1 Exercise 11 May 26 2017 Laser Induced Electron Diffraction and Electron Holography Laser Induced Electron Diffraction In this exercise, we will introduce the basic ideas behind the ultrafast laser-induced electron diffraction (LIED) and electron holography concepts. We will combine several concepts, previously introduced in this class, such as electron-ion elastic scattering combined with strong-laser-field ionization (SFI) and the semi-classical model of the electron propagation in a linearly polarized laser field. For the preparation for this class please read the following articles 1-3 . 1. Huismans, Y. et al. Time-resolved holography with photoelectrons. Science 331, 61–64 (2011). 2. Hickstein, D. D., Ranitovic, P., Witte, S. Tong, X. M. et al. Direct Visualization of Laser-Driven Electron Multiple Scattering and Tunneling Distance in Strong-Field Ionization. Phys Rev Lett (2012). 3. Morishita, T., Le, A.-T., Chen, Z. & Lin, C. D. Accurate retrieval of structural information from laser-induced photoelectron and high-order harmonic spectra by few-cycle laser pulses. Phys Rev Lett 100, 013903 (2008). In the three-step model of strong-field light-matter interactions, the electron ionized slightly after the peak of the laser field can return to its parent ion with the maximum energy of 3.17U p , where U p stands for the ponderomotive potential of the laser field. Compared with the traditional electron diffraction (ED) concepts, used for retrieving atomic and molecular structure, the LIED can be seen as self-imaging tool with a potential for resolving molecular structural dynamics. At the instant of the rescattering the attosecond electron wave packet can elastically scatter from a parent ion, with a certain differential cross section. Since the electrons are born in the continuum at different phases of the laser field and travel different classical trajectories, many of them miss the parent ion, while some of them elastically scatter off the parent ion. The electrons that scatter off the parent ion can be used for self-imaging of the molecular structure with the attosecond temporal an picometer spatial resolution. From left to right, Figure 1 shows a typical few-femtosecond laser electric field, calculated 2D electron momenta ionized from H atom by a strong laser filed, and the electron-ion differential cross section (DCS) calculated from the back- scattering rings (BRRs). Figure 1 Left: Laser electric field (E) and the corresponding vector potential (A) with a two-optical-cycle pulse duration. Middle: 2D electron momenta with two BRRs at P z =-1.8 a.u. and P z =2 a.u., and the electron holography structure in the range from P z =-1.2 a.u. to P z = 1.2 a.u.. Right: Calculated electron diffraction (ED) differential cross-section (DCS) and the LIED DCS (dashed line). As labeled by the red and blue arrows in Fig. 1 a), when the electron tunnels to the right-hand side of the hydrogen atom, along the laser polarization direction (i.e. position “a” in the left panel of Fig. 1), at the instant of the rescattering (position “b” in the left panel of Fig. 1) the electron gets a kick back from laser field (i.e. the
Transcript
1
Exercise11May262017
Laser Induced Electron Diffraction and Electron Holography
LaserInducedElectronDiffractionInthisexercise,wewillintroducethebasicideasbehindtheultrafastlaser-inducedelectrondiffraction(LIED)andelectronholographyconcepts.Wewillcombineseveralconcepts,previouslyintroducedinthisclass,suchaselectron-ionelasticscatteringcombinedwithstrong-laser-fieldionization(SFI)andthesemi-classicalmodeloftheelectronpropagationinalinearlypolarizedlaserfield.Forthepreparationforthisclasspleasereadthefollowingarticles1-3.1. Huismans, Y. et al. Time-resolved holography with photoelectrons. Science 331, 61–64 (2011). 2. Hickstein, D. D., Ranitovic, P., Witte, S. Tong, X. M. et al. Direct Visualization of Laser-Driven Electron Multiple Scattering
