Interference effects in above-threshold ionization from diatomic
  molecules: determining the internuclear separation
Interference effects in above-threshold ionization from diatomic molecules:
determining the internuclear separation
H. Hetzheim,1, 2 C. Figueira de Morisson Faria,3 and W. Becker2
Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
Max-Born-Institut für nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Str. 2A, D-12489 Berlin, Germany
Department of Physics and Astronomy, University College London,
Gower Street, London WC1E 6BT, United Kingdom
(Dated: October 28, 2018)
We calculate angle-resolved above-threshold ionization spectra for diatomic molecules in linearly
polarized laser fields, employing the strong-field approximation. The interference structure resulting
from the individual contributions of the different scattering scenarios is discussed in detail, with
respect to the dependence on the internuclear distance and molecular orientation. We show that,
in general, the contributions from the processes in which the electron is freed at one center and
rescatters off the other obscure the interference maxima and minima obtained from single-center
processes. However, around the boundary of the energy regions for which rescattering has a classical
counterpart, such processes play a negligible role and very clear interference patterns are observed.
In such energy regions, one is able to infer the internuclear distance from the energy difference
between adjacent interference minima.
I. INTRODUCTION
The interaction of matter with an intense laser field
(I & 1013W/cm
) leads to several phenomena, such as
above-threshold ionization (ATI) or high-order harmonic
generation (HHG). Such phenomena owe their existence
to physical mechanisms, in which an electron reaches
the continuum, by tunneling or multiphoton ionization,
at an instant t′. Subsequently, it is accelerated by the
field and driven back towards its parent ion, or molecule,
with which it rescatters or recombines at a later time
t [1]. Such laser-induced recombination or rescattering
processes take place within a fraction of a laser-field cycle.
The period of a typical near-infrared Ti:sapphire laser
pulse is T = 2π/ω ∼ 2.6fs. Thus, HHG and ATI occur on
a time scale of hundreds of attoseconds [2]. Hence, above-
threshold ionization and high-order harmonic generation
may be employed for probing, or even controlling, dy-
namic processes with attosecond and sub-angstrom reso-
lution.
This fact, together with new alignment techniques, has
opened a whole new range of possibilities for studying
molecules in strong laser fields, employing high-energy
photoelectrons or high-order harmonic radiation. Con-
crete examples are the attosecond reconstruction of the
nuclear motion in a molecule [3], the real-time imaging of
vibrational wavepackets [4], the tomographic reconstruc-
tion of molecular orbitals [5], the time-resolved measure-
ment of intramolecular quantum-interference effects [6],
or the determination of internuclear distances [7].
These applications are a direct consequence of the fact
that a molecule possesses a very specific configuration of
ions from which the electron may leave, or off which it
may rescatter causing above-threshold ionization, or re-
combine generating high-harmonics. This leads to char-
acteristic quantum-interference patterns in the HHG or
ATI spectra, in which structural information about the
molecule is hidden. This is true both for polyatomic [8]
and diatomic molecules [6, 7, 9, 10, 11, 13, 14, 15, 16,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. In particular
for diatomic molecules, it has been shown that the high-
order harmonic or ATI spectra exhibit overall maxima
and minima, which are highly dependent on the spatial
separation between both centers in the molecules, and
can be described as the interference between two radi-
ating point sources. In this sense, HHG or ATI by a
diatomic molecule may be viewed as the microscopic ana-
log of a double-slit experiment [6, 10, 11]. Furthermore,
such features depend on the symmetry of the highest oc-
cupied molecular orbitals, and on the alignment angle of
the molecule with respect to the laser-field polarization
[6, 7, 10, 11, 12, 13, 15, 19, 20, 21, 22, 23, 24, 25, 26, 27].
Specifically in the diatomic case, several aspects of
this interference have been extensively studied in the
past few years, such as the influence of the orbital sym-
metry, the internuclear distance, the alignment angles
[6, 7, 10, 11, 12, 13, 15, 19, 20, 21, 22, 23, 27], and
molecular vibration [24, 25, 26], as well as the role of the
laser-field shape [16, 17] or polarization [18]. Further-
more, an adequate modeling of bound molecular states,
in comparison with existing ionization experiments [14],
has also raised considerable debate [15, 19, 20, 27].
