Atomic and Molecular Ionization Dynamics in Strong Laser Fields: From Optical to X-rays

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Atomic and Molecular Ionization Dynamics in Strong Laser Fields: From Optical to X-rays Pierre Agostini and Louis F. DiMauro Department of Physics, The Ohio State University, 191 W Woodruff Ave, Columbus, OH 43210, USA

Contents

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Introduction The First 30 Years of Multiphoton Physics (1963−1993) 2.1 The Genesis of a Field: The Early Days 2.2 Resonant Multiphoton Ionization (MPI) 2.3 Coherence 2.4 Non-Resonant MPI 2.5 Above-Threshold Ionization (ATI): Doorway into the Modern Era 2.6 Non-Perturbative ATI 2.7 Rydberg Resonances and the Role of Atomic Structure 2.8 Multiple Ionization, Anne’s Knee, and the Lambropoulos Curse 2.9 Keldysh Tunneling: Different Mode of Ionization 2.10 Long Wavelength Ionization and the Classical View 2.11 High Harmonics, High-Order ATI and Nonsequential Ionization Lead to the Rescattering/Three-Step Model 2.12 Adiabatic Stabilization Wavelength Scaling of Strong-Field Atomic Physics 3.1 Fundamental Metrics in an Intense Laser–Atom Interaction 3.2 Ionization Experiments in the Strong-Field Limit

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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.

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Low-Energy Structure in Photoelectron Energy Distribution in the Strong-Field Limit 5. Electron Momentum Distribution and Time-Dependent Imaging 5.1 Extracting the Molecular Structure from LIED 6. Non-Sequential Multiple Ionization at Long Wavelengths 7. Strong-Field X-Ray Physics: A Future Path 8. Outlook Acknowledgments References

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We review the progress made over the past 15 years in understanding and exploiting the physical scaling of strong-field atomic processes. These advances were enabled by theory and experiment, but this chapter focuses on the understanding gained by two significant advancements in technology: intense, ultrafast mid-infrared, and X-ray sources. Examining investigations that utilize mid-infrared lasers, we discuss the consequences of a several new phenomena: wavelength scaling of photoelectron energy, a universal low-energy structure in the photoelectron distribution, the ultrafast imaging of the photoelectrons momentum distributions, and multiple ionization. In October 2009, the world’s first X-ray free-electron laser [Linac Coherent Light Source (LCLS)] became operational at SLAC National Laboratory in Menlo Park, California. The LCLS peak power exceeds the performance of previous X-ray sources by 5–6 orders of magnitude. In this review, we discuss some of the initial findings and the implications for future investigations of strong-field nonlinear processes at X-ray frequencies.

1. INTRODUCTION Although the invention of the laser dates back for half a century, intense few-cycle pulses, femtosecond mid-infrared lasers, carrier-to-envelope phase control, XUV, and X-ray free-electron lasers (XFEL), attosecond pulses, in trains or isolated, are the product of the past 15 years. Strong-field ionization of atoms and molecules, the topic of this chapter, is photoionization in a regime where the concept of multiphoton transitions, which could be described by traditional perturbation theory, looses definition since hundreds or thousands of photons are absorbed and emitted in the process. In the last decade or so, the study of photoionization has explored new ranges of parameters, studied novel targets, and benefited from new detection techniques. One of the familiar metrics in strong-field interaction

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(Reiss, 1992) is the ratio of the ponderomotive energy1 to the photon energy z ≡ Up /ω proportional to ω−3 or λ3 . The value of z gives an idea of the number of photons exchanged between the atom and the field. Intense, femtosecond mid-infrared laser (>1 µm) technology became available by the 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 nonlinear 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, been driven by experiments while the theory has followed, often painstakingly. For example, the Schrödinger equation even for a single electron in the presence of a strong electromagnetic field and a Coulomb potential cannot be solved exactly. In this context, low intensity, short wavelength is the regime of perturbation theory. Stark-induced Rydberg resonances, observed and recognized in 1987 (Freeman et al., 1987), involve Stark shifts of the order of the photon energy (z ≈ 1) and theoretical treatment requires methods such as the Floquet theory (Rottke et al., 1994). However, as soon as Stark shifts become large compared to the photon energy, perturbation theory and its extensions become ineffective. Thus, the solution of the Schrödinger equation will depend on other approximations. One such approximation was proposed by Keldysh (1965), and later by Faisal (1973) and Reiss (1980, 1992) [for a detailed comparison between the three approaches, usually dubbed the “KFR” theory see Reiss (1992); for a recent review of the Keldysh approximation see Popov (2004)]. Using the Keldysh approximation, in the high intensity, long wavelength limit, the total ionization rate is shown to behave like the rate of dc-tunneling (Keldysh, 1965). The very simple tunneling expression, and its success (Amosov et al., 1986; Chin, 2011; Perelomov et al., 1965), explains its popularity but the concept of tunneling itself has a limited range of application (Reiss, 2008). Our previous review, 15 years ago (DiMauro & Agostini, 1995), ended with discussions of the Above-Threshold Ionization (ATI) plateau and the non-sequential double ionization. Lasers generating ultrafast light pulses came of age in the following years (Brabec & Krausz, 2000) and the new millennium saw the emergence of new tools for strong-field investigations. Among them, COLTRIMS (or similar) devices (Moshammer et al., 2000), few-cycle pulses (Brabec & Krausz, 2000), and carrier-to-envelope phase 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 1 U = E2 /4ω2 in atomic units is the cycle-average kinetic energy of a free electron in an electromagnetic p field with amplitude E and frequency ω.

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phase effects (Bauer et al., 2005), long-wavelength scaling of ATI (Colosimo et 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 distributions, nonlinear optics in the XUV and the X-ray domains (Meyer et al., 2010), non-sequential multiple ionization, attosecond measurements of the ionization time delay (Schultze et al., 2010), and others. In this chapter, we focus our discussion on wavelength scaling and its implications in strongfield ionization and attosecond generation, the LES, the high-order ATI photoelectron momentum distribution, time-dependent molecular imaging, non-sequential multiple ionization, and finally, on X-ray nonlinear ionization using XFEL. It has occurred to the authors that it would perhaps be useful to include a brief, and undoubtedly incomplete, summary of the first 30 years of this field which, in spite of the recent popularity of the Keldysh “tunnel” ionization that followed the pioneering CO2 laser experiments (Chin et al., 1985; Corkum et al., 1989), the history may not be so familiar to newcomers of this field. Consequently, the outline of the chapter is as follows: the first section following this introduction will be a historical account of watershed contributions. Section 3 establishes the various scaling parameters and the general concept of Keldysh theory and introduces experiments illustrating the wavelength scaling at long wavelengths. Section 4 describes the experimental evidence of the LES and various theoretical interpretations. Section 5 discusses the photoelectron momentum distributions in strong-field ionization, with special application to the imaging of time-dependent molecular structure. Section 6 reviews recent results in non-sequential multiple ionization and, particularly the quantitative analysis based on field-free (e, ne) cross-sections, and Section 7 summarizes experiments conducted using LCLS XFEL at SLAC which demonstrate, among other spectacular results, the complete stripping of electrons or the first evidence of a nonlinear transitions in atoms with hard X-rays. Section 8 will conclude with a brief outlook.

