Two-center interference in high-order
harmonic generation from heteronuclear
diatomic molecules
Xiaosong Zhu,1 Qingbin Zhang,1 Weiyi Hong,1,3,∗ Pengfei Lan,1 and
Peixiang Lu1,2,∗
1
Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and
Technology, Wuhan 430074, China
2 State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai
200062, China
3 [email protected]
∗ [email protected]
Abstract: Two-center interference for heteronuclear diatomic molecules
(HeDM) is investigated. The minimum in the high-order harmonic spectrum, as a consequence of the destructive interference, is shifted to lower
harmonic orders compared with that in a homonuclear case. This phenomenon is explained by performing phase analysis. It is found that, for an
HeDM, the high harmonic spectrum contains information not only on the
internuclear separation but also on the properties of the two separate centers,
which implies the potential application of estimating the asymmetry of
molecules and judging the linear combination of atomic orbitals (LCAO) for
the highest occupied molecular orbital (HOMO). Moreover, the possibility
to monitor the evolution of HOMO itself in molecular dynamics is also
promised.
© 2011 Optical Society of America
OCIS codes: (190.7110) Ultrafast nonlinear optics; (190.4160) Multiharmonic generation;
(300.6560) Spectroscope, x-ray.
References and links
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1.
Introduction
When atoms and molecules are exposed to intense laser irradiation, high-order harmonics are
generated. The high-order harmonic generation (HHG) has been an attractive topic for the
passed two decades. The three-step model can help us understand its physical mechanism: (i)
an atom or molecule emits its electron to continuous state, (ii) the electron is accelerated in the
laser field, (iii) the accelerated electron recombines with the parent ion and high energy photon
is radiated [1]. HHG from atoms is first concerned for the potential applications to serve as a
source of coherent radiation in the EUV band and to produce attosecond pulses [2–6]. When
investigations of HHG are turned to molecules, new challenges and opportunities come forth.
Many novel applications are promised, such as molecular orbital tomographic imaging [7],
attosecond time-resolved electronic and nuclear dynamics probing in molecules [8, 9], polarization manipulation of high harmonics and attosecond pulses [10, 11].
A very important phenomenon for molecular HHG is the ”minimum” in the harmonic spectrum. Lein et al have predicted the minima of H2 and H+
2 theoretically by numerically solving
the time-dependent Schrödinger equation (TDSE) [12, 13]. They demonstrate that this kind
of intensity independent minimum, which is named structural minimum, is a consequence of
the interference between the emissions from the two centers, when the de Broglie wavelength
of the returning electron, the internuclear distance R and alignment angle θ satisfy the relationship Rcos θ =(2m+1)λ /2. This model indicates that the destructive interference requires a
phase difference of π which originates from the recombination processes of the returning wave
packet to separate centers. The two-center interference discussed here can be interpreted as reproduction of Young’s two-slit experiment in the microscopic world. A further insight into this
phenomenon has then been gained theoretically by Kamta et al [14].
The destructive interference has been experimentally observed by Kanai et al [15], where
the HHG from CO2 has been investigated . The highest occupied molecular orbital (HOMO)
of CO2 is mainly contributed by the two oxygen atoms, so CO2 can be treated as a stretched
O2 molecule with two emitting center. It was found that their experimental data agreed well
with the theoretical prediction given by Lein’s model. Almost at the same time, Vozzi et al also
observed this phenomenon in their experiment [16], and the result was also consistent with
the two-center model supposing the recombining electron kinetic energy Ek =h̄ω -I p . Moreover,
Smirnova et al have found intensity dependent minima in their experimental work [9]. They
explain that this kind of minimum, which is named dynamical minimum, results from the interference of multiple channels.
A structural minimum contains very important information, from which structure of objective molecule can be probed and symmetry of the bond can be judged [13, 17]. The two-center
interference also plays an important role in the study of the attosecond nuclear motion [18]
and the imaging of molecular orbital. But till now, researches on the two-center interference
mainly focus on homonuclear diatomic molecule (HoDM), such as H2 , N2 , O2 , or some other
simple molecules with symmetric HOMOs like CO2 [19–23]. Minima from those symmetric molecules are just dependent on internuclear distances and alignment angles, while little
information on the combining atoms is included. For example, minima were observed at the
same position around 55eV in both harmonic spectra from N2 O and CO2 at parallel alignment,
which is determined by the same internuclear distance of 2.3Å. Note that although N2 O is an
asymmetric molecule, it has a very similar HOMO to that of CO2 with a πg symmetry [22, 23].
