Rotational excitation and molecular alignment in intense laser fields
Tamar Seideman
Steacie Institute, National Research Council, Ottawa, Ontario K1A 0R6, Canada
~Received 23 June 1995; accepted 21 July 1995!
Rotational excitation and spatial alignment in moderate intensity radiation fields are studied
numerically and analytically, using time-dependent quantum mechanics. Substantial rotational
excitation is found under conditions typically used in time-resolved spectroscopy experiments. The
broad rotational wave packet excited by the laser pulse is well defined in the conjugate angle space,
peaking along the field polarization direction. Both the rotational excitation and the consequent
spatial alignment can be controlled by the choice of field parameters. Fragment angular distributions
following weak field photodissociation of the rotational wave packet are computed as a probe of the
degree of alignment. In the limit of rapid photodissociation the angular distribution is peaked in the
forward direction, reflecting the anisotropy of the aligned state. Potential applications of the effect
demonstrated range from reaction dynamics of aligned molecules and laser-control to material
deposition and laser-assisted isotope separation. © 1995 American Institute of Physics.
I. INTRODUCTION
Aligning, or even orienting molecules has been one of
the most important goals of modern reaction dynamics.1– 6
Currently employed alignment techniques include focusing
by an electric hexapole field,1 optical pumping with polarized light,2 collisional alignment in a seeded beam,3 and
alignment in a strong dc field.4 – 6
Of particular interest for the present work is the last
technique, which has been extensively analyzed
theoretically4 and is being successfully used by several
groups.4 – 6 Molecular alignment in this experiment results
from the mixing of a large number of total angular momenta
by the electric field. As the field strength increases the hybrid
states resemble less and less free rotors and the motion is
best described in terms of so-called pendular4 states, which
librate over an increasingly limited angular range about the
field axis.
The success of dc field alignment suggests the possibility
of aligning molecules in the ac field of a laser. The latter
offers potentially a significant advantage over the former
since the field required to exert sufficient torque to align the
molecule is trivially obtained by a pulsed laser. Thus, the
goal of aligning light and nonpolar molecules, which in a dc
field is impractical,4 – 6 may be readily achieved.
In multielectron dissociative ionization studies, conducted in high-intensity laser fields ~I'1014 –1016 W cm22!,
it is well established that the ionic fragments are ejected
strongly aligned along the polarization vector.7 There is also
experimental evidence that the molecular ions may align
prior to dissociation at sufficiently long pulse durations.7~a!
The alignment mechanism in these experiments was recently
examined in the far off-resonance limit.8 Time averaging of
the field-matter interaction was shown to reduce the Schrödinger equation to its form in a dc field4 and the attainable
alignment was estimated within the rigid-rotor
approximation.8
A related problem of considerable formal and experimental interest is the possibility of aligning molecules in
moderate-intensity laser fields ~I'109 –1012 W cm22!, where
the intensity is not sufficient to multiply ionize or break the
molecule and the dynamics remains within control. To the
best of our knowledge this method of alignment has not yet
been considered. Radiation sources in the intermediate intensity regime are currently becoming more and more widespread. Indeed, the large majority of gas-phase time-resolved
spectroscopic experiments are conducted in the range of
'109 –1012 W cm22.9 A proper understanding of how ac
fields of intermediate intensity affect molecular systems is
thus becoming increasingly important.
Closely related to laser-induced alignment is the problem
of rotational excitation taking place, e.g., in time-resolved
spectroscopy experiments.9 Theoretical analyses of such experiments often assume that the total angular momentum is
initially zero and remains so. This assumption may not be
realistic. Furthermore, it has been shown10 that rotations introduce qualitative changes in the response of molecules to
intense and moderately intense light. It is thus important to
assess the degree of rotational excitation at typical intensities
and pulse durations used in theoretical and experimental
studies.
In the present work we examine rotational excitation and
spatial alignment in moderate intensity ~I'109 –1012
W cm22! laser fields using a quantum mechanical timedependent theory. We show that substantial rotational excitation takes place under standard experimental conditions. The
resulting broad wave packet in angular momentum space is
spatially well-aligned about the polarization direction. Thus,
laser-induced alignment is qualitatively similar to the excitation, in time-resolved spectroscopic experiments,9 of an energetically broad vibrational wave packet which is welldefined in terms of the conjugate radial coordinate.
We show that alignment survives a certain time-delay
after the turn-off, and that it recurs at controllable times.
Thus, lasers are potentially capable of producing field-free
alignment.
The formalism is applied to the LiH system and to the I2
J. Chem. Phys. 103 (18), 8 November 1995
0021-9606/95/103(18)/7887/10/$6.00
© 1995 American Institute of Physics
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Tamar Seideman: Alignment in intense laser fields
system, representing the limits of a light and a heavy diatomic molecule. The effect demonstrated is quite general,
however, and hence our conclusions are equally applicable to
larger linear and symmetric top molecules. The numerical
calculations are complemented by an analytical model, applicable in the classical limit of short time and large mass.
In the next section we outline the theory and in Sec. III
we describe the analytical approximation. Section IV presents the results and Sec. V summarizes our conclusions.
iĊ jJM K ~ t ! 5
W •«W ~ t ! u j 8 J 8 M 8 K 8 &
3 ^ j JM K u m
J8K8
3exp@ i ~ E JK
j 2E j 8 ! t # ,
~2.4!
where
^ j JM K u mW •«W ~ t ! u j 8 J 8 M 8 K 8 &
W(JM K u J 8 M 8 K 8 ) is an integral over the Euler angles,
We consider an arbitrary molecule possessing a symmetry axis, subject to moderately intense radiation
«W ~ t ! 5«ˆ «« m f ~ t ! cos v t,
~2.1!
where f (t) is a smooth envelope, «ˆ is a unit vector along the
polarization direction, «m is the field amplitude, and v is the
laser frequency. Although the theory described in this section
is general, our major interest is in the case where the field
frequency is at or near resonance with an electronic u0&→u1&
transition, v;v01 .
In order to concentrate on rotational excitation and its
manifestation in spatial alignment, we assume that the pulse
duration has been chosen so as to avoid excitation of more
than two vibronic levels. The practical applicability of this
condition and the case where it is not met are commented on
below.
