Molecular orbital imaging via
above-threshold ionization with
circularly polarized pulses
Xiaosong Zhu,1 Qingbin Zhang,1 Weiyi Hong,1,3 Peixiang Lu,1,2,∗ and
Zhizhan Xu1
1 Wuhan
National Laboratory for Optoelectronics, Huazhong University of Science and
Technology, Wuhan 430074, China
2 School of Science, Wuhan Institute of Technology, Wuhan 430073, China
3 [email protected]
*[email protected]
Abstract:
Above-threshold ionization (ATI) for aligned or orientated
linear molecules by circularly polarized laser pulsed is investigated. It is
found that the all-round structural information of the molecular orbital
is extracted with only one shot by the circularly polarized probe pulse
rather than with multi-shot detections in a linearly polarized case. The
obtained photoelectron momentum spectrum directly depicts the symmetry
and electron distribution of the occupied molecular orbital, which results
from the strong sensitivity of the ionization probability to these structural
features. Our investigation indicates that the circularly polarized probe
scheme would present a simple method to study the angle-dependent
ionization and image the occupied electronic orbital.
© 2011 Optical Society of America
OCIS codes: (020.4180) Multiphoton process; (260.3230) Ionization; (270.6620) Strong-field
process.
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1.
Introduction
Above-threshold ionization (ATI) [1], that atoms or molecules absorb more photons than necessary for ionization, has been an attractive topic for extensive experimental [2, 3] and theoretical [4–7] studies in the past three decades. The development of this area benefits a lot
from advances in laser technology which made possible generation of short and intense laser
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pulses [8]. Lately, strong-field ionization of molecules also attracts a lot of attentions [9–11],
which is more complex due to the additional features such as molecular orbital symmetry and
the anisotropic structure. On the other side, investigations on these additional complexities
and properties offer opportunities for molecular orbital imaging [12–15]. Kamta et al. [12] has
shown that one can image the contour of the electron density of the active molecular orbital with
the help of the strong sensitivity of the angle-dependent ionization probability to the symmetry
and the electron distribution.
To obtain more detailed information from the molecules and achieve the aim of imaging, one
need to fix the molecular axes prior to ionization with alignment [16] or orientation [17, 18]
techniques. After that, a probe pulse is used to initialize the strong-field process and detect the
emitted electrons, fragments or photons. Generally, linearly polarized laser pulse is applied as
the probe pulse. By varying the angle between polarization axis of probe pulse and the main axis
of the alignment or orientation distribution, the all-round structural property of the molecule is
obtained [10, 12, 14]. However, this linearly polarized probe scheme requires multi-shot detections in different laser polarizations, which brings complexity in experiments. Meanwhile, the
fluctuation of experimental parameters may introduce some additional errors in measurements
of the angle-dependent process. In some other works, alternative forms of pulses are employed,
such as orthogonal two-color laser pulses [19–21] and circularly polarized laser pulses [22,23].
In this paper, we have investigated the photoelectron momentum spectra of ATI for aligned
or oriented linear molecules probed with circularly polarized laser pulse. In this circularly polarized probe scheme, the all-round structural information of the objective molecular orbitals
can be simultaneously extracted with only one shot by the probe pulse. It is shown that the
symmetry and corresponding structural features of the occupied molecular orbital are directly
revealed from the obtained photoelectron momentum spectrum, which is associated with the
anisotropic ionization probability. For asymmetric molecules, the asymmetric electronic distributions are also mapped onto the spectra as long as the influence of stark shift is of marginal
importance. This circularly polarized probe scheme presents a simple detecting method to study
the angle-dependent ionization and image the occupied electronic orbital.
2.
