# Extracting continuum electron dynamics from high harmonic emission from molecules.

**ABSTRACT** We show that high harmonic generation is the most sensitive probe of rotational wave packet revivals, revealing very high-order rotational revivals for the first time using any probe. By fitting high-quality experimental data to an exact theory of high harmonic generation from aligned molecules, we can extract the underlying electronic dipole elements for high harmonic emission and uncover that the electron gains angular momentum from the photon field.

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**ABSTRACT:**Complex revival features of rotational wave packets are obtained from the interplay of a molecular rotational distribution and a measured physical observable. The analysis of the measured temporal behavior can be used to retrieve either one or both quantities. We show here the first observation of high order fractional revival (up to 1/12 in CO_{2}) using time-of-flight measurements of ion yields leading to the information required for full reconstruction of the rotational wave packet. We further show via an analysis of higher order fractional revivals in high harmonic generation that new information on the participating ionic channels can be clearly identified, showing the general implication of our results.Physical Review Letters 12/2013; 111(26):263601. · 7.73 Impact Factor - SourceAvailable from: Shungo MiyabeLimor S Spector, Maxim Artamonov, Shungo Miyabe, Todd Martinez, Tamar Seideman, Markus Guehr, Philip H Bucksbaum[Show abstract] [Hide abstract]

**ABSTRACT:**High-order harmonic generation in an atomic or molecular gas is a promising source of sub-femtosecond vacuum ultraviolet coherent radiation for transient scattering, absorption, metrology and imaging applications. High harmonic spectra are sensitive to Ångstrom-scale structure and motion of laser-driven molecules, but interference from radiation produced by random molecular orientations obscures this in all but the simplest cases, such as linear molecules. Here we show how to extract full body-frame high harmonic generation information for molecules with more complicated geometries by utilizing the methods of coherent transient rotational spectroscopy. To demonstrate this approach, we obtain the relative strength of harmonic emission along the three principal axes in the asymmetric-top sulphur dioxide. This greatly simplifies the analysis task of high harmonic spectroscopy and extends its usefulness to more complex molecules.Nature Communications 02/2014; 5:3190. · 10.74 Impact Factor - SourceAvailable from: Zheng Li[Show abstract] [Hide abstract]

**ABSTRACT:**‘‘The noise is the signal” [R. Landauer, Nature (London) 392, 658 (1998)] emphasizes the rich information content encoded in fluctuations. This paper assesses the dynamical role of fluctuations of a quantum system driven far from equilibrium, with laser-aligned molecules as a physical realization. Time evolutions of the expectation value and the uncertainty of a standard observable are computed quantum mechanically and classically. We demonstrate the intricate dynamics of the uncertainty that are strikingly independent of those of the expectation value, and their exceptional sensitivity to quantum properties of the system. In general, detecting the time evolution of the fluctuations of a given observable provides information on the dynamics of correlations in a quantum system.Physical Review A 05/2014; 89(5):052113. · 3.04 Impact Factor

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Extracting Continuum Electron Dynamics from High Harmonic Emission from Molecules

R.M. Lock,1,*S. Ramakrishna,2X. Zhou,1H.C. Kapteyn,1M.M. Murnane,1and T. Seideman2

1Department of Physics and JILA, University of Colorado and NIST, Boulder, Colorado 80309, USA

2Department of Chemistry, Northwestern University, Evanston, Illinois 60208, USA

(Received 5 September 2011; published 26 March 2012)

We show that high harmonic generation is the most sensitive probe of rotational wave packet revivals,

revealing very high-order rotational revivals for the first time using any probe. By fitting high-quality

experimental data to an exact theory of high harmonic generation from aligned molecules, we can extract

the underlying electronic dipole elements for high harmonic emission and uncover that the electron gains

angular momentum from the photon field.

DOI: 10.1103/PhysRevLett.108.133901PACS numbers: 42.65.Ky, 42.65.Re

In recent years, there has been increasing interest in

studying high-order harmonic generation (HHG) from co-

herently aligned molecules as a potential approach for

extracting dynamic molecular structure [1–18]. In these

experiments, a pump pulse creates a rotational coherence

in a molecular gas, inducing nonadiabatic alignment

through the interaction of a short, intense, laser pulse with

the anisotropic polarizability of the molecule. The induced

rotational wave packet manifests itselfin periodic quantum

revivalsandfractionalrevivalsofthealignmentdistribution

that create transient alignment and antialignment of the

molecular sample at certain times after the pump pulse.

