Molecular photoelectron holography by an attosecond XUV pulse in a strong infrared laser field

We have investigated the photoelectron spectra from ionization of diatomic molecules by an attosecond XUV pulse in a strong infrared laser field by quantum calculations. A clear holographic interference structure is observed in the two-dimensional photoelectron momentum spectrum. Moreover, this holographic structure depends sensitively on the electron orbitals and internuclear distance of diatomic molecules. Based on the orbital dependence of the holographic structure, one can identify the symmetries and electron density distributions of molecular orbitals. This indicates that the photoelectron holography by an attosecond XUV pulse in a strong infrared field can be used as an efficient tool for molecular imaging. © 2014 Optical Society of America OCIS codes: (020.4180) Multiphoton process; (260.3230) Ionization; (270.6620) Strong-field process. References and links 1. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163-234 (2009). 2. W. Becker, F. Grasbon, R. 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Introduction
Recent advances in femtosecond laser technology have led to a number of nonlinear phenomena such as high-order harmonic generation (HHG) [1], above-threshold ionization (ATI) [2] and double ionization (DI) [3][4], from which structural and dynamical information on the target system can be extracted.For example, the full three-dimensional (3D) shape of a molecular orbital is reconstructed by a tomographic imaging technique based on HHG [5][6][7].With the laserinduced electron diffraction which was first proposed by Zuo et al. [8], the sub-Ångström structural changes of molecules is imaged with femtosecond time resolution [9].Recently, Huismans et al. reported that the concept of holography can be applied to strong-field ionization to record the temporal and spatial information on the atomic and molecular scale [10].Different from the conventional holography, in strong-field photoelectron holography (PH) the electron beam is created by laser-induced tunnel ionization.After tunneling, the electron can follow two different paths to the detector.One is that the released electron is driven back to the parent ion by the oscillating electric field and scatters off its parent ion, generating a signal wave.The other is that the released electron directly drifts to the detector, generating a reference wave.The interference between the signal and reference waves on a detector creates a hologram in the photoelectron momentum distribution which encodes the temporal and spatial informations of both the parent ion and recolliding electron.With a semiclassical analysis, it was shown that there are four kinds of holographic interferences which include two kinds of interferences between the direct and forward scattered electrons and two kinds of interferences between the direct and backward scattered electrons [11].Recently, experimental and theoretical studies on PH have observed these holographic interference structures in the photoelectron momentum spectra [12][13].
However, previous studies mainly focus on the PH by the infrared (IR) field.In this case, the interference between the direct electron wave packets born at different times within the laser pulse blurs the holographic interference structure in the photoelectron momentum spectrum [14][15][16].Thus, it is difficult to extract the structural information of targets and the dynamical process of ionization electrons from the holographic structure.Recent studies have shown that the single attosecond pulses (SAPs) in the extreme ultraviolet (XUV) regime [17][18][19][20] allows one to restrict the release of the electron wave packet to an attosecond time window by single-photon absorption [21][22].In such a narrow time window, the interference between direct electron wave packets can not occur.Very recently, the study on the ionization of atoms demonstrated that the combination of an attosecond XUV pulse and a strong IR laser pulse can indeed eliminate the interference between direct electron wave packets and thus obtain a clear holographic interference structure in the photoelectron energy spectrum along the laser polarization direction [23].In this paper, we investigate PH of diatomic molecules by an attosecond XUV pulse in a strong IR field.The obtained two-dimensional (2D) photoelectron momentum spectrum shows a clear holographic structure.Furthermore, it is demonstrated that the holographic structure is sensitively dependent on the electron orbitals and internuclear distance of diatomic molecules.This indicates that the photoelectron holography by a attosecond XUV pulse in a strong IR laser field can be applied to image the molecular structure.

