Si-Qi Zhang(張思琪), Qi Zhen(甄琪), Zhi-Jie Yang(楊志杰), Jun Zhang(張軍),Ai-Hua Liu(劉愛華), Kai-Jun Yuan(元凱軍), Xue-Shen Liu(劉學深), and Jing Guo(郭靜)

Institute of Atomic and Molecular Physics,Jilin University,Changchun 130012,China

Keywords: N2 molecules, molecular frame photoelectron momentum distributions, ultrafast photoionization model

1. Introduction

With the rapid advances of ultrashort laser technology,[1–3]electronic dynamics in atoms and molecules on attosecond (10?18s) time scales has attracted great attention.[4–12]Recently, attosecond pulses with a duration of 43 as have been achieved by Gaumnitz et al.,[13]which can track electronic motion. Time-resolved photoelectron momentum and angular distributions based on the pump-probe techniques has been used as an efficient tool to investigate molecular reaction dynamics.[14–16]A transition between the ground state and the excited state of the target is initiated by the ultrafast pump laser pulse,and the dynamics process of the target is monitored by probe pulse.

Cohen and Fano[17]firstly suggested the interference of electron wavepackets in a diatomic molecule in perturbative single-photon ionization,and then Walter and Briggs modeled the interference of electron wavepackets in molecules.[18]Zuo et al. have extended such interference of electron wavepackets to nonperturbative regime.[19]Utilizing the ultrafast laser pulses to the molecules can induce the laser-induced electron interference and laser-induced electron diffraction (LIED),which is applied as an effective tool for imaging molecular orbitals.

Molecular photoelectron diffraction can imagine the time-resolved molecular dynamics.[20–23]Yuan et al.[24]theoretically studied triatomic molecular photoelectron diffraction by circularly x-ray laser pulses, they found that molecular photoelectron diffraction patterns show dependence on the symmetry of the molecular orbitals. Attosecond electron coherences have been tracked by photoelectron spectra in the xray region, which can allow one to monitor attosecond electron migration in atoms and molecules.[25]The photoelectron diffraction patterns in photoelectron spectra encode information about the molecular intrinsic properties including the orbital symmetry of the molecules. Some researchers investigated the dependence of the geometries and orbital structures on the photoelectron diffraction patterns of the N2and CO2molecules in few-cycle laser pulses.[26–28]Experimentally,the diffraction patterns of O2and N2molecules were reported by Meckel et al. in 2008.[29]For experiments,photoelectron momentum distributions (PMDs) and photoelectron angular distributions(PADs)are usually detected in the laboratory frame.Some researches[30,31]have explored the connection between PADs measured in the molecular frame and PADs measured in the laboratory frame.

Recently,the researches about the molecular ultrafast dynamics have been achieved with the single-electron approximation (SEA) frame,[27,32–34]which has been used to study the ionization dynamics of the multi-electron molecules by intense ultrashort linearly polarized laser pulses. Int the present study, we extend the SEA frame for studying photonionization of the diatomic N2molecules to x-ray/XUV regions. The molecular frame photoelectron momentum distributions(MFPMDs) and molecular frame photoelectron angular distributions (MF-PADs) of N2molecules are investigated in circularly polarized attosecond laser fields.

The paper is organized as follows: In Section 2 we describe the numerical method for solving TDSE of the aligned N2molecules within the SEA frame. The results of numerical calculation and theoretical prediction are discussed in Section 3. In Section 4, we summarize our findings. Atomic units(a.u.) (e=ˉh=me=1)are adopted unless specifically stated.

2. Numerical methods

The electron dynamics of the N2molecules within the static nuclei and SEA frame is investigated by solving the twodimensional(2D)time-dependent Schr¨odinger equations(TDSEs)as follows:

The parameters used in the potential of N2molecules are same as those in Ref.[32].

Fig.1. The HOMO orbital of the N2 molecule.

The second-order split-operator method is used in our calculation to obtain the wavefunction propagation in time,[36,37]

3. Results and discussion

We focus on the photoionization process of the N2molecules to describe the effect of wavelength and rotation of laser field on MF-PMDs and MF-PADs by high-frequency soft x-ray and XUV laser pulses.We only study the MF-PMDs and the MF-PADs around the energy ω ?Ip,and other excited states of the system are neglected.

The MF-PMDs and MF-PADs are presented by a circularly polarized x-ray laser pulse with the wavelength λ =5 nm(ω=9.11 a.u.) in Fig.2. The duration is chosen as T =10T0,and T0=2π/ω. From Fig.2 one sees that the MF-PMDs and MF-PADs are mainly localized along the pxdirection. The corresponding photoelectron momentum is about p=4.1 a.u.,two radiation peaks at angles 0°and 180°with large amplitudes appear,and four radiation peaks at angles n×180°±62°(n=0, 1)with small amplitudes also appear. The photoelectron diffraction occurs in the MF-PMDs and MF-PADs. The photoelectron energy is much larger than the ionization potential of the molecules,i.e.,ω ?Ip,the molecular Coulomb potential has little influence on the electron with larger momentum. Thus, the photoelectron diffractions in the MF-PMDs and MF-PADs are determined by the symmetry of the N2molecule.