The complete wave function is thus
(
J M jKj
jJjM jKj
C jj
~ t ! ^ q,R̂ u j J j M j K j &
J K
3exp@ 2iE j j j t # ,
^ R̂ u J j M j K j & 5
A
2J j 11 J j
D K M ~ R̂ ! ,
j j
8p2
~2.2b!
j
D m 8 m being rotation matrices in the notation of Edmonds.11
In Eqs. ~2.2! we indicated explicitly the dependence of J, M ,
and K on the electronic state. Below we omit this index
except when required for clarity.
Substituting Eqs. ~2.2! into the time-dependent Schrödinger equation
]
C ~ q,R̂ u t ! 5HC ~ q,R̂ u t ! ,
]t
T ~ j JK u j 8 J 8 K 8 ! 5 ^ j JK u m ~ q! u j 8 J 8 K 8 &
W •«W ~ t ! , ~2.3!
H5H 0 1 m
we obtain a set of coupled equations for the expansion coefficients,
~2.5c!
W
and the transition dipole operator has been written as m
5 m̂m ~q!.
In Sec. IV we first consider the case where the system is
initially prepared in a single state u 0J i M i K i & and subsequently generalize the discussion to the more common situation, where a thermal average of rotational levels is initially
prepared. In the former case Eq. ~2.4! is supplemented with
the initial conditions
K
C JM
~ t50 ! 5 d j 0 d JJ i d M M i d KK i
j
~2.6a!
and in the latter case with
~ t50 ! 5w J 0 K 0 ~ T ! ,
J M 1K1
C 11
~ t50 ! 50,
~2.6b!
where w JK (T) are Boltzmann weights at temperature T.
In the weak field limit, starting with a pure rotational
state @Eq. ~2.6a!#, first-order perturbation theory gives
J M 0K0
~ t ! ' d J0Jid M 0M id K0Ki,
J M 1K1
~ t ! '2i A2 p
C 00
C 11
W •«W ~ E 11 1 2E 00 0 ! u 0J 0 M 0 K 0 & ,
3 ^ 1J 1 M 1 K 1 u m
J K
J K
where «W (E) is the Fourier transform of Eq. ~2.1! and the
spectroscopic selection rules @implicit in Eq. ~2.5b!# imply
J 1 5J i ,J i 61,
~2.2c!
~2.5b!
T( j JK u j 8 J 8 K 8 ) is an integral over the dynamical coordinates,
J M 0K0
~2.2a!
^ q,R̂ u j J j M j K j & 5 ^ qu j J j K j &^ R̂ u J j M j K j & ,
W ~ JM K u J 8 M 8 K 8 ! 5 ^ JM K u m̂ •«ˆ u J 8 M 8 K 8 & ,
C 00
where J j is the total angular momentum, M j and K j are the
projections thereof onto the space-fixed and body-fixed z
axes, and j is an electronic index, j50,1 where both electronic states are bound. R̂5~f,u,n! denotes the Euler angles
of rotation of the body-fixed (X,Y ,Z) with respect to the
space-fixed (x,y,z) frame, q are the dynamical variables,
and ^q,R̂ u j J j M j K j & are eigenstates of the field-free Hamiltonian H 0 ,
i
J M K
C j 88 8 8 ~ t !
5« m f ~ t ! cos v tW ~ JM K u J 8 M 8 K 8 ! T ~ j JK u j 8 J 8 K 8 ! ,
~2.5a!
II. THEORY
C ~ q,R̂ u t ! 5
(
J8M 8K8
M 1 5M i ,M i 61,
K 1 5K i ,K i 61.
In a nonperturbative field the system will Rabi oscillate between the ground and excited electronic surfaces, exchanging an additional unit of angular momentum with the field on
each transition. The number of rotational states excited is
limited, as discussed below, only by the pulse duration and
the ~J-dependent! detuning from resonance.
From Eq. ~2.5! it is evident that molecular alignment
requires the use of linearly polarized light and, in general, a
parallel transition. The first condition ensures that M is conserved and the second ensures that K is conserved. In this
situation, starting with a rotationally cooled molecular beam
~hence low M 0 , K 0 <J 0 !, a wave packet of high angular
momentum states is populated with small projections onto
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Tamar Seideman: Alignment in intense laser fields
the electric field and the molecular axes. The angular functions in Eq. ~2.2c! are increasingly peaked in the forward
direction as higher rotational levels are accessed. This is
readily appreciated by considering the asymptotic form of
d JKM ~u! for J@K,M . Using Eq. ~4.1.23! of Ref. 11 and Eq.
~22.15.1! of Ref. 12,
J@K,M
S D S
d JKM ——→ cos
uuu&
u
2
K1M
2
u
sin
u
2
D
K2M
J K2M @~ J2K ! u # ,
p
,
2
~2.7!
where J n are Bessel functions of integer order n. While the
precise nodal pattern of d JKM depends on K2M , their overall
envelope is peaked at u50 and p in the classical J@K,M
limit. With a perpendicular transition ~or with unpolarized or
circularly polarized light! K ~or M ! is no longer limited and
the angular dependence of the wave packet takes a more
complicated shape.
In the case of a diatomic molecule the only contribution
to the body-fixed projection of the angular momentum is
from the electronic part. This index ~usually denoted l!,
whether the same on both electronic states ~parallel transition! or differing by one ~perpendicular transition!, remains
constant in a given electronic state and is generally small,
l50,1,... for S,P,... electronic states.13 Hence alignment is
expected regardless of the type of transition but would be
improved in S electronic states. Put alternatively, although
the interaction term is proportional to sin u in the case of a
perpendicular transition ~rather than to cos u, as in the case
of a parallel transition!, the molecular axis is perpendicular
to m̂ and hence the qualitative effect is not altered. This
stands in contrast to the case of an isotropic distribution of
magnetic states.
It is worth noting, however, that individual rotor levels
can be spatially defined only as well as permitted by the
uncertainty principle. As in the case of broad-pulse excitation
of a vibrational wave packet,9 the highest rotational levels
contributing to the wave packets presented below are not
nearly as directed as the ensuing coherent wave packet. Put
alternatively, the expectation value of cos2 u in the angular
functions of Eq. ~2.2c!,
^ JM K u cos2 u u JM K &
5
H
1
@ 3M 2 2J ~ J11 !#@ 3K 2 2J ~ J11 !#
112
3
J ~ J11 !~ 2J13 !~ 2J21 !