For that purpose, both the purely numerical solution
of the time-dependent Schrödinger equation [7, 11, 12],
and the strong-field approximation [10, 13, 15, 19, 20,
22, 23, 24, 25, 26, 27] have been employed. The latter
method allows a transparent physical interpretation of
the phenomena in question as laser-induced rescattering
or recombination processes, and permits a clear space-
time picture, which can be related to the classical orbits
of an electron in a strong laser field [28]. For a diatomic
molecule, there exist two main rescattering or recombi-
nation scenarios: the electron born through ionization at
the center Ci upon its return may recollide and interact
with either the same ion (Ci), or with the other one ( Cj
http://arxiv.org/abs/0704.0712v2
(i 6= j)) Such processes have been taken into account for
high-order harmonic generation employing a two-center
zero-range potential [13, 21], using Bessel function expan-
sions [19], and by means of saddle-point methods [21, 22].
In this paper, we calculate the energy spectra and an-
gular distributions of ATI produced electrons in linearly
polarized laser fields, within the framework of the strong-
field approximation (SFA) and the single-active electron
approximation (SAE). We employ a zero-range poten-
tial model similar to that in [13], and consider both the
direct electrons, which reach the detector without inter-
acting with their parent molecule, and the electrons that
suffer a single act of rescattering before reaching the de-
tector. In the latter case, we put particular emphasis on
interference effects: A final state with given momentum
outside the laser field can be reached via two different
scenarios. An electron can be born at and rescatter off
the same center, or it can be born at one center and
rescatter off the other. We show that the processes in-
volving two centers, in general, obscure the interference
patterns in the ATI spectra, in almost all energy-angle
regions. An exception, however, is the boundary of the
region that after tunneling is classically accessible to the
ionized electron, in other words, the region before the
classical cutoff. Near this boundary, the two-center pro-
cesses yield negligible contributions, and one may identify
very clear interference patterns. This makes it possible to
provide a recipe to determine the internuclear distance R
out of the angle-resolved ATI spectra. Throughout the
article we will use the velocity gauge and atomic units
(e = m = ~ = 1, c = 137).
The paper is organized as follows: In Sec. II we pro-
vide the ATI transition amplitudes for the direct and for
the rescattered electrons, which in Sec. III are employed
to compute the ATI spectra. The interference patterns in
the spectra are analyzed with respect to molecular orien-
tation, internuclear distance, and the position of the de-
tector with respect to the polarization of the laser field
and the molecular axis (Sec III A). In Sec. III B, we
present angle-resolved spectra, from which we infer the
internuclear distance. Finally, in Sec. IV, we summarize
the paper.
II. TRANSITION AMPLITUDES
The transition amplitude for direct ionization, within
the strong-field approximation (SFA) [29], is given by the
Keldysh-Faisal-Reiss amplitude [30]
Mp = −i
dt〈Ψ(V )p (t)|V |Ψ0(t)〉, (1)
where |Ψ0(t)〉 = |Ψ0〉 exp(i|E0|t). The amplitude de-
scribes an electron, initially in the ground state |Ψ0〉, that
is injected in the continuum by the laser field overcom-
ing the ionization potential |E0|, and reaches the detector
with final momentum p. The form of the transition am-
plitude given here, which contains the binding potential
V (r) rather than the interaction with the laser field, was
first presented in Ref. [31]. In the SFA, the final state
with momentum p is described by a Volkov state, which
in velocity gauge has the form
〈r|Ψ(V )p (t)〉 =
(2π)3/2
eip·re
dτ [p+A(τ)]2
. (2)
In the amplitude (1), the electron once ionized does not
interact with the ion (that is, with the binding potential
V ) anymore. If we allow for at most one single act of
rescattering, the amplitude (1) is replaced by
M (0,1)p = −
×〈Ψ(V )p (t)|V U (V )(t, t′)V |Ψ0(t′)〉. (3)
Here, U (V )(t, t′) denotes the Volkov time-evolution oper-
ator, which describes the evolution of the electron in the
presence of the external laser field, ignoring the binding
potential. Equation (3) incorporates direct ionization, as
described by Eq. (1), as well as ionization followed by
rescattering (for details, see, e.g., Ref. [32]).