The model
Here, we investigate the ionization of diatomic molecules by an attosecond XUV pulse in a strong IR laser field by numerically solving time-dependent Schrödinger equation (TDSE).The 2D TDSE is written as (atomic units are used throughout this paper unless otherwise stated): where x, y are the coordinates of the electron.V (x, y) is the soft-coulomb potential which describes the attraction of two nuclei to the electron.It is given by where R is the internuclear distance.Z is the electric charge of the nucleus.Note that comparing with the three-dimensional (3D) model the 2D model underestimates the transverse spreading of the electron wave packets.This results in the fact that in the 2D model the ionized electron wave packets are more easily to be driven back to the parent ion.However, this does not change the phenomena observed in this paper and thus the conclusion is unchanged.
is the total electric field of the IR and XUV pulses.τ indicates the delay between the XUV pulse and the IR field.The electric field of a laser pulse is given by where i =XUV, IR. a i is the amplitude of the laser field.ω i and τ i are the frequency and duration of the laser pulse, respectively.In our study, the carrier frequency of the XUV pulse is ω XUV = 1.33 a.u.(corresponding to the photon energy 36ev) and its duration is τ XUV = 3T XUV (T XUV is the cycle of the XUV pulse).The intensity of the XUV pulse is I XUV = 5 × 10 15 W/cm 2 .The IR field has the wavelength λ IR = 750 nm and its intensity is I IR = 4.2 × 10 13 W/cm 2 .The duration of the IR pulse is τ IR = 4T IR .The delay τ is set to be 0.25T IR , where the XUV pulse is located at the zero cross of the IR electric field.Note that the intensity of the XUV pulse chosen above is extremely intense.In our work, the XUV pulse is used to launch the electron wave packet while the IR pulse only accelerates the ionized electron and then drives it back to the parent ion.To this end, the ionization from the IR pulse need to be suppressed.Thus the intensity of the IR pulse is low and the intensity of the XUV pulse is relative intense.For small internuclear distances, the IR pulse with the intensity of ∼ 10 13 W/cm 2 can hardly ionize the electron and thus the intensity of the XUV pulse could be low [ for example 5×10 13 W/cm 2 ].However, for large internuclear distances the ionization by the IR pulse is greatly enhanced due to the charge resonant enhanced ionization.In order to ensure that the ionization arises mostly from the XUV pulse, we choose such an intense XUV pulse.Although such an intense XUV pulse is not currently available experimentally, it can be achieved in the future with the development of free-electron laser.For the consistency of laser pulse parameters, all the calculations are performed with the XUV pulse intensity of 5×10 15 W/cm 2 .We use the split-operator spectral method [24] to numerically solve the 2D TDSE.Following [25], the electron wave function ψ(t i ) at a given time t i is split into two parts: 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 [26-28].ψ I represents the wave function in the inner region and it is propagated under the full Hamiltonian.ψ II stands for the wave function in the outer region and it is propagated under the Volkov Hamiltonian analytically.We first calculate where A(t i ) = ∞ t i E(τ)dτ is the vector potential of the laser pulse.Then we propagate ψ II from t i to the end of the laser pulse as Finally, the electron momentum distribution is obtained as dP(p) dEdθ where E is the electron energy associated with p and θ is the angle of the emitted electron.
After the end of the pulse, the wave function is allowed to propagate without laser field for an additional time of three optical cycles of the IR pulse in order to collect all photoelectrons with enough low energies.The initial wave function for time propagation is obtained by imaginarytime propagation.

