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Abstract
We comparatively study the effect of orbital symmetry on vortex patterns in photoelectron momentum distributions (PMDs) of perfectly aligned ${{\rm H}_{2}}$ and ${{\rm N}_{2}}$ molecules and their companion atom Ar exposed to a pair of delayed counter-rotating circularly polarized lasers by numerically solving the two-dimensional time-dependent Schrödinger equation. We show that vortex patterns in PMDs strongly depend on the orbital symmetry of atoms and molecules, and numbers of spiral arms in PMDs of ${{\rm N}_{2}}$, ${{\rm H}_{2}}$, and Ar are quite different even though they have nearly identical ionization potentials. We also confirm that vortex structures in PMDs of the highest occupied molecular orbit (HOMO)-2 for ${{\rm N}_{2}}$ are quite different from those of the HOMO but similar to those of Ar. Furthermore, vortex patterns in PMDs of molecules are also sensitive to internuclear distances and alignment angles, which provides more possibilities for controlling the coherent interference of electronic wave packets in comparison with atoms.
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1. INTRODUCTION
Ultrafast polarization shaped femtosecond (fs) pulses provide a robust tool to study quantum coherent control [1–3], which is one of the main goals in physics and chemistry. Furthermore, polarization tailored laser fields have also widely been used for coherently controlling free electron wave packet interference [4,5] and probing the spin–orbit time delay [6] in multiphoton ionization and generating bright circularly polarized high-order harmonics [7–11].
In 2010, Ovchinnikov et al. [12] theoretically demonstrated the possibility for creating and manipulating vortices in the electronic probability density of hydrogen atoms with short electric field pulses. Subsequently, Djiokap et al. [13,14] theoretically predicted vortex interference patterns in photoelectron momentum distributions (PMDs) of helium atoms in time-delayed counter-rotating circularly polarized (CRCP) laser pulses. In 2017, Pengel et al. [15] reported experimentally vortex-shaped PMDs of potassium atoms with a sequence of two CRCP fs laser pulses produced by a polarization shaping technique. In recent years, electron vortices in single photoionization [4,5,16–29] and double photoionization [30] of atoms driven by time-delayed single-color, bichromatic, or multichromatic CRCP laser pulses have been widely investigated. Furthermore, much effort has been made to study theoretically molecular free electron vortices (MFEVs) in single photoionization [31–34] and double photoionization [35,36] by solving the time-dependent Schrödinger equation (TDSE) of molecules by CRCP attosecond (as) or fs laser pulses. For examples, vortex structures in molecular photoelectron distributions (MPMDs) were predicted theoretically in ${\rm H}_{2}^ +$ [31] and ${\rm H}_{3}^{{2} +}$ [32] driven by bichromatic circularly polarized as ultraviolet laser fields and found vortex interference patterns are sensitive to the helicity, time delay, and relative phase among bicircular pulses, molecular geometry, and alignment angles. Then, Guo et al. [33] studied theoretically electron vortices in photoionization of ${{\rm N}_{2}}$ in single-color counter-rotating circularly as pulses by solving the two-dimensional TDSE (2D-TDSE) and found the number of spiral arms in vortex patterns is independent of time delays and carrier envelope phases of these two as pulses. Very recently, Bayer and Wollenhaupt [34] applied the 2D-TDSE model to investigate the influence of the coupled electron–nuclear dynamics on vortex formation dynamics and explored the potential of MFEVs for ultrafast spectroscopy. So far, MFEVs in photoionization have not been reported experimentally.
Many strong-field phenomena of diatomic molecules were compared with those of their companion atoms because they have nearly identical ionization potentials, especially for ionization suppression effects in tunneling ionization [37–41], double ionization [42,43], and high-order harmonic generation [44–46]. In this paper, we comparatively study how orbital symmetry affects vortex structures in PMDs of ${{\rm H}_{2}}$ and ${{\rm N}_{2}}$ molecules and their companion atom Ar in time-delayed CRCP fs laser pulses by solving the 2D-TDSE based on the single-active-electron approximation.
The paper is organized as follows: we first briefly introduce how to solve the 2D-TDSE of atoms and molecules in Section 2. Then, we systematically discuss how vortex structures in PMDs depend on the orbital symmetry of atoms and molecules in Section 3. Finally, a summary is given in Section 4. Atomic units are used throughout this paper unless stated otherwise.
