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Atoms in high intensity mid-infrared pulsesP. Agostini a; L. F. DiMauro a

a Department of Physics, The Ohio State University, Columbus, Ohio, USA

Online Publication Date: 01 May 2008

To cite this Article Agostini, P. and DiMauro, L. F.(2008)'Atoms in high intensity mid-infrared pulses',Contemporary Physics,49:3,179 —197

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Atoms in high intensity mid-infrared pulses

P. Agostini* and L.F. DiMauro

Department of Physics, The Ohio State University, Columbus, Ohio, USA

(Received 27 February 2008; final version received 22 May 2008)

The recent development of intense, ultrashort, table top lasers in the mid-infrared opens new avenues for research instrong field atomic physics. Electrons submitted to such radiation acquire huge quiver energies, even at moderateintensity and interesting properties arise: first, the wavelength offers a convenient experimental knob to tune theionisation regime by controlling the Keldysh parameter. Second, many processes like above-threshold ionisation orhigh harmonic generation, whose characteristics depend directly on this energy, can be pushed to unprecedentedlimits. Third, the wavelength controls the spectral phase of the harmonics and hence the possibility to improve thegeneration of pulses in the attosecond regime. Recent studies of rare gas and alkali atoms’ photoelectron spectra andharmonic generation at 2 and 3.6 mm have begun to confirm the theoretical predictions. However, unexpectedfeatures have also been found showing that strong field interaction still keeps some secrets after more than 40 yearsof investigation.

Keywords: multiphoton ionisation; above-threshold ionisation; high harmonic generation; attosecond pulses

1. Introduction

With the advent of the laser, the early 1960s saw thebirth of a new class of physical problems related to theinteraction of intense laser fields with bound orunbound electrons [1]. The intensity of the laserradiation is measured in power per unit area or Wattsper square cm. At an intensity of about 1016 W cm72,the electric field of the electromagnetic wave ap-proaches that of the Coulomb field experienced by anelectron in the ground state of a hydrogen atom. Whiletoday lasers can e asily produce intensities a thousandto a million times larger, most neutral atoms cannotsurvive long when irradiated with intensities above1015 W cm72 and quickly lose one or several electrons.In a rather narrow range of intensities around 1014 Wcm72, the physics of the interaction between light andatoms or molecules changes rapidly and manyphenomena occur which are too weak to be observedbelow 1013 but are no longer observable above1015 W cm72. This range is that of the so-called‘strong-field’ or ‘high-intensity’ (the two terms will beused indifferently) regime which is discussed in whatfollows. First it is in this range that perturbationtheory breaks down thus challenging theorists. Secondit is in this range that above-threshold ionisation,harmonic generation and more recently routes toattosecond pulses and attophysics have been discov-ered by experimentalists.

For a long time, high power lasers were confined towavelengths in the visible/near infrared range: theRuby laser (0.79 mm), the Nd laser (1.06 mm), the nowpopular Titanium:Sapphire laser (0.8 mm) emit all atthe edge of the infrared. Until recently strong fieldphysics had practically not been explored beyond thisspectral domain. One of the latest laser advances,however, has lead to intense, short pulse, mid-infraredlasers (i.e. with wavelength ranging between 1.5 and4 mm with the current technology) with which physi-cists have started to fill this deficit. Strong field opticalinteraction between lasers and atoms at mid-infraredwavelengths is the subject of this article. This wave-length range has a special significance because of thescaling with l and intensity of many parametersgoverning the physics of both ionisation and harmonicgeneration and the production of attosecond pulses, asdiscussed in the article.

Strong-field physics, as an experimental science,goes back to the first years of the laser era. As soon asthe intensity was boosted around 1013 W cm72 by theinvention of the Q-switch [2], the production of sparksin air was among the first tricks learned by laserphysicists. Sparks meant ionisation. How could a rubylaser whose photon energy is less than 2 eV ionise anatom whose ionisation potential is larger than 10 eV,seemingly breaking Einstein’s photoeffect law? Variousanswers were proposed (see Gold and Bebb’s short

*Corresponding author. Email: [email protected]

Contemporary Physics

Vol. 49, No. 3, May–June 2008, 179–197

ISSN 0010-7514 print/ISSN 1366-5812 online

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review in [3]) including a straightforward extension ofthe theory of two-photon absorption established byMaria Goppert-Mayer in the 1930s [4] and based uponthe time-dependent perturbation theory [5,6]. However,from one of the first measurements [7], the ionisationrate appeared to depend on the intensity exponentiallyand not to follow the power law predicted by theperturbation theory. This result suggested a ratherdifferent mechanism to L.V. Keldysh in Moscow,namely a dc-tunnelling ionisation. In his famous paper[8], he strived to show that the ionisation rate waschanging from power law to exponential (see Equations(3) and (4) below), based on the laser intensity I,wavelength l and the atom ionisation energy I0.The change in the rate dependence on intensityfollows a transition from the multiphoton regime(illustrated in Figure 1(a)) to the tunnelling regime(Figure 1(b)). The former is referred to as ‘perturbative’since it is described by the time-dependent perturbationtheory while the latter is assigned to the theKeldysh approximation usually considered as ‘non-perturbative’ since it includes the laser interaction to allorders.

Keldysh’s view remained quite unpopular in theWest, perhaps because of a number of rough approx-imations in the calculations [9,10], perhaps because it isexperimentally difficult to distinguish between a high-order power law and an exponential growth or, mostlikely, because the tunnelling regime was not easilyreachable with the available equipment. In spite of newtheoretical support through different approaches andimprovements [11–13], Keldysh’s tunnelling regimewas often considered as a fiction until the mid-1980s.The first demonstration [14] was obtained using a highintensity CO2 laser whose photons are so small (0.1 eV)that the lowest order of perturbation required to reachionisation, larger than 100, was manifestly incompa-tible with the experiment. From then on tunnelling

became progressively more accepted (the so-calledADK formula [15] is now adopted by chemists, as wellas, laser plasma physicists) and newer CO2 experiments[16] confirmed the earlier measurements. Interestinglyenough, as experiments became more sophisticatedthanks to new laser technology and numerical solu-tions of the time-dependent Schrodinger equation(TDSE) became accessible [17], signatures of tunnel-ling ionisation were obtained even at wavelengthsmuch shorter than that of the CO2 laser. At this pointtunnelling is considered as the basic mechanism forstrong field, long wavelength ionisation of atoms,although, as will be seen in the following section, manyobjections can still be made to this point of view.

Beside total ionisation, a new aspect of strong fieldinteraction was discovered when people turned tomeasuring the photoelectrons rather than the ionsproduced by the laser [18]. In a process, designated asabove-threshold ionisation or ATI (a name coined firstby Karule [19]), the atom absorbs more photons thanrequired to be ionised, and the excess energy istransferred to the electron whose energy spectra is aseries of peaks separated by the photon energy [20,21].The interpretation was first considered through per-turbation theory [5,19,20]. Faisal [12] and Reiss [13]gave the first non-perturbative expression of thephotoelectron spectrum. A simple physical meaningof this somewhat complicated expression involvinggeneralised Bessel functions did not emerge, however,until the introduction of the so-called ‘Simpleman’stheory’ [22] of ATI. Very simple and intuitiveinterpretations of the high harmonic generation cutoff[23] and strong field multiple ionisation [24] derivedfrom Simpleman’s theory together with the Keldyshtunnelling, form the basis of the current understandingof a large range of strong field phenomena. Keldyshtheory depends on two parameters, discussed here-after, the adiabaticity parameter and the effectiveionisation potential. Many aspects of strong fieldionisation and, particularly their dependence on thelaser wavelength depend on them.

High harmonics generation, or HHG, is as anelementary process, closely related to ATI. It is apractical source of coherent, intense, short pulse XUVto soft X-ray radiation and, in the last few years hasestablished itself as the preferred route to attosecondpulses. Both a large bandwidth and a quasi-linearspectral phase, required for such pulses, have beenproved possible [25]. Currently, attosecond physicsessentially relies on HHG. As will be discussed, longwavelength laser drivers, in principle, can improveattosecond pulses by allowing higher photon energies,or shorter wavelength, and reducing the quadraticspectral phase. It is therefore decisive to explore HHGwith the new mid-infrared sources.

Figure 1. Models of strong field ionisation: (a) N-photonionisation þ S-photon ATI. (b) Keldysh tunnelling energydiagram. (c) Schematic of Simpleman’s ATI: the tunnellingelectron emerges with zero velocity. The rest of the process isdescribed by the motion of a particle in the field.