2. THE FIRST 30 YEARS OF MULTIPHOTON PHYSICS (1963−1993) What is now called strong-field physics, slowly emerged with the advent of the pulsed laser in the early 1960s and since that time, has been strongly coupled to subsequent advances in the technology. What follows is the author’s account of experimental milestones during the period of 1963–1993. For the period preceding the laser, the reader is referred to

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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 commercialized in the US following Maiman’s invention (Maiman, 1960). The available intensities were far from the intensities required to observe QED effects, a theorist’s dream but were sufficient to conduct more modest investigations into razor blade piercing or air breakdown by early experimentalist, as a prelude to more scientific investigations. Perhaps the very first attempt at a measurement of photoionization of a rare gas using ruby laser was the work performed at The Ohio State University by Damon and Tomlinson (1963). It is the first cited reference in the historical Keldysh paper and, interestingly enough, the experimental data is fitted by a straight line in semi-log coordinates, which might have been the inspiration for Keldysh’s tunneling theory. The interest in this problem apparently faded in the US while it was actively investigated in Moscow by N. Delone and his group (Bystrova et al., 1967) at the Lebedev Institute, home of the recent Nobel prizes (N. Basov and A. Prokhorov and of Keldysh himself). In France, at the CEA Saclay, a group directed by F. Bonnal in the department of C. Manus, started some experimental work in 1967 along with the Lebedev group. Soon G. Mainfray took the direction of this group and the first results began to appear (Agostini et al., 1968). The group included M. Trahin and Y. Gontier strong advocates of perturbation theory and the hydrogen atom. Perturbation theory soon became the most popular framework. The only experimental quantity that could be compared to the theory in these early days was the “slope” of log–log plots of the total ion signal versus laser energy, which, according to lowest-order perturbation theory equals the minimum number of photons n0 required to reach the ionization threshold. Unfortunately, the experimental slopes measured in Moscow and Paris were significantly lower than expected. The scenario that was proposed to explain this unsettling finding was based on rate equations assuming an intermediate resonance and a bottleneck which was determining 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 commotion as it showed slopes larger than n0 . It took sometime to realize that the role of the Stark shift and its link with the observed slope (Morellec et al., 1976).

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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 sources were incapable of inducing MPI. Moreover, the connection between the photon statistics and the MPI signal was not clear. A Saclay Ph.D. student in Mainfray’s group, F. Sanchez, who did not believe in photons, was, we believe, the first one to really understand the meaning of . He appreciated that this factor could be controlled by the number of laser cavity modes and hence, the intensity fluctuations of the laser field. He built a very sophisticated neodymium oscillator that based on an adjustable number of modes, thus clearing up the problem (Held et al., 1973), although at the same time removing the mystery and glamor of the subject.

2.4 Non-Resonant MPI Perturbation theory was predicting a deep minima in the MPI cross-sections due to destructive interference between quantum paths. One experiment stirred some controversy (and excitement) in the community with a result that seemed to strongly contradict this prediction (Granneman & Van der Wiel, 1975). It took no less than 5 years to understand 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 experimental approach: a differential measurement of the electron’s energy instead of the total ion/electron yield. The first analyzer was a retarding voltage spectrometer with not much resolution, albeit sufficient to resolve 2.34 eV photon energy from a doubled Nd:YAG. The observation was clear and provided the first evidence of multiphoton ionization with one photon more than required by energy conservation, i.e., included a free–free transition (Agostini et al., 1979). The competition, the FOM group in Amsterdam, had designed and built a magnetic bottle electron spectrometer (which was then commercialized, popularized by many groups, and one variant is being used today in the AMO chamber at the LCLS). In spite of this, they did 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 calibration error in the Saclay result caused by the xenon fine structure. However, the retarding voltage was self-calibrating and the Saclay interpretation was soon bona fide.

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2.6 Non-Perturbative ATI P. Kruit, a Ph.D. student at FOM, and the rest of M. Van der Wiel group were able to equal and actually surpass the experimental capabilities of the Saclay group due to the power of their unique spectrometer. The first evidence of non-perturbative behavior started to emerge from that laboratory as their electron spectra showed (a) many ATI orders and (b) a surprise, a decreased yield in the low-energy part of the electron distribution with increasing 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 that as the intensity rises the atom’s effective binding energy is increased due to the Up Stark shift of Rydberg states which successively closed the lowestorder ATI channels, i.e., the channels requiring n0 +1, n0 +2, . . . photons for ionization (Kruit et al., 1983). If correct, this scenario implies, since Up > ω, that shifted Rydberg states energies coincide during the laser pulse with the energy of an integer number of photons, or, in other words, that the transition becomes transiently resonant. The beauty of this is that all Rydberg states have, under certain conditions, the same shift and therefore their differential energy remains unchanged. However, it was not possible to see it in an experiment until the advent of sub-picosecond amplified lasers in the mid-1980s. The first group to observe these transient resonances was Freeman’s group at AT&T Bell Laboratories (Freeman et al., 1987) and the Stark-induced Rydberg resonances are often referred to as a Freeman resonance. Proper accounting of the Stark shifts required, of course, a more powerful tool than perturbation theory. The Floquet ansatz led to a successfully quantitative theoretical description (Potvliege & Shakeshaft, 1988), in particular, 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. research at Saclay started her investigations of laser produced multiply charged ions and observed that the intensity-dependent ion yields could not be described by a single rate, and hence the discovery of the famous “knee” (L’Huillier et al., 1983) and the possible existence of direct two-electron ionization. Actually, multiphoton double ionization had been studied in alkaline earths in the Ukraine by Suran and Zapesnochnii (1975), who had noted that the probability for double electron ejection was much higher than predicted from perturbative scaling of the cross-sections, based on the number of photons involved. For more on this topic, see DiMauro and Agostini (1995).

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In the early 1980s, intense ultraviolet lasers became available. C. K. Rhodes in Chicago, observing highly charged heavy ions, concluded that an entirely new regime of interaction had been reached (Luk et al., 1983) in which several electrons and perhaps a whole sub-shell could be collectively excited. However, Lambropoulos pointed out that a real laser pulse has a finite rise time during which electrons could be peeled off successively by “normal” multiphoton absorptions (Lambropoulos, 1985). From Karule’s perturbative calculations (Karule, 1975) he also noticed that the saturation intensities were rather quickly going to a limit as a function of the number of absorbed photons and wondered whether the limiting value was connected to Keldysh’ tunneling. The latter question was left unanswered but the victory, although temporary, of perturbation theory remained for a long time as the “Lambropoulos curse” to those who were finding 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 and theory. CO2 lasers became more powerful and reliable. The first observation of rare gases ionization at a wavelength of 10 µm (Chin, 2011; Chin et al., 1985) was a milestone in strong-field physics. The number of photons involved in the transition was so large that any heroic attempt at perturbation theory was hopeless. Actually several attempts in the preceding decade failed at observing tunneling and thus was considered “unfeasible” experimentally (Lompre et al., 1976). However, Chin et al. (1985) 10 µm measurement resulted in good agreement with the tunneling theory of Perelomov et al. (1965) and began an increasingly general reversal of opinion and Keldysh tunneling became something to be taken literally. Let us remark, as discussed by Reiss (1992, 2008) and others (a) tunneling has no meaning in the velocity gauge and (b) it does not account for channel closure. Nevertheless, since the three KFR theories find a similar behavior in the high intensity, long wavelength limit, the tunneling paradigm may be accepted, at least in the range of parameters where it makes sense (see discussion 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 counting the photons was obviously impossible. Ionization, and a fortiori, the ATI spectrum, with such small photons were even more beyond the reach of perturbation theory. van Linden van den Heuvell and Muller (1987) proposed a radical simplification of the theory of ATI, known as “Simpleman’s theory,” by postulating that in the high intensity, long wavelength limit the spectrum was determined by the classical motion of a charge in the laser