Fig.2. Molecular-frame photoelectron diffraction patterns of the N2 molecules at equilibrium internuclear distance RN?N =2.08 a.u. by a circularly polarized laser pulse at λ =5 nm(ω =9.11 a.u.): (a)the MF-PMDs,and(b)the MF-PADs obtained by integrating over energy,(c)theoretical prediction obtained from Eq.(7).

Fig.3. Molecular-frame photoelectron momentum distribution of the N2 molecules by(a)a counter-clockwise and(b)a clockwise circularly polarized laser pulse at λ =30 nm(ω =1.52 a.u.). (c)theoretical prediction obtained from the Eq.(8).

To further understand the rotation direction of the MFPMDs and MF-PADs of N2molecules in the XUV laser field,we present the MF-PMDs and MF-PADs by linearly polarized laser pulses with the wavelength λ =30 nm in Fig.4. The intensity and duration of the laser are the same as those in Fig.3. The MF-PMDs and MF-PADs in the parallel linearly polarized laser pulse for N2molecules are shown in Fig.4(a),four components of the MF-PMDs for N2molecules are distributed in four quadrants respectively, and the intensities of the four components are identical. The distributions cover ranges 28°–79°, 94°–147°, 214°–266°, and 283°–335°. The MF-PADs have four radiation peaks localized at 56°, 123°,236°,303°. The MF-PMDs and MF-PADs of the N2molecule in a perpendicular linearly polarized laser fields are shown in Fig.4(b),the MF-PMDs have six components,and four components with stronger intensity are distributed in four quadrants respectively, the distributions cover the ranges 9°–58°,118°–173°, 188°–239°, and 305°–348°. However two components with weaker intensity are located along the y axis,and the distributions cover the range from 76°to 99°along the upper half of the y axis and the range from 261°to 283°along the bottom half of the y axis. The six radiation peaks of the MF-PADs are localized at 33°, 90°, 146°, 213°, 270°, 328°.The MF-PMDs and MF-PADs of N2molecules by circularly polarized laser pulses with wavelength λ =30 nm are shown in Fig.3,the distributions of the MF-PMDs have four components, and the distributions cover the range from 9°to 96°in the first component,the range from 113°to 165°in the second component, the range from 184°to 275°in the third component, and the range from 296°to 348°in the fourth component. Six radiation peaks of the MF-PADs are localized at 35°, 82°, 140°, 213°, 262°, and 320°. Thus the MF-PMDs for N2molecules in circularly polarized laser fields in Fig.3 are superpositions of the MF-PMDs in both parallel and perpendicular linearly polarized laser fields[40]and the case of the MF-PADs is the same as the MF-PMDs.The rotation direction depends on the helicity of the laser field.[41,42]

Fig.4. The MF-PMDs and MF-PADs of the N2 molecule by linearly polarized laser pulses at λ =30 nm(ω =1.52 a.u.): (a)parallel linear polarization (parallel to the x molecular axis), (b) perpendicular linear polarization(perpendicular to the x molecular axis).

Fig.5. The superpositions of MF-PMDs for the N2 molecule in parallel linearly polarized fields and perpendicular linearly polarized fields: (a)the superpositions for the case of the counter-clockwise circularly polarized laser pulses,(b)the superpositions for the case of the clockwise circularly polarized laser pulses.

4. Conclusions

In summary,we have studied the ultrafast photoionization process of N2molecules by numerically solving 2D TDSE within the frozen-nuclei and SEA frame. The photoelectron momentum and angular distributions of the HOMO electronic state for N2molecules are presented in circularly polarized laser fields. Numerical results show that the wavelengths of the laser pulses have influence on the structure of the MFPMDs and the MF-PADs. Diffraction patterns in the photoelectron distributions are induced by a soft x-ray circularly polarized laser at λ =5 nm,the ultrafast photoionization model can be employed to interpret the MF-PMDs and MF-PADs.[19]The molecular Coulomb potential has little influence on the electron with larger momentum at x-ray high frequency regions. For comparison, we present the MF-PMDs and MFPADs by an XUV laser pulse at λ =30 nm, in this case, the electron with lower momentum is inclined to be influenced by the molecular Coulomb potential in the photoionization process, thus the Coulomb effect cannot be ignored. The angleresolved ionization probability is used to modify the ultrafast photoionization model,the modified ultrafast photoionization model can be employed to interpret the MF-PMDs and MFPADs. It is also found that changing the sign of the helicity will lead to the rotation of the MF-PMDs and MF-PADs,and the rotation direction depends on the helicity of the circularly polarized laser pulses. In addition, we also present the MFPMDs and MF-PADs by both parallel and perpendicular linearly polarized laser pulses with the wavelength λ =30 nm,and the MF-PMDs by the circularly polarized laser pulse are superpositions of the MF-PMDs for the linearly parallel and perpendicular polarized cases.