180 P. Agostini and L.F. DiMauro


Even at moderate intensity (between 1013 and 1014

W cm72), mid-infrared wavelengths mean higherenergy for the photons and electrons resulting fromthe interaction, and thus define a privileged road mapfor the production of soft X-rays. This is because thesource of this energy, the ponderomotive potential orquiver energy of a free electron in an optical field,increases as the square of the wavelength at constantintensity. In addition, a long optical period drives theionisation physics deeper into the tunnelling regime asit approaches the dc situation. In this regime theextremely fast dependence of the rate on the fieldamplitude implies that photoelectrons are released in anarrow range of time close to the peaks of the field.This, in turn has consequences on the energy distribu-tions of the photoelectrons and in particular deter-mines the high energy cutoffs for both ATI and HHG.

Initially CO2 lasers were the only high intensityinfrared sources available to the physicist. The alreadycited pioneering experiments [14,16] and a study ofATI in the limit of long wavelength [26] were all donewith a CO2 laser. However, the CO2 technology hasnot evolved since the late 1980s. Most strong fieldphysics of the last 15 years have been carried out at thewavelength of the Ti:Sapphire laser (0.8 mm). It is onlyrecently that new sources in the mid-infrared havebecome available, giving access to the range between0.8 and 10 mm. The first studies of high harmonics withsuch sources was carried out in alkali atoms by Sheehyet al. [27] and in noble gases by Shan and Chang [28]while the OSU group just reported the first systematicexploration of the dependence of ATI and harmonicgeneration on wavelength [29].

In this article we review the recent advances instrong field physics as a function of the fundamentallaser wavelength. It is true that, forty years after theKeldysh paper, this domain is understood to a largeextent. Nevertheless, as often in Physics when a newtechnology allows one to explore new domains of theparameters space, surprises might be expected (andhoped for). The article will discuss successively thefindings in ATI that fit the current understanding,those which probably do and those which manifestlydon’t. We will then turn to HHG and attosecondpulses and discuss the cutoff, the yield and the spectralphase. However, before the experiments, we brieflyreview the status of the Keldysh approach and the mid-infrared laser technology.

2. Scaling strong field interaction to long wavelengths

2.1. Strong field ionisation

The Keldysh theory [8] amounts to (i) a dc-tunnellingionisation and (ii) a final state which takes into accountonly the interaction with the optical field (Volkov

solution). The field frequency o ¼ 2pc/l is assumed solow that it can be considered dc and the field amplitudeE so high that the potential barrier formed by theCoulomb potential (for a hydrogen atom) –1/r and thedipole interaction –eEr (see Figure 1(b)) thin enoughfor the probability of escaping through it, significant.Both conditions are condensed in the so-calledKeldysh adiabaticity parameter which is the ratio ofthe optical frequency to the tunnelling frequencyot ¼ 2p/tt where the tunnelling time tt is the barrierwidth � I0/eE divided by the electron average velocity� (I0/m)1/2 (m, e, electron mass and charge, respec-tively). Neglecting the influence of the Coulombpotential and ignoring the magnetic field, which isfully justified for the range of field strengths consideredhere, the freed electron undergoes both an oscillatoryand a drift motion. The average kinetic energy of theoscillation with instantaneous velocity v ¼ (E/mo) sinot plays a crucial role in strong field ionisation andhigh harmonic generation. The expression of thisenergy (also known as the ponderomotive energy) is,from elementary classical mechanics:

UP ¼1

2mv2

� �¼ e2E2

4mo2/ l2: ð1Þ

The ponderomotive energy is one of the basicparameters ruling strong-field physics. As discussedbelow, it characterises the photoelectron and harmonicspectra.

Expressing the field as a function of UP, E ¼ (2o/e)(mUP)

1/2 one gets the familiar expression of theKeldysh parameter:

g ¼ oot¼ I0

2UP

� �1=2

; ð2Þ

while another form makes the dependence on thewavelength l and the intensity I apparent and reads inatomic units, g(l, I) ¼ (2pc/l) (2I0/I)

1/2. Depending ong being � 1 or � 1 tunnelling or multiphotonionisation respectively dominates (Figure 1). The twolimiting rates derived by Keldysh can be expressed as:

wtunnel / exp � 2g3o

� �ð3Þ

and

wmultiphoton /E

o

� �2n

; ð4Þ

where n is the minimum number of photons absorbedto reach the ionisation threshold. Equation (3) is

Contemporary Physics 181


identical to the famous tunnelling formula. Equation(4) results from the lowest-order time-dependentperturbation theory probability amplitude in whicheach absorbed photon corresponds to one dipoleinteraction / E.

In principle, the tunnelling regime is valid up tog � 1. This is in a nutshell the Keldysh theory and g isusually taken as an intensity parameter: since UP / I,the higher the intensity the more ionisation is in thetunnelling regime. We will keep here the tunnellingpoint of view although this concept is specific to thelength-gauge (see [13] for a thorough discussion of thispoint) and could be avoided. In this case, otherparameters appear in the theory [30].

If it is intuitively obvious that an increase ofintensity (and hence of field) will decrease the barrierwidth and (exponentially) increase the ionisation rate,the effect of changing the light frequency at constantintensity (i.e. constant barrier width) is much less so,although g is indeed / o. Moreover, as recentlydiscussed by Reiss [30], the concept of tunnelling doesnot go without contradictions. First of all it is clearthat the popular diagram of Figure 1 is meaningfulonly the in the ‘length’ gauge where the interactionenergy is –eE � r. Second the condition g � or � 1 isnot always consistent with the tunnelling picture. Forinstance, at visible wavelengths the hydrogen groundstate is above the saddle point of the potential barrierfor g � 1 and thus the concept of tunnelling ismeaningless just when it should become instrumental.In addition, small values of g imply a completebreakdown of the dipole-length gauge approximationon which the Keldysh picture is built. The Keldyshapproximation implies that the rate is independent ofo in the tunnelling regime. More precisely, the cycle-average ionisation rate is independent of the wave-length at least up to a quasi-static limit defined aso � E/(2I0)

1/2, i.e. for instance, at a typical intensity of1014 W cm72, wavelengths longer than 1.027 mm. Inspite of the long time elapsed since the Keldysh paper,these aspects of the theory have not been system-atically checked hitherto, because of the lack ofconvenient sources, with the exception of the fewworks mentioned in the introduction.

What are the motivations for studying ionisation ofatoms by strong mid-infrared pulses? The first one is totest the predictions of the Keldysh and the Simplemanmodels. The second is to produce more energeticelectrons for the same laser intensity. It is well knownthat laser plasmas can be submitted to arbtitrarily highintensities that can be produced (currently much higherthan 1018 W cm72). However, in atoms there is astrong limitation due to the saturation of the ionisationprobability (wt *1 where w and t are the rate and thepulse length, respectively). Acting on UP through the

wavelength is a way to circumvent this obstacle,particularly as UP / l2 while only / I.

2.2. ATI and Simpleman’s theory

Figure 2 shows the evolution of ATI from low to highintensity. Although not explicitly considered in [8] thephotoelectron energy spectra can be evaluated within asimilar approximation formulated through S-Matrixtheory [12,13]. Incorporating the ATI process, thephotoeffect Einstein’s law for the kinetic energy of thereleased electron KE can be extrapolated to(Figure 1(a)):

KE ¼ ðNþ SÞ�ho� ðI0 þUPÞ: ð5Þ

The UP term in Equation (5) accounts for the increaseof the ionisation potential in the strong field [8] due tothe oscillation kinetic energy that must be provided tothe electron on top of its binding energy to free it. Oneof the questions addressed here is the limit of KE whenN þ S ! ? and o ! 0. Obviously perturbationtheory is of no help. The so-called KFR (Keldysh–Faisal–Reiss), now more often referred to as SFA(strong-field approximation) [30] is derived from the S-Matrix approach and takes into account the interac-tion with the field to all orders and for this reason isusually known as ‘non-pertubative’ allowing one tocalculate the transition probability for any S (seeEquation (5)).