Revivals occur at fractions of the rotational period of the

molecule [19–21], with their positions and nature deter-

mined by symmetry of the molecule and by the excited

angular momentum states. A more intense time-delayed

pulse then generates harmonics from the sample [2–4].

Past work exploring HHG from impulsively aligned

molecules concentrated primarily on extracting structural

information by characterizing variations in HHG intensity

as the rotational distribution experiences low-order rota-

tional revivals (full,1

understood to a first approximation in a semiclassical

three-step model in which a strong laser field ionizes an

electron from the molecule. The electron then propagates

in the laser field, and if it oscillates back to the vicinity of

the parent cation, it can recombine with the molecule and

emit coherent HHG beams [22]. Both the ionization and

recombination steps in HHG are sensitive to the molecular

orbital structure and orientation. Past work demonstrated

that the intensity, phase, and polarization of the HHG

emission are sensitive to the orbital structure of the mole-

cule [10], as well as to the coherence properties and the

revival pattern of the rotational wave packet [23,24].

In this work, through high-fidelity measurements of

HHG emission from molecules, we observe and analyze

higher-order fractional rotational revivals (up to the 1=16)

for the first time. These observations demonstrate that

HHG is the most sensitive probe of rotational wave

packet dynamics, allowing us to uncover new insights

2, and1

4;3

4). HHG emission can be

not observable using conventional probes. Using a two-

center model of the molecule, we can intuitively explain

how the observed high harmonic emission is related to the

alignment distribution, and extract the pump laser intensity

and the rotational temperature of the medium. More re-

markably, by comparing a rigorous theory of HHG from

aligned molecules [23,24] with high signal-to-noise ex-

perimental data, we can extract the underlying electronic

dipole elements of the HHG signal, as well as information

on the continuum electron dynamics. Specifically, we un-

cover that the continuum electron gains angular momen-

tum from the photon field while propagating in the

continuum,and that only a small number of electron partial

waves strongly dominate the HHG dynamics. For mole-

cules with antisymmetric ground state orbitals, electrons

liberated with low electronic angular momentum l ¼ 1

recombine from a higher angular momentum state with

l ¼ 3. Such electronic angular momentum nonconserving

events dominate over angular momentum conserving

events, increasingly so as the harmonic order increases,

forbothCO2andN2O.Thus,attosecondelectrondynamics

manifests itself in the experimental observable of higher-

order fractional rotational revivals and is crucial for their

observation.

In our experiments, we used a Ti:sapphire laser-

amplifier system producing ?25 fs pulses at 800 nm wave-

length, running at a 1 kHz repetition rate. The pump pulse

is stretched to ?120 fs using material dispersion and fo-

cused into a continuous supersonic gas jet of CO2or N2O

to excite a rotational wave packet. The probe beam is then

focusedinto thegas jet to generate harmonics. The focus of

the probe beam is placed slightly before the gas jet [25].

Theintensitiesof the

?2–6 ? 1013W=cm2and ?1–2 ? 1014W=cm2, respec-

tively. The beams are linearly polarized parallel to each

other. The gas jet is a ?150 ?m diameter tube, continu-

ously backed with gas at a pressure of ?700 torr. The

rotational temperature of the molecules is ?60–100 K.

The HHG spectrum is captured using an extreme ultravio-

let (EUV) spectrometer and an EUV CCD camera.

pump andprobe are

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Figures 1(a) and 1(b) plot the intensity of the 31st har-

monic generated in CO2as a function of pump-probe time

delay (solid black line) along with fits using the two-center

interference model [9,15,26,27] (dashed red line) and the

exact theory described in [23,24] (dotted blue line). Small

revivals,previouslyunreportedinCO2,areclearlyvisibleat

time delays corresponding to 1=16, 1=12, and1

tional period, along with the

multiples of those time delays. Figure 2 plots similar mea-

surements and fits for the 25th harmonic from N2O mole-

cules. The1

2revival of N2O has been studied previously

[15,17], but new features are clearly visible at1

and5

6of the rotational period.

