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From Pierre Agostini, Atomic and Molecular Ionization Dynamics in Strong Laser Fields: From Optical toX-rays. In: E. Arimondo, P. R. Berman, C. C. Lin, editor, Advances in Atomic, Molecular, and Optical

Physics. Chennai: Inc.; 2012. p 117-158.ISBN: 978-0-12-396482-3

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CHAPTER 3Atomic and MolecularIonization Dynamics inStrong Laser Fields: FromOptical to X-rays

Pierre Agostini and Louis F. DiMauro

Department of Physics, The Ohio State University, 191 WWoodruff Ave, Columbus, OH 43210, USA

Contents 1. Introduction 1182. The First 30 Years of Multiphoton Physics

(1963−1993) 1202.1 The Genesis of a Field: The Early Days 1212.2 Resonant Multiphoton Ionization (MPI) 1212.3 Coherence 1222.4 Non-Resonant MPI 1222.5 Above-Threshold Ionization (ATI): Doorway

into the Modern Era 1222.6 Non-Perturbative ATI 1232.7 Rydberg Resonances and the Role of

Atomic Structure 1232.8 Multiple Ionization, Anne’s Knee, and the

Lambropoulos Curse 1232.9 Keldysh Tunneling: Different Mode of

Ionization 1242.10 Long Wavelength Ionization and the

Classical View 1242.11 High Harmonics, High-Order ATI and

Nonsequential Ionization Lead to theRescattering/Three-Step Model 125

2.12 Adiabatic Stabilization 1253. Wavelength Scaling of Strong-Field Atomic

Physics 1263.1 Fundamental Metrics in an Intense

Laser–Atom Interaction 1263.2 Ionization Experiments in the Strong-Field

Limit 127

Advances in Atomic, Molecular, and Optical Physics, Volume 61, Copyright © 2012 Elsevier Inc.ISSN 1049-250X, http://dx.doi.org/10.1016/B978-0-12-396482-3.00003-x. All rights reserved.

117

Author’s personal copy

http://dx.doi.org/10.1016/B978-0-12-396482-3.00003-x

118 Pierre Agostini and Louis F. DiMauro

4. Low-Energy Structure in Photoelectron EnergyDistribution in the Strong-Field Limit 131

5. Electron Momentum Distribution andTime-Dependent Imaging 1375.1 Extracting the Molecular Structure

from LIED 1386. Non-Sequential Multiple Ionization at Long

Wavelengths 1417. Strong-Field X-Ray Physics: A Future Path 1468. Outlook 151Acknowledgments 152References 152

Abstract We review the progress made over the past 15 years in under-standing and exploiting the physical scaling of strong-fieldatomic processes. These advances were enabled by theoryand experiment, but this chapter focuses on the understand-ing gained by two significant advancements in technology:intense, ultrafast mid-infrared, and X-ray sources. Examininginvestigations that utilize mid-infrared lasers, we discuss theconsequences of a several new phenomena: wavelength scal-ing of photoelectron energy, a universal low-energy structurein the photoelectron distribution, the ultrafast imaging of thephotoelectrons momentum distributions, and multiple ioniza-tion. In October 2009, the world’s first X-ray free-electron laser[Linac Coherent Light Source (LCLS)] became operational atSLAC National Laboratory in Menlo Park, California. The LCLSpeak power exceeds the performance of previous X-raysources by 5–6 orders of magnitude. In this review, we dis-cuss some of the initial findings and the implications for futureinvestigations of strong-field nonlinear processes at X-rayfrequencies.

1. INTRODUCTION

Although the invention of the laser dates back for half a century, intensefew-cycle pulses, femtosecond mid-infrared lasers, carrier-to-envelopephase control, XUV, and X-ray free-electron lasers (XFEL), attosecondpulses, in trains or isolated, are the product of the past 15 years. Strong-fieldionization of atoms and molecules, the topic of this chapter, is photoion-ization in a regime where the concept of multiphoton transitions, whichcould be described by traditional perturbation theory, looses definitionsince hundreds or thousands of photons are absorbed and emitted in theprocess. In the last decade or so, the study of photoionization has explorednew ranges of parameters, studied novel targets, and benefited from newdetection techniques. One of the familiar metrics in strong-field interaction


Atomic and Molecular Ionization Dynamics in Strong Laser Fields 119

(Reiss, 1992) is the ratio of the ponderomotive energy1 to the photon energyz ≡ Up/�ω proportional to ω−3 or λ3. The value of z gives an idea of thenumber of photons exchanged between the atom and the field. Intense,femtosecond mid-infrared laser (>1 µm) technology became available bythe end of the 1990s, allowing realization of z-values of several hundreds.At the other spectral extreme, the advent of the X-ray free-electron lasers(XFEL) with its unprecedented intensities, opened a route to X-ray non-linear optics, although the corresponding value of z is practically zero,allowing the sequential removal of all the electrons of an atom, like neon,from the inside out.

Strong-field ionization is a field of research which has, all along, beendriven by experiments while the theory has followed, often painstakingly.For example, the Schrödinger equation even for a single electron in the pres-ence of a strong electromagnetic field and a Coulomb potential cannot besolved exactly. In this context, low intensity, short wavelength is the regimeof perturbation theory. Stark-induced Rydberg resonances, observed andrecognized in 1987 (Freeman et al., 1987), involve Stark shifts of the orderof the photon energy (z ≈ 1) and theoretical treatment requires methodssuch as the Floquet theory (Rottke et al., 1994). However, as soon as Starkshifts become large compared to the photon energy, perturbation theoryand its extensions become ineffective. Thus, the solution of the Schrödingerequation will depend on other approximations. One such approximationwas proposed by Keldysh (1965), and later by Faisal (1973) and Reiss(1980, 1992) [for a detailed comparison between the three approaches, usu-ally dubbed the “KFR” theory see Reiss (1992); for a recent review of theKeldysh approximation see Popov (2004)]. Using the Keldysh approxima-tion, in the high intensity, long wavelength limit, the total ionization rateis shown to behave like the rate of dc-tunneling (Keldysh, 1965). The verysimple tunneling expression, and its success (Amosov et al., 1986; Chin,2011; Perelomov et al., 1965), explains its popularity but the concept oftunneling itself has a limited range of application (Reiss, 2008).

Our previous review, 15 years ago (DiMauro & Agostini, 1995), endedwith discussions of the Above-Threshold Ionization (ATI) plateau and thenon-sequential double ionization. Lasers generating ultrafast light pulsescame of age in the following years (Brabec & Krausz, 2000) and the newmillennium saw the emergence of new tools for strong-field investiga-tions. Among them, COLTRIMS (or similar) devices (Moshammer et al.,2000), few-cycle pulses (Brabec & Krausz, 2000), and carrier-to-envelopephase control (Apolonski et al., 2000; Paulus et al., 2001b; Saliéres et al.,2001), intense mid-infrared lasers (Hauri et al., 2007; Schultz et al., 2007;Sheehy et al., 1999). A plethora of discovery quickly followed: absolute

1Up = E2/4ω2 in atomic units is the cycle-average kinetic energy of a free electron in an electromagneticfield with amplitude E and frequency ω.



phase effects (Bauer et al., 2005), long-wavelength scaling of ATI (Colosimoet al., 2008; Tate et al., 2007), attosecond generation (Paul et al., 2001),and attochirp (Mairesse et al., 2003), ion recoil momentum distributions(Ullrich et al., 2003), a universal strong-field low-energy structure (LES)(Blaga et al., 2009), imaging of molecular orbitals from high harmonics(Caillat et al., 2011; Itatani et al., 2004) or photoelectron momentum distri-butions, nonlinear optics in the XUV and the X-ray domains (Meyer et al.,2010), non-sequential multiple ionization, attosecond measurements of theionization time delay (Schultze et al., 2010), and others. In this chapter, wefocus our discussion on wavelength scaling and its implications in strong-field ionization and attosecond generation, the LES, the high-order ATIphotoelectron momentum distribution, time-dependent molecular imag-ing, non-sequential multiple ionization, and finally, on X-ray nonlinearionization using XFEL.

It has occurred to the authors that it would perhaps be useful to includea brief, and undoubtedly incomplete, summary of the first 30 years of thisfield which, in spite of the recent popularity of the Keldysh “tunnel” ioniza-tion that followed the pioneering CO2 laser experiments (Chin et al., 1985;Corkum et al., 1989), the history may not be so familiar to newcomers ofthis field. Consequently, the outline of the chapter is as follows: the firstsection following this introduction will be a historical account of watershedcontributions. Section 3 establishes the various scaling parameters and thegeneral concept of Keldysh theory and introduces experiments illustratingthe wavelength scaling at long wavelengths. Section 4 describes the experi-mental evidence of the LES and various theoretical interpretations. Section5 discusses the photoelectron momentum distributions in strong-field ion-ization, with special application to the imaging of time-dependent molec-ular structure. Section 6 reviews recent results in non-sequential multipleionization and, particularly the quantitative analysis based on field-free(e, ne) cross-sections, and Section 7 summarizes experiments conductedusing LCLS XFEL at SLAC which demonstrate, among other spectacularresults, the complete stripping of electrons or the first evidence of a non-linear transitions in atoms with hard X-rays. Section 8 will conclude witha brief outlook.

2. THE FIRST 30YEARS OF MULTIPHOTON PHYSICS(1963−1993)

What is now called strong-field physics, slowly emerged with the adventof the pulsed laser in the early 1960s and since that time, has been stronglycoupled to subsequent advances in the technology. What follows is theauthor’s account of experimental milestones during the period of1963–1993. For the period preceding the laser, the reader is referred to



Appendix A in Reiss (2009), and for a more specific account of tunneling,see Chin (2011).

2.1 The Genesis of a Field: The Early Days

Q-switched lasers (ruby and neodymium glass) quickly became commer-cialized in the US following Maiman’s invention (Maiman, 1960). Theavailable intensities were far from the intensities required to observe QEDeffects, a theorist’s dream but were sufficient to conduct more modest inves-tigations into razor blade piercing or air breakdown by early experimen-talist, as a prelude to more scientific investigations. Perhaps the very firstattempt at a measurement of photoionization of a rare gas using ruby laserwas the work performed at The Ohio State University by Damon and Tom-linson (1963). It is the first cited reference in the historical Keldysh paperand, interestingly enough, the experimental data is fitted by a straightline in semi-log coordinates, which might have been the inspiration forKeldysh’s tunneling theory. The interest in this problem apparently fadedin the US while it was actively investigated in Moscow by N. Delone andhis group (Bystrova et al., 1967) at the Lebedev Institute, home of the recentNobel prizes (N. Basov and A. Prokhorov and of Keldysh himself). InFrance, at the CEA Saclay, a group directed by F. Bonnal in the depart-ment of C. Manus, started some experimental work in 1967 along with theLebedev group. Soon G. Mainfray took the direction of this group and thefirst results began to appear (Agostini et al., 1968). The group includedM. Trahin and Y. Gontier strong advocates of perturbation theory and thehydrogen atom. Perturbation theory soon became the most popular frame-work. The only experimental quantity that could be compared to the theoryin these early days was the “slope” of log–log plots of the total ion signalversus laser energy, which, according to lowest-order perturbation theoryequals the minimum number of photons n0 required to reach the ionizationthreshold. Unfortunately, the experimental slopes measured in Moscowand Paris were significantly lower than expected. The scenario that wasproposed to explain this unsettling finding was based on rate equationsassuming an intermediate resonance and a bottleneck which was deter-mining the overall slope.

2.2 Resonant Multiphoton Ionization (MPI)

The first well characterized experiments (few-photon ionization of alkali,controlled single-mode laser, and improved detection) caused some com-motion as it showed slopes larger than n0. It took sometime to realizethat the role of the Stark shift and its link with the observed slope(Morellec et al., 1976).



2.3 Coherence

Numerous articles on MPI were mentioning the coherence factor � = n0!,accounting for the photon statistics when using an incoherent light source,but a real measurement of this effect was elusive since incoherent sourceswere incapable of inducing MPI. Moreover, the connection between thephoton statistics and the MPI signal was not clear. A Saclay Ph.D. studentin Mainfray’s group, F. Sanchez, who did not believe in photons, was, webelieve, the first one to really understand the meaning of �. He appreciatedthat this factor could be controlled by the number of laser cavity modes andhence, the intensity fluctuations of the laser field. He built a very sophisti-cated neodymium oscillator that based on an adjustable number of modes,thus clearing up the problem (Held et al., 1973), although at the same timeremoving the mystery and glamor of the subject.

