Ultrafast molecular orbital imaging based on attosecond photoelectron diffraction

Yang Li; Meiyan Qin; Xiaosong Zhu; Qingbin Zhang; Pengfei Lan; Peixiang Lu

doi:10.1364/OE.23.010687

1. Introduction

Advances in the understanding of the interaction of intense laser pulses with matter have opened versatile perspectives in ultrafast optics, for observing new phenomena and developing new technologies [1–4]. Among this, the generation of attosecond extreme ultraviolet (XUV) pulses in atomic and molecular gases based on high harmonic generation (HHG) has attracted considerable attentions for its potential use of probing the structure and ultrafast electron dynamics in matters with unprecedented temporal and spatial resolutions [5–8].

The application of the XUV pulses can be summarized into two categories. In the first category, during the HHG process, the recolliding electron wavepacket encodes information on the generating molecule in the emitted harmonics. Thus the harmonic signal is used to retrieve temporal and structural insight into the generating targets itself, known as ”self-probing” schemes [9–11]. Specifically, it can be used to create a tomographic reconstruction of molecular orbitals, called molecular orbital tomography [12–15]. On the other hand, the availability of isolated attosecond XUV pulses extends the femtosecond spectroscopy and femto-chemistry to the attosecond domain, which is the natural time scale of the electron motion [16–18]. Also, it was recently shown that the attosocond XUV pulses could also provide Ångstörm resolved images of electron wavepackets in molecules [11]. The combination of attosocond and Ångstörm resolution makes it possible to imaging the most fundamental process of physics and chemistry in real time [3, 19].

In the ultrafast imaging experiments, one usually exploits available information of molecular structure in the molecular photonionization spectra. The original idea was proposed by Cohen and Fano [20] more than forty years ago that the photonionization of a diatomic molecule can be viewed as a microscopic version of Young’s double-slit experiment. The emitted electrons, which are initially shared between the two indistinguishable nuclei, would be coherent when their de Broglie wavelengths are comparable to the molecular equilibrium internuclear distance. Such that the electron wavepacket interference effect would appear in the oscillatory behavior of photonionization cross sections in perturbative single-photon ionization. When extending to nonperturbative regime, a new imaging technique called laser induced electron diffraction (LIED) was proposed by Zuo et al [21]. When molecules are exposed to few cycle femtosecond laser pulse, laser induced electrons may rescatter with different nuclei. The interference between the amplitudes arising from different scattering centers would appear in the photonionization momentum distributions (PMD), carrying structure information of the molecule. However, in a femtosecond laser pulse, a train of several electron wavepackets recollides within one pulse and one recollision is convoluted by several recollisions and diffraction events. The overlap of multiple diffraction events, each yielding different energy spectra due to changing laser intensity, leads to extra complications in the diffraction patterns. Besides, the presence of the laser field would distort the electron energy spectra and add an extra dependence on recollision time of the spectra. All these problems make it extremely complicated to retrieve the structure information of the target molecule [22–27].

The fast development of ultrashort free-electron lasers (FELs) and attosecond XUV pulse by HHG make it available of high-energy XUV light sources [28–30]. A fascinating ultrafast imaging technique called molecular attosecond photoelectron diffraction (MAPD) [31–33] emerges, making use of the interaction between and XUV pulses and molecules. The sketch of MAPD is shown in Fig. 1. A molecule is aligned perpendicular to polarization direction of XUV pulse. The XUV photons eject the single electron from molecule. Photonionization takes place when the electron is close to one of the two nuclei, which therefore act as two ”slits”. The interference pattern appears in PMD of the emitted electrons. When the PMD is measured, the internuclear distance can be determined. Further, if the molecular orbital is expressed as linear combination of atomic orbitals (LCAO), the symmetry of the combined atomic orbitals is encoded in the detailed structure of the interference image, which allow us to retrieve the molecular orbital using PMD.

Fig. 1 The sketch of molecular attosecond photoelectron diffraction. The ultrashort XUV laser pulse is polarized perpendicular to the molecular axis and the blue dashed line indicates its propagation direction. The XUV photons eject the electron from the molecule when the electron is close to one of the nuclei. Therefore the two nuclei act as ”slits” in the microscopic double-slit experiment. By measuring the momentum distribution of the ejected electrons, the molecular internuclear distance can be determined and the symmetry of molecular orbital can be retrieved.