The time-dependent molecular alignment distribution

following excitation can be calculated with a high degree

of confidence, with the major uncertainty being the inten-

sity of the exciting pulse and the rotational temperature of

the sample [19,21]. Phenomenological approximations can

be used to describe the rotational angular distribution, by

assuming hcos2?i or hsin22?i alignment parameters, where

? is the angle between the aligning field and the molecular

axis [2,4], and hcos2?i ¼ 1 corresponds to perfect align-

ment. Recent calculations described rotational revivals in

terms of cosine moments hcosN?i [28]. Higher-order co-

sine moments up to N ¼ 8 can well describe the locations

of the new revivals observed in our data [29].

To intuitively understand our data, the rotational angular

distribution can be integrated with the two-center interfer-

ence model, where the dependence of the HHG emission

6of the rota-

1

8revival, and at integer

6,1

4,1

3,2

3,3

4,

on the molecular structure and alignment arises predomi-

nantly from interferences in the recombination step

[26,27]. While by no means complete, this model can

provide useful physical insight. In previous work we

showed that the two-center interference model can well

describe HHG from CO2and N2O during the3

revivals [9,15]. HHG from the full and1

and N2O is anticorrelated with hcos2?i: for the internuclear

distances and molecular orbital shapes in these two mole-

cules, destructive interference between recombination to

the two centers of electron density occurs when the mo-

lecular axis is aligned along the laser polarization direc-

tion. The highest occupied molecular orbitals of CO2and

N2O are both antisymmetric, and thus the HHG amplitude

from a perfectly aligned molecule in the two-center model

is given by

4and1

2

2revivals of CO2

Hð?Þ ¼ Asin

??R

?

cos?

?

;

(1)

where A is a scaling factor, R is the distance between

centers of electron density, ? is the wavelength of the

recombining electron, and ? is the angle between the probe

polarization vector and the molecular axis. To obtain the

harmonic intensity from molecules distributed over a range

of angles, the HHG from a perfectly aligned molecule must

be integrated with the angular distribution, which serves as

a weight factor. The harmonic yield is then

????????

133901-2

IðtÞ ¼

Z?=2

0

Hð?Þ?ð?;tÞsin?d?

????????

2þC;

(2)

FIG. 1 (color online).

(solid black) and fits to the two-center interference model

integrated with the alignment distribution (dashed red) along

with fits to the exact theory of [23,24] (dotted blue) for the 31st

harmonic from CO2for (a) the full rotational period and (b) for

the time period indicated by the dashed black box in (a).

Normalized experimental HHG intensity

FIG. 2 (color online).

(solid black) and fits to the two-center interference model

integrated with the alignment distribution (dashed red) along

with fits to the exact theory of [23,24] (dotted blue) for the 25th

harmonic from N2O for (a) the full rotational period and (b) for

the time period indicated by the dashed black box in (a).

Normalized experimental HHG intensity

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Page 3

where ?ð?;tÞ is the rotational angular distribution, t is the

pump-probe time delay, and C is included for background

compensation. Although the model omits the angular de-

pendence of the ionization matrix element and the evolu-

tion of the continuum electron subject to the strong field,

we account for the antisymmetry of the ionization matrix

element with respect to ? ! ?-? by limiting the integra-

tion in Eq. (2) to the 0 to ?=2 range. Fits to this model for

CO2and N2O are shown as the dashed red lines in Figs. 1

and 2, respectively. The fits were obtained through a least-

squares fitting procedure with A, Bð¼ R=?Þ, and C as the

fit parameters. Using the values of B extracted from the

fitting procedure and ? for each harmonic order, we ex-

tracted R for each molecule. For CO2, R ¼ 0:241 ?

0:010 nm, while for N2O R ¼ 0:229 ? 0:021 nm. These

values agree with the known distances between centers of

electron density 0.232 and 0.231 nm for CO2and N2O,

respectively. The wavelength of the recombining electron

? was obtained using the dispersion relationship Ek¼

nh?, where Ekis the energy of the recombining electron

and nh? is the harmonic energy. Thus, the two-center

model can be fit to the general features of all revivals

present. It can also extract the rotational temperature and

pump laser intensity [29].

