Probing molecular frame photoionization via laser generated high-order harmonics from aligned molecules

0.13 MB·English

by  Anh-Thu Le

Probing molecular frame photoionization via laser generated high-order harmonics
from aligned molecules
Anh-Thu Le,1 R. R. Lucchese,2, M. T. Lee,3 and C. D. Lin1
1Department of Physics, Cardwell Hall, Kansas State University, Manhattan, KS 66506, USA
2Department of Chemistry, Texas A&M University, College Station, Texas 77843-3255, USA
3 Departamento de Qu´ımica, Universidade Federal de S˜ao Carlos, 13565-905, S˜ao Paulo, Brazil
(Dated: January 9, 2009)
Present photoionization experimentscannot measuremolecular frame photoelectron angular dis-
tributions (MFPAD) from the outermost valence electrons of molecules. We show that details of
9
the MFPAD can be retrieved with high-order harmonics generated by infrared lasers from aligned
0
molecules. Usingaccuratelycalculatedphotoionizationtransitiondipolemomentsforfixed-in-space
0
molecules, we show that the dependence of the magnitude and phase of the high-order harmonics
2
on the alignment angle of the molecules observed in recent experiments can be quantitatively re-

n produced. This result providestheneeded theoretical basis forultrafast dynamicchemical imaging
a using infrared laser pulses.
J

9 PACSnumbers: 33.80.Eh,42.65.Ky

]
h
Photoionization (PI) is the basic process that allows has to be performed, thus losing much valuable informa-
p
- the most direct investigation of molecular structure. tion on the structure of the molecule.
m Measurements of total, partial and differential PI cross
InthisLetterweshowthatrecentadvancesinthegen-
o sections have a long history. Until recently, however,
eration of high-order harmonics from molecules by in-
t almost all the experiments are performed from an en-
a tense infrared lasers has made it possible to probe fixed-
s. semble of randomly oriented molecules. Thus the rich in-spacemolecularPIfromHOMO.Gasphasemolecules
c dynamical structure of photoelectron angular distribu- can be impulsively aligned by a short sub-picosecondin-
i tionforfixed-in-spacemoleculespredictedinthe seminal
s frared laser [6]. After the pulse is turned off, molecules
y paper by Dill [1] more than 30 years ago still remains
will be partially aligned or anti-aligned at the time in-
h largelyunexplored. Inrecentyears,fixed-in-spacePIhas
tervals of rotational revival [7, 8]. During these revivals
p
been investigated with x-ray or VUV photons by using
[ which lastfor tens or hundreds of femtoseconds,another
the photoelectron-photoion coincidence technique [2, 3]. probe pulse can be used to illuminate molecules to ob-
1 Followinginner-shellorinner-valence-shellionizationthe
serve the emission of high-order harmonics. The first
v moleculariondissociates. Themolecularaxisatthetime experiment of this type was reported by Kanai et al. [8]
1
of ionization is inferred from the direction of motion of
1 usingN2,O2 andCO2 molecules. Since thenmanymore
the fragment ion if the dissociation time is short com-
3 experiments have been reported[7, 9, 10, 11, 12, 13]. To
1 pared to the rotational period. Clearly this method is studythedependenceofhigh-orderharmonicsgeneration
. notapplicabletoPIfromthehighestoccupiedmolecular
1 (HHG)onthealignmentofmolecules,theprobelasercan
0 orbital (HOMO). beappliedatthedifferenttimeasthemoleculesevolve,or
9 Consider photoionization of a linear molecule. For a
by changing the direction of the probe laser polarization
0 polarized light, the differential cross section can be ex-
at a fixed time delay. More recently, HHG from mixed
v: pressed in the general form [4] (atomic units are used gases[10,11]andinterferometrytechnique[12]havealso
i throughout, unless otherwise indicated)
X been used such that the phase of the HHG can also be
r 2 measured.