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In this paper, We present a systematic study of ultrafast imaging of molecular orbitals with different symmetries using MAPD technique. Three typical molecules (and molecular ions), i.e., $H_{2}^{+}$ , CO₂ and N₂ are chosen. The PMD of each molecule is obtained by solving the two-dimensional time-dependent Schrödinger equation (TDSE) within Born-Oppenheimer (BO) approximation. The effective single-active-electron (SAE) potential is adopted to correctly reproduce the highest occupied molecular orbitals (HOMO) of each molecule. Clear diffraction patterns appear in PMD. The internuclear distance can be determined with respect to the position of interference fringes. By stretching the molecules, the diffraction patterns change accordingly which in turn determine the change of internuclear distance with high precision. Furthermore, the detailed structure of the interference image, i.e., the relative heights of the interference fringes reflects the structure information of molecular orbitals. Using a simple two-center interference model, we successfully reproduced the difference pattern and it shows good agreement with the TDSE calculations. These results enable us to retrieve the molecular orbital for PAD with an inversion algorithm invoking the symmetry of the initial molecular orbital.

2. Theoretical model

To simulate the photoionization process in attosecond XUV pulse, we use the split-operator spectral method [34] to numerically solve the corresponding two-dimensional TDSE (atomic units are used unless stated otherwise)

i \frac{\partial}{\partial t} ψ (\vec{r}, t) = ℋ (\vec{r}, t) ψ (\vec{r}, t),

where r⃗ ≡ (x, y) denotes the electron position in (x, y) plane. Below we assume that the molecular orbital is aligned in the x direction. The Hamiltonian reads

ℋ (\vec{r}, t) = - \frac{1}{2} \nabla^{2} + V (\vec{r}) + \vec{r} \cdot \vec{E} (t) .

E⃗(t) is the electronic field which is polarized along the y direction, perpendicular to the molecular axis. We use a ten-cycle linearly polarized XUV pulse with wavelength of 5 nm. The electron field with a sine-squared envelope is expressed as

\vec{E} (t) = E_{0} {sin}^{2} (\frac{π t}{T}) cos (ω t) {\vec{e}}_{y},

in which E₀, ω and T is the amplitude, angular frequency and the duration of the XUV pulse, respectively. The laser intensity is chosen to be 1.0 × 10¹⁵W/cm². The high intensity and short wavelength of the XUV pulse enable us to generate photoelectrons with energy as high as 200eV in the nonperturbative single-photon ionization process. The corresponding de Broglie wavelength of the high-energy electron is about 1.8 a.u., which is less than the molecular equilibrium internuclear distance. Thus the diffraction of high-energy electrons can achieve accurate measurement of the molecular internuclear distance. The effective SAE soft-core potential [35–37] is in the form of

V (\vec{r}) = - \sum_{j = 1}^{N} \frac{Z_{j}^{\infty} + (Z_{j}^{0} - Z_{j}^{\infty}) exp (- {| {\vec{r}}_{j} |}^{2} / σ_{j}^{2})}{\sqrt{{| {\vec{r}}_{j} |}^{2} + ζ_{j}^{2}}} .

The subscript j = 1,...,N labels the nuclei at fixed positions ρ⃗_j and r⃗_j = r⃗ − ρ⃗_j.

Z_{j}^{\infty} + (Z_{j}^{0} - Z_{j}^{\infty}) exp (- {| {\vec{r}}_{j} |}^{2} / σ_{j}^{2})

denotes the position-dependent screened effective charge for the j-th nucleus, where

Z_{j}^{\infty}

is the effective nuclear charge of the nucleus j as seen by an electron at infinite distance and

Z_{j}^{0}

is the bare charge of nucleus j. Parameter σ_j characterizes the decrease of the effective charge with the distance to the nucleus. ζ_j is the soft-core parameter. For

H_{2}^{+}

, the effective charge is chosen to be constant and the ordinary soft-core potential [38] is reproduced. For CO₂ and N₂, the values of

Z_{j}^{\infty}

were derived from a Mulliken analysis carried out in ab initio study. The values of the parameters of three molecules are summarized in Tab. 1.

Table 1. Values of all parameters used for soft-core potentials of $H_{2}^{+}$ , CO₂ and N₂ [37].

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The initial normalized wavefuntion is obtained by imaginary-time propagation and orthogonalization under symmetry conditions corresponding to HOMO of each molecule. After that, the initial wavefunction is served as the initial condition for the real-time propagation. The time-dependent wavefuntion ψ(r⃗, t) is propagated using the second-order split-operator spectral method [34] in real time on a Cartesian grid. Due to the linearity of the Schrödinger equation, the electron wave function ψ(t_i) (r⃗ is omitted for simplicity) at a given time t_i can be split into two parts:

ψ (t_{i}) = ψ (t_{i}) [1 - F_{s} (R_{c})] + ψ (t_{i}) F_{s} (R_{c}) = ψ_{I} (t_{i}) + ψ_{I I} (t_{i}) .