2. THEORETICAL METHODS
Based on the dipole approximation and length gauge, the 2D-TDSE of atoms and molecules in CRCP laser pulses can be written as
where $\hat H(x,y,t) = {\hat H_0}(x,y) + {\hat H_{\rm{int}}}(x,y,t)$, and the field-free Hamiltonian ${\hat H_0}(x,y) = \hat T(x,y) + {V_c}(x,y)$. The kinetic operator is expressed asFor the Ar atom, the soft-core Coulomb potential can be given by [27]
where $Z(x,y) = 1 + 17{\rm exp}[- ({x^2} + {y^2})]$, and the soft-core parameter $a = 12.159$, which can give the ionization potential of 15.76 eV.The soft-core Coulomb potential of ${{\rm H}_{2}}$ and ${{\rm N}_{2}}$ molecules can be written as [47,48]
Table 1. Values of All Parameters Used in Soft-Core Coulomb Potentials for ${{\rm H}_{2}}$ and ${{\rm N}_{2}}$ at Several Internuclear Distancesa
The term ${\hat H_{\rm{int}}}(x,y,t)$ is the laser–electron interaction in the length gauge, which can be written as
with the right $(+)$ or left $(-)$ handed circularly polarized laser field [27]:After the laser field is finished, wave functions are further propagated for an additional eight optical cycles to ensure that all the ionized components are away from the core; we then filter out the wave function in the range $\sqrt {{x^2} + {y^2}} \lt 60\,{\rm a.u}.$, which is regarded as the unionized wave packet. Finally, the ionized wave packet can be obtained by ${\psi _{\rm{ion}}}(x,y) = [(1 - M({r_b})]{\psi _{\rm{final}}}(x,y)$. Here, ${\psi _{\rm{final}}}(x,y)$ is the wave packet at the final time, and $M({r_b})$ is a mask function:
In this paper, we use $\alpha = 1\,{\rm a. u}.$ and ${r_b} = 60\,{\rm a. u}.$ Wave functions in momentum space can be obtained by applying the Fourier transformation to the ionized wave function as
The PMD is calculated by
3. RESULTS AND DISCUSSION
A. Comparison of PMDs in Single-Photon and Two-Photon Ionizations of ${{\textbf H}_{\textbf 2}}$ and ${{\textbf N}_{\textbf 2}}$ at Equilibrium Distance and Their Companion Atom Ar
Figures 1(a)–1(c) show initial wave functions of the highest occupied molecular orbit (HOMO) of ${{\rm H}_{2}}$ and ${{\rm N}_{2}}$ together with their companion atom Ar. The present calculated ionization potentials are 15.5 eV, 15.6 eV, and 15.76 eV for ${{\rm H}_{2}}$, ${{\rm N}_{2}}$, and Ar, respectively. We can see that orbital symmetries of these three systems are quite different even though they have nearly identical ionization potentials. Since the generation of electron vortices comes from the coherent interference of time-delayed electronic wave packets, one can expect that vortex patterns in PMDs should sensitively depend on the orbital symmetry of initial wave functions of atoms and molecules. Figures 1(d)–1(i) compare the PMDs resulting from photoionization of ${{\rm H}_{2}}$ and ${{\rm N}_{2}}$ and their companion atom Ar with a pair of time-delayed CRCP laser pulses. In Figs. 1(d) and 1(g), we can see that the number of spiral arms in a vortex pattern is four for single-photon ionization and six for two-photon ionization, namely, the number of spiral arms that satisfies ${{ C}_{{2n} + {2}}}$ ($n$ is the number of absorbed photons) for Ar atoms. Furthermore, the active electron preferably ionizes along the direction parallel (perpendicular) to the largest electron density distributions for single- (two)-photon ionization processes. For ${{\rm H}_{2}}$ molecules, the number of spiral arms is two (four) for single- (two)-photon ionization, which meets ${{ C}_{{2n}}}$ rotational symmetry as shown in Figs. 1(e) and 1(h), which is quite similar to the case of the 1s orbit of hydrogen atoms [19]. For ${{\rm N}_{2}}$ molecules, the situation is quite different, that is, the number of spiral arms does not depend on the number of absorbed photons [see Figs. 1(f) and 1(i)], which agrees well with the conclusion in Ref. [33]. Clearly, orbital symmetries are imprinted in the PMDs of atoms and molecules. To check our PMDs of diatomic molecules, we also present the photoelectron angular distributions (PADs) obtained from the attosecond perturbation ionization theory (APIT) [33,50]. We can see that our PMDs calculated by the TDSE method agree well with those PADs from the APIT.