Theoretical model
We carry out the calculation with the semiclassical model under the strong field approximation
(SFA) [24–27]. The photoelectron momentum spectrum is given by
b(p) = i
∞
−∞
a(t )F(t ) · d[p + A(t )]exp{−i
∞
t
[(p + A(t ))2 /2 + Ip ]dt }dt ,
(1)
where Ip is the ionization energy of the objective atom or molecule, p(t) is the momentum
of electron, F(t) is the electric field of the laser pulse and A(t) is the vector potential. The
ground-state amplitude is calculated by
a(t) = exp[−
t
−∞
w(t )dt ],
(2)
where the ionization rate w(t) is obtained with the molecular tunneling ionization (MO-ADK)
theory [9].
We integrate through the whole time when the laser pulse interacts with atoms or molecules,
so the final momentum distribution after the interaction is obtained. For noble gas atoms, the
dipole moment takes a simple analytic form
d(p) = i(
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p
27/2 α 5/4
) 2
,
π
(p + α )3
(3)
Received 21 Apr 2011; revised 26 May 2011; accepted 17 Jun 2011; published 1 Jul 2011
18 July 2011 / Vol. 19, No. 15 / OPTICS EXPRESS 13724
where α =2Ip . For molecules, the dipole moment is calculated by [28, 29]
d(p) = i∇p Ψ(p).
(4)
Ψ(p)
is the Fourier transform of the highest occupied molecular orbital (HOMO) Ψ(r) in coordinate space, which is obtained with the Gaussian 03 ab initio code [30]. Equation (4) is derived
from the formula
1
d 3 rrexp[−ip · r]Ψ(r),
(5)
d(p) =
(2π )3/2
where plane waves are used as the continuum state. The plane wave approximation is appropriate to some extent that only the overall angular distribution of the momentum spectrum is
concerned in this work. For investigations of fine structures, Coulomb continuum wave functions are needed [31].
3.
Result and discussion
In this paper, we consider 780 nm 10-cycle circularly polarized laser pulses, where a trapezoidal
profile with 4-cycle linear ramp is adopted. Through this paper, we consider left circularly
polarized (LCP) pulses if not specified.
Above all, we will study the ionization of the Ar atoms by such probe pulse. The ionization
energy of Ar is 15.8 eV, and the peak intensity of the laser pulse is 0.3 PW/cm2 . The Keldysh
parameter γ = I p /2Up = 0.96, where Up = F02 /4ω 2 with F0 and ω being the amplitude
and angular frequency of laser. The obtained photoelectron momentum spectrum is shown in
Fig. 1(a), and Fig. 1(b) presents the slice cut along the +px axis from the spectrum. In Fig. 1(a)
we can see that the spectrum shows ring structures corresponding to the separate ATI peaks. As
shown in Fig. 1(b), the spacing of the peaks turn out to be 0.059 a.u. in energy, which is right
the energy of the photon. The ring shaped stripes indicate the angle-independent ionization and
the isotropic structure of Ar is directly revealed by this one-shot detection.
Fig. 1. (a) Photoelectron momentum spectrum for Ar, I p =15.8 eV, I=0.3 PW/cm2 . (b) The
slice cut along the +px axis from the spectrum.
Now we proceed to molecules with anisotropic structures. In this work, all the intensities
applied for various molecules are below the saturation intensity.
We first discuss the molecule N2 , whose HOMO typically has a σg symmetry. The molecules
are aligned along the x axis in the laboratory frame prior to ionization. Sectional view of HOMO
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of N2 is shown in Fig. 2(a), where the black dots denote the position of the two N nuclei.
Figure 2(b) presents the obtained photoelectron momentum spectrum. A peak intensity of 0.3
PW/cm2 is used and the ionization energy Ip =15.6 eV.
Fig. 2. (a) Sectional view of the HOMO of N2 with black dots denoting the positions
of nuclei. (b) Photoelectron momentum spectrum for N2 , I p =15.6 eV, I=0.3 PW/cm2 . (c)
Integration over momentum along the px direction. (d) Integration over momentum along
the py direction.