Here, F_s(R_c) = 1/(1+e^{−(r−R_c)/Δ}) is a split function that separates the whole space into the inner (0 → R_c) and outer (R_c → R_max) regions smoothly [39, 40]. ψ_I represents the wave function in the inner region and it is propagated under the full Hamiltonian using the split-operator algorithm. ψ_II stands for the wave function in the outer region and it is propagated under the Volkov Hamiltonian analytically. We first calculate

𝒞 (\vec{p}, t_{i}) = \int ψ_{II} (t_{i}) \frac{e^{- i [\vec{p} + \vec{A} (t_{i})] \cdot \vec{r}}}{2 π} d^{2} \vec{r},

where

\vec{A} (t_{i}) = \int_{t_{i}}^{\infty} \vec{E} (τ) d τ

is the vector potential of the laser pulse. Then we propagate ψ_II from t_i to the end of the laser pulse as

ψ (\infty, t_{i}) = \int \bar{𝒞} (\vec{p}, t_{i}) \frac{e^{i \vec{p} \cdot \vec{r}}}{2 π} d^{2} \vec{p},

with

\bar{𝒞} (\vec{p}, t_{i}) = \exp (- i \int_{t_{i}}^{\infty} \frac{1}{2} {[\vec{p} + \vec{A} (τ)]}^{2} d τ) 𝒞 (\vec{p}, t_{i})

. Finally, the electron momentum distribution is obtained as

\frac{d^{2} 𝒫 (\vec{p})}{d E d Ω} = {| \sum_{i} \bar{𝒞} (\vec{p}, t_{i}) |}^{2},

where E is the electron energy associated with p⃗ as E = |p⃗|²/2 and Ω is the angle of the emitted electron. After the end of the pulse, the wave function is propagated for additional few femtoseconds to collect the ”slow” photoelectrons. In our simulations, the grid size is the same for x and y directions: Δx = Δy = 0.1 a.u. and the grid ranges from −300 a.u. to 300 a.u. in each direction. The boundary of the inner region R_c is 100 a.u.. The real time step is chosen as Δt = 0.001 a.u. which satisfies the convergence condition Δt/Δx² < 0.5. After the pulse, the wave function propagates freely for 3 fs.

3. Results and discussions

Figure 2 shows the molecular orbitals of $H_{2}^{+}$ and the corresponding electron diffraction patterns for three different internuclear distances under the nonperturbative single-photon ionization process. With high intensity XUV pulse, there are also multiphoton absorptions. However, electrons due to multiphoton ionization process appear at higher momentum region and have no influence for electrons with low momentum. Below we will concentrate on the single-photon ionized electrons. Clear diffraction fringes are observed in the electron momentum distributions and these fringes are markedly well resolved in the ponderomotive motion ”circle” corresponding to the momentum region of the ejected electrons by single-photon ionization. Moreover, by stretching the molecules from R = 2 a.u. to R = 6 a.u., the number of diffraction fringes increase with the internuclear distance, which is similar to Young’s double-slit experiment [20].

Fig. 2 Upper panels: the molecular orbitals of $H_{2}^{+}$ for three different internuclear distances of (a) R = 2 a.u., (b) R = 4 a.u. and (c) R = 6 a.u. Lower panels: the corresponding electron diffraction patterns (wider logarithmic scale) for different internuclear distances. The wavelength of the driving laser pulse is 5 nm and the intensity is 1.0 × 10¹⁵W/cm².

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The variation of diffraction patterns as the increase of the internuclear distance is one of the key point of MAPD, which allows us to extract the geometrical information, i.e., the internuclear distance of the molecules.

Another feature can be observed in PMDs of $H_{2}^{+}$ is that the diffraction stripes are always perpendicular to the molecular axis and there is a interference maximum for p_x = 0, independent of the internuclear distance. This can be attributed to the symmetry of molecular orbital of $H_{2}^{+}$ . We show that the diffraction pattern can be well reproduced by one step model of photoemission (PE). For laser wavelength as short as 5 nm used in our study, the ejected electrons energy is as high as 200 eV such that the effect of the parent ion can be neglected. The single-photon ionization can be treated as a single transition process from a molecular orbital to the final state, where the final state is approximated by plane waves. The transition amplitude reads

𝒞_{PE} (\vec{p}) = i \int_{- \infty}^{\infty} e^{- i S (\vec{p}, t)} E (t) d_{y} (\vec{p} + \vec{A} (t)) d t .