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

2.11 High Harmonics, High-Order ATI and Nonsequential Ionization Lead to the Rescattering/Three-Step Model The late 1980s witnessed the discovery of high harmonics plateau by the Saclay (Ferray et al., 1988) and Chicago (McPherson et al., 1987) groups. At the same time, the demonstration of amplification and subsequent recompression of optically chirped pulse (CPA) (Strickland & Mourou, 1985) opened the road to the development of intense femtosecond lasers and allowed for a wealth of new results. Another important advance in strongfield investigations occurred in the early 1990s: DiMauro and coworkers introduced kilohertz repetition rate lasers in their experiments (Saeed et al., 1990), reaching unprecedented sensitivity. Three observations, the harmonic cutoff, the high-order ATI plateau (Paulus et al., 1994), the nonsequential double ionization of helium (Fittinghoff et al., 1992) beckoned for a more complete view of the laser–matter interaction, beyond the “Simpleman” model, namely the need for accounting for the quivering continuum electron’s re-interaction (or rescattering) with the core. Initiated by Kuchiev (1987) this culminated with the works of Kulander and co-workers (Schafer et al., 1993) on the one hand and Corkum (Corkum, 1993) on the other hand and 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 the ionization probability with increasing intensity in the high intensity, high frequency limit. This ultimate non-perturbative effect, dubbed adiabatic stabilization, created a lot of excitement. The experiment though is extremely 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 threedecade old, was considered mature and perhaps coming to an end. However, novel tools and the influx of new ideas propelled the field toward new discoveries, in a perfect illustration of the futility of predicting the future in fundamental research. This review is meant to highlight some of these recent advances.

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3. WAVELENGTH SCALING OF STRONG-FIELD ATOMIC PHYSICS 3.1 Fundamental Metrics in an Intense Laser–Atom Interaction The purpose of the following section is to develop the background for the reader of the basic wavelength scaling principles and metrics guiding strong-field physics. The scaling provides a understanding of the lowfrequency behavior and allows an extrapolation into the unfamiliar realm of 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 and XFELs can easily exceed 1 a.u. by orders of magnitude, but the maximum intensity experience by an atom will be determined by the depletion (saturation) of the initial state. For low-frequency ionization (ω < Ip , where Ip is the ionization potential), the saturation intensity (Isat) can be estimated by classical over-the-barrier ionization (Augst et al., 1989) given as IOTB = cIp /128πZ2 (Z ≡ effective charge) or ADK tunneling (Amosov et al., 1986). In this scenario, all ground state neutral atoms experience intensities less 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 (see Section 7). The ponderomotive or quiver energy, Up , is an important quantity since it can easily reach values several times the electron-binding energy, especially at long wavelength. At an intensity of 1 PW/cm2 (helium saturation intensity) for a typical titanium sapphire femtosecond pulse, Up ∼ 60 eV. However by increasing the wavelength to 4 µm, Up will increase 25-fold producing a ponderomotive energy of 1.5 keV! Therefore, this simple example stresses the advantage of the λ2 -scaling of Up toward long wavelength for studying neutral atoms in unprecedented large ponderomotive potentials. In Stark contrast, a 1 PW/cm2 , 1 keV X-ray pulse only produces a 0.1 meV quiver energy. Although Up appears as an negligible quantity for X-rays, it does prove useful in the design of two-color streaking metrology (Düsterer et al., 2011) for temporal XFEL characterization. Keldysh defined an adiabaticity parameter, γ , as the ratio of the tunnel time to the optical period (Keldysh, 1965). In the limit γ  1 the ionization is 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 electron velocity Keldysh derived an expression for γ in terms of Ip , and Up , as γ = (Ip /2Up )1/2 . Using the expression for Up , γ is found to be proportional 2 The Coulomb field in the hydrogen atom ground state equates to an equivalent light intensity of 35 PW/cm2 .

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to λ−1 (2Ip /I)1/2 . Thus in the optical regime γ  1, while for the X-rays γ  1. Up appears naturally in the KFR theory (Faisal, 1973; Keldysh, 1965; Reiss, 1980). Thus in Reiss (1980), z = Up /ω > 1, dubbed the continuum intensity parameter, is one of the conditions for strong-field ionization and z ∝ 1/ω3 . Comparing Up to the electron-binding energy Ip or to the electron rest energy mc2 provides a parameter (Reiss, 1992, 2008) which defines the strong-field or relativistic limits, respectively: z1 =

2Up e2 I/ω2 = = 1/γ 2 , Ip 2mIp

(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 intensity parameter, z1 , does not involve any tunneling concept. It can be easily shown that z1 is also the ratio of the interaction Hamiltonian to the unperturbed Hamiltonian related to perturbation theory (Reiss & Smirnov, 1997). Consequently, a perturbative treatment of the atom–field interaction requires that γ > 1, and z and z1 < 1. Figure 1 is a plot of intensity versus frequency for z = 1 (dot-dash linecontinua), z1 = 1 (solid line-bound), and the perturbative limit (dashed line), below which perturbation theory is valid. For the long wavelengths used in our laboratory for strong-field studies and attosecond generation, the photon energy varies between 0.35 and 1.6 eV. For valence electrons at these frequencies and Up  Ip , the distortion of the continua dominates the dynamics and the strong-field condition is satisfied for z1 , z, and γ at intensities; 10−3 < I < 10−1 PW/cm2 . In this regime the quasi-classical model 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 wavelengths, e.g., 0.8 µm, by increasing the intensity with femtosecond pulses. However, the intensity knob is limited by depletion effects, so studies are solely confined to helium and neon atoms, i.e., neutrals with the highest binding energy. Beginning in the late 1990s the emergence of intense ultrafast sources operating at wavelengths longer than 1 µm enabled broader exploration of the strong-field limit in atoms and molecules. These longer wavelength sources are mainly based on parametric processes such as optical parametric amplifies (OPA), optical parametric chirped-pulse amplification (OPCPA), and difference frequency generation (DFG) (Colosimo et al., 2008; Dichiara et al., 2012; Nisoli et al., 2003). Different configurations can achieve gigawatt-level output that provides coverage from 1.3–2.3 µm to 2.8–4 µm and pulse durations as short as three-cycles (Hauri et al., 2007; Schmidt et al., 2010). Figure 2 shows

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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 atomic units (a.u.) and standard units. The dashed line is the perturbative limit defined as 10% of the continuum z-parameter and the double line is 1 a.u. field. At optical frequencies, the continua distortion defines the perturbative limit.