Considerable further insight is gained by accounting for

angular momentum exactly using the theory of [23,24],

which allows us to extract the electron dipole moments,

and also uncover new insight into the electron dynamics

of molecular HHG. We start with the expression for

the expectation value of the time-dependent induced

dipole in a direction parallel to the polarization of the

probe pulse [23],

hcðtÞj ~ ? ? ^ njcðtÞi

¼

Z

þ sin2?X

d^R?ð^R;?Þ

?

cos2?X

Ylklð^RÞY?

l;l0;kl

Ylklð^RÞY?

l0klð^RÞFjjðl;l0;kl;tÞ

?

l;l0;kl

l0klð^RÞF?ðl;l0;kl;tÞþ c:c:

(3)

In Eq. (3), ?ð^R:?Þ is the rotational density matrix, where

^R ¼ ð?;’Þ are the Eulerangles of rotationof the molecular

frame with respect to the laboratory frame and ? is the

pump-probe time delay. The Yl;klare spherical harmonics,

where the indices l and kldenote the quantum numbers of

the continuum electron angular momentum and its projec-

tion on the molecular axis, respectively. The exact expres-

sionsforthe electronicdipole matrixelements parallel(Fk)

and (F?) perpendicular to the molecular axis are provided

in Refs. [23,24]. Physically, each electronic dipole matrix

element Fkð?Þðl;l0;kl;?Þ can be understood in terms of the

familiar three-step process of HHG. It signifies an electron

ionized either parallel (k) or perpendicular (?) to the

molecular axis into a continuum state that has an angular

momentum value of l0about the space fixed z axis, its

propagation and eventual recombination into the ground

state from a continuum state of angular momentum l, while

the projection of the angular momentum of the continuum

electron about the molecular axis given by klis conserved.

While the theory of [23,24] is exact and explicitly shows

the existence and origin of high-order fractional rotational

revivals, the numerical calculation of the electronic dipole

matrix elements is very difficult and hence all existing

calculations of HHG from aligned molecules involve sig-

nificant approximations.

For the purpose of fitting to data, the amplitude of

harmonics emitted by molecules with a ?gsymmetry,

such as CO2and N2O, is expressed as

Z

¼

n

? hcos2n?sin6?i;

where ? is the harmonic frequency, A, B, C, and D are the

fit parameters to be extracted, and the measured signal

is given as the squared absolute value of Eq. (4). The

coefficients are related to the electronic dipole matrix

elements as

X

þ g2n

X

where f, g, and h are known constants, containing the

transformation of the spherical harmonics into geometric

functions. Angular momentum selection rules allow only

for odd values of l and l0and for ?gsymmetry, klcan take

values ?1 in Eq. (5) and ?2 in Eq. (6).

In principle, for a given value of n, one can have an

infinitenumberofpartialwave(l)contributions.Inpractice,

however, the series of partial waves converges rapidly and

higher-order angular momentum values become progres-

sivelysmaller.Thisresult,

Refs. [23,24], is substantiated by our experiments below.

The constraints l þ l0? 2n and l þ l0? 2n þ 4, conse-

quently, lead to truncation also of the series expansion in

n. A fit to reliable data will determine which partial wave

continuumfunctionsparticipateintheelectronicdynamics.

FromFigs.1and2welearnthat,inourpresentexperiments,

values of n up to n ¼ 3 participate in the dynamics while C

andDinEq.(4)arenegligible.Asaroughinitialestimateof

the number of partial waves making sufficiently large con-

tributions to be observable, we use calculations within the

molecularstrongfieldapproximation(SFA).Ourresultsare

independent of the validity of the SFA, depending only on

the accuracy of the experimental data, but the electronic

matrixelementsextracted canserve totestthe resultsof the

SFA. This procedure yields

dtexpði?tÞhcðtÞj ~ ? ? ^ njcðtÞi

X

ðA2nþ iB2nÞhcos2n?sin2?i þX

n

ðC2nþ iD2nÞ

(4)

A2nþ iB2n¼

lþl0?2n

X

kl¼?1

f2n

ll0Fkðl;l0;kl;?Þ

ll0F?ðl;l0;0;?Þ;

X

(5)

C2nþ iD2n¼

lþl0?2nþ4

kl¼?2

h2n

ll0F?ðl;l0;kl;?Þ;