Here, S(p⃗, t) is the semiclassical action

S (\vec{p}, t) = \int_{t}^{\infty} [\frac{1}{2} {(\vec{p} + \vec{A} (t^{'}))}^{2} + I_{p}] d t^{'}

and d_y(p⃗) is the dipole moment which takes the form

d_{y} (\vec{p}) = 〈 ϕ_{\vec{p}} (\vec{r}) | y | ψ_{i} (\vec{r}) 〉 = \frac{1}{{(2 π)}^{3 / 2}} \int e^{- i \vec{p} \cdot \vec{r}} y ψ_{i} (\vec{r}) d \vec{r},

where the initial molecular orbital is obtained with the ab initio Gaussian 03 code. The finally PMD can be obtained by

\frac{d^{2} 𝒫 (\vec{p})}{d E d Ω} = {| 𝒞_{PE} (\vec{p}) |}^{2} .

Figure 3(d)–3(f) display the PMDs of different internuclear distance calculated by PE and the the results of TDSE are also presented in Fig. 3(a)–3(c) for comparison. One can see that the these two results agree perfectly with each other. The PMDs calculated by PE well reproduce all the features of diffraction images, including the positions and shapes of the interference fringes. This indicates that the distortion of the molecular orbital due to the XUV pulse is ignorable. Slightly difference appears at larger p_x, which can be attributed to the fact that PE predicts a little higher ionization probability of the molecules thus the high-order interference peaks in the TDSE calculation are not well resolved as the PE results. Nevertheless, the good agreement between the PE and TDSE calculations indicates the validity of classical double-slit predictions when de Broglie wavelength of the ejected electrons is less than the internuclear distance in a MAPD experiment [31, 33].

Fig. 3 Photonelectron momentum distributions of $H_{2}^{+}$ for three different internuclear distances of R = 2 a.u. (left column), R = 4 a.u. (middle column) and R = 6 a.u. (right column). The upper panels (same as Fig. 1 (d)–(f) but a narrower logarithmic scale) are the results of TDSE calculations and the lower panels are calculated by PE.

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By integrating the PMDs over p_y, the corresponding electron momentum spectra along p_x direction are obtained, as displayed in Fig. 4. Again, the agreement between TDSE and PE calculations is achieved. The number of interference peaks is related to the internuclear distance. For R = 2 a.u., there are three peaks within the interval [−5 a.u., 5 a.u.]. For R = 4 a.u., six peaks appear (although the highest-order peaks are weak), twice as the interference peaks for R = 2 a.u.. For R = 6 a.u. one can observe nine peaks, triple as those for R = 2 a.u.. This observation can be summarized in accordance with the classical double-slit experiment as

R = 2 π / Δ p_{x},

in which Δp_x denotes the momentum separation between the adjacent interference peaks. The internuclear distance derived from Eq. (13) is summarized in Tab. 2 and Δp_x is obtained by the momentum separation between the zero-order and first-order interference maximum. The measured internuclear distance from both TDSE and PE shows good agreement with the exact internuclear distance. The maximum error for R = 2 a.u. is 4.5%. For R = 4 a.u. and 6 a.u., we also calculated the internuclear distance averaged over high-order interference peaks. The error of measured internuclear distance is less than 3%. All these results imply that the MAPD serves as a efficient tool to measure the internuclear distance of

H_{2}^{+}

.

Fig. 4 Electron momentum spectra of $H_{2}^{+}$ along the p_x direction by integration over p_y for R = 2 a.u. (left column), R = 4 a.u. (middle column) and R = 6 a.u. (right column). The upper panels are the results of TDSE calculations and the lower panels are calculated by PE.

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Table 2. Measured internuclear distance of $H_{2}^{+}$ from TDSE as well as PE in comparison with the exact internuclear distance. The internuclear distance is calculated using the zero-order and first-order ionization peaks

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Next, we will turn to a slightly more complicated molecule, CO₂. Figure 5(b) and 5(e) display the electron momentum distributions of CO₂ from equilibrium to stretched internuclear distance. Similar as $H_{2}^{+}$ , The PMDs of CO₂ also show clear diffraction patterns on the ponderomotive motion circle. By stretching the molecule from R = 2.24 a.u. to R = 3 a.u., the number diffraction fringes increase with the internuclear distance. However, there is a clear difference of the PMDs between CO₂ and $H_{2}^{+}$ . For $H_{2}^{+}$ it is an interference maximum at p_x = 0, while for CO₂ it is an interference minimum. This fact can be attributed to the symmetry properties of HOMO of CO₂. Figure 5(a) and 5(d) show the HOMO of CO₂ for R = 2.24 a.u. and R = 3 a.u., respectively. The HOMO of CO₂, which is of Π_g symmetry, is antisymmetric with respect to two nodal planes: one containing the molecular axis and another containing the carbon ion and perpendicular to the molecular axis. During the interaction between the molecule and the XUV pulse, the structure properties of molecular orbital are converged in the propagation of the electron wavepacket and the initial symmetry persists to some extend under the force of laser fields [37]. Thereby, the symmetry properties are imprinted onto the PMDs, leading to distinct diffraction patterns for molecule with different symmetries. This fact is also confirmed by PE calculations, as presented in Fig. 5(c) and 5(f). The electron momentum distributions calculated by PE show clear diffraction patterns and interference minimum appears at p_x = 0. Although the high-order diffraction fringes are not well resolved for TDSE calculations compared with PE, The information contained in low-order fringes is enough for reading the internuclear distance and retrieving the symmetry properties of the molecular orbital. This issue only becomes critical for molecules with short internuclear separations. For these molecules, the number of interference fringes is limited, so it is hard to measure the internuclear distance from the diffraction pattern. This shortcoming can be overcome by using XUV pulse with shorter wavelength, such that more energetic electron can be ejected and the radius of the ponderomotive motion circle becomes larger to include more interference fringes. Thereby the short internuclear distance can be readily read out from the diffraction pattern.