a summary of experimental results from our group illustrating the impact of wavelength on the strong-field processes of (a) electron production and (b, c) high harmonic generation. Figure 2a shows a series of photoelectron energy distributions at constant intensity for xenon atoms exposed to different fundamental wavelengths. The distributions show a clear increase in the photoelectron energy at longer wavelengths. Within the quasi-classical interpretation (Corkum, 1993; Schafer et al., 1993; Sheehy et al., 1998; Walker et al., 1996) both the electron and the harmonic distributions are intimately linked by the rescattering process. Electrons with energy beyond the “Simpleman” 2Up cutoff prediction result from elastic scattering of the returning field-driven electron from the core potential thus allowing energies as high as 10Up . The distributions at 0.8 µm, 2 µm, and 4 µm have values of Up of 5, 30, and 120 eV, respectively, for a constant intensity of 80 TW/cm2 . Consequently, the measured maximum energies at 50, 300, and 1200 eV reflect the 10Up cutoff predicted by a simple classical λ2 -scaling (Up ∝ λ2 ) of the isolated atom responding to the strong-field. For completeness, the high harmonic frequency comb generated in argon 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 rescattering process by a recombination of the field-driven electron wave packet with the initial state (one-photon dipole matrix element). The highest

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Figure 2 (Color online) Effect of the λ2 -scaling with fundamental wavelength in an intense laser–atom interaction at constant intensity. (a) Comparison of the measured xenon 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 energy extends to 1 keV. (b) The measured high harmonic (HHG) spectral distributions resulting from 0.8 µm (yellow shaded) to 2 µm (black/gray shaded) excitation of argon atoms at a constant intensity (180 TW/cm2 ). The 2 µm HHG extend to the Al-filter absorption edge at 75 eV, while the 0.8 µm HHG cutoff is 50 eV. Plot (c), under the same conditions as (b) except using a Zr-filter, shows that the 2 µm energy cutoff is at 220 eV. In both cases, the particle energies dramatically increase with longer wavelength and analysis verifies the expected λ2 -scaling of Up .

photon energy is determined by the maximum classical return energy of 3.17Up (Corkum, 1993; Schafer et al., 1993) thus yielding a cutoff law of 3.17Up + Ip (Krause et al., 1992). These experiments are typically conducted at high gas density(1016–18 −1 cc ) making comparison to single-atom models difficult since macroscopic 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 different wavelengths in scaled Up units. Plotted in this manner, the global transition of the xenon ionization dynamics is revealed. A comparison of the distributions uncovers some meaningful differences: the 2 µm and 4 µm spectra are similar but different from the 0.8 µm case, while the 1.3 µm result shows a transitional behavior. These global features provide clear evidence of an evolution: the 0.8 µm distribution shows electron peaks separated by the photon energy (ATI), Rydberg structure near zero energy, a broad resonant enhancement near 4.5Up (Hertlein et al., 1997; Nandor et al., 1999; Paulus et al., 2001a) and a slowly modulated distribution decaying to 10Up energy. The 2 µm and 4 µm distributions have a distinctly different structure indicative of the strong-field limit; no ATI peaks and a rapid decay from zero to near 2Up energy (Simpleman limit), followed by a plateau extending to 10Up (rescattering limit). This result emphasizes the Keldysh picture: since the potential barrier remains

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(b)

Figure 3 (a) Comparison of photoelectron distributions of argon for different excitations 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) and intensity is 0.08 PW/cm2 . For this intensity Up is approximately 5 eV, 13 eV, 30 eV, and 100 eV for wavelengths of 0.8 µm, 1.3 µm, 2 µm, and 3.6 µm, respectively. Also shown 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 between experiment and theory is very good and reproduces the loss of resonant structure at the 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 the TDSE calculations (closed circle) as a function of wavelength. The error bars reflect the uncertainty in the intensity used in the evaluation of Up . The dashed line illustrates a λ−4.4 dependence. (b) Electron energy distribution (scale in keV) of helium at 0.8 µm (Up = 56 eV and γ ∼ 0.5) and 2 µm (Up = 400 eV and γ ∼ 0.17) at 1 PW/cm2 . As the rescattering cross-section become less effective at high-impact energies, the distribution tends toward the “Simpleman” atom, i.e., most electrons confined below 2Up . All the curves are normalized for unit amplitude.

constant at fixed intensity, the dramatic evolution can only be attributed to a change of the fundamental frequency becoming slow compared to the tunneling frequency. Inspection of Table 1 shows that these observations are consistent with the behavior expected from the strong-field metrics. It should be noted that, as the wavelength is increased and the metric values tend toward the strong-field limit, the majority of photoelectrons become progressively confined below 2Up , consistent with the Simpleman view. 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 energy of 0.5 keV (10Up ) while the 2 µm distribution extends to 3 keV. For 2 µm excitation (Up = 500 eV, γ = 0.15, z = 806 and z1 = 42), the distribution shows that by 2Up , the electron yield has decreased by 7-orders of magnitude and in fact the 10Up cutoff is unobserved in the experiment due to detector noise. This is quantified for xenon in Figure 3a inset which shows that the wavelength dependence of the ratio of electrons with energy >2Up

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

Up (eV)

γ

z

z1

0.8 1.3

5 13

1.1 0.67

3 14

0.83 2

2.0

30

0.45

44

5

4.0

120

0.22

390

21

(rescattered) to those <2Up (direct) decreases as ∼λ−4 . One physical interpretation of this scaling is that electrons with energy >2Up resulted from elastic rescattering. Thus, the number of returning electrons with a given kinetic energy decreases by λ−2 and the wave packet’s transverse spread scales in area also as λ−2 resulting in a total decrease in the fraction of rescattered electrons that scale as λ−4 .

4. LOW-ENERGY STRUCTURE IN PHOTOELECTRON ENERGY DISTRIBUTION IN THE STRONG-FIELD LIMIT The understanding of strong-field ionization in the high intensity/low frequency limit made a giant step forward with the Simpleman ansatz which treats classically the motion of the photoelectron in the laser field and neglects the influence of the ionic potential. The photoelectron dynamics following from the 1D Newton equation with initial conditions derived from tunneling (initial position at the outer turning point of the Stark potential and zero initial velocity) is the mechanism underlying the Keldysh approximation and the KFR theories. As discussed above, other ionization features like the ATI plateau, the rescattering rings (Yang et al., 1993), double ionization, etc. require elastic or inelastic scattering of the laser-driven electron with the ionic core. The smaller the Keldysh parameter, the better the ansatz is expected to work. Classical mechanics describes the motion of an electron in the laser field but the so-called Strong-Field Approximation (SFA)3 reduces the ensemble of possible quantum paths connecting initial and final states to a small number of “trajectories” along which the quasiclassical action is stationary. The SFA is more computer-effective than the full numerical solution of the Schrödinger equation and yields, in most cases, excellent agreement. The efficacy of SFA has been widely demonstrated for both ATI and high harmonic generation. A strong discrepancy with SFA in experimental 3 In 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 (Lewenstein et al., 1994; Milosevic et al., 2006). In this chapter, SFA refers only to the latter.