(6)

found numericallyin

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A6þ iB6? 2f6

33Fkð3;3;?1;?Þ;

(7)

A4þ iB4? 2f4

31Fkð3;1;?1;?Þ þ 2f4

þ g4

33Fkð3;3;?1;?Þ

31F?ð3;1;0;?Þ;

(8)

A2þ iB2? 2f2

11Fkð1;1;?1;?Þ þ 2f2

þ 2f2

þ g2

Using the fact that Fkð1;1;?1;?Þ ¼ F?ð1;1;0;?Þ and

Fkð3;1;?1;?Þ ¼1

one can solve the system of Eqs. (7)–(9) and obtain esti-

mates for the dominant electronic dipole matrix elements.

(The second relationship arises from the fact that the angu-

lar parts of the dipole matrix elements depend on the

symmetry of the ground state only, while its radial part is

influenced by the molecular potential and the ionizing

field.)

This theory fully explains the fractional revivals in CO2

shown in Fig. 1 and reveals their physical content: angular

momentum theory indicates that hcos2?i exhibits only the1

and1

2revivals, since it allows only second-order rotational

coherences, whereas hcos2?sin2?i allows fourth-order rota-

tional coherences and thus also exhibits the1

Higher moments, such as hcos4?sin2?i, can be shown to

involve rotational coherences of order six; hence, they

express themselves as additional revivals such as the

1=12 and1

6, while the 1=16 and 3=16 revivals emerge

from hcos6?sin2?i, which involves rotational coherences

of order eight. As the highest order fractional revival

features seen in the present experimental signal are that

of the 1=16 and 3=16, we find that the electronic dynamics

is determined by partial waves up to l þ l0¼ 6.

Similarly, for the revivals in N2O shown in Fig. 2, the

rotational expectation value hcos4?sin2?i is the lowest

order in the series that yields fractional revivals at1

1

3, whereas the element hcos6?sin2?i is responsible for the

observed rotational revivals at1

period. Most interestingly, our ability to apply analytical

angular momentum algebraic considerations to correlate

each fractional revival with a specific rotational expecta-

tion value, and hence a specific electronic dipole element,

provides a route to extract the continuum electronic dy-

namics that underlie the signal from the experimental data.

Figures 3(a) and 3(b) show the magnitudes of the elec-

tronic dipole elements extracted from our experimental

data for N2O and CO2as functions of the harmonic order.

Several interesting features of the electronic matrix ele-

ments responsible for HHG are apparent. First, while the

expansion in Eq. (5) is in principle infinite, only a small

number of partial waves l, l0¼ 1, 3 dominate. Second,

for both CO2 and N2O the dominant element is

jFkð3;1;kl;?Þj—signifying ionization into a continuum

electronic wave function with angular momentum l0¼ 1

31Fkð3;1;?1;?Þ

11F?ð1;1;0;?Þ

33Fkð3;3;?1;?Þ þ g2

31F?ð3;1;0;?Þ:

(9)

pF?ð3;1;0;?Þ, where p is a constant,

4

8and3

8revivals.

6and

8and3

8of the rotational

followed by propagation in the field, during the course of

which angular momentum is gained from the photon field,

and recombination of partial wave l ¼ 3. Third, the

dominance of the jFkð3;1;kl;?Þj element over the

jFkð1;1;kl;?Þj and jFkð3;3;kl;?Þj angular momentum

conserving events increases with the harmonic order. We

note that the experimental dipole elements follow the same

order of dominance that was theoretically found for O2

(which has the same symmetry as CO2and N2O in its

ground state) within the strong field approximation,

namely jFkð3; 3; j1j; ?Þj < jFkð1; 1; j1j; ?Þj < jFkð3; 1;

j1j; ?Þj [23,24]. The involvement of non-negligible elec-

tronic angular momenta (here l up to 3 with significant

amplitude) in the continuum electronic dynamics is the

reason why fractional revivals such as 1=12 and1

and1

6for N2O, are conspicuous in the HHG signals.

These are associated with higher-order rotational moments

(such as hcos4?sin2?i in the present example), which allow

for high-order rotational coherences and hence high-order

fractional rotational revivals.