2.4 Non-Resonant MPI

Perturbation theory was predicting a deep minima in the MPIcross-sections due to destructive interference between quantum paths. Oneexperiment stirred some controversy (and excitement) in the communitywith a result that seemed to strongly contradict this prediction(Granneman & Van der Wiel, 1975). It took no less than 5 years tounderstand the problem and restore confidence in perturbation theory(Morellec et al., 1980).

2.5 Above-Threshold Ionization (ATI): Doorway into the Modern Era

Above-Threshold Ionization resulted from a slight shift in the experimen-tal approach: a differential measurement of the electron’s energy insteadof the total ion/electron yield. The first analyzer was a retarding voltagespectrometer with not much resolution, albeit sufficient to resolve 2.34 eVphoton energy from a doubled Nd:YAG. The observation was clear andprovided the first evidence of multiphoton ionization with one photonmore than required by energy conservation, i.e., included a free–free transi-tion (Agostini et al., 1979). The competition, the FOM group in Amsterdam,had designed and built a magnetic bottle electron spectrometer (whichwas then commercialized, popularized by many groups, and one variantis being used today in the AMO chamber at the LCLS). In spite of this, theydid not yet succeeded in seeing ATI and (this is a personal memory of Prof.Agostini from the reaction to a FOM seminar) strongly suspected a calibra-tion error in the Saclay result caused by the xenon fine structure. However,the retarding voltage was self-calibrating and the Saclay interpretation wassoon bona fide.



2.6 Non-Perturbative ATI

P. Kruit, a Ph.D. student at FOM, and the rest of M. Van der Wiel groupwere able to equal and actually surpass the experimental capabilities of theSaclay group due to the power of their unique spectrometer. The first evi-dence of non-perturbative behavior started to emerge from that laboratoryas their electron spectra showed (a) many ATI orders and (b) a surprise,a decreased yield in the low-energy part of the electron distribution withincreasing laser intensity (Kruit et al., 1983).

2.7 Rydberg Resonances and the Role of Atomic Structure

The FOM result was explained by a channel closure model which stated thatas the intensity rises the atom’s effective binding energy is increased due tothe Up Stark shift of Rydberg states which successively closed the lowest-order ATI channels, i.e., the channels requiring n0+1, n0 +2, . . . photons forionization (Kruit et al., 1983). If correct, this scenario implies, since Up > �ω,that shifted Rydberg states energies coincide during the laser pulse with theenergy of an integer number of photons, or, in other words, that the tran-sition becomes transiently resonant. The beauty of this is that all Rydbergstates have, under certain conditions, the same shift and therefore their dif-ferential energy remains unchanged. However, it was not possible to seeit in an experiment until the advent of sub-picosecond amplified lasers inthe mid-1980s. The first group to observe these transient resonances wasFreeman’s group at AT&T Bell Laboratories (Freeman et al., 1987) and theStark-induced Rydberg resonances are often referred to as a Freeman res-onance. Proper accounting of the Stark shifts required, of course, a morepowerful tool than perturbation theory. The Floquet ansatz led to a success-fully quantitative theoretical description (Potvliege & Shakeshaft, 1988), inparticular, a beautiful experiment in hydrogen (Rottke et al., 1994).

2.8 Multiple Ionization, Anne’s Knee, and the Lambropoulos Curse

It was at about the same time that L’Huillier performing her Ph.D. researchat Saclay started her investigations of laser produced multiply chargedions and observed that the intensity-dependent ion yields could not bedescribed by a single rate, and hence the discovery of the famous “knee”(L’Huillier et al., 1983) and the possible existence of direct two-electronionization. Actually, multiphoton double ionization had been studied inalkaline earths in the Ukraine by Suran and Zapesnochnii (1975), who hadnoted that the probability for double electron ejection was much higherthan predicted from perturbative scaling of the cross-sections, based onthe number of photons involved. For more on this topic, see DiMauro andAgostini (1995).



In the early 1980s, intense ultraviolet lasers became available.C. K. Rhodes in Chicago, observing highly charged heavy ions, concludedthat an entirely new regime of interaction had been reached (Luk et al.,1983) in which several electrons and perhaps a whole sub-shell could becollectively excited. However, Lambropoulos pointed out that a real laserpulse has a finite rise time during which electrons could be peeled offsuccessively by “normal” multiphoton absorptions (Lambropoulos, 1985).From Karule’s perturbative calculations (Karule, 1975) he also noticed thatthe saturation intensities were rather quickly going to a limit as a functionof the number of absorbed photons and wondered whether the limitingvalue was connected to Keldysh’ tunneling. The latter question was leftunanswered but the victory, although temporary, of perturbation theoryremained for a long time as the “Lambropoulos curse” to those who werefinding the perturbative approach irrelevant.

2.9 Keldysh Tunneling: Different Mode of Ionization

The 1980s were indeed a turning point in the field for both experiment andtheory. CO2 lasers became more powerful and reliable. The first observa-tion of rare gases ionization at a wavelength of 10 µm (Chin, 2011; Chinet al., 1985) was a milestone in strong-field physics. The number of photonsinvolved in the transition was so large that any heroic attempt at pertur-bation theory was hopeless. Actually several attempts in the precedingdecade failed at observing tunneling and thus was considered “unfea-sible” experimentally (Lompre et al., 1976). However, Chin et al. (1985)10 µm measurement resulted in good agreement with the tunneling the-ory of Perelomov et al. (1965) and began an increasingly general reversal ofopinion and Keldysh tunneling became something to be taken literally. Letus remark, as discussed by Reiss (1992, 2008) and others (a) tunneling hasno meaning in the velocity gauge and (b) it does not account for channelclosure. Nevertheless, since the three KFR theories find a similar behaviorin the high intensity, long wavelength limit, the tunneling paradigm maybe accepted, at least in the range of parameters where it makes sense (seediscussion below).

2.10 Long Wavelength Ionization and the Classical View

Interaction in the microwave region first explored by Bayfield and Koch(1974) and then by Gallagher (1988) was another case where countingthe photons was obviously impossible. Ionization, and a fortiori, the ATIspectrum, with such small photons were even more beyond the reach ofperturbation theory. van Linden van den Heuvell and Muller (1987) pro-posed a radical simplification of the theory of ATI, known as “Simpleman’stheory,” by postulating that in the high intensity, long wavelength limit thespectrum was determined by the classical motion of a charge in the laser



field. In a completely different direction that same year, exact solutions ofthe time-dependent Schrödinger equation (TDSE) for a complex atom in astrong laser field were implemented by Kulander (1987). With the develop-ment of cheaper, more powerful computers, TDSE became a popular toolin 1990s (Muller, 1999).

2.11 High Harmonics, High-Order ATI and Nonsequential IonizationLead to the Rescattering/Three-Step Model

The late 1980s witnessed the discovery of high harmonics plateau by theSaclay (Ferray et al., 1988) and Chicago (McPherson et al., 1987) groups. Atthe same time, the demonstration of amplification and subsequent recom-pression of optically chirped pulse (CPA) (Strickland & Mourou, 1985)opened the road to the development of intense femtosecond lasers andallowed for a wealth of new results. Another important advance in strong-field investigations occurred in the early 1990s: DiMauro and coworkersintroduced kilohertz repetition rate lasers in their experiments (Saeed etal., 1990), reaching unprecedented sensitivity. Three observations, the har-monic cutoff, the high-order ATI plateau (Paulus et al., 1994), the nonse-quential double ionization of helium (Fittinghoff et al., 1992) beckoned fora more complete view of the laser–matter interaction, beyond the “Simple-man” model, namely the need for accounting for the quivering continuumelectron’s re-interaction (or rescattering) with the core. Initiated by Kuchiev(1987) this culminated with the works of Kulander and co-workers (Schaferet al., 1993) on the one hand and Corkum (Corkum, 1993) on the other handand eventually became what is known as the three-step model.

2.12 Adiabatic Stabilization

By the end of 1980s and the early 1990s a calculation by Pont and Gavrila(1990) predicted a strange and spectacular effect: the decrease of theionization probability with increasing intensity in the high intensity, highfrequency limit. This ultimate non-perturbative effect, dubbed adiabaticstabilization, created a lot of excitement. The experiment though isextremely difficult and only one result hitherto (DeBoer et al., 1994)provides a beginning of support.

By the mid-1990s the study of strong-field ionization, which was three-decade old, was considered mature and perhaps coming to an end.However, novel tools and the influx of new ideas propelled the field towardnew discoveries, in a perfect illustration of the futility of predicting thefuture in fundamental research. This review is meant to highlight some ofthese recent advances.



3. WAVELENGTH SCALING OF STRONG-FIELDATOMIC PHYSICS

3.1 Fundamental Metrics in an Intense Laser–Atom Interaction

The purpose of the following section is to develop the background forthe reader of the basic wavelength scaling principles and metrics guid-ing strong-field physics. The scaling provides a understanding of the low-frequency behavior and allows an extrapolation into the unfamiliar realmof intense X-ray physics.

A simple metric of an intense laser–atom interaction is the atomic unit(a.u.) of field2 (50 V/Å). Focused amplified ultrafast laser systems andXFELs can easily exceed 1 a.u. by orders of magnitude, but the maximumintensity experience by an atom will be determined by the depletion (sat-uration) of the initial state. For low-frequency ionization (�ω < Ip, whereIp is the ionization potential), the saturation intensity (Isat) can be esti-mated by classical over-the-barrier ionization (Augst et al., 1989) given asIOTB = cIp/128πZ2 (Z ≡ effective charge) or ADK tunneling (Amosov et al.,1986). In this scenario, all ground state neutral atoms experience intensitiesless than an atomic unit, e.g., for helium (Ip = 24 eV), Isat ∼ 1 PW/cm2.However, the situation is different for X-rays ionizing core electrons (seeSection 7).

The ponderomotive or quiver energy, Up, is an important quantity sinceit can easily reach values several times the electron-binding energy, espe-cially at long wavelength. At an intensity of 1 PW/cm2 (helium satura-tion intensity) for a typical titanium sapphire femtosecond pulse, Up ∼60 eV. However by increasing the wavelength to 4 µm, Up will increase25-fold producing a ponderomotive energy of 1.5 keV! Therefore, this sim-ple example stresses the advantage of the λ2-scaling of Up toward longwavelength for studying neutral atoms in unprecedented large pondero-motive potentials. In Stark contrast, a 1 PW/cm2, 1 keV X-ray pulse onlyproduces a 0.1 meV quiver energy. Although Up appears as an negligiblequantity for X-rays, it does prove useful in the design of two-color streakingmetrology (Düsterer et al., 2011) for temporal XFEL characterization.

Keldysh defined an adiabaticity parameter, γ , as the ratio of the tunneltime to the optical period (Keldysh, 1965). In the limit γ � 1 the ionizationis tunneling through the barrier of the Stark potential while for γ � 1,for which the field changes rapidly compared with the tunneling time,ionization is multiphoton. Using the width of the barrier and the electronvelocity Keldysh derived an expression for γ in terms of Ip, and Up, asγ = (Ip/2Up)1/2. Using the expression for Up, γ is found to be proportional

2The Coulomb field in the hydrogen atom ground state equates to an equivalent light intensity of35 PW/cm2.



to λ−1(2Ip/I)1/2. Thus in the optical regime γ � 1, while for the X-raysγ � 1.

Up appears naturally in the KFR theory (Faisal, 1973; Keldysh, 1965;Reiss, 1980). Thus in Reiss (1980), z = Up/�ω > 1, dubbed the continuumintensity parameter, is one of the conditions for strong-field ionization andz ∝ 1/ω3. Comparing Up to the electron-binding energy Ip or to the electronrest energy mc2 provides a parameter (Reiss, 1992, 2008) which defines thestrong-field or relativistic limits, respectively:

z1 = 2Up

Ip= e2I/ω2

2mIp= 1/γ 2, (1)

where I is the intensity and γ the familiar Keldysh parameter. In Keldysh(1965) γ is related to the tunneling time while the bound-state intensityparameter, z1, does not involve any tunneling concept. It can be easilyshown that z1 is also the ratio of the interaction Hamiltonian to the unper-turbed Hamiltonian related to perturbation theory (Reiss & Smirnov, 1997).Consequently, a perturbative treatment of the atom–field interactionrequires that γ > 1, and z and z1 < 1.

