* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download ABOVE-THRESHOLD IONIZATION: FROM CLASSICAL FEATURES
Quantum key distribution wikipedia , lookup
Coherent states wikipedia , lookup
Path integral formulation wikipedia , lookup
Renormalization group wikipedia , lookup
Renormalization wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Canonical quantization wikipedia , lookup
Matter wave wikipedia , lookup
X-ray fluorescence wikipedia , lookup
Double-slit experiment wikipedia , lookup
Wave–particle duality wikipedia , lookup
Particle in a box wikipedia , lookup
Auger electron spectroscopy wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Tight binding wikipedia , lookup
Atomic theory wikipedia , lookup
X-ray photoelectron spectroscopy wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Atomic orbital wikipedia , lookup
Hydrogen atom wikipedia , lookup
Electron configuration wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 48 ABOVE-THRESHOLD IONIZATION: FROM CLASSICAL FEATURES TO QUANTUM EFFECTS W. BECKER 1 , F. GRASBON 2 , R. KOPOLD 1 , D.B. MILOŠEVIĆ 3 , G.G. PAULUS 2 and H. WALTHER 2,4 1 Max-Born-Institut, Max-Born-Str. 2A, 12489 Berlin, Germany; 2 Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany; 3 Faculty of Science, University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Hercegovina; 4 Ludwig-Maximilians-Universität München, Germany I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . B. Theoretical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . II. Direct Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Classical Model . . . . . . . . . . . . . . . . . . . . . . . . . . B. Quantum-mechanical Description of Direct Electrons . . . . C. Interferences of Direct Electrons . . . . . . . . . . . . . . . . . . III. Rescattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Classical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . B. Quantum-mechanical Description . . . . . . . . . . . . . . . . . . IV. ATI in the Relativistic Regime . . . . . . . . . . . . . . . . . . . . . . A. Basic Relativistic Kinematics . . . . . . . . . . . . . . . . . . . . B. Rescattering in the Relativistic Regime . . . . . . . . . . . . . . V. Quantum Orbits in High-order Harmonic Generation . . . . . . . A. The Lewenstein Model of High-order Harmonic Generation B. Elliptically Polarized Fields . . . . . . . . . . . . . . . . . . . . . . C. HHG by a Two-color Bicircular Field . . . . . . . . . . . . . . . D. HHG in the Relativistic Regime . . . . . . . . . . . . . . . . . . . VI. Applications of ATI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Characterization of High Harmonics . . . . . . . . . . . . . . . . B. The “Absolute Phase” of Few-cycle Laser Pulses . . . . . . . VII. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 38 38 40 40 43 47 50 50 53 73 73 75 76 77 78 78 84 86 86 90 91 91 I. Introduction With the discovery of above-threshold ionization (ATI) by Agostini et al. (1979) intense-laser atom physics entered the nonperturbative regime. These experiments recorded the photoelectron kinetic-energy spectra generated by laser irradiation 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 series of 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 Schrödinger equation (Paulus, 1996). by way of counting ions, and the data were well described by lowest-order perturbation theory (LOPT) with respect to the electron–field interaction. This LOPT regime was already highly nonlinear (see, e.g., Mainfray and Manus, 1991), the lowest order being the minimal number N of photons necessary for ionization. An ATI spectrum consists of a series of peaks separated by the photon energy, see Fig. 1. They reveal that an atom may absorb many more photons than the 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 peaks at odd harmonics of the laser frequency (McPherson et al., 1987; Wildenauer, 1987). The spectra of this high-order harmonic generation (HHG) display a plateau (Ferray et al., 1988), i.e., the initial decrease of the harmonic yield with increasing harmonic order is followed by a flat region where the harmonic intensity is more or less independent of its order. This plateau region terminates at 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 by ionization. Thereafter, the laser’s linearly polarized electric field accelerates the electron away from the atom. However, when the field changes direction, then, depending on the initial time of ionization, it may drive the electron back to its parent ion, where it may recombine into the ground state, emitting its entire energy – 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 beautifully explains the cutoff energy of the plateau, as well as the fact that the yield of HHG strongly decreases when the laser field is elliptically polarized. In this event, the electron 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. Elastic scattering should contribute to ATI. Indeed, the corresponding characteristic features in the angular distributions were observed by Yang et al. (1993), and an extended plateau in the energy spectra due to this mechanism, much like the plateau of HHG, was identified by Paulus et al. (1994c). Under the same conditions, a surprisingly large yield of doubly charged ions was recorded (l’Huillier et al., 1983; Fittinghoff et al., 1992) that was incompatible with a sequential ionization process. A potential mechanism causing this nonsequential ionization (NSDI) is inelastic scattering. It was only recently, however, that this inelastic-scattering scenario emerged as the dominant mechanism of NSDI, through analysis of measurements of the momentum distribution of the doubly charged 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 in fully quantum-mechanical descriptions of HHG (Lewenstein et al., 1994; Becker et al., 1994b) and ATI (Becker et al. 1994a; Lewenstein et al., 1995a). This work has led to the concept of “quantum orbits,” a fully quantum-mechanical generalization of the classical orbits of the simple-man model that retains the intuitive appeal of the former, but allows for interference and incorporates quantum-mechanical tunneling. The quantum orbits arise naturally in the context of Feynman’s path integral (Salières 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 for elliptical polarization. Alongside with theory, we will provide a review of the experimental status of ATI. We also give a brief survey of recent applications of ATI. High-order harmonic generation is considered only insofar as it provides further illustrations of the concept and application of quantum orbits. We do not deal with the important collective aspects of HHG, and no attempt is made to represent the vast literature on HHG. For this purpose, we refer to the recent reviews by Salières et al. (1999) and Brabec and Krausz (2000). Earlier reviews pertinent to ATI have been given by Mainfray and Manus (1991), DiMauro and Agostini (1995), and Protopapas et al. (1997). The entire field of laser–atom physics has been compactly surveyed by Kulander and Lewenstein (1996) and, recently, by Joachain et al. (2000). Both of these reviews concentrate on the theory. Nonsequential double ionization is well covered in a recent focus issue of 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 such intensities, atoms may ionize so quickly that complete ionization has taken place before the laser pulse has reached its maximum. This calls, on the one 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 in femtosecond laser technology, in particular since the invention of titanium– sapphire (Ti:Sa) femtosecond lasers (Spence et al., 1991), generation of laser fields with strengths comparable to inner atomic fields has become routine. The prerequisite of detailed investigations of ATI, however, has been the development of femtosecond laser systems with high repetition rate. Owing to the latter, the detection of faint but qualitatively important features of ATI spectra with low statistical noise has become possible. This holds, in particular, if multiply differential 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 the classically allowed regions. State-of-the-art pulses are as short as 5 fs (Nisoli et 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 simultaneously triggers a high-resolution clock. The electrons drift in a field-free flight tube of known length towards an electron detector, which then gives the respective stop pulses to the clock. Now, their kinetic energy can easily be calculated from their time of flight. This approach has by far the highest energy resolution and is comparatively simple. However, the higher the laser repetition rate, the more demanding 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 coldtarget recoil-ion-momentum spectroscopy (COLTRIMS) technology (Dörner et al., 2000), which is capable of providing complete kinematic determination of the 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 disadvantage of COLTRIMS is the poor energy resolution for the electrons and the exacting technology. B. Theoretical Methods The single-active-electron approximation (SAE) replaces the atom in the laser field by a single electron that interacts with the laser field and is bound by an effective potential so optimized as to reproduce the ground state and singly excited states. Up to now, in single ionization no qualitative effect has been identified 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 pulses at 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 in argon with spectra calculated by numerical solution of the three-dimensional time-dependent Schrödinger equation (TDSE) (Nandor et al., 1999); see Fig. 2. The agreement between theory and experiment is as remarkable as it has been achieved for low-order ATI in hydrogen; cf. Dörr et al. (1990) for the Sturmian– Floquet calculation and Rottke et al. (1990) for the experiment. For helium, a comparison of total ionization rates with and without the SAE in the abovebarrier regime has lent further support to the SAE (Scrinzi et al., 1999). Numerical solution of the one-particle TDSE in one dimension was instrumental for the understanding of ATI in its early days; for a review, see Eberly et al. (1992). For the various methods of solving the TDSE in more than one dimension we refer to Joachain et al. (2000). Comparatively few papers have dealt with high-order ATI in three (that is, in effect, two) dimensions. This is particularly challenging since the emission of plateau electrons is caused by very small changes in the wave function, and the large excursion amplitudes of free-electron motion in high-intensity low-frequency fields necessitate a large spatial grid. This is exacerbated for energies above the cutoff and for elliptical polarization. Expansion of the radial wave function in terms of a set of B-spline functions was used by Paulus (1996), by Cormier and Lambropoulos (1997), and by Lambropoulos et al. (1998). Matrix-iterative methods were employed by Nurhuda and Faisal (1999). The most detailed calculations have been carried out by Nandor et al. (1999) and by Muller (1999a,b, 2001a,b). The techniques are detailed by Muller (1999c). To our knowledge, no results for high-order ATI for elliptical polarization based on numerical solution of the TDSE have been published 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 (Vázquez de Aldana and Roso, 1999; Vázquez de Aldana et al., 2001) and various 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 have been 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 socalled Volkov solution (Volkov, 1935), which is available for the Schrödinger equation as well as for relativistic wave equations, and considers the binding potential 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 the TDSE, 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 into several 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 electron enters 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 highly nonlinear function of the laser electric field E(t0 ). For example, the quasistatic Ammosov–Delone–Krainov (ADK) tunneling rate (Perelomov et al., 1966a,b; Ammosov et al., 1986) is given by (in atomic units) 2n∗ − |m| − 1 3 3 4 2EIP 4 2EIP , G(t) = AEIP exp − |E(t)| 3|E(t)| (1) where E(t) is the instantaneous √ electric field, EIP > 0 is the ionization potential 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 momentum on the direction of the laser polarization. The constant A depends on the actual and the effective quantum numbers. The rate G(t) was derived on the assumption that the laser √ frequency is low, excited states play no role, and the Keldysh parameter g = EIP / 2UP is small compared with unity [UP is the ponderomotive potential of the laser field; see Eq. (3) below]. Instantaneous rates that hold for 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 the instantaneous ionization rate G(t) is that it develops a sharp maximum at times when 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 will be weighted according to the value of the rate G(t0 ). Classically, an electron born by tunneling will start its orbit with a velocity of zero at the classical “exit of the tunnel” at r ~ EIP / |eE|, which, for strong fields, is a few atomic units away from the position of the ion. We will, usually, ignore this small offset and have the electronic orbit start at x(t = t0 ) = 0 (the position of the ion) with ẋ(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 be considered below in Sect. III, where the electron is driven back to the ion and rescatters. 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 trajectory in the strong laser field. During this step, the influence of the atomic potential is neglected. For an intense laser field, the electron’s oscillation amplitude is much larger than the atomic diameter, and so this is well justified. For a vector potential A(t) that is chosen so that its cycle average A(t)T is zero, the electron’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 term p ≡ eA(t0 ), which is the drift momentum measured at the detector, and a term that oscillates in phase with the vector potential A(t). The kinetic energy of this electron, averaged over a cycle T of the laser field, is m p2 e2 v(t)2 T = + A(t)2 T ≡ Edrift + UP . 2 2m 2m The ponderomotive energy E= (3) e2 (4) A(t)2 T , 2m viz. the cycle-averaged kinetic energy of the electron’s wiggling motion, is frequently employed to characterize the laser intensity. A useful formula is UP = Up [eV] = 0.09337I [W/ cm2 ] l2 [m] for a laser with intensity I and wavelength l. If the electron is to have a nonzero velocity v0 at time t0 , one has to replace eA(t0 ) by eA(t0 ) + mv0 ≡ p in the velocity (2). 42 W. Becker et al. [II Most of the time, we will be concerned with the monochromatic elliptically polarized laser field (−1 x 1) wA (x̂ sin wt − x ŷ cos wt) E(t) = 1 + x2 (5) with vector potential A(t) = A 1 + x2 (x̂ cos wt + x ŷ sin wt) (6) and ponderomotive energy UP = (eA)2 / 4m. The drift energy Edrift = (eA(t0 ))2 / 2m is restricted to the interval 2x 2 2 UP Edrift UP . 