In order to apply Eqs. (1) and (3) to a diatomic
molecule, we consider the two-center binding potential
V (r) = V0(r−R1) + V0(r−R2), (4)
where Ri (i = 1, 2) denote the coordinates of the centers
Ci (i = 1, 2). For the ground-state wave function, we
employ a linear combination of atomic orbitals (LCAO):
Ψ(r) = c1Ψ0(r−R1) + c2Ψ0(r−R2). (5)
Specifically, we will use the zero-range potential
V (r) =
r, (6)
whose single bound state is described by the wave func-
Ψ0(r) =
)1/2 1
e−κr, (7)
with κ =
2|E0|. The regularization operator ∂/∂r r
acts on the wave function to its right in order to satisfy
the proper boundary conditions at the origin [34]. For
direct ionization by a monochromatic linearly polarized
laser field
A(t) = A0 cosωt e, (8)
the evaluation of the amplitude (1) is straightforward.
Taking R1 = R/2 and R2 = −R/2, so that R is the
internuclear distance of the two centers, one obtains by
expanding the exponent in the Volkov wave function in
Eq. (2) into Bessel functions
M0p =
+ Up + |E0| −Nω
J−(2l+N)
2p · e
, (9)
0 0.5 1 1.5 2 2.5
energy in units of U
s) atom
molecule R=2 a.u.
molecule R=6 a.u.
FIG. 1: (Color online) ATI spectra of the direct electrons in
the laser polarization direction in the atomic and molecular
case, for a molecule aligned parallel to a laser field of frequency
ω = 0.058 a.u. and ponderomotive potential Up = 2.08 a.u.
We consider a symmetric combination of atomic orbitals [c1 =
c2 = 1/
2 in Eq. (5)], with ionization potential |E0| = 0.9
a.u. In order to facilitate the comparison, the same ionization
potential V0(r) was chosen in the atomic and molecular cases.
The arrows mark the various destructive interference energies
(n = 0, 1, 2) for R = 6 a.u..
where Up = A
0/4 denotes the ponderomotive energy of
the laser field (8). The prefactor, which is proportional
F = −κ+ exp(−κR)/R, (10)
is of no relevance, since we do not attempt to calculate
total ionization rates. It is, however, worth mentioning
that the limit of R → 0 is not straightforward. Below we
will not face this limit. For a more detailed discussion,
see, e.g., Refs. [13, 35].
The only difference between the matrix element (9) for
a molecule and the corresponding matrix element for an
atom, besides the R-dependent prefactor, is the presence
of the term cos(p ·R/2). This term describes the inter-
ference of electron orbits with momentum p originating
from one or the other center of the two-center potential
(4). The cosine term comes from assuming a symmetric
combination of orbitals in the ground-state wave func-
tion (5), so that c1 = c2 = 1/
2. For an antisymmetric
combination, so that c1 = −c2, the cosine is replaced by
a sine, leading to suppression of electrons with low mo-
menta due to destructive interference [9, 10, 11]. The
interference factor cos(p ·R/2) = cos(pR cos θ/2) yields
destructive interference for electrons with energies
E ≡ p
(2n+ 1)π
R cos θ
for integer n. An illustration of the interference effect
is given in Fig. 1, which shows the spectrum of the di-
rect electrons for an atom and for a symmetric diatomic
molecule aligned parallel to the laser-field polarization.