Figure 1 is a plot of intensity versus frequency for z = 1 (dot-dash line-continua), z1 = 1 (solid line-bound), and the perturbative limit (dashedline), below which perturbation theory is valid. For the long wavelengthsused in our laboratory for strong-field studies and attosecond generation,the photon energy varies between 0.35 and 1.6 eV. For valence electrons atthese frequencies and Up � Ip, the distortion of the continua dominatesthe dynamics and the strong-field condition is satisfied for z1, z, and γ atintensities; 10−3 < I < 10−1 PW/cm2. In this regime the quasi-classicalmodel of rescattering is extremely useful.

3.2 Ionization Experiments in the Strong-Field Limit

The strong-field limit (γ < 1 and z, z1 > 1) is reached at near-visible wave-lengths, e.g., 0.8 µm, by increasing the intensity with femtosecond pulses.However, the intensity knob is limited by depletion effects, so studies aresolely confined to helium and neon atoms, i.e., neutrals with the highestbinding energy. Beginning in the late 1990s the emergence of intense ultra-fast sources operating at wavelengths longer than 1 µm enabled broaderexploration of the strong-field limit in atoms and molecules. These longerwavelength sources are mainly based on parametric processes suchas optical parametric amplifies (OPA), optical parametric chirped-pulseamplification (OPCPA), and difference frequency generation (DFG)(Colosimo et al., 2008; Dichiara et al., 2012; Nisoli et al., 2003). Different con-figurations can achieve gigawatt-level output that providescoverage from 1.3–2.3 µm to 2.8–4 µm and pulse durations as shortas three-cycles (Hauri et al., 2007; Schmidt et al., 2010). Figure 2 shows



Figure 1 A plot of the z (continua-state: dot-dash line) and z1 (hydrogen bound-state:solid line) intensity scaling parameter in the optical regime as a function of intensity.The lines are for the z-parameters equal to unity and the axis are plotted in atomicunits (a.u.) and standard units. The dashed line is the perturbative limit defined as10% of the continuum z-parameter and the double line is 1 a.u. field. At opticalfrequencies, the continua distortion defines the perturbative limit.

a summary of experimental results from our group illustrating the impactof wavelength on the strong-field processes of (a) electron production and(b, c) high harmonic generation. Figure 2a shows a series of photoelectronenergy distributions at constant intensity for xenon atoms exposed to dif-ferent fundamental wavelengths. The distributions show a clear increase inthe photoelectron energy at longer wavelengths. Within the quasi-classicalinterpretation (Corkum, 1993; Schafer et al., 1993; Sheehy et al., 1998; Walkeret al., 1996) both the electron and the harmonic distributions are intimatelylinked by the rescattering process. Electrons with energy beyond the “Sim-pleman” 2Up cutoff prediction result from elastic scattering of the returningfield-driven electron from the core potential thus allowing energies as highas 10Up. The distributions at 0.8 µm, 2 µm, and 4 µm have values of Up of5, 30, and 120 eV, respectively, for a constant intensity of 80 TW/cm2. Con-sequently, the measured maximum energies at 50, 300, and 1200 eV reflectthe 10Up cutoff predicted by a simple classical λ2-scaling (Up ∝ λ2) of theisolated atom responding to the strong-field.

For completeness, the high harmonic frequency comb generated inargon is shown for 0.8 µm and 2 µm at a constant intensity (180 TW/cm2)in Figure 2b and c. Odd-order high harmonics are generated in the rescat-tering process by a recombination of the field-driven electron wave packetwith the initial state (one-photon dipole matrix element). The highest



Figure 2 (Color online) Effect of the λ2-scaling with fundamental wavelength in anintense laser–atom interaction at constant intensity. (a) Comparison of the measuredxenon electron energy distribution at different wavelengths but constant intensity(80 TW/cm2). At 0.8 µm the maximum energy is 50 eV while for 4 µm the energyextends to 1 keV. (b) The measured high harmonic (HHG) spectral distributionsresulting from 0.8 µm (yellow shaded) to 2 µm (black/gray shaded) excitation of argonatoms at a constant intensity (180 TW/cm2). The 2 µm HHG extend to the Al-filterabsorption edge at 75 eV, while the 0.8 µm HHG cutoff is 50 eV. Plot (c), under thesame conditions as (b) except using a Zr-filter, shows that the 2 µm energy cutoff is at220 eV. In both cases, the particle energies dramatically increase with longerwavelength and analysis verifies the expected λ2-scaling of Up.

photon energy is determined by the maximum classical return energy of3.17Up (Corkum, 1993; Schafer et al., 1993) thus yielding a cutoff law of3.17Up + Ip (Krause et al., 1992).

These experiments are typically conducted at high gas density(1016–18

cc−1) making comparison to single-atom models difficult since macro-scopic contributions cannot be neglected (Gaarde et al., 2008). Nonetheless,the distributions show that the high harmonic comb is more dense (∝ λ)and extend to higher photon energy (∝ λ2) as the wavelength increases.

To illustrate the evolution from multiphoton to the strong-field limit,Figure 3a plots the constant intensity photoelectron distributions at dif-ferent wavelengths in scaled Up units. Plotted in this manner, the globaltransition of the xenon ionization dynamics is revealed. A comparisonof the distributions uncovers some meaningful differences: the 2 µm and4 µm spectra are similar but different from the 0.8 µm case, while the1.3 µm result shows a transitional behavior. These global features pro-vide clear evidence of an evolution: the 0.8 µm distribution shows elec-tron peaks separated by the photon energy (ATI), Rydberg structure nearzero energy, a broad resonant enhancement near 4.5Up (Hertlein et al.,1997; Nandor et al., 1999; Paulus et al., 2001a) and a slowly modulateddistribution decaying to 10Up energy. The 2 µm and 4 µm distributionshave a distinctly different structure indicative of the strong-field limit; noATI peaks and a rapid decay from zero to near 2Up energy (Simplemanlimit), followed by a plateau extending to 10Up (rescattering limit). Thisresult emphasizes the Keldysh picture: since the potential barrier remains



(a) (b)

Figure 3 (a) Comparison of photoelectron distributions of argon for differentexcitations at constant intensity plotted in energy units of Up. Excitation is at 0.8 µm(solid line), 1.3 µm (dot-dashed line), 2 µm (dashed line) and 3.6 µm (dotted line) andintensity is 0.08 PW/cm2. For this intensity Up is approximately 5 eV, 13 eV, 30 eV, and100 eV for wavelengths of 0.8 µm, 1.3 µm, 2 µm, and 3.6 µm, respectively. Alsoshown in (a) are the TDSE calculations (gray lines) at 0.8 µm (solid line) and 2 µm(dashed line), simulating the experimental conditions. The agreement betweenexperiment and theory is very good and reproduces the loss of resonant structure atthe longer wavelength. The inset in (a) is a plot of the electron yield ratio(> 2Up/ < 2Up) determined by the experiment (open circle, with error bars) and theTDSE calculations (closed circle) as a function of wavelength. The error bars reflectthe uncertainty in the intensity used in the evaluation of Up. The dashed lineillustrates a λ−4.4 dependence. (b) Electron energy distribution (scale in keV) ofhelium at 0.8 µm (Up = 56 eV and γ ∼ 0.5) and 2 µm (Up = 400 eV and γ ∼ 0.17) at1 PW/cm2. As the rescattering cross-section become less effective at high-impactenergies, the distribution tends toward the “Simpleman” atom, i.e., most electronsconfined below 2Up. All the curves are normalized for unit amplitude.

constant at fixed intensity, the dramatic evolution can only be attributedto a change of the fundamental frequency becoming slow compared to thetunneling frequency. Inspection of Table 1 shows that these observationsare consistent with the behavior expected from the strong-field metrics.

It should be noted that, as the wavelength is increased and the metricvalues tend toward the strong-field limit, the majority of photoelectronsbecome progressively confined below 2Up, consistent with the Simplemanview. This is illustrated in Figure 3b for helium ionized by a 1 PW/cm2,0.8 µm and 2 µm pulse, the former case has a maximum electron energyof 0.5 keV (10Up) while the 2 µm distribution extends to 3 keV. For 2 µmexcitation (Up = 500 eV, γ = 0.15, z = 806 and z1 = 42), the distributionshows that by 2Up, the electron yield has decreased by 7-orders of mag-nitude and in fact the 10Up cutoff is unobserved in the experiment due todetector noise. This is quantified for xenon in Figure 3a inset which showsthat the wavelength dependence of the ratio of electrons with energy >2Up



Table 1 Summary of scaling parameters for xenon atomsexposed to different wavelengths at constant intensity(80 TW/cm2).

λ (µm) Up (eV) γ z z1

0.8 5 1.1 3 0.831.3 13 0.67 14 22.0 30 0.45 44 54.0 120 0.22 390 21

(rescattered) to those <2Up (direct) decreases as ∼λ−4. One physical inter-pretation of this scaling is that electrons with energy >2Up resulted fromelastic rescattering. Thus, the number of returning electrons with a givenkinetic energy decreases by λ−2 and the wave packet’s transverse spreadscales in area also as λ−2 resulting in a total decrease in the fraction ofrescattered electrons that scale as λ−4.

4. LOW-ENERGY STRUCTURE IN PHOTOELECTRON ENERGYDISTRIBUTION INTHE STRONG-FIELD LIMIT

The understanding of strong-field ionization in the high intensity/low fre-quency limit made a giant step forward with the Simpleman ansatz whichtreats classically the motion of the photoelectron in the laser field andneglects the influence of the ionic potential. The photoelectron dynam-ics following from the 1D Newton equation with initial conditions derivedfrom tunneling (initial position at the outer turning point of the Stark poten-tial and zero initial velocity) is the mechanism underlying the Keldyshapproximation and the KFR theories. As discussed above, other ionizationfeatures like the ATI plateau, the rescattering rings (Yang et al., 1993), dou-ble ionization, etc. require elastic or inelastic scattering of the laser-drivenelectron with the ionic core. The smaller the Keldysh parameter, the betterthe ansatz is expected to work. Classical mechanics describes the motion ofan electron in the laser field but the so-called Strong-Field Approximation(SFA)3 reduces the ensemble of possible quantum paths connecting initialand final states to a small number of “trajectories” along which the quasi-classical action is stationary. The SFA is more computer-effective than thefull numerical solution of the Schrödinger equation and yields, in mostcases, excellent agreement.

The efficacy of SFA has been widely demonstrated for both ATI andhigh harmonic generation. A strong discrepancy with SFA in experimental

3In the literature, SFA refers to two different theories. One is the Keldysh approach in the velocity gauge(Reiss, 1980), the other treats the Keldysh amplitude through the saddle-point approximation (Lewensteinet al., 1994; Milosevic et al., 2006). In this chapter, SFA refers only to the latter.



Figure 4 Comparison of experimental and theoretical photoelectron energy spectra.The low-energy region of the spectra for Ar, N2, and H2 produced by 150 TW/cm2,2000 nm pulses. Inset: The entire photoelectron energy distribution showing thedirect and rescattered electrons. The intensity-averaged KFR calculation reproducesneither the rescattered part of the spectrum (inset) nor the LES (shaded region). Thefigure is reproduced, with permission, from Blaga et al. (2009).

observations (Blaga et al., 2009; Quan et al., 2009) therefore came as a majorsurprise, since the disagreement appeared to become more pronounced atlonger wavelengths despite the smaller Keldysh parameter. Briefly, thephotoelectron energy spectra reveal a low-energy peak or structure (LES)that is completely absent from the SFA calculation (see Figure 4). In oneexperiment (Blaga et al., 2009), a single broad peak (width > �ω) appears inall the spectra recorded from atoms and molecules for which γ � 0.8. In theother (Quan et al., 2009) a second feature is observed and both experimentsreport that the LES becomes more prominent the longer the wavelength.It is likely that the LES was previously undetected because the majority ofexperiments were conducted at near-visible wavelengths.

Figure 5 reveals the universality of the LES: a log–log plot of the LESwidth versus the Keldysh parameter for all investigated targets and laserwavelengths shows a striking scaling of ≈γ −2. A third important observa-tion was the absence of LES for circularly polarized light. Finally, angulardistributions of the photoelectrons (Catoire et al., 2009) provide anothersignature of the LES.