In this paper, two-center interference for heteronuclear diatomic molecule (HeDM), which
has seldom been studied to the best of our knowledge, is investigated. We focus on the structural
minimum caused by this two-center interference and find that the minimum shows new characteristics. The phase difference between the two emissions in an HeDM consists of two parts: the
spatial phase difference and the additional recombining phase difference. The latter part arises
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from the unequal recombining processes, which will induce a shift of the interference minimum
in the spectrum. This implies that information not only on the internuclear separation but also
on the properties of the two separate centers is packed in the harmonic spectrum. Our study
indicates that the two-center interference offers opportunities to estimate the asymmetry of the
objective HeDM and to judge the linear combination of diatomic orbital (LCAO) for molecular
orbitals. Moreover, the possibility to monitor the evolution of the HOMO itself in molecular
dynamics is promised.
2.
Theoretical model
We apply CO as the target molecule, because effective destructive interference could be observed due to the comparable contributions to HOMO from atoms C and O. As shown in Fig.
1, (X,Y,Z) is the molecule frame, and (x,y,z) is the laboratory frame. The orientation angle θ
is defined as the angle between the directions of the intrinsic dipole moment of CO (Z axis in
the molecule frame) and the electric field (at the ionization time). The length |RC | and |RO | are
0.65 Å and 0.48 Å respectively, so the internuclear distance R=|RO -RC |=1.13 Å. Assume that
the incident pulse is linearly polarized along x axis, and the accelerated electron returns in the
direction of polarization with particular momentum p (i.e. particular de Broglie wavelength λ ).
Fig. 1. The schematic for the process of recombination. (X,Y,Z) is the molecule frame,
and (x,y,z) is the laboratory frame. θ is defined as the angle between the directions of the
intrinsic dipole moment of CO and the electric field. |RC | and |RO | are 0.65 Å and 0.48 Å
respectively.
The HOMO of CO is σ -type, schematic of which can be seen in Fig. 1. According to the
LCAO approximation,
ΨCO (r) = ∑ cμ ψμ (r − Rμ ) = ΨC (r − RC ) + ΨO (r − RO ),
μ
(1)
where ΨCO is the HOMO of model CO constructed by the linear combination of atomic orbitals ψμ . These atomic orbitals are approximated by Gaussian basis set. As the recombination
at respective center is concerned, the linear summation is divided into 2 groups according to
different contributing atoms [19, 20]:
ΨC (r − RC ) = ∑ cu ψu (r − RC ),
(2)
u
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ΨO (r − RO ) = ∑ cv ψv (r − RO ),
(3)
v
where ΨC and ΨO represent the sum of orbitals contributed from atoms C and O respectively.
The coefficients cμ (cu , cv ) for the weights of contributions are obtained with the Gaussian
03 ab initio code [24]. Here ΨC and ΨO are two portions of the HOMO of CO rather than
independent atomic orbitals of C and O.
Sectional view of the constructed HOMO is shown in Fig. 2(b), the portions ΨC and ΨO are
also shown in Fig. 2(a) and Fig. 2(c). We can see that ΨC is mainly an combination of 2s and
2pz orbitals, while ΨO is purely 2pz approximately.
Fig. 2. Sectional view of (a)ΨC , (b)ΨCO , (c)ΨO . The black dots represent the nuclear positions.
We employ the Lewenstein model [1] to calculate the harmonic radiation. The timedependent dipole moment is given by
3/2
π
dt
x(t) = i
ε + i(t − t )/2
−∞
×drec pst (t ,t) − A(t) dion pst (t ,t) − A(t )
×exp −iSst (t ,t) E(t ) + c.c..
t
(4)
In this equation, E(t) is the electric field of the laser pulse, A(t) is the corresponding vector
potential, ε is a positive regularization constant. pst and Sst are the stationary momentum and
quasiclassical action, which are given by
pst (t ,t) =
1
t − t
t
t
A(t )dt ,
(5)
1
1 t 2 Sst (t ,t) = (t − t )I p − p2st (t ,t)(t − t ) +
A (t )dt ,
(6)
2
2 t
where Ip =14.014 eV is the ionization energy.