2 1+x 1 + x2 (7) For linear polarization, it can acquire any value between 0 and 2UP , while for circular polarization it is restricted to the value UP . Quantum mechanics considerably softens these classical bounds. However, these bounds are useful as benchmarks 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 proportional to the vector potential, both at the time t0 of ionization. The probability of a certain drift momentum is weighted with the ionization rate at time t0 . The electron is preferably ionized when the absolute value of the electric field is near its maximum. Then, for linear polarization, the vector potential and, hence, the drift momentum are near zero. In order to reach the maximal drift energy of 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 ATI electron spectrum for increasing energy, see Fig. 1. Sometimes, this interplay between the instantaneous ionization rate and the drift momentum has surprising consequences, notably for fields where the connection between the two is less straightforward than for a linearly polarized sinusoidal field, e.g. for a two-color field (Paulus et al., 1995; Chelkowski and Bandrauk, 2000; Ehlotzky, 2001). Another illustration is the dodging phenomenon for the direct ATI electrons in an elliptically polarized laser field (Paulus et al., 1998; Goreslavskii and Popruzhenko, 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 the focused 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 II] ATI: CLASSICAL TO QUANTUM 43 Fig. 3. Dependence of the photoelectron yield as a function of the ellipticity x of the elliptically polarized laser field (5) for electrons with an energy of 16.1 eV. Only electrons emitted parallel to the major axis of the polarization ellipse are recorded. The ATI spectrum corresponding to linear polarization (x = 0) is shown in the inset. The laser intensity was 0.8×1014 W/cm2 at a wavelength of 630 nm. The figure illustrates the dodging effect mentioned in Sect. II.A.1: ionization primarily takes place when the electric field is near an extremum. For elliptical polarization, the electric field then points in the direction of the major axis of the polarization ellipse, and the vector potential in the direction of the minor axis. Hence, the electron’s drift momentum p = eA(t0 ) is in the direction of the minor axis. It is the larger, the larger the ellipticity x is. Consequently, emission in the direction of the large component of the field decreases with increasing ellipticity: the electron dodges the strong 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-consistent description, returned to the field when the electron is left behind by the trailing edge of the pulse. In the opposite case, where the electron escapes from the pulse perpendicularly to its direction of propagation, the wiggling energy is converted into drift energy (Muller et al., 1983). The effects of a space-dependent ponderomotive potential UP were observed in the “surfing” experiments of Bucksbaum et al. (1987). B. Quantum-mechanical Description of Direct Electrons There is an enormous body of work on the quantum-mechanical description of laser-induced ionization. For reviews, we refer to Delone and Krainov (1994, 1998). Here we want to concentrate on the analytical approach dating back to Keldysh (1964) and Perelomov, Popov and Terent’ev (Perelomov et al. 1966a,b; 44 W. Becker et al. [II Perelomov and Popov, 1967). The goal is to find a suitable approximation to the probability amplitude for detecting an ATI electron with drift momentum p that originates from laser irradiation of an atom that was in its ground state |y0 before the laser pulse arrived: Mp = lim t → ∞, t → −∞ yp (t) |U (t, t )| y0 (t ) . (8) Here, U (t, t ) is the time-evolution operator of the Hamiltonian (à = 1) 1 (9) H (t) = − ∇2 − er · E(t) + V (r), 2m 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 atom without the field and for a free electron in the laser field without the atom, 1 (10) Ha ≡ Hatom = − ∇2 + V (r), 2m 1 (11) Hf (t) ≡ Hfield (t) = − ∇2 − er · E(t). 2m The corresponding time-evolution operators are denoted by Ua and Uf , respectively. In Eq. (8), |yp and |y0 are a scattering state with asymptotic momentum p and the ground state, respectively, of the atomic Hamiltonian Ha . The eigenstates of the time-dependent Schrödinger equation with the Hamiltonian Hf (t) are known as the Volkov states and are of compact analytical form. In the length gauge, one has |yp(Vv) (t) = |p − eA(t)e−iSp (t) , (12) exp i[(p − eA(t)) · r] with |p − eA(t) a plane-wave state [r|p − eA(t) = (2p ) and t 1 Sp (t) = dt [p − eA(t)]2 . (13) 2m The lower limit of the integral is immaterial. It introduces a phase that does not contribute to any observable. The time-evolution operator U (t, t ) satisfies integral equations (Dyson equations) that are convenient if one wants to generate perturbation expansions with respect to either the interaction HI (t) = −er · E(t) with the laser field, t (14) U (t, t ) = Ua (t, t ) − i dt U (t, t) HI (t) Ua (t, t ), −3/ 2 t or the binding potential V (r), t U (t, t ) = Uf (t, t ) − i dt Uf (t, t) V U (t, t ). (15) t Equation (14) also holds if U and Ua in the second term on the right-hand side are interchanged. The equivalent is true of Eq. (15). With the help of the integral II] ATI: CLASSICAL TO QUANTUM 45 equation (14), using the orthogonality of the eigenstates of Ha , we rewrite Eq. (8) in the form t dt yp (t) |U (t, t)HI (t)| y0 (t) , Mp = −i lim t→∞ (16) −∞ which is still exact. A crucial simplification occurs if we now introduce the strong-field approximation. That is, we make the substitutions |yp → |yp(Vv) and U → Uf , with the result ∞ Mp = −i dt0 yp(Vv) (t0 ) |HI (t0 )| y0 (t0 ) . (17) −∞ The physical content of this substitution is that, after the electron has been promoted 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 above definition 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 see Reiss (1992). In the amplitude (17), one may write − er · E(t0 ) = Hf (t0 ) − Ha + V (r) = −i ←−− −−→ ð ð −i + V (r). ðt0 ðt0 (18) Via integration by parts, the amplitude (17) can then be rewritten as ∞ Mp = −i dt0 yp(Vv) (t0 ) |V (r)| y0 (t0 ) . (19) −∞ This form is particularly useful for a short-range or zero-range potential, since these 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 Keldysh parameter g), the saddle-point method (method of steepest descent) can be invoked (Dykhne, 1960). This consists in expanding the phase of the integrand about the points where the phase is stationary. Given the form of the Volkov wave functions (12) and the time dependence of the ground-state wave function, |y0 (t) = exp(iEIP t)|y0 , this amounts to determining the solutions of d [EIP t + Sp (t)] = EIP + 12 [p − eA(t)]2 = 0. dt (20) 46 W. Becker et al. [II Let us consider a periodic (not necessarily monochromatic) vector potential with period T = 2p / w. In terms of the solutions ts of Eq. (20), the amplitude (19) can then be written as p2 Mp ∝ d + EIP + UP − nw 2m n 1/ 2 (21) 2p i × ei[EIP ts + Sp (ts )] p − eA(ts )|V |y0 , (t ) S s p s where Sp 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 of the 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 retrieve the 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 consists of the discrete energies Ep ≡ p2 = nw − UP − EIP 2m (22) can be attributed to interference of the contributions from different periods. This interference is destructive, unless the energy corresponds to one of the discrete peaks (22). Depending on the shape of the vector potential A(t) and its symmetries, there will be several solutions ts (for a sinusoidal field, there are two in the upper half plane and two in the lower, which are complex conjugate to the former) within one period of the field. Their interference creates a beat pattern in the calculated spectrum. This is, however, difficult to observe due to its sensitive dependence on the laser intensity, which is not very well controlled in 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 is better defined than the intensity, these interferences have been observed (Paulus et 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 pulse of finite extent, the saddle points are still determined by Eq. (20). They have, however, no longer any periodicity. Hence, the discreteness of the spectrum is lost. Interference from different parts of the pulse may lead to unexpected effects (Raczyński and Zaremba, 1997). II] ATI: CLASSICAL TO QUANTUM 47 While for an infinitely long monochromatic pulse the spectrum is symmetric upon p → −p, this forward–backward symmetry no longer holds for a finite pulse. Analysis of the spatial asymmetry of the spectrum may aid in determination 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 = px̂ with |p| eA, there are two possible ionization times wt01 = p / 2 + d and wt02 = 3p / 2 − d. The corresponding classical orbits are illustrated in Fig. 4. As discussed above, while A(t01 ) = A(t02 ), the field satisfies E(t01 ) = −E(t02 ). Hence, electrons ionized at t01 and t02 depart in opposite directions right after the instant of ionization. As illustrated in Fig. 4, the electric field changes sign soon after t01 . Hence, the electron ionized at this time turns around at a later time and acquires the same drift momentum as the electron ionized at time t02 , which keeps its original direction. We expect quantum-mechanical interference of the contribution of these 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 of the quantum-mechanical amplitude always has more than one solution. For the √ field (6) and p = 2mE x̂, the solutions are cos wts = z 1 + x2 √ − û ± ûx 2 − ûIP (1 − x 2 ) − x 2 z , (23) where z = (1 − x 2 )/ (1 + x 2 ), û = E/ 2UP , and ûIP = EIP / 2UP . Obviously, the solutions 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 solid line is the effective potential V (x) − exE(t) at times t01 (left) and t02 (right). The electron ionized at t01 is turned around by the field shortly after ionization. In contrast, the electron ionized at t02 maintains its original direction. This is a strongly simplified picture of the physics underlying the interferences of direct electrons. 48 W. Becker et al. [II Fig. 5. (a) Positions of the saddle points wts in the upper half of the complex wt plane in the interval 1 p Re wt 3 p , calculated from Eq. (23) for E = 17w − E − U , where E IP p IP = 15.76 eV, 2 2 UP = 3.68 eV, and àw = 1.96 eV. The arrows indicate the motion of the saddle points for increasing ellipticity x. The two branches meet at x0 = 0.755. For several values of the ellipticity, insets depict the ellipse traced out by the electric-field vector, and the positions of the latter at the emission times Re ts are marked by solid dots. (b) The function Re F, which determines the magnitude of the amplitude 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 (Im F + y), whose oscillations are caused by constructive and 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 square root on the right-hand side of Eq. (23) is imaginary, and the solutions are symmetric with respect to Re wts = p . For |x| x0 , this square root is real, and all solutions have Re wts = p . This has important consequences for the saddle-point amplitude (21). In the first case, both solutions contribute to the amplitude and an interference pattern results. This corresponds to the case of linear polarization discussed above. In the second case, inspection of the integration contour in the complex plane shows that it has to be routed only through 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 the form Mp ~ exp(Re F) cos(Im F + 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 ellipticity distribution, which moves to smaller ellipticity for increasing energy. The same tendency can be observed in the numerical evaluation of the amplitude (21); see Paulus 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 II] ATI: CLASSICAL TO QUANTUM 49 Fig. 6. ATI spectra in xenon for an intensity of 1.2×1014 W/cm2 for various energies in the direction of the large component of the elliptically polarized field as a function of the ellipticity. The inset shows the energy spectrum for linear polarization. The three traces for the lower energies display the interference phenomenon of the direct electrons discussed in Sect. II.C; the one for the highest energy 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 (Bashkansky et al., 1988). Mending this deficiency requires improved treatment of the binding potential (Krstić and Mittleman, 1991). More discussion of this point has been provided elsewhere by Becker et al., 1998, who also give further references. The spatial dependence introduced by Coulomb–Volkov solutions in place of the usual Volkov solutions (12) already suffices to destroy the fourfold symmetry, and angular distributions have been calculated with their help by Jaroń et al. (1999). However, even for a zero-range potential the fourfold symmetry is broken provided the effects of the finite binding energy are treated beyond the KFR 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, once detached, has the choice of starting its subsequent travel either against or with the direction of the electric field, by close analogy with the opposite directions of 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 photodetachment microscope 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 magnetic field 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 of ionization 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 into account. To this end, we integrate the electron’s velocity (2) to obtain its t trajectory e x(t) = (t − t0 ) A(t0 ) − dt A(t) . (24) m t0 The condition that the electron return to the ion at some time t1 > t0 is x(t1 ) = 0. For linear polarization in the x-direction, this implies x(t1 ) = 0, and y(t) ≡ z(t) ≡ 0. This yields t1 as a function of t0 . We defer discussion of elliptical polarization 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 the ionization energy in the form of one photon. This process is responsible for the plateau of high-order harmonic generation. (2) The electron may scatter inelastically off the ion. In particular, it may dislodge a second electron (or more) from the ionic ground state. This process is now believed to constitute the dominant contribution to nonsequential double ionization. (3) The electron may scatter elastically. In this process, it can acquire drift energies 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 harmonic generation. From Eq. (2), the kinetic energy of the electron at the time of its return is Eret = e2 [A(t1 ) − A(t0 )]2 . 2m (25) Maximizing this energy with respect to t0 under the condition that x(t1 ) = 0 yields Eret, max = 3.17UP for wt0 = 108◦ and wt1 = 342◦ (Corkum, 1993; Kulander et 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): The return condition x(t1 ) = 0 can be written in the form F(t1 ) = F(t0 ) + (t1 − t0 ) F (t0 ), where the function F(t) = dt A(t) ~ sin wt (solid curve) is an integral of the vector potential A(t) ~ cos wt (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 straight lines both yield the same kinetic energy Eret < Eret,max . The figure shows that one starts earlier and returns later while the other one starts later and returns earlier. Obviously, there can be many more intersections 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◦ , so that 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 is again given by Eq. (2), but with px = e[2A(t1 ) − A(t0 )] so that Ebs = e2 [2A(t1 ) − A(t0 )]2 . 