Fig. 5 Left column: HOMO of CO₂ for two different internuclear distances of (a) R = 2.24 a.u. and (d) R = 3 a.u. The corresponding photonelectron momentum distributions are calculated by TDSE (middle column) and PE (right column), respectively. Laser parameters are the same as Fig. 2.

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The integrated electron momentum spectra for different internuclear distances over p_y are shown in Fig. 6, calculated by TDSE (left column) and PE (right column), respectively. These two results also agree well with each other. For both internuclear distances, there is a interference minimum at p_x = 0, so a direct analogy to the double-slit experiment is not feasible for CO₂, due to the symmetry properties of HOMO. Using the linear combination of atomic orbitals (LCAO) scheme, the HOMO of CO₂, which possess Π_g symmetry, can be expressed as the antisymmetric combination of two 2p_y atomic orbitals of oxygen that are oriented perpendicular to the molecular axis. The two oxygens’ 2p_y orbital is antisymmetric, which can be also viewed as two ”slits”. The difference is that one slit is masked by a phase function, changing the phase of the electron wavepacket ejected from this slit by π. As a consequence, the interference maximum and minimum exchange their positions. We calculate the internuclear separations of the two oxygen ions using the zero-order and first-order interference minimum and then the C-O bond length is obtained by dividing the measured internuclear separations by two. The results are summarized in Tab. 3. The measured C-O bond length from TDSE and PE shows good agreement with the exact one. The maximum error is less than 3% for both cases. More precisely, we calculated the averaged C-O bond length over three interference minimums for R = 2.24 a.u. and four minimums for R = 3 a.u.. The results are R = 2.197 a.u. and R = 3.045 a.u., respectively, with error less than 2%. Therefore, it is apparent to see that MAPD is also useful for probing molecules with antisymmetry properties.

Fig. 6 Electron momentum spectra of CO₂ along the p_x direction by integration over p_y for R = 2.24 a.u. (upper panels) and R = 4 a.u. (Lower panels). The left column is the results of TDSE calculations and the right column is calculated by PE.

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Table 3. Measured C-O bond length of CO₂ from TDSE as well as PE in comparison with the exact bond-length from equilibrium to stretched geometries. The C-O bond length is calculated using the zero-order and first-order interference minimum.

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Another molecule for the demonstration of MAPD is N₂. In the case of N₂, the HOMO, of Σ_g symmetry, is symmetry about the molecular axis and with two nodal planes, each of them containing one nitrogen ion, perpendicular to the molecular axis. The upper panels of Fig. 7 shows HOMO of N₂ for three different internuclear separations 2.05 a.u., 4 a.u. and 6 a.u.. The HOMO is a bonding orbital which can be expressed as a combination as two p_x nitrogen orbitals aligned along the molecular axis. These structure properties make the corresponding PMDs a little more complicated, as shown in Fig. 7 (middle panels for TDSE and lower panels for PE). The general trend of the PMDs is the same as $H_{2}^{+}$ and CO₂ : as the internuclear distance stretched from equilibrium 2.05 a.u. to 6 a.u., the number of diffraction stripes increases accordingly. However, the detailed structure is different. At p_x = 0, there is a small maximum, which becomes invisible for large internuclear separations. Then the intensity becomes more pronounced for higher-order fringes. This characteristic is attributed to the structure properties for N₂ orbital. The HOMO of N₂ shows alternating positive and negative lobes along the molecular axis and electron wavepackets ejected from different part of the orbital interfere with each other. So it is not a trivial task to directly relate to the double-slit experiment without additional assumptions. Nevertheless, we can at least measure the value of the internuclear distance from the period of the oscillations in its fringe pattern.