and Tunneling Distance in Strong-Field Ionization. Phys Rev Lett (2012). 3. Morishita, T., Le, A.-T., Chen, Z. & Lin, C. D. Accurate retrieval of structural information from laser-induced photoelectron and
high-order harmonic spectra by few-cycle laser pulses. Phys Rev Lett 100, 013903 (2008). Inthethree-stepmodelofstrong-fieldlight-matterinteractions,theelectronionizedslightlyafterthepeakofthe laser field can return to its parent ionwith themaximum energy of 3.17Up, whereUp stands for theponderomotivepotentialofthelaserfield.Comparedwiththetraditionalelectrondiffraction(ED)concepts,usedforretrievingatomicandmolecularstructure,theLIEDcanbeseenasself-imagingtoolwithapotentialforresolvingmolecularstructuraldynamics.Attheinstantoftherescatteringtheattosecondelectronwavepacketcanelasticallyscatterfromaparention,withacertaindifferentialcrosssection.Sincetheelectronsareborninthecontinuumatdifferentphasesofthelaserfieldandtraveldifferentclassicaltrajectories,manyofthemmisstheparention,whilesomeofthemelasticallyscatterofftheparention.Theelectronsthatscatterofftheparentioncanbeusedforself-imagingofthemolecularstructurewiththeattosecondtemporalanpicometerspatialresolution.Fromlefttoright,Figure1showsatypicalfew-femtosecondlaserelectricfield,calculated2DelectronmomentaionizedfromHatombyastrong laser filed,andtheelectron-iondifferentialcrosssection (DCS)calculated fromtheback-scatteringrings(BRRs).
As labeledbytheredandbluearrows inFig.1a),whentheelectrontunnels to theright-handsideof thehydrogenatom,alongthelaserpolarizationdirection(i.e.position“a”intheleftpanelofFig.1),attheinstantoftherescattering(position“b”intheleftpanelofFig.1)theelectrongetsakickbackfromlaserfield(i.e.the
2
vectorpotentialisatitsmaximumatthispoint)inthesamedirectionwheretheelectroncamefrom.Ontheother hand, at the position “a’”, an electron is released on to the left-hand side of the atom, and it canbackscattertotheleftbygettingthemomentumkickfromthelaserfield(i.e.vectorpotentialatinstant“b’”).
2) Use the calculated BRRmomenta for the highest energy ring in Fig. 1b) to approximate the laserintensityusedinthiscalculation.Hint:usethecut-offenergyalongthelaserpolarizationfromFig.1b).
ElectronHolographyThe complex structure in the electron-momentum distribution between the BRRs in Fig. 1 c), taken fromreference3,wasconsideredtobetoocomplextointerpretindetail,andexplainedintermsofdirectionizationfromthe laser field. In thispartof theexercise,wewill showthat themajor featuresof these2Delectronmomentacanbequantitativelyreproducedusingtheconceptsandmodelsintroducedinthisclass.Inreference1,itwasshownthattheelectronmomentacanbeenseenas2Delectronholographypatterns,wherethedirectandrescatteredelectronsinterfereontheelectrondetector.Inreference2,itwasshownthattheelectronholographicpatternscanbequantitativelyreproducedbyusingasphericalandplanewave(SPW)interferencemodelwhichassumesthattheelectronwavepacket,freedfromtheionbythestronglaserfield,canbeseenas a planewave, and the rescattered electrons can be seen as a spherical wave. The interference of thesphericalandplanewavescreatescomplexelectron2Dmomentawithseveralminimaandmaximaalongthelaserpolarization.Suchstructurewasnamedspiderstructureinref.3.Thedistancebetweentheinterferenceminimadependssolelyontherelativephasesofthesphericalandplaneelectronwavepackets,andisgivenbythetimetheelectronspendsinthecontinuumbeforetheinstantoftherescattering.