As displayed in Fig. 2(b), the spectrum shows pairs of crescent stripes with spacing corresponding to the photon energy, which indicates strong anisotropic properties. There is a deep
sink along the px axis in the momentum spectrum, while momentum distribution along the py
axis is favored. In order to further elucidate this phenomena, in Fig. 2(c) and 2(d) we present
the 1D momentum distribution integrated over the momentum along px and py directions respectively. It is obvious to see the dip at around py =0 in Fig. 2(c) and the peak at around px =0
in Fig. 2(d). Moreover, the curves for both 1D momentum distributions are symmetric.
To compare with N2 , a typical molecule with a HOMO of πg symmetry, CO2 , is investigated.
The ionization energy is 13.8 eV, and the same intensity of 0.3 PW/cm2 is used. Correspondingly, sectional view of HOMO together with positions of nuclei is presented in Fig. 3(a). Figure 3(b) shows the obtained photoelectron momentum spectrum. Figures 3(c) and 3(d) are 1D
momentum distribution integrated over the momentum along px and py directions respectively.
Butterfly-like structures are observed in Fig. 3(b). Although the sinks in Fig. 3(b) are not as
obvious as in Fig. 2(b), deep dips are still easily found in Fig. 3(c) and 3(d). The collected photoelectrons are predominated by those with momentum oblique to the internuclear axis rather
than parallel or perpendicular to it. A more detailed observation is that the photoelectrons escape along the ±x directions are still less than those along the ±y directions.
The sectional view of the HOMO of N2 shows that, according to the σ symmetry, the electron distribution concentrates along the internuclear axis, while the perpendicular extension is
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Fig. 3. (a) Sectional view of the HOMO of CO2 with black dots denoting the positions of
nuclei. (b) Photoelectron momentum spectrum for CO2 , I p =13.8 eV, I=0.3 PW/cm2 . (c)
Integration over momentum along the px direction. (d) Integration over momentum along
the py direction.
limited. It is concluded that the ionization probability maximizes when the laser polarization
axis overlaps with a major axis of electron distribution in the orbital, provided that the orbital is
spatially symmetric with respect to this axis [12]. So electrons will preferentially ionize when
the electric field is parallel to the internuclear axis.
After ionized at time t0 , the emitted electrons are still driven by the remaining part of probe
pulse, and the final momentum is given according to the simple-man model [22, 23, 32]
p = −|e|
∞
t0
F(t)dt = −|e|A(t0 ).
(6)
As shown in Fig. 4(a), when the transient electric field points to +x direction parallel to the
molecule axis at t1 , the photoelectron yield is highest. For a LCP probe pulse, the corresponding
transient vector potential points to the perpendicular direction +y and the final momentum of
the escaped electron p = −|e|A(t1 ) points to the -y direction, which are indicated by solid green
arrows. The case is similar when the electric field points to the -x direction and these preferred
ionizations lead to maximum distribution along the py axis in the photoelectron momentum
spectrum. A quarter of optical circle later, at t2 , the electric field rotates a 90 degree to the
+y direction. The decreased ionization probability results in the sink along the px axis in the
momentum spectrum, which is explained by the dashed blue arrows.
In contrast, for the HOMO of CO2 , there are four lobes separated by two orthogonal nodal
planes. The nodal plane parallel to the internuclear axis is due to nodal planes of the two component p orbitals from each O nucleus, while the perpendicular one is formed because of the
anti-bonding combination of the two p orbitals separated by an internuclear distance of 2.3Å.
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Considering its πg symmetry that the orbital is spatially antisymmetric relative to the nodal
planes, ionizations both along and perpendicular to the internuclear axis are suppressed due to
the destructive molecular interference discussed in [12, 33], which is indicated by the dashed
blue arrows in Fig. 4(b). This kind of angle-dependent ionization at last results in the butterfly
shaped momentum spectrum.
Fig. 4. (Color online) Illustration for the photoelectron momentum spectra for (a) N2 and
(b) CO2 respectively.