Although the LES structure was reproduced by TDSE calculations, whenavailable, no physical explanation was identified. General arguments byBlaga et al. (2009) and Faisal (2009) suggested that the LES is due forwardrescattering. Quan et al. (2009) compared their data to semi-classical cal-culation and concluded that the LES had a purely classical interpretationand stressed the role of the ionic potential (which is neglected in standardSFA). Thus, improvement to the SFA theory should include the influence



Figure 5 LES universal mapping for various atoms and wavelengths versus theKeldysh parameter γ for He, Ar, Xe together with TDSE calculations. The solid lineis a fit which suggests a simple scaling law. The figure is reproduced, withpermission, from Blaga et al. (2009).

of the Coulomb potential and more precisely account for the initial longitu-dinal and transverse momentum of the photoelectron (Delone & Krainov,1991). This is achieved by the “trajectory-based Coulomb SFA” procedure(Yan et al., 2010). The probability amplitude to a final state with momentum�p is evaluated by the saddle-point approximation leading to an expressionof the form:

˜M(SFA)

�p ∑

s

C�ps e−iS(�p,Ip,s,t), (2)

where C�ps is a prefactor, S(�p, Ip, s, t) = ∫ Tpulset�ps

(

12 [p + A(t)]2 + Ip

)

dt, thequasi-classical action and A the laser vector potential. The sum runs overthe saddle point times t�ps solutions of the saddle-point equation ∂S

∂t |�ps = 0.Yan et al. (2010) analysis sorts the saddle-point trajectories ending with thesame final momentum at the detector into four types. Calling zt the tunnelexit abscissa, pz the initial momentum along the laser polarization, p0x andpx the initial and final transverse momenta, the four types are defined by:

(1) ztpz > 0 and pxp0x > 0.(2) ztpz < 0 and pxp0x > 0.(3) ztpz < 0 and pxp0x < 0.(4) ztpz > 0 and pxp0x < 0.

The first two are the usual “short” and “long” trajectories, well known inharmonic generation theory (Antoine et al., 1996). Type (1) corresponds toelectrons moving directly toward the detector, type (2) are electrons thatfirst move away from the detector but are turned around by the laser drift



momentum A(t0). Type (3) electrons move away from the detector and havean initial transverse momentum pointing in the “wrong” direction and type(4) are electrons tunneling toward the detector with a “wrong” transversemomentum. A study of the partial contributions to the LES according tothe type of trajectory shows that the LES is due to type (3) (Figure 6).

Based on the argument that the saddle-point trajectories are close to clas-sical, the above analysis strongly suggests that a semi-classical approach(tunneling plus classical motion in the combined laser and Coulomb fields)should yield similar results. Monte Carlo calculations, already investigatedby Chen and Nam (2002) and Quan et al. (2009), were revisited by Liu et al.(2010). In the high intensity, long wavelength regime where the Keldyshparameter is small, the quiver amplitude is much larger than the lengthof the tunnel zt and the transverse motion in one period, hence the num-ber of rescattering may be large. The interplay between multiple forwardrescattering and the distortion of the trajectories by the Coulomb potentialnear the core are at the origin of the LES according to Liu et al. (2010). Intheir approach the energy spectra, for direct comparison with the experi-ment, is calculated as a function of the coordinates in the phase-space: ini-tial transverse momentum, initial laser phase (p⊥, ϕi) for hydrogen atoms.

(a) (b)

Figure 6 (a) Partial spectra due to trajectories of types (1)–(4) (see text). (b) Dominanttrajectories in the zx-plane contributing to the final momentum in the range of theLES. Note the different scales in x and z . The figure is reproduced, with permission,from Yan et al. (2010). Copyright (2010) by the American Physical Society.



The analysis confirms that the LES is directly related to the transversemomentum modified by the Coulomb potential, ruling out the chaoticmotion observed for some initial conditions (near the initial phase with therange of 1.66–1.74 rad for a field maximum at π/2) as the origin of the LES.Calculations of the simulated LES characteristics as a function of the laserparameters clearly agree with the experiment (see Figure 7).

Another Monte Carlo simulation, compared to 3D TDSE results (Lemellet al., 2012), reveals the correlation between the LES and the photoelectronangular momentum (see Figure 8). The trajectories identified as responsiblefor the LES, low energy, high angular momentum island of Figure 8, agreewith those identified by Yan et al. (2010).

Another classical approach (Kästner et al., 2012) sheds additional lighton the conclusions of the numerical investigations. A study of the finalmomentum as a function of the initial phase and transverse momentumpz(φ

′, p′ρ) shows 2D saddle-points which appear after the second laser

period. They are responsible for an infinite series of peaks (one per sub-sequent optical cycle) in the energy spectrum converging to pz = 0. Thepeaks are not related to head-on, back-scattering events responsible for theATI plateau nor to the chaotic dynamics which occur in the same regionof the phase-space. The consequence of this model is that the LES com-posed of a series of peaks which could explain the double hump struc-ture reported by Quan et al. (2009). Interestingly, the saddle-points existin 1D and can be determined analytically. The energy bunching due to the

Figure 7 Distribution of ionized electrons in the energy, angular momentum (E , L)plane observed within an angular cone of θ = 10◦ around the polarization axis. Theisland at large values of L is identified as the origin of the LES as demonstrated by theprojection onto the energy axis. Inset: electric field of laser pulse (2.2 µm,1014 W/cm2, eight cycles). The figure is reproduced, with permission, from Lemellet al. (2012). Copyright (2012) by the American Physical Society.



Figure 8 The dependence of the LES on (a) laser intensity I0 at λ = 2 µm, (b)wavelength at an intensity of 9.0 × 1013 W/cm2, and (c) wavelength at γ = 0.534. Thesolid line is a fit to all of the LES data (see Liu et al. (2010) for more information). Thefigure is reproduced, with permission, from Liu et al. (2010). Copyright (2010) by theAmerican Physical Society.

saddle-point and the corresponding 1D trajectory are shown in Figure 9and are comprises forward scattering and a soft-backward collision. Thisanalysis gives a new and general perspective on understanding the dynam-ics of a point charge in the combined fields of a laser and Coulomb potential.Above all, it predicts an infinite series of peaks for the LES and shows theuniversality of the phenomenon which does not depend on the details ofthe potential (long or short range). Future experiments are needed to testthese predictions.

Quantitative comparison with experiment is still limited, as well as defi-ciency of new measurements. However, the classical mechanics underlyingthe quantum propagation is still very enlightening. It would be interest-ing to understand quantitatively the amplitude of the LES. Contrary to theATI plateau which is less than 1% of the direct electrons, the LES appearsto be comparable in magnitude (10–50%). It is likely that this is due, inpart, to the fact that low-energy electrons are created near the peak of thelaser field cycle and, in part, to the large forward scattering cross-section.



Figure 9 Classical mechanics view of the LES: 1D calculation showing the deflectionfunction (left) and corresponding photoelectron spectrum (right) for a Gaussianpotential. The calculation predicts the LES as a series of peaks converging to p = 0.The figure is reproduced, with permission, from Kästner et al. (2012). Copyright (2012)by the American Physical Society.

However since it has been shown that the trajectories involve both forwardand backward scattering, the reason for the LES’s amplitude is not obvious.Some further answers are expected from a full quantum calculation usinga second-order KFR theory (Guo et al., submitted for publication).

5. ELECTRON MOMENTUM DISTRIBUTION ANDTIME-DEPENDENT IMAGING

The determination of the atomic positions in molecules, plays an essen-tial role in physics and chemistry. X-ray and electron diffraction methodsroutinely achieve sub-Angstrom spatial resolutions. However, in the timedomain they are limited to picoseconds and although recent developmentsin femtosecond XFELs (Emma et al., 2010) and electron beams (Eichbergeret al., 2010; Zewail & Thomas, 2009) are changing this perspective, ultra-fast filming of molecules “in action” is still beyond the reach of the sci-entists. In contrast, self-probing of the molecular structure by the electronwave packet produced by strong-field interaction with the molecule underinterrogation, can in principle reach high spatial and, as will be shownhere, temporal resolution. The initial idea (Lein et al., 2002) was first pro-posed using the high harmonics plateau and tested by the NRC group(Itatani et al., 2004). The electron wave packet extracted from the HOMOand driven back to the molecular core by the oscillating laser field, in firstapproximation as a plane wave, can either recombine, emitting a high har-monic photon or scatter. In both cases information about the molecularstructure can be retrieved from the process. For instance, from the highharmonic spectra generated by molecules aligned at different angles with



respect to the laser polarization, tomographic images of the molecule canbe reconstructed (Haessler et al., 2010; Itatani et al., 2004).

The photoelectron momentum distribution from strong-field ionizationof molecules will also convey information on the molecular structure. Thisis the principle of laser-induced electron diffraction (LIED) (Meckel et al.,2008). In the rescattering model, the tunnel ionized electron wave packetwill elastically scatter from the core after approximately a half of an opticalperiod T. Since T = λ/2πc, changing the wavelength allows a variation ofthis time interval. In other words, recording the photoelectron momentumdistributions at different wavelengths amounts to taking snapshots of themolecule at different times (Blaga et al., 2012). The spatial resolution isdetermined by the kinetic energy of the photoelectron, ∝ Iλ2. Using mid-infrared lasers this energy can be quite large and thus LIED is capable ofachieving sub-Angstrom spatial resolution with a temporal resolution ofbetter than 10 fs.

5.1 Extracting the Molecular Structure from LIED

The Quantitative rescattering theory (QRS) (Chen et al., 2009; Lin et al.,2010; Morishita et al., 2008) has now been validated by experiments onvarious atoms and molecules (Cornaggia, 2009; Ray et al., 2008) and pro-vides the framework for interpreting an LIED measurement. In the strong-field approximation, the detected electron momentum, �p, is the sum of themomentum at recollision, �pr, and the field vector potential, �Ar, at the timeof rescattering, tr.

�p = �pr + �Ar. (3)

The analysis of the momentum distribution along the circumferencedefined at constant |�pr| is equivalent to the field-free differentialcross-section measurement performed using conventional electron diffrac-tion at fixed electron energy (Figure 10). In this view, the laser generatesa flux of electrons with momentum �pr that undergo elastic scattering attime tr. In principle, alignment of the molecular sample is required. How-ever in the strong-field ionization regime, the contribution of moleculesaligned with the laser polarization is largely dominant and alignment isnot necessary. As an example, the distribution of momentum parallel andperpendicular to the laser polarization for unaligned N2 molecules ionizedby a linearly polarized, 20 fs pulse is shown in Figure 10a. The differen-tial cross-section extracted from this distribution for a constant recollisionenergy of 100 eV is shown in Figure 10b and can be compared, on the onehand, to the cross-section obtained from electron diffraction (Dubois &Rudd, 1976) and, on the other hand, to the theoretical cross-section. Thegood agreement between the three values validates the LIED approach forthe nitrogen and oxygen molecules.



Figure 10 (a) Momentum distribution from strong-field ionization. The laserpolarization is along the horizontal axis. The black circle shows the limitingmomentum for direct electrons. �A is the field vector potential. After backscattering anelectron gains additional momentum from the field, resulting in a larger detectedmomentum (magenta circle). The detected momentum is �p = �pr �A(tr). The insetdefines the rescattering angle, θ , and the momentum transfer, �q. (b) Comparison ofthe DCS extracted from the N2 distribution in a (260 TW/cm2, 50 fs, 2.0 µm pulses) for|�pr| = 2.71 a.u. to the data from CED measurements and a calculation using theindependent atom model (IAM). (c) Calculated MCFs plotted as a function ofmomentum transfer for three different N2 bond distances. The fringes shift from ≈6to 8 Å−1. The figure is reproduced, with permission, from Blaga et al. (2012).



5.1.1 Method for Retrieving the Internuclear Distance

Using mid-infrared lasers to promote core penetrating (hard) collisionsallows a number of simplifications to the LIED approach. The diffractionpattern is assumed to arise from a molecule that is treated as a set of inde-pendent atoms and during the sub-cycle interval the inter-atomic distancesare fixed. Any chemical properties that arise from molecular bonding elec-trons, e.g., HOMO, make negligible contributions. In the special case ofhomonuclear diatomic molecules, the scattering differential cross-sectionfor electrons colliding with unaligned molecules can be expressed as

dσM

d = 2σA

(

1 + sin �q · �R|�q · �R|

)

, (4)

where �q is the momentum transfer and R the inter-atomic distance. Theso-called molecular contrast factor MCF = sin(qR)/qR, is very sensitive tosmall deviations in R for large values of q. This allows good spatial res-olution for collisions with large momentum transfer. Equation (4) reflectsthe interference between the electron waves scattered by the two atomsof a diatomic molecule. The MCF, or equivalently, the position of the firstminimum in a two-slit interference pattern, as a function of q is shownin Figure 10c. The experimental MCF is extracted from the momentumdistribution measurement. The differential cross-section and the intranu-clear distance R are retrieved using a genetic algorithm. The N2 analysisat |�pr| = 2.97 a.u. (kinetic energy = 120 eV), shown in Figure 11, yields aN–N distance of 1.14 Å in good agreement with the value extracted fromthe independent atom model, conventional electron diffraction, and theknown value for neutral N2 bond length.