a d2σ 2(2π)3E (cid:12) (cid:12)
dΩkdΩn = 3c (cid:12)(cid:12)(cid:12)XIlmµYl∗m(Ωk)Y1∗µ(Ωn)(cid:12)(cid:12)(cid:12) . (1) stoHoHdGusiisnag htihgehltyhnreoen-lsitneeparmpordoecles[s14a]n.d cEalnecbtreounnsdaerre-
(cid:12)lmµ (cid:12)
(cid:12) (cid:12) first released from the molecule by tunnel ionization
In this expression, the molecular axis is fixed, the direc- and thrown into the laser field. As the laser’s electric
tionsoflightpolarizationandelectronemissionaregiven field changesdirection,the electrons maybe drivenback
by Ωn and Ωk, respectively. This is called the molecular to the ion core after gaining kinetic energy from the
frame photoionization angular distribution (MFPAD). field. When the electrons recombine with the ion, high-
−
WecalculatethecontinuumwavefunctionΨf,klm andthe order harmonic photons are emitted. Since the photo-
dipole I = k1/2hΨ |r |Ψ− i (with r = z for lin- recombination (PR) in the last step is the time-reversed
lmµ i µ f,klm µ
earpolarization)usingtheiterativeSchwingervariational process ofPI, it is thus expected that HHG spectra hold
method [4] within the complete-active-space configura- valuable information on the molecule. Indeed, HHG
tion interaction scheme [5]. To compare with measure- spectra had been used to extract the HOMO in N2 us-
ments, integration of Eq. (1) over unobserved variables ing tomographic method [15]. However the tomographic
2
method relies exclusively on approximating the contin- 1.5 3.5
ugacsspioeutqnmaimuocepnaerelanmedlltexiifocptrltaHooertmlcoHieuvnGteldseh(rbeωcedaysP,incpθIpao)bstlla=tteeuenerdemPxiinpweoglsrmma.etvµeshIesnneIsetlod[mwr1Dyaµ6hsY,((iωc1lQa∗mh7,Rp]θ(irΩw)Sso)kedkfr)funhoYocoamr1t∗wvµHeo(naΩfHdttenGfiohv)xeefwel(atodwhrip-laeiietnnirhnde-- Photoionization cross section (MB)0.15 HHHHHH223333151579 (CaO)2 Phase of transition dipole (radians)−−01210.....55555 (b)234456C000500ooooooO2
kkn)andthe returningelectronwave-packetW(Ek,θ), 00 10 20Align30men4t 0angl5e0 (deg6r0ees)70 80 90 −2.515 20 25 3H0armo35nic o4rd0er 45 50 55
1 3.5
raTdHenihepsdrpioseeωlecmθ=tmoiIstdootem+hlteeEhDensea(tt,lωapiwwbg,oinlθlitiat)mshhrh=ietezωnhsaWtetttihahto(erneEnagprnkldhee,siloθiraott)etifooci×onttnihnodedenn(biωpmeeor,otogfwθlely)tee.hcfeoouenfrltaptPhhrreRoeabhxi(enaiosrdrlmuawPsco(ieIet2nr)dh)-., HHG yield (arb. units)0000....2468 HHHHHH223333151579 (Cc)O2 Relative phase (radians)012...123555 (d) PPPPPPCaeaeaeOrrrrrrapapap2l.l.l.lll eee(((49elll 00x(((49epKK00x.,,)pKK 00.,,).. 6500xx..65II00xx))II00))
p k 0
ics, Ip the ionization potential, and Ek = k2/2 the “in- 00 10 20 30 40 50 60 70 80 90 −0.517 19 21 23 25 27 29 31 33 35
cident” energy of the returning electron. The validity of Alignment angle (degrees) Harmonic order
this equationhasbeenestablishedforatomictargets(no
θ dependence foratoms)[16, 17]aswellasfor H+ target FIG. 1: (Color online) CO2 differential photoionization cross
2 section(a)andphaseofthetransitiondipole(b). Thephoton
[18]. Inbothcases,HHGspectracanbecalculatedaccu-
energiesareexpressedinunitsofharmonicordersfor800-nm
rately by solving the time-dependent Schr¨odinger equa-
laser. (c) HHGyield as function of angle between pump and
tion(TDSE).InthisLetter,weobtainedtransitiondipole probe laser polarization directions. The laser intensity and
d(ω,θ) from the state-of-the-art PI calculations. The duration are of 0.55×I0 (I0 = 1014 W/cm2), and 120 fs for
wave-packet W(E ,θ) is obtained from the strong-field the pump pulse, and 2.5×I0 and 25 fs for the probe pulse.