Results and discussions
Figures 1(a)-1(c) show the two-dimensional projections of molecular orbitals 1sσ g , 2pσ u and 2sσ g , respectively.Figures 1(d)-1(f) show the corresponding photoelectron momentum distributions.The internuclear distance of molecules is fixed to be 3 a.u.For the molecular orbital 1sσ g , the photoelectron momentum distribution exhibits an arc-like structure, which results from the interference between the direct and scattered electron wave packets.For the molecular orbital 2pσ u , there is a nodal axis perpendicular to the molecular axis, which divides the electron wave function into two parts with opposite signs.For this molecular orbital, the arc-like structure  is tremendously suppressed and a fork-like structure appears in the photoelectron momentum spectrum, as shown in Fig. 1(e).For the molecular orbital 2sσ g , there are two nodal axes which divide the electron wave function into three part.In this case, the resulting holographic pattern is a fork-like structure, as shown in Fig. 1(f).These results indicate that for molecular orbitals with σ symmetry the holographic pattern in the photoelectron momentum distribution changes gradually from the arc-like structure to the fork-like structure as the number of the nodal axis in the molecular orbital increases.
In order to elucidate the dependence of the holographic structure on the molecular orbitals, we perform a semiclassical analysis [23,[29][30].Firstly, the electron is ionized by the XUV pulse.Then, the ionized electron oscillates in the IR field, which is governed by the classical mechanics.The electron motion along the laser polarization direction is described by  3. The photoelectron spectrum as a function of the recollision momentum p r for molecular orbitals 1sσ g , 2pσ u and 2sσ g , respectively.
where x 0 and v 0x are initial coordinate and velocity of the electron at the instant of ionization, respectively.The distribution of v 0x is determined by the energy spectrum of the XUV pulse.t 0 is the ionization time.A IR (t) is the vector potential of the IR pulse which is given by A IR (t) = − t −∞ E IR (τ)dτ.Equation ( 9) indicates that the photoelectron with certain v 0x can be driven back to the parent ion by the IR field.The recollision momentum p r of the electron at the returning time t r is given by After the electron is scattered elastically by the parent ion, it will achieve a final momentum p f = p r − A IR (t r ).Following the procedure described in [9], we resolve the interference between the scattered electron with different recollision momenta and the direct electron.Figure 2 shows the photoelectron spectra resulting from the electron interference involving the scattered electron with the recollision momentum p r = 0.4 a.u.for molecular orbitals 1sσ g , 2pσ u and 2sσ g , respectively.It is seen from Fig. 2(a) that for the molecular orbital 1sσ g the photoelectron spectrum exhibits a four-peak structure.As the molecular orbital changes from 1sσ g to 2pσ u , the holographic pattern in the photoelectron spectrum changes from a four-peak structure to a five-peak structure, as shown in Fig. 2(b).When the molecular orbital is 2sσ g , the holographic pattern in the photoelectron spectrum is an eight-peak structure, as shown in Fig. 2(c).In order to get insight into the dependence of the holographic structure on the recollision momentum p r of the electron, we display the photoelectron spectra as a function of the recollision momentum p r in Fig. 3 for molecular orbitals 1sσ g , 2pσ u and 2sσ g , respectively.It is seen that for these three molecular orbitals the numbers of interference peaks is independent on the recollision momentum p r .However, comparing with molecular orbitals 2pσ u and 2sσ g , for the molecular orbital 1sσ g the locations of the maxima in the interference structure change more quickly as the recollision momentum p r increases.
For molecular orbitals 1sσ g , 2pσ u and 2sσ g , the electron cloud is concentrated along the molecular axis.Now, we consider ionization for the diatomic molecules with molecular orbitals 2pπ u [see Fig. 4(a)] and 2pπ g [see Fig. 4(b)], where the electron cloud is localized off the nuclei due to the symmetry restriction.For the molecular orbital 2pπ u , the wave function has a nodal axis that coincides with the molecular axis.This nodal axis divide the wave function into two parts which are concentrated along the y axis, as shown in Fig. 4 with a nodal line along the axis P y = 0 appears in the photoelectron momentum spectrum.For the molecular orbital 2pπ g , there is another nodal axis perpendicular to the molecular axis in addition to the nodal axis along the molecular axis, as shown in Fig. 4(b).These two nodal axes divide the electron wave function into four parts.For this molecular orbital, the holographic pattern in the photoelectron momentum spectrum is a clear fork-like structure, as shown in Fig. 4(d).Similar to the case for the molecular orbital 2pπ u , there is also a nodal line along the axis P y = 0 in the photoelectron momentum distribution.In order to explore the dependence of the holographic structure on the recollision momentum, we show the photoelectron spectrum as a function of the recolliding momentum for molecular orbitals 2pπ u and 2pπ g in Fig. 5.For the molecular orbital 2pπ u , it is seen that for the recollision momentum p r = 0.36 a.u. the holographic pattern in the photoelectron spectrum is a four-peak structure, as shown in Fig. 5(a).As the recollision momentum p r increases to 0.4 a.u., the holographic pattern changes to be a six-peak structure.This indicates that for the molecular orbital 2pπ u the holographic pattern depends on the recollision momentum p r .For the molecular orbital 2pπ g , the holographic pattern is an eight-peak structure in the recollision momentum range from 0.36 a.u to 0.4 a.u.As the recollision momentum increases from 0.4 a.u. to 0.46 a.u., the holographic pattern is a six-peak structure.When the recollision momentum further increases from 0.46 a.u. to 0.5 a.u., the holographic pattern is again an eight-peak structure.
The results above focus on the molecular orbital dependence of the holographic pattern.Now, we investigate the dependence of the holographic pattern on the internuclear distance.The molecular orbital is fixed to be 1sσ g .Figure 6 shows the 2D photoelectron momentum distributions from the ionization of diatomic molecules at the internuclear distance R = 2 a.u., 4 a.u., 6 a.u. and 8 a.u., respectively.For these internuclear distances, the photoelectron mo- mentum spectra exhibit a similar arc-like structure.However, a closer look reveals that the photoelectron momentum distributions along the laser polarization direction depend on the internuclear distance.In Fig. 7, we show the photoelectron momentum distributions along the laser polarization direction.For R = 2 a.u., there are four peaks in the photoelectron momentum spectrum, as shown in Fig. 7(a).As the internuclear distance R increases to 4 a.u., no obvious interference structure appears in the photoelectron momentum spectrum, as shown in Fig. 7 (b).When the internuclear distance further increases to 6 a.u., there are five peaks in the photoelectron momentum spectrum, as shown in Fig. 7(c).Comparing with the interference peaks for the internuclear distance R = 2 a.u., these interference peaks move to the zero momentum.For the internuclear distance R = 8 a.u., there is also five peaks in the photoelectron momentum spectrum, as shown in Fig. 7(d).These results indicate that the numbers and locations of the interference peaks in the holographic structure along the laser polarization direction depend sensitively on the internuclear distance.