2m (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 very close to those that afford the maximal return energy. It is important to keep in mind that for maximal return energy or backscattering energy, the electron has to start its orbit shortly after a maximum of the electric field strength. As a consequence, it returns or rescatters near a zero of the field, see Fig. 7. This also provides an intuitive explanation of the energy 52 W. Becker et al. [III Fig. 8. Maximum drift energy after rescattering (ATI plateau cutoff) upon the mth return to the ion core during the ionization process. Electrons with the shortest orbits (m = 1) can acquire the highest energy, 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 and backscatters by 180◦ , then it will be accelerated by another half-cycle of the field. 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 returns to the ion many times. These solutions having long “travel times” t1 − t0 are very important for the intensity-dependent quantum-mechanical enhancements of the ATI plateau to be discussed in Sect. III.B.7. Here we will be satisfied with mentioning another property of the classical orbits: obviously, the return energy will have extrema, e.g. the maximum of Ebs, max = 10.007UP mentioned above, 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 times that will lead to this energy: one earlier than t0, max , the other one later. From the graphical construction of Fig. 7 it is easy to see that the former has a longer travel time than the latter. In the closely related case of HHG, these correspond to the “long” and the “short” orbit (Lewenstein et al., 1995b). The cutoffs of the solutions 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 energy since part of the maximal energy 3.17UP of the returning electron will go into the III] ATI: CLASSICAL TO QUANTUM 53 transverse motion. This implies that, for fixed energy Ebs , there is a cutoff in the angular distribution; in other words, rescattering events will only be recorded for angles 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 following equations (Paulus et al., 1994a): (27) Ebs = 12 A(t0 )2 + 2A(t1 ) [A(t1 ) − A(t0 )] (1 ± cos q0 ) , A(t1 ) cot q = cot q0 − . (28) sin q0 |A(t0 ) − A(t1 )| Here q0 is the scattering angle at the instant of rescattering, which may have any value between 0 and p , as opposed to the observed scattering angle q at the detector (outside the field). In Eq. (27), the upper (lower) sign holds for A(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 the energy 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 of the Schrödinger equation (Paulus et al., 1995) and experimentally for He at intensities around 1015 W/cm2 . Indeed, the latter spectra show an extended plateau for energies between 2UP and 10UP (Walker et al., 1996; Sheehy et al., 1998). For comparatively low intensities, angular distributions have been recorded in xenon with very high precision by Nandor et al. (1998). They also show the effects just discussed, but with much additional structure that appears to be attributable to quantum-mechanical interference and to multiphoton resonance with ponderomotively upshifted Rydberg states (Freeman resonances; Freeman et al., 1987). B. Quantum-mechanical Description In order to incorporate the possibility of rescattering into the quantummechanical description, we have to allow the freed electron once again to interact with 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 we did in Sect. II.B, we replace the exact scattering state |yp by a Volkov state and the 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. This procedure corresponds to adopting the Born approximation for the rescattering process. 54 W. Becker et al. [III 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 that explained in Eq. (18) the two terms can be combined into one, ∞ Mp = −i t1 dt1 −∞ dt0 yp(Vv) (t1 ) |VUf (t1 , t0 )V | y0 (t0 ) , (29) −∞ which now describes both the direct and the rescattered electrons. The physical content of the amplitude (29) corresponds to the recollision scenario: The electron is promoted into the continuum at some time t0 ; it propagates in the continuum subject to the laser field until at the later time t1 it returns to within the range of the binding potential, whereupon it scatters into its final Volkov state. Exact numerical evaluation of the amplitude (29) for a finite-range binding potential is very cumbersome. For a zero-range potential, however, the spatial integrations in the matrix element become trivial, and the computation is rather straightforward. If the field dependence of the Volkov wave function and the Volkov time-evolution operator is expanded in terms of Bessel functions, one of the temporal integrations in the amplitude (29) can be carried out analytically and yields the same d function as in Eq. (21), specifying the peak energies. The remaining quadrature with respect to the travel time t1 − t0 has to be carried out numerically; see Lohr et al. (1997) and Milošević 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 may be done first, and the integral over the return time t1 is then evaluated by Fourier transformation (Milošević 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 interactions with the binding potential were already discussed by Reiss (1980). The first explicit calculations of angular-resolved energy spectra were carried out by Becker et al. (1994a, 1995) and by Bao et al. (1996). Closely related rescattering models were presented by Smirnov and Krainov (1998) and by Goreslavskii and Popruzhenko (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 electron is 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 by Weingartshofer et al. (1977, 1983). Some quantum features of electron scattering in intense laser fields are remarkably similar to HATI; see Kull et al. (2000) and Görlinger et al. (2000). III] ATI: CLASSICAL TO QUANTUM 55 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 direct amplitude (19). This procedure provides much more physical insight than Bessel-function expansions, and establishes the connection with Feynman’s path integral, 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 ) = d 3 k yk(Vv) (t1 ) yk(Vv) (t0 ) , (30) so that the amplitude Mp is represented by the five-dimensional integral t1 ∞ Mp ~ dt1 dt0 d 3 k 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) = − 1 2m 1 − 2m ∞ dt [p − eA(t)]2 t1 t1 dt [k − eA(t)] + t0 (33) t0 2 dt EIP −∞ in the exponent consists of three parts, according to the three stages discussed above. 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 assume that the function mp (t1 , t0 , k) depends only weakly on its arguments. Indeed, for a zero-range potential, it is a constant. We now have to determine the stationary points with respect to the five variables t1 , t0 and k. They are given by the solutions of the three conditions (Lewenstein et al., 1995a) [k − eA(t0 )]2 = −2mEIP , t0 (t1 − t0 ) k = dt eA(t), (34) (35) t1 [k − eA(t1 )]2 = [p − eA(t1 )]2 , (36) respectively. The first condition (34) attempts to enforce energy conservation at the time of tunneling. The second condition (35) ensures that the electron 56 W. Becker et al. [III 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 equations have several solutions (t1s , t0s , ks ), (s = 1, 2, . . .), of which only those are relevant for which Re t1s > Re t0s , such that the recollision is later than ionization. The matrix element can be written as Mp ~ s (2p ià)5 det(ð 2 Sp /ðqj(s) ðqk(s) )j, k = 1, ..., 5 1/ 2 eiSp (t1s ,t0s ,ks ) mp (t1s , t0s , ks ), (37) where qi(s) (i = 1, . . . , 5) runs over the five variables t1s , t0s and ks . As we noted already for the direct electrons in the context of Eq. (21), the sum has to 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 points within 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 we apply the saddle-point approximation to the probability amplitude for given final momentum p, and not to the complete wave function of the final state. This is the reason why only few solutions contribute, while a semiclassical computation of the wave function, which contains all possible outcomes, requires consideration of 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, for a linearly polarized field, the first saddle-point equation (34) implies that the electron starts on its orbit with a speed of zero. Provided the final momentum p is classically accessible, the resulting solutions are entirely real. They correspond to 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 Keldysh parameter g 2 = EIP / (2UP ) is small compared with unity, the imaginary parts of the solutions of Eqs. (34)–(36) are still not too large, and the real parts are still close to these simple-man solutions. In this case, approximate analytical solutions to the saddle-point equations can be written down, which yield an analytical approximation to the amplitude (31) (Goreslavskii and Popruzhenko, 2000). On the other hand, for elliptical polarization, the solutions are always complex, even when 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 will never return to the point where it was released. Equation (34) then only implies that k − eA(t0 ) is a complex null vector. III] ATI: CLASSICAL TO QUANTUM 57 With the solutions (t1s , t0s , ks ) (s = 1, 2, . . .) of Eqs. (34)–(36), the sth quantum orbit has the form t (t − t0s )ks − t0s dt eA(t) (Re t0s t Re t1s ), (38) mx(t) = t (t − t1s )p − t1s dt eA(t) (t Re t1s ). We regard the orbit as a function of the real time t. The conditions x(t0 ) = 0 and x(t1 ) = 0, however, are satisfied for the complex times t0 and t1 . As a consequence, the quantum orbit (38) as a function of real time does not depart from the origin but, rather, from the “exit of the tunnel.” This is clearly visible in Figs. 14, 15, 17 and 20 below. In contrast to the start time t0 , the return time t1 is real to a good approximation, see Fig. 10 (below). In consequence, the orbits return almost exactly to the origin. B.2. Connection with Feynman’s path integral Any quantum-mechanical transition amplitude, such as the ionization amplitude (8), can also be represented in terms of Feynman’s path integral. To this end, we recall the path-integral representation of the complete time-evolution operator of the system atom + field, U (rt, r t ) = D[r(t)]eiS(t,t ) , (39) (rt) ↔ (r t ) t where S(t, t ) = t 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 all classical 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, we have reduced the exact transition amplitude to the form (31). In implementing the strong-field approximation, we have approximated the exact action of the system appropriately at the various stages of the process: before the initial ionization, in between ionization and rescattering, and after rescattering, as in the decomposition (33) of the action. This still left us with a five-dimensional variety of paths. Out of those, finally, the saddle-point approximation (37) selects the handful of “relevant paths” (Antoine et al., 1997; Kopold et al., 2000a; Salières et al., 2001). These are essentially the orbits of the classical model, yet quantum mechanics is fully present: Their coherent superposition as expressed in 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. 58 W. Becker et al. [III B.3. Connection with closed-orbit theory There appears to be a close similarity to the concepts of periodic-closed-orbit theory, see, e.g., Du and Delos (1988), Gutzwiller (1990), and Delande and Buchleitner (1994). The photoabsorption cross section s (E) of an atom in the state |yi with energy Ei can be expressed in the form (Du and Delos, 1998) ∞ e2 iEt s (E) = 4p Re dt e yi |DU (t, 0)D| yi , (40) àc 0 where D = r · û is the dipole operator responsible for photoabsorption of the field with polarization û and E ≈ Ei + àw. The quantity U (t, 0) is the timeevolution operator in the presence of the binding potential as well as additional static external electric and magnetic fields that may be present. In effect, the timeevolution operator propagates wave packets at constant energy that emanate from the atom and are reflected by the caustics of the potential back to the atom where they interfere with each other and with the starting wave packets. This leads to oscillations in the photoabsorption spectrum. In a semiclassical approximation, the time-evolution operator can be expanded in terms of classical closed orbits that start from and return to the vicinity of the atom, defined by the spatial range of the wave function |yi . Since the classical problem is chaotic, there are more and more such orbits when the energy nears zero. Fourier transformation of the photoabsorption 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 as a coherent source of electrons and to causing rescattering, while in closed-orbit theory the interplay of its spatial shape with the external static fields generates the rich structure of the closed orbits. In our case, closed orbits are entirely due to the time dependence of the laser field. The most important difference is that closed-orbit theory is concerned with total photoabsorption rates as a function of frequency, while we consider differential electron spectra for a laser field with fixed frequency. In other words, our orbits depend on the final state of the electron. From Eq. (19), in view of the completeness of the Volkov states, the total ionization probability due to direct electrons is d 3 p |Mp |2 ∞ t1 2 (Vv) = 2e Re dt1 dt0 y0 (t1 ) r · E(t1 )U (t1 , t0 )r · E(t0 ) y0 (t0 ) . −∞ −∞ (41) This differs from the photoabsorption cross section (40) only by the presence of the Volkov time-evolution operator U (Vv) (t1 , t0 ), which reflects the strongfield approximation, instead of the exact time-evolution operator U (t, 0) of the III] ATI: CLASSICAL TO QUANTUM 59 time-independent problem [for which the time-evolution operator U (t, t ) only depends on the time difference t − t ]. In the total ionization probability (41), via the same partial integration (18) as above, the electron–field interaction r · E(t) can be replaced by V (r). The result then looks like the differential HATI amplitude (29) except that it is sandwiched by the ground state. This correspondence 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 an arbitrary 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 its orbit is outside the range of the binding potential. This is trivially guaranteed for the three-dimensional binding potential of zero range, V (r) = ð 2p d(r) r. mú ðr (42) This potential supports a single (s-wave) bound state at the energy −ú 2 / 2m and a continuum that is undistorted from the free continuum except for the s wave, as required by completeness (Demkov and Ostrovskii, 1989). Without the regularization operator (ð/ðr) r, which acts on the subsequent state, the potential does not admit any bound state. There are several possibilities to adjust the one parameter ú to an individual atom or ion. In most cases one will determine it so 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 its saddle-point approximation (37), hold for a much wider class of potentials. Regardless of the potential, the saddle-point equations (34)−(36) have the electron start from and return to the center of the binding potential, which is the origin, and do not depend on its shape. The potential only enters