(the black line). The phase of the plane wave is given by , and the phase of the spherical wave is given by
quantum phase accumulated by the electrons during theiroscillatory path between ionization and rescattering isdirectly imprinted onto the photoelectron angular distribu-tion. Thus, the shape and spacing of the interference struc-tures directly correspond to the specific number of times
the electron reencounters its parent ion before scatteringstrongly. Through an analysis of the energy cutoff of thesenewly observed structures, we show that when an atom isionized by an intense laser field, the electron emerges at afinite distance from its parent ion. This distance corre-sponds to the far side of the quantum tunneling barrier.Our simple model allows us to uncover valuable physicalinsight into strong-field electron dynamics and provides anexplanation for many of the features previously seen inexperimental photoelectron distributions.Our experimental apparatus consists of an ultrafast 30 fs
Ti:sapphire laser and a velocity-map imaging photoelec-tron spectrometer. Nonlinear crystals and an optical para-metric amplifier generate ultrafast pulses with wavelengthsspanning 0.26 to 2 !m and durations of 40 fs. Spectra wererecorded at various wavelengths between 0.26 and 2:0 !m,with intensities ranging from 5! 1013 to 5! 1014 W=cm2
for xenon and argon gases. For driving wavelengths be-tween 0.39 and 2:0 !m, we observe interference structuresaligned along the laser polarization axis [Fig. 1(a)].Huismans and co-workers [17] observed these structuresand interpreted them in terms of an interference betweenlaser-driven electron trajectories that recollide with the ionand electron trajectories that do not. In this Letter, we referto these angular features as ‘‘spider structures’’ because oftheir resemblance to the body and legs of a spider. Fordriving laser wavelengths of 1.3 and 2 !m, we observeadditional ‘‘inner spider’’ structures [Fig. 1(b)] at lowfinal momenta that have more closely spaced fringes.Unlike other low-energy angular structures reported inthe literature [27–29], these inner spider structures remain
FIG. 1 (color online). (a) The photoelectron angular distribu-tion resulting from the ionization of argon gas with a 1300 nm,7:5! 1013 W=cm2, 40 fs laser pulse. The primary spider struc-ture (minima shown with white lines) results from the interfer-ence between directly ionized electrons and those that are drivenback to scatter from the Coulomb potential. (b) The newlyobserved inner structure (minima shown with red lines) resultsfrom electron trajectories that scatter from the Coulomb poten-tial on their second reencounter with the ion and interfere withunscattered trajectories. (a)–(e) Simple laser-driven electrondynamics during ionization explain the primary and inner spiderstructures.
FIG. 2 (color online). The PSW model. The superposition of aplane wave (a) and a spherical wave (b) generates an interferencepattern (c) that has the same shape as the spider structures. Thephase of the plane wave is given by !plane / eikPk , while the
phase of the spherical wave is given by !spherical / eikPtotal ,
where Pk is the momentum parallel to the laser polarization,
lation frequency of the plane and spherical waves. Only the realpart of the complex wave is shown in panels (a) and (b), whilej!j2 is shown in panel (c). (d) The minima of thespider structure calculated using the plane-spherical model(white lines) match the experimental data across a broadrange of wavelengths and intensities (shown: 1300 nm,5! 1013 W=cm2, 40 fs driving laser).
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5) Assumingthatattheinstantoftherescatteringtheelectronisbacktoitsorigin(i.e.z(tr)=0),solvetheaboveequationnumericallyandplot the initial versus returningphase foran800nmwavelength.Confirmthattheelectronbornat189owouldreturn2/3oftheoptical-cyclelater.
Figure 3 Action “S” vs the parallel momenta for three different laser wavelengths. The given slopes define the modulationfrequenciesthatdependonlyonthelaserwavelengthforthegivenlaserintensity.Thephaseoftheelectroninthelaserfieldisgivenbye-iS/h.