The spectra envelopes calculated from SFA show amore or less clearly defined cutoff at an energy of 2UP.Little notice was taken of this aspect until experimentsstarted measuring differential cross-sections [18] and apioneering paper [22] revealed the meaning of thiscutoff in terms of classical mechanics and shed acompletely new light on the strong field ionisationprocess, especially at long wavelength [26]. Classicalaspects of ATI have been recently reviewed in [31]. Theessence of the Simpleman’s theory can be summarisedas follows: the electron energy spectrum is assimilatedto the kinetic energy distribution of the drift motion ofa classical particle (e, m) in a field E sin ot with theinitial conditions t ¼ t0, v ¼ 0, x ¼ 0 (v, velocity; x,position) corresponding to an electron tunnelling outnear the core with zero velocity. Integrating theNewton equation €x ¼ E sinot yields _x ¼ (E/o)(1 7 cos ot) where the drift velocity vdrift ¼ (E/o)results from the conservation of the canonical mo-mentum. The corresponding drift kinetic energy is:

KEdrift ¼ 2UP cos 2ot0: ð6Þ

This immediately accounts for a cutoff at 2UP forot0 ¼ 0, i.e. for electrons released at the zero-crossing



of the field while those ‘born’ at the peaks of the field(ot0 ¼ (2k þ 1)p/2) have no drift motion. Experi-ments [32,33] confirmed that a large proportion ofthe photoelectrons have an energy less than 2UP butalso revealed a long plateau extending between 2UP

and 10UP. The plateau can be easily explained byallowing the electron to backscatter from the core attime t1(t0) of the oscillating electron on the parent ion,followed by a drift motion whose new kinetic energy isgiven by [32]:

KEbsdriftðt0Þ ¼ 2UPð2cosot1ðt0Þ � cosot0Þ2 ð7Þ

yielding a cutoff at 10UP. Both cutoffs are proportionalto UP and therefore to l2. For instance, at a typicalintensity of 1014 W cm72 and a wavelength of 4 mm,10UP represents the absorption of almost 5000photons. Note that the initial phase ot0 ¼ 1.88 rad

corresponding to the maximum kinetic energy at thereturn to the position x ¼ 0 is different from the initialphase ot0 ¼ 1.83 corresponding to the maximum driftKEbs

drift ¼ 10UP energy after a backscattering collision[31]. Actually there are two electron trajectoriesleading to the same return energy as will be discussedin the next sub-section but the second one yields alower drift energy of 9.169UP. The ATI ‘plateau’ hasattracted a lot of interest [31] and some controversy[34]. We will come back to this point when discussingthe experimental results hereafter. To conclude thisshort summary of ATI it is perhaps useful to remindthe reader that this process has found in the recentyears a number of interesting applications in themetrology of ultrafast XUV pulses [35,36] and in thecontrol of the carrier-to-envelope phase of few-cyclepulses [37]. As far as the wavelengths scaling isconcerned, ATI in the mid-IR is expected to show a

Figure 2. ATI spectra in the (upper panel) low and (lower panel) high intensity limits: in (a) the characteristic photon energystructure dominates the spectrum. In (b) this structure tends to be washed out while some features of the envelope can bepredicted classically.



huge increase in electron energies (typically from tensto thousands of eV for wavelengths from 0.8 to 4 mm).Since ionisation, at the same time, is evolving moredeeply into the tunnelling regime, the electron isreleased (with zero kinetic energy) over a narrowtime interval within the optical cycle and the spectrumprogressively loses its quantum character and getscloser to the classical distribution of drift energyimplied by the distribution of the initial times.

2.3. High harmonics and attosecond pulses

By focusing an intense, short pulse (fs) laser in anargon (rare gases are the most common targets) jet,odd harmonics of the fundamental frequency aregenerated essentially in the same direction as that ofthe fundamental beam. High harmonics generation(HHG) is a process akin to ATI [38] (Figure 3): theatom/molecule starts by absorbing a (large) number ofphotons. However, instead of leaving its parent ion (orelastically scattering from it before moving away) as inATI, the electron recombines to the atom ground statewhile emitting a hard photon. Experiments on HHGgo back to the beginning of the 1980s in plasmas andlater (1987) in atoms [39,40]. Quantum mechanicallythe energy of this photon is equal to the summedenergy of the absorbed photons. Classically, the activeelectron is accelerated in the field and it is this kineticenergy which is converted into a photon energy at therecombination. Since the initial and final atomic statesare the same state, dipole selection rules imply that atotal even number of photons are exchanged, hence theharmonics are odd order. Simpleman’s theory can beextended to harmonic generation. (This looks prettyobvious now but it took a few years before this was

actually achieved.) The oscillating electron instanta-neous velocity when it re-collides with the core asderived from the equation of motion is v(t1(t0)) ¼ (E/o)(1 – cos ot1(t0)). The resulting kinetic energy12 vðt1ðt0ÞÞ

2 as a function of the initial time t0 turnsout to have a maximum equal to 3.17UP. Since thisenergy is the source of the high harmonic energy, theso-called harmonic cutoff is therefore bounded to3.17UP þ I0 [23,24] (the maximum energy, againproportional to l2 promises a huge increase of photonenergy with mid-infrared drivers).

Theory of HHG can be divided into a microscopicand a macroscopic parts. The former is the ‘single-atom’ response or the time-dependent dipole whoseFourier transform is the harmonic spectrum [41].The latter (phase-matching) includes the propagationinside the generating medium, the space and timedependence of the laser pulse etc. [42]. Beyond theclassical model, quantum theories of HHG include thestrong field approximation [41] and simulations fromnumerical solutions of the time-dependent Schrodingerequation. In spite of computing times increasingrapidly with the laser wavelength, the latter cangenerate spectra like the ones shown in Figure 4 inwhich indeed the increase in cutoff is manifest [43].

With the most common driving wavelength of0.8 mm (Ti:Sapphire laser) and targets (rare gases), theHHG efficiency is of the order of 1076. BothLewenstein’s strong field approximation and TDSEsimulations indicate that the single-atom dipoledecreases rapidly with the laser wavelength [43]. Thisis most likely due to a combination of the wave packetspreading during the electron trajectory in the con-tinuum and the decrease of the recombination cross-section as the electron energy increases (Figure 5). Asdiscussed below, TDSE calculations show indeed thatthe single atom power spectrum yield scales asl75.5 + 0.5 at constant intensity [43] (see also [44]).However, a complete simulation including phase-matching is difficult and still under progress. Conse-quently experiment becomes all the more interestingfor an assessment of this dependence.

High harmonics are currently the preferred route tothe synthesis of attosecond pulses first predicted at thebeginning of the 1990s [45,46] and observed atthe dawn of the 21st century (for a review see [25]).The current world record of 130 as was obtained in2006 [47]. From this perspective, the longer wave-lengths have a certain advantage: theory (both classicaland quantum models) predicts that for each harmonicenergy there are two electron trajectories a short one(�1/2 cycle) and a long one (�3/4 cycle) [48]. Eachtrajectory is defined by an initial phase and a returnphase. The difference between these phases representsthe duration of the electron trajectory under the action

Figure 3. (a) ATI versus (b) high harmonic generation:quantum processes (arrows) and classical limits (wavy lines).Both points of view assert the kinship between the twoprocesses.



of the field between emission and recombination times.As a function of the harmonic energy it defines a groupdelay dispersion (or ‘attochirp’) which imposes a limitto the duration of the Fourier-synthesised pulses. Theattochirp is a decreasing function of wavelength(Figure 6) as discussed below.

3. Mid-infrared laser technology

The practical generation of intense, femtosecondpulses in the mid-infrared, depending on the spectralrange is based on the nonlinear optics processes [49] ofdifference frequency generation (DFG) or OpticalParametric Amplification (OPA). A 0.8 mm Ti:Sap-phire (Ti:S) and a 1.053 mm YLF lasers are used in thedesigns described hereafter. The first one is a commer-cial 3 kHz repetition rate chirped pulse amplificationlaser. The second one is a passively mode-lockedneodymium-doped yttrium lithium fluoride (Nd:YLF)oscillator that directly seeds a 1 kHz regenerativeamplifier. The output energy is typically of 700 mJavailable for MIR generation.

3.1. The 3.6 lm source

The 3.6 mm radiation is obtained through DFG: thenonlinear crystal converts the wavelengths of 0.8 mm(‘pump’) and that at 1.053 mm (‘idler’) into the ‘signal’

at 3.6 mm, subject to the energy and phase matchingconstraints:

o800 ¼ o1053 þ o3600; ð8Þ

k800 ¼ k1053 þ k3600; ð9Þ

where the o’s are the angular frequencies and the k’sare the wavevectors of the radiations. Equations (8)and (9) are depicted in Figure 7. The exchange ofenergy between the pump (p), signal (s), and idler (i)during the optical parametric amplification process isdescribed within the slowly varying amplitude approx-imation by a set of coupled nonlinear equations [49]involving the phase and group velocities at the variousfrequencies present in the nonlinear medium and thewave vector mismatch Dk ¼ j kp – ks – kij.

With a KTA (Potassium Titanyl Arsenate, KTiOA-sO4) nonlinear crystal and by a suitable choice of thecentre wavelength of Ti:S light, tunable MIR radiationwith centre wavelengths in the range of 3–4 mm can beproduced, limited by the transmission dropping from

Figure 4. The nonlinear dipole created by the interaction of the atom with the field can be calculated by numerically solving theSchrodinger equation. The figure shows the dipole spectra for l ¼ 0.8 and 2 mm.

Figure 5. (a) Effects responsible for the harmonic yielddecrease as a function of the wavelength: wave-packetspread; (b) kinetic energy versus wavelength; (c)recombination cross-section versus energy.

