethe classical drift momentum (2) provided p is such that p = eA(t) at some time t.The imaginary part of t0 can be related to a tunneling time (Hauge and Støvneng,1989).In Eq. (21), the ionization amplitude is represented as the coherent sum over all

saddle points within one period of the field. The fact that the spectrum consistsof the discrete energies

Ep ≡ p2

2m= nw − UP − EIP (22)

can be attributed to interference of the contributions from different periods.This interference is destructive, unless the energy corresponds to one of thediscrete peaks (22). Depending on the shape of the vector potential A(t) andits symmetries, there will be several solutions ts (for a sinusoidal field, there aretwo in the upper half plane and two in the lower, which are complex conjugateto the former) within one period of the field. Their interference creates a beatpattern in the calculated spectrum. This is, however, difficult to observe due toits sensitive dependence on the laser intensity, which is not very well controlledin an experiment so that the interference effects are usually washed out.Interferences also exist for an elliptically polarized laser field for fixed electron

momentum as a function of the ellipticity. Since, in experiments, the ellipticity isbetter defined than the intensity, these interferences have been observed (Pauluset al., 1998); see next subsection.The amplitude (19) admits a vector potential A(t) of arbitrary shape; it is

by no means restricted to a monochromatic field of infinite extent. For a pulseof finite extent, the saddle points are still determined by Eq. (20). They have,however, no longer any periodicity. Hence, the discreteness of the spectrum islost. Interference from different parts of the pulse may lead to unexpected effects(Raczynski and Zaremba, 1997).


While for an infinitely long monochromatic pulse the spectrum is symmetricupon p → −p, this forward–backward symmetry no longer holds for afinite pulse. Analysis of the spatial asymmetry of the spectrum may aid indetermination of the pulse length or the absolute carrier phase (Dietrich et al.,2000; Hansen et al., 2001; Paulus et al., 2001b); see Sec. VI.B.

C. Interferences of Direct Electrons

For linear polarization and a drift momentum p = px with |p| � eA, thereare two possible ionization times wt01 = p /2 + d and wt02 = 3p /2 − d. Thecorresponding classical orbits are illustrated in Fig. 4. As discussed above, whileA(t01) = A(t02), the field satisfies E(t01) = −E(t02). Hence, electrons ionized att01 and t02 depart in opposite directions right after the instant of ionization. Asillustrated in Fig. 4, the electric field changes sign soon after t01. Hence, theelectron ionized at this time turns around at a later time and acquires the samedrift momentum as the electron ionized at time t02, which keeps its originaldirection. We expect quantum-mechanical interference of the contribution ofthese two ionization channels.For elliptical polarization, classically, there is at most one ionization time

for given drift momentum. However, Eq. (20) for the complex saddle points ofthe quantum-mechanical amplitude always has more than one solution. For thefield (6) and p =

√2mEx, the solutions are

coswts =z√1 + x2

[−√û±

√ûx2 − ûIP(1 − x2) − x2z

], (23)

where z = (1 − x2)/ (1 + x2), û = E/2UP , and ûIP = EIP/2UP . Obviously, thesolutions ts come in complex conjugate pairs. Those in the upper half-plane

Fig. 4. Classical trajectories (dashed lines) of electrons having the same drift momentum. The solidline is the effective potential V (x) − exE(t) at times t01 (left) and t02 (right). The electron ionizedat t01 is turned around by the field shortly after ionization. In contrast, the electron ionized at t02maintains its original direction. This is a strongly simplified picture of the physics underlying the

interferences of direct electrons.


Fig. 5. (a) Positions of the saddle points wts in the upper half of the complex wt plane in the interval12p � Rewt � 3

2p , calculated from Eq. (23) for E = 17w − EIP − Up, where EIP = 15.76 eV,UP = 3.68 eV, and àw = 1.96 eV. The arrows indicate the motion of the saddle points for increasingellipticity x. The two branches meet at x0 = 0.755. For several values of the ellipticity, insets depictthe ellipse traced out by the electric-field vector, and the positions of the latter at the emissiontimes Re ts are marked by solid dots. (b) The function ReF, which determines the magnitude of theamplitude Mp. The existence of the valley near the ellipticity x0 is related to the effect of dodging,illustrated in Fig. 3. (c) The function cos2(ImF + y), whose oscillations are caused by constructiveand destructive interference. The essential physics behind this interference is sketched in Fig. 4.

From Paulus et al. (1998).

enter the amplitude (21). The solutions are plotted in Fig. 5. For −x0 < x � x0[with x0 given by the zero of the square root in Eq. (23)], the second squareroot on the right-hand side of Eq. (23) is imaginary, and the solutions aresymmetric with respect to Rewts = p . For |x| � x0, this square root isreal, and all solutions have Rewts = p . This has important consequences forthe saddle-point amplitude (21). In the first case, both solutions contribute tothe amplitude and an interference pattern results. This corresponds to the caseof linear polarization discussed above. In the second case, inspection of theintegration contour in the complex plane shows that it has to be routed onlythrough the one solution that is closest to the real axis (Leubner, 1981). Hence,there is no interference. In the first case, the amplitude can be written in theform Mp ~ exp(ReF) cos(ImF + y), in the second case the cosine is absent.The two arguments are also plotted in Fig. 5.The corresponding interferences have been observed by Paulus et al. (1998);

see Fig. 6. They are responsible for the undulating pattern in the ellipticitydistribution, which moves to smaller ellipticity for increasing energy. The sametendency can be observed in the numerical evaluation of the amplitude (21); seePaulus et al. (1998) for an example.For elliptical polarization, the KFR amplitude (19) must be applied with

due caution: it predicts fourfold symmetry of the angular distribution, while


Fig. 6. ATI spectra in xenon for an intensity of 1.2×1014 W/cm2 for various energies in the directionof the large component of the elliptically polarized field as a function of the ellipticity. The insetshows the energy spectrum for linear polarization. The three traces for the lower energies displaythe interference phenomenon of the direct electrons discussed in Sect. II.C; the one for the highestenergy belongs to a plateau electron. The interference dips are related to Fig. 5c. From Paulus et al.

(1998).

the experimental distributions only display inversion symmetry (Bashkanskyet al., 1988). Mending this deficiency requires improved treatment of the bindingpotential (Krstic and Mittleman, 1991). More discussion of this point has beenprovided elsewhere by Becker et al., 1998, who also give further references.The spatial dependence introduced by Coulomb–Volkov solutions in place of theusual Volkov solutions (12) already suffices to destroy the fourfold symmetry,and angular distributions have been calculated with their help by Jaron et al.(1999). However, even for a zero-range potential the fourfold symmetry isbroken provided the effects of the finite binding energy are treated beyond theKFR approximation (Borca et al., 2001).Very similar interferences have been seen by Bryant et al. (1987) in the

photodetachment of H− in a constant electric field. Here the electron, oncedetached, has the choice of starting its subsequent travel either against or withthe direction of the electric field, by close analogy with the opposite directionsof initial travel for the ionization times t01 and t02 in the present case; see Fig. 4.A spatial resolution of the same effect is observed by the photodetachmentmicroscope of Blondel et al. (1999). The theoretical description reproduces the

50 W. Becker et al. [III

observed patterns. Additional bottle-neck structures develop when a magneticfield is applied parallel to the electric field (Kramer et al., 2001).

III. Rescattering

Thus far, we have dealt with “direct” electrons, which after the first step ofionization leave the laser focus without any additional interaction with the ion.Next, we will consider the consequences of one such additional encounter.

A. The Classical Theory

The classical model becomes much richer if rescattering effects are taken intoaccount. To this end, we integrate the electron’s velocity (2) to obtain itstrajectory

x(t) =e

m

((t − t0)A(t0) −

∫ t

t0

dt A(t)). (24)

The condition that the electron return to the ion at some time t1 > t0 isx(t1) = 0. For linear polarization in the x-direction, this implies x(t1) = 0, andy(t) ≡ z(t) ≡ 0. This yields t1 as a function of t0. We defer discussion of ellipticalpolarization to a later time.When the electron returns, one of the following can happen (Corkum, 1993):

(1) The electron may recombine with the ion, emitting its energy plus theionization energy in the form of one photon. This process is responsiblefor the plateau of high-order harmonic generation.

(2) The electron may scatter inelastically off the ion. In particular, it may dis-lodge a second electron (or more) from the ionic ground state. This process isnow believed to constitute the dominant contribution to nonsequential doubleionization.

(3) The electron may scatter elastically. In this process, it can acquire driftenergies much higher than otherwise.In the following, we will concentrate on this high-order above-threshold

ionization (HATI). We will, however, also briefly discuss high-order harmonicgeneration.From Eq. (2), the kinetic energy of the electron at the time of its return is

Eret =e2

2m[A(t1) − A(t0)]

2 . (25)

Maximizing this energy with respect to t0 under the condition that x(t1) = 0 yieldsEret, max = 3.17UP for wt0 = 108◦ and wt1 = 342◦ (Corkum, 1993; Kulanderet al., 1993). It is easy to see that after rescattering the electron can attain a

III] ATI: CLASSICAL TO QUANTUM 51

Fig. 7. Graphical solution of the return time t1 for given start time t0; cf. Paulus et al. (1995): Thereturn condition x(t1) = 0 can be written in the form F(t1) = F(t0) + (t1 − t0)F

′(t0), where thefunction F(t) =

∫dt A(t) ~ sinwt (solid curve) is an integral of the vector potential A(t) ~ coswt

(dotted curve). The thick solid straight line, which is the tangent to F(t) at t = t0, intersects F(t)for the first time at t = t1. The start (ionization) time t0 was chosen such that the kinetic energy Eret(Eq. 25) at the return time t1 is maximal and equal to Eret,max = 3.17UP . The two adjacent straightlines both yield the same kinetic energy Eret < Eret,max. The figure shows that one starts earlier andreturns later while the other one starts later and returns earlier. Obviously, there can be many moreintersections with larger values of t1 provided the start times are near the extrema of F(t). They

correspond to the orbits with longer travel times.

much higher energy: Suppose that at t = t1 the electron backscatters by 180◦, sothat mv(t1 − 0) = e[A(t0) − A(t1)] just before and mv(t1 + 0) = −e[A(t0) − A(t1)]just after the event of backscattering. Then, for t > t1, the electron’s velocity isagain given by Eq. (2), but with px = e[2A(t1) − A(t0)] so that

Ebs =e2

2m[2A(t1) − A(t0)]

2. (26)

Maximizing Ebs under the same condition as above yields Ebs, max = 10.007UP(Paulus et al., 1994a) for wt0 = 105◦ and wt1 = 352◦. These values are veryclose to those that afford the maximal return energy.It is important to keep in mind that for maximal return energy or backscat-

tering energy, the electron has to start its orbit shortly after a maximum of theelectric field strength. As a consequence, it returns or rescatters near a zero ofthe field, see Fig. 7. This also provides an intuitive explanation of the energy


Fig. 8. Maximum drift energy after rescattering (ATI plateau cutoff) upon the mth return to the ioncore during the ionization process. Electrons with the shortest orbits (m = 1) can acquire the highestenergy, whereas electrons that pass the ion core once before rescattering at the second return (m = 2)have a rather low energy. Each return corresponds to two quantum orbits: the mth return corresponds

to the quantum orbits 2m + 1 and 2m + 2.