quantum phase accumulated by the electrons during theiroscillatory path between ionization and rescattering isdirectly imprinted onto the photoelectron angular distribu-tion. Thus, the shape and spacing of the interference struc-tures directly correspond to the specific number of times
the electron reencounters its parent ion before scatteringstrongly. Through an analysis of the energy cutoff of thesenewly observed structures, we show that when an atom isionized by an intense laser field, the electron emerges at afinite distance from its parent ion. This distance corre-sponds to the far side of the quantum tunneling barrier.Our simple model allows us to uncover valuable physicalinsight into strong-field electron dynamics and provides anexplanation for many of the features previously seen inexperimental photoelectron distributions.Our experimental apparatus consists of an ultrafast 30 fs
Ti:sapphire laser and a velocity-map imaging photoelec-tron spectrometer. Nonlinear crystals and an optical para-metric amplifier generate ultrafast pulses with wavelengthsspanning 0.26 to 2 !m and durations of 40 fs. Spectra wererecorded at various wavelengths between 0.26 and 2:0 !m,with intensities ranging from 5! 1013 to 5! 1014 W=cm2
for xenon and argon gases. For driving wavelengths be-tween 0.39 and 2:0 !m, we observe interference structuresaligned along the laser polarization axis [Fig. 1(a)].Huismans and co-workers [17] observed these structuresand interpreted them in terms of an interference betweenlaser-driven electron trajectories that recollide with the ionand electron trajectories that do not. In this Letter, we referto these angular features as ‘‘spider structures’’ because oftheir resemblance to the body and legs of a spider. Fordriving laser wavelengths of 1.3 and 2 !m, we observeadditional ‘‘inner spider’’ structures [Fig. 1(b)] at lowfinal momenta that have more closely spaced fringes.Unlike other low-energy angular structures reported inthe literature [27–29], these inner spider structures remain
FIG. 1 (color online). (a) The photoelectron angular distribu-tion resulting from the ionization of argon gas with a 1300 nm,7:5! 1013 W=cm2, 40 fs laser pulse. The primary spider struc-ture (minima shown with white lines) results from the interfer-ence between directly ionized electrons and those that are drivenback to scatter from the Coulomb potential. (b) The newlyobserved inner structure (minima shown with red lines) resultsfrom electron trajectories that scatter from the Coulomb poten-tial on their second reencounter with the ion and interfere withunscattered trajectories. (a)–(e) Simple laser-driven electrondynamics during ionization explain the primary and inner spiderstructures.
FIG. 2 (color online). The PSW model. The superposition of aplane wave (a) and a spherical wave (b) generates an interferencepattern (c) that has the same shape as the spider structures. Thephase of the plane wave is given by !plane / eikPk , while the
phase of the spherical wave is given by !spherical / eikPtotal ,
where Pk is the momentum parallel to the laser polarization,
lation frequency of the plane and spherical waves. Only the realpart of the complex wave is shown in panels (a) and (b), whilej!j2 is shown in panel (c). (d) The minima of thespider structure calculated using the plane-spherical model(white lines) match the experimental data across a broadrange of wavelengths and intensities (shown: 1300 nm,5! 1013 W=cm2, 40 fs driving laser).
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quantum phase accumulated by the electrons during theiroscillatory path between ionization and rescattering isdirectly imprinted onto the photoelectron angular distribu-tion. Thus, the shape and spacing of the interference struc-tures directly correspond to the specific number of times
the electron reencounters its parent ion before scatteringstrongly. Through an analysis of the energy cutoff of thesenewly observed structures, we show that when an atom isionized by an intense laser field, the electron emerges at afinite distance from its parent ion. This distance corre-sponds to the far side of the quantum tunneling barrier.Our simple model allows us to uncover valuable physicalinsight into strong-field electron dynamics and provides anexplanation for many of the features previously seen inexperimental photoelectron distributions.Our experimental apparatus consists of an ultrafast 30 fs