The above results demonstrate that we can easily tell the symmetry of the HOMO of objective
linear molecules from the photoelectron momentum spectra, which is obtained with only one
shot by the circularly polarized pulse. Apparently, the symmetry has close relationship with
orbital structural features like the electron distribution, nodal planes and corresponding lobes
of different parity, which have notable influence on the angular dependence of ionization. Now
that we apply a circularly polarized pulse long enough to ensure that the objective molecule
is exposed to electric field in various directions equally, the anisotropic ionization will only
results from the angle-dependent properties of the orbital itself. At the end of the laser-molecule
interaction, the final momentum of the photoelectron is turned a 90 degree relative to that when
ionized, while the ionization probability distribution vs. angle is not blurred by the remaining
part of the probe pulse. Finally, the structural information is printed in the detected spectrum,
by the help of which the symmetry and corresponding structural features are revealed.
Based on the above discussion, the πu symmetry can be identified corresponding to a final
momentum distribution maximum along the internuclear axis and minimum perpendicular to
it. This method still can not in more detail distinguish the σ orbitals (the σu and σg orbitals),
although a strong suppression will occur in the perpendicular direction for the anti-symmetric
σu orbitals which promises a lower ionization probability than that in a σg case [12]. It should
also be noted that, all the molecules investigated in this paper are linear molecules with 2D
orbitals. For more complicated molecules, one-shot detection might not be enough, but the
probe pulses needed are still much less than that in a linearly polarized probe scheme.
N2 and CO2 are both typically symmetric molecules. In the following we will study on
asymmetric molecules. The Stark shift is an important phenomenon for asymmetric molecules,
which will lead a time-dependent ionization energy in respond to the electric field F and meanwhile an angle-dependent ionization energy in the circularly polarized probe scheme. The linear
Stark shift of the HOMO is determined by the permanent dipole of HOMO μh and the external field F(t): Ip (t) = I p0 + μh · F(t), where Ip0 is the field free ionization energy [11]. The
Stark shift introduces additional factors for the angle-dependent ionization, which might confuse reading the electron distribution from the photoelectron momentum spectra.
To investigate the response of asymmetric molecules to circularly polarized pulse with the
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Stark shift of marginal importance, we consider the molecule NO. The HOMO of NO is combined by two 2p orbitals from N and O respectively with a πg symmetry, which is very similar
to that of CO2 . But the two p orbitals contribute unequally, which leads week asymmetry in
electron distribution shown in Fig. 5(a). The permanent dipole of HOMO is evaluated to be
only 0.33 a.u. (0.84 D) by [11]
μh = −
drrρ H (r),
(7)
where ρ H (r) is the electron density of the HOMO calculated by ρ H (r) = drΨ∗ (r)Ψ(r). Such
low permanent dipole and applied laser intensity I=0.12 PW/cm2 will only lead an maximum
Stark shift of 4% of the field-free ionization energy Ip0 =9.26 eV at the instant when the electric
field is parallel to the dipole.
We applied both the left and right circularly polarized (LCP and RCP) probe pulses corresponding to the top and bottom rows of Fig. 5 respectively. Figures 5(a) and 5(d) show the
sectional view of HOMO, the positions of the nuclei and illustration of the laser-molecule interaction. The middle column gives the two photoelectron momentum spectra with respect to
LCP and RCP pulses. And the last column presents integrations over momentum along py , px
for Figs. 5(b) and 5(e) respectively.
Fig. 5. (Color online) (a) and (d) The sectional view of the HOMO of NO, the positions of
the nuclei and illustration of the laser-molecule interaction. (b) and (e) Photoelectron momentum spectra for NO, I p =9.26 eV, I=0.12 PW/cm2 . (c) and (f) Integrations over momentum along the py (dashed green lines), px (solid blue lines) directions. For all the panels,
the top row corresponds to the LCP case while the bottom row corresponds to the RCP
case.