gain through backscattering: if the electron returns near a zero of the field andbackscatters by 180◦, then it will be accelerated by another half-cycle of thefield.In general, the equation x(t1) = 0 for fixed t0 may have any number of

solutions. This becomes evident from the graphical solution presented in Fig. 7.If the electron starts at a time t0 just past an extremum of the field, it returnsto the ion many times. These solutions having long “travel times” t1 − t0 arevery important for the intensity-dependent quantum-mechanical enhancementsof the ATI plateau to be discussed in Sect. III.B.7. Here we will be satisfiedwith mentioning another property of the classical orbits: obviously, the returnenergy will have extrema, e.g. the maximum of Ebs, max = 10.007UP mentionedabove, which is assumed for a certain time t0,max (t0,max = 108◦ in the example).If we are interested in a fixed energy Ebs < Ebs, max, there are two start timesthat will lead to this energy: one earlier than t0, max, the other one later. Fromthe graphical construction of Fig. 7 it is easy to see that the former has a longertravel time than the latter. In the closely related case of HHG, these correspondto the “long” and the “short” orbit (Lewenstein et al., 1995b). The cutoffs of thesolutions with longer and longer travel times are depicted in Fig. 8.If we consider rescattering into an arbitrary angle q with respect to the

direction of the linearly polarized laser field, we expect a lower maximal energysince part of the maximal energy 3.17UP of the returning electron will go into the


transverse motion. This implies that, for fixed energy Ebs, there is a cutoff in theangular distribution; in other words, rescattering events will only be recorded forangles such that 0 � q � qmax(Ebs). This is a manifestation of rainbow scattering(Lewenstein et al., 1995a). All of this kinematics is contained in the followingequations (Paulus et al., 1994a):

Ebs = 12

[A(t0)

2 + 2A(t1) [A(t1) − A(t0)] (1± cos q0)], (27)

cot q = cot q0 −A(t1)

sin q0 |A(t0) − A(t1)| . (28)

Here q0 is the scattering angle at the instant of rescattering, which may haveany value between 0 and p , as opposed to the observed scattering angle q atthe detector (outside the field). In Eq. (27), the upper (lower) sign holds forA(t0) > A(t1) (A(t0) < A(t1)).Pronounced lobes in the angular distributions about the polarization direction

were first observed by Yang et al. (1993), while the rescattering plateau in theenergy spectrum with its cutoff at 10UP was identified by Paulus et al. (1994b,c).These spectra prominently display the classical cutoffs at qmax and Ebs,max.The classical features become the better developed the higher the intensity is.Hence, they are particularly conspicuous in the strong-field tunneling limit.This has been shown theoretically by comparison with numerical solutions ofthe Schrodinger equation (Paulus et al., 1995) and experimentally for He atintensities around 1015 W/cm2. Indeed, the latter spectra show an extendedplateau for energies between 2UP and 10UP (Walker et al., 1996; Sheehyet al., 1998). For comparatively low intensities, angular distributions have beenrecorded in xenon with very high precision by Nandor et al. (1998). They alsoshow the effects just discussed, but with much additional structure that appears tobe attributable to quantum-mechanical interference and to multiphoton resonancewith ponderomotively upshifted Rydberg states (Freeman resonances; Freemanet al., 1987).

B. Quantum-mechanical Description

In order to incorporate the possibility of rescattering into the quantum-mechanical description, we have to allow the freed electron once again to interactwith the ion (Lohr et al., 1997). To this end, we return to the exact equation (16)and insert the Dyson integral equation (15). This yields two terms. Next, as wedid in Sect. II.B, we replace the exact scattering state |yp〉 by a Volkov state andthe exact time-evolution operator U by the Volkov-time evolution operator Uf .In other words, we disregard the interaction with the binding potential V (r),except for the one single interaction that is explicit in the Dyson equation. Thisprocedure corresponds to adopting the Born approximation for the rescatteringprocess.


Of the two terms, the first is identical with the “direct” amplitude (17) or (19).The second describes rescattering. Via integration by parts similar to thatexplained in Eq. (18) the two terms can be combined into one,

Mp = −i∫ ∞

−∞dt1

∫ t1

−∞dt0⟨y (Vv)p (t1) |VUf (t1, t0)V |y0(t0)

⟩, (29)

which now describes both the direct and the rescattered electrons. The physicalcontent of the amplitude (29) corresponds to the recollision scenario: Theelectron is promoted into the continuum at some time t0; it propagates in thecontinuum subject to the laser field until at the later time t1 it returns to withinthe range of the binding potential, whereupon it scatters into its final Volkovstate.Exact numerical evaluation of the amplitude (29) for a finite-range binding

potential is very cumbersome. For a zero-range potential, however, the spatialintegrations in the matrix element become trivial, and the computation is ratherstraightforward. If the field dependence of the Volkov wave function and theVolkov time-evolution operator is expanded in terms of Bessel functions, one ofthe temporal integrations in the amplitude (29) can be carried out analyticallyand yields the same d function as in Eq. (21), specifying the peak energies. Theremaining quadrature with respect to the travel time t1 − t0 has to be carriedout numerically; see Lohr et al. (1997) and Milosevic and Ehlotzky (1998a),where explicit formulas can be found; for elliptical polarization see Becker et al.(1995) and Kopold (2001). Alternatively, the integral over the travel time maybe done first, and the integral over the return time t1 is then evaluated by Fouriertransformation (Milosevic and Ehlotzky, 1998b).The relevance of the rescattering mechanism to ATI and multiple ionization

was suggested early by Kuchiev (1987) and by Beigman and Chichkov (1987).Improvements of the customary KFR theory by including further interactionswith the binding potential were already discussed by Reiss (1980). The firstexplicit calculations of angular-resolved energy spectra were carried out byBecker et al. (1994a, 1995) and by Bao et al. (1996). Closely related rescatteringmodels were presented by Smirnov and Krainov (1998) and by Goreslavskii andPopruzhenko (1998, 2000).The physics of high-order ATI is related to electron scattering at atoms in the

presence of a strong laser field. In the former case, the initial state of the electronis a wave packet created by tunneling, while in the latter it is a plane-wave state.This latter problem was studied theoretically by Bunkin and Fedorov (1966)and by Kroll and Watson (1973). Corresponding experiments were done byWeingartshofer et al. (1977, 1983). Some quantum features of electron scatteringin intense laser fields are remarkably similar to HATI; see Kull et al. (2000) andGorlinger et al. (2000).


B.1. Saddle-point methods

For sufficiently high intensity, the temporal integrations in the amplitude (29)can be carried out by the saddle-point method, as in the case of the directamplitude (19). This procedure provides much more physical insight thanBessel-function expansions, and establishes the connection with Feynman’s pathintegral, to be discussed below.In this context, rather than taking advantage of the explicit form of the Volkov

time-evolution operator, we expand it in terms of the Volkov states (12),

Uf (t1, t0) =∫d3k

∣∣∣y (Vv)k (t1)⟩⟨y (Vv)k (t0)

∣∣∣ , (30)

so that the amplitude Mp is represented by the five-dimensional integral

Mp ~∫ ∞

−∞dt1

∫ t1

−∞dt0

∫d3k exp[iSp(t1, t0,k)]mp(t1, t0,k) (31)

with the function

mp(t1, t0,k) = 〈p − eA(t1) |V |k − eA(t1)〉〈k − eA(t0) |V |y0〉 . (32)

For ATI, the action

Sp(t1, t0,k) = −12m

∫ ∞

t1

dt [p − eA(t)]2

−12m

∫ t1

t0

dt [k − eA(t)]2 +∫ t0

−∞dt EIP

(33)

in the exponent consists of three parts, according to the three stages discussedabove.As above in Eq. (20), we approximate the amplitude (31) by expanding the

phase (33) of the integrand about its stationary points. In this process, we assumethat the function mp(t1, t0,k) depends only weakly on its arguments. Indeed, fora zero-range potential, it is a constant. We now have to determine the stationarypoints with respect to the five variables t1, t0 and k. They are given by thesolutions of the three conditions (Lewenstein et al., 1995a)

[k − eA(t0)]2 = −2mEIP, (34)

(t1 − t0)k =∫ t0

t1

dt eA(t), (35)

[k − eA(t1)]2 = [p − eA(t1)]

2, (36)

respectively. The first condition (34) attempts to enforce energy conservationat the time of tunneling. The second condition (35) ensures that the electron


returns to its parent ion, and the third one (36) expresses that, on this occasion,it rescatters elastically into its final state. In general, the saddle-point equationshave several solutions (t1s, t0s,ks), (s = 1, 2, . . .), of which only those are relevantfor which Re t1s > Re t0s, such that the recollision is later than ionization. Thematrix element can be written as

Mp ~∑s

((2p ià)5

det(ð2Sp/ðq(s)j ðq

(s)k )j, k = 1, ..., 5

)1/2eiSp(t1s ,t0s ,ks)mp(t1s, t0s,ks), (37)

where q(s)i (i = 1, . . . , 5) runs over the five variables t1s, t0s and ks. As wenoted already for the direct electrons in the context of Eq. (21), the sum hasto be extended only over a subset of the solutions of the saddle-point equations(34)−(36). However, in the present case, determining this subset may be tricky(Kopold et al., 2000a). For a periodic field, the sum over the periods in Eq. (37)can be carried out by Poisson’s formula. This leaves a sum over the saddle pointswithin one period and produces a d function as in Eq. (21).The computation of ATI now consists of two separate tasks. First, the solutions

of the saddle-point equations (34)−(36) have to be determined and, second,the appropriate subset has to be inserted into expression (37). Note that weapply the saddle-point approximation to the probability amplitude for given finalmomentum p, and not to the complete wave function of the final state. This is thereason why only few solutions contribute, while a semiclassical computation ofthe wave function, which contains all possible outcomes, requires considerationof a very large number of trajectories (van de Sand and Rost, 2000).Since EIP > 0, the condition (34) of “energy conservation” at the time of

ionization cannot be satisfied for any real time t0. As a consequence, all solutions(t1s, t0s,ks) become complex. If the ionization potential EIP is zero, then, fora linearly polarized field, the first saddle-point equation (34) implies that theelectron starts on its orbit with a speed of zero. Provided the final momentum pis classically accessible, the resulting solutions are entirely real. They correspondto the so-called “simple-man model” (van Linden van den Heuvell and Muller,1988; Kulander et al., 1993; Corkum, 1993). For EIP Ñ 0, so long as the Keldyshparameter g2 = EIP/ (2UP) is small compared with unity, the imaginary parts ofthe solutions of Eqs. (34)–(36) are still not too large, and the real parts are stillclose to these simple-man solutions. In this case, approximate analytical solutionsto the saddle-point equations can be written down, which yield an analyticalapproximation to the amplitude (31) (Goreslavskii and Popruzhenko, 2000). Onthe other hand, for elliptical polarization, the solutions are always complex, evenwhen EIP = 0. This reflects the fact that, for any polarization other than linear,an electron set free at any time during the optical cycle with velocity zero willnever return to the point where it was released. Equation (34) then only impliesthat k − eA(t0) is a complex null vector.


With the solutions (t1s, t0s,ks) (s = 1, 2, . . .) of Eqs. (34)–(36), the sth quantumorbit has the form

mx(t) =

{(t − t0s)ks −

∫ tt0sdt eA(t) (Re t0s � t � Re t1s),

(t − t1s)p −∫ tt1sdt eA(t) (t � Re t1s).

(38)

We regard the orbit as a function of the real time t. The conditions x(t0) = 0and x(t1) = 0, however, are satisfied for the complex times t0 and t1. As aconsequence, the quantum orbit (38) as a function of real time does not departfrom the origin but, rather, from the “exit of the tunnel.” This is clearly visible inFigs. 14, 15, 17 and 20 below. In contrast to the start time t0, the return time t1is real to a good approximation, see Fig. 10 (below). In consequence, the orbitsreturn almost exactly to the origin.