via the form factors in Eq. (32). For the SFA to be applicable, they must depend on time only weakly. The procedure corresponds to the Born approximation. It will be the better justified, the shorter the range of the potential is, so that the form factor depends only weakly on the momenta. Excited bound states do not enter the 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 threedimensional TDSE for hydrogen, see Cormier and Lambropoulos (1997) for the latter and Kopold and Becker (1999) for the former. There is good qualitative agreement within the ATI plateau; in particular, the positions of the dips in the spectrum that are due to destructive interference agree within a few percent. The comparison confirms that the detailed shape of the potential has only 60 W. Becker et al. [III minor significance for the HATI spectrum. Clearly, however, the real physical systems best described by a zero-range potential are negative ions with a s-wave ground state. B.5. A homogeneous integral equation An alternative route to the standard KFR matrix element (19) and its improved version (29) starts from the homogeneous integral equation t |Y(t) = −i dt Uf (t, t) V |Y(t), (43) −∞ which holds for the state that develops out of the unperturbed ground state due to its interaction with the laser field. This integral equation can be derived immediately from the Dyson equation (15) if one applies both sides of the latter to the atomic ground state |y0 (t ) in the limit where t → −∞. By inspection, one may convince oneself that the term Uf (t, t )|y0 t ) makes no contribution for t − t → ∞ so that Eq. (43) is left. This equation was first introduced in the context of the quasi-energy formalism by Berson (1975) and by Manakov and Rapoport (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 unperturbed atomic ground state |y0 (t), one can read off the matrix element (19) for direct ionization. Iterating Eq. (43) one gets t t |Y(t) = − dt 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 potential (42), since in this case it allows one to calculate the wave function in all space provided it is known at the origin. For the latter, to a first approximation, one may employ the unperturbed wave function. Better approximations are obtained by using more accurate expressions. These incorporate the possibility that the ionized 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 obeys Y(r, t) ∝ (1/r − ú) exp(−iEt) for all times. The complex quasi-energy E has to 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 (Manakov and Fainshtein, 1980; Manakov et al., 2000). The interaction with a laser field for a finite period of time was considered along similar lines by Faisal et al. III] ATI: CLASSICAL TO QUANTUM 61 (1990) and Filipowicz et al. (1991); see also Gottlieb et al. (1991) and Robustelli et al. (1997). The integral equation (43) was also used for two-center potentials in order to model negative molecular ions (Krstić 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 typical of a high laser intensity, cf. the data of Walker et al. (1996). The solid circles that make up the topmost curve of the upper panel were calculated from the amplitude (29) by means of a zero-range potential, while the other curves give the results of including an increasing number of quantum orbits in the saddle-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 contribution of the second pair comes close, in particular near its cutoff around 7UP . The contribution 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 exact calculation. 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 the shortest travel times. These parameters uniquely specify the quantum orbits in space and time. Their behavior is very different for energies below and above the classical cutoff at 10UP . Below the cutoff, the imaginary parts of the parameters are only weakly dependent on the energy. Both orbits have to be included in the sum (37), and their interference leads to the beat pattern, which is visible in the spectrum of Fig. 9. Notice that the imaginary parts of both the return times t1s and the momenta kxs are small. In contrast, the imaginary part of the travel 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 has forgotten 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 the artifact of the small spikes visible in Fig. 9. For energies above the cutoff, just one orbit contributes and, as a consequence, the spectrum smoothly decreases without any trace of interferences. The real part of the parameters stays approximately constant, while the imaginary part increases strongly with increasing energy. This is 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, 62 W. Becker et al. [III Fig. 9. Upper panel: ATI spectrum in the direction of the laser field for linear polarization for 1015 W/cm2 , àw = 0.0584 a.u., and a binding energy of EIP = 0.9 a.u. The electron energy is given in multiples of UP . The curve at the top (solid circles) is the exact result from Eq. (29). The other curves were calculated from the saddle-point approximation (37). From bottom to top, more and more quantum orbits are taken into account; the results are displaced with respect to each other for visual convenience. The curve at the bottom incorporates just the pair of orbits with the shortest travel times, the next one up includes in addition the pair with the next-to-shortest travel times, and so on. The occasional small spikes are artifacts of the saddle-point approximation, cf. Goreslavskii and Popruzhenko (1999) and Kopold et al. (2000a). Lower panel: The envelopes of the contributions of the individual pairs are shown all on the same scale so that the quantitative relevance of the various pairs is put in perspective. The cutoffs of the various orbits agree with those displayed in Fig. 8. From Kopold et al. (2000a). III] ATI: CLASSICAL TO QUANTUM 63 Fig. 10. Saddle points (ts , ts , 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 ts the start time (elsewhere denoted by t0s ). The figure shows a comparison of elliptical polarization (x = 0.5, solid circles) and linear polarization (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 thescaling of k given 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 and Popruzhenko (2000). A uniform approximation was described in a different context by Schomerus and Sieber (1997). It reproduces the spectra of Fig. 9 without the spikes (Schomerus and Faria, 2002). B.7. Enhancements in ATI spectra In several experiments, pronounced enhancements of groups of ATI peaks in the plateau region (by up to an order of magnitude) have been observed upon a 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 resonant process. Near the resonances, the contrast of the spectra is remarkably reduced (Cormier et al., 2001). For the experiments reported so far, the effect is most pronounced for argon. This holds not only for a laser wavelength of 800 nm but also for 630 nm (Paulus et al., 1994c). The enhancements are so strong that in experiments implying significant focal averaging the observed spectral 64 W. Becker et al. [III Fig. 11. ATI spectra in argon at 800 nm recorded in the direction of the linearly polarized field for various intensities rising by increments of 0.1 I0 from 0.5 I0 (bottom curve) to 1.0 I0 (top curve). The horizontal lines mark the maxima of the ATI plateaus for each intensity. For intensities I > 0.8 I0 a 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 actual peak intensity. In this sense, ATI in toto has been called a resonant process (Muller, 1999b). A big step towards understanding the physical origin of the enhancements was made in theoretical studies that reproduced the enhancements in the single-active-electron approximation by numerical solution of the oneparticle time-dependent Schrödinger equation in three dimensions (Muller and Kooiman, 1998; Muller, 1999a,b; Nandor et al., 1999), thereby ruling out any mechanism 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 intensity interval of 0.3 to 1.0 × I0 in steps of 0.1 × I0 are displayed. The maximum intensity I0 was calibrated by using the cutoff energy of 10UP . This leads to I0 ≈ 8×1013 W/cm2 . There is a striking difference between the spectra for I 0.8I0 and those for higher intensity: within a small intensity interval a group of ATI peaks corresponding to energies between about 15 eV and 25 eV grows very quickly. In the figure this is emphasized by horizontal lines drawn at the maximal heights of the plateaus. Increasing the intensity above 0.9I0 leads to a smaller 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 is made up of contributions from all intensities I I0 that are contained within the spatio-temporal pulse profile. This means that a spectrum for a fixed intensity would show the enhanced group of ATI peaks only at that intensity where it first III] ATI: CLASSICAL TO QUANTUM 65 Fig. 12. Comparison of the intensity dependence of ATI electrons with different energies. For visual convenience, the overall increase in yield with increasing intensity has been subtracted. The electrons at 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 of the subtraction of the overall increase, the resonance-like behavior corresponds to those intensities where the respective curves start rising. It is evident that for the plateau electrons this does not happen at those intensities where the atomic states shift into resonance. Quite to the contrary, the intensity 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 other words, the enhancement happens at a well-defined intensity or at least within a very 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 with ponderomotively upshifted Rydberg states. In some cases, in particular for electrons with rather low energy, one particular Rydberg state could be definitely identified as responsible. In others, notably for the strong enhancement that for appropriate 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 fs as 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 enhanced ATI peaks in the plateau and of the low-energy ATI peaks, see Fig. 12. The latter 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 66 W. Becker et al. [III matrix element (29), which does not incorporate any excited states, produces much the same enhancements (Paulus et al., 2001a; Kopold et al., 2001). An example is shown in Fig. 13. In these calculations, the enhancements occur for intensities 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 the minimum number of photons required for ionization in place of k. Such channel closings 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 in the low-energy ATI spectrum (Faisal and Scanzano, 1992). Comparison of the channel-closing condition (45) with the ATI energy spectrum (22) shows that at a channel closing electrons may be produced with zero drift momentum p. In this event, the energy of the k photons is entirely used to overcome the binding potential raised by the ponderomotive energy, and no energy is left for a drift motion. An electron having a drift momentum near zero has many recurring opportunities to rescatter. Indeed, the quantum-orbit analysis of the spectra of Fig. 13 shows that at the channel closings, and only there, an exceptionally large number of orbits are required to reproduce the exact result. All of these orbits conspire to interfere constructively to produce the observed enhancements. In the tunneling 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 about the ion for many cycles of the field before the final act of rescattering. A detailed comparison between Muller’s numerical simulations and results based on Eq. (29) has been made by Kopold et al. (2001). This paper also includes an assessment of the consequences of focal averaging. It is noteworthy that both approaches predict ATI enhancements also for helium deeply in the tunneling regime, in spite of the obvious multiphoton character of the channel-closing condition (45). Unfortunately, the helium data of Walker et al. (1996) and Sheehy et al. (1998) do not allow one to draw conclusions about the presence or absence of enhancements. The interference interpretation just given requires the existence of a sufficient number of orbits to contribute to the energy considered. The lower panel of Fig. 9 shows that too few orbits contribute for energies above about 8UP . Indeed, the enhancements observed experimentally are restricted to the lower two-thirds of the plateau. The interpretation also implies that the enhancements should disappear for ultrashort pulses, where late returns do not occur. This has been observed in experiments by Paulus et al. (2002). In numerical simulations of HHG based on the three-dimensional TDSE, the same effect has noticed by de 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) III] ATI: CLASSICAL TO QUANTUM 67 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 (solid symbols) 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 actual onset of the continuum is hard to see and may better be replaced by an effective 68 W. Becker et al. [III value. This is illustrated, for example, by the photoabsorption spectra of Garton and 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 dependence of the enhancements on the shape of the potential and the presence or absence of excited bound states has been investigated (Faria et al., 2002). The results are largely 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 and Ivanov, 2001a): orbits that would miss the ion in the absence of the binding potential are refocused to the ion in its presence. This emphasizes the importance of 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 laser pulse, the rate of NSDI should decrease. Indeed, this has been experimentally confirmed 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 describe rescattering for an elliptically polarized laser field meets with difficulties. The problem is that an electron that starts with zero velocity almost never returns exactly 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-projection of Eq. (35). Real solutions are obtained, provided the final momentum p is classically accessible. In contrast, for elliptical polarization, three equations are left since now both the x-projection and the y-projection of Eq. (35) have to be considered. Hence, there is no simple-man model for elliptical polarization, even when EIP = 0. The same situation occurs for HHG. This does not mean that there is no HATI or HHG for elliptical polarization: quantum-mechanical wave-function spreading assures overlapping of the wave packet of the returning electron with the ion (Dietrich et al., 1994; Gottlieb et al., 1996). The complete absence of HHG for a circularly polarized laser field is sometimes taken as confirmation of the rescattering mechanism. This conclusion is not rigorous since there is still sufficient overlapping. Rather, the absence of HHG is due to angularmomentum 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 the ion or by admitting a nonzero initial velocity, but in doing so a large amount of arbitrariness is unavoidable. Instead, we will just solve the saddle-point equations (34)−(36) and accept and interpret the complex solutions. III] ATI: CLASSICAL TO QUANTUM 69 Fig. 14. ATI spectrum in the direction of the large component of the elliptically polarized driving laser field (5) for x = 0.5 (see the field ellipse in the upper right corner of the figure) and electron energies between 2.5 and 10.5UP . The other parameters are w = 1.59 eV, EIP = 24.5 eV, and I = 5×1014 W/cm2 . The open circles give the yields of the individual ATI peaks calculated from the integral (29) for the zero-range potential. The other curves represent the contributions to the quantum-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 the spectrum, viz. 1 and 2, 3 and 4, and 5 and 6, are presented near the margins of the figures. The position of the ion is marked by a cross; notice that the orbits do not depart from there, but rather from a point several atomic units away from it. This is the point where the electron tunnels into the continuum. 