J
is maximized for M 5K50 ~if the case of 6M '6K'J is
excluded! and approaches the limit of 21 from below as J→`.
Excluding the case of large u M u and u K u 'J, which is not of
interest here, single rotor states cannot be truly well-defined
spatially.
After the laser field has been turned off, the rotational
composition of the wave packet remains constant but
dephasing will take place on a time scale which depends on
the rotational energies accessed by the pulse and hence its
energy spread and intensity. The process of dephasing ~and,
at later times, rephasing! is well understood14 and its precise
occurrence in time can be readily calculated with knowledge
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of the rotational constants. Since rotational periods are long,
dephasing takes place on a long ~but system dependent! time
scale and hence alignment is expected to survive a substantial time delay during which the Hamiltonian is field free.
A direct technique for probing the alignment experimentally is weak-field photodissociation, used in the past to
probe molecular orientation by an electric hexapole1 and a dc
field.4 – 6 As shown in Sec. IV, provided that the photodissociation is simple and rapid, the angular distribution of the
photodissociation fragments reflects the angular anisotropy
of the aligned state.
Using Eq. ~2.2a!, the photodissociation cross section is
given as
s ~ k̂,t f ! 5
5
S
S
D
DU (
4 p 2 v 21
W 21 •«ˆ 21 u C ~ q,R̂ u t f ! & u 2
u ^ kW 2 u m
c
4 p 2 v 21
c
J1M 1K1
J M 1K1
C 11
~tf!
U
2
W 21 •«ˆ 21 u 1J 1 M 1 K 1 & exp~ 2iE J11 K 1 t f ! ,
3 ^ kW 2 u m
~2.8!
W2
where ^q,R̂ u k & are scattering wave functions of momentum
kW 5(k̂,k), k̂ being the scattering angles k̂5( u k , f k ) and k
5 A2mE. In Eq. ~2.8! we chose the frequency of the photodissociation field to couple the upper electronic state with the
continuum. Alternatively, the alignment of the ground level
can be probed by tuning the weak field to v20 . For simplicity
we omitted all quantum indices specifying the continuum
state save for the scattering angles. We note, however, that
for dissociation into molecular fragments, ^q,R̂ u kW 2& ~and
hence the angular distribution! depends on the photofragment magnetic state.15 Nonetheless, the latter is not resolved
in the large majority of experiments. Following summation
over final magnetic states the form of the angular distribution
is readily shown15,16 to depend only on the total angular momentum in the scattering state, J 2 and its space-fixed and
body-fixed z projections, M 2 and K 2 . The weak field selection rules J 2 5J 1 , J 161, M 2 5M 1 , M 161, K 2 5K 1 , K 161
ensure that in the limit of rapid photodissociation the dependence of the scattering amplitude on k̂ closely resembles the
dependence of the aligned sample on R̂. The partial wave
amplitudes ^ kW 2 u m̂21 •«ˆ 21 u 1J 1 M 1 K 1 & in Eq. ~2.8! are computed using the method of Ref. 17.
To gain better insight into the parameters which determine the alignment, it is instructive to rewrite Eq. ~2.4! in the
form
iḞ Jj ~ t ! 5 e Jj F Jj ~ t ! 1exp@ i v 01 t ~ d j 1 2 d j 0 !#
W •«W ~ t ! u j 8 J 8 & ,
3^ j J u m
J
F j 88 ~ t !
(
J
8
~2.9!
where F Jj (t)[C Jj (t)exp@2i e Jj t#, v015E 01 2E 00 and we set
E 0050. e Jj 'B e j [J(J11)2K 2 ] is the rotational energy and
B e j 5[2mR 2e j ] 21 . In Eq. ~2.9! and below we assume that the
field is linearly polarized and the transition is parallel,
M 5M i , K5K i , and for clarity of notation we omit the M i
and K i indices.
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Tamar Seideman: Alignment in intense laser fields
At optical frequencies Eqs. ~2.4! and ~2.9! show that
rotational excitation and alignment would be dramatically
enhanced on resonance, v'v01 , since for large detuning the
right-hand side is highly oscillatory. The extent of rotational
excitation in this case is determined either by the pulse duration t or by the relative magnitudes of the detuning from
resonance
D JJ 8 [ e j 88 2 e Jj
J
~2.10a!
and the effective Rabi coupling,
8 1
V J,J
R 5 2« mW ~ J u J 8 ! T ~ j J u j 8J 8 !
~2.10b!
@see Eqs. ~2.5!#. For short pulse durations J is restricted by
the requirement of enough Rabi oscillations,
8
J& t V JJ
R
~2.10c!
while for long pulses it is restricted by the requirement of
sufficient intensity
J,J 8
8
.
V J,J
R *D
~2.10d!
Perhaps counterintuitively, DJ,J 8 in Eqs. ~2.10! is not the partial detuning for a J→J 8 5J,J61 transition, which scales as
2JB e , but rather a combination of the partial with all accumulated detunings, and scales more closely to B e J(J11).
The two qualitative criteria for rotational excitation of Eq.
~2.10! are made more rigorous in Sec. IV by numerical calculations.
Defining an effective time variable x5B e t, Eq. ~2.9! assumes the form
i
d J
F ~ x ! 5 @ J ~ J11 ! 2K 2 # F Jj ~ x !
dx j
1exp@ i v 01 x /B e ~ d j 1 2 d j 0 !#
W •«W /B e ! u j 8 J 8 & ,
3^ j J u m
J
F j 88 ~ x !
(
J
8
~2.11a!
where «W /B e takes the role of an effective field variable. Thus,
increasing the reduced mass by a given factor is equivalent
to simultaneously increasing the field strength and decreasing the pulse duration by the same factor,
m→n3m;« m →n3« m ,
t → t /n.
~2.11b!
The scaling law of Eq. ~2.11b! is essentially exact, independent of the rigid-rotor approximation of the rotational energy
in Eq. ~2.11a!. Physically Eqs. ~2.11! signify that a heavier
molecule requires a longer time to respond to the field. With
increased mass, however, the rotational spacing decreases,
and a given number of rotational levels is accessed by the
field at lower intensities.
III. ANALYTICALLY SOLVABLE MODEL
For near- or on-resonance excitation, in particular at optical frequencies, the rotating wave approximation is essentially exact. In this situation Eq. ~2.9! reduces to
iḞ Jj ~ t ! 'B e @ J ~ J11 ! 2K 2 # F Jj ~ t !