Fig. 7 Upper panels: HOMO of N₂ from equilibrium to stretched internuclear distances (a) R = 2.05 a.u., (b) R = 4 a.u. and (c) R = 6 a.u. The corresponding photonelectron momentum distributions are calculated by TDSE (middle panels) and PE (lower panels), respectively. Laser parameters are the same as Fig. 2.

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The integrated electron momentum spectra of N₂ for different internuclear distances over p_y are shown in Fig. 8, calculated by TDSE (upper panels) and PE (lower panels), respectively. The relative heights of the diffraction fringes can be more clearly viewed from Fig. 8. The height of the interference peaks increases for high-order fringes, which originate from the interplay between the interference of the positive and negative lobes of HOMO. For our purpose of measuring the internuclear distance, We calculated it from one period of the oscillation located at moderate p_x, as indicated by the dashed lines in Fig. 8. The value of internuclear distances are obtained from Eq. (13), summarized in Tab. 4. For stretched R = 4 a.u. and 6 a.u., the measured internuclear distances show good agreement with the exact ones, including the TDSE and PE results. While for equilibrium R = 2.05 a.u., The maximum error is 12%, which is due to the complicated diffraction pattern of N₂ and the limited numbers of interference fringes. Our results suggest that in order to precisely measure the equilibrium internuclear distance of N₂ one should use XUV pulses with shorter wavelength.

Fig. 8 Electron momentum spectra of N₂ along the p_x direction by integration over p_y for R = 2.05 a.u. (left column), R = 4 a.u. (middle column) and R = 6 a.u. (right column). The upper panels are the results of TDSE calculations and the lower panels are calculated by PE. The black dashed lines in each panel indicate one oscillation period used for measuring the internuclear distance.

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Table 4. Measured internuclear distance of N₂ from TDSE as well as PE in comparison with the exact internuclear distance. The internuclear distance is calculated using the zero-order and first-order ionization peaks

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Above discussions are based the assumption that the molecule is perfectly aligned perpendicular to the electric field. Next we will go one step further to investigate the influence of molecular alignment effects. In reality, the alignment is achieved by exciting the molecule into a rotational wave packet by a moderately intense pump pulse. In this case, perfect alignment can not be achieved and the effect of angular dispersion should be taken into account in the experiment of MAPD. We take CO₂ for example to investigate the alignment effect on the PMD. The effect of molecular partial alignment is simulated by incoherent superposition of PMD of different angles given by

𝒮 (\vec{p}) = \int {| \sum_{i} \bar{𝒞} (\vec{p}, t_{i}, θ) |}^{2} ρ (θ; τ) d θ,

where 𝒞̄(p⃗, t_i, θ) is calculated by Eq. (6) for different angles and ρ(θ; τ) is the weighted angular distribution written as

ρ (θ; τ) = (1 / Z) \sum_{J_{i}} Q (J_{i}) \sum_{M_{i} = - J_{i}}^{J_{i}} \int {| Ψ^{J_{i} M_{i}} (θ, ϕ; τ) |}^{2} d ϕ .

Here Q(J_i) = exp(−BJ_i(J_i + 1)/(k_BT)) is the Boltzmann distribution function of the initial field-free state |J_i, M_i〉 at temperature T,

Z = \sum_{J = 0}^{J_{\max}} (2 J + 1) Q (J)

is the partition function, k_B and B are the Boltzmann constant and the rotational constant of the molecule, respectively. Ψ^J_iM_i(θ, ϕ; τ) is the time-dependent rotational wave packet excited from the initial state |J_i, M_i〉 by the pump pulse, and is obtained by solving the TDSE within the rigid-rotor approximation [41, 42]

i \frac{\partial Ψ (θ, ϕ; τ)}{\partial t} = [B J^{2} - \frac{E_{p} {(τ)}^{2}}{2} (α_{‖} {cos}^{2} θ + α_{⊥} {sin}^{2} θ)] Ψ (θ, ϕ; τ),

where α_‖ and α_⊥ are the anisotropic polarizabilities in parallel and perpendicular directions with respect to the molecular axis, respectively. The degree of alignment is characterized by the alignment parameter <cos²θ> yielding

< {cos}^{2} θ > (τ) = (1 / Z) \sum_{J_{i}} Q (J_{i}) \sum_{M_{i} = - J_{i}}^{J_{i}} 〈 Ψ^{J_{i} M_{i}} (θ, ϕ; τ) | {cos}^{2} θ | Ψ^{J_{i} M_{i}} (θ, ϕ; τ) 〉 .