The clearly visible sharp dips in the spectrum, due to
destructive interference, are indicated by the arrows in
the figure. Next, we turn to the evaluation of the ma-
trix element (3), which allows rescattering. With the
two-center potential (4) and the symmetric ground-state
wave function (5), the matrix element reads
(2π)3/2
d3r′ e−ip·re
dτ(p+A(τ))2 [V0(r+R/2) + V0(r−R/2)]
× U (V )(rt; r′t′) [V0(r′ +R/2) + V0(r′ −R/2)] [Ψ0(r′ +R/2) + Ψ0(r′ −R/2)] ei|E0|t
. (12)
For the zero-range potential (6), the integrations over
space can be carried out easily [32], which leaves a
two-dimensional integral over the ionization time t′ and
the rescattering time t. For finite-range potentials, one
may proceed by introducing form factors and employing
saddle-point methods. In this case, the single-center pre-
factors cause an overall decrease of the yield for increas-
ing photoelectron energy. There are, however, no signif-
icant changes in the interference patterns in comparison
with the zero-range case, since these prefactors do not
influence the action or the cosine factor. For a detailed
discussion of the single-atom case, see, e.g., Ref. [36].
We split the eight integrals into those where the elec-
tron rescatters off the same center from which it was ion-
ized (r = r′ = ±R/2) and those where it rescatters off
the opposite center (r = −r′ = ±R/2). We refer to the
respective terms by M ij where i = +,− denote the cen-
ter of ionization and j = +,− the center of rescattering.
The integrals M++ and M−−, which specify the elec-
trons coming from and rescattering off the same center,
are essentially identical to the corresponding results for
an atom [32]. The structure of the molecule is reflected
in the integrals M+− and M−+, which characterize the
electrons that experience the presence of both centers.
Evaluating the remaining two integrals over t and t′, we
substitute t′ = t − τ . The doubly infinite integral over t
then yields a δ-function expressing energy conservation,
while the semi-infinite integration over τ has to be cal-
culated numerically. Expanding all oscillating exponents
in terms of Bessel functions, we obtain for the transition
amplitudes
M++p +M
p = 2F cos
+ Up + |E0| −Nω
J−(2l+N)
2p · e
)3/2{
e−i(|E0|τ+lα)e
−iUpτ [1−(
sinωτ/2
= 2F cos
M (atom)p , (13)
M+−p = Fe
+ Up + |E0| −Nω
2τ e−i[|E0|τ+lα+(2l+N)β−]e
−iUpτ [1−(
sinωτ/2
J−(2l+N)
, (14)
M−+p = Fe
+ Up + |E0| −Nω
2τ e−i[|E0|τ+lα+(2l+N)β+]e
−iUpτ [1−(
sinωτ/2
J−(2l+N)
. (15)
The real quantities A, B± and the phases α and β± are
defined by
Ae−iα = e−2iωτ +
e−iωτ , (16)
−iβ± = p · e± R · e
[i sinωτ − (1− cosωτ)].(17)
Upon R → −R, we have B± exp(−iβ±) →
B∓ exp(−iβ∓). Consequently, the matrix element Mp
does not change when R → −R. The complete matrix
element is the sum of the terms (13) – (15),
Mp = M
p . (18)
The first two terms describe electrons originating from
and rescattering off the same center. They are pro-
portional to the atomic ionization amplitude M
(atom)
[32] multiplied by the wave-function overlap F and the
two-center interference factor cos(p ·R/2), which we ob-
served for the direct electrons in Eq. (9). The behavior of
the exchange terms M+−p and M
p is more complicated
and will be discussed below.
The transition amplitude Mp simplifies enormously
when the molecule is aligned perpendicularly to the field
so that R · e = 0. Equation (17) shows that in this case
B+ = B− and β+ = β− = 0. Hence, the integrals on
the right-hand side of Eqs. (14) and (15) are equal and
M+−p +M
p becomes proportional to 2 cos(p ·R/2) just
like M++p +M
p . If, in addition, the electron is emitted
perpendicularly to the field so that also p · e = 0, then
B+ = B− = 0 and we have Mp = 0 unless N is even.
Substituting τ → τ/ω in Eqs. (13)–(17) one can see
that the amplitudes M ijp and their sum Mp depend on
the parameters of the problem through the dimension-
less quantities p2/ω, Up/ω, and ωR
2 when the relative
orientations of the vectors R, p, and e are kept fixed.