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 to be the rescattering plateau for linear polarization has turned into a staircase for elliptical polarization. Each step can be attributed to one particular pair of orbits, and for each step the real parts of such orbits are displayed in the figure. The orbits are closely related to their analogs in the case of linear polarization, exhibited in Fig. 9. In particular, their cutoffs oscillate with increasing travel time as illustrated in Figs. 8 and 9. The main difference is that for elliptical polarization the orbits are two-dimensional and encircle the ion. The pair of orbits with the shortest travel times generates the part of the spectrum preceding the final (highest-energy) cutoff. However, this part is very weak in 70 W. Becker et al. [III relation to the yields at lower energies. The latter are generated by orbits with longer travel times, whose contributions for linear polarization are marginal, see Fig. 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. This velocity is largely in the direction of the small component of the elliptically polarized field. The larger this velocity is, the smaller is the contribution that the associated orbit makes to the spectrum. [This can be compared with the distribution of transverse momenta in a Gaussian wave packet (Dietrich et al., 1994; Gottlieb et al., 1996).] For the shortest orbit, while the large component of the field changes sign so that the electron is driven back to the core in this direction, the small component has the same sign for the entire duration of the orbit. Hence, a particularly large initial velocity in this direction is required in order to compensate the drift induced by the small field component. For the longer orbits, the small component changes direction, too, during the travel time and, consequently, a smaller initial transverse velocity suffices to allow the electron to return to the ion. Support for these qualitative statements can be found 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, the momentum ks has two nonzero components, kxs and kys . Both have substantial imaginary parts, in particular kys . This is a consequence of the lack of a classical simple-man model for elliptical polarization, as discussed above. For the orbits (s = 3, 4) (not shown), the imaginary parts are much smaller, in keeping with 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 one corresponding to energies below 10 eV) is due to direct electrons and does not concern us here. The other ones correspond to the steps of Fig. 14. The real parts of representative orbits, calculated from Eqs. (34)−(36), are shown in the figure. In order to reach a maximum contrast for the steps, the spectrum was recorded 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 either directly or after rescattering so that one expects interference of these two paths. However, Fig. 9 shows that, for linear polarization and high intensity, the transition region where both paths make a contribution of comparable magnitude is very narrow. The situation is more favorable for elliptical polarization: since the plateau turns into a staircase (Fig.14), the yields of the two paths remain comparable over a larger energy region. This has permitted experimental III] ATI: CLASSICAL TO QUANTUM 71 Fig. 15. ATI spectrum in xenon for an elliptically polarized laser field with ellipticity x = 0.36 and intensity 0.77×1014 W/cm2 for emission at an angle with respect to the polarization axis as indicated in the upper right. The spectrum has a staircase-like appearance. The respective steps are shaded differently. For each step, the real parts of the responsible quantum orbits are displayed. The dots with the crosses mark the position of the atom, and the length scale is given in the upper left of the figure. From Salières 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 plateau in the direction of the laser polarization is very noticeable. The side lobes corresponding 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 the direction of the major axis of the field and another weaker one to its right. The lower panel of Fig. 16 exhibits (on the right) EAD’s of a sequence of ATI peaks where the interference is best developed and (on the left) compares them with theoretical calculations from the amplitude (29). The parameters underlying the calculation do not exactly match the experiment. This is mostly attributable to the insufficient description of the direct electrons for elliptical polarization. The theoretical results, however, show the same interference pattern. In order to make sure that this pattern is really due to interference between direct and rescattered electrons, the two contributions have been displayed separately for one of the peaks (s = 17): neither one shows a pronounced dip, only their coherent superposition does. For more details of the theory we refer to Kopold (2001). 72 W. Becker et al. [III Fig. 16. Upper panels: density plots of measurements of the energy-resolved angular distributions for 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 polarization ellipse is at 0º. Dark means high electron yield. Yields can only be compared horizontally, not vertically, 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 the angular distribution for the ATI peaks s = 11, . . . , 21 for elliptical polarization (x = 0.48). The other parameters are EIP = 0.436 a.u. (just below the binding energy of xenon in order to stay away from a channel closing) and I = 5.7×1013 W/cm2 . For the ATI peak s = 17, the contributions of the direct electrons (dashed line) and the rescattered electrons (dotted line) are displayed separately. The slight 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 distribution extracted 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 relativistic velocities |v| ~ c within one cycle. Such intensities are characterized by the ponderomotive energy UP becoming comparable with or exceeding the electron’s rest energy mc2 . We will briefly summarize the kinematics of an otherwise free electron in the presence of such a field. In other words, we will generalize the simple-man model of Sect. II.A.1 to the case of “relativistic intensity.” The changes are surprisingly few. A. Basic Relativistic Kinematics For a four-vector potential Am = (A0 , A), the electron’s four-vector velocity is (Jackson, 1999) mũm = pm − eAm , (46) where ũm = g(c, v) with v = v(t) the ordinary velocity dx/dt, and the usual relativistic factor g = [1 − (v/c)2 ]−1/ 2 (not to be confused with the Keldysh parameter). The four-velocity satisfies ũ2 ≡ ũ · ũ ≡ ũm ũm = c2 so that the four-vector mũ is on the mass shell. Equation (46) is the analog of the nonrelativistic Eq. (2). We will consider a plane-wave field of arbitrary polarization, Am = 2 m ai (k · x)ei (47) i=1 with the four-dimensional wave vector k m = (w/c, k) so that k 2 = 0 and k · ei = 0. The field (47) differs from the field (6) by the fact that the wave fronts are now given by k · x = const. in place of t = const., that is, we do no longer make the dipole approximation. We will assume that the laser field propagates in the z-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 the physical meaning of the drift momentum as in the nonrelativistic case. Since the 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 direction as well as p · k = w( p0 − pz )/c are constants of the motion inside the field (47). If we 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 assume that 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 ũ2 = c2 , using the conditions (49), we obtain the energy as a function of u, g= E e2 (A(u) − A(u0 ))2 . = 1 + mc2 2m2 c2 (50) This yields the cycle-averaged kinetic energy Ekin = E − mc2 = p2T + UP . 2m (51) This is exactly the same decomposition into drift energy and ponderomotive energy as in the nonrelativistic case, Eq. (3). The ponderomotive energy is still defined by Eq. (4), and the classical bounds of the spectrum discussed in Sect. II.A.1 are unchanged. However, velocity and canonical momentum are connected by the relativistic expresion mgv = pT − eA, and the cycle average was performed with respect to u rather than the time t. Since it can be shown that u is proportional to the electron’s proper time, this was, actually, the proper thing 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 a momentum in the direction of its propagation or, alternatively, that the magnetic field via the Lorentz force causes a drift in the propagation direction or, alternatively, that the laser field exerts radiation pressure. All three statements are essentially equivalent. As a consequence, electrons born with zero velocity in a 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, for circular polarization, they are emitted in a cone given by the angle |vT (t)| 2 2m tan q = = (53) = |vz (t)| ∞ |pT | g∞ − 1 with respect to the propagation direction. The subscript ∞ characterizes quantities 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 = 1.053 mm and was used to draw conclusions regarding the actual (nonzero) value of the initial velocity (McKnaught et al., 1997), which can be introduced as discussed in the nonrelativistic case in Sect. II.A.1. There are, however, still some unresolved issues in the interpretation of these experiments (Taı̈eb et al., 2001). The cycle average of Eq. (50) can be written in the covariant form p2 = m2 c2 − (eA)2 ≡ m2∗ c2 > m2 c2 , (54) where p2 and (eA)2 < 0 are invariant four-dimensional scalar products. This relation 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 − m2 c2 Y = 0, (55) which explicitly displays the effective mass. However, one has to keep in mind that this apparently increased mass is just due to the transverse wiggling motion of the electron, viz. the ponderomotive energy, and that there is nothing especially relativistic about it. All the same, envisioning the ponderomotive energy as a mass increase makes sense since, like the rest mass, it is an energy reservoir 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-field approximation (17), taking the relativistic instead of the nonrelativistic Volkov wave function (Reiss, 1990; Faisal and Radożycki, 1993; Crawford and Reiss, 1997). In particular, the stationary-phase approximation is well justified, leading to a form similar to Eq. (21) (Krainov and Shokri, 1995; Popov et al., 1997; Mur et 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 rest mass – is the drift momentum (52) in the direction of propagation of the laser field, which can be traced to the Lorentz force. This has virtually no effect on the initial process of ionization where the electron’s velocity is low, but since it is always positive it prevents the electron from returning to the ion. Therefore, with increasing intensity it gradually eliminates the significance of rescattering processes. This effect can be estimated by calculating the distance by which the 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 ≈ UP l 2mc2 (56) with l the wavelength of the laser field. Obviously, this distance can exceed the width of the wave packet of the returning electron to the point where it does not overlap 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 the plateau. 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 multiplenonsequential-ionization experiments of neon at 2 × 1018 W/cm2 a conspicuous suppression of the highest charge state has been attributed to the magnetic-fieldinduced 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 highorder harmonic generation differ only in the third step: elastic scattering versus recombination. Correspondingly, the description in terms of quantum orbits can be applied to HHG as well; in fact, quantum orbits were introduced for the first time in the analysis of HHG by Lewenstein et al. (1994). It is from the practical point of view that the two processes differ greatly: HHG by one single atom has never been observed, only HHG by an ensemble of atoms. This introduces phase matching as an additional consideration, equal in significance to the single-atom behavior (Salières 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 laser field. A bichromatic elliptically polarized laser field was considered by Milošević et 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 is extended into the relativistic regime. A bichromatic linearly polarized laser field was investigated by Faria et al. (2000), and a simplified version of the quantumorbits formalism was used to deal with problems in the presence of a laser field and an additional static electric field (Milošević and Starace, 1998, 1999c) or a laser field and an additional magnetic field (Milošević 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 û in the HHG process in the context of the strong-field approximation (Lewenstein et al., 1994), Mû (W) ~ ∞ dt1 −∞ t1 dt0 −∞ d 3 k exp [iSW (t1 , t0 , k)] mû (t1 , t0 , k), (57) has the same structure as the corresponding expression (31) for ATI. The function 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 t0 due to interaction with the laser field, and another one at time t1 that corresponds to recombination into the ground state followed by emission of the high-order harmonic photon having the polarization û. The difference to ATI is mostly in the first term of the action: ∞ SW (t1 , t0 , k) = t1 dt (EIP − W) − 1 2m t1 t0 dt [k − eA(t)]2 + t0 dt EIP , (59) −∞ which now refers to the emitted photon. The corresponding saddle-point approximation of Eq. (57) is like the HATI approximation (37), except that the summation 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 time of 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 analysis and interpretation (Salières et al., 1999). Conversely, numerical solutions of the TDSE 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 be spatially resolved in an experiment by Bellini et al. (1998). Spectral resolution was achieved by exploiting the dependence of phase matching on the position of the atomic jet with respect to the laser focus by Lee et al. (2001) and by Salières et al. (2001). 