In our simulations, a 100-fs (FWHM) 800-nm linearly polarized pump pulse with intensity of 4.0 × 10¹³W/cm² is used to nonadiabatically align the CO₂ molecule. The initial rotational temperature is taken to be 40 K. The time evolution of the alignment factor <cos²θ> (τ) is presented in Fig. 9(a). We choose delay of 21.1 ps indicate by point A. The corresponding weighted angular distribution ρ(θ; τ) is shown in Fig. 9(b). One can see that strong alignment of CO₂ is achieved by the pump pulse. After that, a delayed XUV pulse polarized perpendicular to the pump pulse is used to ionize the molecule. The angular averaged photoelectron momentum distribution is calculated by Eq. (14), as displayed in Fig. 9(c). In comparison with the perfect alignment case shown in Fig. 5(b), the angular averaged PMD successfully reproduces all the fringes of the diffraction pattern, except that the contrast of fringes becomes weak. But we are still able to exact the internuclear distance and symmetry properties of the molecule from the slightly less contrasted diffraction pattern. As shown in Fig. 9(d), the integrated electron momentum spectrum is well resolved and the positions of the interference fringes do not shift compared to Fig. 6(a). The less contrast of the fringes can be overcome by high-repetition and high-accuracy data recording for experimental realization. Thus we conclude that angular dispersion do not affect the extraction of internuclear distance and symmetry properties from the diffraction pattern.

Fig. 9 (a) The alignment factor <<cos²θ>> for linearly polarized laser pulse with intensity of 4.0 × 10¹³W/cm². The point A indicates the delay of 21.1 ps that the simulation is performed at. (b) The polar plots of the angular distributions at point A. (c) Angular averaged photoelectron momentum distribution for CO₂ at equilibrium R = 2.24 a.u. and (d) the corresponding electron momentum spectra along the p_x direction.

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Below we aim at retrieving the molecular orbital from the diffraction pattern using a simple two-center interference model [33]. The two-center interference model can be derived from PE, i.e., Eq. (9). In our case, the typical electron momentum on the ponderomotive motion circle is 4.2 a.u. and the amplitude of the vector potential A⃗(t) for the 5-nm XUV pulse with intensity of 1.0 × 10¹⁵W/cm² is E₀/ω = 0.04 a.u.. Such that |p⃗| ≫ |A⃗(t)|. Under this condition, the laser dressed effect of the ionized electrons and the distortion of molecular orbital can be neglected and we obtain p⃗ + A⃗(t) ≃ p⃗. Thus Eq. (9) can be approximated by

𝒞_{PE} (\vec{p}) ≃ d_{y} (\vec{p}) i \int_{- \infty}^{\infty} e^{- i S (\vec{p}, t)} E (t) d t .

The dipole moment d_y(p⃗) can be viewed as the Fourier transform of ψ_i(r⃗) weight by y, yielding

𝒞_{PE} (\vec{p}) \propto ℱ [y ψ_{i} (\vec{r})] .

And therefore the electron momentum spectra can be written as

𝒯 (p_{x}) \propto \int {| ℱ [y ψ_{i} (\vec{r})] |}^{2} d p_{y} .

For

H_{2}^{+}

, the initial orbital is expressed as linear combinations of two hydrogenic 1s orbital located at ±R/2, i.e.,

ψ_{H_{2}^{+}} = ψ_{1 s} (x - R / 2, y) + ψ_{1 s} (x + R / 2, y)

. By expressing ψ_1s(x, y) as α exp(−βr), the electron momentum spectra for

H_{2}^{+}

in Eq. (20) is

𝒯_{H_{2}^{+}} (p_{x}) \propto α \frac{{sin}^{2} (p_{x} R / 2)}{{(p_{x}^{2} + β^{2})}^{2}} .

In this equation, the measured internuclear distance R is obtained from Tab. 2 as 2.091 a.u.. β serves as a fitting parameter to reproduce the diffraction pattern of Fig. 4(a). For CO₂, The HOMO is an antisymmetric combination of two 2p_y atomic orbitals of oxygen that are oriented perpendicular to the molecular axis, i.e., ψ_CO₂ = ψ_{2p_y} (x − R, y) −ψ_{2p_y} (x + R, y), where ψ_{2p_y} (x, y) is αyexp(−βr). Thus the electron momentum spectra for CO₂ is

𝒯_{{CO}_{2}} (p_{x}) \propto α \frac{{cos}^{2} (p_{x} R)}{{(p_{x}^{2} + β^{2})}^{9 / 2}} .