III. PHOTOELECTRON SPECTRA
In this section we discuss the ATI spectra computed
employing the transition matrix elements (1) and (3),
and a symmetric combination of equivalent centers [c1 =
c2 = 1/
2 in Eq. (5)]. For the sake of simplicity, in
the comparison between the atomic and molecular case
the same ionization potential V0 is chosen. Specifically,
we take |E0| = 0.9 a.u. in Eqs. (5) and (7) through-
out [37]. In Sec. III A, we perform a detailed analysis of
the interference patterns with respect to the molecular
orientation, rescattering scenarios, and the direction of
electron emission, while in Sec. III B we provide a recipe
for measuring the internuclear distance from an analysis
of the interference patterns in the angle-resolved spectra.
A. Analysis of the interference patterns
As a first step, we investigate how the interference pat-
terns are influenced by the orientation of the molecule
with respect to the laser-polarization direction. Such re-
sults are displayed in Figs. 2 and 3 for parallel and per-
pendicular orientations, respectively. In both cases, we
compare the entire ATI spectrum consisting of the di-
rect and the rescattered electrons in the atomic and the
molecular case. Unless stated otherwise (cf. Fig. 5), we
consider electron emission in the laser-polarization direc-
tion.
As expected from Eq. (9), for energies smaller than
2Up, the main contributions to the yield come from the
direct-ionization matrix element (9). Apart from the
interference-related factor of cos(p ·R/2), the transition
matrix element is identical to that obtained for a sin-
gle atom (cf. Fig. 1). This factor is responsible for the
sharp interference dips at the positions given by Eq. (11).
In the plateau energy region, however, the spectra de-
pend on the laser-field polarization in a more complex
way, as will be discussed next. In case the molecule is
aligned parallel to the laser-field polarization (Fig. 2),
the plateau is strongly enhanced in the molecular case,
and the structure of the spectrum is very different from
the atomic case and dependent on the internuclear dis-
tanceR. Indeed, inspection of the exchange integrals (14)
and (15) does not reveal any simple dependence on the
internuclear distance. Generally, for the molecular case
there are more pathways into a given final state. For our
case of a two-center potential, there are four pathways in
place of one for the atomic case. If they add coherently,
a significant enhancement can result, ideally by a factor
of 16, which is roughly what is observed in Fig. 2 be-
fore the cutoff. The structure caused by the cosine factor
is suppressed in the plateau region. This is caused by
the contribution of the processes in which the electron
is ejected from one center and rescatters off the other.
Such processes correspond to the transition amplitudes
M−+p andM
p , which do not exhibit the proportionality
to the cosine that is characteristic of the the one-center
scattering amplitudes M++p and M
p . A further par-
ticular feature observed in this case is the displacement
of the cut-off to higher energies with increasing internu-
clear distance. This can be understood by the fact that
an electron that moves from one center to the other may
gain more energy from the field since it may be acceler-
ated over a longer distance before it recollides.
A strikingly different behavior is observed if the
molecule is aligned perpendicular to the direction of the
laser field so that R ·e = 0. In Fig. 3 we consider the case
where the electron is emitted in the direction of the laser
polarization so that p ·R = 0, too. In this case, there is
a general enhancement of the ATI yield in comparison to
that of a single atom by roughly a factor of two for the di-
rect electrons and a much larger factor for the rescattered
electrons. Notice that the molecular spectrum is practi-
cally independent of the internuclear distance [21], since
the R dependence of the prefactor (10) is weak for R & 2
and the exponential of R2/(2τ) is small for the values
of R that we consider and the values of τ that give the
dominant contributions to the integral. The entire ATI
spectrum does not show any interference structure, since
the contributions from the two centers add constructively
for p ·R = 0. In fact, the cosine term in the matrix el-
ements M0p, M
p and M
p simply reduces to one and
the spectrum therefore looks like that of an atom. Specif-
ically within the plateau, by symmetry the contributions
from the two centers always interfere constructively. This
results in a spectrum that is largely independent on the
internuclear separation R, except that the plateau is en-
hanced, compared with the atomic case, by the existence
of four pathways. Formally, this can be understood as
discussed above at the end of Sec. II.