Figure 6. The ‘attochirp’ becomes apparent when plottingthe time elapsed between ejection and recombination as afunction of the harmonic order. The slopes of the tangents tothe curves representing a group delay dispersion or‘attochirp’ are shown here for two wavelengths ascalculated from the classical kinematics of the electron inthe laser field.



about 90% at 3.7 mm to 50% around 4 mm (Figure 8).Given the small energy of an MIR photon, this tuningrange only requires that the Ti:S be adjusted from 0.78to 0.83 mm. A sketch of beam routing in the KTAcrystal is shown in Figure 8(b). Typically, 2.5 mJ, 100fs Ti:S pulses centred at 0.815 mm and 500 mJ, 17 psNd:YLF pulses centred at 1.053 mm are used in a5 mm long KTA crystal to produce 160 mJ of MIRcorresponding to 28% of the theoretical maximumpossible MIR energy. At the mixing crystal, the Ti:Sand Nd:YLF FWHM spot sizes were � 2.5 mm.Mixing the amplified Ti:S and Nd:YLF beams requireslocking the repetition rates of the two oscillators. To

do so, the cavity length of one oscillator must bereferenced to the other (some problems and theirsolutions related to this locking can be found in [50]).One of the serious difficulties met in the MIR is thelack of flexible detectors available in the visible/IRrange. A thermal camera (Electrophysics M/N PV320)can be used in the 3–4 mm range. Typical spot imagesrecorded with such a camera is displayed inFigure 9(a).

3.2. The 2 lm source

Generation of the 2 mm radiation uses a differentarrangement. The source used in the experimentalresults presented hereafter is based upon a commercialoptical parametric amplifier (OPA) (model HE-TO-PAS-5/800 built by Light Conversion). The 2 mm OPAis pumped by 5 mJ, 50 fs Ti:S pulses derived from a Ti:Sfs system. The TOPAS starts by using a small amountof pump light to generate a broadband spectrum viasuperfluorescence. A narrow spectral portion is thenused as a seed with the remaining 0.8 mm light. After sixpasses in the two BBO (beta-barium borate) nonlinearcrystals 400 mJ of 2 mm radiation is available (ontarget). Because this is a parametric amplifier thatsplits one 0.8 mm photon into two lower energy ones,conservation of energy implies that 600 mJ of 1.3 mmidler radiation is also generated. The focus is about 2times larger than the diffraction limit.

4. Experiments

The experiments to be discussed fall into two maincategories depending on whether electrons or ions

Figure 7. (a) Energy level diagram showing the photonabsorption and emission (arrows) at the pump np signal nsand idler ni frequencies. Wavevectors relation for phase-matching in (b) non-collinear and (c) collinear differencefrequency generation.

Figure 8. (a) KTA transmission. (b) Schematic representation of the two pump beams and the signal beam in the KTA crystal(from [50]). Reproduced with permission from P. Colosimo.



(subsection 4.1) or photons (subsection 4.2) aredetected. The effect of l on the particle energy atconstant intensity is spectacular. Moreover, it is inprinciple possible, by scaling the intensity I, thewavelength l and the ionisation energy I0, to buildcompletely different physical situations with the sameKeldysh parameter g ¼ (2pc/l)(2I0/I)

1/2. Will this‘Keldysh scaling’ work? In setting out to answer thisquestion we have in mind the various properties whichcharacterise the ionisation regime of the system understudy. Target atoms were rare gases and alkalis and thelasers were the Ti:S 0.8 mm and the 2 and 3.6 mmsources described above.

4.1. Ionisation/ATI

One could ask the experiment to answer two questions:(i) how does the ionisation probability change withwavelength at constant laser intensity? (ii) How is thephotoelectron energy spectrum affected by the wave-length change? The first question is difficult as itrequires either absolute measurements or to keep allthe experimental parameters (intensity, target density,interaction volume) constant while changing thewavelength. However, the latter is not done by turninga knob but by discrete changes of wavelength obtainedby different lasers. In that context the second questionis much more accessible than the first one.

Two different time-of-flight (TOF) electron/ionspectrometers, abundantly described in the past, havebeen used: a Wiley–McLaren ion time-of-flight spec-trometer (built with a special accelerating field

configuration allowing a better focusing of the iontrajectories) and a coincidence machine in whichelectrons are allowed to fly field free from the laser–gas interaction region to a multichannel plate detectorwhile ions are accelerated toward another detector by astatic electric field. All ions are accelerated to the sameenergy due to the (� 100 V cm71) static electric fieldapplied to the interaction region. The electron spectro-meter was energy calibrated by using a long pulse (�200 ps) xenon ATI spectrum. A separate machine wasused for the alkali atom studies. Details can be foundin [50].

4.1.1. l2-scaling

To study the evolution of the ionisation/photoelectronenergy spectra as a function of the wavelength, argon isa convenient target atom [29]. Argon ionisation at theTi:S wavelength of 0.8 mm is extremely well documen-ted [34,51–55] thus providing a benchmark for thelonger wavelength results.

The change of atomic response with increasingwavelength is clearly revealed in the measured photo-electron energy distributions displayed in Figure 10(a).The maximum electron energy scales approximately asl2 as expected with values of 50, 130, 300 and 1 keVwhile UP is *5, 13, 30 and 100 eV at 0.8, 1.3, 2 and3.6 mm, respectively. Note that the maximum numberof absorbed photons changes from *40 at 0.8 mm to*3000 at 3.6 mm with only a factor 4.5 change in thefundamental wavelength. We have a first indicationthat the physics indeed tends to follow Simpleman’s

Figure 9. (a) Far-field intensity distribution from the 3.6 mm source. (b) Interferometric two-photon autocorrelation of thepulse. The pulse FWHM is 170 + 10 fs.



theory and that the gross features of the photoelectronspectrum behave classically. This is not a surprise: asthe photon energy ! 0 and the quantum numbers! ? the principle of correspondence implies a changeto classical mechanics. However, this is the first testover a significant range of wavelength.

The evolution of the physics according to Keldysh’stheory is more immediately evident in Figure 10(b)plotted as a function of the energy scaled in values ofUP which make clear the evolution toward thestructure defined by the tunnelling ionisation (aspectrum with a cutoff at 2UP) followed by a plateaudescending to the 10UP cutoff caused by the rescatter-ing of the photoelectron and re-acceleration by thefield. Also shown in (b) are the calculations (grey lines)at 0.8 mm (solid) and 2 mm (dash) simulating theexperimental conditions. The agreement is very goodand reproduces the loss of resonant structure at thelonger wavelength. The inset is a plot of the electronyield ratio (Ne (42UP)/Ne(52UP) determined by theexperiment (open circles with error bars) and theTDSE calculations (solid circles) as a function ofwavelength. The dashed line illustrates a l74.4

dependence. This decrease of the plateau is connectedto the increase of the scattering electron energymatching the increase of UP with l. Measurements inhelium as well as xenon yield comparable results andthus confirm that the l2-scaling is a universal propertyof ATI. This is consistent with the Keldysh approachin which the specific nature of the atom is basicallyabsent, except for the ground state binding energy andwave function.

Confirmation of the changing ionisation dynamicsis provided by the 1.3 mm distribution which showsboth multiphoton (ATI and broad resonance,although diminishing) and tunnelling (developingclassical distribution) character. Once again, it appearsthat the photoelectron evolves closer to the Simplemanbehaviour, which is consistent with ionisation increas-ingly in the tunnelling regime.

It should be noted that, as the wavelength isincreased, the majority of photoelectrons becomeprogressively confined below 2UP. This is quantifiedin the Figure 1(a) inset which shows that thewavelength dependence of the ratio of electrons withenergy 4 2UP (rescattered) to those 52UP (direct)decreases as *l74. One physical interpretation of thisscaling is that electrons with energy 42UP must haverescattered and this cross-section declines rapidly withenergy and impact parameter. The number of return-ing electrons with a given kinetic energy decreases byl72 and the wave packet’s transverse spread scales inarea also as l72. Consequently, the total decrease inthe fraction of rescattered electrons scales as l74.Some additional insight into the changing ionisationdynamics is provided by the 1.3 mm distributionwhich shows both multiphoton (ATI and broadresonance, although diminishing) and tunnelling(developing classical distribution) character. It there-fore appears that the photoelectron spectra becomecloser to the Simpleman’s behaviour, which isconsistent with increased tunnelling character of theionisation regime, as predicted by the Keldyshparameter.

Figure 10. Photelectron energy spectra from argon at 8 6 1013 W cm72 for 0.8 mm (solid), 1.3 mm (dash-dot), 2 mm (dash) and3.6 mm (dotted) excitation. The energy is in eV in (a) and in units of the ponderomotive energy which makes the scaling apparentin (b).