k
approximation (SFA)[16, 18, 19]. We then use Eq. (2) Rotational temperature is taken to be 105 K. (d) Harmonic
phase(relativetothephasefromKr)forparallelalignedand
to obtain D(ω,θ) for each fixed-in-space molecule. To
perpendicular aligned ensembles, under two sets of parame-
compare with experiments, proper coherent convolution
ters(solidanddashedlines)thatleadtotwodifferentdegrees
with the partialalignmentofmoleculeshas tobe carried of alignment distributions. Experimentaldata from Boutu el
out [see Eq. (2) of Ref. [10]]. al. [13] are shown as symbols.
In Fig. 1(a), we show the theoretical CO2 PI differen-
tialcrosssectionfortheemissiondirectionalongthelight
polarization (k kn), which are to be compared to HHG cal amplitude and phase of d(ω,θ) shown in Fig. 1 have
data below. By changing the orientation of the fixed-in- never been observed directly in PI experiments. Below
space molecule, complementary information on the an- we show that they have been observed in recent HHG
gular distribution of photoelectrons from fixed-in-space measurements.
molecules is obtained. In order to compare with HHG InFig.1(c)weshowthe simulatedtypicalHHGyields
data, the photon energies are expressed in units of pho- fromalignedCO2,asafunctionoftheanglebetweenthe
tonenergyof800-nmlaser(1.55eV).First,we notethat pumpandprobelaserspolarizationdirections. Thesere-
the differential cross section is large when the molecule sults are consistent with recent experiments [13, 20, 21],
is aligned at a large angle. The cross sections vanish at and with earlier theoretical results [22, 23, 24]. The re-
θ =0andπ/2duetotheπ symmetryoftheHOMOand sults also resemble the data for the induced dipole re-
g
the dipole selection rule for the final state. Second, the trieved from mixed gases experiments by Wagner et al.
cross section shows near-zero minima for angles around [10](seetheirFig.4). Inoursimulation,thelaserparam-
◦ ◦
35 to 45 for harmonics above the 25th order, or H25. eters are taken from the experiment of [12]. The align-
The position ofthe minimum movesto a largerangle,as ment distribution is obtained from numerical solution of
energyincreases. Theseminimahavebeeninterpretedas the TDSE within the rotor model [6, 22, 23]. At the de-
evidenceofinterferenceofelectronwavesfromtheatomic lay time corresponding to the maximum alignment near
centers. In Fig. 1(b), we show the phase of the PI dipole half-revival(1/2 of the rotationalperiod), a probe beam
◦ ◦ ◦
vs photon energy, for alignment angles from 20 to 60 . with polarizationdirection varying from 0 to 90 is used
◦ ◦
Between 30 and 45 the dipole phase undergoes rapid to generate high-order harmonics. Comparing Fig. 1(c)
change between H25 and H39. This is the region where with Fig. 1(a), we note that the HHG yields follow the
thecrosssectionsareneartheminima[seeFig.1(a)]. The general angular and photon energy dependence of the
phasechangeisabout2.0to2.5,insteadofπasshouldbe thedifferentialPIcrosssections. Bothfiguresshowlarge
◦
expectedif the crosssectionindeedgoes to zero. Foran- yieldsatlargeangles,andminima near35 forharmonic
◦ ◦
glesbelow20 andabove60 thephaseevolvessmoothly orders above H31. Due to the averaging over the molec-
vs photon energy. The “rich” structures in the theoreti- ular alignment distributions, the angular dependence of
3
1
HHG is smoother.