For arbitrary p · R, if the electron is emitted perpen-
dicular to the laser polarization so that p · e = 0, then
we see from Eqs. (16) and (17) that B+ = B−, while
β− = β+ + π. The sum of the two exchange terms then
goes like cos(p ·R/2) for even N and like sin(p ·R/2)
for odd N . This holds regardless of the orientation of
the molecule. If the laser polarization is perpendicular to
both the electron momentum and the internuclear axis,
then B+ = B− = 0. This implies that Mp is nonzero
only for integer N . Each other electron peak is missing.
Next, we discuss the interference pattern in more detail
by analyzing the individual contributions to the transi-
tion matrix element. In Fig. 4, we separately investigate
the individual contributions to the amplitude (18). If
only M++p and M
p are taken, a very pronounced min-
inum is observed near 5Up. These matrix elements cor-
respond to the case in which the electron is ejected from
and rescatters off the same center, so that the minimum is
due to the term cos(p ·R/2). In the full spectrum |Mp|2,
however, this minimum is absent because it is filled by the
contributions from the exchange terms M−+p and M
For a given orientation of the molecule with respect to
the laser field and for fixed momentum p, Fig. 4 shows
that the contribution |M−+p |2 of the scenario in which
the electron is freed at the center C1 and rescatters off
at the center C2 is different from that of |M+−p |2 where
it is released at C2 and rescatters at C1. The same has
been observed for high-order harmonic generation in a
two-center system [13].
The imprints of interference can still be observed if
the electron is emitted away from the laser polarization
direction. This will cause, however, an overall decrease
in the photoelectron energies for both the direct and the
rescattered electrons. Examples are presented in Fig. 5.
This behavior is known from atomic ionization, and its
origin is the same in the molecular case; for a discussion,
see, e.g., Ref. [32].
0 2 4 6 8 10
energy in units of U
s) atom
molecule R=2 a.u.
molecule R=4 a.u.
FIG. 2: (Color online) Comparison of the complete ATI spec-
tra consisting of direct and rescattered electrons in the atomic
and the molecular case for internuclear distances of R = 2
a.u. and R = 4 a.u.. The molecule is aligned parallel to
the laser-polarization direction, and the electrons are emitted
in the same direction. The arrows mark the destructive in-
terferences (n = 0, 1) of the molecule for R = 2 a.u.. The
destructive interference for n = 1 is already in the plateau
region, where the role of exchange terms becomes important.
The remaining parameters are the same as in Fig. 1.
0 2 4 6 8 10
energy in units of U
s) atom
molecule R=2 a.u.
molecule R=4 a.u.
FIG. 3: (Color online) The same as Fig. 2 but with the
molecular axis perpendicular to the laser polarization. The
electrons are emitted parallel to the laser polarization.
B. Determining the internuclear distance
9 10 11
0 2 4 6 8 10
energy in units of U
FIG. 4: (Color online) Individual contributions of the var-
ious rescattering scenarios to the total amplitude (18) for
a diatomic molecule with internuclear distance R = 2 a.u.
aligned parallel to the laser-field polarization, for the same
molecular and field parameters as in Fig. 2. The electrons are
emitted in the polarization direction. The arrows mark the
respective cutoff energies for the various transition amplitude
matrix elements. The inset at the lower left is an enlargement
of the region near the cutoff where the direct terms and the
exchange terms differ in a characteristic fashion, allowing for
the determination of the internuclear separation.
-8 Ψ=0
0 2 4 6 8 10 12
energy in units of U
A || R
A ⊥ R
FIG. 5: (Color online) Electron yield for the parameters of
Fig. 1 and internuclear distance R = 2 a.u. for different
emission angles ψ with respect to the polarization of the laser
field. The molecule is aligned parallel (perpendicular) to the
laser field in the upper (lower) panel.
FIG. 6: (Color online) Angle-resolved ATI spectra on a log-
arithmic scale for a diatomic molecule with internuclear dis-
tances R = 2 a.u. and R = 3 a.u. (middle and bottom
panels, respectively), aligned parallel to the laser-field polar-
ization, compared to the single-atom case (upper panel). The
binding energy is E0 = 0.9 a.u. in all cases, and the laser fre-
quency and the ponderomotive potential are ω = 0.058 a.u.
and Up = 2.08 a.u., respectively. The plotted lines depict the
minima (solid lines) and the maxima (dashed lines) of the
energy distribution given by Eq. (11).