B.2. Connection with Feynman’s path integral

Any quantum-mechanical transition amplitude, such as the ionization ampli-tude (8), can also be represented in terms of Feynman’s path integral. To thisend, we recall the path-integral representation of the complete time-evolutionoperator of the system atom + field,

U (rt, r′t′) =∫(rt)↔ (r′t′)

D[r(t)]eiS(t,t′), (39)

where S(t, t′) =∫ tt′ dt L[r(t)], t] is the action calculated along a system path, and

the integral measure D[r(t)] mandates summation over all paths that connect(rt) and (r’ t′) (see, e.g., Schulman, 1977).The path integral (39) sums over the functional set of all continuous

paths. In the quasi-classical limit, this can be reduced to a sum over allclassical paths, which are those for which the action S(t, t′) is stationary.For quadratic Hamiltonians, this WKB approximation is exact. In our case,motivated by the success of the classical three-step model of Sect. III.A, wehave reduced the exact transition amplitude to the form (31). In implementingthe strong-field approximation, we have approximated the exact action of thesystem appropriately at the various stages of the process: before the initialionization, in between ionization and rescattering, and after rescattering, as inthe decomposition (33) of the action. This still left us with a five-dimensionalvariety of paths. Out of those, finally, the saddle-point approximation (37) selectsthe handful of “relevant paths” (Antoine et al., 1997; Kopold et al., 2000a;Salieres et al., 2001). These are essentially the orbits of the classical model, yetquantum mechanics is fully present: Their coherent superposition as expressedin the form (37) allows for interference of the contributions of different orbits,and the fact that they are complex accounts for their origin via tunneling.


B.3. Connection with closed-orbit theory

There appears to be a close similarity to the concepts of periodic-closed-orbittheory, see, e.g., Du and Delos (1988), Gutzwiller (1990), and Delande andBuchleitner (1994). The photoabsorption cross section s (E) of an atom in thestate |yi〉 with energy Ei can be expressed in the form (Du and Delos, 1998)

s (E) = 4pe2

àcRe

[∫ ∞

0dt eiEt 〈yi |DU (t, 0)D|yi〉

], (40)

where D = r · û is the dipole operator responsible for photoabsorption of thefield with polarization û and E ≈ Ei + àw. The quantity U (t, 0) is the time-evolution operator in the presence of the binding potential as well as additionalstatic external electric and magnetic fields that may be present. In effect, the time-evolution operator propagates wave packets at constant energy that emanate fromthe atom and are reflected by the caustics of the potential back to the atom wherethey interfere with each other and with the starting wave packets. This leads tooscillations in the photoabsorption spectrum. In a semiclassical approximation,the time-evolution operator can be expanded in terms of classical closed orbitsthat start from and return to the vicinity of the atom, defined by the spatial rangeof the wave function |yi〉. Since the classical problem is chaotic, there are moreand more such orbits when the energy nears zero. Fourier transformation of thephotoabsorption spectrum reveals the recurrence times of the classical orbits.There are several differences to the quantum orbits we are considering here.

In our case, the role of the binding potential is, in effect, reduced to acting asa coherent source of electrons and to causing rescattering, while in closed-orbittheory the interplay of its spatial shape with the external static fields generatesthe rich structure of the closed orbits. In our case, closed orbits are entirely dueto the time dependence of the laser field. The most important difference is thatclosed-orbit theory is concerned with total photoabsorption rates as a functionof frequency, while we consider differential electron spectra for a laser fieldwith fixed frequency. In other words, our orbits depend on the final state ofthe electron.From Eq. (19), in view of the completeness of the Volkov states, the total

ionization probability due to direct electrons is∫d3p |Mp|2

= 2e2 Re

[∫ ∞

−∞dt1

∫ t1

−∞dt0⟨y0(t1)

∣∣r · E(t1)U (Vv)(t1, t0)r · E(t0)∣∣y0(t0)⟩

].

(41)This differs from the photoabsorption cross section (40) only by the presenceof the Volkov time-evolution operator U (Vv)(t1, t0), which reflects the strong-field approximation, instead of the exact time-evolution operator U (t, 0) of the


time-independent problem [for which the time-evolution operator U (t, t′) onlydepends on the time difference t − t′]. In the total ionization probability (41),via the same partial integration (18) as above, the electron–field interactionr · E(t) can be replaced by V (r). The result then looks like the differentialHATI amplitude (29) except that it is sandwiched by the ground state. Thiscorrespondence is a manifestation of the optical theorem.

B.4. The role of the binding potential

The improved Keldysh approximation (29) has been written down for anarbitrary binding potential V (r). The expansion in terms of the binding potential,introduced via the Dyson equation (15), is a strong-field approximation (SFA),which is valid when the electron’s quiver amplitude is so large that most of itsorbit is outside the range of the binding potential. This is trivially guaranteedfor the three-dimensional binding potential of zero range,

V (r) =2pmúd(r)

ð

ðrr. (42)

This potential supports a single (s-wave) bound state at the energy −ú2/2mand a continuum that is undistorted from the free continuum except for thes wave, as required by completeness (Demkov and Ostrovskii, 1989). Without theregularization operator (ð/ðr) r, which acts on the subsequent state, the potentialdoes not admit any bound state. There are several possibilities to adjust the oneparameter ú to an individual atom or ion. In most cases one will determine itso as to reproduce the ionization potential; see, however, Sect. III.B.7.The zero-range potential (42) underlies many of the explicit results exhibited

in this chapter. However, we emphasize that the amplitude (29), as well as itssaddle-point approximation (37), hold for a much wider class of potentials.Regardless of the potential, the saddle-point equations (34)−(36) have theelectron start from and return to the center of the binding potential, which isthe origin, and do not depend on its shape. The potential only enters via theform factors in Eq. (32). For the SFA to be applicable, they must depend ontime only weakly. The procedure corresponds to the Born approximation. It willbe the better justified, the shorter the range of the potential is, so that the formfactor depends only weakly on the momenta. Excited bound states do not enterthe amplitude (29) regardless of the potential used.For a comparison of a high-order ATI spectrum calculated for the zero-range

potential (42) with the same spectrum extracted from a solution of the three-dimensional TDSE for hydrogen, see Cormier and Lambropoulos (1997) for thelatter and Kopold and Becker (1999) for the former. There is good qualitativeagreement within the ATI plateau; in particular, the positions of the dips in thespectrum that are due to destructive interference agree within a few percent.The comparison confirms that the detailed shape of the potential has only


minor significance for the HATI spectrum. Clearly, however, the real physicalsystems best described by a zero-range potential are negative ions with a s-waveground state.

B.5. A homogeneous integral equation

An alternative route to the standard KFR matrix element (19) and its improvedversion (29) starts from the homogeneous integral equation

|Y(t)〉 = −i∫ t

−∞dt Uf (t, t)V |Y(t)〉, (43)

which holds for the state that develops out of the unperturbed ground statedue to its interaction with the laser field. This integral equation can be derivedimmediately from the Dyson equation (15) if one applies both sides of the latterto the atomic ground state |y0(t′)〉 in the limit where t′ → −∞. By inspection,one may convince oneself that the term Uf (t, t′)|y0t′)〉 makes no contributionfor t − t′ →∞ so that Eq. (43) is left. This equation was first introduced in thecontext of the quasi-energy formalism by Berson (1975) and by Manakov andRapoport (1975) for circular polarization and Manakov and Fainshtein (1980)for arbitrary polarization.Inserting on the right-hand side of the integral equation (43) the expansion (30)

of Uf in terms of Volkov states and replacing |Y(t)〉 by the unperturbedatomic ground state |y0(t)〉, one can read off the matrix element (19) for directionization. Iterating Eq. (43) one gets

|Y(t)〉 = −∫ t

−∞dt∫ t

−∞dt ′ Uf (t, t)VUf (t , t ′)V |Y(t ′)〉, (44)

which yields the improved KFR amplitude (29) in the same fashion.The integral equation (43) is particularly useful for the zero-range poten-

tial (42), since in this case it allows one to calculate the wave function in all spaceprovided it is known at the origin. For the latter, to a first approximation, onemay employ the unperturbed wave function. Better approximations are obtainedby using more accurate expressions. These incorporate the possibility that theionized electron revisits the core, as illustrated by Eq. (44).For the zero-range potential and a monochromatic plane wave with circular

polarization, it can be shown that the wave function near the origin exactly obeysY(r, t) ∝ (1/r − ú) exp(−iEt) for all times. The complex quasi-energy E hasto be determined as the eigenvalue of a nonlinear integral equation (Berson,1975; Manakov and Rapoport, 1975). For any polarization other than circular,the time dependence at the origin is given by a Floquet expansion (Manakovand Fainshtein, 1980; Manakov et al., 2000). The interaction with a laser fieldfor a finite period of time was considered along similar lines by Faisal et al.


(1990) and Filipowicz et al. (1991); see also Gottlieb et al. (1991) and Robustelliet al. (1997). The integral equation (43) was also used for two-center potentialsin order to model negative molecular ions (Krstic et al., 1991; Kopold et al.,1998).

B.6. Quantum orbits for linear polarization

For linear polarization, Fig. 9 presents a calculated ATI spectrum that is typicalof a high laser intensity, cf. the data of Walker et al. (1996). The solid circlesthat make up the topmost curve of the upper panel were calculated from theamplitude (29) by means of a zero-range potential, while the other curvesgive the results of including an increasing number of quantum orbits in thesaddle-point approximation (37). The spectra that result from just the sth pair(which comprises the orbits 2s − 1 and 2s) are displayed in the lower panel.Quantitatively, the first pair dominates the entire spectrum, but the contributionof the second pair comes close, in particular near its cutoff around 7UP . Thecontribution of the third pair is already weaker by almost one order of magnitude,and the subsequent pairs hardly play a role anymore. Indeed, in the upper panel,already the third curve from bottom virtually agrees with the result of the exactcalculation.The dependence of the parameters t1s, t1s − t0s (the travel time), and ks on the

electron energy Ep is illustrated in Fig. 10 for the two orbits (s = 1, 2) having theshortest travel times. These parameters uniquely specify the quantum orbits inspace and time. Their behavior is very different for energies below and above theclassical cutoff at 10UP . Below the cutoff, the imaginary parts of the parametersare only weakly dependent on the energy. Both orbits have to be included inthe sum (37), and their interference leads to the beat pattern, which is visiblein the spectrum of Fig. 9. Notice that the imaginary parts of both the returntimes t1s and the momenta kxs are small. In contrast, the imaginary part of thetravel times t1s − t0s, which are related via t0s to the tunneling rate, is substantial;see Fig. 5, where the ionization time ts is plotted for the direct electrons. Hence,after rescattering, the orbits are real for all practical purposes: the electron hasforgotten its origin via tunneling. The parameter values of the two orbits (s = 1, 2)approach each other closely near the cutoff. At some point, one of the two orbits(drawn dashed in the figure) has to be dropped from the sum (37). This causes theartifact of the small spikes visible in Fig. 9. For energies above the cutoff, just oneorbit contributes and, as a consequence, the spectrum smoothly decreases withoutany trace of interferences. The real part of the parameters stays approximatelyconstant, while the imaginary part increases strongly with increasing energy. Thisis responsible for the steep drop of the spectrum after the cutoff. Similar behavior,as a function of ellipticity, occurs in Fig. 5.The procedure of dropping one of the orbits of each pair after its cutoff

can be replaced by a more rigorous method. In the vicinity of the cutoffs,


Fig. 9. Upper panel: ATI spectrum in the direction of the laser field for linear polarization for1015 W/cm2, àw = 0.0584 a.u., and a binding energy of EIP = 0.9 a.u. The electron energy is givenin multiples of UP . The curve at the top (solid circles) is the exact result from Eq. (29). The othercurves were calculated from the saddle-point approximation (37). From bottom to top, more andmore quantum orbits are taken into account; the results are displaced with respect to each other forvisual convenience. The curve at the bottom incorporates just the pair of orbits with the shortesttravel times, the next one up includes in addition the pair with the next-to-shortest travel times, andso on. The occasional small spikes are artifacts of the saddle-point approximation, cf. Goreslavskiiand Popruzhenko (1999) and Kopold et al. (2000a). Lower panel: The envelopes of the contributionsof the individual pairs are shown all on the same scale so that the quantitative relevance of the variouspairs is put in perspective. The cutoffs of the various orbits agree with those displayed in Fig. 8.