5.1.2 Control of the Bond Length Time Dependence

Repeating the experiment at different laser wavelength should yield differ-ent results. At any wavelength the electron wave packet is released in thecontinuum close to the peaks of the laser field (the tunneling rate dependsexponentially on the field). The rescattering takes place about half a cyclelater, a time which varies with the wavelength. Note that this is not a pump–probe scheme in the traditional sense since the experiment does not involvescanning a delay. However, changing the wavelength is equivalent to vary-ing the delay between the tunneling and rescattering times during whichthe molecule can evolve. In the case of nitrogen, measurements at 1.7 µm,2.0 µm, and 2.3 µm yield essentially the same result. This is not surprisingsince removal of a HOMO electron results in a little change in the bondcharacter and thus a small change <5 pm in the bond length between neu-tral and the ion (see Figure 11).



Figure 11 Bond change versus time for N2: measurements for three wavelength(squares: 2.3 µm, circles: 2 µm, diamonds: 1.7 µm) compared to the calculated one(red line) and the equilibrium length for the neutral and ion. The equilibrium bondlength is indicated by the solid (neutral) and dashed gray (ion) lines. (Forinterpretation of the references to color in this figure legend, the reader is referred tothe web version of this book.) The figure is reproduced, with permission, from Blagaet al. (2012).

In contrast, the oxygen anti-bonding HOMO results in a significant bondlength change over a time interval of a few femtosecond [1.21 Å (neutral) to1.10 Å (ion)]. The LIED measurement at the three wavelengths yield bondlengths of 1.10, 1.11, and 1.02 Å. The extracted O–O distances deviate by2–4 standard deviations from the equilibrium value, and agree with a bondthat has contracted during the 4–6 fs interval after tunneling [see Figure 12and Blaga et al. (2012) for more details].

This measurement is a proof-of-principle experiment and there is signif-icant prospect for improving the procedure. For example, few-cycle pulses,bichromatic fields, additional wavelengths, and improved counting ratescould enhance the temporal resolution. Femtosecond pump–probe meth-ods can be conducted using LIED to interrogate molecules undergoingconformal transformations. Application to more complex systems, alignedor oriented targets, and excited molecules (vibrational or electronic) wouldbe of great interest. Compared to conventional electron diffraction meth-ods, LIED’s ability to coherently control the returning electron wave packetcould open new paths in electron diffraction.

6. NON-SEQUENTIAL MULTIPLE IONIZATION AT LONGWAVELENGTHS

Even though 40 years have pasted since the first investigations(Suran & Zapesnochnii, 1975), the question of how several electrons are



Figure 12 Bond change versus time for O2: lines. For O2 (same conventions as inFigure 11). For the three wavelengths the measurements are consistently shorter(≈0.1 Å) than the neutrals equilibrium length and statistically meaningful. Theanalysis suggests that this is a consequence of the 4–6 fs bond evolution followingionization. The x -symbol is the bond length derived by Meckel et al. (2008). The figureis reproduced, with permission, from Blaga et al. (2012).

simultaneously ejected during an intense laser–atom interaction is still alively debate. Theoretical approaches, ranging from numerical modelingof the electron–electron correlation by a time-dependent potential (Parkeret al., 2006; Watson et al., 1997) to many-body S-Matrix theory (Becker &Faisal, 2005) and purely classical mechanics (Wang & Eberly, 2010) and ref-erences therein, show that the process cannot be understood without elec-tron correlation (Becker & Rottke, 2008) and one size may not fit all. Oftenhowever ionization experiments in a strong laser field can be understoodqualitatively by a simple three-step process: production of photoelectrons,acceleration by the laser field, and (in)elastic recollisions (Corkum, 1993;Kuchiev, 1987). The recollision channel, inelastic scattering, discussed inthis section is observed through the charge distribution spectrum recordedwhen atoms, e.g., rare gases, are exposed to intense mid-infrared pulses.

Laser-driven multiple ionization can be the result of (e, 2e) (Mosham-mer et al., 2000; Walker et al., 1994) or higher (e, ne) processes (Feuersteinet al., 2000), or excitation followed by field ionization (Yudin & Ivanov,2001). Most experiments have been performed at near-infrared (NIR) wave-lengths (λ � 1 µm). The majority of these investigations were limited tomeasurements of double NSI since depletion of the neutral ground staterestricts the maximum intensity, and thus the return energy, to the (e, 2e)inelastic channel. This limitation holds for all neutral atoms ionized by aNIR field. In addition, extraction of the inelastic NSI channel can be maskedby the presences of photon-assisted processes (Schins et al., 1994). The cen-tral point of what follows is to discuss the understanding of NSI and itsquantitative interpretation using field-free scattering cross-sections.



The experiment was conducted using intense long wavelength pulses3.6 µm and 3.2 µm and the corresponding λ2 increase in the electron’srescattering energy (Colosimo et al., 2008) can drive (e, ne) processes forn = 2–6 (Dichiara et al., 2012). At these wavelengths the field strength canbe kept low enough to avoid depletion of the neutral ground state andat the same time open a large number of (e, ne) channels. It was shownthat the results could be interpreted by known (field-free) electron impactcross-sections (Achenbach et al., 1983, 1984) and thus the link between NSIand laser-driven recollision was addressed over a broad energy range.

The main result is that laser-driven scattering is indeed comparable toconventional electron impact ionization albeit broadened by the nearly“spectrally white” returning wave packet which according to classicalmechanics, spans from 0 to 3.2Up. For comparison, conventional (e, ne)scattering experiments are performed with nearly monochromatic beams.A large Up and modest binding potential, Ip, implies tunnel ionization(small Keldysh parameter). Our mid-infrared laser provides: 25 eV > Up >120 eV, γ � 0.5, and electron wave packets with up to 400 eV of rescatter-ing energy.

The experiment is briefly described here, for details see DiChiara et al.,2011, 2012. The laser consists of a difference frequency generation (DFG) ina KTA crystal amplifier fed by two independent laser systems: a 100 fspump pulse generated by a Ti:Sapphire chirped pulse amplifier and a16 ps pulse generated by a regeneratively amplified Nd:YLF laser (Saeedet al., 1990). The idler beam has a central wavelength tunable to 3.6 µmor 3.2 µm, a 100 fs pulse duration and a peak power exceeding 1 GW. Thebeam is focused into a time-of-flight mass spectrometer. The mass resolu-tion of the spectrometer fully resolves all xenon isotopes up to the high-est charge state observed. The intensity calibration ±10%, was based onphotoelectron spectra characteristic change of slope at 2Up (Colosimo et al.,2008. Figure13 shows the measured ion yields of xenon at 3.6 µm wherethe highest charge state observed is Xe6+. When compared to the sequen-tial yields calculated using the ADK tunneling rate (Amosov et al., 1986),for charge states up to Xe6+ strong discrepancies appear while evidence ofextreme non-sequential (e, ne) processes (where over 700 MIR photons arecoupled to neutral xenon liberating six electrons bound by a net potentialof 243.6 eV) are identified. In contrast, non-sequential yields, shown withdashed lines in Figure 13, are found in excellent agreement with the data.

Our analysis is based on defining effective, energy-averaged cross-sections for laser-driven multiple ionization. The known experimentalcross-sections (Achenbach et al., 1984), when available, are used in the cal-culation (Dichiara et al., 2012). The spectral width of the returning electrondistribution are expected to be very broad, due to the distribution of ion-ization times (Micheau et al., 2009). In addition, laser experiments are nec-essarily integrated over the spatial and temporal intensity distributions of



Figure 13 Ionization yields of xenon at 3.6 µm versus intensity or Keldyshparameter. Solid lines: ADK sequential tunneling yields. Dashed lines: non-sequentialyields (see text). Table: ionization potentials for Xe+ to Xe6+ ions. The figureis reproduced, with permission, from Dichiara et al. (2012).

the focused beam. Hence, we define an effective, energy-averaged, cross-section (function of intensity I) as:

σ̃ (I) =∫

dE′ σ(E′)Wp(E′(I))∫

dE′ Wp(E′(I)), (5)

where Wp is the net return distribution and σ(E) is the experimental cross-section. The expansion of the wave packet (Delone & Krainov, 1991) istaken into account to ensure that each classical trajectory represents a wavepacket with the correct area at recollision and the calculation is summedfor individual volume elements of a Gaussian focus.

In this approach, an energetically broad wave packet is related to thelaser-driven measured branching ratio and a field-free measured electronimpact cross-section. On the other hand, the lateral expansion of the wavepacket quantifies the observed yield. The observed multiple ionizationchannels agree with our assumption of a direct (e, ne) ionization event.



Figure 14 (Color online) (a) Photoelectron energy distributions are shown for 100(red), 80 (green), 64 (blue), and 51 (black) TW/cm2. (b) Collisional excitation andionization cross-sections used in our calculation. (c, d) Equation (5) for 3.6 µm and3.2 µm, respectively compared to the data. The figure is reproduced, with permission,from Dichiara et al. (2012). Copyright (2010) by the American Physical Society.

Experimental and calculated NSI yields for Xe3+ and higher charge-statesagree within a 10% uncertainty in our intensity calibration and the over-all experimental error (see Figure 13). In addition, the excellent agreementbetween observed branching ratios and effective cross-sections for chan-nels (e, ne) (n � 3) (see Figure 14) justifies the correspondence betweenlaser-driven inelastic scattering and field-free impact ionization and con-firms the key role of the “white” recolliding wave packet. Moreover, itvalidates retaining only channels that couple higher charge states to pho-toionization of the neutral ground state. A qualitative argument justifiesthe surprising result that field-free cross-sections are sufficient to interpreta process taking place in a high-intensity laser pulse because the most ener-getic recolliding electrons return near the zero of the laser field the Coulomb



potential suffers only a small distortion at that moment. In addition, evenat the field maximum, the region of the potential near the ion ionizationenergy is not significantly distorted.

7. STRONG-FIELD X-RAY PHYSICS: A FUTURE PATH

The last decade has seen the emergence of a novel class of ultrafast sources,dubbed fourth-generation, that provide light with unprecedented pulseduration and/or energy at high photon frequency. Various approacheshave been pursued that produce different attributes for investigating newregimes of light–matter interactions. For example, high harmonic gener-ation (Brabec & Krausz, 2000; Saliéres et al., 1999), using the physics dis-cussed above, has enabled the attosecond frontier (Agostini & DiMauro,2004). However, free-electron lasers operating in the soft and hard X-rayregime are defining unique capabilities and applications that are unrivaledby previous electron-based sources. In the context of this review, X-ray free-electron lasers (XFEL) are allowing the first experimental realization aimedat addressing intense X-ray–matter interactions, thus we will limit our dis-cussion to only this aspect. In the next paragraph, the reader is referred toa number of recent reviews that provide a more comprehensive treatmentof these devices and their application.

The first step toward a user-based facility was initiated in Hamburg,Germany with the commissioning of the Free electron LASer in Hamburg(FLASH) in 2005 at DESY. Initially the FLASH FEL produced ∼100 µJ,50–10 fs pulses at fundamental photon energies of 25–95 eV while anupgraded, FLASH II, in 2007 pushed operation to 200 eV. The reader isreferred to a recent review on the machine capabilities (Tiedtke et al., 2009)and science program (Meyer et al., 2010). A parallel project at StanfordLinear Accelerator Center (SLAC) in Menlo Park, California pursued thedevelopment of a millijoule, hard X-ray (0.8–8 keV fundamental operation)FEL. In the summer of 2009, the Linac Coherent Light Source (LCLS) wascommissioned and rapidly achieved full output specifications (Emma et al.,2010), while a user scientific program began that Autumn (Bucksbaum etal., 2011).

In sharp contrast with our discussion on optical ionization wherevalence shell ionization dominates, X-rays preferentially ionize inner shellelectrons. Examination of the absorption spectrum of neon shows that lessthan 10% of the total cross-section (0.35 Mb) at the K-edge (0.8701 keV) isattributable to valence shell excitation. Consequently, photoionization cre-ates an inner shell vacancy that is an excited state which spontaneouslyrelaxes by emitting a photon or Auger decay. For neon, and other low-Zatoms, the ultrafast (2.6 fs) Auger process dominates [one valence elec-tron fills the 1s-hole and one is freed (∼0.82–0.85 keV energy)] leaving a



Figure 15 Neon charge-state yields for X-ray energies below, above and far abovethe 1s-shell binding energy, 0.87 keV. (a) Experimental charge-state distribution for0.8 keV (top), 1.05 keV (middle) and 2.0 keV (bottom). (b) Comparison of experimentalcharge-state yields, corrected for detection efficiency, with rate equation simulations.The cross-sections in the model are based on perturbation theory. The figure isreproduced, with permission, from Young et al. (2010).

doubly-charged neon ion. Ionization with unity probability by linearabsorption of a 10 fs, 0.8701 keV pulse requires a flux of 3 × 1032

photons/cm2 s or a peak intensity of 4×1016 W/cm2. This intensity is orderof magnitudes larger than that needed to drive near-threshold one-photonionization of a 2p-valence electron (21.6 eV binding energy) with unit prob-ability, in fact it even exceeds that required to tunnel ionize helium at opti-cal frequencies. However, this extraordinary soft X-ray intensity produceslittle quiver of the continuum electron due to the λ2 scaling of Up ∼ meV.