Tophysicallyinterpretourresults,eachelectronicdipole

matrix element Fkð?Þðl;l0;kl;?Þ can be translated into an

underlying dynamical event. For molecules such as N2, the

bound ?gstate can be written as a superposition of elec-

tronic angular momentum states l ¼ 0, l ¼ 2, l ¼ 4, where

the l ¼ 0 state strongly dominates and the l ¼ 4 contribu-

tion is very small. This bound state undergoes ionization

primarily to the l ¼ 1 continuum state. Even if the free

electrongainsangularmomentuminthestrongfield,dipole

selection arguments in the recombination step strongly

favorprocesseswherethel ¼ 1continuumelectronrecom-

bines to the l ¼ 0 electronic angular momentum state (i.e.,

angular momentum conserving trajectories). In molecules

such as O2, N2O, and CO2, the bound (?g) state can be

written as a superposition of electronic angular momentum

states l ¼ 2, l ¼ 4, and the l ¼ 2 state strongly dominates.

This state ionizes to make predominantly the l ¼ 1 contin-

uum state (with smaller contributions from l ¼ 3 and 5).

While traveling in the continuum, the electron can gain

angular momentum from the photon field. In this case,

because the l ¼ 0 state is not occupied, recombination

must occur predominantly into l ¼ 2, from either the

l ¼ 1 or l ¼ 3 states (i.e., angular momentum conserving

andnonconserving trajectories).

Fðl0¼ 1;l ¼ 3Þ dipole therefore includes the interference

oftwoquantumpathways.Thus,inthecaseofO2,N2O,and

CO2, the dipole recombination selection rules do not con-

ceal the continuum electron dynamics, allowing us to see

that what the electron really wants to do is to gain angular

momentum from the photon field [see Fig. 3(c)].

The picture that emerges thus associates with the high-

order rotational expectation values an interesting interplay

between strong field electron dynamics and angular mo-

mentum propensities. In particular, the high-order rota-

tional expectation values are enabled by transitions from

higher angular momentum states of the continuum states to

6for CO2,

3and1

Thedominating

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Page 5

the ground state of the molecule. Each of these high-order

moments, in turn, can be linked to a given high-order

fractional revival in the HHG spectrum. In this way the

electronic dynamics expresses itself in the experimental

observable. Thus, the rotational revival structure contains

rich information not only about the rotational coherences

but also about the attosecond electronic motions, which

can be extracted from a fit of an exact theory to high-

quality experimental data.

Our results suggest a range of new and fascinating

opportunities that can be explored. For example, more

information about electronic dynamics could be gained

from the phases of the electronic matrix elements. Our

theory can readily extract these phases from data, but this

requires measurements not only of the harmonic intensities

butalso of their phasesat the high-order fractionalrevivals,

which are challenging. Moreover, an interesting contro-

versy in HHG from aligned molecules is the role played by

multiple orbitals. For CO2, the conclusions of Ref. [30] are

in striking contradiction with those of Refs. [31,32]. Our

approach could be readily extended to explore the role of

multiple orbitals.

The authors gratefully acknowledge the help of

Dr. Maxim Artamonov. They thank the DOE Office of

Basic Energy Sciences (Grants No. DE-FG02-99ER14982

and No. DE-FG02-04ER15612) and the AFOSR (Grant

No. P.O. 217178/01 // FA9550-11-1-0001) for support,

and the NSF EUV ERC for use of facilities.

*Corresponding author.

Robynne.Lock@jila.colorado.edu

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athttp://link.aps.org/

for

FIG. 3 (color online).

(black squares), jFkð3;1;kl;?Þj (red circles), and jFkð3;3;kl;?Þj (green triangles). Fkðl;l0;kl;?Þ signifies ionization parallel to the

molecular axis into a continuum state with angular momentum l0, propagation in the field, followed by recombination into the ground

state from a continuum state of angular momentum l. (c) Schematic illustrating an electron trajectory in the continuum where an

electron undergoes tunnel ionization from the l ¼ 2 electronic state of the molecule, emerges in the continuum in an l ¼ 1 state where

it gains angular momentum, and then finally recombines from an l ¼ 3 state back into the ground electronic state of the molecule.

Fundamental electronic dipole elements versus the harmonic order for (a) CO2and (b) N2O for jFkð1;1;kl;?Þj

PRL 108, 133901 (2012)

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