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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 the direct and rescattered electrons. The intensity-averaged KFR calculation reproduces neither the rescattered part of the spectrum (inset) nor the LES (shaded region). The figure is reproduced, with permission, from Blaga et al. (2009).

observations (Blaga et al., 2009; Quan et al., 2009) therefore came as a major surprise, since the disagreement appeared to become more pronounced at longer wavelengths despite the smaller Keldysh parameter. Briefly, the photoelectron energy spectra reveal a low-energy peak or structure (LES) that is completely absent from the SFA calculation (see Figure 4). In one experiment (Blaga et al., 2009), a single broad peak (width > ω) appears in all the spectra recorded from atoms and molecules for which γ  0.8. In the other (Quan et al., 2009) a second feature is observed and both experiments report that the LES becomes more prominent the longer the wavelength. It is likely that the LES was previously undetected because the majority of experiments were conducted at near-visible wavelengths. Figure 5 reveals the universality of the LES: a log–log plot of the LES width versus the Keldysh parameter for all investigated targets and laser wavelengths shows a striking scaling of ≈γ −2 . A third important observation was the absence of LES for circularly polarized light. Finally, angular distributions of the photoelectrons (Catoire et al., 2009) provide another signature of the LES. Although the LES structure was reproduced by TDSE calculations, when available, no physical explanation was identified. General arguments by Blaga et al. (2009) and Faisal (2009) suggested that the LES is due forward rescattering. Quan et al. (2009) compared their data to semi-classical calculation and concluded that the LES had a purely classical interpretation and stressed the role of the ionic potential (which is neglected in standard SFA). Thus, improvement to the SFA theory should include the influence

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

of the Coulomb potential and more precisely account for the initial longitudinal 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 expression of the form:   (SFA) M Cps e−iS(p,Ip ,s,t) , (2) p s

  Tpulse  1 where Cps is a prefactor, S(p, Ip , s, t) = tp [p + A(t)]2 + Ip dt, the 2 s quasi-classical action and A the laser vector potential. The sum runs over the saddle point times tps solutions of the saddle-point equation ∂S ∂t |ps = 0. Yan et al. (2010) analysis sorts the saddle-point trajectories ending with the same final momentum at the detector into four types. Calling zt the tunnel exit abscissa, pz the initial momentum along the laser polarization, p0x and px the initial and final transverse momenta, the four types are defined by: (1) zt pz (2) zt pz (3) zt pz (4) zt pz

> 0 and px p0x < 0 and px p0x < 0 and px p0x > 0 and px p0x

> 0. > 0. < 0. < 0.

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

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momentum A(t0 ). Type (3) electrons move away from the detector and have an initial transverse momentum pointing in the “wrong” direction and type (4) are electrons tunneling toward the detector with a “wrong” transverse momentum. A study of the partial contributions to the LES according to the 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 classical, 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 investigated by 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 Keldysh parameter is small, the quiver amplitude is much larger than the length of the tunnel zt and the transverse motion in one period, hence the number of rescattering may be large. The interplay between multiple forward rescattering and the distortion of the trajectories by the Coulomb potential near the core are at the origin of the LES according to Liu et al. (2010). In their approach the energy spectra, for direct comparison with the experiment, is calculated as a function of the coordinates in the phase-space: initial 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) Dominant trajectories in the zx-plane contributing to the final momentum in the range of the LES. 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.

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The analysis confirms that the LES is directly related to the transverse momentum modified by the Coulomb potential, ruling out the chaotic motion observed for some initial conditions (near the initial phase with the range 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 laser parameters clearly agree with the experiment (see Figure 7). Another Monte Carlo simulation, compared to 3D TDSE results (Lemell et al., 2012), reveals the correlation between the LES and the photoelectron angular momentum (see Figure 8). The trajectories identified as responsible for the LES, low energy, high angular momentum island of Figure 8, agree with those identified by Yan et al. (2010). Another classical approach (Kästner et al., 2012) sheds additional light on the conclusions of the numerical investigations. A study of the final momentum as a function of the initial phase and transverse momentum pz (φ  , pρ ) shows 2D saddle-points which appear after the second laser period. They are responsible for an infinite series of peaks (one per subsequent optical cycle) in the energy spectrum converging to pz = 0. The peaks are not related to head-on, back-scattering events responsible for the ATI plateau nor to the chaotic dynamics which occur in the same region of the phase-space. The consequence of this model is that the LES composed of a series of peaks which could explain the double hump structure reported by Quan et al. (2009). Interestingly, the saddle-points exist in 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. The island at large values of L is identified as the origin of the LES as demonstrated by the projection 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 Lemell et al. (2012). Copyright (2012) by the American Physical Society.

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Figure 8 The dependence of the LES on (a) laser intensity I 0 at λ = 2 µm, (b) wavelength at an intensity of 9.0 × 1013 W/cm2 , and (c) wavelength at γ = 0.534. The solid line is a fit to all of the LES data (see Liu et al. (2010) for more information). The figure is reproduced, with permission, from Liu et al. (2010). Copyright (2010) by the American Physical Society.

saddle-point and the corresponding 1D trajectory are shown in Figure 9 and are comprises forward scattering and a soft-backward collision. This analysis gives a new and general perspective on understanding the dynamics 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 the universality of the phenomenon which does not depend on the details of the potential (long or short range). Future experiments are needed to test these predictions. Quantitative comparison with experiment is still limited, as well as deficiency of new measurements. However, the classical mechanics underlying the quantum propagation is still very enlightening. It would be interesting to understand quantitatively the amplitude of the LES. Contrary to the ATI plateau which is less than 1% of the direct electrons, the LES appears to be comparable in magnitude (10–50%). It is likely that this is due, in part, to the fact that low-energy electrons are created near the peak of the laser field cycle and, in part, to the large forward scattering cross-section.

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Figure 9 Classical mechanics view of the LES: 1D calculation showing the deflection function (left) and corresponding photoelectron spectrum (right) for a Gaussian potential. 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 forward and backward scattering, the reason for the LES’s amplitude is not obvious. Some further answers are expected from a full quantum calculation using a second-order KFR theory (Guo et al., submitted for publication).

5. ELECTRON MOMENTUM DISTRIBUTION AND TIME-DEPENDENT IMAGING The determination of the atomic positions in molecules, plays an essential role in physics and chemistry. X-ray and electron diffraction methods routinely achieve sub-Angstrom spatial resolutions. However, in the time domain they are limited to picoseconds and although recent developments in femtosecond XFELs (Emma et al., 2010) and electron beams (Eichberger et al., 2010; Zewail & Thomas, 2009) are changing this perspective, ultrafast filming of molecules “in action” is still beyond the reach of the scientists. In contrast, self-probing of the molecular structure by the electron wave packet produced by strong-field interaction with the molecule under interrogation, can in principle reach high spatial and, as will be shown here, temporal resolution. The initial idea (Lein et al., 2002) was first proposed using the high harmonics plateau and tested by the NRC group (Itatani et al., 2004). The electron wave packet extracted from the HOMO and driven back to the molecular core by the oscillating laser field, in first approximation as a plane wave, can either recombine, emitting a high harmonic photon or scatter. In both cases information about the molecular structure can be retrieved from the process. For instance, from the high harmonic spectra generated by molecules aligned at different angles with