From Kopold et al. (2000a).


Fig. 10. Saddle points (ts, t′s,ks) for the orbits (s = 1, 2) having the two shortest travel times. In

this figure, ts is the return time (elsewhere denoted by t1s), and t′s the start time (elsewhere denoted

by t0s). The figure shows a comparison of elliptical polarization (x = 0.5, solid circles) and linearpolarization (open squares). The values of the other parameters are those of Fig. 9 (eA = 2.04 a.u.).The symbols identify electron energies of 11.5, 10.4, 8.92, 6.01, and 2.49, all in multiples of UP .The dashed orbits have to be dropped from the sum (37) after the cutoff. With the scaling of kgiven on the ordinate, the saddle points depend only on the Keldysh parameter g =

√|E0|/2UP .an approximation in terms of Airy functions was used by Goreslavskii andPopruzhenko (2000). A uniform approximation was described in a differentcontext by Schomerus and Sieber (1997). It reproduces the spectra of Fig. 9without the spikes (Schomerus and Faria, 2002).

B.7. Enhancements in ATI spectra

In several experiments, pronounced enhancements of groups of ATI peaks inthe plateau region (by up to an order of magnitude) have been observed upona change of the laser intensity by just a few percent (Hertlein et al., 1997;Hansch et al., 1997; Nandor et al., 1999). This behavior suggests a resonantprocess. Near the resonances, the contrast of the spectra is remarkably reduced(Cormier et al., 2001). For the experiments reported so far, the effect is mostpronounced for argon. This holds not only for a laser wavelength of 800 nmbut also for 630 nm (Paulus et al., 1994c). The enhancements are so strongthat in experiments implying significant focal averaging the observed spectral


Fig. 11. ATI spectra in argon at 800 nm recorded in the direction of the linearly polarized field forvarious intensities rising by increments of 0.1 I0 from 0.5 I0 (bottom curve) to 1.0 I0 (top curve). Thehorizontal lines mark the maxima of the ATI plateaus for each intensity. For intensities I > 0.8 I0a group of ATI peaks between 15 eV and 25 eV quickly grows. (The spectra shown here represent

only a fraction of those actually measured.) From Paulus et al. (2001a).

intensity may well be dominated by these enhancements, regardless of the actualpeak intensity. In this sense, ATI in toto has been called a resonant process(Muller, 1999b). A big step towards understanding the physical origin of theenhancements was made in theoretical studies that reproduced the enhancementsin the single-active-electron approximation by numerical solution of the one-particle time-dependent Schrodinger equation in three dimensions (Muller andKooiman, 1998; Muller, 1999a,b; Nandor et al., 1999), thereby ruling out anymechanism that invokes electron–electron correlation.In Fig. 11 we show results of a measurement of the same effect, but for

a shorter pulse length of 50 fs (Paulus et al., 2001a). Spectra in an intensityinterval of 0.3 to 1.0 × I0 in steps of 0.1 × I0 are displayed. The maximumintensity I0 was calibrated by using the cutoff energy of 10UP . This leads toI0 ≈ 8×1013 W/cm2. There is a striking difference between the spectra forI � 0.8I0 and those for higher intensity: within a small intensity interval a groupof ATI peaks corresponding to energies between about 15 eV and 25 eV growsvery quickly. In the figure this is emphasized by horizontal lines drawn at themaximal heights of the plateaus. Increasing the intensity above 0.9I0 leads to asmaller growth rate of these peaks. The plateau, however, preserves its shape.For an interpretation, it should be kept in mind that a measured ATI spectrum ismade up of contributions from all intensities I � I0 that are contained within thespatio-temporal pulse profile. This means that a spectrum for a fixed intensitywould show the enhanced group of ATI peaks only at that intensity where it first


Fig. 12. Comparison of the intensity dependence of ATI electrons with different energies. For visualconvenience, the overall increase in yield with increasing intensity has been subtracted. The electronsat 6.4 eV and 7.3 eV are due to the strongest Freeman resonances, i.e. resonance with atomic states.Those labeled “plateau” are electrons in the plateau region of the spectra. As a consequence ofthe subtraction of the overall increase, the resonance-like behavior corresponds to those intensitieswhere the respective curves start rising. It is evident that for the plateau electrons this does nothappen at those intensities where the atomic states shift into resonance. Quite to the contrary, theintensity at which the yield of the plateau electrons starts its rise is reflected in the yield of the

low-energy electrons by a brief halt in their rise. This is indicated by the dashed circles.

appears in our measurement, namely at I ≈ 0.85I0 = 7×1013 W/cm2. In otherwords, the enhancement happens at a well-defined intensity or at least within avery narrow intensity interval.Analyzing the wave function of the atom in the laser field, Muller (1999a)

suggested that the enhancements are related to multiphoton resonances withponderomotively upshifted Rydberg states. In some cases, in particular forelectrons with rather low energy, one particular Rydberg state could be definitelyidentified as responsible. In others, notably for the strong enhancement that forappropriate intensities dominates the middle of the plateau, this was not possible(Muller, 2001a).A closer look at the data of the measurement shown in Fig. 11 reveals

that under the conditions of this experiment (i.e. a pulse duration of 50 fsas compared with more than 100 fs in the other measurements mentioned)resonantly enhanced multiphoton ionization does not play an essential role.This can be deduced from the different intensity dependence of the enhancedATI peaks in the plateau and of the low-energy ATI peaks, see Fig. 12. Thelatter are known to originate from atomic resonances (Freeman et al., 1987).In Muller’s numerical simulations, the existence of excited bound states

appears to be instrumental for the enhancements. Yet, the modified KFR


matrix element (29), which does not incorporate any excited states, producesmuch the same enhancements (Paulus et al., 2001a; Kopold et al., 2001). Anexample is shown in Fig. 13. In these calculations, the enhancements occur forintensities for which an ATI channel closes. This is the case when

EIP + Up = kàw. (45)

For an intensity slightly higher than specified by this condition, k + 1 is theminimum number of photons required for ionization in place of k . Such channelclosings are very visible in the multiphoton-detachment yields of negative ions(Tang et al., 1991) and have been shown to produce a separate comb of peaks inthe low-energy ATI spectrum (Faisal and Scanzano, 1992). Comparison of thechannel-closing condition (45) with the ATI energy spectrum (22) shows that ata channel closing electrons may be produced with zero drift momentum p. Inthis event, the energy of the k photons is entirely used to overcome the bindingpotential raised by the ponderomotive energy, and no energy is left for a driftmotion. An electron having a drift momentum near zero has many recurringopportunities to rescatter. Indeed, the quantum-orbit analysis of the spectra ofFig. 13 shows that at the channel closings, and only there, an exceptionally largenumber of orbits are required to reproduce the exact result. All of these orbitsconspire to interfere constructively to produce the observed enhancements. In thetunneling regime, this can be proved analytically (Popruzhenko et al., 2002).In Muller’s numerical simulations, inspection of the temporal evolution reveals

that at the intensities that produce the enhancements electrons linger aboutthe ion for many cycles of the field before the final act of rescattering.A detailed comparison between Muller’s numerical simulations and results basedon Eq. (29) has been made by Kopold et al. (2001). This paper also includesan assessment of the consequences of focal averaging. It is noteworthy that bothapproaches predict ATI enhancements also for helium deeply in the tunnelingregime, in spite of the obvious multiphoton character of the channel-closingcondition (45). Unfortunately, the helium data of Walker et al. (1996) and Sheehyet al. (1998) do not allow one to draw conclusions about the presence or absenceof enhancements.The interference interpretation just given requires the existence of a sufficient

number of orbits to contribute to the energy considered. The lower panel ofFig. 9 shows that too few orbits contribute for energies above about 8UP . Indeed,the enhancements observed experimentally are restricted to the lower two-thirdsof the plateau. The interpretation also implies that the enhancements shoulddisappear for ultrashort pulses, where late returns do not occur. This has beenobserved in experiments by Paulus et al. (2002). In numerical simulations ofHHG based on the three-dimensional TDSE, the same effect has noticed byde Bohan et al. (1998).When the modified KFR matrix element (29) is used to describe data for real

atoms, the ionization potential EIP has to be replaced by an effective (lower)


Fig. 13. ATI spectra for EIP = 14.7 eV, w = 1.55 a.u., and three intensities: at a channel closing(h = UP/w = 2.526, middle panel), below the channel closing (h = 2.326, lower panel), and above(h = 2.626, upper panel). In each panel, the exact result calculated from Eq. (29) is shown (solidsymbols) and approximations involving the first 2 (dashed line), 6 (dot-dashed line), and 40 (solid

line) quantum orbits in Eq. (37). From Kopold et al. (2001).

value that corresponds to the de facto onset of the continuum (Paulus et al.,2001a; Kopold et al., 2001). It is a fact that, for a Coulomb potential, the actualonset of the continuum is hard to see and may better be replaced by an effective


value. This is illustrated, for example, by the photoabsorption spectra of Gartonand Tomkins (1967).Numerical simulations predict very similar enhancements in high-order

harmonic spectra (Toma et al., 1999) and in nonsequential double ionization(NSDI) of helium (Muller, 2001c). In HHG in one dimension, the dependenceof the enhancements on the shape of the potential and the presence or absenceof excited bound states has been investigated (Faria et al., 2002). The results arelargely compatible with the quantum-orbit picture.In a semiclassical framework, the binding potential can be incorporated into

the orbits. This leads to Coulomb refocusing (Ivanov et al., 1996; Yudin andIvanov, 2001a): orbits that would miss the ion in the absence of the bindingpotential are refocused to the ion in its presence. This emphasizes the importanceof late returns and leads to a substantial increase of rescattering effects without,however, resonant behavior. If late returns are cut off due to an ultrashort laserpulse, the rate of NSDI should decrease. Indeed, this has been experimentallyconfirmed by comparison of 12-fs and 50-fs pulses (Bhardwaj et al., 2001).

B.8. Quantum orbits for elliptical polarization

Formulation of a classical model of the simple-man variety to describerescattering for an elliptically polarized laser field meets with difficulties. Theproblem is that an electron that starts with zero velocity almost never returnsexactly to its starting point if the laser field has elliptical polarization. Formally,this shows in the saddle-point equations (34)−(36) as follows. For EIP = 0,Eq. (34) yields k = eA(t0) if real solutions are sought. For linear polarization,this leaves two equations to be solved for t0 and t1: Eq. (36) and the x-projectionof Eq. (35). Real solutions are obtained, provided the final momentum p isclassically accessible. In contrast, for elliptical polarization, three equations areleft since now both the x-projection and the y-projection of Eq. (35) have tobe considered. Hence, there is no simple-man model for elliptical polarization,even when EIP = 0. The same situation occurs for HHG. This does not meanthat there is no HATI or HHG for elliptical polarization: quantum-mechanicalwave-function spreading assures overlapping of the wave packet of the returningelectron with the ion (Dietrich et al., 1994; Gottlieb et al., 1996). The completeabsence of HHG for a circularly polarized laser field is sometimes taken asconfirmation of the rescattering mechanism. This conclusion is not rigorous sincethere is still sufficient overlapping. Rather, the absence of HHG is due to angular-momentum selection rules or, equivalently, destructive interference.One might try to formulate a simple-man model for elliptical polarization by

relaxing the requirement that the electron return exactly to the position of theion or by admitting a nonzero initial velocity, but in doing so a large amount ofarbitrariness is unavoidable. Instead, we will just solve the saddle-point equations(34)−(36) and accept and interpret the complex solutions.