78 W. Becker et al. [V At the end of Sect. II.B we remarked that the quantum-orbit formalism is not restricted to periodic fields, but can equally well be applied to finite pulses. For a periodic field, interference of contributions from different cycles generates a discrete spectrum. For a finite pulse, it enhances or suppresses particular frequency intervals. This was dubbed “intra-atomic phase matching” by Christov et al. (2001) and has been calculated in terms of quantum orbits; in the context of the TDSE, see Watson et al. (1997). This mechanism underlies the engineering of a HHG spectrum by tailoring the pulse shape in a feedbackcontrolled experiment (Bartels et al., 2000, 2001). Individual HHG peaks could be 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 threestep nature of HHG – direct ATI followed by continuum propagation followed by laser-assisted recombination – is particularly emphasized in the approach of Kuchiev and Ostrovsky (1999, 2001), where the integration over the intermediate momentum k is replaced by a discrete summation over ATI channels. The latter is carried out by a variant of the saddle-point approach, which is reminiscent of Regge poles and leads to a complex effective channel number. B. Elliptically Polarized Fields High-order-harmonic generation by an elliptically polarized field is of great interest for applications such as the generation of sub-femtosecond pulses (Corkum et al., 1994). For theoretical calculations in the context of the SFA, see Becker et al. (1994, 1997) and Antoine et al. (1996); for a fairly comprehensive list of references, see Milošević (2000). Fields having polarization other than linear generate particularly appealing quantum orbits since they allow them to unfold in a plane. As an example, Fig. 17 shows a HHG spectrum for the elliptically polarized laser field (5) (Kopold et al., 2000b; Milošević, 2000). The figure confirms that the “exact results” are well approximated by the contributions of only the six shortest orbits. This figure is the analog of Fig. 14 for HATI. The spectrum exhibits the same staircase structure, and everything said there applies here as well. C. HHG by a Two-color Bicircular Field The bichromatic w–2w laser field E(t) = 12 i E1 e+ e−iwt + E2 e− e−2iwt + c.c., (61) whose two components √ are circularly polarized and counter-rotating in the same plane (e± = (x̂ ± iŷ)/ 2), is known to generate high harmonics very efficiently; V] ATI: CLASSICAL TO QUANTUM 79 Fig. 17. High-order harmonic spectrum for an elliptically polarized laser field with the same parameters as in Fig. 14 and harmonic orders between 25 and 77. The open circles are calculated from the integral (57), and the curves labeled 1 through 6 represent the individual contributions to the quantum-orbit approximation of the six shortest quantum orbits, numbered as in Fig. 14. The contributions from quantum orbits 2, 4 and 6 have to be dropped above their intersections with curves 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) for a theoretical description. We will call this field “bicircular.” This high efficiency was surprising because, for a monochromatic field, the harmonic emission rate decreases with increasing ellipticity (cf. the preceding subsection) and a circularly polarized laser field does not produce any harmonics at all. A more detailed analysis, based on the quantum-orbits formalism, gives an explanation of this effect (Milošević et al., 2000, 2001a,b). The harmonics produced this way can be of a practical importance because of their high intensity (Milošević and Sandner, 2000) and temporal characteristics (attosecond pulse trains; Milošević and Becker, 2000). The more general case of an rw−sw (with r and s integers) bicircular field was considered by Milošević 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). 80 W. Becker et al. [V 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 − 12 T t 12 T , 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 and 43w are marked by asterisks and solid circles, respectively. These times and harmonics correspond to the dominant saddle-point solution 2 in Fig. 19. In between the ionization time (asterisks) and the recombination time (solid circles) the x-component of the electric field changes from its negative maximum to its positive maximum, whereas its y-component remains small and does not change sign. From Milošević et al. (2000). Figure 18 presents an example of the harmonic spectrum for the bicircular field (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 comparably smooth. Furthermore, the cutoff is less pronounced and there are small oscillations after the cutoff. These features can be explained in terms of the quantum orbits. Figure 19a shows the first eleven solutions (those having the shortest travel times) of the system of the saddle-point equations (34), (35) and (60), while Fig. 19b shows the individual contributions to the harmonic emission rate of the first eight of these solutions (Milošević et al., 2000). Obviously, in the plateau region the contribution of a single orbit, corresponding to solution 2, is dominant by 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 the monochromatic linearly polarized field (Lewenstein et al., 1995b) where essentially two orbits contribute in the plateau and just one in the cutoff region. Figure 19a tells which solutions are dominant. The probability of harmonic emission V] ATI: CLASSICAL TO QUANTUM 81 Fig. 19. Saddle-point analysis of the results of Fig. 18. (a) The imaginary part of the recombination time t1 as a function of the real part of the travel time t1 − t0 , obtained from the solutions of the saddle-point equations (34), (35) and (60). Each point on the curves corresponds to a specific value of 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 boldface italics. Values of the harmonic order that approximately determine the cutoffs for each particular solution are marked by stars with the corresponding harmonic numbers next to them. Those values of the harmonic order for which | Im t1 | is minimal are identified as well. (b) The partial contributions to the harmonic-emission rate of each of the first eight solutions of the saddle-point equations. From Milošević et al. (2000). 82 W. Becker et al. [V Fig. 20. Real parts of the quantum orbits for the same parameters as in Fig. 18 and for the harmonic W = 43w. Five orbits are shown that correspond to the saddle-point solutions 2, 3, 4, 5 and 8 in Fig. 19. The direction of the electron’s travel is given by the arrows. In each case, the electron is “born” a few atomic units away from the position of the ion (at the origin), where its orbit almost exactly terminates. The dominant contribution to the 43rd harmonic intensity comes from the shortest orbit number 2, whose shape closely resembles the orbit in the case of a linearly polarized monochromatic field. From Milošević et al. (2000). decreases with increasing absolute value of the imaginary part of the recombination time t1 . The possible cutoff of the harmonic spectrum can be defined as the value 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 in Fig. 19a, these points are marked by asterisks and by the corresponding harmonic order. As a consequence of wave-function spreading, the emission rate decreases with increasing travel time t1 − t0 . This gives an additional reason why the contribution 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 solutions 2, 3, 4, 5 and 8 in Fig. 19. The dominant contribution comes from the shortest orbit 2 (thick line). It starts at the point (4.06, 0.66) by setting off in the negative y-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 polarized field. This behavior can be understood by inspection of the driving bicircular field depicted in the inset of Fig. 18, where the start time and the recombination time of the orbit are marked. During the entire length of the orbit, the field exerts a force in the positive y-direction. The effect of this force is canceled by the electron’s initial velocity in the negative y-direction. The force in the x-direction is much like that in the case of a linearly polarized driving field. Since HHG by a V] ATI: CLASSICAL TO QUANTUM 83 linearly polarized field is most efficient, this makes plausible the high efficiency of 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 the 43rd harmonic is smaller. The other orbits are still longer and more complicated so 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 the field (61), see the inset of Fig. 18. If the group of harmonics includes harmonics of either parity, then the field consists of a sequence of essentially linearly polarized, strongly chirped attosecond pulses, each rotated by 120º with respect to the previous one. If, on the other hand, one were able to select harmonics of definite helicity, i. e. either W = (3n + 1) w or W = (3n − 1) w, then one would obtain 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 cycle of the bicircular field (61) on an arbitrary isotropic scale. The position of the origin is indicated in the upper and the left margin. The parameters are I1 = I2 = 9.36×1014 W/cm2 , àw = 1.6 eV, and EIP = 24.6 eV. The plot displays two traces: The circular trace is generated by the ten harmonics W = (3n + 1)w with n = 10, . . . , 19, all having positive helicity. The starlike trace is generated by all harmonics W = (3n ± 1)w between the orders 31 and 59, regardless of their helicity. The curve at the bottom represents the x-component of the field of the latter group over one cycle, the time scale being given on the horizontal axis. It shows that the field is strongly chirped. The black blob at the center is due to the fact that the field is near zero throughout most of the cycle, cf. the trace of the x-component. From Milošević and Becker (2000). 84 W. Becker et al. [V D. HHG in the Relativistic Regime Quantum orbits can also be employed in the relativistic regime starting from the Klein–Gordon equation (55). Milošević et al. (2001c, 2002) found that the relativistic harmonic-emission matrix element has a form similar to that in Eq. (57), but with the relativistic action (à = c = 1) ∞ t1 t0 SW (t1 , t0 , k) = du (EIP − m − W) − du ek (u) + du (EIP − m), (62) t1 t0 −∞ where ek (u) = Ek + eA(u) · k + 2e A(u) Ek − ẑ · k (63) and Ek = (k 2 + m2 )1/ 2 , u = (t − z)/ w. Solving the classical Hamilton–Jacobi equation for Hamilton’s principal function it can be shown that ek (u) is the classical 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 part is responsible for the emission of odd harmonics W = (2n + 1) w, while the other one originates from the intensity-dependent drift momentum of the electron in the field and allows for emission of even harmonics W = 2nw. Similarly to the nonrelativistic case, the integral over the intermediate electron momentum k can be calculated by the saddle-point method. The stationarity t condition t01 du 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 from and returns to the nucleus. As above, the start time and, to a lesser degree, the recombination 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 the photon’s direction of propagation ẑ is given by t0 (t1 − t0 )ks⊥ = du eA(u). (64) Introducing M = e 2 2 t1 t0 t1 2 2 du A (u)/ (t1 − t0 ) − ks⊥ > 0, one has 2 ks2 = ks⊥ + 2 2 ) (M2 − ks⊥ , 2 2 4(m + M ) (65) which yields eks as a function of t0 and t1 . The two stationarity equations connected with the integrals over t0 and t1 are eks (t0 ) = m − EIP , W = eks (t1 ) + EIP − m. (66) (67) As in the nonrelativistic case, they express energy conservation at the time of tunneling t0 and at the time of recombination t1 , respectively. The final expression V] ATI: CLASSICAL TO QUANTUM 85 Fig. 22. Harmonic-emission rate as a function of the harmonic order for ultrahigh-order harmonic generation by an Ar8+ ion (EIP = 422 eV) in the presence of an 800-nm Ti:Sa laser having the intensity 1.5×1018 W/ cm2 . Both the nonrelativistic and the relativistic results are shown. The corresponding relativistic electron orbit with the shortest travel time that is responsible for the emission of the harmonic W = 100 000w is shown in the inset. The arrows indicate which way the electron 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 Milošević 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 saddle points (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-high harmonic orders, this is practically impossible, so that the saddle-point method is the only way to produce reasonable results. Figure 22 presents an example. The nonrelativistic result is obtained from Eq. (37) where the summation is over the solutions of the system of the nonrelativistic saddle-point equations (34), (35) and (60). It is, of course, inapplicable for the high intensity of 1.5×1018 W/cm2 at 800 nm and is only shown to demonstrate the dramatic impact of relativistic kinematics. For the relativistic result, the summation in Eq. (37) is over the relativistic solutions of Eqs. (64)−(67). The relativistic harmonic-emission rate assumes a convex shape, and the difference between the relativistic and nonrelativistic results reaches several hundred orders of magnitude in the upper part of the nonrelativistic plateau. The origin of this dramatic 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 Kulyagin et al. (1996). This is illustrated in the inset of Fig. 22, which shows the real part of the dominant shortest orbit for the harmonic W = 100 000w. In order to counteract this drift so that the electron is able to return to the ion, the electron has to take off with a very substantial initial velocity in the direction opposite to 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 intensity of Fig. 22, it is a real effect for lower laser-field intensities where relativistic effects are still small (Walser et al., 2000; Kylstra et al., 2001; Milošević et al., 2001c, 2002). In this case, the three plateaus visible in the nonrelativistic curve of Fig. 22 are related to the three pairs of orbits, whose contribution to the harmonic emission rate is dominant in the particular spectral region (see Figs. 2 and 3 of Milošević et al., 2001c). These are very similar to the pairs of orbits that we 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 these orbits becomes so dominant that the multiplateau and the interference-related oscillatory structure disappear completely. The reason is that the effect of the v × B drift increases with increasing travel time; see Eqs. (52) and (56) in Sect. 