> (a) CO
Wenextshowthatthephaseoffixed-in-spacePItran- 2θs 0.5 2
o
c
sition dipole can also be probed by comparing to the <
0
phase of the harmonics. Experimentally the harmonics 3 H21 (Theory)
(b) H25 (Theory)
phase can be extracted from measurements of HHG us- H25 (Exp.)
2
ingmixedgases[11,13]orinterferometrymethod[12]. In
Fig.1(d),weshowtherecentexperimentaldataofBoutu
1
et al. [13] where the harmonics phases (relative to that d)
e
from Kr) are obtained for the parallel aligned and per- mailz 80 H31 (Theory)
pendicularly aligned ensembles, shown by black and red or (c) H35 (Theory)
symbols, respectively. For the latter, the phase does not y (n 6 H35 (Exp.)
changemuchwithinH17toH31. Fortheparallelaligned ensit 4
nt
molecules,thephasejumpfromH17toH31wasreported c i 2
ni
to be 2.0±0.6 radians (in the figure it was shown at 2.6 mo 0
radians). Our simulation results are shown for two dif- Har 8 (d) HH3379 ((TThheeoorryy))
ferent ensembles, with alignment distributions confined 6 H37 (Exp.)
◦ ◦
in a cone angle of 25 and 35 at half maximum, re- 4
sultingfromdifferentpumpbeamsandgastemperatures 2
(see Caption). From Fig. 1(b), we note that for a fixed
0
harmonic, the phase from the parallel aligned ensemble 31.6 32.1 32.6 33.1 33.6
Time delay (ps)
is larger if the angular spread of the molecular distri-
butions is smaller. Therefore, the position of the phase
FIG.2: (Coloronline)NormalizedHHGyield(vsisotropically
jump for the less aligned ensemble (dashed line) slightly distributed molecules) from CO2 for different harmonics as
shifted toward higher energies. For the perpendicularly function of pump-probe delay time near 3/4-revival. Laser
alignedmolecules,thephaseissmallanddoesnotchange parameters are the same as in Fig. 1(c). The experimental
much with the harmonic order. This is consistent with dataaretakenfromZhouetal[12]. Thealignmentparameter
hcos2θi is plotted for reference (a).
the phase shown in Fig. 1(b). Thus the phase of HHG
in Fig. 1(d) is seen to mimic the phase of the PI dipole
shown in Fig. 1(b).
Next we show in Fig. 2 how the HHG yields change are shown in Fig. 3(c) and (b), respectively. Our sim-
with the time delay near the 3/4-revival where the ulations (solid lines) are also in good agreement with
molecule can be most strongly aligned. Our simulations these measurements. The HHG yield is small when the
were carried out with the laser parameters from Zhou molecules are maximally aligned. This is in agreement
et al. [12]. The degree of alignment (top panel), as with H31 seen in Fig. 2(c) for the delay time near 3/4-
measured by hcos2θi is maximum when molecules are revival. However, we note that the phase is maximum
maximallyaligned(verticallyinthefigure)andminimum when the molecules are maximally aligned. This can be
when molecules are anti-aligned. Our results are shown understood from the phase in Fig. 1(b) where it shows
for the same harmonics that had been analyzed in [12] thatthephaseislargewhenthealignmentangleissmall.
(see their Fig. 2 - according to a new assignment by the Italsoshowsthat the phase is smallwhenthe alignment
authors of [12], the harmonic orders should be properly angle is large, thus for anti-aligned molecules the phase
shifted down by two orders, compared to the ones given ofthe HHG shouldbe small, as seenin the experimental
in their original paper [21]). For the lower harmonics data and in the simulation.