Fig. 14. ATI spectrum in the direction of the large component of the elliptically polarized drivinglaser field (5) for x = 0.5 (see the field ellipse in the upper right corner of the figure) and electronenergies between 2.5 and 10.5UP . The other parameters are w = 1.59 eV, EIP = 24.5 eV, andI = 5×1014 W/cm2. The open circles give the yields of the individual ATI peaks calculated fromthe integral (29) for the zero-range potential. The other curves represent the contributions to thequantum-path approximation (37) of the shortest trajectories 1 and 2 (dot-dashed line), 3 and 4(long-dashed line), and 5 and 6 (short-dashed line), as well as the sum of all six (solid line). Please,note that some of these curves overlap partly or entirely. The orbits responsible for each part of thespectrum, viz. 1 and 2, 3 and 4, and 5 and 6, are presented near the margins of the figures. Theposition of the ion is marked by a cross; notice that the orbits do not depart from there, but ratherfrom a point several atomic units away from it. This is the point where the electron tunnels into thecontinuum. The electron travels the orbits in the direction of the arrows. Experimental data for a

similar situation are shown in Fig. 15. From Kopold et al. (2000b).

The results of such a calculation are presented in Fig. 14. What used tobe the rescattering plateau for linear polarization has turned into a staircasefor elliptical polarization. Each step can be attributed to one particular pairof orbits, and for each step the real parts of such orbits are displayed in thefigure. The orbits are closely related to their analogs in the case of linearpolarization, exhibited in Fig. 9. In particular, their cutoffs oscillate withincreasing travel time as illustrated in Figs. 8 and 9. The main difference is thatfor elliptical polarization the orbits are two-dimensional and encircle the ion. Thepair of orbits with the shortest travel times generates the part of the spectrumpreceding the final (highest-energy) cutoff. However, this part is very weak in


relation to the yields at lower energies. The latter are generated by orbits withlonger travel times, whose contributions for linear polarization are marginal, seeFig. 9.Intuitively, this staircase structure can be understood as follows. Return of the

electron to the ion is possible if the electron has a nonzero initial velocity. Thisvelocity is largely in the direction of the small component of the ellipticallypolarized field. The larger this velocity is, the smaller is the contribution thatthe associated orbit makes to the spectrum. [This can be compared with thedistribution of transverse momenta in a Gaussian wave packet (Dietrich et al.,1994; Gottlieb et al., 1996).] For the shortest orbit, while the large componentof the field changes sign so that the electron is driven back to the core in thisdirection, the small component has the same sign for the entire duration of theorbit. Hence, a particularly large initial velocity in this direction is required inorder to compensate the drift induced by the small field component. For thelonger orbits, the small component changes direction, too, during the travel timeand, consequently, a smaller initial transverse velocity suffices to allow theelectron to return to the ion. Support for these qualitative statements can befound in the orbits depicted in Fig. 14.The parameters of the two shortest quantum orbits can be read from Fig. 10

and compared with the case of linear polarization. For elliptical polarization, themomentum ks has two nonzero components, kxs and kys. Both have substantialimaginary parts, in particular kys. This is a consequence of the lack of aclassical simple-man model for elliptical polarization, as discussed above. For theorbits (s = 3, 4) (not shown), the imaginary parts are much smaller, in keepingwith the fact that they make a larger contribution to the spectrum.Figure 15 presents a corresponding measurement of an ATI spectrum and

displays the staircase structure predicted by the theory. The first step (the onecorresponding to energies below 10 eV) is due to direct electrons and does notconcern us here. The other ones correspond to the steps of Fig. 14. The realparts of representative orbits, calculated from Eqs. (34)−(36), are shown in thefigure. In order to reach a maximum contrast for the steps, the spectrum wasrecorded at 30º to the major axis of the polarization ellipse.

B.9. Interference between direct and rescattered electrons

In the lower part of the plateau, the electron can reach a given energy eitherdirectly or after rescattering so that one expects interference of these two paths.However, Fig. 9 shows that, for linear polarization and high intensity, thetransition region where both paths make a contribution of comparable magnitudeis very narrow. The situation is more favorable for elliptical polarization:since the plateau turns into a staircase (Fig.14), the yields of the two pathsremain comparable over a larger energy region. This has permitted experimental


Fig. 15. ATI spectrum in xenon for an elliptically polarized laser field with ellipticity x = 0.36and intensity 0.77×1014 W/cm2 for emission at an angle with respect to the polarization axis asindicated in the upper right. The spectrum has a staircase-like appearance. The respective steps areshaded differently. For each step, the real parts of the responsible quantum orbits are displayed. Thedots with the crosses mark the position of the atom, and the length scale is given in the upper left

of the figure. From Salieres et al. (2001).

observation of this interference effect in the energy-resolved angular distribution(EAD) (Paulus et al., 2000).Figure 16 shows a comparison of the EAD’s for linear and for elliptical

polarization at the same intensity. For linear polarization, the standard plateauin the direction of the laser polarization is very noticeable. The side lobescorresponding to rainbow scattering, mentioned in Sect. III.A, are also visible.For elliptical polarization, the plateau has split into two, one to the left of thedirection of the major axis of the field and another weaker one to its right. Thelower panel of Fig. 16 exhibits (on the right) EAD’s of a sequence of ATI peakswhere the interference is best developed and (on the left) compares them withtheoretical calculations from the amplitude (29). The parameters underlying thecalculation do not exactly match the experiment. This is mostly attributableto the insufficient description of the direct electrons for elliptical polarization.The theoretical results, however, show the same interference pattern. In orderto make sure that this pattern is really due to interference between direct andrescattered electrons, the two contributions have been displayed separately forone of the peaks (s = 17): neither one shows a pronounced dip, only theircoherent superposition does. For more details of the theory we refer to Kopold(2001).


Fig. 16. Upper panels: density plots of measurements of the energy-resolved angular distributionsfor Xe at an intensity of 7.7×1013 W/cm2 and a wavelength of 800 nm for (a) linear polarization and(b) elliptical polarization with ellipticity x = 0.36. The direction of the major axis of the polarizationellipse is at 0º. Dark means high electron yield. Yields can only be compared horizontally, notvertically, since the data were normalized separately for each ATI peak. For linear polarization,the cutoff is at 10UP = 46 eV. Lower panels: (a) theoretical calculation from Eq. (29) of theangular distribution for the ATI peaks s = 11, . . . , 21 for elliptical polarization (x = 0.48). Theother parameters are EIP = 0.436 a.u. (just below the binding energy of xenon in order to stay awayfrom a channel closing) and I = 5.7×1013 W/cm2. For the ATI peak s = 17, the contributions of thedirect electrons (dashed line) and the rescattered electrons (dotted line) are displayed separately. Theslight variation in the former is unrelated to the interference pattern of the total yield (solid line),which results from the coherent sum of the two contributions. (b) Experimental angular distributionextracted from the upper panel (b) of the ATI peaks s = 15, . . . , 25. From Paulus et al. (2000).

IV] ATI: CLASSICAL TO QUANTUM 73

IV. ATI in the Relativistic Regime

A sufficiently intense laser field accelerates an electron from rest to relativisticvelocities |v| ~ c within one cycle. Such intensities are characterized by theponderomotive energy UP becoming comparable with or exceeding the electron’srest energy mc2. We will briefly summarize the kinematics of an otherwise freeelectron in the presence of such a field. In other words, we will generalizethe simple-man model of Sect. II.A.1 to the case of “relativistic intensity.” Thechanges are surprisingly few.

A. Basic Relativistic Kinematics

For a four-vector potential Am = (A0,A), the electron’s four-vector velocity is(Jackson, 1999)

mum = pm − eAm , (46)

where um = g(c, v) with v = v(t) the ordinary velocity dx/dt, and the usualrelativistic factor g = [1 − (v/c)2]−1/2 (not to be confused with the Keldyshparameter). The four-velocity satisfies u2 ≡ u · u ≡ um um = c2 so thatthe four-vector mu is on the mass shell. Equation (46) is the analog of thenonrelativistic Eq. (2).We will consider a plane-wave field of arbitrary polarization,

Am =2∑i = 1

ai(k · x)emi (47)

with the four-dimensional wave vector km = (w/c,k) so that k2 = 0 and k · ei = 0.The field (47) differs from the field (6) by the fact that the wave fronts are nowgiven by k · x = const. in place of t = const., that is, we do no longer makethe dipole approximation. We will assume that the laser field propagates in thez-direction so that k = |k|ez.The four-vector pm = (E/c,p) is the canonical momentum. For a vector

potential whose cycle average vanishes, its spatial components p have thephysical meaning of the drift momentum as in the nonrelativistic case. Sincethe electron–field interaction depends only on

u ≡ k · x /w = t − z/c, (48)

the canonical momentum pT ≡ ( px, py, 0) transverse to the propagation directionas well as p · k = w( p0 − pz)/c are constants of the motion inside the field (47). Ifwe assume that the laser field (47) is turned on and off as a function of u = t − z/c,then pT and p · k are also conserved when the electron enters and leaves the field.

74 W. Becker et al. [IV

As we did for the nonrelativistic simple-man model in Sect. II.A.1, we assumethat the electron is initially, at some space–time instant u0, at rest. Then

pT = eA(u0) and p0 − pz = mc for all times. (49)

From the condition that u2 = c2, using the conditions (49), we obtain the energyas a function of u,

g =E

mc2= 1 +

e2

2m2c2(A(u) − A(u0))

2 . (50)

This yields the cycle-averaged kinetic energy

〈Ekin〉 = 〈E〉 − mc2 = p2T2m

+ UP. (51)

This is exactly the same decomposition into drift energy and ponderomotiveenergy as in the nonrelativistic case, Eq. (3). The ponderomotive energy isstill defined by Eq. (4), and the classical bounds of the spectrum discussedin Sect. II.A.1 are unchanged. However, velocity and canonical momentum areconnected by the relativistic expresion mgv = pT − eA, and the cycle averagewas performed with respect to u rather than the time t. Since it can be shownthat u is proportional to the electron’s proper time, this was, actually, the properthing to do (Kibble, 1966).The fact that p · k is a conserved quantity implies that the electron’s velocity

in the propagation direction of the laser field is given by

pz = mgvz = mc(g − 1) = Ekin/c. (52)

The presence of this momentum reflects the fact that a laser photon has amomentum in the direction of its propagation or, alternatively, that the magneticfield via the Lorentz force causes a drift in the propagation direction or,alternatively, that the laser field exerts radiation pressure. All three statementsare essentially equivalent. As a consequence, electrons born with zero velocity ina relativistic laser field are no longer emitted in the direction of its polarization,but acquire a component in the propagation direction of the laser so that, forcircular polarization, they are emitted in a cone given by the angle

tan q =|vT (t)||vz(t)|

∣∣∣∣∞=2m|pT | =

√2

g∞ − 1(53)

with respect to the propagation direction. The subscript ∞ characterizesquantities outside of the laser field. In the derivation, Eqs. (49)–(52) were used.The angle q has been observed by Moore et al. (1995, 1999) for intensities