4.1.2. Keldysh-scaling

The Keldysh parameter with values g(0.8 mm) ¼ 1.3,g(1.3 mm) ¼ 0.8, g(2 mm) ¼ 0.5 and g(3.6 mm) ¼ 0.3 at0.08 PW cm72 is qualitatively consistent with theevolution of the spectra described in the previoussubsection. Since the potential barrier width remainsconstant, this evolution must be attributed to a changeof the fundamental frequency becoming slow com-pared to the tunnelling frequency, as postulated byKeldysh. This point is addressed again at the end ofthis subsection.

Besides the change of the l2 scaling of the electronenergy at constant laser intensity for a given atom onecould ask what would happen if two atoms withdifferent ionisation potentials are irradiated by laserlight whose intensity and wavelength are scaled to keepthe Keldysh parameter g constant. Let us take forinstance argon at 0.8 mm and 8 6 1013 W cm72. Whatintensity should be applied to an alkali atom, sayrubidium, at a wavelength of 3.6 mm so that it wouldnot differ from argon at 0.8 mm? From the expressionof g (Equation (2)) one gets:

IðRbÞIðArÞ ¼

I0ðRbÞI0ðArÞ �

lðArÞlðRbÞ ð10Þ

or, plugging the values above, I(Rb)/I(Ar) ¼ 1.3 61072. In other words rubidium at 3.6 mm and anintensity of 1.04 6 1012 W cm72 may simulate argonat 0.8 mm from the Keldysh’s point of view and becharacterised by the same g ¼ 1.28. The experiment [56]indeed confirms that the photoelectron spectrum(Figure 11) has the strong field behaviour, at leastqualitatively, with visible 2UP and 10UP limits.

At this point it is perhaps worthwhile discussing thevalue itself of the Keldysh parameter. Reiss [30] haspointed out that the limiting value g ¼ 1 supposed toseparate the tunnelling from the multiphoton regime isbasically meaningless since in for instance in hydrogenat 0.8 mm a value of g ¼ 0.91 actually corresponds to azero-length barrier while at g ¼ 0.3 when tunnellingshould occur the potential barrier is below the groundstate and the electron can freely escape to the con-tinuum over the barrier. For lower values of g the laserfield is so intense that the dipole approximation breaksdown. So g has clearly only a qualitative value as asignal of tunnelling. Let us note in addition that the dcpicture that defines the potential barrier and hence, thetunnelling, is only part of the definition of g whichdepends also on the frequency (Equation (2)). In thatsense g is not just an intensity parameter. Let usremark that one of the parameters introduced in theReiss strong field approximation [13,30], namelyz1 ¼ 2UP/I0 ¼ 1/g2, although totally independent ofthe tunnelling concept is related to g and while anotherone z ¼ UP/h�o introduces the frequency through thecomparison of UP to the photon energy.

It is a certain dependence on frequency (orwavelength) which was tested (for the first time) bythe experiments reported here above, carried out atconstant intensity, i.e. at constant barrier width ortunnelling probability. From the Kelysh’ point of viewthis is a test of the influence of the time left to theelectron to cross the potential barrier before the fieldreverses. From Reiss’ point of view, alternatively, it isthe ratio of the quiver energy to the photon energy thatrules the strong versus the low field behaviour.Whether one takes UP as the value of the ac-Starkshift of Rydberg states [57] or as the oscillating energyof a free electron in the field, z is a pure intensityparameter as it is intended to be in [13] reflecting howmany photons is the shift or the oscillating energy inthe field and does not imply any notion of time. Since gis linear in o and 1/z / o3 one may think that someexperiment will succeed in deciding which of theKeldysh or the Reiss condition describes moreaccurately the physics and whether Keldysh tunnellinggoes beyond a convenient fiction.

4.1.3. Plateau-smoothing

A comparison of the experimental distributions ofFigure 10 reveals a number of differences: the 2 and3.6 mm spectra are similar but clearly different from the0.8 mm case while the 1.3 mm result shows, notsurprisingly, a transitional behaviour: (1) in the0.8 mm distribution (g ¼ 1.3 the ATI characteristicstructure (peaks separated by the photon energy) isclearly visible. Moreover, a Rydberg structure of

Figure 11. Rubidium photoelectron energy spectrum at3.6 mm. The 2UP and 10UP values are indicated by the arrows.



Freeman resonances [58] near zero energy is resolved.A broad enhancement near 4.5UP [32,54,55] and,finally, a slowly modulated distribution decaying to10UP energy can be identified. The subject of thissection is to discuss the evolution of the enhancement.The 2 mm and 3.6 mm distributions appear to have adistinctly different structure: no ATI peaks, a rapiddecay from zero to near 2UP energy (Simpleman limit),followed by a plateau extending to the 10UP rescatter-ing limit. The broad resonant enhancement near 4.5UP

so conspicuous at 0.8 mm is either completely absent orstretched over so large an energy range that it ispractically washed out. Now, this enhancement isextremely well documented in the literature[31,32,34,52,54,55] and has, at least, two possibleinterpretations [34]: either a multiphoton resonanceinvolving an excited state or an effect connected tochannel-closure. Wassaf et al. [34] have analysed thissituation with the help of the strong field Floquetapproach and/or TDSE simulations and conclude tothe resonance interpretation. It is not our intentionhere to debate this point but rather to argue that thetwo interpretations become rapidly indistinguishablewhen the wavelength increases.

The argument is straightforward: in both scenariosthe condition for the enhancement is obtained byequating a quantity scaling as I 6 l2 (the ac-Starkshift of the resonant states or the effective ionisationpotential I0 þ UP) to the energy of an integer numberof photons. This can be figured as the intersection inthe intensity-energy Floquet plane (see Figure 2 in

[34]) of one horizontal straight line with a line ofslope ¼ 7UP / 7l2 representing the dressed groundstate. From this perspective the difference between thetwo scenarios, experimentally accessible, is given bythe difference between the two intensities satisfying oneor the other of the two equations (in atomic units)(Figure 12):

I1 ¼ 4o2ðko� I0Þ; ð11Þ

I2 ¼ 4o2ðko� ERÞ; ð12Þ

where ER is the energy of a resonant state, I0 theionisation potential and k an integer. The sketch inFigures 12(b) and (c) show how the difference I1 –I2 ! 0 when the wavelength increases. The intensity-induced resonances repeat when UP has increased byone photon energy. Moreover, while a single Rydbergstate induces a well-defined resonance (as in thecalculation of [34]), several Rydberg states wouldinduce a group of nearby resonances. The interactionvolume intensity distribution easily covers a largenumber of photon energy in the mid-infrared.Although a complete calculation has not yet beencarried out it is intuitive that the combination of thesetwo effects would give rise to a progressively morespread out and less prominent enhancement. To sumup, the enhancement in the photoelectron spectrum ofargon between 15 and 30 eV at 0.8 mm tends to moveto higher energies and to smooth out at longer

Figure 12. (a) Argon plateau at three wavelengths (0.8 mm upper, 2 mm middle, 3.6 mm lower). Schematic illustrating theargument in Section 4.1.3: (Intensity-Energy) Floquet diagram for two wavelengths (b) l and (c) l0 4 l. The intensities I1 and I2are the solutions of Equations (11) and (12).



wavelengths. At constant peak intensity this is a directeffect of the l2-scaling of UP, the decreasing of thephoton size and the averaging over the laser intensitydistribution.

4.1.4. Low-energy enhancement

Whether or not the argon enhancement is consistentwith Keldysh theory (the channel closure scenario iswhile the resonance one is not, since it implies excitedstates absent from the Keldysh model) is perhaps stillan open question although, at this time, TDSE andFloquet calculations [34,52] appear to support theresonance view. In this subsection we discuss anotherexperimental observation that points to an effect ofRydberg states. The effect is still under investigation atthe time of this writing and therefore will be onlybriefly described here. The basic signature of the effectis illustrated in Figure 13 in xenon although it seems tobe quite universal. Qualitatively analysed, the effectconsists of an enhancement in the photoelectronspectrum at long wavelength (or at least becomingmore visible at long wavelength) in the lowest part ofthe spectrum: in scaled units of energy it occurs inxenon around 0.1–0.2UP. Such a bump in the spectrumenvelope is clearly incompatible with the classicalspectrum resulting from the initial phase distributionfrom tunnelling ionisation [59] and the Keldysh theory,at variance with most of the findings reported here.Attempts to simulate such spectra using numericalTDSE are difficult for several reasons: (i) even a single-intensity calculation places ever increasing demands oncomputational power as the wavelength increases; (ii)

realistic simulations must include a spatial-temporalaveraging over the interaction volume/laser pulseintensity distribution. Because of the large value ofUP/h�o a large number of calculations must be carriedout with extremely small intensity steps due to thesmall photon size. Such calculations are in progress[60]. Although definitive theoretical spectra are not yetavailable it appears that Keldysh model simulationscannot reproduce the enhancement of Figure 13 whileTDSE calculations can. It is likely that excited states,ignored in the Keldysh approximation, play a rolehere. The angular distributions of the low energyenhancement part of the spectrum have also perhapsrevealing characteristics [61].