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respect to the laser polarization, tomographic images of the molecule can be reconstructed (Haessler et al., 2010; Itatani et al., 2004). The photoelectron momentum distribution from strong-field ionization of molecules will also convey information on the molecular structure. This is the principle of laser-induced electron diffraction (LIED) (Meckel et al., 2008). In the rescattering model, the tunnel ionized electron wave packet will elastically scatter from the core after approximately a half of an optical period T. Since T = λ/2πc, changing the wavelength allows a variation of this time interval. In other words, recording the photoelectron momentum distributions at different wavelengths amounts to taking snapshots of the molecule at different times (Blaga et al., 2012). The spatial resolution is determined by the kinetic energy of the photoelectron, ∝ Iλ2 . Using midinfrared lasers this energy can be quite large and thus LIED is capable of achieving sub-Angstrom spatial resolution with a temporal resolution of better 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 on various atoms and molecules (Cornaggia, 2009; Ray et al., 2008) and provides the framework for interpreting an LIED measurement. In the strongfield approximation, the detected electron momentum, p, is the sum of the  r , at the time momentum at recollision, pr , and the field vector potential, A of rescattering, tr .  r. p = pr + A

(3)

The analysis of the momentum distribution along the circumference defined at constant |pr | is equivalent to the field-free differential cross-section measurement performed using conventional electron diffraction at fixed electron energy (Figure 10). In this view, the laser generates a flux of electrons with momentum pr that undergo elastic scattering at time tr . In principle, alignment of the molecular sample is required. However in the strong-field ionization regime, the contribution of molecules aligned with the laser polarization is largely dominant and alignment is not necessary. As an example, the distribution of momentum parallel and perpendicular to the laser polarization for unaligned N2 molecules ionized by a linearly polarized, 20 fs pulse is shown in Figure 10a. The differential cross-section extracted from this distribution for a constant recollision energy of 100 eV is shown in Figure 10b and can be compared, on the one hand, to the cross-section obtained from electron diffraction (Dubois & Rudd, 1976) and, on the other hand, to the theoretical cross-section. The good agreement between the three values validates the LIED approach for the nitrogen and oxygen molecules.

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

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5.1.1 Method for Retrieving the Internuclear Distance Using mid-infrared lasers to promote core penetrating (hard) collisions allows a number of simplifications to the LIED approach. The diffraction pattern is assumed to arise from a molecule that is treated as a set of independent atoms and during the sub-cycle interval the inter-atomic distances are fixed. Any chemical properties that arise from molecular bonding electrons, e.g., HOMO, make negligible contributions. In the special case of homonuclear diatomic molecules, the scattering differential cross-section for electrons colliding with unaligned molecules can be expressed as    dσM sin q · R , (4) = 2σA 1 +  d |q · R| where q is the momentum transfer and R the inter-atomic distance. The so-called molecular contrast factor MCF = sin(qR)/qR, is very sensitive to small deviations in R for large values of q. This allows good spatial resolution for collisions with large momentum transfer. Equation (4) reflects the interference between the electron waves scattered by the two atoms of a diatomic molecule. The MCF, or equivalently, the position of the first minimum in a two-slit interference pattern, as a function of q is shown in Figure 10c. The experimental MCF is extracted from the momentum distribution measurement. The differential cross-section and the intranuclear distance R are retrieved using a genetic algorithm. The N2 analysis at |pr | = 2.97 a.u. (kinetic energy = 120 eV), shown in Figure 11, yields a N–N distance of 1.14 Å in good agreement with the value extracted from the independent atom model, conventional electron diffraction, and the known 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 different results. At any wavelength the electron wave packet is released in the continuum close to the peaks of the laser field (the tunneling rate depends exponentially on the field). The rescattering takes place about half a cycle later, 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 involve scanning a delay. However, changing the wavelength is equivalent to varying the delay between the tunneling and rescattering times during which the 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 surprising since removal of a HOMO electron results in a little change in the bond character and thus a small change <5 pm in the bond length between neutral and the ion (see Figure 11).

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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 bond length is indicated by the solid (neutral) and dashed gray (ion) lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this book.) The figure is reproduced, with permission, from Blaga et al. (2012).

In contrast, the oxygen anti-bonding HOMO results in a significant bond length change over a time interval of a few femtosecond [1.21 Å (neutral) to 1.10 Å (ion)]. The LIED measurement at the three wavelengths yield bond lengths of 1.10, 1.11, and 1.02 Å. The extracted O–O distances deviate by 2–4 standard deviations from the equilibrium value, and agree with a bond that has contracted during the 4–6 fs interval after tunneling [see Figure 12 and Blaga et al. (2012) for more details]. This measurement is a proof-of-principle experiment and there is significant prospect for improving the procedure. For example, few-cycle pulses, bichromatic fields, additional wavelengths, and improved counting rates could enhance the temporal resolution. Femtosecond pump–probe methods can be conducted using LIED to interrogate molecules undergoing conformal transformations. Application to more complex systems, aligned or oriented targets, and excited molecules (vibrational or electronic) would be of great interest. Compared to conventional electron diffraction methods, LIED’s ability to coherently control the returning electron wave packet could open new paths in electron diffraction.

6. NON-SEQUENTIAL MULTIPLE IONIZATION AT LONG WAVELENGTHS Even though 40 years have pasted since the first investigations (Suran & Zapesnochnii, 1975), the question of how several electrons are

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

simultaneously ejected during an intense laser–atom interaction is still a lively debate. Theoretical approaches, ranging from numerical modeling of the electron–electron correlation by a time-dependent potential (Parker et al., 2006; Watson et al., 1997) to many-body S-Matrix theory (Becker & Faisal, 2005) and purely classical mechanics (Wang & Eberly, 2010) and references therein, show that the process cannot be understood without electron correlation (Becker & Rottke, 2008) and one size may not fit all. Often however ionization experiments in a strong laser field can be understood qualitatively 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 in this section is observed through the charge distribution spectrum recorded when atoms, e.g., rare gases, are exposed to intense mid-infrared pulses. Laser-driven multiple ionization can be the result of (e, 2e) (Moshammer et al., 2000; Walker et al., 1994) or higher (e, ne) processes (Feuerstein et al., 2000), or excitation followed by field ionization (Yudin & Ivanov, 2001). Most experiments have been performed at near-infrared (NIR) wavelengths (λ  1 µm). The majority of these investigations were limited to measurements of double NSI since depletion of the neutral ground state restricts the maximum intensity, and thus the return energy, to the (e, 2e) inelastic channel. This limitation holds for all neutral atoms ionized by a NIR field. In addition, extraction of the inelastic NSI channel can be masked by the presences of photon-assisted processes (Schins et al., 1994). The central point of what follows is to discuss the understanding of NSI and its quantitative interpretation using field-free scattering cross-sections.