J11
1 f ~t!
J
8
F j 88 ~ t ! V JJ
(
R ,
J 5J21
8
~3.1!
where we used Eqs. ~2.1!, ~2.5!, and ~2.10b!. Cast in the
form of a system of equations linking nearest neighbors, the
problem of rotational excitation in intense field is formally
equivalent to a variety of physically different problems arising in various areas of physics.18 –21 These include multiphoton excitation of atoms18 and molecules,19 resonant Raman
transitions of atoms20 and, interestingly, diffusion processes
between electrodes.21 It is thus of interest to examine the
general properties of this system within an analytically solvable approximation.
To zero order in the inverse coupling parameter in Eq.
8
~2.11!, B e /« m !1, assuming V JJ
R ' V R , independent of J,
we approximate Eq. ~3.1! by
J11
iḞ 0j ,J ~ t ! ' f ~ t ! V R
0
F j 8 ,J 8 ~ t ! .
(
J 5J21
8
~3.2!
Both the neglect of B e and the neglect of the J dependence
of the coupling become better approximations the heavier the
molecule. In that case B e is small ~B e '231027 a.u. for I2 for
8
instance! and the dependence of V JJ
R on J is primarily due
to W(J u J 8 ) in Eq. ~2.10b!. T( j J u j 8 J 8 ) depends on J only
through the small centrifugal potential ;B e J(J11).
It is worth stressing, however, that the analytical model
developed below is intended to provide complementary
qualitative insight, rather than quantitative results. We note,
moreover, that Eq. ~3.2! cannot describe reliably the case of
high rotational excitation. The neglect of the rotational energy implies that J-dependent detuning from resonance is not
accounted for and hence the number of rotational levels excited is limited only by the pulse duration. Equations ~2.10!
show that the neglect of B e restricts Eq. ~3.2! to the classical
limit of large mass and short time.
A WKB-type correction to Eq. ~3.2! can be obtained by
replacing VJR by V R [J̄(t)], where J̄(t) is the averaged angular momentum at time t,
J̄ ~ t ! 5 A^ C ~ q,R̂ u t ! u J 2 u C ~ q,R̂ u t ! &
5
A( uF ~ t !u J~ J11 !.
J
j
2
~3.3!
To first order, F Jj (t) in Eq. ~3.3! are approximated by F 0j ,J (t),
the solutions of Eq. ~3.2! derived below. Thus,
J11
iḞ 1j ,J ~ t ! ' f ~ t ! V R @ J̄ ~ t !#
1
F j 8 ,J 8 ~ t ! .
(
J 5J21
8
~3.4!
In the appendix we suggest that the effect of detuning can
also be accounted for in an approximate manner by introducing a second correction to the model.
Proceeding to express F 0j J explicitly, we first transform
Eq. ~3.2! to the form
iḞ 0j J ~ z ! '
0
F j 8 ,J 8 ~ z ! ,
(
J
8
~3.5a!
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Tamar Seideman: Alignment in intense laser fields
7891
where
z 5V R
E
t
f ~ t 8 ! dt 8
~3.5b!
and differentiation is with respect to z. Equations ~3.5! show
explicitly that the approximation introduced in Eq. ~3.2!
eliminates all coherent pulse-shape effects from the model:
The linear dependence on the field implies that the expansion
coefficients depend only on the time integral of the laser
pulse, Eq. ~3.5b!.
Assuming for simplicity that J i 50, Eq. ~3.5a! is supplemented by the initial conditions
F 0j J ~ t50 ! 5 d j 0 d J0
~3.6a!
and the selection rules implicit in Eq. ~2.5b! show that the
diagonal term is absent,
0
0
iḞ 0j ,J ~ z ! 'F j 8 ,J21 ~ z ! 1F j 8 ,J11 ~ z ! .
~3.6b!
@In the more general case the diagonal term can be eliminated by defining F 0j ,J 5e 2i z A j ,J where A j ,J satisfy Eq.
~3.6b!#. Noting the recurrence relation relating Bessel functions
d
J ~ 2z ! 5J J21 ~ 2z ! 2J J11 ~ 2z !
dz J
@Ref. 12, Eq. ~9.1.27b!#, we find
F 0j ,J 5e 2i p J/2 $ J J ~ 2 z ! 1J J12 ~ 2 z ! %
5e 2i p J/2
J11
J J11 ~ 2 z !
z
~3.7!
with J50,2,... for j50, J51,3,... for j51. In obtaining the
second equality we noted Eq. ~9.1.27! of Ref. 12.
Using Eqs. ~9.1.76!, ~9.1.77!, and ~9.1.7! of Ref. 12, it is
readily verified that F 0j J are complete and satisfy the initial
conditions ~3.6b!. It is worth pointing out that Eq. ~3.7! is a
unique solution of the system ~3.2! for all J, including the
ground level which is coupled to only a single state. This
follows from the symmetry property of the Bessel
functions12 J 2n 5(21) n J n . The first-order coefficients, F 1j J
of Eq. ~3.4!, take a similar form to Eq. ~3.7! with z replaced
by * t V R (t 8 ) f (t 8 )dt 8 .
Although pertaining to a limiting situation, Eq. ~3.7! provides physical insight and helps in visualizing several more
general features of the expansion coefficients. The properties
of the Bessel functions12 show that the Jth partial wave is
significantly populated at time 'J/V R . Following the first
peak at t'(J1 21 ) p /2V R , the expansion coefficients oscillate at the Rabi frequency with the amplitude of oscillations
decaying monotonically as ~VR t!23/2. In Sec. IV we show
that the exact coefficients behave similarly in the classical
limit of large mass and short time. We compare the numerical coefficients with their analytical approximation and provide quantitative criteria for the range of applicability of the
analytical model.
It is interesting to note that a formally similar ~although
physically quite different! problem was solved analytically in
the past using a different approach.20 Although the final form
FIG. 1. Rotational wave packet for the ground ~a! and excited ~b! electronic
states of LiH vs time and the Euler angle, measured with respect to the
polarization direction. The laser pulse is Gaussian with s21'100 fs and the
intensity is 1011 W cm22. The insert in ~a! shows the time evolution of the
average angular momentum, Eq. ~3.3!, for the ground ~solid curve! and
excited ~dashed curve! electronic states wave packets. The insert in ~b!
shows the angular distribution of the photofragments following weak field
photodissociation of the excited wave packet.
derived in Ref. 20 appears very different from Eq. ~3.7!, the
two forms are rigorously equivalent, as we show in the appendix.