One can remark that strong field interaction, inspite of its respectable age still keeps in reserve somesurprises.

4.2. High harmonic generation

High harmonics, with orders of several hundredcannot be hoped to be described in terms of theperturbative expansion central to nonlinear optics [49].As a non-perturbative effect HHG has exerted acontinuing fascination on theorists. Its attraction forexperimentalists as a coherent, intense, short pulsesource of XUV radiation was fuelled at the beginningof the 1990s by the recognition of its potential toproduce attosecond pulses [45].

Achieving the atomic scale for both time (25 as)and length (0.5 A) is a distant frontier for time-resolved imaging, spectroscopy, metrology and, moregenerally, optical science. The atomic scale of time is

Figure 13. (a) Xenon photoelectron energy spectra at 2 mm. (b) Blow-up of the spectrum low energy part showing the so farunexplained low energy enhancement.



defined by the motion of electrons in their orbitals.Resolving this temporal scale with light pulses wouldenable a new class of time-resolved investigations ofelectron dynamics by freezing and ultimately control-ling electronic motion. The atomic length scale ischaracterised by the size of the orbital. The grandchallenge of resolving both scales promises images andmovies of complex molecular objects. At this time, theonly demonstrated method for making pulses ap-proaching the atomic time scale [47] is HHG in raregases. From the perspective of attosecond generationat shorter durations and wavelengths, evaluating theefficacy for producing high harmonic combs using longwavelength fundamental fields is pivotal.

4.2.1. Setup

This subsection briefly outlines the experimental setupshown in Figure 14. The high harmonics are generatedby focusing the laser pulses into a gas cell, operatingwith a continuous gas flow. The cell is a thin (1.5 mm)rectangular cross-section metal nozzle wrapped inTeflon (a trick first invented and tested in Lund [62]),backed with a pressure of up to 250 torr (significantlyhigher than usual with visible/near infrared drivingwavelengths). The laser drills holes through theTeflon, ensuring minimal gas load on the pumps andlimiting reabsorption of the XUV radiation in thegeneration chamber where the pressure is kept to1073–1074 torr.

The 0.5 m Hettrick design (Hettrick Scientific) softX-ray spectrometer is differentially pumped withrespect to the harmonic source chamber with the gascell placed at the spectrometer’s entrance plane. Itoperates at a 28 grazing angle and has three different

dispersion gratings. A Kirkpatrick–Baez mirror geo-metry collects and focuses the soft X-ray radiation onthe exit plane where it is detected on a back-illuminated, soft X-ray CCD camera (Andor Inc.).Different metal filters (Al or Zr depending on thewavelength range studied) are used to suppress thefundamental and low-order harmonic light.

Currently, harmonic generation is possible onlywith our 0.8 and 2 mm sources. The focusing optics arechosen to produce the same Rayleigh range at thetwo wavelengths and the same intensity (�1.8 6 1014

W cm72).

4.2.2. High frequency cutoff

The harmonic comb is a continuous series of odd-orderharmonics of the fundamental frequency that termi-nates at an energy defined by the cutoff law. Compar-ing the cutoff to the standard law (I0 þ 3.2UP) is thefirst goal of the experiment [29]. This is not as easy as itmay sound. The harmonic spectra at 2 mm arerecorded in three independent measurements withdifferent filters which have been separately normalised:aluminium (Figure 15) and zirconium for the low-energy portion and aluminum for high-energy one(Figure 16). First, Figure 15 shows the argon harmonicdistribution in the EUV region driven by 0.8 mm and2 mm fundamental fields, transmitted by an Al filter[63]. The 0.8 mm distribution shows a 50 eV cutoff (i.e.order 31) consistent with previous measurements, time-dependent numerical results [43] and the cutoff law. Incontrast, as seen in Figure 15 excitation with the longerwavelength produces a denser harmonic comb

Figure 14. High harmonic setup.

Figure 15. High harmonic comb produced in argon for anintensity of 1.8 6 1014 W cm72 with 0.8 mm (dashed) and2 mm (solid line) pulses through an Al filter (transmission indotted line). The cutoff at 75 eV is due to the Altransmission. The harmonic cutoff is visible in thefollowing figure.



(consequence of the smaller photon energy) extendingto the Al L-edge at *70 eV energy. Using a Zr filterinstead, the same argon harmonic comb is observed toextend over the entire Zr filter transmission window(60–200 eV) (Figure 16). The fast structure is the 2h�o(1.2 eV) spacing while the overall shape is a convolutedinstrumental response. Obviously, no harmonics aregenerated in this region using 0.8 mm excitation. A thirdset of measurements utilising the second Al-transmis-sion window is necessary to establish that the 2 mmharmonic cutoff is *220 eV because the measurementusing the zirconium filter is obviously spectrally limitedby the transmission drop at short wavelengths. Thiscutoff value is consistent with the TDSE calculations[43] and extends the cutoff in argon beyond the valuefound in a previous work at 1.5 mm [28] (Figure 16) asexpected from the longer driving wavelength.

4.2.3. Yield

While the above distributions clearly assert theeffectiveness of the l2-scaling for the production ofhigher energy harmonic photons, the dependence ofthe yield on the wavelength, albeit crucial information,is more difficult to predict theoretically. As alreadymentioned, numerical solutions of the TDSE show thatthe ‘single atom’ dipole power spectrum scales asl75.5 + 0.5 at constant intensity [43] (Figure 17). As inordinary nonlinear optics, the actual HHG depends

not only on this atomic response but also on themacroscopic effects, e.g. propagation and phase-matching. Difficult numerical simulations incorporat-ing both are in progress [64,65]. Although individualdispersive effects from free electrons, the Gouy phase(i.e. the p phase shift that a Gaussian beam undergoesat focus) and the intrinsic dipole phase [42] mostgenerally result in an even more rapid decrease of themacroscopic yield scaling with wavelength [28], controlparameters (focusing and target geometry, density)enable means for a global optimisation. Independentlyof theory, experiment can provide a valuable metricand, in some cases, good surprises. The change inharmonic yield in argon between 0.8 and 2 mm, wasmonitored while keeping the other conditions fixed,e.g. intensity, argon density, focusing and collectionsystem. Confining the comparison to a spectralbandwidth (35–50 eV) common to both wavelengths,the 2 mm harmonics are observed to be 1000-timesweaker than with 0.8 mm and 6-times less than thatpredicted by calculations [43]. Naturally, in an experi-ment one has the resource of independent optimisationof the parameters. Optimising only the argon densityfor maximum harmonic yield at both wavelengthsresults in a 2 mm yield that is only 85-times weakerthan the 0.8 mm one. This preliminary, not fullyoptimised result (pump intensity, focusing geometryare still available for optimisation), shows thatprobably the brightness of the 2 mm-pumped harmonicsource can be made comparable with that of 0.8 mmdriver in the EUV (�50 eV) and obviously superior athigher energies (50 � h�o � 200 eV).

4.2.4. Attochirp

The spectral goup delay dispersion is one of thefundamental limits of ultrashort pulses [66]. In thecontext of attosecond pulses Fourier-synthesised fromhigh harmonics, an inherent group delay dispersion

Figure 16. High harmonic spectra produced in argon for anintensity of 1.8 6 1014 W cm72 at 2 mm through Zr (filled)and Al (unfilled) filters together with the filter transmissioncurves (dot-dash and dot, respectively). The graphs are notcorrected for the grating efficiency [63]. The real cutoff (*220eV) is seen in the unfilled curve. Reprinted figure withpermission from P. Colosimo et al., Nature Phys. 4, 386,2008. Copyright (2008) by Nature Publishing Group.

Figure 17. Yield versus wavelength from strong fieldapproximation (SFA) (crosses) and TDSE calculations(dots) from [43]. The strong decrease is discussed in thetext. Reprinted figure with permission from J. Tate et al.,Phys. Rev. Lett., 98, 013901, 2007. Copyright (2007) by theAmerican Physical Society.



exists, due to relationship between one harmonic (orrather a group of contiguous harmonics) and thelength of the electron trajectories that generates them[67] (Section 2.3). Hence the spectral amplitude alone(Figure 15 for instance), is not sufficient information tomodel the corresponding time domain pulse train butmust be completed by the spectral phase.

A number of pump–probe methods able to measurethe spectral phase have been demonstrated. A firstgroup including RABBITT, atto-streak and CRAB isbased on the analysis of the photoelectron spectraproduced by the superposition of the group ofharmonics under examination and the fundamentalfrequency. In RABBITT [36,37] the spectral phase isretrieved from a series of spectra recorded as afunction of the delay between harmonic and funda-mental pulses. In the atto-streak: an attosecondphotoelectron pulse is measured by the change ofmomentum induced by the laser optical field [68]while CRAB is a phase-only variant of the FROGmethod [47].