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The experiment was conducted using intense long wavelength pulses 3.6 µm and 3.2 µm and the corresponding λ2 increase in the electron’s rescattering energy (Colosimo et al., 2008) can drive (e, ne) processes for n = 2–6 (Dichiara et al., 2012). At these wavelengths the field strength can be kept low enough to avoid depletion of the neutral ground state and at the same time open a large number of (e, ne) channels. It was shown that the results could be interpreted by known (field-free) electron impact cross-sections (Achenbach et al., 1983, 1984) and thus the link between NSI and laser-driven recollision was addressed over a broad energy range. The main result is that laser-driven scattering is indeed comparable to conventional electron impact ionization albeit broadened by the nearly “spectrally white” returning wave packet which according to classical mechanics, 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 rescattering energy. The experiment is briefly described here, for details see DiChiara et al., 2011, 2012. The laser consists of a difference frequency generation (DFG) in a KTA crystal amplifier fed by two independent laser systems: a 100 fs pump pulse generated by a Ti:Sapphire chirped pulse amplifier and a 16 ps pulse generated by a regeneratively amplified Nd:YLF laser (Saeed et al., 1990). The idler beam has a central wavelength tunable to 3.6 µm or 3.2 µm, a 100 fs pulse duration and a peak power exceeding 1 GW. The beam is focused into a time-of-flight mass spectrometer. The mass resolution of the spectrometer fully resolves all xenon isotopes up to the highest charge state observed. The intensity calibration ±10%, was based on photoelectron spectra characteristic change of slope at 2Up (Colosimo et al., 2008. Figure13 shows the measured ion yields of xenon at 3.6 µm where the highest charge state observed is Xe6+ . When compared to the sequential yields calculated using the ADK tunneling rate (Amosov et al., 1986), for charge states up to Xe6+ strong discrepancies appear while evidence of extreme non-sequential (e, ne) processes (where over 700 MIR photons are coupled to neutral xenon liberating six electrons bound by a net potential of 243.6 eV) are identified. In contrast, non-sequential yields, shown with dashed lines in Figure 13, are found in excellent agreement with the data. Our analysis is based on defining effective, energy-averaged crosssections for laser-driven multiple ionization. The known experimental cross-sections (Achenbach et al., 1984), when available, are used in the calculation (Dichiara et al., 2012). The spectral width of the returning electron distribution are expected to be very broad, due to the distribution of ionization times (Micheau et al., 2009). In addition, laser experiments are necessarily integrated over the spatial and temporal intensity distributions of

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

the focused beam. Hence, we define an effective, energy-averaged, crosssection (function of intensity I) as:  dE σ (E )Wp (E (I))  , (5) σ˜ (I) = dE Wp (E (I)) where Wp is the net return distribution and σ (E) is the experimental crosssection. The expansion of the wave packet (Delone & Krainov, 1991) is taken into account to ensure that each classical trajectory represents a wave packet with the correct area at recollision and the calculation is summed for individual volume elements of a Gaussian focus. In this approach, an energetically broad wave packet is related to the laser-driven measured branching ratio and a field-free measured electron impact cross-section. On the other hand, the lateral expansion of the wave packet quantifies the observed yield. The observed multiple ionization channels agree with our assumption of a direct (e, ne) ionization event.

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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 and ionization cross-sections used in our calculation. (c, d) Equation (5) for 3.6 µm and 3.2 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-states agree within a 10% uncertainty in our intensity calibration and the overall experimental error (see Figure 13). In addition, the excellent agreement between observed branching ratios and effective cross-sections for channels (e, ne) (n  3) (see Figure 14) justifies the correspondence between laser-driven inelastic scattering and field-free impact ionization and confirms the key role of the “white” recolliding wave packet. Moreover, it validates retaining only channels that couple higher charge states to photoionization of the neutral ground state. A qualitative argument justifies the surprising result that field-free cross-sections are sufficient to interpret a process taking place in a high-intensity laser pulse because the most energetic recolliding electrons return near the zero of the laser field the Coulomb

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potential suffers only a small distortion at that moment. In addition, even at the field maximum, the region of the potential near the ion ionization energy 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 pulse duration and/or energy at high photon frequency. Various approaches have been pursued that produce different attributes for investigating new regimes of light–matter interactions. For example, high harmonic generation (Brabec & Krausz, 2000; Saliéres et al., 1999), using the physics discussed above, has enabled the attosecond frontier (Agostini & DiMauro, 2004). However, free-electron lasers operating in the soft and hard X-ray regime are defining unique capabilities and applications that are unrivaled by previous electron-based sources. In the context of this review, X-ray freeelectron lasers (XFEL) are allowing the first experimental realization aimed at addressing intense X-ray–matter interactions, thus we will limit our discussion to only this aspect. In the next paragraph, the reader is referred to a number of recent reviews that provide a more comprehensive treatment of 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 an upgraded, FLASH II, in 2007 pushed operation to 200 eV. The reader is referred to a recent review on the machine capabilities (Tiedtke et al., 2009) and science program (Meyer et al., 2010). A parallel project at Stanford Linear Accelerator Center (SLAC) in Menlo Park, California pursued the development of a millijoule, hard X-ray (0.8–8 keV fundamental operation) FEL. In the summer of 2009, the Linac Coherent Light Source (LCLS) was commissioned and rapidly achieved full output specifications (Emma et al., 2010), while a user scientific program began that Autumn (Bucksbaum et al., 2011). In sharp contrast with our discussion on optical ionization where valence shell ionization dominates, X-rays preferentially ionize inner shell electrons. Examination of the absorption spectrum of neon shows that less than 10% of the total cross-section (0.35 Mb) at the K-edge (0.8701 keV) is attributable to valence shell excitation. Consequently, photoionization creates an inner shell vacancy that is an excited state which spontaneously relaxes by emitting a photon or Auger decay. For neon, and other low-Z atoms, the ultrafast (2.6 fs) Auger process dominates [one valence electron fills the 1s-hole and one is freed (∼0.82–0.85 keV energy)] leaving a

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Figure 15 Neon charge-state yields for X-ray energies below, above and far above the 1s-shell binding energy, 0.87 keV. (a) Experimental charge-state distribution for 0.8 keV (top), 1.05 keV (middle) and 2.0 keV (bottom). (b) Comparison of experimental charge-state yields, corrected for detection efficiency, with rate equation simulations. The cross-sections in the model are based on perturbation theory. The figure is reproduced, with permission, from Young et al. (2010).

doubly-charged neon ion. Ionization with unity probability by linear absorption 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 order of magnitudes larger than that needed to drive near-threshold one-photon ionization of a 2p-valence electron (21.6 eV binding energy) with unit probability, in fact it even exceeds that required to tunnel ionize helium at optical frequencies. However, this extraordinary soft X-ray intensity produces little quiver of the continuum electron due to the λ2 scaling of Up ∼ meV. The first LCLS experiment studied the photoionization of neon atoms using the AMO end-station, the reader is referred to a review paper (Bozek, 2009) for a detailed description. In summary, the LCLS could deliver 100–200 fs, millijoule pulses over a 0.8–2 keV photon energy producing a ∼3 µm focus in the AMO end-station main interaction chamber. In the experiment (Young et al., 2010), neon ion m/q-distributions were investigated at different intensities, pulse duration, and photon energies. Figure 15 shows the (a) measured ion yields and (b) a comparison between theory and experiment at three different photon energies. Clearly, the large flux of