IV. RESULTS AND DISCUSSION
A. The LiH system
In this subsection we apply the theory to the
X( 1 S 1 )→A( 1 S 1 )→C( 1 S 1 ) transition of the LiH system.
The pump field is tuned to the X( v 50)→A( v 50) transition
frequency and, subsequent to a time delay, the weak probe
excites the ~field-free! A-state wave packet to the dissociative C state.
We use the ab initio data of Partridge and Langhoff22 for
the X and A potential energy curves and the transition dipole
function coupling them. The C state is taken from the work
of Gatti et al.23 The ab initio data 22,23 are fit to cubic splines.
Figure 1~a! shows the rotational wave packet on the
ground electronic state as a function of time and the Euler
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Tamar Seideman: Alignment in intense laser fields
angle u, measured with respect to the polarization axis. The
laser field is Gaussian in time and frequency domains,
f ~ t ! 5exp@ 2 ~ t2t 0 ! 2 s 2 # ,
~4.1!
where s215100 fs is chosen sufficiently large to rule out the
possibility of exciting vibrations: s550 cm21<vv , where
the vibrational frequency vv is 1353 cm21 for the X and 296
cm21 for the A state.22 The intensity is 1011 W cm22, corresponding to Rabi coupling of 6 – 831024 @see Eq. ~2.10b!#
and the initial conditions are given by Eq. ~2.6a! with J i 50.
By the end of the pulse, t*t 0 1 s 21 , the wave packet concentrates in the forward direction, making small amplitude
vibrational motion about the field axis. The insert in Fig. 1~a!
shows the expectation value of the angular momentum vector, Eq. ~3.3! providing an approximate measure of the rotational excitation induced by the field. As a function of time
during the laser pulse, J̄(t) increases gradually as the intensity grows, reaches its maximum at the peak of the pulse,
t't 0 and remains constant thereafter. The expectation value
of cos2 u in the wave packet, ^C~q,R̂ u t!ucos2 uuC~q,R̂ u t!&, is
similarly a smooth, monotonically increasing function of
time, reaching '0.6 by the end of the pulse. At t*t 0 1 s 21
the wave packet consists of 20 rotational levels with
u C Jj (t) u .1026.
We have chosen the field parameters such that the rotational excitation is limited by the pulse duration, Eq. ~2.10c!.
However, the limit imposed by the balance between the detuning and the intensity, Eq. ~2.10d!, is reached with only a
small increase in the time duration. As discussed in more
detail below, this choice ensures that J̄(t) and cos2 u(t) increase smoothly and monotonically. When the field is left on
for a much longer time than that required for excitation of
the number of levels permitted by Eq. ~2.10d!, the expansion
coefficients recur, due to reflection from the edge of the rotational basis. Consequently J̄(t) and cos2 u(t) oscillate in
time and, with a sufficiently slow turn-off, the wave packet
returns adiabatically to the ground state.
Figure 1~b! shows the corresponding wave packet on the
excited A state, consisting ~due to the chosen initial conditions! of odd rotational levels J51,3,...,21. Similar to the
probability density on the ground electronic state,
uCj51(t.t 0 )u2 is strongly peaked about the field polarization
vector. The time-dependent expectation value of the angular
momentum vector is shown as a dashed curve in the insert of
Fig. 1~a!. Excitation of the wave packet to the dissociative C
state produces the angular distribution shown as an insert in
Fig. 1~b!. We have chosen the frequency of the probe field
sufficiently large to ensure that the photodissociation is rapid
and simple. Under these conditions, as illustrated in Fig.
1~b!, the angular distribution of the photofragments reflects
the angular anisotropy of the A-state wave packet, peaking
strongly in the forward direction. Qualitatively similar alignment and rotational excitation have been obtained with a
variety of pulse shapes. The details of the time evolution,
however, are sensitive to the turn-on and turn-off of the laser
pulse as discussed below.
The effect of varying the intensity of the pump field is
illustrated in Fig. 2. We have chosen a pulse of the form
FIG. 2. ~a! Excited state rotational wave packet for LiH following the pump
pulse and a time delay. The laser pulse is given by Eq. ~4.2! with s21
1 '200
10
W cm22, ~---! 1011 W cm22, ~—! 1012
fs and s21
2 '100 fs. ~•–•! 10
W cm22. ~b! Corresponding photofragment angular distribution.
f ~ t ! 5 $ 11exp@~ t 1 2t ! s 1 # 1exp@~ t2t 2 ! s 2 # % 21
~4.2!
with a slow turn-on, s21
1 '200 fs and a fairly rapid turn-off
s21
'100
fs.
Pulses
of
the form ~4.2! can be constructed, for
1
instance, as described in Ref. 24, and have been used in a
variety of applications. Figure 2~a! shows the excited state
wave packet as a function of the Euler angle u following the
pump pulse and a time delay t d '300 fs. Figure 2~b! shows
the photodissociation cross section following excitation of
the aligned sample to the repulsive C state. As the intensity
increases, the number of rotational levels populated increases
and the alignment is improved. More precisely,
J̄(t*t 1 )53.0, 4.5, and 7.5 and cos2 u(t * t1) 5 0.74, 0.79, and
0.82 at 1010, 1011, and 1012 W cm22, respectively. Equation
~2.11! shows that isotopic substitution would have a very
similar effect provided that the pulse duration is increased
simultaneously with the reduced mass. We note that the pulse
shape of Eq. ~4.2! produces better alignment than the Gaussian envelope of Eq. ~4.1!. This may be expected since it
allows a longer pulse duration without making the turn-off
adiabatically slow.
In Figs. 1 and 2 we assumed that the system has been
initially prepared in a single rotational state, J i 50. In reality
a rotationally cooled molecular beam consists in general of
several rotational levels, each populated according to BoltzJK
mann weight exp( 2 Eji i/kBT) and consisting of an isotropic
distribution of magnetic states, 2J i <M i <J i . Figure 3 examines the effect of increasing the temperature on the excited wave packet @Fig. 3~a!# and on the photofragment angular distribution @Fig. 3~b!#. The pulse envelope is given by
Eq. ~4.1! with s215200 fs, the intensity is 1012 W/cm22, and
the temperature increases from 0 to 50 K from bottom to top.