Again, the C-O bond length R is obtained from Tab. 3 as 2.189 a.u. and the parameter β is determined by fitting the diffraction pattern of Fig. 5(a). For N₂. the HOMO is combined of two 2p_y atomic orbitals of oxygen aligned along the molecular axis, which can be expressed as ψ_N₂ = ψ_{2p_x} (x − R/2, y) − ψ_{2p_x} (x + R/2, y), where ψ_{2p_x} (x, y) is αxexp(−βr). It is hard to provide an analytical expression for the electron momentum spectra for N₂, so we numerically calculate it using Eq. (20) with R obtained from Tab. 4. The fitting parameter β is determined to best reproduce Fig. 8(a). For all the three molecules, the parameter α is calculated by normalization of the molecular orbital. The reconstructed molecular orbitals are displayed in Fig. 10. They all reproduce the main features of each target molecule and show good agreement with the real molecular orbitals. The results demonstrate that the MAPD technique permits the retrieval of not only the geometrical information but also the orbital structure of the molecule from observable photoelectron momentum distributions. We should remind that the knowledge of the symmetry properties needs to be known a priori in order to retrieve the shape of the orbitals.

Fig. 10 The reconstructed molecular orbitals of (a) $H_{2}^{+}$ , (b) CO₂ and (c) N₂.

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4. Conclusion

In summary, we have presented a systematic theoretical analysis of the ionization dynamics of $H_{2}^{+}$ , CO₂ and N₂ molecules in intense ultrashort linearly polarized XUV pulses within the framework of molecular attosecond photoelectron diffraction. The momentum distributions of the emitted photoelectrons are calculated by solving the two-dimensional time-dependent Schrödinger equation as well as the photoemission method. We have shown that, if the molecule is initially aligned perpendicular to the field polarization, the internuclear distance of the molecule can be determined from the diffraction pattern of the photoelectron momentum distribution with a high accuracy of a few percent. The partial alignment effect has been considered and the robustness of internuclear distance measurement with respect to the rotational motions has been demonstrated. Besides, we have also shown that the symmetry properties of molecules are also imprint into PMD and the molecular orbitals can be successfully reconstructed using an inversion algorithm based on two-center interference model. Our results provide an theoretical guide for ultrafast attosecond photoelectron imaging of molecules.

All the discussions in this paper are based on single-active-electron approximations. Recent experiments have shown that molecules interacting with a strong laser field could be ionized from several valence orbitals with different symmetries simultaneously [11]. Thus an overlap of multiple diffraction events could appear in the momentum distributions. Due to different symmetries of the orbitals, the contribution of multichannel contributions can be disentangled with a fitted procedure. This implies that the molecular attosecond photoelectron diffraction also offers possibility to study multielectron effect.

Acknowledgments

This work was supported by the 973 Program of China under Grant No. 2011CB808103, the NNSF of China under Grants No. 11404123, 11234004, and 61275126, the Doctoral fund of Ministry of Education of China under Grant No. 20100142110047. Numerical simulations presented in this paper were carried out using the High Performance Computing Center experimental testbed in SCTS/CGCL (see http://grid.hust.edu.cn/hpcc).

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Molecule Nucleus	$H_{2}^{+}$		CO₂			N₂
Molecule Nucleus	H	H	O	C	O	N	N
ζ_j (a.u.)	1.0	1.0	1.0	1.0	1.0	1.2	1.2
$σ_{j}^{2}$ (a.u.)	∞	∞	0.577	0.750	0.577	0.700	0.700
$Z_{j}^{0}$	1.0	1.0	8	6	8	7	7
$Z_{j}^{\infty}$	1.0	1.0	0.173	0.654	0.173	0.500	0.500

Internuclear distance of $H_{2}^{+}$ R(a.u.)	Measured R(a.u.)from TDSE	Measured R(a.u.)from PE

2.0	2.091	2.068
4.0	4.069	4.042
6.0	6.064	6.013

C-O bond length of CO₂ R(a.u.)	Measured R(a.u.) from TDSE	Measured R(a.u.) from PE

2.24 (equilibrium)	2.189	2.254
3.00 (stretched)	2.989	3.062

Internuclear distance of N₂ R(a.u.)	Measured R(a.u.) from TDSE	Measured R(a.u.) from PE

2.05	2.291	2.314
4.00	4.093	4.022
6.00	6.097	6.086

Molecule Nucleus	$H_{2}^{+}$		CO₂			N₂
Molecule Nucleus	H	H	O	C	O	N	N
ζ_j (a.u.)	1.0	1.0	1.0	1.0	1.0	1.2	1.2
$σ_{j}^{2}$ (a.u.)	∞	∞	0.577	0.750	0.577	0.700	0.700
$Z_{j}^{0}$	1.0	1.0	8	6	8	7	7
$Z_{j}^{\infty}$	1.0	1.0	0.173	0.654	0.173	0.500	0.500

Ultrafast molecular orbital imaging based on attosecond photoelectron diffraction

Abstract

1. Introduction

2. Theoretical model

3. Results and discussions

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (10)

Tables (4)

Equations (22)

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