Ti:sapphire laser and a velocity-map imaging photoelec-tron spectrometer. Nonlinear crystals and an optical para-metric amplifier generate ultrafast pulses with wavelengthsspanning 0.26 to 2 !m and durations of 40 fs. Spectra wererecorded at various wavelengths between 0.26 and 2:0 !m,with intensities ranging from 5! 1013 to 5! 1014 W=cm2
for xenon and argon gases. For driving wavelengths be-tween 0.39 and 2:0 !m, we observe interference structuresaligned along the laser polarization axis [Fig. 1(a)].Huismans and co-workers [17] observed these structuresand interpreted them in terms of an interference betweenlaser-driven electron trajectories that recollide with the ionand electron trajectories that do not. In this Letter, we referto these angular features as ‘‘spider structures’’ because oftheir resemblance to the body and legs of a spider. Fordriving laser wavelengths of 1.3 and 2 !m, we observeadditional ‘‘inner spider’’ structures [Fig. 1(b)] at lowfinal momenta that have more closely spaced fringes.Unlike other low-energy angular structures reported inthe literature [27–29], these inner spider structures remain
FIG. 1 (color online). (a) The photoelectron angular distribu-tion resulting from the ionization of argon gas with a 1300 nm,7:5! 1013 W=cm2, 40 fs laser pulse. The primary spider struc-ture (minima shown with white lines) results from the interfer-ence between directly ionized electrons and those that are drivenback to scatter from the Coulomb potential. (b) The newlyobserved inner structure (minima shown with red lines) resultsfrom electron trajectories that scatter from the Coulomb poten-tial on their second reencounter with the ion and interfere withunscattered trajectories. (a)–(e) Simple laser-driven electrondynamics during ionization explain the primary and inner spiderstructures.
FIG. 2 (color online). The PSW model. The superposition of aplane wave (a) and a spherical wave (b) generates an interferencepattern (c) that has the same shape as the spider structures. Thephase of the plane wave is given by !plane / eikPk , while the
phase of the spherical wave is given by !spherical / eikPtotal ,
where Pk is the momentum parallel to the laser polarization,
lation frequency of the plane and spherical waves. Only the realpart of the complex wave is shown in panels (a) and (b), whilej!j2 is shown in panel (c). (d) The minima of thespider structure calculated using the plane-spherical model(white lines) match the experimental data across a broadrange of wavelengths and intensities (shown: 1300 nm,5! 1013 W=cm2, 40 fs driving laser).
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1. High-Harmonic Generation
where the electric field amplitude, or so called carrier envelope E0(t) varies very slowly in
comparison with carrier phase so we can take it to be a constant. Approximating that v(t =
t0)=0, from equation 1.14 we find the initial electron velocity or so called drift momentum to
be:
v0 = �e E0(t)
me!sin(!t0) and (1.16)
v(t) =e E0(t)
me![sin(!t)� sin(!t0)], (1.17)
as a function of the initial phase !t0. Substituting further 1.16 into 1.15 and approximating
again that the electron tunnels out of the potential barrier at z(t = t0)=0, from 1.15 we find
the electron’s amplitude time dependence to be:
z0 =e E0(t)
me!sin(!t0)t0 +
e E0(t)
me!2
cos(!t0) and (1.18)
z(t) =e E0(t)
me!2
[sin(!t0)(!t0 � !t) + cos(!t0)� cos(!t)]. (1.19)
If the electron revisits its parent ion at instant ⌧ , where ⌧ > t0, then z(⌧) = 0 gives us how
the returning electron phase !⌧ depends on its initial phase !t0. From 1.19 we find:
A numerical solution of this equation is given in figure 1.2 b).
Using equation 1.17 we can determine the maximum kinetic energy an electron can acquire
in the laser electric field by plotting its kinetic energy at instant ⌧ versus its initial phase. The
kinetic energy is:
Ekin(t0, ⌧) =e2
E
20
2me!2[sin(!⌧)� sin(!t0)]
2, or (1.21)
Ekin(t0, ⌧) = 2Up
[sin(!⌧)� sin(!t0)]2 and (1.22)
from figure 1.2 a) we see that an electron born at !t0 = 197� returns to its parent ion with a
maximum kinetic energy which is just a function of the laser field intensity and its wave length.
The ponderomotive energy U
p
is defined in equation 1.5. From this classical approximation we
can determine that the maximum kinetic energy an electron can gain in the laser field before
revisiting its parent ion for the first time is:
Ekin,max = 3.17Up. (1.23)
10
remarkably robust to changes in the intensity of the drivinglaser.