IV] ATI: CLASSICAL TO QUANTUM 75

of several 1018 W/cm2 and l = 1.053mm and was used to draw conclusionsregarding the actual (nonzero) value of the initial velocity (McKnaught et al.,1997), which can be introduced as discussed in the nonrelativistic case inSect. II.A.1. There are, however, still some unresolved issues in the interpretationof these experiments (Taıeb et al., 2001).The cycle average of Eq. (50) can be written in the covariant form

p2 = m2c2 − 〈(eA)2〉 ≡ m2∗c2 > m2c2, (54)

where p2 and (eA)2 < 0 are invariant four-dimensional scalar products. Thisrelation is often used to introduce the so-called “relativistic effective mass” m∗.It occurs very naturally in the context of the Klein–Gordon equation

[(iðm − eAm)

2 − m2c2]Y = 0, (55)

which explicitly displays the effective mass. However, one has to keep inmind that this apparently increased mass is just due to the transverse wigglingmotion of the electron, viz. the ponderomotive energy, and that there is nothingespecially relativistic about it. All the same, envisioning the ponderomotiveenergy as a mass increase makes sense since, like the rest mass, it is an energyreservoir that is not easily tapped.The classical kinematics just discussed are embedded in quantum-mechanical

calculations, which can be carried out along the lines of the strong-fieldapproximation (17), taking the relativistic instead of the nonrelativistic Volkovwave function (Reiss, 1990; Faisal and Radozycki, 1993; Crawford and Reiss,1997). In particular, the stationary-phase approximation is well justified, leadingto a form similar to Eq. (21) (Krainov and Shokri, 1995; Popov et al., 1997; Muret al., 1998; Krainov, 1999; Ortner and Rylyuk, 2000).

B. Rescattering in the Relativistic Regime

With increasing laser intensity, the first relativistic effect to become significant –before the ponderomotive potential becomes comparable with the electronic restmass – is the drift momentum (52) in the direction of propagation of the laserfield, which can be traced to the Lorentz force. This has virtually no effect onthe initial process of ionization where the electron’s velocity is low, but since itis always positive it prevents the electron from returning to the ion. Therefore,with increasing intensity it gradually eliminates the significance of rescatteringprocesses. This effect can be estimated by calculating the distance by whichthe electron misses the ion in the z-direction when it returns to the ion in the

76 W. Becker et al. [V

x−y plane (approximately at the time tret ≈ T/2). From Eqs. (52) and (51) (where,for simplicity, we only kept UP), we obtain

〈vz〉T/2 ≈ UP2mc2

l (56)

with l the wavelength of the laser field. Obviously, this distance can exceed thewidth of the wave packet of the returning electron to the point where it does notoverlap anymore with the ion, even when UP/mc2 � 1.In HHG the consequences have been investigated in a number of recent

theoretical works (see Sect. V.D) and were found to cause a dramatic drop of theplateau. The same should be expected for high-order ATI, but to our knowledge,this has not been explored in detail. However, in the analysis of multiple-nonsequential-ionization experiments of neon at 2× 1018 W/cm2 a conspicuoussuppression of the highest charge state has been attributed to the magnetic-field-induced drift (Dammasch et al., 2001).

V. Quantum Orbits in High-order Harmonic Generation

According to the rescattering model, the physics of high-order ATI and high-order harmonic generation differ only in the third step: elastic scattering versusrecombination. Correspondingly, the description in terms of quantum orbits canbe applied to HHG as well; in fact, quantum orbits were introduced for the firsttime in the analysis of HHG by Lewenstein et al. (1994). It is from the practicalpoint of view that the two processes differ greatly: HHG by one single atom hasnever been observed, only HHG by an ensemble of atoms. This introduces phasematching as an additional consideration, equal in significance to the single-atombehavior (Salieres et al., 1999; Brabec and Krausz, 2000).Below we will consider examples of a quantum-orbit analysis of HHG for

several nonstandard situations. The first example is an elliptically polarized laserfield. A bichromatic elliptically polarized laser field was considered by Milosevicet al. (2000), and in Sect. V.C we concentrated on a special case of such a field:a two-color bicircular field. Finally, in Sect. V.D the quantum-orbits formalism isextended into the relativistic regime. A bichromatic linearly polarized laser fieldwas investigated by Faria et al. (2000), and a simplified version of the quantum-orbits formalism was used to deal with problems in the presence of a laser fieldand an additional static electric field (Milosevic and Starace, 1998, 1999c) ora laser field and an additional magnetic field (Milosevic and Starace, 1999a,b,2000).

V] ATI: CLASSICAL TO QUANTUM 77

A. The Lewenstein Model of High-order Harmonic Generation

The matrix element for emission of a photon with frequency W and polarization ûin the HHG process in the context of the strong-field approximation (Lewensteinet al., 1994),

Mû(W) ~∫ ∞

−∞dt1

∫ t1

−∞dt0

∫d3k exp [iSW(t1, t0,k)]mû(t1, t0,k), (57)

has the same structure as the corresponding expression (31) for ATI. Thefunction

mû(t1, t0,k) = 〈y0 |er · û|k − eA(t1)〉〈k − eA(t0) |er · E(t0)|y0〉 (58)

is the product of two matrix elements: one that describes the ionization at time t0due to interaction with the laser field, and another one at time t1 that correspondsto recombination into the ground state followed by emission of the high-orderharmonic photon having the polarization û. The difference to ATI is mostly inthe first term of the action:

SW(t1, t0,k) =∫ ∞

t1

dt (EIP − W) −12m

∫ t1

t0

dt [k − eA(t)]2 +∫ t0

−∞dt EIP, (59)

which now refers to the emitted photon. The corresponding saddle-pointapproximation of Eq. (57) is like the HATI approximation (37), except that thesummation is now over saddle points that are solutions of the system of equations(34), (35) and (Lewenstein et al., 1995b, Kopold et al., 2000b)

[k − eA(t1)]2 = 2m(W − EIP). (60)

The last equation corresponds to the condition of energy conservation at the timeof recombination and replaces the condition (36) of elastic rescattering in HATI.For a linearly polarized monochromatic field, quantum orbits were employed

from the very beginning for the evaluation of HHG in the Lewenstein model(Lewenstein et al., 1994, 1995b) and routinely applied in the theoretical analysisand interpretation (Salieres et al., 1999). Conversely, numerical solutions of theTDSE were analyzed in terms of the short (t1) and the long (t2) quantum orbit,and the dominance of these two orbits was corroborated (Gaarde et al., 1999;Kim et al., 2001). The contributions of the long and the short orbit could bespatially resolved in an experiment by Bellini et al. (1998). Spectral resolutionwas achieved by exploiting the dependence of phase matching on the position ofthe atomic jet with respect to the laser focus by Lee et al. (2001) and by Saliereset al. (2001).


At the end of Sect. II.B we remarked that the quantum-orbit formalismis not restricted to periodic fields, but can equally well be applied to finitepulses. For a periodic field, interference of contributions from different cyclesgenerates a discrete spectrum. For a finite pulse, it enhances or suppressesparticular frequency intervals. This was dubbed “intra-atomic phase matching”by Christov et al. (2001) and has been calculated in terms of quantum orbits; inthe context of the TDSE, see Watson et al. (1997). This mechanism underliesthe engineering of a HHG spectrum by tailoring the pulse shape in a feedback-controlled experiment (Bartels et al., 2000, 2001). Individual HHG peaks couldbe enhanced by up to an order of magnitude.A description of HHG that is practically equivalent to the Lewenstein

model is based on the integral equation (43) and the zero-range potential (42)(Becker et al., 1990, 1994b). The equivalence implies that the contribution of“continuum–continuum terms” is insignificant (Becker et al., 1997). The three-step nature of HHG – direct ATI followed by continuum propagation followedby laser-assisted recombination – is particularly emphasized in the approach ofKuchiev and Ostrovsky (1999, 2001), where the integration over the intermediatemomentum k is replaced by a discrete summation over ATI channels. The latteris carried out by a variant of the saddle-point approach, which is reminiscent ofRegge poles and leads to a complex effective channel number.

B. Elliptically Polarized Fields

High-order-harmonic generation by an elliptically polarized field is of greatinterest for applications such as the generation of sub-femtosecond pulses(Corkum et al., 1994). For theoretical calculations in the context of the SFA, seeBecker et al. (1994, 1997) and Antoine et al. (1996); for a fairly comprehensivelist of references, see Milosevic (2000). Fields having polarization other thanlinear generate particularly appealing quantum orbits since they allow them tounfold in a plane. As an example, Fig. 17 shows a HHG spectrum for theelliptically polarized laser field (5) (Kopold et al., 2000b; Milosevic, 2000).The figure confirms that the “exact results” are well approximated by thecontributions of only the six shortest orbits. This figure is the analog of Fig. 14for HATI. The spectrum exhibits the same staircase structure, and everythingsaid there applies here as well.

C. HHG by a Two-color Bicircular Field

The bichromatic w–2w laser field

E(t) = 12 i(E1e+e−iwt + E2e−e−2iwt) + c.c., (61)

whose two components are circularly polarized and counter-rotating in the sameplane (e± = (x± iy)/

√2), is known to generate high harmonics very efficiently;


Fig. 17. High-order harmonic spectrum for an elliptically polarized laser field with the sameparameters as in Fig. 14 and harmonic orders between 25 and 77. The open circles are calculatedfrom the integral (57), and the curves labeled 1 through 6 represent the individual contributions tothe quantum-orbit approximation of the six shortest quantum orbits, numbered as in Fig. 14. Thecontributions from quantum orbits 2, 4 and 6 have to be dropped above their intersections withcurves 1, 3 and 6, respectively. The coherent sum of all six orbits is represented by the solid line.Typical orbits responsible for each part of the spectrum are depicted as in Fig. 14. From Kopold

et al. (2000b).

see Eichmann et al. (1995) for experimental results and Long et al. (1995) fora theoretical description. We will call this field “bicircular.” This high efficiencywas surprising because, for a monochromatic field, the harmonic emissionrate decreases with increasing ellipticity (cf. the preceding subsection) and acircularly polarized laser field does not produce any harmonics at all. A moredetailed analysis, based on the quantum-orbits formalism, gives an explanationof this effect (Milosevic et al., 2000, 2001a,b). The harmonics produced this waycan be of a practical importance because of their high intensity (Milosevic andSandner, 2000) and temporal characteristics (attosecond pulse trains; Milosevicand Becker, 2000). The more general case of an rw−sw (with r and s integers)bicircular field was considered by Milosevic et al. (2001a).For the laser field (61), selection rules only permit emission of circularly

polarized harmonics with frequencies W = (3n ± 1)w and helicities ±1.Similar selection rules govern harmonic generation by a ring-shaped molecule(Ceccherini and Bauer, 2001) or a carbon nanotube (Alon et al., 2000).


Fig. 18. Harmonic-emission rate as a function of the harmonic order for the bicircular laser field (61)with w = 1.6 eV and intensities I1 = I2 = 4×1014 W/cm2. The ionization potential is EIP = 15.76 eV(argon). The inset shows the laser electric-field vector in the x−y plane for times − 12T � t �

12T ,

with T = 2p /w being the period of the field (61). The arrows indicate the time evolution of the field.The ionization time t0 and the recombination time t1 of the three harmonics W = 19w, 31w and43w are marked by asterisks and solid circles, respectively. These times and harmonics correspondto the dominant saddle-point solution 2 in Fig. 19. In between the ionization time (asterisks) and therecombination time (solid circles) the x-component of the electric field changes from its negativemaximum to its positive maximum, whereas its y-component remains small and does not change

sign. From Milosevic et al. (2000).