A second type of method [69] is purely optical:harmonics are generated by a superposition of afundamental driving laser and its second harmonic.For a small proportion of second harmonic, even orderharmonics are generated, e.g. by absorption of an evennumber of fundamental photons and one secondharmonic photon. (The total number of exchangedphotons, including the harmonic one, must be evensince the atom is making a transition between twostates of the same parity, namely from and to theground state.) Although standard nonlinear opticsexplains the generation of even-order harmonics thisway, the perturbative calculation is simply impossiblefor the orders of Figure 15. A very clever method ofcalculation is proposed in [69]. It is based onevaluating the asymmetry of the electron trajectoriesimposed by the ‘right–left’ asymmetry of the totalpump field for linearly polarised fields and assimilatingthe strength of the even harmonics with this asymme-try which, in first approximation, can be calculatedfrom classical mechanics. The schematic of theexperiment as it is implemented at OSU is shown inFigure 18.

After a test at l ¼ 0.8–0.4 mm, the method hasbeen implemented at 2–1 mm. Figure 19 shows apreliminary result where one can get a hint of theobserved dispersion and quadratic spectral phase.Processing and tests are in progress. Although at themoment it is not yet possible to assert the wavelengthdependence of the ‘attochirp’, it will be in theimmediate future. This measurement is of the utmostimportance for the future of attosecond generation atlong wavelength. Moreover the attochirp is of centralimportance in orbital tomography based on high

harmonic generation by aligned molecules [70] (see[71] for a recent review).

5. Conclusion

Strong-field interactions using the emerging longwavelength laser technology has potential impact inboth fundamental and applied investigations. Thebasic quasi-classical principles that provide the foun-dation of strong-field physics can be explored and theirscaling laws tested. In contrast to varying the intensity,

Figure 18. Setup of the o – 2o method. A small part of thefundamental beam is taken for second harmonic generation.The two beams are then superposed and a fine-tuning timedelay is introduced before the harmonic cell.

Figure 19. Typical graph of the harmonic spectrum versusthe delay between the o and 2o pulses. Each even harmonic(order indicated on the left) is strongly modulated. Thespectral phase difference of the two odd harmonics adjacentto it can be retrieved from the phase of this modulation [69].



wavelength offers the experimentalist the ability tocontinuously tune the interaction from tunnelling tomultiphoton regimes, explore a broader class ofatomic/molecular systems by exploiting the Keldyshscaling and realise novel target configurations.Furthermore, the two-boost in the ponderomotiveenergy can be exploited to achieve unprecedentedinteraction energies in the quasi-classical limit or inprinciple, achieve relativistic electron motion in neutralatom ionisation.

From an applied perspective, long wavelength pulsesdriving a high harmonic source may be the preferredroute towards brighter and more energetic attosecondbursts. In addition, the ponderomotive boost can alsoresult in better and less intrusive metrology.

All of these potentially interesting directions wouldbe academic if the prospect for generating longwavelength pulses was improbable. As demonstratedin this review, nonlinear optics provides one obvioussolution although ultimately a long wavelength sourcecompetitive with the best ultra-fast titanium:sapphirelaser system would be optimal. The advent of a newlaser amplifier architecture called optical parametricchirp pulse amplification has the potential of filling thisvoid and thus enabling a new class of strong-fieldinvestigations.

Acknowledgement

The authors are grateful for the dedication of The Ohio StateUniversity group for the investigations presented in thisreview. In particular, we thankCosmin Blaga, Fabrice Catoire,Phil Colosimo, Gilles Doumy, Anne Marie March, JenniferTate and Jonathan Wheeler. We also acknowledge the fruitfulcollaboration with Professor Gerhard Paulus at the Universityof Jena. LFD acknowledges support from the Edward andSylvia Hagenlocker chair endowment. The authors acknowl-edge support from the US Department of Energy, BasicEnergy Sciences and the National Science Foundation.

Notes on contributors

Pierre Agostini is Professor of Physics at the Ohio StateUniversity since 2003. He was for more than 30 years at theCEA Saclay (France). He has been the host of USC (LosAngeles), FOM-Amolf (Amsterdam), Laval University(Quebec) and Max Born Institute (Berlin). He is a Humboldtand OSA fellow. His current research interest in laser–matterinteraction includes strong-field ionisation, nonlinear opticsand high harmonic generation, X-ray ultrafast metrologyand molecular dynamical imaging.

Louis F. DiMauro is the Edward and Sylvia HagenlockerChair and Professor of Physics at The Ohio State Universityand a Fellow of AAAS, APS and OSA. He completed hisPh.D. in 1980 at the University of Connecticut and was apostdoctoral fellow at SUNY at Stony Brook until 1982 underthe supervision of Professor Harold Metcalf. His researchinterest is in experimental ultra-fast and strong-field physics.His current work is focused on the generation, measurementand application of attosecond X-ray pulses, and investigations

of the fundamental scaling properties of strong-field atomicphysics using lasers and X-ray free-electron lasers.

References

[1] Z. Fried and J.H. Eberly, Scattering of a High-Intensity,Low-Frequency Electromagnetic Wave by an UnboundElectron, Phys. Rev. 136 (1964), pp. B871–B887; J.H.Eberly, Interaction of very intense light with free electron,Progr. Opt. 7 (1969), pp. 361–415.

[2] F.J. McClung and R.W. Hellwarth, Giant pulsationsfrom Ruby, J. Appl. Phys. 33 (1962), pp. 828–829.

[3] A. Gold and H.B. Bebb, Theory of multiphotonionization, Phys. Rev. Lett. 14 (1965), pp. 60–63.

[4] M. Goppert-Mayer, ber Elementarakte mit zwei Quan-tensprngen, Ann. Phys. 401 (1931), pp. 273–294.

[5] W. Zernik, Two-photon ionization of atomic hydrogen,Phys. Rev. 135 (1964), pp. A51–A57.

[6] A. Gold and H.B. Bebb, Multiphoton ionization ofhydrogen and rare-gas atoms, Phys. Rev. 143 (1966), pp.1–24.

[7] E.K. Damon and R.G. Tomlinson, Observation ofIonization of Gases by a Ruby Laser, Appl. Opt. 2(1963), pp. 546–547.

[8] L.V. Keldysh, Ionization in the field of a strongelectromagnetic wave, Sov. Phys. JETP 20 (1965), pp.1307–1314.

[9] P.W. Milonni, Shortcomings of the Keldysh approxima-tion, Phys. Rev. A 38 (1988), pp. 2682–2685.

[10] M. Fedorov, in Atomic and Free Electrons in a StrongLight Field (World Scientific, Singapore 1991).

[11] A.M. Perelomov, V.S. Popov, and M.V. Terent’ev,Ionization of atoms in an alternating electric field, Sov.Phys. JETP 23 (1966), pp. 924–934.

[12] F.H. Faisal, Multiple absorption of laser photonsby atoms, J. Phys. B: At. Mol. Phys. 6 (1973), pp.L89–L92.

[13] H.R. Reiss, Effect of an intense electromagnetic field on aweakly bound system, Phys. Rev. A 22 (1980), pp. 1786–1813.

[14] S.L. Chin, F. Yergeau, and P. Lavigne, Tunnel ionisationof Xe in an ultra-intense CO, laser field (1014 Wcm72,with multiple charge creation), J. Phys. B: At. Mol Phys18 (1985), pp. L213–L215.

[15] M.V. Amosov, N.B. Delone, and V.P. Krainov, Tunnelionization of complex atoms and of atomic ions in analternating electromagnetic field, Sov. Phys. JETP 64(1986), pp. 1191–1194.

[16] F.A. Ilkov, J.E. Decker, and S.L. Chin, Ionization ofatoms in the tunneling regime with experimental evidenceusing Hg, J. Phys. B: At. Mol Phys 25 (1992), pp. 4005–4020.

[17] K.C. Kulander, Time-dependent Hartree-Fock theory ofmultiphoton ionization: helium, Phys. Rev. A 36 (1987),pp. 2726–2738.

[18] P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N.K.Rahman, Free-Free Transitions Following Six-PhotonIonization of Xenon Atoms, Phys. Rev. Lett. 42 (1979),pp. 1127–1130.

[19] E. Karule, Two-photon ionisation of atomic hydrogensimultaneously with one-photon ionisation, J. Phys. B: At.Mol Phys. 11 (1978), pp. 441–447.

[20] Y. Gontier, M. Poirier, and M. Trahin, Multiphotonabsorptions above the ionisation threshold, J. Phys. B: At.Mol. Phys. 13 (1980), pp. 1381–1387.



[21] P. Agostini and G. Petite, Photoelectric effect understrong irradiation, Contemp. Phys. 29 (1988), pp. 57–77.

[22] H.R. Reiss, Inherent contradictions in the tunnellingmultipphoton dichotomy, Phys. Rev. A 75 (2007), pp.031404-1–4.