As expected from the discussion following Eq. ~2.7!, the
alignment survives Boltzmann averaging although the angular distribution broadens with increasing temperature. For the
relatively light LiH system at 50 K, the initial state consists
of J i <5 with w J i . 0.03, and 2J i >M i >J i . The high level
of rotational excitation, however @J̄(t*t 0 1 s 21 )517 under
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Tamar Seideman: Alignment in intense laser fields
FIG. 3. ~a! Excited state rotational wave packet for LiH following the pump
pulse and a time delay. The laser pulse is Gaussian with s21'60 fs and the
intensity is 1012 W cm22. The temperature increases from 0 to 50 K from
bottom to top. ~b! Corresponding photofragment angular distribution.
the conditions of Fig. 3# ensures that the angular momentum
remains large with respect to its z projection.
Clearly, for a heavier molecule more rotational levels J i
would be initially populated at comparable temperatures.
However, a smaller rotational constant implies that higher
total angular momenta would be accessed by the field at the
same intensity. Hence the conclusions of Fig. 3 are expected
to remain qualitatively applicable.
B. The I2 system
In this subsection we consider the limit of a heavy molecule and compare excitation of a parallel band with that of a
perpendicular band. We note briefly the effect of offresonance excitation and examine the validity of the analytical solution of Sec. III in the near classical ~t→0, B e →0!
limit. We use approximate ~Morse! potential energy curves
3 1 25
and the perpendicular
for the parallel X( 1 S 1
g )→B( P 0u )
1 1
3
26
X( S g )→A( P 1u ) transitions of I2 .
The qualitative effect of the increased mass may be
again anticipated from Eq. ~2.11!. The quantitative details,
however, depend on system properties including the shift of
the equilibrium distance between the ground and excited surfaces and the difference between the ground and excited
states rotational constants. Both properties affect the magnitude and J dependence of the detuning from resonance and
Rabi coupling.
Figures 4~a! and 4~b! show, respectively, the ground and
the excited states rotational wave packets subsequent to the
pump pulse and a time delay. The intensity is 109 W cm22 in
all traces and the pulse envelope is given by Eq. ~4.1! with
s2152.4 ps. The solid curves correspond to on-resonance
excitation of the parallel X→B transition v5E(B, v 50)
2E(X, v 50). The excited state wave packet @Fig. 4~b!# may
be qualitatively compared with the corresponding wave
packet for the LiH system, shown at 1–3 orders of magnitude higher intensity in Fig. 2~a!. As expected, rotational
excitation is facilitated due to the decreased rotational spacing, J̄(t)513 at t*t 0 for the wave packet of Fig. 4. Consequently the alignment is improved; cos2 u(t) reaches 0.81 at
7893
FIG. 4. Ground ~a! and excited ~b! rotational wave packets for I2 following
the pump pulse and a time delay. The laser pulse is Gaussian with s21'2.4
ps and the intensity is 109 W cm22. ~—! On resonance excitation of a parallel transition, v 5E(B, v 50)2E(X, v 50). ~---! On resonance excitation
of a perpendicular transition, v 5E(A, v 50)2E(X, v 50). ~•–•! offresonance excitation of a parallel transition, v 5(E(B, v 50)2E(X, v 50))/
2.
the end of the pulse. In accord with Eq. ~2.11!, however, a
longer pulse is required to align the heavy molecule. Classically speaking, the increased rotational period implies that
torque need be exerted over a longer period of time in order
to rotate the randomly oriented sample into the field direction.
The dashed curves in Fig. 4 show the rotational wave
packets on the I 2 , X, and A states following on-resonance
excitation of the perpendicular transition, v5E(A, v 50)
2E(X, v 50). The angular functions in the excited electronic state have a node in the forward direction since the
projection of the electronic angular momentum on the molecular axis is l51 in the A state @Eqs. ~2.2c! and ~2.7!#.
Nevertheless, the overall envelope of the wave packet in
both states is fairly well directed along the field axis due to
the higher level of rotational excitation.
The dot-dashed curve in Fig. 4~a! examines the effect of
detuning from resonance while keeping all other pulse parameters as in the case shown as solid curves. With the laser
frequency chosen as half the X→B transition frequency, the
B state is excited only transiently. At the end of the pulse the
excited state population is negligible. Nonzero angular momentum states of the ground electronic surface are populated
to some extent through Raman-type transitions. Thus, the
ground state wave packet shown as a dotted curve in Fig.
4~a! is not isotropic and possesses a shallow maximum in the
forward direction. Nevertheless, the rotational excitation is
very modest at the intensity of Figs. 4, J̄(t>t 0 )50.04 for the
ground state wave packet.
Figures 5~a!–5~d! show the time evolution of several of
the expansion coefficients in the wave packet given as solid
curves in Fig. 4. The abscissa in Fig. 5 are labeled by the
time integral of the pulse, h (t)5 * t f (t 8 )dt 8 . The Jth partial
wave acquires significant amplitude at time t J 'J/V JR and
subsequently oscillates in time at its J-dependent Rabi frequency. The amplitude of oscillations decays monotonically
with time as population migrates to higher rotational states.
J. Chem. Phys., Vol. 103, No. 18, 8 November 1995
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7894
Tamar Seideman: Alignment in intense laser fields
Before concluding this section it is worth commenting
on the practical applicability of the two vibronic states approximation. In both Secs. IV A and IV B we chose the pulse
duration sufficiently large to ensure that higher vibrational
levels are well outside the energetic width of the field. The
electronic transitions were chosen such that multiphoton
transitions to dissociative or ionic states will not be on resonance. In practice the latter condition is easier to satisfy for a
light molecule, where the manifold of electronic states is
sparse. While off-resonance multiphoton ionization or dissociation may be possible at the considered intensities, they are
not expected to affect the alignment but only to deplete the
population, and that to a small ~but system-dependent! extent. We note that off-resonant excitation of a third electronic
state, while changing the detailed dynamics, is not expected
to alter qualitatively the conclusions of this section.