Figure 18 presents an example of the harmonic spectrum for the bicircularfield (61). The results are obtained by numerical integration from Eq. (57).Compared with the spectrum of a monochromatic linearly polarized field (see,for example, the nonrelativistic curve in Fig. 22), the spectrum is comparablysmooth. Furthermore, the cutoff is less pronounced and there are small oscilla-tions after the cutoff. These features can be explained in terms of the quantumorbits. Figure 19a shows the first eleven solutions (those having the shortest traveltimes) of the system of the saddle-point equations (34), (35) and (60), whileFig. 19b shows the individual contributions to the harmonic emission rate of thefirst eight of these solutions (Milosevic et al., 2000). Obviously, in the plateauregion the contribution of a single orbit, corresponding to solution 2, is dominantby one order of magnitude, while in the cutoff region more solutions are relevant(in particular solution 5). This is just the opposite of the standard situation of themonochromatic linearly polarized field (Lewenstein et al., 1995b) where essen-tially two orbits contribute in the plateau and just one in the cutoff region. Fig-ure 19a tells which solutions are dominant. The probability of harmonic emission


Fig. 19. Saddle-point analysis of the results of Fig. 18. (a) The imaginary part of the recombinationtime t1 as a function of the real part of the travel time t1 − t0, obtained from the solutions of thesaddle-point equations (34), (35) and (60). Each point on the curves corresponds to a specific valueof the harmonic frequency W, which is treated as a continuous variable. For the interval of Re(t1 − t0)covered in the figure, eleven solutions were found, which are labeled with the numbers in boldfaceitalics. Values of the harmonic order that approximately determine the cutoffs for each particularsolution are marked by stars with the corresponding harmonic numbers next to them. Those values ofthe harmonic order for which | Im t1| is minimal are identified as well. (b) The partial contributionsto the harmonic-emission rate of each of the first eight solutions of the saddle-point equations. From

Milosevic et al. (2000).


Fig. 20. Real parts of the quantum orbits for the same parameters as in Fig. 18 and for the harmonicW = 43w. Five orbits are shown that correspond to the saddle-point solutions 2, 3, 4, 5 and 8 inFig. 19. The direction of the electron’s travel is given by the arrows. In each case, the electronis “born” a few atomic units away from the position of the ion (at the origin), where its orbitalmost exactly terminates. The dominant contribution to the 43rd harmonic intensity comes fromthe shortest orbit number 2, whose shape closely resembles the orbit in the case of a linearly polarized

monochromatic field. From Milosevic et al. (2000).

decreases with increasing absolute value of the imaginary part of the recombina-tion time t1. The possible cutoff of the harmonic spectrum can be defined as thevalue of the harmonic order after which | Im t1| becomes larger than (say) 0.01T .The probability of HHG is maximal when | Im t1| is minimal. For each solution inFig. 19a, these points are marked by asterisks and by the corresponding harmonicorder. As a consequence of wave-function spreading, the emission rate decreaseswith increasing travel time t1 − t0. This gives an additional reason why thecontribution of solution 2 is dominant in the plateau region.Let us now consider the quantum orbits. In Fig. 20 for the fixed harmonic

W = 43w, we present the five orbits that correspond to saddle-point solutions2, 3, 4, 5 and 8 in Fig. 19. The dominant contribution comes from the shortestorbit 2 (thick line). It starts at the point (4.06, 0.66) by setting off in the negativey-direction, but soon turns until it travels at an angle of 68º to the negative y-axis.Thereafter, it is essentially linear, as would be the case for a linearly polarizedfield. This behavior can be understood by inspection of the driving bicircular fielddepicted in the inset of Fig. 18, where the start time and the recombination timeof the orbit are marked. During the entire length of the orbit, the field exertsa force in the positive y-direction. The effect of this force is canceled by theelectron’s initial velocity in the negative y-direction. The force in the x-directionis much like that in the case of a linearly polarized driving field. Since HHG by a


linearly polarized field is most efficient, this makes plausible the high efficiencyof HHG by the bicircular field.The orbit that corresponds to solution 3 has a shape similar to that of

orbit 2, but is much longer. The corresponding travel time is longer, too,and, consequently, the contribution of solution 3 to the emission rate of the43rd harmonic is smaller. The other orbits are still longer and more complicatedso that their contribution is negligible.The electric field of a group of plateau harmonics is displayed in Fig. 18. It

shows interesting behavior, which again reflects the threefold symmetry of thefield (61), see the inset of Fig. 18. If the group of harmonics includes harmonicsof either parity, then the field consists of a sequence of essentially linearlypolarized, strongly chirped attosecond pulses, each rotated by 120º with respectto the previous one. If, on the other hand, one were able to select harmonics ofdefinite helicity, i. e. either W = (3n + 1)w or W = (3n − 1)w, then one wouldobtain a sequence of attosecond pulses with approximately circular polarization.Both cases are illustrated in Fig. 21.

Fig. 21. Parametric polar plot of the electric-field vector of a group of harmonics during one cycleof the bicircular field (61) on an arbitrary isotropic scale. The position of the origin is indicated inthe upper and the left margin. The parameters are I1 = I2 = 9.36×1014 W/cm2, àw = 1.6 eV, andEIP = 24.6 eV. The plot displays two traces: The circular trace is generated by the ten harmonicsW = (3n + 1)w with n = 10, . . . , 19, all having positive helicity. The starlike trace is generated by allharmonics W = (3n ± 1)w between the orders 31 and 59, regardless of their helicity. The curve atthe bottom represents the x-component of the field of the latter group over one cycle, the time scalebeing given on the horizontal axis. It shows that the field is strongly chirped. The black blob at thecenter is due to the fact that the field is near zero throughout most of the cycle, cf. the trace of the

x-component. From Milosevic and Becker (2000).


D. HHG in the Relativistic Regime

Quantum orbits can also be employed in the relativistic regime starting fromthe Klein–Gordon equation (55). Milosevic et al. (2001c, 2002) found thatthe relativistic harmonic-emission matrix element has a form similar to that inEq. (57), but with the relativistic action (à = c = 1)

SW(t1, t0,k) =∫ ∞

t1

du (EIP − m − W) −∫ t1

t0

du ek (u) +∫ t0

−∞du (EIP − m), (62)

whereek (u) = Ek + eA(u) ·

k + e2A(u)

Ek − z · k (63)

and Ek = (k2 + m2)1/2, u = (t − z)/w. Solving the classical Hamilton–Jacobiequation for Hamilton’s principal function it can be shown that ek (u) is theclassical relativistic electron energy in the laser field. In the relativistic case,the function mû(t1, t0,k) in Eq. (57) consists of two parts: the dominant partis responsible for the emission of odd harmonics W = (2n + 1)w, whilethe other one originates from the intensity-dependent drift momentum of theelectron in the field and allows for emission of even harmonics W = 2nw.Similarly to the nonrelativistic case, the integral over the intermediate electronmomentum k can be calculated by the saddle-point method. The stationaritycondition

∫ t1t0du ðek (u)/ðk = 0, with ðek /ðk = dr/dt, implies r(t0) = r(t1), so

that the stationary relativistic electron orbit is such that the electron starts fromand returns to the nucleus. As above, the start time and, to a lesser degree, therecombination time are complex.In the relativistic case, the stationary momentum k = ks is introduced in

the following way. For fixed t0 and t1, its component ks⊥ perpendicular to thephoton’s direction of propagation z is given by

(t1 − t0)ks⊥ =∫ t0

t1

du eA(u). (64)

IntroducingM2 = e2∫ t1t0duA2(u)/ (t1 − t0) − k2s⊥ > 0, one has

k2s = k2s⊥ +(M2 − k2s⊥)

2

4(m2 +M2), (65)

which yields eks as a function of t0 and t1. The two stationarity equationsconnected with the integrals over t0 and t1 are

eks (t0) = m − EIP, (66)

W = eks (t1) + EIP − m. (67)

As in the nonrelativistic case, they express energy conservation at the time oftunneling t0 and at the time of recombination t1, respectively. The final expression


Fig. 22. Harmonic-emission rate as a function of the harmonic order for ultrahigh-order harmonicgeneration by an Ar8+ ion (EIP = 422 eV) in the presence of an 800-nm Ti:Sa laser having theintensity 1.5×1018 W/cm2. Both the nonrelativistic and the relativistic results are shown. Thecorresponding relativistic electron orbit with the shortest travel time that is responsible for theemission of the harmonic W = 100000w is shown in the inset. The arrows indicate which way theelectron travels the orbit. The laser field is linearly polarized in the x-direction and the v×B electron

drift is in the z-direction. From Milosevic et al. (2002).

for the relativistic harmonic-emission matrix element has the form (37) with (62),where the summation is now over the appropriate subset of the relativistic saddlepoints (t1s, t0s,ks) that are the solutions of the system of equations (64)−(67).In the relativistic case it is very difficult to evaluate the harmonic-emission

rates by numerical integration. For very high laser-field intensities and ultra-highharmonic orders, this is practically impossible, so that the saddle-point methodis the only way to produce reasonable results. Figure 22 presents an example.The nonrelativistic result is obtained from Eq. (37) where the summation isover the solutions of the system of the nonrelativistic saddle-point equations(34), (35) and (60). It is, of course, inapplicable for the high intensity of1.5×1018 W/cm2 at 800 nm and is only shown to demonstrate the dramaticimpact of relativistic kinematics. For the relativistic result, the summation inEq. (37) is over the relativistic solutions of Eqs. (64)−(67). The relativisticharmonic-emission rate assumes a convex shape, and the difference betweenthe relativistic and nonrelativistic results reaches several hundred orders ofmagnitude in the upper part of the nonrelativistic plateau. The origin of thisdramatic suppression is the magnetic-field-induced v× B drift. The significance

86 W. Becker et al. [VI

of this drift for the rescattering mechanism was emphasized early by Kulyaginet al. (1996). This is illustrated in the inset of Fig. 22, which shows the realpart of the dominant shortest orbit for the harmonic W = 100000w. In order tocounteract this drift so that the electron is able to return to the ion, the electronhas to take off with a very substantial initial velocity in the direction oppositeto the laser propagation. The probability of such a large initial velocity is low,and this is the reason for the strong suppression. As in the nonrelativistic case,the electron is “born” at a distance of 7.5 a.u. from the nucleus.The nonrelativistic harmonic yield shows a pronounced multiplateau structure.

While this is an artifact of the nonrelativistic approximation for the intensityof Fig. 22, it is a real effect for lower laser-field intensities where relativisticeffects are still small (Walser et al., 2000; Kylstra et al., 2001; Milosevic et al.,2001c, 2002). In this case, the three plateaus visible in the nonrelativistic curve ofFig. 22 are related to the three pairs of orbits, whose contribution to the harmonicemission rate is dominant in the particular spectral region (see Figs. 2 and 3of Milosevic et al., 2001c). These are very similar to the pairs of orbits thatwe have discussed for the elliptically polarized laser field in Fig. 17. However,for the very high intensity of Fig. 22, the contribution of the shortest of theseorbits becomes so dominant that the multiplateau and the interference-relatedoscillatory structure disappear completely. The reason is that the effect of thev × B drift increases with increasing travel time; see Eqs. (52) and (56) inSect. IV.A. This is in contrast to the nonrelativistic case of elliptical polarization,where longer orbits may be favored because the minor component of the fieldoscillates and, therefore, for a longer orbit a smaller initial velocity may besufficient to allow the electron to return.