Fig. 1. Initial wave functions of (a) Ar, (b) the HOMO of ${{\rm H}_{2}}$ at equilibrium distance, and (c) the HOMO of ${{\rm N}_{2}}$ at its equilibrium distance. Corresponding PMDs and PADs of (e), (h) ${{\rm H}_{2}}$ and (f), (i) ${{\rm N}_{2}}$ and (d), (g) their companion atom Ar by a pair of CRCP laser pulses at different wavelengths of 30 nm (single-photon ionization, middle row) and 100 nm (two-photon ionization, bottom row) and time delay ${T_d} = 4\,{\rm o. c}.$ The PADs of ${{\rm H}_{2}}$ and ${{\rm N}_{2}}$ are obtained from the attosecond perturbation ionization theory.
B. Effect of Internuclear Distance on Vortex Structures in PMDs of ${{\textbf H}_{\textbf 2}}$ and ${{\textbf N}_{\textbf 2}}$
In Ref. [34], Bayer and Wollenhaupt found that vortex structures in PMDs of the rigid singly charged potassium dimer ${\rm K}_{2}^ +$ by a CRCP laser field depend on molecular internuclear distances. Figure 2 presents initial wave functions of the HOMO of ${{\rm H}_{2}}$ and corresponding MPMDs in single- (two)-photon ionization processes at three different internuclear distances. Using the molecular potential parameters tabulated in Table 1, the calculated ionization potentials of ${{\rm H}_{2}}$ are 10.07 eV and 7.62 eV for internuclear distances of 4 a.u. and 8 a.u., respectively. By stretching the ${{\rm H}_{2}}$ molecule from ${R} = {1.4}\;{\rm a.u}.$ to ${\rm R} = {8}\;{\rm a.u}.$, we can clearly see that vortex patterns sensitively depend on molecular internuclear distances, and the numbers of spiral arms change gradually from ${{C}_{{2n}}}$ to ${{C}_{{2n} + {2}}}$ for single- (two)-photon ionization. This is because spherical symmetries are further broken as internuclear distance increases. Furthermore, as can be seen, the spatial distribution of initial wave functions of ${{\rm H}_{2}}$ at ${R} = {8}\;{\rm a.u}.$ has a dumbbell shape, which is very similar to that of Ar. As a result, the number of spiral arms in vortex patterns of ${{\rm H}_{2}}$ at ${R} = {8}\;{\rm a.u}.$ is exactly the same as that of Ar. This is clear evidence that vortex interference patterns depend sensitively on the orbital symmetry of atoms and molecules.
Fig. 2. Initial wave functions of the HOMO of ${{\rm H}_{2}}$ at three different internuclear distances of (a) 1.4 a.u., (b) 4 a.u., and (c) 8 a.u. Corresponding MPMDs resulting from single-photon ionization at laser wavelengths of (d) 30 nm, (e) 100 nm, and (f) 120 nm, and two-photon ionization at laser wavelengths of (g) 100 nm, (h) 140 nm, and (i) 170 nm. Time delay ${T_d} = 4\,{\rm o.c}.$ is used.
Figure 3 shows the initial wave functions of the HOMO of ${{\rm N}_{2}}$ and corresponding MPMDs resulting from single- (two)-photon ionization at three different internuclear distances. With molecular potential parameters tabulated in Table 1, our calculated ionization potentials of ${{\rm N}_{2}}$ are 10.07 eV and 8.71 eV for internuclear distances of 4 a.u. and 6 a.u., respectively. It is well known that a molecular orbit can be constructed by a linear combination of atomic orbits (LCAO) based on molecular orbit theory [51,52] and the HOMO (i.e., $3{\sigma _g}$) of ${{\rm N}_{2}}$ can be considered simply as the combination of $2{p_0}$ orbits of two ${\rm N}$ atoms if orbital hybridization is ignored. It can be clearly seen that the molecular orbit becomes more elongated along the molecular axis as internuclear distance increases, and the $3{\sigma _g}$ orbit of ${{\rm N}_{2}}$ is combined from two almost separated $2{p_0}$ orbits of ${\rm N}$ atoms at ${R} = {6}\;{\rm a.u}.$ [see Figs. 3(a)–3(c)]. In Figs. 3(d)–3(i), we can see that vortex structures in MPMDs depend on internuclear distances, and the number of spiral arms varies from four to six as internuclear distance increases for both single-photon and two-photon ionizations, which can be well understood from the remarkable change of molecular orbital symmetry.