V. CONCLUSIONS
FIG. 5. Expansion coefficients u C jJ u as a function of the time integral of the
laser pulse, h (t)5 * t f (t 8 )dt 8 . ~a!–~d! correspond to the wave packets
shown as solid curves in Fig. 4: In ~e! and ~f! the pulse duration is increased
to s21'6 ps, see the text for detail. The upper scale corresponds to ~a!–~d!
and the lower one to ~e! and ~f!. The dotted curves show the analytical
approximation, uF j0u of Eq. ~3.7!. ~a! J50, ~b! J51, ~c! J55, ~d! J58, ~e!
J51, ~f! J54.
The dashed curves in Fig. 5 compare the numerical results with their analytical approximation, F 0j J of Eq. ~3.7!.
The short time behavior is reproduced fairly well by the
simple analytical model, although the period of oscillations
and the rate of decay of their amplitude deviate from their
numerical values. Both features result from the neglect of the
J dependence of the coupling since VR determines both the
oscillations frequency and their time decay @see Eq. ~3.7! and
the discussion below#. The neglect of the rotational energy in
Eq. ~3.2! has a negligible effect on the short-time dynamics,
where the rotational excitation is modest.
In Figs. 5~e! and 5~f! we increase the pulse duration to
s2156 ps so as to display the long-time behavior. The pulse
envelope and the intensity are as in Figs. 4 and 5~a!– 5~d!. At
sufficiently long times the rotational excitation is limited by
the balance between the intensity and the detuning, rather
than by the pulse duration as in the examples discussed
above. As a consequence, the numerical coefficients recur at
characteristic times, due to reflection from the edge of the
rotational basis, determined by Eq. ~2.10d!. The recurrence at
long times cannot be reproduced by the analytical model due
to its neglect of detuning effects. Considering its simplicity,
however, it is remarkable that Eq. ~3.7! describes fairly well
the short and intermediate time dynamics in the case of a
heavy molecule.
In this work we studied rotational excitation and its
manifestation as spatial alignment in moderate intensity,
pulsed fields. We showed that substantial rotational excitation takes place under typical conditions used in timeresolved spectroscopy experiments. The resulting broad
wave packet in angular momentum space is well defined in
the conjugate angle space, approaching, as the intensity increases, the limit of a minimum uncertainty wave packet.
Our results indicate that the field-induced alignment survives
a significant time delay before dephasing ~and rephasing! of
the wave packet takes place.
Most of the discussion was limited to the case of a light
molecule, LiH, and to on-resonance excitation of a parallel
transition. The limit of a heavy molecule and the effects of
pumping a perpendicular transition and of off-resonance excitation were considered briefly. We noted the formal analogy of the rotational excitation problem to a range of physically different phenomena, arising in various fields.18 –21 By
solving a simple analytical model, applicable in the classical
limit of short time and large mass, we developed a readily
visualized picture of the rotational excitation dynamics.
Since the related problem of alignment in a far offresonance, highly intense laser field has already been demonstrated experimentally,7 it is of interest to briefly compare
the two modes of laser-induced alignment. Formally offresonance alignment, where real population resides on a
single electronic state, is quite different from the two electronic states problem discussed above. As shown in Ref. 8
the former mechanism is rather similar to alignment in a dc
field.4
The two modes of alignment differ also in practice and
may prove complementary. As suggested by the results of
Sec. IV B, the intensity required in the on-resonance case is
several orders of magnitude below that required at offresonance frequencies. The risk of ionizing and/or dissociating the molecule7 is thus avoided. On the other hand, offresonance excitation is not subject to the availability of
sources in the appropriate wavelength regime for a given
molecule, and hence different molecules may be aligned with
the same laser. Further, population loss due to coincident
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Tamar Seideman: Alignment in intense laser fields
multiphoton excitation of dissociative or ionic states need
not be taken into consideration in choosing the system.
For light molecules, such as H2 , where the polarizability
is typically small, we found off-resonance alignment to be
impractical since field ionization takes place before the intensity is sufficient to align the molecule. In this case nearresonance alignment is of advantage since the manifold of
electronic states is typically sparse. By contrast, for heavy
molecules with a dense electronic spectrum off-resonance
laser-induced alignment may prove of advantage provided
the initial temperature is sufficiently low. Further theoretical
study of this mode of alignment is currently underway.
To conclude we note briefly several exciting extensions
and applications of laser-induced alignment. These include a
study of the role played by the direction of impact in bimolecular reactions, laser focusing of molecules onto a substrate, the atomic analogue of which has already been
demonstrated,27 laser-assisted isotope separation, currently
under investigation, laser-induced trapping of molecules,8
and the possibility of directing current in a field-aligned
chain molecule, analogous to current control in multiple
quantum wells.28 The theoretical modeling of these processes
will be the goal in future research.
F 0j J @ z ~ t !# 5
2
p
7895
E
p
0
dx cos@~ N11 ! x # sin@~ N2J ! x #
3sin x exp@ 2i2 z cos x # .
~A2!
Separating Eq. ~A2! into its real and imaginary parts and
using Eqs. ~3.71.13! and ~3.71.18! of Ref. 29 we find
F 0j J ~ t ! 5exp@ 2i ~ 2N2J ! p /2 #
2exp@ 2iJ p /2 #
2N2J11
J 2N2J11 ~ 2 z !
z
J11
J J11 ~ 2 z ! .
z
~A3!
In the limit N→` the first of the two Bessel functions
decays as A Ne 24&N/3 @Ref. 12, Eqs. ~9.3.4!# and Eq. ~A3!
reduces to Eq. ~3.7!. Thus, the unphysical dependence of Eq.
~A1! on the size of the basis is eliminated.
Given an estimate of the limit on the basis size, introduced at long pulse durations by the J-dependent detuning,
Eq. ~A3! can be employed to build-in recurrences of the
coefficients in an artificial way. Such an estimate can be
found numerically, as the zero of Eq. ~2.10d! or through
analytical arguments. It is clear, however, that the artificial
cutoff solution can only provide a crude, zero order approximation to the long time dynamics.
ACKNOWLEDGMENTS
I am grateful to Dr. M. Yu. Ivanov, Dr. P. B. Corkum, Dr.
A. Stolow, and Dr. H. Stapelfeldt for many fascinating discussions about this and other topics. I would also like to
thank Dr. W. Siebrand, Dr. M. Yu. Ivanov, and Dr. H.
Stapelfeldt for reading the manuscript and making useful
comments.