H21 and H25, the yields follow the inverse of the align- We remark that the laser intensity used in the JILA
mentparameter. ThisiseasilyunderstoodfromFig.1(a) experimentsisquitehigh,thustheeffectofthedepletion
whichshowsthatthePIcrosssectionsatlargeanglesare of the ground state by the laser should be considered
much larger than at small angles. For the higher orders, [22, 23, 25]. For CO2, the alignment dependence of the
Fig. 1(a) indicates that the PI cross sections show two ionization rate is still not fully settled yet. The results
humps, with the one at smaller angles only a factor of from the MO-ADK theory [26], the SFA theory and the
about two to three times smaller than the other. Qual- experiment [27] all disagree with each other, with the
itatively, this explains why the HHG yields for H31 and SFA predicting a peak near 40◦, compared to 30◦ from
up show a pronounced peak for the parallel alignment. MO-ADK theory, and 45◦ from experiment [27]. Using
We note a quantitatively good agreement between our ionization rates from MO-ADK, we have not been able
calculations and the experimental data. to simulate experimental data accurately. We thus used
Evolution of the HHG yields has also been studied by ionizationratesfromSFA,exceptthatwerenormalizethe
Wagner et al. [10] using the mixed gases technique. The SFA rate to that of the MO-ADK rate at laser intensity
yield and phase for H31 vs time delay near half-revival of I0, which gives a factor of 10. Note that the same
4
1 7 0.8
2θ<cos>0.05 (a) CO2 6 (a) N2 HHHH12227159 2θ<cos>000...246 (b) N2
Magnitude (normalized)Phase (radians) 120123 ((bc)) ETEThhxxpepe.o.orryy Photoionization cross section (MB)12345 H35 HHG yield (normalized)012...1230555 (c) HHHH23231515−−−−23235757 ((((eetthhxxeeppoo..))rryy))
021 21.5 22 22.5 00 10 20 30 40 50 60 70 80 90 03.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2
Time delay (ps) Alignment angle (degrees) Time delay (ps)
FIG. 3: (Color online) Alignment parameter hcos2θi (a), FIG.4: (Coloronline)(a)SameasinFig.1(a),butforN2. (b)
phase(b)andmagnitude(c)ofthe31thorderharmonicfrom and (c): Alignment parameter hcos2θi and harmonic yields
CO2,asfunctionsofpump-probedelaytimenearhalf-revival. H21-H25 and H35-H37 from N2. The experimetal data and
TheexperimentaldataaretakenfromWagneretal[10]. The laser parameters are taken from Itatani et al [7].
laser intensity and duration are of 0.38×I0 and 140 fs for
the pump, and 2.5×I0 and 30 fs for the probe. Rotational
temperature is taken to be70 K.
varyinglaserintensity,HHGspectrafromlower-lyingor-
bitals can also be similarly probed [28]. Having estab-
correction factor has been found for the SFA ionization lished the connection between HHG and the PI dipole
from Kr,which has almostthe same ionizationpotential matrix elements, see Eq. (2), time-resolved PI cross sec-
as for CO2. With the corrected SFA rate, we found that tions can be extracted from HHG spectra in a typical
oursimulationsgiveabetterquantitativeagreementwith pump-probeexperimentalsetup,thusopeninguptheop-
the JILA data [12]. portunity of using infrared laser pulses for ultrafast dy-
WehavealsostudiedN2 molecules. Thefixed-in-space namic chemical imaging.
differential PI cross sections are shown in Fig. 4(a). For We thank X. Zhou, N. Wagner, M. Murnane, H.
H17 to H21 the cross sections are quite large since this Kapteyn,andP.Salieresforcommunicatingtheirresults
is in the tail region of the famous shape resonance in to us and the valuable discussions. This work was sup-
N2. For these orders, the cross sections are highly for- portedinpartbytheChemicalSciences,Geosciencesand
wardly peaked. For energies above H25, the cross sec- BiosciencesDivision,OfficeofBasicEnergySciences,Of-
tions become much smaller, and they are of the same fice of Science, U. S. Department of Energy.
order of magnitude between parallel and perpendicular
alignments. However, note that the cross sections have
◦ ◦
minima at large angles around 60 – 70 . The (normal-
ized) HHG spectra of N2 has been measured vs the time
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5
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