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X-rays is sufficient not only to saturate the neutral neon ionization, as estimated above, but also essentially strip off all the electrons. The three different wavelengths provide a window into the ionization sequence. At the low photon energy (0.8 keV), K-shell ionization cannot occur via one-photon ionization, thus charge state production proceeds through a sequence of valence shell excitation terminating at Ne8+ since only the 8, n = 2 electrons can be ionized. At 2 keV photon energy all 10-electrons are removed through a sequence of 1s-photoionization followed by Auger decay resulting in a propensity in even-charge state production (see bottom graphs in Figure 15). This is the first observation of sequential atomic ionization from the inside out, as opposed to sequential valence ionization (outside-in) by strong optical fields. At intermediate photon energy (1.05 keV) ionization of the low-charge states proceeds via photoionization/Auger decay until the K-shell binding energy (0.1096 keV for Ne6+ ) exceeds 1.05 keV. At this point, valence photoionization proceeds and terminates with the production of Ne8+ . The theoretical calculations, shown in Figure 15b, are based on a rate equation model that includes only sequential single-photon absorption and Auger decay processes, which produces good agreement with the measurement. This investigation provided a preview of the richness of new phenomena that can arise from matter interacting with an intense X-ray pulse but equally important, it showed that the primary ionization mechanisms were well within the framework of perturbation theory, even at intensities of 1017 W/cm2 . The potential to study laser–matter interaction involving X-ray intensities near 1017 W/cm2 led naturally to the objective of observing for the first time nonlinear, multiphoton X-ray absorption. Both ion and electron measurements were performed to study the ionization of a K-shell electron by simultaneous absorption of two X-ray photons (Doumy et al., 2011). An estimate based on the expected X-ray fluxes and a value of two-photon ionization cross-section (Novikov & Hopersky, 2001, 2002) given by perturbation theory suggested that the effect would be weak, but observable. One caveat stems from the fact that the X-ray radiation can be accompanied by harmonics of the fundamental frequency, including a small (<1%) amount of second harmonic that was verified by ionization measurements using the helium atoms. At the photon energies required to avoid direct K-shell ionization of neon (<0.87 keV), the second harmonic falls in a region where the LCLS transport mirrors reflectivity is high, so the contamination on target is too large to be able to isolate the nonlinear contribution. The LCLS AMO end-station mirror reflectivity was specifically designed to provide a sharp cutoff above 2.0 keV. Thus, working at photon energies >1.1 keV insured good discrimination against harmonic content. Furthermore, the neon investigations discussed above showed that at 1.05 keV photon energy that the atom sequentially absorbs a large number of photons, leaving the ion stripped of all n = 2 valence electrons

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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 bars result from an improved model that accounts for shake-up processes. The striped bar in (b) incorporates the adjusted two-photon cross-section needed to reproduce the Ne9+ production below threshold. The figure is reproduced, with permission, from Doumy et al. (2011). Copyright (2011) by the American Physical Society.

(Young et al., 2010). However, the experiment did observe a small production of Ne9+ that was not accountable with the linear absorption model calculations. Based on this evidence, the experiment focused on production of 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 ionized by a single-photon/Auger sequence is Ne6+ resulting in the creation of Ne8+ which cannot be ionized by one-photon absorption. Thus, the lowest-order channel is a two-photon process. In contrast, Ne8+ ionization is possible with 1225 eV photons, resulting in the direct production of Ne 9+ . Studying the production of Ne9+ at these two-photon energies as a function of pulse energy (intensity) should then reveal the linearity/nonlinearity of the process. Figure 16 shows a typical ion time-of-flight spectra obtained at both wavelengths. The first thing to note is that the LCLS easily ionizes 8-electrons from neon at both photon energies. The presence of Ne9+ is small but obvious at 1.11 keV, while at 1.225 keV Ne9+ the larger production proceeds by allowed single-photon ionization. Figure 17 shows the evolution of the ratio of Ne9+ /Ne8+ production as a function of the measured X-ray pulse energy. In order to facilitate the comparison, each ratio has been normalized at 0.8 mJ pulse energy. The ratios 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 different in both cases: linear at 1.225 keV and quadratic at 1.11 keV, a signature of second-order nonlinearity.

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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 at 1.11 keV (solid line) and linear behavior at 1.225 keV (dashed line). The figure is reproduced, with permission, from Doumy et al. (2011). Copyright (2011) by the American Physical Society.

The study concludes, that neon atoms subjected to ultra-intense, 1 keV X-rays undergo a complex sequence of excitation, ionization, and relaxation processes. Experiments combined with more complete theoretical analysis have identified and characterized the nonlinear response resulting from two channels: (1) the dominant direct two-photon, one-electron ionization and (2) a sequence of transient excited states that competes with the Auger decay clock. They also conclude that due to near resonance contributions the two-photon multiphoton absorption cross-section is 2–3 orders magnitude higher than the earlier calculation (Novikov & Hopersky, 2001). However at the time of this writing, all the X-ray observations at the LCLS are well described as expected by perturbative framework. This leads to the obvious query: where is the strong-field nonperturbative limit for X-ray frequencies. The scaling laws developed in Section 3 may provide some appreciation of this challenge. Figure 18 scales the 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 surprisingly the difference in frequency scaling of z1 and z results in the core-state distortion dominating over the continuum-state above 500 eV photon energy. This is in sharp contrast to the usual picture at optical frequencies of a quivering continuum wave packet. Within this understanding, entrance into the strong-field regime will require intensities of nearly 1020 W/cm2 ,

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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 atomic units (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 frequencies compared to optical since the distortion of the bound-state dominates and defines the breakdown of perturbation theory. The dashed line shows the z1 parameter for the neon valence L-shell which distorts at lower intensity than the core.

which are beyond the performance of current XFELs. However, the total bound-state contributions (core + valence) result in a dichotomy since the strong-field limit defined by the these states are clearly at odds. Therefore, the breakdown of the perturbation theory may happen at significantly lower intensities than expected and within access to the current XFEL operational parameters.

8. OUTLOOK In the forthcoming years one can expect significant progress in mid-infrared lasers in wavelength, intensity, and time resolution. Wavelengths extending to 10 µm, pulse energy of a few mJ, and pulse duration down to one cycle are likely and will allow to further explore the scaling properties of strong-field physics. In particular it should be possible to investigate relativistic interactions without the need of ultrahigh intensity 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 spectrometers and other techniques associated with pump–probe scheme will allow to interrogate complex molecules under conformal transformation

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

ACKNOWLEDGMENTS We wish to thank the post-doctoral researchers and graduate students, past and present, who have tirelessly carried out the research discussed in this review: Cosmin Blaga, Fabrice Catoire, Phil Colosimo, Anthony DiChiara, Gilles Doumy, Anne Marie March, Chris Roedig, Kevin Schultz, Emily Sistrunk, Jennifer Tate, and Kaikai Zhang. Special appreciation is expressed to our colleagues, Chii-Dong Lin, Junliang Xu, whose fruitful discussions on LIED have resulted in an exciting collaboration. Both authors acknowledge support from the US Department of Energy and supported by its Division of Chemical Science, Office of Basic Energy Sciences, the National Science Foundation, and the Army Research Office. LFD gratefully acknowledges the continued support of Dr. Edward and Sylvia Hagenlocker.

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