To physically interpret both the inner and primary spiderstructures, we begin with the three-step (recollision) modelof strong-field high-harmonic generation, which starts withtunneling ionization of an electron from the neutral atom inthe presence of a strong laser field [Fig. 1(c)]. Next, theelectron moves under the influence of the driving field, firstbeing accelerated away from the parent ion and then drivenback toward it [Fig. 1(d)]. Depending on the phase of thelaser field when the electron tunnels, the electron mayreturn to the vicinity of the parent ion, or it may driftaway. Huismans and co-workers showed that the spiderstructures can be reproduced by assuming that electronsborn with low transverse momentum can scatter elasticallyfrom the ion [Fig. 1(e)], a method referred to as the‘‘generalized strong-field approximation’’ [17].
The plane-spherical wave (PSW) model (Fig. 2) furthersimplifies the generalized strong-field approximationmodel to its essential physics by treating the revisitingelectrons as a simple plane wave and the rescattered elec-trons as a spherical wave. In the PSWmodel, the only thingthat needs to be calculated is the spatial modulation fre-quency of the plane and spherical waves on the detector,i.e., the dependence of the phase of the electron on the final
momentum. In the Lewenstein model [26,30], the phase ofan ionized electron is e!iS=@, where S is the quasiclassicalaction, given by
Sðp; t; tbÞ ¼Z t
tb
!pðt0Þ22me
þ Ip
"dt0; (1)
where IP is the ionization potential of the atom or mole-cule, me is the mass of the electron, tb is the time theelectron tunnels into the continuum, and pðt0Þ is the mo-mentum. Thus, the probability of observing an electron atsome position on the detector can be found by integratingall of the classical trajectories that reach the same finalmomentum and adding them coherently. In practice,Eq. (1) needs only be integrated up until the time whenthe electron rescatters from the ion, after which bothelectrons take identical paths to the detector. In the PSWmodel, the situation is further simplified because the cal-culation can be completed in just one dimension: along thecomponent of electron momentum in the direction of thelaser polarization direction (Pk). By plotting the actionversus Pk [Fig. 3(b)], we can estimate the modulationfrequency as the slope of this line. This modulation fre-quency, in turn, determines the spacing of the interferencefringes in the spider structure.
FIG. 3 (color online). (a) Depending on when an electron tunnels, it will be driven past the ion a specific number of times.(b) Quasiclassical action obtained by integrating each electron trajectory from when it is born to when it revisits the ion for the second-to-last time versus the final momentum parallel to the laser polarization (Pk). The frequency of the plane and spherical waves is givenby the slope of this graph. The ‘‘steps’’ at low momenta correspond to regions of multiple rescattering. (c) Experimental photoelectrondistribution from argon ionized by a 1:3 !m, 7:5& 1013 W=cm2 laser shows a clear boundary between the high momentum region(one revisit) and the low-momentum region where scattering can take place during a subsequent revisit. (d) Photoelectron distributionfrom xenon using a 2:0 !m, 5& 1013 W=cm2 laser displays two boundaries that correspond to the second and fourth revisits.(e) Time-dependent Schrodinger equation (TDSE) calculations reproduce the low-momenta structures in xenon. (f) Single-cycle PSWmodel, including scattering on the first revisit (outer spiders) as well as the second and fourth revisits (inner spider structures).