Fig. 3. Initial wave functions of the HOMO of ${{\rm N}_{2}}$ at three different internuclear distances of (a) 2.08 a.u., (b) 4 a.u., and (c) 6 a.u. Corresponding MPMDs resulting from single-photon ionization at laser wavelengths of (d) 30 nm, (e) 60 nm, and (f) 80 nm, and those in two-photon ionization at laser wavelengths of (g) 100 nm, (h) 150 nm, and (i) 180 nm. Time delay ${T_d} = 4\,{\rm o.c}.$ is used in the calculations.
C. Effect of Alignment Angle on Vortex Structures in PMDs of ${{\textbf H}_{\textbf 2}}$ and ${{\textbf N}_{\textbf 2}}$
In Ref. [34], Bayer and Wollenhaupt demonstrated that vortex structures in PMDs of the dimer ${\rm K}_{2}^ +$ by a bichromatic CRCP laser field depend on molecular alignment angles and the number of spiral arms changes from five for the aligned molecule at $\theta {= 0^ \circ}$ to three for the isotropic ensemble of molecules. To study the effect of alignment angles on vortex patterns in MPMDs of diatomic molecules, we can fix the laser field and rotate the molecule instead. In Figs. 4(a)–4(d), we exhibit initial wave functions of the HOMO of ${{\rm H}_{2}}$ at four different alignment angles and find that the spatial distribution of wave functions is rotated as expected. In Figs. 4(e)–4(l), one can see that the number of spiral arms in vortex patterns does not depend on alignment angles for either single-photon or two-photon ionization. Furthermore, the MPMDs in single-photon ionization rotate as alignment angle increases, and the electron prefers to ionize along the direction perpendicular to the molecular axis [see Figs. 4(e)–4(h)], which is very similar to the conclusion for ${\rm H}_{2}^ +$ [48]. However, the situation is different for the two-photon ionization case; MPMDs have a slight rotation, and probabilities on spiral arms depend on alignment angles [see Figs. 4(i)–4(l)].
Fig. 4. Initial wave functions of the HOMO of ${{\rm H}_{2}}$ at four different alignment angles of (a) 0°, (b) 30°, (c) 60°, and (d) 90°. Corresponding PMDs resulting from single-photon ionization ($\lambda = 30\;{\rm nm}$) at different alignment angles of (e) 0°, (f) 30°, (g) 60°, and (h) 90°. PMDs resulting from two-photon ionization ($\lambda = 100\;{\rm nm}$) at different alignment angles of (i) 0°, (j) 30°, (k) 60°, and (l) 90°. Here, $R = {1.4}\;{\rm a.\rm u}.$ and ${T_d} = 4\,{\rm o. c}.$ are used in the simulations.
Figures 5(a)–5(d) show initial wave functions of the HOMO of ${{\rm N}_{2}}$ at four alignment angles. In Figs. 5(e)–5(h), we can clearly see that vortex structures in the single-photon ionization of ${{\rm N}_{2}}$ strongly depend on alignment angles; the number of spiral arms remains at four, and two broken spiral arms show up as the alignment angle increases. Furthermore, the MPMDs rotate as alignment angle increases. However, vortex structures in MPMDs for two-photon ionization of ${{\rm N}_{2}}$ have a weaker dependence on alignment angles, and the number of spiral arms remains at four.
Fig. 5. Initial wave functions of the HOMO of ${{\rm N}_{2}}$ at four different alignment angles of (a) 0°, (b) 30°, (c) 60°, and (d) 90°. Corresponding PMDs resulting from single-photon ionization ($\lambda = 30\;{\rm nm}$) at different alignment angles of (e) 0°, (f) 30°, (g) 60°, and (h) 90°. PMDs resulting from two-photon ionization ($\lambda = 100\;{\rm nm}$) at different alignment angles of (i) 0°, (j) 30°, (k) 60°, and (l) 90°. Here, $R = {2.08}\;{\rm a. u}.$ and ${T_d} = 4\,{\rm o. c}.$ are used in the calculations.
Fig. 6. Initial wave functions of (a) Ar and the HOMO-2 of ${{\rm N}_{2}}$ at four different alignment angles of (b) 0°, (c) 30°, (d) 60°, and (e) 90°. Corresponding PMDs resulting from single-photon ionization ($\lambda = 30\;{\rm nm}$) of (f) Ar and the HOMO-2 for ${{\rm N}_{2}}$ at different alignment angles of (g) 0°, (h) 30°, (i) 60°, and (j) 90°. PMDs resulting from two-photon ionization ($\lambda = 100\;{\rm nm}$) of (k) Ar and the HOMO-2 for ${{\rm N}_{2}}$ at different alignment angles of (l) 0°, (m) 30°, (n) 60°, and (o) 90°. Time delay ${T_d} = 6\,{\rm o.c}.$ is used.