The first LCLS experiment studied the photoionization of neon atomsusing the AMO end-station, the reader is referred to a review paper (Bozek,2009) for a detailed description. In summary, the LCLS could deliver100–200 fs, millijoule pulses over a 0.8–2 keV photon energy producinga ∼3 µm focus in the AMO end-station main interaction chamber. In theexperiment (Young et al., 2010), neon ion m/q-distributions were investi-gated at different intensities, pulse duration, and photon energies. Figure15shows the (a) measured ion yields and (b) a comparison between theoryand experiment at three different photon energies. Clearly, the large flux of



X-rays is sufficient not only to saturate the neutral neon ionization, as esti-mated above, but also essentially strip off all the electrons. The three differ-ent wavelengths provide a window into the ionization sequence. At the lowphoton energy (0.8 keV), K-shell ionization cannot occur via one-photonionization, thus charge state production proceeds through a sequence ofvalence shell excitation terminating at Ne8+ since only the 8, n = 2 elec-trons can be ionized. At 2 keV photon energy all 10-electrons are removedthrough a sequence of 1s-photoionization followed by Auger decay result-ing in a propensity in even-charge state production (see bottom graphs inFigure 15). This is the first observation of sequential atomic ionization fromthe inside out, as opposed to sequential valence ionization (outside-in) bystrong optical fields. At intermediate photon energy (1.05 keV) ionizationof the low-charge states proceeds via photoionization/Auger decay untilthe K-shell binding energy (0.1096 keV for Ne6+) exceeds 1.05 keV. At thispoint, valence photoionization proceeds and terminates with the produc-tion of Ne8+. The theoretical calculations, shown in Figure 15b, are based ona rate equation model that includes only sequential single-photon absorp-tion and Auger decay processes, which produces good agreement withthe measurement. This investigation provided a preview of the richnessof new phenomena that can arise from matter interacting with an intenseX-ray pulse but equally important, it showed that the primary ionizationmechanisms were well within the framework of perturbation theory, evenat intensities of 1017 W/cm2.

The potential to study laser–matter interaction involving X-ray inten-sities near 1017 W/cm2 led naturally to the objective of observing for thefirst time nonlinear, multiphoton X-ray absorption. Both ion and electronmeasurements were performed to study the ionization of a K-shell electronby simultaneous absorption of two X-ray photons (Doumy et al., 2011). Anestimate based on the expected X-ray fluxes and a value of two-photonionization cross-section (Novikov & Hopersky, 2001, 2002) given by per-turbation theory suggested that the effect would be weak, but observable.One caveat stems from the fact that the X-ray radiation can be accom-panied by harmonics of the fundamental frequency, including a small(<1%) amount of second harmonic that was verified by ionization mea-surements using the helium atoms. At the photon energies required toavoid direct K-shell ionization of neon (<0.87 keV), the second harmonicfalls in a region where the LCLS transport mirrors reflectivity is high, so thecontamination on target is too large to be able to isolate the nonlinear con-tribution. The LCLS AMO end-station mirror reflectivity was specificallydesigned to provide a sharp cutoff above 2.0 keV. Thus, working at pho-ton energies >1.1 keV insured good discrimination against harmonic con-tent. Furthermore, the neon investigations discussed above showed that at1.05 keV photon energy that the atom sequentially absorbs a large numberof photons, leaving the ion stripped of all n = 2 valence electrons



Figure 16 Neon experimental (black bars) and simulated (white and gray bars)charge-state distributions produced by (a) 1.225 keV and (b) 1.11 keV X-ray beams.The white bars are based on the model used in Young et al. (2010) while the gray barsresult from an improved model that accounts for shake-up processes. The striped barin (b) incorporates the adjusted two-photon cross-section needed to reproduce theNe9+ production below threshold. The figure is reproduced, with permission, fromDoumy et al. (2011). Copyright (2011) by the American Physical Society.

(Young et al., 2010). However, the experiment did observe a small pro-duction of Ne9+ that was not accountable with the linear absorption modelcalculations. Based on this evidence, the experiment focused on productionof Ne9+ at two-photon energies (1.11 keV and 1.225 keV).

At the lower photon energy, the last ground state ion that can be ion-ized by a single-photon/Auger sequence is Ne6+ resulting in the creationof Ne8+ which cannot be ionized by one-photon absorption. Thus, thelowest-order channel is a two-photon process. In contrast, Ne8+ ionizationis possible with 1225 eV photons, resulting in the direct production of Ne9+.Studying the production of Ne9+ at these two-photon energies as a functionof pulse energy (intensity) should then reveal the linearity/nonlinearity ofthe process. Figure 16 shows a typical ion time-of-flight spectra obtainedat both wavelengths. The first thing to note is that the LCLS easily ionizes8-electrons from neon at both photon energies. The presence of Ne9+ issmall but obvious at 1.11 keV, while at 1.225 keV Ne9+ the larger produc-tion proceeds by allowed single-photon ionization.

Figure 17 shows the evolution of the ratio of Ne9+/Ne8+ production asa function of the measured X-ray pulse energy. In order to facilitate thecomparison, each ratio has been normalized at 0.8 mJ pulse energy. Theratios vary from 0.5% to 2.5% at 1.11 keV and 15% to 60% at 1.225 keV.The analysis clearly shows that the intensity-dependence is very differentin both cases: linear at 1.225 keV and quadratic at 1.11 keV, a signature ofsecond-order nonlinearity.



Figure 17 Normalized Ne9+/Ne8+ ratio as a function of pulse energy for 1.11 keV(filled circles) and 1.225 keV (open squares). Fits yield a quadratic response at1.11 keV (solid line) and linear behavior at 1.225 keV (dashed line). The figure isreproduced, with permission, from Doumy et al. (2011). Copyright (2011) by theAmerican Physical Society.

The study concludes, that neon atoms subjected to ultra-intense, 1 keVX-rays undergo a complex sequence of excitation, ionization, and relax-ation processes. Experiments combined with more complete theoreticalanalysis have identified and characterized the nonlinear response result-ing from two channels: (1) the dominant direct two-photon, one-electronionization and (2) a sequence of transient excited states that competes withthe Auger decay clock. They also conclude that due to near resonancecontributions the two-photon multiphoton absorption cross-section is 2–3orders magnitude higher than the earlier calculation (Novikov & Hoper-sky, 2001). However at the time of this writing, all the X-ray observationsat the LCLS are well described as expected by perturbative framework.

This leads to the obvious query: where is the strong-field non-perturbative limit for X-ray frequencies. The scaling laws developed inSection 3 may provide some appreciation of this challenge. Figure 18 scalesthe bound, z1, and continuum, z, intensity parameters into the X-ray regime.For this case, the z1 = 1 curves are plotted for both the K-shell (solid line:core) and L-shell (dash-dot: valence) electrons of neon. Not surprisinglythe difference in frequency scaling of z1 and z results in the core-state dis-tortion dominating over the continuum-state above 500 eV photon energy.This is in sharp contrast to the usual picture at optical frequencies of aquivering continuum wave packet. Within this understanding, entranceinto the strong-field regime will require intensities of nearly 1020 W/cm2,



Figure 18 A plot of the z (continua-state: dot-dash line) and z1 (neon K -shell state:solid line) intensity scaling parameter in the X-ray regime as a function of intensity.The lines are for the z-parameters equal to unity and the axis are plotted in atomicunits (a.u.) and standard units. The dash line is the perturbative limit defined as 10%of the K -shell z1-parameter. The situation is very different at X-ray frequenciescompared to optical since the distortion of the bound-state dominates and defines thebreakdown of perturbation theory. The dashed line shows the z1 parameter for theneon valence L-shell which distorts at lower intensity than the core.

which are beyond the performance of current XFELs. However, the totalbound-state contributions (core + valence) result in a dichotomy since thestrong-field limit defined by the these states are clearly at odds. There-fore, the breakdown of the perturbation theory may happen at significantlylower intensities than expected and within access to the current XFEL oper-ational parameters.

8. OUTLOOK

In the forthcoming years one can expect significant progress inmid-infrared lasers in wavelength, intensity, and time resolution.Wavelengths extending to 10 µm, pulse energy of a few mJ, and pulseduration down to one cycle are likely and will allow to further explore thescaling properties of strong-field physics. In particular it should be pos-sible to investigate relativistic interactions without the need of ultrahighintensity lasers thanks to the λ-scaling of the electrons velocities in the field(Salamin et al., 2006). On the detection side the introduction of VMI spec-trometers and other techniques associated with pump–probe scheme willallow to interrogate complex molecules under conformal transformation



with better time and angular resolution and to control the returning elec-tron wave packet. Various approaches to the attosecond domain of ioniza-tion should also reveal new aspects of electron–electron correlation (Eckleet al., 2008; Mercouris et al., 2004; Schultze et al., 2010) and shed new light innon-relativistic laser–matter interaction. At the other end of the spectrum,experiments will benefit from improvements in the control of the XFELspulse characteristics, while the multiplication of such devices in the worldwill make the access easier and provide new opportunities to explore anddevelop X-ray nonlinear optics.

ACKNOWLEDGMENTS

We wish to thank the post-doctoral researchers and graduate students, pastand present, who have tirelessly carried out the research discussed in thisreview: Cosmin Blaga, Fabrice Catoire, Phil Colosimo, Anthony DiChiara,Gilles Doumy, Anne Marie March, Chris Roedig, Kevin Schultz, EmilySistrunk, Jennifer Tate, and Kaikai Zhang. Special appreciation is expressedto our colleagues, Chii-Dong Lin, Junliang Xu, whose fruitful discussionson LIED have resulted in an exciting collaboration. Both authors acknowl-edge support from the US Department of Energy and supported by itsDivision of Chemical Science, Office of Basic Energy Sciences, the NationalScience Foundation, and the Army Research Office. LFD gratefullyacknowledges the continued support of Dr. Edward and SylviaHagenlocker.

REFERENCESAchenbach, C., Mller, A., Salzborn, A., & Becker, R. (1983). Electron-impact double ionization

of I1+ and Xeq+ (q = 1, . . . , 4) ions: Role of 4d electrons like in photoionization. PhysicalReview Letters, 50, 2070–2073.

Achenbach, C., Mller, A., Salzborn, A., & Becker, R. (1984). Single ionisation of multiplycharged xenon ions by electron impact. Journal of Physics B, 17, 1405–1425.

Agostini, P., Barjot, G., Bonnal, J. F., Mainfray, G., Manus, C., Morellec, J. (1968). Multiphotonionization of hydrogen and rare gases. IEEE Journal of Quantum Electronics, 4, 667–672.

Agostini, P., Fabre, F., Mainfray, G., Petite, G., & Rahman, N. K. (1979). Free–free transitionsfollowing six-photon ionization of xenon atoms. Physical Review Letters, 42, 1127–1130.

Agostini, P., & DiMauro, L. F. (2004). The physics of attosecond pulses. Reports on Progress inPhysics, 67, 813–855.

Amosov, M. V., Delone, N. B., & Krainov, V. P. (1986). Tunnel ionization of complex atoms andof atomic ions in an alternating electromagnetic field. Soviet Physics – Journal of Experimentaland Theoretical Physics, 64, 1191–1194.

Antoine, Ph., L’Huillier, A., & Lewenstein, M. (1996). Attosecond pulse trains using high-orderharmonics. Physical Review Letters, 77, 1234–1237.

Apolonski, A., Poppe, A., Tempea, G., Spielmann, Ch., Udem, Th., Holzwarth, R., Hnsch, T.W., & Krausz, F. (2000). Controlling the phase evolution of few-cycle light pulses. PhysicalReview Letters, 85, 740–743.



Augst, S., Strickland, D., Meyerhofer, D. D., Chin, S. L., & Eberly, J. H. (1989). Tunnelingionization of noble gases in a high-intensity laser field. Physical Review Letters, 63, 2212–2215.