The harmonic spectrum is then obtained by Fourier transforming the time-dependent dipole
acceleration a(t) [25]:
Sq ∼|
a(t)exp(−iqωLt) |2 ,
(7)
where a(t) = ẍ(t), ωL is the frequency of the driving pulse and q corresponds to the harmonic
order.
The transition dipole moment between the ground and the continuum state is calculated by
dCO (p) = ΨCO (r)|r|e−ip·r .
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(8)
Received 8 Nov 2010; revised 17 Dec 2010; accepted 18 Dec 2010; published 3 Jan 2011
17 January 2011 / Vol. 19, No. 2 / OPTICS EXPRESS 440
The linear combination for HOMO of CO has been divided into two groups, so the transition
dipole moment can be written as
dCO (p) = ΨC (r − RC )|r|e−ip·r + ΨO (r − RO )|r|e−ip·r =
ΨC (r1 )|r1 + RC |e
−ip·
r1
e
−ip·RC
(9)
+ ΨO (r2 )|r2 + RO |e
−ip·
r2
e
−ip·RO
.
(10)
In the above equation, the terms growing linearly with length RC and RO are artifacts coming
from the lack of orthogonality between the Volkov states and the field-free bound states, and
actually they have little influence on the result [19, 20]. We neglect these two terms and get
dCO (p)
= dC (p)e−ip·RC + dO (p)e−ip·RO
=
iφC −ip·RC
|dC (p)|e
(11)
iφO −ip·RO
êc + |dO (p)|e
êo ,
(12)
where êc and êo are unit vectors of dC and dO respectively. The dipole moments are obtained
by dC = ΨC (r)|r|e−ip·r and dO = ΨO (r)|r|e−ip·r respectively, where the integrations are
performed numerically. The two terms in Eq. (12) describe the recombination process of each
center respectively. -p · RC and -p · RO represent phases gained by the returning electrons before
recombination. The phases φC and φO , which arise from the recombination, are obtained by the
phase angles of the transition dipole moments dC and dO respectively. The total phase difference
between the emissions from the two centers is defined as
Δφ
= [φC + (−p · RC )] − [φO + (−p · RO )]
= (φC − φO ) + pR cos θ ,
(13)
(14)
where two terms are included. The former term, called the recombining phase difference, represents the phase difference from the unequal recombination processes; while the latter one,
called the spatial phase difference, represents the phase difference that originates when the
wave packet of returning electron travels through the projection of the internuclear distance on
the laser polarization axis.
For a bonding HoDM, Eq. (12) will reduce to [18–20, 26]
dHoDM (p) = 2datom (p) cos(p · R/2),
(15)
and only the spatial term in Eq. (14) remains. An antibonding HoDM can be seen as a special
kind of HeDM, for which an intrinsic phase difference of π exists. Unless stated otherwise,
HoDM refers to the bonding kind in this paper.
For the recombination,
a modification should be applied, that is, drec (p)=dCO (pk ) and effec2
tive momentum pk = p + 2Ipp/|p| [19, 26]. This modification implies that the recombination
occurs within the potential well and the acceleration effect of the binding potential is considered in the modified Lewenstein model. As discussed in previous works, the returning electron
is accelerated and the corresponding wavelength decreases when the electron enters the potential well before recombination near the nuclei [13, 14]. The interference is determined by the
wavelength where the electron is faster in the core region instead of the wavelength corresponding to the asymptotic energy far away [17]. The results with and without this modification have
been compared: if the modification is not applied, the minimum position estimated is far away
from the exact numerical result [27].
To confirm this modified model, we compare our result obtained by this model with Lein’s
obtained by solving TDSE numerically. As shown in Fig. 3, the two results agree well considering the positions of minima [12, 13]. Consistent results with Kamta’s are also obtained in
various angles [14], which confirms that this modified model is appropriate.
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0
0
10
o
(a) 0
Harmonic intensity (arb. units)
Hamonic intensity (arb. units)
10
−5
10
−10
10
0
25
50
75
Harmonic order
o
(b) 30
−5
10
−10
10
100
0
25
50
75
Harmonic order
100
Fig. 3. Harmonic spectra of H2 obtained using the Lewenstein model with the modified effective momentum pk . 780nm laser is used with intensity I=5×1014 W/cm2 and a duration
of 26 fs. Internuclear distance of H2 is 1.4 a.u. and the alignment angle is (a) 0o , and (b)
30o respectively.