FIG. 7: (Color online) Angle-resolved ATI spectra on a log-
arithmic scale for a diatomic molecule with ionization poten-
tial E0 = 0.9 a.u. and internuclear distances R = 4 a.u.,
R = 6 a.u. and R = 8 a.u. (upper, middle and bottom
panels, respectively), aligned parallel to the laser-field polar-
ization. The field parameters are the same as in the previous
figure. The plotted lines depict the minima (solid lines) and
the maxima (dashed lines) of the energy distribution given by
Eq. (11).
For a complete picture of the angle-resolved ATI spec-
trum, not restricted to emission in particular directions,
we will now present density plots. While they invariably
imply loss of fine details and depend on the positioning
and gradient of the false-color scale, they give a compre-
hensive overview of the general structure. We restrict
ourselves to the case of parallel alignment [41]. In this
case the spectrum is symmetrical with respect to the in-
ternuclear axis. It is obvious that the electrons with max-
FIG. 8: (Color online) Enlargement of a limited energy-angle
region of Fig. 7 for R = 8 a.u. (lowest panel) with increased
resolution. The indents of Fig. 7 are distinctly visible as val-
leys deeply cut into the high ridge that precedes the cutoff.
imal kinetic energy will be detected in the direction of the
laser field.
The angle-resolved spectra displayed in Figs. 6 and 7
are very intricate and do not exhibit any simple struc-
tures. They depend strongly on the internuclear sep-
aration but do not, on a first inspection, lend them-
selves in any obvious way to the assignment of a specific
value of R to a given spectrum. Especially, owing to the
presence and magnitude of the exchange terms (14) and
(15), the two-center interference, which is expressed in
the cos(p ·R/2) term, is not immediately visible. How-
ever, looking more closely, one can observe a very distinct
manifestation of this term just near the classical bound-
ary of the spectrum. Roughly, the latter agrees with the
boundaries of the colored areas in the various panels of
Figs. 6 and 7. We observe well-defined indents in the
overall smooth curve that defines the classical boundary.
The positions of these indents and, especially, their sepa-
rations agree quite well with the interference minima pre-
dicted by Eq. (11). The figures show that the separation
δE between the indents (on the scale of Up) monotoni-
cally decreases with increasing R. Hence, by comparing a
measured angle-resolved spectrum with Figs. 6 and 7 we
can infer the internuclear separation. For the parameters
underlying Figs. 6 and 7, the resulting function R(δE) is
given in Table I. Fig. 8 exhibits an enlargement of the
relevant area around the classical cutoff for the case of
R = 8 a.u. with a higher resolution of the electron yield.
The indents are very clearly visible like valleys that cut
into the drop of a plateau on a topographical map.
An analytical formula for R(δE) can be gained from
an analytical formula for the classical cutoff energy E(θ)
as a function of the angle θ. Intersecting this with the
energies of the interference minima given by Eq. (11) al-
lows one to determine the function R(δE) in dependence
of the parameters of the problem. For the case of an
atom, such a formula for E(θ) is actually known [39]. At
least for R ≤ 6 a.u., Figs. 6 and 7 show that this classical
boundary does not depend very strongly on the inter-
nuclear separation R, so that the atomic result could be
employed. However, even with this simplification, the re-
sulting formula is quite complicated and we refrain from
presenting it here.
The question arises of why near the classical bound-
ary the interference term cos(p ·R/2) roughly multiplies
the angle-resolved spectrum, like it does for the direct
electrons. The answer can be inferred from Fig. 4. The
total ionization amplitude Mp is the superposition (13)
of four different scenarios such that the electron starts
from and rescatters off one or the other center. The two
contributions (13) where they start from and rescatter
off the same center are identical except for the geomet-
rical phase, which leads to the cosine in Eq. (13). In
contrast, the other two contributions (14) and (15) are
uncorrelated since they are generated by geometrically
different scenarios. Their magnitudes are different and
almost nowhere do they exhibit a significant construc-
tive interference. The two contributions (13) are individ-
ually large when the long orbit and the short orbit add
constructively, as is the case specifically just before the
classical cutoff. In this case, they dominate the other
two terms (14) and (15) by a factor of the order of 2 to
4. Hence the complete spectrum distinctly exhibits the
geometrical interference, which is expressed in the factor
cos(p ·R/2).