APPENDIX
Reference 20 formulated an analytical model describing
the migration of population from low l to high l states of the
hydrogen atom through degenerate Raman coupling. Within
a certain set of approximations, the dynamical equations
were reduced to a similar form to Eq. ~3.2!, where, however,
V R f (t) is replaced by u«(t)u2 due to the Raman nature of the
interaction. Laplace transformation with respect to the laser
energy yields a tridiagonal system in the conjugate space
whose solution vector is backtransformed to the time domain
using Cauchy’s formula. The final solution of Ref. 20 in
terms of the variables of Sec. III takes the form,
F 0j J @ z ~ t !# 5
2
N11
N
(
n51
F
F
3sin
~ 21 ! n
G F G
S DG
N2J
np
n p sin
N11
N11
3exp 22i z cos
np
N11
,
~A1!
where now z (t)5 * t u «(t 8 ) u 2 dt 8 and N is the number of levels ~N5J max11!.
Defining x5n p /(N11) and replacing the n summation
by integration over x gives
1
See, for example, V. A. Cho and R. B. Bernstein, J. Phys. Chem. 95, 8129
~1991!, and references therein; P. R. Brooks, J. S. McKillop, and H. G.
Pippin, Chem. Phys. Lett. 66, 144 ~1979!; S. Stolte, in Atomic and Molecular Beam Methods, edited by G. Scoles ~Oxford University Press,
New York, 1988!, p. 631; D. A. Baugh, D. Y. Kim, V. A. Cho, L. C. Pipes,
J. C. Petteway, and C. D. Fuglesang, Chem. Phys. Lett. 219, 207 ~1994!.
2
See, for example, R. N. Zare, J. Chem. Phys. 69, 5199 ~1978!; M. A.
Treffers and J. Korving, Chem. Phys. Lett. 97, 342 ~1983!; K. Bergman
Atomic and Molecular Beam Methods, edited by G. Scoles ~Oxford University Press, New York, 1988!, p. 293; M. Hoffmeister, R. Schleysing,
and H. J. Loesch, J. Phys. Chem. 91, 5441 ~1987!.
3
See, for example, M. P. Sinha, C. D. Caldwell, and R. N. Zare, J. Chem.
Phys. 61, 491 ~1974!; M. J. Weida and D. J. Nesbitt, J. Chem. Phys. 100,
6372 ~1994!.
4
See, for example, B. Friedrich and D. R. Herschbach, Z. Phys. D At. Mol.
Clusters 18, 153 ~1991!; B. Friedrich, D. R. Herschbach, J.-M. Rost, H.-G.
Rubahn, M. Renger, and M. Verbeek, J. Chem. Soc. Faraday Trans. 89,
1539 ~1993!; B. Friedrich, A. Slenczka, and D. R. Herschbach, Chem.
Phys. Lett. 221, 333 ~1994!.
5
H. J. Loesch and J. Remscheid, J. Chem. Phys. 93, 4779 ~1990!; H. J.
Loesch and J. Möller, ibid. 97, 2158 ~1993!.
6
M. Wu, R. J. Bemish, and R. E. Miller, J. Chem. Phys. 101, 9447 ~1994!.
7
~a! D. Normand, L. A. Lompre, and C. Cornaggia, J. Phys. B 25, L497
~1992!; ~b! P. Dietrich, D. T. Strickland, M. Laberge, and P. B. Corkum,
Phys. Rev. A 47, 2305 ~1993!.
8
B. Friedrich and D. R. Herschbach, Phys. Rev. Lett. 74, 4623 ~1995!.
9
See, for example, A. H. Zewail, Femtochemistry ~World Scientific, Singapore, 1994!, Vols. I and II, and references therein; T. Baumert, M.
Groser, R. Thalweiser, and G. Gerber, Phys. Rev. Lett. 67, 3753 ~1991!; I.
Fischer, D. M. Villeneuve, M. J. J. Vrakking, and A. Stolow, J. Chem.
Phys. ~in press!.
10
E. E. Aubanel, A. Conjusteau, and A. D. Bandrauk, Phys. Rev. A 48,
R4011 ~1993!; E. E. Aubanel, J-M. Gauthier, and A. D. Bandrauk, ibid.
48, 2145 ~1993!.
11
A. R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd ed.
~Princeton University, Princeton, 1960!.
12
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions
~Dover, New York, 1965!.
13
l is replaced by V5ul1Su for nonzero spin.
14
P. M. Felker, J. Phys. Chem. 96, 7844 ~1992!, and references therein.
15
T. Seideman, J. Chem. Phys. 102, 6487 ~1995!.
J. Chem. Phys., Vol. 103, No. 18, 8 November 1995
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7896
Tamar Seideman: Alignment in intense laser fields
G. G. Balint-Kurti and M. Shapiro, Chem. Phys. 61, 137 ~1981!; 72, 456
~1982!.
17
T. Seideman, J. Chem. Phys. 98, 1989 ~1993!.
18
Z. Bialynicka-Birula, I. Bialynicka-Birula, J. H. Eberly, and B. W. Shore,
Phys. Rev. A 16, 2048 ~1977!.
19
A. Nauts and R. E. Wyatt, Phys. Rev. Lett. 25, 2238 ~1983!.
20
R. Grobe, G. Leuchs, and K. Rzazewski, Phys. Rev. A 34, 1188 ~1986!.
21
T. C. Kavanaugh, M. S. Friedrichs, R. A. Friesner, and A. J. Bard, J.
Electroanal. Chem. 283, 1 ~1990!.
22
H. Partridge and S. R. Langhoff, J. Chem. Phys. 74, 2361 ~1981!.
16
23
C. Gatti, S. Polezzo, M. Raimondi, and M. Simonetta, Mol. Phys. 41,
1259 ~1980!.
24
See, for instance, A. Weiner, J. P. Heritage, and E. M. Kirschner, J. Opt.
Soc. Am. B 5, 1563 ~1988!.
25
R. S. Mulliken, J. Chem. Phys. 55, 288 ~1971!.
26
S. Gerstenkorn and P. Luc, J. Phys. 46, 867 ~1985!.
27
J. J. McClelland, R. E. Scholten, E. C. Palm, and R. J. Celotta, Science
262, 877 ~1993!.
28
E. Dupont, P. B. Corkum, H. C. Liu, M. Buchanan, and Z. R. Wasilewski,
Phys. Rev. Lett. 74, 3596 ~1995!.
29
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products
~Academic, Orlando, 1980!.
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