3.
Result and discussion
Parallel alignment with the orientation angle θ =0◦ is investigated first. According to the twocenter model,
(16)
R| cos θ | = (2m + 1)λ /2, m = 0, 1, . . .
and
λ = h/p,
(17)
destructive interference takes place when the electron recombines with energy 29.54 eV or
265.87 eV for an HoDM with internuclear distance R=1.13 Å. To extend the cutoff to higher
harmonic orders with relative low laser intensity, mid-infrared driving pulse at 1300 nm is
used [22], which implies the first and second interference minimum should locate around the
31st and 279th harmonic orders. Figure 4 shows the harmonic spectrum of CO at θ =0◦ . Calculation is performed for laser intensity I=3×1014 W/cm2 and a duration of 30 fs full width
at half maximum, which can be achieved by the optical parametric amplifier (OPA) technique
nowadays [28]. The sine squared envelope is employed as f(t)=sin2 ( Tπpt ), where T p determines
the pulse duration. A minimum is observed around the 131st harmonic, which is far away from
the 31st or 279th harmonic in the HoDM case.
To explain this phenomenon and understand the underlying mechanism, phase analysis is
performed. The recombining phases for the two centers versus returning electron energy are
shown in Fig. 5(a), as indicated by the yellow and blue curves respectively. Then the recombining phase difference is obtained, which is indicated by the green curve in Fig. 5(a). In Fig.
5(b), both the recombining phase difference and the spatial phase difference (blue curve) are
presented, the sum of which turns to be the total phase difference (red curve) according to Eq.
(14). All the phases are shifted into the interval from -π to π .
The two-center model indicates that the interference takes place when the total phase difference equals π (or -π ). As depicted by Eq. (14), in an HoDM with the same internuclear
distance, the total phase difference is dominated by the spatial phase difference, so the minima
will locate where the blue curve in Fig. 5(b) equals π . The positions are right 29.54 eV and
265.87 eV, just the same as the prediction from two-center model. While for the CO molecule,
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Harmonic intensity (arb. units)
−5
10
−10
10
0
50
100
150
200
250
Harmonic order
Fig. 4. Harmonic spectrum of CO at θ =0◦ , with laser intensity I=3×1014 W/cm2 , wavelength λ =1300 nm and a duration of 30 fs.
the recombining phase difference is added (green curve). This induces a shift of the position
where the total phase difference reaches π to 14.99 eV and 121.6 eV (red curve), which imply
the 15th and the 127th harmonics for 1300 nm. The latter well explains the observed minimum
in Fig. (4). But there is no minimum around the 15th harmonics. The discrepancy arises from
the inaccuracy at low orders caused by the strong-field approximation (SFA) in Lewenstein
model, and from the fact that the amplitude of dC and dO are not comparable enough to form
efficient destructive interference there.
φ
C
φ
O
φ −φ
C
O
φ −φ
(b)
C
1
1
0.5
0.5
phase (π)
phase (π)
(a)
0
−0.5
−1
0
O
pRcosθ
Δφ
0
−0.5
100
200
300
Electron kin. energy (eV)
−1
0
100
200
300
Elecron kin. energy (eV)
Fig. 5. Phase analysis for the recombination of CO.
The result illustrates that minima in HeDMs are not dependent on the internuclear separation
alone. Shift of the position of minimum, which originates from the difference between the two
recombination processes, is an important measurement for the asymmetry of the molecule. Note
that the asymmetry is probed by the different phase properties of each recombination process,
and the asymmetry of the phase property is probed actually. Further, this phase asymmetry is
a consequence of the asymmetric LCAO for HOMO, since the phase of each recombination
process is affected by the combination of atomic orbitals for each center. So, the asymmetry of
the HOMO of objective molecule is probed finally. An antibonding HoDM is the opposite case
of bonding HoDM, where a constant phase shift of π (or -π ) exists in full energy region. The
amplitude difference will not shift the minimum, but the degree of destructive interference will
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17 January 2011 / Vol. 19, No. 2 / OPTICS EXPRESS 443
be reduced.