Internuclear distances Energy differences of the indents
of the molecule at the spectral boundary
R = 2 a.u. δE ≈ 4.5Up
R = 3 a.u. δE ≈ 3.2Up
R = 4 a.u. δE ≈ 2.4Up
R = 6 a.u. δE ≈ 1.3Up
R = 8 a.u. δE ≈ 1.1Up
TABLE I: Energy differences between adjacent indents
around the classical boundary of the angle-resolved spectrum
of Fig. 6. The differences are taken, for each internuclear
separation, by starting with the first indent for Ψ ≥ 0◦ as a
function of the energy.
IV. CONCLUSIONS
We have analyzed ATI spectra for a two-center
molecule in a linearly polarized laser field. The terms
of the two-center wave function contributing to the in-
terference structure within the SFA formalism could be
identified as well as the absence of the interference struc-
ture throughout most of the plateau region. We have
shown that the angle-resolved spectra can be used to de-
termine the internuclear distance of a molecule aligned
with the laser field, by reading off the energy differences
between subsequent interference minima at the classical
boundary of the spectrum.
The validity of this method depends upon how close
to reality is the angle-resolved spectrum calculated for
our model molecule. Certainly, the spectrum of the di-
rect electrons cannot be trusted for this purpose. How-
ever, high-order ATI of an atom is well described by the
SFA and a zero-range potential, especially near the classi-
cal boundary [32]; for a comparison of spectra calculated
from the SFA with the solution of the time-dependent
Schrödinger equation, see Ref. [40] for the case of an
atom. Experimentally, application of the method re-
quires a high detection efficiency that allows one to ob-
tain a sufficient number of counts down to the classical
boundary.
The problem of how to extract the internuclear sepa-
ration from a diffraction pattern has been addressed by
a different method in Ref. [7], employing the numerical
solution of the time-dependent Schrdinger equation. In
[7], however, it is necessary to compute a radial distribu-
tion function from the diffraction intensity, whereas, with
the method discussed in this paper, one may determine
the internuclear distances directly from the spectra. The
method suggested in Ref. [7] has the advantage that it
analyzes direct electrons and, therefore, does not require
exceptionally high detection efficiency.
Finally, in a real physical system, there exist additional
effects, which have not been incorporated in this model
and may alter the interference patterns. Molecular vi-
bration, for instance, causes an intensity loss in the high-
harmonic signal [25], which may lead to a blurring in the
patterns. However, recently, numerical ATI computa-
tions in which such an effect is included have shown that
for H+2 the angle-dependent interference patterns related
to the double-slit physical picture remain distinguishable
in the case considered [17]. Generally, the amount of
blurring depends on the rigidity of the vibrational poten-
tial. Since the period of vibrations is much longer than
the laser period, with a few-cycle laser pulse our method
could be used to track a vibrational wave packet or the
dissociation of a molecule [4]. Another feature which has
not been incorporated in our model is the dependence of
the ionization potential on the internuclear distance. In
fact, we have taken E0 to be constant, whereas, in reality,
it decreases with R [13]. This feature, however, will only
cause an overall energy shift in the spectra. Therefore,
it will not affect the distance between two consecutive
minima or maxima in the patterns for constant internu-
clear distance (Figs. 6 and 7). Therefore, we expect our
method to be applicable to real physical systems and a
wide parameter range.
Acknowledgments
C.F.M.F. would like to thank L.E. Chipperfield, R.
Torres, and J.P. Marangos for useful discussions and the
UK Engineering and Physical Sciences Research Council
(Advanced Fellowship, grant No. EP/D07309X/1) for
financial support
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will move over the entire angle-resolved ATI spectrum.