Conclusion
In conclusion, we have investigated the photoelectron holography of diatomic molecules by an attosecond XUV pulse in a strong IR laser field by numerically solving TDSE.A clear holographic structure is obtained in the 2D photoelectron momentum distribution.For the molecular orbitals with σ symmetry where the electron cloud is concentrated along the molecular axis,  the numbers of the interference peaks in the holographic structure increase as the number of the nodal axis increases.Moreover, the numbers and locations of the interference peaks in the photoelectron momentum spectrum along the laser polarization direction are sensitively dependent on the internuclear distance.For the molecular orbitals with π symmetry where the electron cloud is localized off the nuclei, there is a nodal line along the laser polarization direction in the 2D photoelectron momentum distribution.These results show that the holographic structure depends on the molecular structure.This indicates that the photoelectron holography by an attosecond XUV pulse in a strong IR field can be used as an efficient tool for molecular imaging.

Fig. 1 .
Fig.1.The photoelectron momentum distributions from the ionization of diatomic molecules by an attosecond XUV pulse in the presence of a strong IR field for molecular orbitals 1sσ g , 2pσ u and 2sσ g , respectively.The internuclear distance of molecules is fixed to be 3 a.u.

Fig. 2 .
Fig.2.The photoelectron spectra resulting from the electron interference with the recollision momentum p r = 0.4 a.u.for molecular orbitals 1sσ g , 2pσ u and 2sσ g , respectively.
Fig.3.The photoelectron spectrum as a function of the recollision momentum p r for molecular orbitals 1sσ g , 2pσ u and 2sσ g , respectively.

Fig. 4 .
Fig.4.The photoelectron momentum distributions from the ionization of diatomic molecules by an attosecond XUV pulse in the presence of a strong IR field for molecular orbitals 2pπ u and 2pπ g , respectively.The internuclear distance of molecules is fixed to be 3 a.u.

Fig. 5 .
Figure2shows the photoelectron spectra resulting from the electron interference involving the scattered electron with the recollision momentum p r = 0.4 a.u.for molecular orbitals 1sσ g , 2pσ u and 2sσ g , respectively.It is seen from Fig.2(a) that for the molecular orbital 1sσ g the photoelectron spectrum exhibits a four-peak structure.As the molecular orbital changes from 1sσ g to 2pσ u , the holographic pattern in the photoelectron spectrum changes from a four-peak structure to a five-peak structure, as shown in Fig.2(b).When the molecular orbital is 2sσ g , the holographic pattern in the photoelectron spectrum is an eight-peak structure, as shown in Fig.2(c).In order to get insight into the dependence of the holographic structure on the recollision momentum p r of the electron, we display the photoelectron spectra as a function of the recollision momentum p r in Fig.3for molecular orbitals 1sσ g , 2pσ u and 2sσ g , respectively.It is seen that for these three molecular orbitals the numbers of interference peaks is independent on the recollision momentum p r .However, comparing with molecular orbitals 2pσ u and 2sσ g , for the molecular orbital 1sσ g the locations of the maxima in the interference structure change more quickly as the recollision momentum p r increases.For molecular orbitals 1sσ g , 2pσ u and 2sσ g , the electron cloud is concentrated along the molecular axis.Now, we consider ionization for the diatomic molecules with molecular orbitals 2pπ u [see Fig.4(a)] and 2pπ g [see Fig.4(b)], where the electron cloud is localized off the nuclei due to the symmetry restriction.For the molecular orbital 2pπ u , the wave function has a nodal axis that coincides with the molecular axis.This nodal axis divide the wave function into two parts which are concentrated along the y axis, as shown in Fig.4(a).The corresponding photoelectron momentum distribution is shown in Fig.4(c).It is seen that an arc-like structure