Bauer, D., Milosevic, D. B., & Becker, W. (2005). Strong-field approximation for intense-laseratom processes: The choice of gauge. Physical Review A, 72, 023415.

Bayfield, J. K., & Koch, P. M. (1974). Multiphoton ionization of highly excited hydrogen atoms.Physical Review Letters, 33, 258–261.

Becker, A., & Faisal, F. H. (2005). Intense-field many-body S-matrix theory. Journal of PhysicsB, 38, R1–R56.

Becker, W., & Rottke, H. (2008). Many-electron strong-field physics. Contemporary Physics, 49,199–223.

Blaga, C. I., Catoire, F., Colosimo, P., Paulus, G. G., Muller, H. G., Agostini, P., et al. (2009).Strong-field photoionization revisited. Nature Physics, 5, 335–338.

Blaga, C. I., Xu, J., DiChiara, A. D., Sistrunk, E., Zhang, K., Agostini, P., et al. (2012). Imagingultrafast molecular dynamics with laser-induced electron diffraction. Nature, 483, 194.

Bozek, J. (2009). AMO instrumentation for the LCLS X-ray FEL. Europhysics Journal, 169, 129–132.

Brabec, T., & Krausz, F. (2000). Intense few-cycle laser fields: Frontiers of nonlinear optics.Reviews of Modern Physics, 72, 545–591.

Bucksbaum, P. H., Coffee, R., & Berrah, N. (2011). In Arimondo, E., Berman, P. R., & Lin, C. C.(Eds.), Advances in atomic molecular and optical physics (Vol. 60). San Diego: Academic Press.

Bystrova, T. B., Voronov, G. S., Delaone, G. A., & Delone, N. B. (1967). Multiphoton ionizationof xenon and krypton atoms at wavelength λ = 1.06 µm. Journal of Experimental andTheoretical Physics, 5, 223–225.

Caillat, J., Maquet, A., Haessler, S., Fabre, B., Ruchon, T., Salires, P. (2011). Attosecond resolvedelectron release in two-color near-threshold photoionization of N2. Physical Review Letters,106, 093002.

Catoire, F. et al. (2009). Midinfrared strong field ionization angular distributions. Laser Physics,19, 1574–1580.

Chen, Z. J., Le, A.-T., Morishita, T., & Lin, C. D. (2009). Quantitative rescattering theory forlaser-induced high-energy plateau photoelectron spectra. Physical Review A, 79, 033409.

Chen, J., & Nam, C. H. (2002). Ion momentum distributions for He single and double ionizationin strong laser fields. Physical Review A, 66, 053415.

Chin, S. L. (2011). From MPI to tunnel ionization. In S. H. Lin, A. A. Villayes & Y. Fujimura(Eds.),. Advances in multi-photon processes and spectroscopy (Vol. 20). London: World Scien-tific.

Chin, S. L., Yergeau, F., & Lavigne, P. (1985). Tunnel ionisation of Xe in an ultra-intense CO2laser field (1014 W cm−2) with multiple charge creation. Journal of Physics B, 18, L213–L215.

Colosimo, P., Doumy, G., Blaga, C. I., Wheeler, J., Hauri, C., Catoire, F., et al. (2008). Scalingstrong-field interactions towards the classical limit. Nature Physics, 4, 386–389.

Corkum, P. B. (1993). Plasma perspective on strong-field multiphoton ionization. PhysicalReview Letters, 71, 1994–1997.

Corkum, P. B., Burnett, N. H., & Brunel, F. (1989). Above-threshold ionization in the long-wavelength limit. Physical Review Letters, 62, 1259–1262.

Cornaggia, C. (2009). Electron ion elastic scattering in molecules probed by laser-inducedionization. Journal of Physics B, 42, 161002.

Damon, E. K., & Tomlinson, R. G. (1963). Observation of ionization of gases by a ruby laser.Applied Optics, 2, 546–547.

de Boer, M. P., Hoogenraad, J. H., Vrijen, R. B., Constantinescu, R. C., Noordam, L. D., Muller,H. G. (1994). Adiabatic stabilization against photoionization: An experimental study. Phys-ical Review A, 50, 4085–4098.

Delone, N. B., & Krainov, V. P. (1991). Energy and angular electron spectra for the tunnelionization of atoms by strong low-frequency radiation. Journal of the Optical Society ofAmerica B – Optical Physics, 8, 1207–1211.

DiChiara, A. D., Ghimire, S., Blaga, C. I., Sistrunk, E., Power, E. P., March, A. M., et al. (2011).Scaling of high-order harmonic generation in the long wavelength limit of a strong laserfield. IEEE Journal of Selected Topics in Quantum Electronics, 99, 419–433.



Dichiara, A. D., Sistrunk, E., Blaga, C. I., Szafruga, U. B., Agostini, P., DiMauro, L. F. (2012).Inelastic scattering of broadband electron wave packets driven by an intense midinfraredlaser field. Physical Review Letters, 108, 033002.

DiMauro, L. F., & Agostini, P. (1995). In Bederson, B., & Walther, H. (Eds.), Advances in atomicmolecular and optical physics (Vol. 35). San Diego: Academic Press.

Doumy, G., Roedig, C., Son, S. K., Blaga, C. I., DiChiara, A. D., Santra, R., et al. (2011). Nonlinearatomic response to intense ultrashort X-rays. Physical Review Letters, 106, 083002.

Dubois, R. D., & Rudd, M. E. (1976). Differential cross sections for elastic scattering of electronsfrom argon, neon, nitrogen and carbon monoxide. Journal of Physics B, 9, 2657–2667.

Düsterer, S., Radcliffe, P., Bostedt, C., Bozek, J., Cavalieri, A. L., Coffee, R., et al. (2011).Femtosecond X-ray pulse length characterization at the Linac coherent light source free-electron laser. New Journal of Physics, 13, 093024.

Eckle, P., Pfeiffer, A. N., Cirelli, C., Staudte, A., Drner, R., Muller, H. G., et al. (2008). Attosecondionization tunneling delay time measurements in helium. Science, 322, 1525–1529.

Eichberger, M., Schfer, H., Krumova, M., Beyer, M., Demsar, J., Berger, H., et al. (2010). Snap-shots of cooperative atomic motions in the optical suppression of charge density waves.Nature, 468, 799–802.

Emma, P., Akre, R., Arthur, J., Bionta, R., Bostedt, C., Bozek, J., et al. (2010). First lasing andoperation of an angstrom-wavelength free-electron laser. Nature Photonics, 4, 641–644.

Faisal, F. H. (1973). Multiple absorption of laser photons by atoms. Journal of Physics B, 6,L89–L92.

Faisal, F. H. (2009). Ionization surprise. Nature Physics, 5, 319–320.Ferray, M., L’Huillier, A., Li, X. F., Lompre, L. A., Mainfray, G., & Manus, C. (1988). Multiple-

harmonic conversion of 1064 nm radiation in rare gases. Journal of Physics B, 21, L31–L35.Fittinghoff, D. N., Bolton, P. P., Chang, N., & Kulander, K. C. (1992). Observation of non-

sequential double ionization of helium with optical tunneling. Physical Review Letter, 69,2642–2645.

Feuerstein, B., Moshammer, R., & Ullrich, J. (2000). Nonsequential multiple ionization inintense laser pulses: Interpretation of ion momentum distributions within the classical‘rescattering’ model. Journal of Physics B, 33, L823–L830.

Freeman, R. R., Bucksbaum, P. H., Milchberg, H., Darack, S., Schumacher, D., & Geusic, M.E. (1987). Above-threshold ionization with sub-picosecond pulses. Physical Review Letters,59, 1092–1095.

Gaarde, M. B., Tate, J. L., & Schafer, K. J. (2008). Macroscopic aspects of attosecond pulsegeneration. Journal of Physics B, 41, 132001.

Gallagher, T. F. (1988). Above-threshold ionization in low-frequency limit. Physical ReviewLetters, 61, 2304–2307.

Granneman, E. H. A., & Van der Wiel, M. J. (1975). Two-photon ionization measurements onatomic caesium by means of an argon ion laser. Journal of Physics B, 8, 1617–1626.

Guo, L., Han S. S., Liu X., Cheng Y., Xu Z. Z., Fan, J., et al. (submitted for publication). Scalingof the low-energy structure in above-threshold ionization in the tunneling regime: Theoryand experiment.

Haessler, S., Caillat, J., Boutu, W., Giovanetti-Teixeira, C., Ruchon, T., Auguste, T., et al. (2010).Attosecond imaging of molecular electronic wave packets. Nature Physics, 6, 200–206.

Hauri, C. P., Lopez-Martens, R. B., Blaga, C. I., Schultz, K. D., Cryan, J., Chirla, R., et al. (2007).Intense self-compressed, self-phase-stabilized few-cycle pulses at 2 µm from an opticalfilament. Optics Letters, 32, 868–870.

Held, B., Mainfray, G., Manus, C., Morellec, J., & Sanchez, F. (1973). Resonant multiphotonionization of a caesium atomic beam by a tunable wavelength q-switched neodymiumglass laser. Physical Review Letters, 30, 423–426.

Hertlein, M. P., Bucksbaum, P. H., & Muller, H. G. (1997). Evidence for resonant effects inhigh-order ATI spectra. Journal of Physics B, 30, L197–L205.

Itatani, J., Levesque, J., Zeidler, D., Niikura, H., Ppin, H., Kieffer, J. C., et al. (2004). Tomo-graphic imaging of molecular orbitals. Nature, 432, 867–870.

Karule, E. M. (1975). In Atomic processes report of the Latvian Academy of Science (pp. 5–24). Paperno. Y�K539.188, Riga.

Kästner, A., Saalmann, U., & Rost, J. M. (2012). Electron-energy bunching in laser-driven softrecollisions. Physical Review Letters, 108, 033201.



Keldysh, V. S. (1965). Ionization in the field of a strong electromagnetic wave. Soviet Physics –Journal of Experimental and Theoretical Physics, 20, 1307–1314.

Krainov, V. P., Reiss, H. H., & Smirnov, B. M. (1997). Radiative processes in atomic physics. NewYork: John Wiley.

Krause, J. L., Schafer, K. J., & Kulander, K. C. (1992). Calculation of photoemission from atomssubject to intense laser fields. Physical Review A, 45, 4998–5010.

Kruit, P., Kimman, J., Muller, H. G., & van der Wiel, M. J. (1983). Electron spectra from mul-tiphoton ionization of xenon at 1064, 532 and 355 nm. Physical Review A, 28, 248–255.

Kuchiev, M. Yu. (1987). Atomic antenna. Journal of Experimental and Theoretical Physics Letters,45, 404–406.

Kulander, K. C. (1987). Hartree–Fock theory of multiphoton ionization: Helium. PhysicalReview A, 36, 2726–2738.

Lambropoulos, P. (1985). Mechanism for multiple ionization of atoms by strong pulsed lasers.Physical Review Letters, 55, 2141–2144.

Lein, M., Hay, N., Velotta, R., Marangos, J. P., & Knight, P. L. (2002). Interference effects inhigh-order harmonic generation with molecules. Physical Review A, 66, 023805.

Lemell, C., Dimitriou K. I., Tong, X.-M., Nagele, S., Kartashov, D. V., Burgdrfer, J., et al. (2012).Low-energy peak structure in strong-field ionization by mid-infrared laser-pulses: Two-dimensional focusing by the atomic potential. Physical Review A, 85, 011403.

Lewenstein, M., Balcou, Ph., Ivanov, M. Y., L’Huillier, A., & Corkum, P. B. (1994). Theory ofhigh harmonic generation by low frequency laser fields. Physical Review A, 49, 2117–2132.

L’Huillier, A., Lompre, L. A., Mainfray, G., & Manus, C. (1983). Multiply charged ions inducedby multiphoton ionization in rare gases at 0.53 µm. Physical Review A, 27, 2503–2512.

Lin, C. D., Le, A.-T., Chen, Z., Morishita, T., & Lucchese, R. (2010). Strong-field rescatteringphysics: Self-imaging of a molecule by its own electrons. Journal of Physics B, 43, 122001.

Liu, C. P., & Hatsagortsyan, K. Z. (2010). Origin of unexpected low energy structure in pho-toelectron spectra induced by midinfrared strong laser fields. Physical Review Letters, 105,113003.

Lompre, L. A., Mainfray, G., Manus, C., Repoux, S., & Thebault, J. (1976). Multiphoton ion-ization of rare gases at very high laser intensity (1015 W/cm2) by a 30 psec Laser Pulse at1.06 µm. Physical Review Letters, 36, 949–952.