Compared with Fig. 3, one can see basic properties associated with πg symmetry from the
obtained spectra of NO, which is explained above. Besides, obvious contrast between the upper and lower halves can be observed, see Fig. 5(b) and solid blue curve in Fig. 5(c), which
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implies different ionization probabilities for opposite electric fields. In the LCP case as indicated in Fig. 5(a), since the electron distribution concentrates more at the N atomic center than
O, electrons prefer to ionize from the N end when the Fx > 0 (see the solid blue arrows). The
corresponding vector potential −Ay points to the -y direction at these ionization moments and
the emitted electrons will gain final momentums with their py components to the negative direction. This explains why probability with minus py is higher than that with positive one in
the obtained spectrum in Fig. 5(b). Similarly, in an RCP case, obvious contrast is also found
in Fig. 5(e) and 5(f), except that the escaped electrons prefer momentums with py components
to the positive direction. Such difference results from the reason that, for an RCP pulse, −Ay
points to the +y direction when Fx > 0.
We have also compared these results for NO with those leaving Stark shift out of account,
and find that the effect of Stark shift just quite weakly offsets the unequal distributions in the
momentum spectra, which would hardly confuse reading the electron distribution of the orbital
via ionization.
Our investigation has implied that, besides the above mentioned structural information about
orbital symmetry, the asymmetric electric distribution of HOMO can also be detected with
this circularly polarized probing scheme. By comparing Fig. 5(a) and 5(b) (and Fig. 5(d) and
5(e)), it can be seen that the photoelectron momentum spectra give direct descriptions of the
orbital. The obtained spectrum by an LCP (RCP) probe pulse is just like mapping of orbital
distribution after a 90 degree rotation counter-clockwise (clockwise). This conclusion is also
true to the previously studied N2 and CO2 (see Fig. 2 and 3).
We further discuss the generality of this scheme to asymmetric molecules with big permanent dipoles, such as CO with μh =1.72 a.u. (4.37 D) [11] calculated by Eq. (5). As we have
discussed above, an additional strong Stark shift might make the detection for the orbital distribution ambigous. A direct method to reduce this influence is to apply a low laser intensity.
For example, the ionization energy of CO is 14.014 eV, so an I=0.1 PW/cm2 circularly polarized laser pulse will lead a maximum Stark shift of 12.6% of Ip0 . Stark shifts will be still lower
with respect to further reduced intensities, which implies this scheme has the potential to be
applied to molecules with big permanent dipoles.
4.
Conclusion
We have investigated the ATI of aligned or oriented linear molecules by circularly polarized
laser pulse. Compared with a linearly polarized probe scheme where multi-shot detections are
required, this circularly polarized probe scheme enables one to extract the all-round structural
information of objective orbital with only one shot by the probe pulse. We have shown that the
detected photoelectron momentum spectrum offers the possibility to directly infer the symmetry and corresponding structural features of the occupied molecular orbital, which is due to the
influence of these structural properties on the anisotropic ionization probability. Besides, asymmetric electron distribution in asymmetric molecule is also mapped onto the spectrum as long
as the influence of stark shift is of marginal importance. Actually, through investigations on N2 ,
CO2 , and NO, we can see that the spectra are just like mappings of the orbital electron distributions after a 90 degree rotation counter-clockwise (clockwise) with respect to LCP (RCP) probe
pulses. Our investigation indicates that the circularly polarized probe scheme would present a
simple detecting method to study the angle-dependent strong-field ionization and image the
occupied electronic orbital.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grants
No. 60925021, 10904045, 10734080 and the National Basic Research Program of China under
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(C) 2011 OSA
Received 21 Apr 2011; revised 26 May 2011; accepted 17 Jun 2011; published 1 Jul 2011
18 July 2011 / Vol. 19, No. 15 / OPTICS EXPRESS 13730
Grant No. 2011CB808103. This work was partially supported by the State Key Laboratory of
Precision Spectroscopy of East China Normal University.
#146256 - $15.00 USD
(C) 2011 OSA
Received 21 Apr 2011; revised 26 May 2011; accepted 17 Jun 2011; published 1 Jul 2011
18 July 2011 / Vol. 19, No. 15 / OPTICS EXPRESS 13731