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 field oscillates and, therefore, for a longer orbit a smaller initial velocity may be sufficient to allow the electron to return. VI. Applications of ATI Experimental and theoretical advances in understanding ATI – some of which have been treated in this review – permit its application to the investigation of other effects. One obvious idea is to exploit the nonlinear properties of ATI. This is particularly relevant to characterization of high-order harmonics and measurement of attosecond pulses in the soft-X-ray regime. In this spectral region (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 effect of 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 a cross-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 kinetic energy Eq = qàw − EIP . Simultaneous presence of a fraction of the fundamental laser beam in the near infrared (NIR) produces sidebands, i.e. electrons with energies qàw − EIP ± màw (m q). The strength of the sidebands can be changed by temporally delaying the fundamental with respect to the harmonic by a time t. Optimal overlapping of the pulses (t = 0) leads to a maximum in the strength of the sidebands, whereas complete separation entirely eliminates them. The strength of the sidebands as a function of t can be used to determine the 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 a positive initial energy Ei , which will be identified with Eq . For Up àw, which is well satisfied for the weak field we will consider, the differential ionization rate in the field direction is given by (in atomic units) ∞ ð 2G Jm2 ∝ |p| · ðEðW m = −∞ Ef 2(Ei + mw) d(E − mw − Ei ). w2 (68) m is the order Here, Ef is the amplitude of the electric field of the fundamental, of 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 bands are not, in general, symmetric. However, for sufficiently weak fields, both fields can be treated by lowest-order perturbation theory. It follows that a sideband of 2|m| order m is proportional to Eh2 Ef , where Eh is the field strength of the harmonic radiation. In this case, the cross correlation for a sideband of order m can be calculated as ∞ 2|m| Eh2 (t) · Ef (t − t) dt. (69) Cm (t) = −∞ Figure 23 (overleaf) shows the result of a corresponding calculation, which is compared with results from a numerical solution of the appropriate onedimensional Schrödinger 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-correlation functions. Numerical and experimental investigations of this problem were made by Véniard 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 determine harmonic-pulse durations significantly shorter than that of the fundamental in the NIR spectral region. In 1996 already, Véniard et al. pointed out that 88 W. Becker et al. [VI Fig. 23. Cross-correlation of near-infrared and soft-X-ray pulses. A harmonic of order q creates photoelectrons at the kinetic energy qàw − EIP . Sidebands are created by simultaneous irradiation with the fundamental of frequency w. Plotted are the heights of the sidebands for various side-band orders m versus the delay t between the fundamental and the harmonic. The solid line represents the analytical approximation (69), whereas the points were calculated by numerically solving the appropriate (one-dimensional) Schrödinger 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 the relative phase of neighboring harmonics. This is an extremely important insight because the phase dependence of the harmonics as a function of their order (or frequency) determines whether they are mode-locked and whether the corresponding pulses – which would constitute attosecond pulses in the softX-ray region if bandwidth-limited – are chirped. In fact, Paul et al. (2001) used this 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 generated with appreciable amplitude. This calls for intensities of the NIR beam below 1012 W/cm2 . Along with the fact that the NIR field generates only oddorder harmonics this ensures that only two adjacent harmonics contribute to each sideband. 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 emission of one NIR photon (Eq = (q + 1)àw − àw). Each of these two channels receives contributions from two different quantum paths, which are related to the temporal order of the interaction with the harmonic and the NIR field. (In contrast to the quantum orbits we considered elsewhere in this chapter, the quantum paths here 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 harmonics q = 11, . . . , 19. The attosecond pulses are separated by 1.35 fs, which is half the cycle time of the driving laser. The latter is represented by the dashed cosine function. Reprinted with permission from 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 the amplitudes of all four quantum paths. Due to the fact that two paths represent absorption from the NIR field whereas the other two represent emission into it, the interference term between these two contributions is essentially proportional to cos(÷q−1 − ÷q+1 + 2àwt). By varying the delay t between the harmonic and the NIR radiation, the difference ÷q−1 − ÷q+1 of the phases of the two harmonics can 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 modelocked and make up a train of attosecond pulses of 250 as FWHM duration, see Fig. 24. A.2. Isolated attosecond pulses With respect to applications, isolated attosecond pulses appear more useful than a train of pulses separated by half the period of the fundamental. Isolated attosecond 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-cycle regime). Then, however, the spectral width of the harmonics will be so broad that it is no longer possible to identify individual sidebands as necessary for the method 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 of 1.8 fs and 150 as, respectively. The experimental setup, in principle, resembles that of Paul et al. with the difference that higher intensities of the NIR ra- 90 W. Becker et al. [VI diation are used for the photoionization cross correlation. In addition, only photoelectrons ejected perpendicularly to the laser polarization are detected. The motivation for choosing these conditions can be deduced from a classical analysis of trajectories of electrons that were injected into the electric field of the few-cycle NIR pulse by absorption of a harmonic photon. If the duration of the X-ray pulse is shorter than the optical period T in the NIR, then the final kinetic energy of the photoelectrons depends on the phase wt0 when the injection took place, i. e. it exhibits a modulation with a period of T/ 2. By delaying the 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 distribution also exhibits such a modulation, and is measureable with much higher precision than the center of mass of the distribution. For the two approaches, it is not the envelope of the fundamental that enters the correlation function, but rather the optical period. The restriction to photoelectrons emitted perpendicularly to the laser polarization suppresses the influence of effects related to the emission and absorption of photons from the laser field, i.e. the sidebands which were crucial for 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 areas of the natural sciences is driving the development to shorter and shorter laser pulses. At a FWHM duration shorter than a few optical cycles the time variation of the pulse’s electric field depends on the phase f of the carrier frequency with respect to the center of the envelope, the so-called “absolute phase.” The electric field 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 f can 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, will influence various effects of the laser–atom interaction. This is one reason for the significance of this new parameter of laser pulses. The precise knowledge and control of the absolute phase will pave the way to new regimes in coherent X-ray generation and attosecond generation; for an overview see Krausz (2001). In addition, such extremely well-defined laser pulses are likely to have applications for the coherent control of chemical reactions and other processes. Another reason is that phase control of femtosecond laser pulses has already had a huge impact on frequency metrology. This is because phase-stabilized femtosecond lasers can be viewed as ultra-broadband frequency combs that can be used to measure optical frequencies 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 phasestabilized (Reichert et al., 1999; Apolonski et al., 2000), which is sufficient for frequency metrology. Strong-field effects require amplified laser pulses. Nisoli et al. (1997) demonstrated that it is possible to generate powerful (>500 mJ) 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 due to the absolute phase by performing a shot-to-shot analysis of the number of photoelectrons emitted in opposite directions. To this end, a field-free drift tube is placed symmetrically around the target gas. Each end of the tube is equipped with an electron detector. Because of its characteristic appearance, this was dubbed a stereo-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 vector potential) is different in the positive and negative x-directions. Recall from Eq. (2) that the electron’s drift momentum depends on the vector potential at its time of birth. Therefore, depending on the value of the absolute phase, such a laser pulse creates more electrons in one direction than in the other. A theoretical analysis of the photoelectrons’ angular distribution was given by Dietrich et al. (2000) and Hansen et al. (2001). An equivalent statement is that the number of 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 to produce only a few that go left, and vice versa. This can be proved by correlation analysis. Each laser shot is sorted into a contingency map according to the number of electrons recorded at both detectors. Anticorrelations can then be seen in structures perpendicular to the diagonal, see Fig. 25 (overleaf). VII. Acknowledgments We learned a lot in discussions with S.L. Chin, M. Dörr, 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 in part by Deutsche Forschungsgemeinschaft and Volkswagen Stiftung. VIII. References Agostini, P., Fabre, F., Mainfray, G., Petite, G., and Rahman, N. (1979). Phys. Rev. Lett. 42, 1127. Agostini, P., Antonetti, A., Breger, P., Crance, M., Migus, A., Muller, H.G., and Petite, G. (1989). J. Phys. B 22, 1971. Alon, O.E., Averbukh, V., and Moiseyev, N. (2000). Phys. Rev. Lett. 85, 5218. Ammosov, M.V., Delone, N.B., and Krainov, V.P. (1986). Zh. Eksp. Teor. Fiz. 91, 2008 [Sov. Phys.JETP 64, 1191]. 92 W. Becker et al. [VIII Fig. 25. Evidence of absolute-phase effects from few-cycle laser pulses. In this contingency map, every laser shot is recorded according to the number of photoelectrons measured in the left and the right arm of the “stereo” ATI spectrometer. The number of laser shots with electron numbers according to the coordinates of the pixel is coded in grey shades. For visual convenience the darkest shades were chosen for medium numbers of laser shots. (The most frequent result of the laser pulses was about 5 electrons in each of both arms.) The signature of the absolute phase is an anticorrelation in the number of electrons recorded with the left and the right detector. In the contingency map they form a structure perpendicular to the diagonal. Shown here is a measurement with krypton atoms for circular laser polarization, a pulse duration of 6 fs, and an intensity of 5×1013 W/cm2 . From Paulus et al. (2001b). Antoine, P., L’Huillier, A., Lewenstein, M., Salières, P., and Carré, B. (1996). Phys. Rev. A 53, 1725. Antoine, Ph., Gaarde, M., Salières, P., Carré, B., L’Huillier, A., and Lewenstein, M. (1997). In “Multiphoton Processes 1996” (P. Lambropoulos, H. Walther, Eds.), Institute of Physics Conference Series No. 154. Institute of Physics Publishing, Bristol, p. 142. Apolonski, A., Poppe, A., Tempea, G., Spielmann, Ch., Udem, Th., Holzwarth, R., Hänsch, T.W., and Krausz, F. (2000). Phys. Rev. Lett. 85, 740. Bao, D., Chen, S.G., and Liu, J. (1995). Appl. Phys. B 62, 313. Bartels, R., Backus, S., Zeek, E., Misoguti, L., Vdovin, G., Christov, I.P., Murnane, M.M., and Kapteyn, H.C. (2000). Nature (London) 406, 164. Bartels, R., Backus, S., Christov, I., Kapteyn, H., and Murnane, M. (2001). Chem. Phys. 267, 277. Bashkansky, M., Bucksbaum, P.H., and Schumacher, D.W. (1988). Phys. Rev. Lett. 60, 2458. Becker, W., Schlicher, R.R., and Scully, M.O. (1986). J. Phys. B 19, L785. Becker, W., Schlicher, R.R., Scully, M.O., and Wódkiewicz, K. (1987). J. Opt. Soc. Am. B 4, 743. Becker, W., Long, S., and McIver, J.K. (1990). Phys. Rev. A 42, 4416. Becker, W., Long, S., and McIver, J.K. (1992). Phys. Rev. A 46, R5334. Becker, W., Lohr, A., and Kleber, M. (1994a). J. Phys. B 27, L325. Corrigendum: 28, 1931. Becker, W., Long, S., and McIver, J.K. (1994b). Phys. Rev. A 50, 1540. VIII] ATI: CLASSICAL TO QUANTUM 93 Becker, W., Lohr, A., and Kleber, M. (1995). Quantum Semiclass. Opt. 7, 423. Becker, W., Lohr, A., Kleber, M., and Lewenstein, M. (1997). Phys. Rev. A 56, 645. Becker, W., Kleber, M., Lohr, A., Paulus, G.G., Walther, H., and Zacher, F. (1998). Laser Phys. 8, 56. Beigman, I.L., and Chichkov, B.N. (1987). Pis’ma Zh. Eksp. Teor. Fiz. 46, 314 [JETP Lett. 46, xxx]. Bellini, M., Lyngå, C., Tozzi, A., Gaarde, M.B., Hänsch, T.W., L’Huillier, A., and Wahlström, C.-G. (1998). Phys. Rev. Lett. 81, 297. Berson, I.J. (1975). J. Phys. B 8, 3078. Bhardwaj, V.R., Aseyev, S.A., Mehendale, M., Yudin, G.L., Villeneuve, D.M., Rayner, D.M., Ivanov, M.Yu., and Corkum, P.B. (2001). Phys. Rev. Lett. 86, 3522. Blondel, C., Delsart, C., Dulieu, F., and Valli, C. (1999). Eur. Phys. J. D 5, 207. Borca, B., Frolov, M.V., Manakov, N.L., and Starace, A.F. (2001). Phys. Rev. Lett. 87, 133001. Bordas, C., Paulig, F., Helm, H., and Huestis, D.L. (1996). Rev. Sci. Instrum. 67, 2257. Brabec, T., and Krausz, F. (2000). Rev. Mod. Phys. 72, 545. Bryant, H.C., Mohagheghi, A., Stewart, J.E., Donahue, J.B., Quick, C.R., Reeder, R.A., Yuan, V., Hummer, C.R., Smith, W.W., Cohen, C., Reinhardt, W.P., and Overman, L. (1987). Phys. Rev. Lett. 58, 2412. Bucksbaum, P.H., Bashkansky, M., Freeman, R.R., McIlrath, T.J., and DiMauro, L.F. (1986). Phys. Rev. Lett. 56, 2590. Bucksbaum, P.H., Bashkansky, M., and McIlrath, T.J. (1987). Phys. Rev. Lett. 58, 349. Bunkin, F.V., and Fedorov, M.V. (1966). Sov. Phys.-JETP 22, 844. Ceccherini, F., and Bauer, D. (2001). Phys. Rev. A 64, 033423. Chelkowski, S., and Bandrauk, A.D. (2000). Laser Phys. 10, 216. Chin, S.L., Yergeau, F., and Lavigne, P. (1985). J. Phys. B 18, L213. Christov, I.P., Bartels, R., Kapteyn, H.C., and Murnane, M.M. (2001). Phys. Rev. Lett. 86, 5458. Corkum, P.B. (1993). Phys. Rev. Lett. 71, 1994. Corkum, P.B., Burnett, N.H., and Ivanov, M.Y. (1994). Opt. Lett. 19, 1870. Cormier, E., and Lambropoulos, P. (1997). J. Phys. B 30, 77. Cormier, E., Garzella, D., Breger, P., Agostini, P., Chériaux, G., and Leblanc, C. (2001). J. Phys. B 34, L9. Crawford, D.P., and Reiss, H.R. (1997). Opt. Express 2, 289. Dammasch, M., Dörr, M., Eichmann, U., Lenz, E., and Sandner, W. (2001). Phys. Rev. A 64, 0614xx(R). de Bohan, A., Antoine, P., Milošević, D.B., and Piraux, B. (1998). Phys. Rev. Lett. 81, 1837. Delande, D., and Buchleitner, A. (1994). Adv. At. Mol. Opt. Phys. 34, 85. Delone, N.B., and Krainov, V.P. (1994). “Multiphoton Processes in Atoms.” Springer, Berlin. Delone, N.B., and Krainov, V.P. (1998). Usp. Fiz. Nauk. 168, 531 [Phys. Usp. 41, 469]. Demkov, Yu., and Ostrovskii, V.N. (1989). “Zero-Range Potentials and their Applications in Atomic Physics.” Plenum, New York. Dietrich, P., Burnett, N.H., Ivanov, M., and Corkum, P.B. (1994). Phys. Rev. A 50, R3585. Dietrich, P., Krausz, F., and Corkum, P.B. (2000). Opt. Lett. 25, 16. DiMauro, L.F., and Agostini, P. (1995). Adv. At. Mol. Opt. Phys. 35, 79. Dörner, R., Mergel, V., Jagutzki, O., Spielberger, L., Ullrich, J., Moshammer, R., and SchmidtBöcking, H. (2000). Phys. Rep. 330, 96. Dörr, M., Potvliege, R.M., and Shakeshaft, R. (1990). Phys. Rev. A 41, 558. Drescher, M., Hentschel, M., Kienberger, R., Tempea, G., Spielmann, Ch., Reider, G.A., Corkum, P.B., and Krausz, F. (2001). Science 291, 1923. Du, M.L., and Delos, J.B. (1988). Phys. Rev. A 38, 1896, 1913. Dykhne, A.M. (1960). Zh. Eksp. Teor. Fiz. 38, 570 [Sov. Phys.-JETP 11, 411]. Eberly, J.H., Grobe, R., Law, C.K., and Su, Q. (1992). Adv. At. Mol. Opt. Phys. Suppl. 1, 301. Ehlotzky, F. (2001). Phys. Rep. 345, 175. 