Luk, T. S., Pummer, H., Boyer, K., Shahidi, M., Egger, H., Rhodes, C. K. (1983). Anomalouscollision-free multiple ionization of atoms with intense picosecond ultraviolet radiation.Physical Review Letters, 51, 110–113.

Maiman, T. H. (1960). Stimulated optical radiation in ruby. Nature, 187, 493–494.Mairesse, Y., de Bohan, A., Frasinski, L. J., Merdji, H., Dinu, L. C., Monchicourt, P., & Breger,

P., et al. (2003). Attosecond synchronization of high-harmonic soft X-rays. Science, 302,1540–1543.

McPherson, A., Gibson, G., Jara, H., Johann, U., Luk, T. S., McIntyre, I. A., et al. (1987). Studiesof multiphoton production of vacuum-ultraviolet radiation in the rare gases. Journal ofOptical Society of America B, 4, 595–601.

Meckel, M., Comtois, D., Zeidler, D., Staudte, A., Pavii, D., Bandulet, H. C., et al. (2008).Laser-induced electron tunneling and diffraction. Science, 320, 1478–1482.

Mercouris, Th., Komninos, Y., & Nicolaides, C. A. (2004). Theory and computation of theattosecond dynamics of pairs of electrons excited by high-frequency short light pulses.Physical Review A, 69, 032502.

Meyer, M., Costello, J. T., Dusterer, S., Li, W. B., Radcliffe, P., and other references in the 2010Topical Special Issue. Intense X-ray science: The first 5 years of FLASH. Journal of PhysicsB, 43(19), 194006.

Micheau, S., Chen, Z. J., Le, A.-T., & Lin, C. D. (2009). Quantitative rescattering theory fornonsequential double ionization of atoms by intense laser pulses. Physical Review A, 79,013417.

Milosevic, D. B., Paulus, G. G., Bauer, D., & Becker, W. (2006). Above-threshold ionization byfew-cycle pulses. Journal of Physics B, 39, R203–R262.

Morellec, J., Normand, D., Mainfray, G., & Manus, C. (1980). First experimental evidenceof destructive interference effects in two-photon ionization of Cs atoms. Physical ReviewLetters, 44, 1394–1397.



Morellec, J., Normand, D., & Petite, G. (1976). Resonances shifts in multiphoton ionization ofCs atoms Phys. Physical Review A, 14, 300–312.

Morishita, T., Le, A.- T., Chen, Z., & Lin, C. D. (2008). Accurate retrieval of structural infor-mation from laser-induced photoelectron and high-order harmonic spectra by few-cyclelaser pulses. Physical Review Letters, 100, 013903.

Moshammer, R., Feuerstein, B., Schmitt, W., Dorn, A., Schrter, C. D., Ullrich, J., et al. (2000).Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse. PhysicalReview Letters, 84, 447–450.

Muller, H. G. (1999). An efficient propagation scheme of the resolution of the Schrö dingerequation in the velocity gauge. Laser Physics, 9, 138–151.

Nandor, M. J., Walker, M. A., VanWoerkom, L. D., & Muller, H. G. (1999). Detailed com-parison of above-threshold-ionization spectra from accurate numerical integrations andhigh-resolution measurements. Physical Review A, 60, R1771–R1774.

Nisoli, M., Sansone, G., Stagira, S., De Silvestri, S., Vozzi, C., Pascolini, M., et al. (2003). Effectsof carrier-envelope phase differences of few-optical-cycle light pulses in single-shot high-order-harmonic spectra. Physical Review Letters, 91, 213905.

Novikov, S. A., & Hopersky, A. N. (2001). Two-photon excitation-ionization of the 1s shell ofhighly charged positive atomic ions. Journal of Physics B, 34, 4857–4863.

Novikov, S. A., & Hopersky, A. N. (2002). Two-photon knocking-out of two electrons of the1s2-shell of the neon atom. Journal of Physics B, 35, L339–L343.

Parker, J. S., Doherty, B. J. S., Taylor, K. T., Schultz, K. D., Blaga, C. I., & DiMauro, L. F.(2006). High-energy cutoff in the spectrum of strong-field nonsequential double ionization.Physical Review Letters, 96, 133001.

Paul, P. M., Toma, E. S., Breger, P., Mullot, G., Auge, F., Balcou, Ph., et al. (2001). Observationof a train of attosecond pulses from high harmonic generation. Science, 292, 1689–1692.

Paulus, G. G., Grasbon, F., Walther, H., Kopold, R., & Becker, W. (2001a). Channel-closing-induced resonances in the above-threshold ionization plateau. Physical Review A, 64,021401.

Paulus, G. G., Grasbon, F., Walther, H., Villoresi, P., Nisoli, M., Stagira, S., et al. (2001b).Absolute-phase phenomena in photoionization with few-cycle laser pulses. Nature, 414,182–184.

Paulus, G. G., Nicklich, W., Xu, H., Lambropoulos, P., Walther, H., et al. (1994). Plateau inabove threshold ionization spectra. Physical Review Letters, 72, 2851–2854.

Perelomov, A. M., Popov, V. S., & Terente’ev, M. V. (1965). Ionization of atoms in an alternatingfield. Journal of Experimental and Theoretical Physics, 23, 924–934.

Pont, M., & Gavrila, M. (1990). Stabilization of atomic hydrogen in superintense, high-frequency laser fields of circular polarization. Physical Review Letters, 65, 2362–2365.

Popov, V. S. (2004). Tunnel and multiphoton ionization of atoms and ions in a strong laserfield (Keldysh theory). Physics – Uspekii, 47, 855–885.

Potvliege, R. M., & Shakeshaft, R. (1988). Time-independent theory of multiphoton ionizationof an atom by an intense field. Physical Review A, 38, 4597–4621.

Quan, W., Lin, Z., Wu, M., Kang, H., Liu, H., Liu, X., et al. (2009). Classical aspects in above-threshold ionization with a midinfrared strong laser field. Physical Review Letters, 103,093001.

Ray, D., Ulrich, B., Bocharova, I., Maharjan, C., Ranitovic, P., Gramkow, B., et al. (2008). Large-angle electron diffraction structure in laser-induced rescattering from rare gases. PhysicalReview Letters, 100, 143002.

Reiss, H. R. (1980). Effect of an intense electromagnetic-field on a weakly bound system.Physical Review A, 22, 1786–1813.

Reiss, H. R. (1992). Theoretical methods in quantum optics—S-matrix and Keldysh techniquesfor strong-field processes. Progress in Quantum Electronics, 16, 1–71.

Reiss, H. R. (2008). Limits on tunneling theories of strong-field ionization. Physical ReviewLetters, 101, 043002.

Reiss, H. R. (2009). Special analytical properties of ultrastrong coherent fields. European Phys-ical Journal D, 55, 365–374.

Rottke, H., Wolff-Rottke, B., Feldmann, D., Welge, K. H., Dorr, M., Potvliege, R. M., et al.(1994). Atomic hydrogen in a strong optical field. Physical Review A, 49, 4837–4851.



Saeed, M., Kim, D. W., & DiMauro, L. F. (1990). Optimization and characterization of a high-repetition rate, high-powered Nd:YLF regenerative amplifier. Applied Optics, 29, 1752–1757.

Salamin, Y. I., Hu, S. X., Hatsagortsyan, K. Z., & Keitel, C. H. (2006). Relativistic high-powerlaser–matter interactions. Physics Reports – Review Section of Physics Letters, 427, 41–155.

Saliéres, P., Carre, B., Le Déeroff , L., Grasbon, F., Paulus, G. G., Walther, H., et al. (2001).Feynman’s path-integral approach for intense-laser/atom interactions. Science, 292, 402–405.

Saliéres, P., L’Huillier, A., Antoine, P., & Lewenstein, M. (1999). In Bederson, B., & Walther,H. (Eds.), Advances in atomic molecular and optical physics (Vol. 41). San Diego: AcademicPress.

Schafer, K. J., Yang, B., DiMauro, L. F., & Kulander, K. C. (1993). Above threshold ionizationbeyond the high harmonic cutoff. Physical Review Letters, 70, 1599–1602.

Schins, J., Breger, P., Agostini, P., Constantinescu, R. C., & Muller, H. G., et al. (1994). Obser-vation of laser-assisted Auger decay in argon. Physical Review Letters, 73, 2180–2183.

Schmidt, B. E., Bjot, P., Giguere, M., Shiner, A. D., Trallero-Herrero, C., et al. (2010). Compres-sion of 1.8 µm laser pulses to sub two optical cycles with bulk material. Applied PhysicsLetters, 96, 121109.

Schultz, K. D., Blaga, C. I., Chirla, R., Colosimo, P., Cryan, J., March, A. M., et al. (2007).Strong-field physics with mid-infrared lasers. Journal of Modern Optics, 54, 1075–1085.

Schultze, M., Fiess, M., Karpowicz, N., Gagnon, J., Korbman, M., Hofstetter, M., et al. (2010).Delay in photoemission. Science, 328, 1658–1662.

Sheehy, B., Lafon, R., Widmer, M., Walker, B., DiMauro, L. F., Agostini, P., et al. (1998). Singleand multiple electron dynamics in the strong field tunneling limit. Physical Review A, 58,3942–3952.

Sheehy, B., Martin, J. D. D., DiMauro, L. F., Agostini, P., Schafer, K., Gaarde, M., & Kulander,K. C. (1999). High harmonic generation at long wavelengths. Physical Review Letters, 83,5270–5273.

Strickland, D., & Mourou, G. (1985). Compression of amplified chirped optical pulses. OpticsCommunications, 56, 219.

Suran, V. V., & Zapesnochnii, I. P. (1975). Observation of Sr2+ in multiple-photon ionizationof strontium. Soviet Technical Physics Letters, 1, 420–425.

Tate, J., Auguste, T., Muller, H. G., Salieres, P., Agostini, P., & DiMauro, L. F. (2007). Thescaling of wave packet dynamics in an intense mid-infrared field. Physical Review Letters,98, 013901.

Tiedtke, K., Azima, A., von Bargen, N., Bittner, L., Bonfigt, S., & Dusterer, S., et al. (2009). Thesoft X-ray free-electron laser FLASH at DESY: Beamlines, diagnostics and end-stations.New Journal of Physics, 11, 023029.

Ullrich, J., Moshammer, R., Dorn, A., Dorner, R., Schmidt, L. Ph. H., & Schmidt-Bocking, H.(2003). Recoil-ion and electron momentum spectroscopy: Reaction-microscopes. Reportson Progress in Physics, 66, 1463–1545.

van Linden van den Heuvell, H. B., & Muller, H. G. (1987). In Knight, P. L., & Smith, S. J.(Eds.), Multiphoton processes. Cambridge: Cambridge University Press.

Walker, B., Sheehy, B., DiMauro, L. F., Agostini, P., Schafer, K. J., & Kulander, K. C. (1994).Precision measurement of strong field double ionization of helium. Physical Review Letters,73, 1227–1230.

Walker, B., Sheehy, B., Kulander, K. C., & DiMauro, L. F. (1996). Elastic rescattering in thestrong field tunneling limit. Physical Review Letters, 77, 5031–5034.

Wang, X., & Eberly, J. H. (2010). Elliptical polarization and probability of double ionization.Physical Review Letters, 105, 083001.

Watson, J. B., Sanpera, A., Lappas, D. G., Knight, P. L., & Burnett, K. (1997). Nonsequentialdouble ionization of helium. Physical Review Letters, 78, 1884–1887.

Yan, T.-M., Popruzhenko, S. V., Vrakking, M. J. J., & Bauer, D. (2010). Low-energy structuresin strong field ionization revealed by quantum orbits. Physical Review Letters, 105, 253002.

Yang, B., Schafer, K. J., Walker, B., Kulander, K. C., Agostini, P., & DiMauro, L. F. (1993).Intensity-dependent scattering rings in high order above-threshold ionization. PhysicalReview Letters, 71, 3770–3773.



Young, L., Kanter, E. P., Krassig, B., Li, Y., March, A. M., Pratt, S. T., et al. (2010). Femtosecondelectronic response of atoms to ultraintense X-rays. Nature, 466, 56–61.

Yudin, G. L., & Ivanov, M. Y. (2001). Physics of correlated double ionization of atoms in intenselaser fields: Quasistatic tunneling limit. Physical Review A, 63, 033404.

Zewail, A. H., & Thomas, J. M. (2009). 4D electron microscopy: Imaging in space and time. London:Imperial College Press.


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