According to the molecular orbital theory, only atomic orbitals with close energy levels can
efficiently combine to molecular orbitals. For an HoDM, pairs of identical atomic orbitals
overlap to form molecular orbitals with equal contributions. But the case is more complex
in HeDMs. It is not that easy to say how an HOMO is formed in an HeDM. The HOMO is
combined by some atomic orbitals with various coefficients, or some HOMOs are non-bonding
orbitals for lone pairs only from one atom. Our investigation has offered an opportunity to
identify the molecular orbitals, with the help of the relationship between the minimum and the
orbital combination. Since the position of minimum is dependent on the orbital combination,
the LCAO can be known by comparing the observation of the position of minimum with the
theoretical predictions [29].
For example, we assume a linear combination for CO:
√
√
√
3
3
3
ψC,2s (r − RC ) +
ψC,2pz (r − RC ) −
ψO,2pz (r − RO ).
(18)
ΨCO (r) =
3
3
3
We then calculate the harmonic spectrum of this artificial CO, which predicts a minimum
around the 119th order. As shown in Fig. 6, harmonic spectra from the real and artificial HOMO
are both presented. To obtain a clearer observation, thick curves are plotted to represent the
smoothed harmonic spectra where fine structure has been eliminated by convolution with a
Gaussian of appropriate width [13]:
Ssmooth (ω ) =
S(ω̃ )exp(−(ω̃ − ω )2 /σ 2 )d ω̃ ,
(19)
where σ = 5ωL . The minima of the two curves locate at different frequencies in the spectra.
By observing the minimum position, we will be able to judge which combination the HOMO
is really combined of. Moreover, once the relationship between the harmonic spectra and the
LCAO is already known, it will be possible to monitor molecular dynamics. How the combination of the orbitals changes or how the HOMO evolves can be deduced by reading the harmonic
spectra.
−4
Harmonic intensity (arb. units)
10
−5
10
−6
10
−7
10
−8
10
artificial
real
artificial, smoothed
real, smoothed
−9
10
50
100
150
200
Harmonic order
Fig. 6. Comparison of HHG from two probable HOMO of CO, a real one (blue) and an
artificial one with orbital combination described in the text (red). The thick curves represent
the smoothed harmonic spectra.
The electric field is alternating in an laser pulse, and the asymmetric molecule will experience opposite electric field in adjacent half optical cycle. This means that harmonics at θ and
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π − θ orientations will be generated in one shot. The HHG and phase analysis for CO oriented at θ =180◦ have been investigated, shown in Fig. 7. The applied laser parameters and the
representations of each curve are the same as in Fig. 4 and Fig 5.
0
(b) 1
phase (π)
(a)
−5
10
0
−1
0
−10
(c) 1
−15
phase (π)
Harmonic intensity (arb. units)
10
10
10
0
100
200
Harmonic order
100
200
300
Electron kin. energy (eV)
0
−1
0
100
200
300
Electron kin. energy (eV)
Fig. 7. The HHG and phase analysis for CO oriented at θ =180◦ . The applied laser parameters and the representations of each curve are the same as in Fig. 4 and Fig 5.
It is shown in Fig. 7 that the reversal of electric field does not change the shape of the
harmonic spectra, and all the phases are just reversed. According to the definition of total phase
difference in Eq. (14), in the reversed electric field
Δφrev = (−φC + φO ) + pR cos(π − θ ) = −Δφ .
(20)
The total phase difference has just changed its sign, but the position where it reaches π or π does not change. So the minimum position will not change and the shape of the spectrum
remains the same.
Note that we are emphasizing the ”shape” of the spectrum. The two-center interference is insensitive to the reversal of electric field. But the ionization rate might be increased or decreased
in opposite electric field. The symmetry between half-cycles in HHG is broken due to the preferred direction of ionization or Column focusing effect [30]. This system-induced gating can
be utilized to manipulate the HHG to obtain a wide supercontinuum [31]. Recently, a research
has been done for the HHG of CO as a polar molecule [32]. The author indicates that the effect
of system-induced gating of CO will be weakened by the Stark shift. As we are focusing on the
two-center interference effect in this paper and the ionization-induced asymmetry is limited by
the stark shift, the assumption that HHG of CO is insensitive to the reversal of electric field is
employed in this paper.
Next, we extend our investigation to all the orientation angles. Phase analysis for spatial
phase difference alone is first performed, shown in Fig. 8. The boundaries of the red and blue
colors represent where the phase differences reach π (or −π ). For an HoDM which has the same
internuclear distance as CO, the minima will locate in harmonic spectra along the boundary
lines in Fig. 8. The two pairs of boundary lines also match with the prediction by Eq. (16).