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Accurate Retrieval of Structural Information from Laser-Induced Photoelectronand High-Order Harmonic Spectra by Few-Cycle Laser Pulses
Toru Morishita,1,2 Anh-Thu Le,1 Zhangjin Chen,1 and C. D. Lin1
1Department of Physics, Cardwell Hall, Kansas State University, Manhattan, Kansas 66506, USA2Department of Applied Physics and Chemistry, University of Electro-Communications,
1-5-1 Chofu-ga-oka, Chofu-shi, Tokyo, 182-8585, Japanand PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan
(Received 19 July 2007; published 8 January 2008)
By analyzing accurate theoretical results from solving the time-dependent Schrodinger equation ofatoms in few-cycle laser pulses, we established the general conclusion that laser-generated high-energyelectron momentum spectra and high-order harmonic spectra can be used to extract accurate differentialelastic scattering and photo-recombination cross sections of the target ion with free electrons, respectively.Since both electron scattering and photoionization (the inverse of photo-recombination) are the conven-tional means for interrogating the structure of atoms and molecules, this result implies that existing few-cycle infrared lasers can be implemented for ultrafast imaging of transient molecules with temporalresolution of a few femtoseconds.
Electron diffraction and x-ray diffraction are the con-ventional methods for imaging molecules to achieve spa-tial resolution of better than sub-Angstroms, but they areincapable of achieving temporal resolutions of femto- totens of femtoseconds, in order to follow chemical andbiological transformations. To image such transient events,unique facilities like ultrafast electron diffraction method[1] or large facilities such as x-ray free-electron lasers(XFELs) are being developed. Instead of pursuing theseevolving technologies, here we provide the needed quanti-tative analysis to show that existing few-cycle infraredlasers may be implemented for ultrafast imaging of tran-sient molecules.
When an atom is exposed to an infrared laser, the atom isfirst tunnel ionized with the release of an electron. Thiselectron is placed in the oscillating electric field of the laserand may be driven back to revisit its parent ion. Thisreencounter incurs various elastic and inelastic electron-ion collision phenomena where the structural informationof the target is embedded [2,3]. The possibility of usingsuch laser-induced returning electrons for self-imagingmolecules has been discussed frequently in the past.Theoretical studies of laser-induced electron momentumimages of simple molecules do show interference maximaand minima typical of diffraction images, but they areobserved only for large internuclear distances [4–8].Furthermore, the role of laser fields on these diffractionimages has been shown to be quite complicated [5].Recently, it was reported that the outermost molecularorbital of the N2 molecule can be extracted from thehigh-order harmonic generation (HHG) spectra using thetomographic procedure [9]. This interesting result hasgenerated a lot of excitement, but the reported results areobtained based on a number of assumptions [10–12]. Tomake dynamic chemical imaging with infrared lasers as apractical tool, general theoretical considerations, espe-
cially the validity of the extraction procedure, should beexamined carefully.
In this Letter, we show that elastic scattering crosssections of the target ion by free electrons can be accu-rately extracted from laser-induced photoelectron mo-mentum spectra. We also show that accurate photo-recombination cross sections of the target ion can beextracted from the HHG spectra. Our conclusions arebased on accurate theoretical results by solving the time-dependent Schrodinger equations (TDSE) of atoms in in-tense laser fields. While these conclusions are derived fromatomic targets, the same conclusions are expected to applyto molecular targets as well (where accurate TDSE calcu-lations are very difficult). For molecules, these results havefar-reaching implications. Both elastic scattering and pho-toionization are the standard tools for studying the struc-ture of atoms and molecules in conventional energydomain measurements; thus, high-energy photoelectronsand high harmonics generated by infrared lasers offer thepromise for revealing the structure of the target, with theadded advantage of temporal resolution down to a fewfemtoseconds offered by few-cycle pulses.
Consider a typical few-cycle laser pulse, with meanwavelength of 800 nm and peak intensity of1014 W=cm2. The electric field F!t" # $@A!t"=@t andthe vector potential A!t" of such a laser pulse are depictedin Fig. 1(a). By placing a hydrogen atom in such a laserpulse, we solved the TDSE to obtain the photoelectronenergy and momentum distributions, shown in Figs. 1(b)and 1(c), respectively. Figure 1(d) shows the electronmomentum image of Ar in the same laser pulse. Thetheoretical method for solving the TDSE has been de-scribed previously [13,14].
In Fig. 1(b), two particular energies, 2Up and 10Up, aremarked, where Up # A2
0=4 is the ponderomotive energy,with A0 being the peak value of the vector potential of the
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