Molecular orbital imaging via above-threshold ionization with circularly polarized pulses

Xiaosong Zhu; Qingbin Zhang; Weiyi Hong; Peixiang Lu; Zhizhan Xu

doi:10.1364/OE.19.013722

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 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 \int_{- \infty}^{\infty} a (t') F (t') \cdot d [p + A (t')] exp {- i \int_{t'}^{\infty} [{(p + A (t ″))}^{2} / 2 + I_{p}] d t ″} d t',

where I_p 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 [- \int_{- \infty}^{t} w (t') d t'],

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 (\frac{2^{7 / 2} α^{5 / 4}}{π}) \frac{p}{{(p^{2} + α)}^{3}},

where α =2I_p. For molecules, the dipole moment is calculated by [28, 29]

d (p) = i \nabla_{p} \tilde{Ψ} (p) .

\tilde{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

d (p) = \frac{1}{{(2 π)}^{3 / 2}} \int d^{3} r r exp [- i p \cdot r] Ψ (r),

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 $γ = \sqrt{I_{p} / 2 U_{p}} = 0.96$ , where $U_{p} = F_{0}^{2} / 4 ω^{2}$ with F ₀ 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 +p_x 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 +p_x axis from the spectrum.

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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 σ_g symmetry. The molecules are aligned along the x axis in the laboratory frame prior to ionization. Sectional view of HOMO 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 I_p=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 p_x direction. (d) Integration over momentum along the p_y direction.

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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 p_x axis in the momentum spectrum, while momentum distribution along the p_y 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 p_x and p_y directions respectively. It is obvious to see the dip at around p_y=0 in Fig. 2(c) and the peak at around p_x=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 π_g 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 p_x and p_y 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.

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 p_x direction. (d) Integration over momentum along the p_y direction.

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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 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 t ₀, 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 | \int_{t_{0}}^{\infty} F (t) d t = - | e | A (t_{0}) .

As shown in Fig. 4(a), when the transient electric field points to +x direction parallel to the molecule axis at t ₁, 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(t ₁) 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 p_y axis in the photoelectron momentum spectrum. A quarter of optical circle later, at t ₂, the electric field rotates a 90 degree to the +y direction. The decreased ionization probability results in the sink along the p_x axis in the momentum spectrum, which is explained by the dashed blue arrows.

Fig. 4 (Color online) Illustration for the photoelectron momentum spectra for (a) N₂ and (b) CO₂ respectively.

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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. Considering its π_g 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.

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 π_u 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 σ_u and σ_g orbitals), although a strong suppression will occur in the perpendicular direction for the anti-symmetric σ_u orbitals which promises a lower ionization probability than that in a σ_g 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 μ_h and the external field F(t): I_p(t) = I_p ₀ + μ_h · F(t), where I_p ₀ 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 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 π_g 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]

μ_{h} = - \int d rr ρ^{H} (r),

where ρ ^H(r) is the electron density of the HOMO calculated by ρ^H(r) = d rΨ^*(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 I_p ₀=9.26 eV at the instant when the electric field is parallel to the dipole.

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 p_y (dashed green lines), p_x (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.

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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 p_y, p_x for Figs. 5(b) and 5(e) respectively.

Compared with Fig. 3, one can see basic properties associated with π_g 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 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 F_x > 0 (see the solid blue arrows). The corresponding vector potential –A_y points to the -y direction at these ionization moments and the emitted electrons will gain final momentums with their p_y components to the negative direction. This explains why probability with minus p_y 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 p_y components to the positive direction. Such difference results from the reason that, for an RCP pulse, –A_y points to the +y direction when F_x > 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 μ_h=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 I_p ₀. 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 Grant No. 2011CB808103. This work was partially supported by the State Key Laboratory of Precision Spectroscopy of East China Normal University.

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Molecular orbital imaging via above-threshold ionization with circularly polarized pulses

Abstract

1. Introduction

2. Theoretical model

3. Result and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (7)

Optics Express