94 W. Becker et al. [VIII Eichmann, H., Egbert, A., Nolte, S., Momma, C., Wellegehausen, B., Becker, W., Long, S., and McIver, J.K. (1995). Phys. Rev. A 51, R3414. Faisal, F.H.M. (1973). J. Phys. B 6, L89. Faisal, F.H.M., and Radożycki, T. (1993). Phys. Rev. A 47, 4464. Faisal, F.H.M., and Scanzano, P. (1992). Phys. Rev. Lett. 68, 2909. Faisal, F.H.M., Filipowicz, P., and Rz˛ażewski, K. (1990). Phys. Rev. A 41, 6176. Faria, C. Figueira de Morisson, Milošević, D.B., and Paulus, G.G. (2000). Phys. Rev. A 61, 063415. Faria, C. Figueira de Morisson, Kopold, R., Becker, W., and Rost, J.M. (2002). Phys. Rev. A, to be published. Ferray, M., l’Huillier, A., Li, X.F., Lompré, L.A., Mainfray, G., and Manus, C. (1988). J. Phys. B 21, L31. Filipowicz, P., Faisal, F.H.M., and Rz˛ażewski, K. (1991). Phys. Rev. A 44, 2210. Fittinghoff, D.N., Bolton, P.R., Chang, B., and Kulander, K.C. (1992). Phys. Rev. Lett. 69, 2642. Freeman, R.R., Bucksbaum, P.H., Milchberg, H., Darack, S., Schumacher, D., and Geusic, M.E. (1987). Phys. Rev. Lett. 59, 1092. Gaarde, M.B., Salin, F., Constant, E., Balcou, Ph., Schafer, K.J., Kulander, K.C., and L’Huillier, A. (1999). Phys. Rev. A 59, 1367. Gaarde, M.B., Schafer, K.J., Kulander, K.C., Sheehy, B., Kim, D., and DiMauro, L.F. (2000). Phys. Rev. Lett. 84, 2822. Gallagher, T.F., and Scholz, T.J. (1989). Phys. Rev. A 40, 2762. Garton, W.R.S., and Tomkins, F.S. (1967). Astrophys. J. 158, 839. Goreslavskii, S.P., and Popruzhenko, S.V. (1996). Zh. Eksp. Teor. Fiz. 110, 1200 [JETP 83, 661]. Goreslavskii, S.P., and Popruzhenko, S.V. (1999a). Phys. Lett. A 249, 477. Goreslavskii, S.P., and Popruzhenko, S.V. (1999b). J. Phys. B 32, L531. Goreslavskii, S.P., and Popruzhenko, S.V. (2000). Zh. Eksp. Teor. Fiz. 117, 895 [JETP 90, 778]. Görlinger, J., Plagne, L., and Kull, H.-J. (2000). Appl. Phys. B 71, 331. Gottlieb, B., Kleber, M., and Krause, J. (1991). Z. Phys. A 339, 201. Gottlieb, B., Lohr, A., Becker, W., and Kleber, M. (1996). Phys. Rev. A 54, R1022. Gribakin, G.F., and Kuchiev, M.Yu. (1997). Phys. Rev. A 55, 3760. Gutzwiller, M. (1990). “Chaos in Classical and Quantum Mechanics.” Springer, Berlin. Hansch, P., Walker, M.A., and Van Woerkom, L.D. (1997). Phys. Rev. A 55, R2535. Hansen, J.P., Lu, J., Madsen, L.B., and Nilsen, H.M. (2001). Phys. Rev. A 64, 033418. Hauge, E.H., and Støvneng, J.A. (1989). Rev. Mod. Phys. 59, 917. Hentschel, M., Kienberger, R., Spielmann, Ch., Reider, G.A., Milošević, N., Brabec, T., Corkum, P.B., Heinzmann, U., Drescher, M., and Krausz, F. (2002). Nature (London), in press. Hertlein, M.P., Bucksbaum, P.H., and Muller, H.G. (1997). J. Phys. B 30, L197. Hu, S.X., and Keitel, C.H. (2001). Phys. Rev. A 63, 053402. Ivanov, M.Yu., Brabec, Th., and Burnett, N. (1996). Phys. Rev. A 54, 742. Jackson, J.D. (1999). “Classical Electrodynamics,” 3rd edition. Wiley, New York. Jaroń, A., Kamiński, J.Z., and Ehlotzky, F. (1999). Opt. Commun. 163, 115. Joachain, C.J., Dörr, M., and Kylstra, N. (2000). Adv. At. Mol. Opt. Phys. 42, 225. Jones, D.J., Diddams, S.A., Ranka, J.K., Stentz, A., Windeler, R.S., Hall, J.L., and Cundiff, S.T. (2000). Science 288, 635. Keldysh, L.V. (1964). Zh. Eksp. Teor. Fiz. 47, 1945 [Sov. Phys.–JETP 20, 1307]. Kibble, T.W.B. (1966). Phys. Rev. 150, 1060. Kim, J.-H., Lee, D.G., Shin, H.J., and Nam, C.H. (2001). Phys. Rev. A 63, 063403. Kopold, R. (2001). Ph.D. Dissertation. Munich Technical University. In German. Kopold, R., and Becker, W. (1999). J. Phys. B 32, L419. Kopold, R., Becker, W., and Kleber, M. (1998). Phys. Rev. A 58, 4022. Kopold, R., Becker, W., and Kleber, M. (2000a). Opt. Commun. 179, 39. Kopold, R., Milošević, D.B., and Becker, W. (2000b). Phys. Rev. Lett. 84, 3831. VIII] ATI: CLASSICAL TO QUANTUM 95 Kopold, R., Becker, W., Kleber, M., and Paulus, G.G. (2001). to be published. Krainov, V.P. (1999). J. Phys. B 32, 1607. Krainov, V.P., and Shokri, B. (1995). Laser Phys. 5, 793. Kramer, T., Bracher, C., and Kleber, M. (2001). Europhys. Lett., to be published. Krausz, F. (2001). Phys. World 14, 41. Kroll, N.M., and Watson, K.M. (1973). Phys. Rev. A 8, 804. Krstić, P., and Mittleman, M.H. (1991). Phys. Rev. A 44, 5938. Krstić, P.S., Milošević, D.B., and Janev, R.K. (1991). Phys. Rev. A 44, 3089. Kruit, P., Kimman, J., Muller, H.G., and van der Wiel, M.J. (1983). Phys. Rev. A 28, 248. Kuchiev, M.Yu. (1987). Pis’ma Zh. Eksp. Teor. Fiz. 45, 319 [JETP Lett. 45, 404]. Kuchiev, M.Yu., and Ostrovsky, V.N. (1999). J. Phys. B 32, L189. Kuchiev, M.Yu., and Ostrovsky, V.N. (2001). J. Phys. B 34, 405. Kulander, K.C., and Lewenstein, M. (1996). In “Atomic, Molecular, & Optical Physics Handbook” (G.W. Drake, Ed.). American Institute of Physics Press, Woodbury, p. 828. Kulander, K.C., Schafer, K.J., and Krause, K.L. (1993). In “Super-Intense Laser–Atom Physics,” (B. Piraux, A. L’Huillier, K. Rz˛ażewski, Eds.), Vol. 316 of NATO Advanced Studies Institute, Series B: Physics. Plenum, New York, p. 95. Kull, H.-J., Görlinger, J., and Plagne, L. (2000). Laser Phys. 10, 151. Kulyagin, R.V., Shubin, N.Yu., and Taranukhin, V.D. (1996). Laser Phys. 6, 79. Kylstra, N.J., Worthington, R.A., Patel, A., Knight, P.L., Vázquez de Aldana, J.R., and Roso, L. (2000). Phys. Rev. Lett. 85, 1835. Kylstra, N.J., Potvliege, R.M., and Joachain, C.J. (2001). J. Phys. B 34, L55. Lambropoulos, P., Maragakis, P., and Cormier, E. (1998). Laser Phys. 8, 625. Lee, D.G., Shin, H.J., Cha, J.H., Hong, K.H., Kim, J.-H., and Nam, C.H. (2001). Phys. Rev. A 63, 021801(R). Lein, M., Gross, E.K.U., and Engel, V. (2001). Phys. Rev. A 64, 023406. Leubner, C. (1981). Phys. Rev. A 23, 2877. Lewenstein, M., Balcou, Ph., Ivanov, M.Yu., L’Huillier, A., and Corkum, P.B. (1994). Phys. Rev. A 49, 2117. Lewenstein, M., Kulander, K.C., Schafer, K.J., and Bucksbaum, P.H. (1995a). Phys. Rev. A 51, 1495. Lewenstein, M., Salières, P., and L’Huillier, A. (1995b). Phys. Rev. A 52, 4747. l’Huillier, A., Lompré, L.A., Mainfray, G., and Manus, C. (1983). Phys. Rev. A 27, 2503. Lindner, F., Dreischuh, A., Grasbon, F., Paulus, G.G., and Walther, H. (2001). to be published. Lohr, A., Kleber, M., Kopold, R., and Becker, W. (1997). Phys. Rev. A 55, R4003. Long, S., Becker, W., and McIver, J.K. (1995). Phys. Rev. A 52, 2262. Mainfray, G., and Manus, C. (1991). Rep. Prog. Phys. 54, 1333. Manakov, N.L., and Fainshtein, A.G. (1980). Zh. Eksp. Teor. Fiz. 79, 751 [Sov. Phys.-JETP 52, 382]. Manakov, N.L., and Rapoport, L.P. (1975). Zh. Eksp. Teor. Fiz. 69, 842 [Sov. Phys.-JETP 42, 430]. Manakov, N.L., Frolov, M.V., Starace, A.F., and Fabrikant, I.I. (2000). J. Phys. B 33, R141. McKnaught, S.J., Knauer, J.P., and Meyerhofer, D.D. (1997). Phys. Rev. Lett. 78, 626. McPherson, A., Gibson, G., Jara, H., Johann, U., Luk, T.S., McIntyre, I.A., Boyer, K., and Rhodes, C.K. (1987). J. Opt. Soc. Am. B 4, 595. Milošević, D.B. (2000). J. Phys. B 33, 2479. Milošević, D.B., and Becker, W. (2000). Phys. Rev. A 62, 011403(R). Milošević, D.B., and Ehlotzky, F. (1998a). Phys. Rev. A 57, 5002. Milošević, D.B., and Ehlotzky, F. (1998b). Phys. Rev. A 58, 3124. Milošević, D.B., and Sandner, W. (2000). Opt. Lett. 25, 1532. Milošević, D.B., and Starace, A.F. (1998). Phys. Rev. Lett. 81, 5097. Milošević, D.B., and Starace, A.F. (1999a). Phys. Rev. Lett. 82, 2653. Milošević, D.B., and Starace, A.F. (1999b). Phys. Rev. A 60, 3160. Milošević, D.B., and Starace, A.F. (1999c). Phys. Rev. A 60, 3943. 96 W. Becker et al. [VIII Milošević, D.B., and Starace, A.F. (2000). Laser Phys. 10, 278. Milošević, D.B., Becker, W., and Kopold, R. (2000). Phys. Rev. A 61, 063403. Milošević, D.B., Becker, W., and Kopold, R. (2001a). In “Atoms, Molecules and Quantum Dots in Laser Fields: Fundamental Processes” (N. Bloembergen, N. Rahman, A. Rizzo, Eds.), Conference Proceedings Vol. 71. Italian Physical Society/Editrice Compositori, Bologna) p. 239. Milošević, D.B., Becker, W., Kopold, R., and Sandner, W. (2001b). Laser Phys. 11, 165. Milošević, D.B., Hu, S., and Becker, W. (2001c). Phys. Rev. A 63, 011403(R). Milošević, D.B., Hu, S.X., and Becker, W. (2002). Laser Phys. 12, xxx. Mohideen, U., Sher, M.H., Tom, H.W.K., Aumiller, G.D., Wood II, O.R., Freeman, R.R., Bokor, J., and Bucksbaum, P.H. (1993). Phys. Rev. Lett. 71, 509. Moore, C.I., Knauer, J.P., and Meyerhofer, D.D. (1995). Phys. Rev. Lett. 74, 2439. Moore, C.L., Ting, A., McNaught, S.J., Qiu, J., Burris, H.R., and Sprangle, P. (1999). Phys. Rev. Lett. 82, 1688. Moshammer, R., Feuerstein, B., Schmitt, W., Dorn, A., Schröter, C.D., Ullrich, J., Rottke, H., Trump, C., Wittmann, M., Korn, G., Hoffmann, K., and Sandner, W. (2000). Phys. Rev. Lett. 84, 447. Muller, H.G. (1999a). Phys. Rev. A 60, 1341. Muller, H.G. (1999b). Phys. Rev. Lett. 83, 3158. Muller, H.G. (1999c). Laser Phys. 9, 138. Muller, H.G. (2001a). Opt. Express 8, 44. Muller, H.G. (2001b). Opt. Express 8, 86. Muller, H.G. (2001c). Opt. Express 8, 417. Muller, H.G., and Kooiman, F.C. (1998). Phys. Rev. Lett. 81, 1207. Muller, H.G., Tip, A., and van der Wiel, M.J. (1983). J. Phys. B 16, L679. Mur, V.D., Karnakov, B.M., and Popov, V.S. (1998). Zh. Eksp. Teor. Fiz. 114, 798 [J. Exp. Theor. Phys. 87, 433]. Mur, V.D., Popruzhenko, S.V., and Popov, V.S. (2001). Zh. Eksp. Teor. Fiz. 119, 893 [J. Exp. Theor. Phys. 92, 777]. Nandor, M.J., Walker, M.A., and Van Woerkom, L.D. (1998). J. Phys. B 31, 4617. Nandor, M.J., Walker, M.A., Van Woerkom, L.D., and Muller, H.G. (1999). Phys. Rev. A 60, R1771. Nisoli, M., De Silvestri, S., Svelto, O., Szipöcs, R., Ferencz, K., Spielmann, Ch., Sartania, S., and Krausz, F. (1997). Opt. Lett. 22, 522. Nurhuda, M., and Faisal, F.H.M. (1999). Phys. Rev. A 60, 3125. Ortner, J., and Rylyuk, V.M. (2000). Phys. Rev. A 61, 033403. Parker, J.S., Moore, L.R., Meharg, K.J., Dundas, D., and Taylor, K.T. (2001). J. Phys. B 34, L69. Patel, A., Protopapas, M., Lappas, D.G., and Knight, P.L. (1998). Phys. Rev. A 58, R2652. Paul, P.M., Toma, E.S., Breger, P., Mullot, G., Augé, F., Balcou, Ph., Muller, H.G., and Agostini, P. (2001). Science 292, 1689. Paulus, G.G. (1996). “Multiphotonenionisation mit intensiven, ultrakurzen Laserpulsen.” Utz, München. Paulus, G.G., Becker, W., Nicklich, W., and Walther, H. (1994a). J. Phys. B 27, L703. Paulus, G.G., Nicklich, W., and Walther, H. (1994b). Europhys. Lett. 27, 267. Paulus, G.G., Nicklich, W., Xu, H., Lambropoulos, P., and Walther, H. (1994c). Phys. Rev. Lett. 72, 2851. Paulus, G.G., Becker, W., and Walther, H. (1995). Phys. Rev. A 52, 4043. Paulus, G.G., Zacher, F., Walther, H., Lohr, A., Becker, W., and Kleber, M. (1998). Phys. Rev. Lett. 80, 484. Paulus, G.G., Grasbon, F., Dreischuh, A., Walther, H., Kopold, R., and Becker, W. (2000). Phys. Rev. Lett. 84, 3791. Paulus, G.G., Grasbon, F., Walther, H., Kopold, R., and Becker, W. (2001a). Phys. Rev. A 64, 021401(R). VIII] ATI: CLASSICAL TO QUANTUM 97 Paulus, G.G., Grasbon, F., Walther, H., Villoresi, P., Nisoli, M., Stagira, S., Priori, E., and De Silvestri, S. (2001b). Nature (London), in press. Paulus, G.G., Grasbon, F., Walther, H., Nisoli, M., Stagira, S., Sansine, G., and De Silvestri, S. (2002). to be published. Perelomov, A.M., and Popov, V.S. (1967). Zh. Eksp. Teor. Fiz. 52, 514 [Sov. Phys.-JETP 25, 336]. Perelomov, A.M., Popov, V.S., and Terent’ev, M.V. (1966a). Zh. Eksp. Teor. Fiz. 50, 1393 [Sov. Phys.-JETP 23, 924]. Perelomov, A.M., Popov, V.S., and Terent’ev, M.V. (1966b). Zh. Eksp. Teor. Fiz. 51, 309 [Sov. Phys.-JETP 24, 207]. Popov, V.S., Mur, V.D., and Karnakov, B.M. (1997). Pis’ma Zh. Eksp. Teor. Fiz. 66, 213 [JETP Lett. 66, 229]. Popruzhenko, S.V., Goreslavskii, S.P., Korneev, P.A., and Becker, W. (2002). to be published. Protopapas, M., Keitel, C.H., and Knight, P.L. (1997a). Rep. Progr. Phys. 60, 389. Protopapas, M., Lappas, D.G., and Knight, P.L. (1997b). Phys. Rev. Lett. 79, 4550. Raczyński, A., and Zaremba, J. (1997). Phys. Lett. A 232, 428. Reichert, J., Holzwarth, R., Udem, Th., and Hänsch, T.W. (1999). Opt. Commun. 172, 59. Reiss, H.R. (1980). Phys. Rev. A 22, 1786. Reiss, H.R. (1990). J. Opt. Soc. Am. B 7, 574. Reiss, H.R. (1992). Prog. Quantum Electron. 16, 1. Robustelli, D., Saladin, D., and Scharf, G. (1997). Helv. Phys. Acta 70, 96. Rottke, H., Wolff, B., Brickwedde, M., Feldmann, D., and Welge, K.H. (1990). Phys. Rev. Lett. 64, 404. Salières, P., L’Huillier, A., Antoine, Ph., and Lewenstein, M. (1999). Adv. At. Mol. Opt. Phys. 41, 83. Salières, P., Carré, B., le Déroff, L., Grasbon, F., Paulus, G.G., Walther, H., Kopold, R., Becker, W., Milošević, D.B., Sanpera, A., and Lewenstein, M. (2001). Science 292, 902. Schins, J.M., Breger, P., Agostini, P., Constantinescu, R.C., Muller, H.G., Bouhal, A., Grillon, G., Antonetti, A., and Mysyrowicz, A. (1996). J. Opt. Soc. Am. B 13, 197. Schomerus, H., and Faria, C. Figueira de Morisson (2002). unpublished. Schomerus, H., and Sieber, M. (1997). J. Phys. A 30, 4537. Schulman, L. (1977). “Techniques and Applications of Path Integration.” Benjamin, New York. Scrinzi, A., Geissler, M., and Brabec, Th. (1999). Phys. Rev. Lett. 83, 706. Sheehy, B., Lafon, R., Widmer, M., Walker, B., DiMauro, L.F., Agostini, P.A., and Kulander, K.C. (1998). Phys. Rev. A 58, 3942. Smirnov, M.B., and Krainov, V.P. (1998). J. Phys. B 31, L519. Smyth, E.S., Parker, J.S., and Taylor, K.T. (1998). Comput. Phys. Commun. 114, 1. Spence, D.E., Kean, P.N., and Sibbett, W. (1991). Opt. Lett. 16, 42. Taı̈eb, R., Véniard, V., and Maquet, A. (2001). Phys. Rev. Lett. 87, 053002. Tang, C.Y., Bryant, H.C., Harris, P.G., Mohagheghi, A.H., Reeder, R.A., Sharifian, H., Tootoonchi, H., Quick, C.R., Donahue, J.B., Cohen, S., and Smith, W.W. (1991). Phys. Rev. Lett. 66, 3124. Toma, E.S., Antoine, Ph., de Bohan, A., and Muller, H.G. (1999). J. Phys. B 32, 5843. van de Sand, G., and Rost, J.M. (2000). Phys. Rev. A 62, 053403. van Linden van den Heuvell, H.B., and Muller, H.G. (1988). In “Multiphoton Processes”, Vol. 8 of Cambridge Studies in Modern Optics. Cambridge University Press, Cambridge, p. 25. Vázquez de Aldana, J.R., and Roso, L. (1999). Opt. Express 5, 144. Vázquez de Aldana, J.R., Kylstra, N.J., Roso, L., Knight, P.L., Patel, A., and Worthington, R.A. (2001). Phys. Rev. A 64, 013411. Véniard, V., Taı̈eb, R., and Maquet, A. (1995). Phys. Rev. Lett. 74, 4161. Véniard, V., Taı̈eb, R., and Maquet, A. (1996). Phys. Rev. A 54, 721. Volkov, D.M. (1935). Z. Phys. 94, 250. Walker, B., Sheehy, B., Kulander, K.C., and DiMauro, L.F. (1996). Phys. Rev. Lett. 77, 5031. Walser, M.W., Keitel, C.H., Scrinzi, A., and Brabec, T. (2000). Phys. Rev. Lett. 85, 5082. 98 W. Becker et al. [VIII Walsh, T.D.G., Ilkov, F.A., and Chin, S.L. (1994). J. Phys. B 27, 3767. Watson, J.B., Sanpera, A., Burnett, K., and Knight, P.L. (1997). Phys. Rev. A 55, 1224. Weber, Th., Giessen, H., Weckenbrock, M., Urbasch, G., Staudte, A., Spielberger, L., Jagutzki, O., Mergel, V., Vollmer, M., and Dörner, R. (2000a). Nature (London) 404, 608. Weber, Th., Weckenbrock, M., Staudte, A., Spielberger, L., Jagutzki, O., Mergel, V., Afaneh, F., Urbasch, G., Vollmer, M., Giessen, H., and Dörner, R. (2000b). Phys. Rev. Lett. 84, 443. Weingartshofer, A., Holmes, J.K., Caudle, G., Clarke, E.M., and Krüger, H. (1977). Phys. Rev. Lett. 39, 269. Weingartshofer, A., Holmes, J.K., Sabbagh, J., and Chin, S.L. (1983). J. Phys. B 16, 1805. Wildenauer, J. (1987). J. Appl. Phys. 62, 41. Yang, B., Schafer, K.J., Walker, B., Kulander, K.C., Agostini, P., and DiMauro, L.F. (1993). Phys. Rev. Lett. 71, 3770. Yergeau, F., Chin, S.L., and Lavigne, P. (1987). J. Phys. B 20, 723. Yudin, G.L., and Ivanov, M.Yu. (2001a). Phys. Rev. A 63, 033404. Yudin, G.L., and Ivanov, M.Yu. (2001b). Phys. Rev. A 64, 013409.