[23] H.B. van Linden, van den Heuvell, and H.G. Muller,Limiting cases of excess-photon ionization, in Multi-photon Processes, S.J. Smith, and P.L. Knight, eds.,Cambridge University Press, 1987, pp. 25–34.

[24] K.J. Schafer, B Yang, L.F. DiMauro, and K.C.Kulander, Above threshold ionization beyond the highharmonic cutoff, Phys. Rev Lett. 70 (1993), pp. 1599–1602.

[25] P.B.Corkum,Plasmaperspective on strongfieldmultiphotonionization, Phys. Rev. Lett. 71 (1993), pp. 1994–1997.

[26] P. Agostini and L.F. DiMauro, The physics of attosecondlight pulses, Rep. Prog. Phys. 67 (2004), pp. 813–855.

[27] T.F. Gallagher, Above-Threshold Ionization in Low-Frequency Limit, Phys. Rev. Lett. 61 (1988), pp. 2304–2307; P.B. Corkum, N.H. Burnett, and F. Brunel,Above-threshold ionization in the long-wavelength limit,Phys. Rev. Lett. 62 (1989), pp. 1259–1962.

[28] B. Sheehy, J.D.D. Martin, L.F. DiMauro, P. Agostini,K.J. Schafer, M.B. Gaarde, and K.C. Kulander, HighHarmonic Generation at Long Wavelengths, Phys. Rev.Lett. 83 (1999), pp. 5270–5273.

[29] B. Shan and Z. Chang, Dramatic extension of the high-order harmonic cutoff by using a long-wavelength drivingfield, Phys. Rev. A 65 (2001), p. 011804-1–4.

[30] P. Colosimo,G. Doumy, C.I. Blaga, J.Wheeler, C.Hauri,F. Catoire, J. Tate, R. Chirla, A.M. March, G.G. Paulus,H.G. Muller, P. Agostini, and L.F. DiMauro, Scalingstrong-field interactions towards the classical limit, NaturePhys. 4 (2008), pp. 386–389.

[31] W. Becker, et al., Above-threshold ionization: Fromclassical features to quantum effects, Adv. At. Mol.Opt. Phys. 48 (2002), pp. 35–98.

[32] G.G. Paulus, et al., Rescattering effects in above-thresh-old ionization: a classical model, J. Phys. B At. Mol. Opt.Phys. 27 (1994), pp. L703–L708.

[33] B. Walker, et al., Elastic Rescattering in the Strong FieldTunnelingLimit, Phys.Rev.Lett. 77 (1994), pp. 5031–5034.

[34] J. Wassaf, et al., Strong Field Atomic Ionization: Originof High-Energy Structures in Photoelectron Spectra,Phys. Rev. Lett. 90 (2003), pp. 013303–013306.

[35] J. Schins, et al.,Observation ofLaser-AssistedAugerDecayin Argon, Phys. rev. Lett. 73 (1994), pp. 2181–2184.

[36] P.M. Paul, et al., Observation of a Train of AttosecondPulses from High Harmonic Generation, Science 292(2001), pp. 1689–1692.

[37] G.G. Paulus, et al., Absolute-phase phenomena inphotoionization with few-cycle laser pulses, Nature 414(2001), pp. 182–184.

[38] B.W. Shore and P.L. Knight, Enhancement of highoptical harmonics by excess-photon ionisation, J. Phys. BAt. Mol. Opt. Phys. 20 (1987), pp. 413–423.

[39] R.L. Carman, C.K. Rhodes, and R.F. Benjamin,Observation of harmonics in the visible and ultravioletcreated in CO2-laser-produced plasmas, Phys. Rev. A 24(1981), pp. 2649–2663.

[40] A. L’Huillier, et al., Multiple-harmonic generation in raregases at high laser intensity, Phys. Rev. A 39 (1989), pp.5751–5761.

[41] M. Lewenstein, et al., Theory of high-harmonic genera-tion by low-frequency laser fields, Phys. Rev. A 49 (1994),pp. 2117–2132.

[42] P. Salieres, A. L’Huillier, and M. Lewenstein, CoherenceControl of High-Order Harmonics, Phys. Rev. Lett. 74(1995), pp. 3776–3779.

[43] J. Tate, et al., Scaling of Wave-Packet Dynamics in anIntense Midinfrared Field, Phys. Rev. Lett. 98 (2007), pp.013901–013904.

[44] W. Becker, S. Long, and J.K. McIver, Modelingharmonic generation by a zero-range potential, Phys.Rev. A 50 (1994), pp. 1540–1560.

[45] G. Farkas and S. Toth, Proposal for attosecond pulsegeneration using laser-induced multiple-harmonic conver-sion processes in rare gases, Phys. Lett. A 168 (1992), pp.447–450; S.E. Harris, et al., Atomic scale temporalstructure inherent to high-order harmonic generation,Opt. Commun. 100 (1993), pp. 487–490.

[46] P. Antoine, A. L’Huillier, and M. Lewenstein, Attose-cond Pulse Trains Using HighOrder Harmonics, Phys.Rev. Lett. 77 (1996), pp. 1234–1237.

[47] G. Sansone, et al., Isolated Single-Cycle AttosecondPulses, Science 314 (2006), pp. 443–446.

[48] M. Lewenstein, et al., Phase of the atomic polarization inhigh-order harmonic generation, Phys. Rev. A 52 (1995),pp. 4747–4754.

[49] R.W. Boyd, Nonlinear Optics, Academic Press, 1992.[50] P. Colosimo, A Study of Wavelength Dependence of

Strong-Field Optical Ionization. Ph.D. dissertation,Stony Brook, August 2007.

[51] G.G. Paulus, et al., Channel-closing-induced resonancesin the above-threshold ionization plateau, Phys. Rev. A 64(2001), pp. 021401–021404.

[52] H.G. Muller and F.C. Kooiman, Bunching and Focusingof Tunneling Wave Packets in Enhancement of High-Order Above-Threshold Ionization, Phys. Rev. Lett. 81(1998), pp. 1207–1210.

[53] R.M. Potvliege and S. Vu�cic, High-order above-thresholdionization of argon: Plateau resonances and the Floquetquasienergy spectrum, Phys. Rev. A 74 (2006), pp.023412–023423.

[54] M.P. Hertlein, P.H. Bucksbaum, and H.G. Muller,Evidence for resonant effects in high-order ATI spectra, J.Phys. B 30 (1997), pp. L197–L206.

[55] M.J. Nandor, et al., Detailed comparison of above-threshold-ionization spectra from accurate numericalintegrations and high-resolution measurements, Phys.Rev. A 60 (1999), pp. R1771–R1774.

[56] P. Colosimo, et al., unpublished.[57] P. Avan, et al., Effect of high frequency irradiation on the

dynamical properties of weakly bound electrons, J. Phys.(Paris) 37 (1976), pp. 993–1010.

[58] R. Freeman, et al., Above-threshold ionization withsubpicosecond laser pulses, Phys. Rev. Lett. 59 (1987),pp. 1092–1095.

[59] L.F. Di Mauro and P. Agostini, Ionization Dynamics instrong laser fields, Adv. At. Mol. Opt. Phys. 35 (1995),pp. 79–120.

[60] F. Catoire, et al., Non classical electron energyspectra in strong field ionization. Nat. Phys (inpreparation).

[61] G. Paulus, et al., Above-threshold ionization spectra atmid-infrared wavelengths, Phys. Rev. (in preparation).

[62] A. L’Huillier, private communication.[63] G. Doumy, et al., Strong-field physics using mid-infrared

lasers, Proceedings SPIE Conference, San Jose, 2007,Invited paper 6460-24.

[64] T. Auguste, et al., Harmonic yield scaling with drivinglaser wavelength, Phys. Rev. (in press).



[65] V.S. Yakovlev, M. Ivanov, and F. Krausz, Enhancedphase-matching for generation of soft X-ray harmonicsand attosecond pulses in atomic gases, Opt. Express 15(2007), pp. 15351–15364.

[66] J.C. Diels and W. Rudolf, Ultrashort Laser PulsePhenomena, Academic Press, 1996.

[67] Y. Mairesse, et al., Attosecond Synchronization of High-Harmonic Soft X-rays, Science 302 (2003), pp. 1540–1543.

[68] M. Hentschel, et al., Attosecond metrology, Nature 414(2001), pp. 509–513.

[69] N. Dudovich, et al., Measuring and controlling thebirth of attosecond XUV pulses, Nature Phys. 2 (2006),pp. 781–786.

[70] J. Itatani, et al., Tomographic imaging of molecularorbitals, Nature 432 (2004), pp. 867–871.

[71] J.P. Marangos, et al., Dynamic imaging of moleculesusing high order harmonic generation, Phys. Chem.Chem. Phys 10 (2008), pp. 35–48.



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