VI. Applications of ATI

Experimental and theoretical advances in understanding ATI – some of whichhave been treated in this review – permit its application to the investigation ofother effects. One obvious idea is to exploit the nonlinear properties of ATI.This is particularly relevant to characterization of high-order harmonics andmeasurement of attosecond pulses in the soft-X-ray regime. In this spectralregion (vacuum UV) virtually all bulk non-linear media are opaque. ATI,in contrast, is usually studied under high- or ultra-high-vacuum conditions.Another advantage over conventional nonlinear optics is that the nonlinear effectof photoelectron emission can be observed from more or less any direction,whereby different properties of the effect can be exploited.

A. Characterization of High Harmonics

The most straightforward approach to characterize high-order harmonics is across-correlation scheme: An (isolated) harmonic of frequency qw, where q is

VI] ATI: CLASSICAL TO QUANTUM 87

an odd integer, produces electrons by single-photon ionization with a kineticenergy Eq = qàw − EIP. Simultaneous presence of a fraction of the fundamentallaser beam in the near infrared (NIR) produces sidebands, i.e. electrons withenergies qàw − EIP ± màw (m � q). The strength of the sidebands can bechanged by temporally delaying the fundamental with respect to the harmonicby a time t . Optimal overlapping of the pulses (t = 0) leads to a maximum inthe strength of the sidebands, whereas complete separation entirely eliminatesthem. The strength of the sidebands as a function of t can be used to determinethe duration of the harmonic pulse.For theoretical modeling, the simple ansatz of Becker et al. (1986) can be used,

which assumes that an electron is born in the presence of the laser field with apositive initial energy Ei, which will be identified with Eq. For Up � àw, whichis well satisfied for the weak field we will consider, the differential ionizationrate in the field direction is given by (in atomic units)

ð2GðEðW

∝ |p| ·∞∑

m = −∞J 2m

(Ef√2(Ei + mw)w2

)d(E − mw − Ei). (68)

Here, Ef is the amplitude of the electric field of the fundamental, m is the orderof the sideband, p is the momentum of the photoelectron (|p| =√2(Ei + mw)),and Jm is the Bessel function of the first kind. The intensities of the side bandsare not, in general, symmetric. However, for sufficiently weak fields, both fieldscan be treated by lowest-order perturbation theory. It follows that a sideband oforder m is proportional to E2hE2|m|

f , where Eh is the field strength of the harmonicradiation. In this case, the cross correlation for a sideband of order m can becalculated as

Cm(t) =

∞∫−∞E2h (t) · E2|m|

f (t − t) dt. (69)

Figure 23 (overleaf) shows the result of a corresponding calculation, whichis compared with results from a numerical solution of the appropriate one-dimensional Schrodinger equation. The agreement is nearly perfect.Hence, if the NIR pulse is precisely known, the pulse duration of the harmonic

(and even its shape) can be determined by deconvolution of the cross-correlationfunctions. Numerical and experimental investigations of this problem were madeby Veniard et al. (1995) and Schins et al. (1996), respectively.

A.1. Measurement of attosecond pulses

Apparently, an experiment as discussed above will not be able to determineharmonic-pulse durations significantly shorter than that of the fundamentalin the NIR spectral region. In 1996 already, Veniard et al. pointed out that


Fig. 23. Cross-correlation of near-infrared and soft-X-ray pulses. A harmonic of order q createsphotoelectrons at the kinetic energy qàw − EIP. Sidebands are created by simultaneous irradiationwith the fundamental of frequency w. Plotted are the heights of the sidebands for various side-bandorders m versus the delay t between the fundamental and the harmonic. The solid line representsthe analytical approximation (69), whereas the points were calculated by numerically solving theappropriate (one-dimensional) Schrodinger equation. In each case, the analytical approximation was

normalized to the maximum of the numerical result.

the cross correlation of harmonic and NIR radiation provides access to therelative phase of neighboring harmonics. This is an extremely important insightbecause the phase dependence of the harmonics as a function of their order(or frequency) determines whether they are mode-locked and whether thecorresponding pulses – which would constitute attosecond pulses in the soft-X-ray region if bandwidth-limited – are chirped. In fact, Paul et al. (2001) usedthis scheme for the first observation of a train of attosecond pulses.In order to achieve phase measurement of adjacent harmonics, the conditions

have to be chosen such that only sidebands of order m = ±1 are generatedwith appreciable amplitude. This calls for intensities of the NIR beam be-low 1012 W/cm2. Along with the fact that the NIR field generates only odd-order harmonics this ensures that only two adjacent harmonics contribute to eachsideband. An electron with energy Eq = qàw − EIP, with q an even integer,can be generated by absorption of the lower harmonic plus one NIR photon(Eq = (q − 1)àw + àw) or by absorption of the upper harmonic and emissionof one NIR photon (Eq = (q + 1)àw − àw). Each of these two channels receivescontributions from two different quantum paths, which are related to the temporalorder of the interaction with the harmonic and the NIR field. (In contrast tothe quantum orbits we considered elsewhere in this chapter, the quantum pathshere are defined in state space rather than position space.) The photoelectron

VI] ATI: CLASSICAL TO QUANTUM 89

Fig. 24. Reconstruction of a train of attosecond pulses synthesized from the five harmonicsq = 11, . . . , 19. The attosecond pulses are separated by 1.35 fs, which is half the cycle time ofthe driving laser. The latter is represented by the dashed cosine function. Reprinted with permissionfrom Paul et al. (2001), Science 292, 1689, fig. 4. © 2001 American Association for the Advancement

of Science.

yield at energy Eq is proportional to the square of the (coherent) sum of theamplitudes of all four quantum paths. Due to the fact that two paths representabsorption from the NIR field whereas the other two represent emission into it,the interference term between these two contributions is essentially proportionalto cos(÷q−1 − ÷q+1 + 2àwt). By varying the delay t between the harmonic andthe NIR radiation, the difference ÷q−1 − ÷q+1 of the phases of the two harmonicscan be recorded. The result of the corresponding experiment (Paul et al., 2001)is that the phase of the harmonics depends almost linearly on their frequency.Hence, the harmonics considered in the experiment (q = 11 to 19) are mode-locked and make up a train of attosecond pulses of 250 as FWHM duration, seeFig. 24.

A.2. Isolated attosecond pulses

With respect to applications, isolated attosecond pulses appear more useful thana train of pulses separated by half the period of the fundamental. Isolatedattosecond pulses could be generated by sufficiently short fundamental pulses,i.e. pulses of about 5 fs, which consist of less than two optical cycles (few-cycleregime). Then, however, the spectral width of the harmonics will be so broadthat it is no longer possible to identify individual sidebands as necessary for themethod of Paul et al. (2001).Nevertheless, Drescher et al. (2001) and Hentschel et al. (2002) succeeded

in performing measurements of the harmonic-pulse length with a resolution of1.8 fs and 150 as, respectively. The experimental setup, in principle, resemblesthat of Paul et al. with the difference that higher intensities of the NIR ra-


diation are used for the photoionization cross correlation. In addition, onlyphotoelectrons ejected perpendicularly to the laser polarization are detected.The motivation for choosing these conditions can be deduced from a classicalanalysis of trajectories of electrons that were injected into the electric field ofthe few-cycle NIR pulse by absorption of a harmonic photon. If the duration ofthe X-ray pulse is shorter than the optical period T in the NIR, then the finalkinetic energy of the photoelectrons depends on the phase wt0 when the injectiontook place, i. e. it exhibits a modulation with a period of T/2. By delayingthe fundamental with respect to the harmonic, the modulation can be recorded.This was done in the experiment of Drescher et al. (2001). Hentschel et al.(2002) relized that the width of the photoelectrons’ kinetic energy distributionalso exhibits such a modulation, and is measureable with much higher precisionthan the center of mass of the distribution. For the two approaches, it is not theenvelope of the fundamental that enters the correlation function, but rather theoptical period. The restriction to photoelectrons emitted perpendicularly to thelaser polarization suppresses the influence of effects related to the emission andabsorption of photons from the laser field, i.e. the sidebands which were crucialfor the experiment of Paul et al. (2001).

B. The “Absolute Phase” of Few-cycle Laser Pulses

The need for highest intensities and extremely broad bandwidths in several areasof the natural sciences is driving the development to shorter and shorter laserpulses. At a FWHM duration shorter than a few optical cycles the time variationof the pulse’s electric field depends on the phase f of the carrier frequency withrespect to the center of the envelope, the so-called “absolute phase.” The electricfield should be written as

E(t) = E0(t) ex cos(wt + f), (70)

where the function E0(t) is maximal at t = 0. Clearly, for a long pulse the phase fcan be practically eliminated by resetting the clock. For a short pulse, however,the shape of the field (70) strongly depends on this phase, which, therefore, willinfluence various effects of the laser–atom interaction. This is one reason for thesignificance of this new parameter of laser pulses. The precise knowledge andcontrol of the absolute phase will pave the way to new regimes in coherent X-raygeneration and attosecond generation; for an overview see Krausz (2001). In ad-dition, such extremely well-defined laser pulses are likely to have applications forthe coherent control of chemical reactions and other processes. Another reason isthat phase control of femtosecond laser pulses has already had a huge impact onfrequency metrology. This is because phase-stabilized femtosecond lasers can beviewed as ultra-broadband frequency combs that can be used to measure opticalfrequencies with atomic-clock precision; see, e.g., Jones et al. (2000).

VIII] ATI: CLASSICAL TO QUANTUM 91

With current laser technology, only femtosecond laser oscillators can be phase-stabilized (Reichert et al., 1999; Apolonski et al., 2000), which is sufficient forfrequency metrology. Strong-field effects require amplified laser pulses. Nisoliet al. (1997) demonstrated that it is possible to generate powerful (>500mJ)laser pulses in the few-cycle regime. However, these are not stabilized and,accordingly, the absolute phase changes in a random fashion from pulse to pulse.In a recent experiment, Paulus et al. (2001) were able to detect effects dueto the absolute phase by performing a shot-to-shot analysis of the number ofphotoelectrons emitted in opposite directions. To this end, a field-free drift tube isplaced symmetrically around the target gas. Each end of the tube is equipped withan electron detector. Because of its characteristic appearance, this was dubbed astereo-ATI spectrometer.A characteristic feature of few-cycle pulses such as (70) is that, depending

on the absolute phase, the peak electric-field strength (and thus also the vectorpotential) is different in the positive and negative x-directions. Recall fromEq. (2) that the electron’s drift momentum depends on the vector potential at itstime of birth. Therefore, depending on the value of the absolute phase, such alaser pulse creates more electrons in one direction than in the other. A theoreticalanalysis of the photoelectrons’ angular distribution was given by Dietrich et al.(2000) and Hansen et al. (2001). An equivalent statement is that the numberof electrons emitted to the left vs. those emitted to the right is anticorrelated:A laser shot for which many electrons are seen at the right detector is likely toproduce only a few that go left, and vice versa. This can be proved by correlationanalysis. Each laser shot is sorted into a contingency map according to thenumber of electrons recorded at both detectors. Anticorrelations can then be seenin structures perpendicular to the diagonal, see Fig. 25 (overleaf).

VII. Acknowledgments

We learned a lot in discussions with S.L. Chin, M. Dorr, C. Faria, S.P.Goreslavskii, C.J. Joachain, M. Kleber, V.P. Krainov, M. Lewenstein, A. Lohr,H.G. Muller, S.V. Popruzhenko, and W. Sandner. This work was supported inpart by Deutsche Forschungsgemeinschaft and Volkswagen Stiftung.

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