Phase analysis for total phase difference is shown in Fig. 9. The boundaries of the red and
blue colors represent where the phase differences reach π (or −π ) too. Compare Fig. 9 with
Fig. 8, the boundary lines, which predict the interference minima, shift to lower energy. This
phenomenon is due to the added recombining phase difference as discussed above. The conclusion that all the phases just change the signs in opposite electric field can also be extended to
all orientation angles.
Moreover, new characteristic is observed. For an HoDM, there is no minimum at angles close
to 90◦ , i.e., perpendicular alignment (Fig. 8). The position of the minimum tends to infinity
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Fig. 8. Phase analysis for spatial phase difference. The boundaries of the red and blue
colors represent where the phase differences reach π (or −π ). All the phases are shifted in
the interval from -1 to 1 (in unit of π ).
Fig. 9. Phase analysis for total phase difference. The boundaries of the red and blue colors
represent where the phase differences reach π (or −π ). All the phases are shifted in the
interval from -1 to 1 (in unit of π ).
according to the calculation of Eq. (16). But for CO, the boundary lines bend to 90◦ . It is
because although the spatial phase difference decreases rapidly as the angle tends to 90◦ , the
recombining phase difference remains. At these angles, minima are mainly affected by the
recombining phase difference, which reflect the properties of the atoms.
Harmonic spectra at all orientation angles are obtained in Fig. 10. To observe the entire
variation of minima, a stronger intensity I=6×1014 W/cm2 is applied to locate most of the
minima in the plateau region. The other parameters are kept the same as in Fig. 4. Two pairs
of ”minimum grooves” are observed, which is consistent to the prediction of phase analysis
in Fig. 9. The symmetry with respect to 90◦ is observed, which has been discussed in the
parallel condition. The reason for the blurring and disappearing of the first minimum grooves
at small angles and angles tending to 180◦ has been discussed above too. The HHG efficiency
decreases as θ tends to 90◦ due to the nodal structure of the HOMO [33]. As an extension
of the investigation for parallel alignment, it is possible to judge the LCAO information about
the HOMO and monitor molecular dynamics by reading the shift and the bend of the minima
curves rather than by measuring the shift of one minimum only. For a highly asymmetric HeDM
like HF, whose non-bonding HOMO is dominantly contributed by the fluorine atom, angle
independent result is obtained.
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Fig. 10. Harmonic spectra at all orientation angles in arbitrary unit. I=6×1014 W/cm2 , the
other parameters are the same as in Fig. 4.
4.
Conclusion
Two-center interference for a typical HeDM CO is investigated. Harmonic spectra are obtained
using modified Lewenstein model. The positions of minima in the spectra are found to be
shifted to lower harmonic orders compared with that from an HoDM with the same internuclear
distance. Phase analysis is performed to explain this phenomenon. It shows that the shift results
from the additional recombining phase difference, which pushes the position where the value
of total phase difference equals π to lower harmonic orders. It is found that the result of such
interference is not influenced by the reversal of the electric field. The investigation is also
extended to all orientation angles. In addition to the shift of minima, new phenomenon that the
angle-dependent minimum curves bend to 90◦ is observed. No minimum will be found at angles
close to perpendicular alignment in the harmonic spectra from an HoDM, but in the HeDM
case minimum appears at these angles where the recombining phase difference dominates the
interference. Our investigation indicates that information on not only the molecular structure
but also the atomic orbital combination for HOMO at the two centers is packed in the harmonic
spectra of an HeDM, which can be judged by probing the HHG signals. Moreover, our study
also implies the possibility to monitor the evolution of the HOMOs in molecular dynamics.
Finally, besides diatomic molecules, our analysis method for the two-center interference is
applicable to any asymmetric molecule whose HOMO is mainly contributed by two centers.
Acknowledgment
This work was supported by the National Natural Science Foundation of China under Grants
No. 60925021, 10904045 and the National Basic Research Program of China under Grant No.
2011CB808101.
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Received 8 Nov 2010; revised 17 Dec 2010; accepted 18 Dec 2010; published 3 Jan 2011
17 January 2011 / Vol. 19, No. 2 / OPTICS EXPRESS 447