D. Comparison of PMDs of HOMO-2 for ${{\textbf N}_{\textbf 2}}$ at Equilibrium Distance and Ar
Tunneling ionization from inner orbits of molecules by lasers has been studied experimentally and theoretically [53–57]. In Figs. 6(b)–6(e), we can see that spatial distributions of the HOMO-2 (i.e., $2{\sigma _u}$) of ${{\rm N}_{2}}$ at several alignment angles exhibit a dumbbell shape, which is very similar to that of Ar [see Fig. 6(a)]. Therefore, it is very interesting to compare vortex patterns in PMDs of the HOMO-2 for ${{\rm N}_{2}}$ at several alignment angles and Ar even if they have different ionization potentials. The calculated ionization potential is 18.70 eV for the HOMO-2 of ${{\rm N}_{2}}$ with the soft-core parameter ${a_j}$ of 2.60, and other potential parameters are kept the same as those of the HOMO. As can be seen in Figs. 6(f)–6(o), the number of spiral arms in PMDs is four (six) in single- (two)-photon ionization for both the HOMO-2 of ${{\rm N}_{2}}$ at several alignment angles and Ar. Furthermore, the electrons in both the HOMO-2 of ${{\rm N}_{2}}$ and Ar preferably ionize along the direction parallel to the largest electron density distributions for single-photon ionization, which agrees well with Fig. 1(a) in Ref. [58].
4. CONCLUSION
In summary, we comparatively studied the effect of orbital symmetry on vortex patterns in PMDs of the perfectly aligned ${{\rm H}_{2}}$ and ${{\rm N}_{2}}$ molecules and their companion atom Ar driven by a pair of delayed CRCP laser pulses by numerically solving the 2D-TDSE. First, we found that vortex patterns in PMDs strongly depend on the orbital symmetries of atoms and molecules, and the numbers of spiral arms in PMDs of ${{\rm N}_{2}}$, ${{\rm H}_{2}}$, and Ar are quite different even though they have nearly identical ionization potentials. Then, we investigated how vortex patterns in PMDs depend on orbital symmetries by artificially stretching or rotating ${{\rm H}_{2}}$ and ${{\rm N}_{2}}$ and confirmed that vortex structures in PMDs are sensitive to molecular internuclear distances and alignment angles. Finally, we further explored vortex patterns in PMDs of the HOMO-2 for ${{\rm N}_{2}}$ at different alignment angles and demonstrated that vortex structures of the HOMO-2 are quite different from those of the HOMO but similar to those of Ar. Therefore, we conclude that vortex structures in PMDs strongly depend on the orbital symmetry of atoms and molecules. In the future, we plan to study electron vortices in vibrating molecules by polarization shaped laser pulses. Hopefully, MFEVs can also be experimentally measured and used for probing molecular structures and dynamics in the near future.
Funding
National Natural Science Foundation of China (12164044, 11765018, 11864037); Natural Science Basic Research Program of Shaanxi Province (2022JM-015).
Acknowledgment
We thank Prof. Jing Guo of Jilin University for providing the photoelectron angular distributions of H2 and N2 obtained from the attosecond perturbation ionization theory.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
REFERENCES
1. K. Misawa, “Applications of polarization-shaped femtosecond laser pulses,” Adv. Phys. X 1, 544–569 (2016). [CrossRef]
2. H. X. Qi, Z. Z. Lian, D. H. Fei, Z. Chen, and Z. Hu, “Manipulation of matter with shaped-pulse light field and its applications,” Adv. Phys. X 6, 194390 (2021). [CrossRef]
3. T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 288301 (2004). [CrossRef]
4. S. Kerbastadt, K. Eickhoff, T. Bayer, and M. Wollenhaupt, “Control of free electron wave packets by polarization-tailored ultrashort bichromatic laser fields,” Adv. Phys. X 4, 1672583 (2019). [CrossRef]
5. K. Eickhoff, L. Englert, T. Bayer, and M. Wollenhaupt, “Multichromatic polarization-controlled pulse sequences for coherent control of multiphoton ionization,” Front. Phys. 9, 675258 (2021). [CrossRef]
6. P. P. Ge, Y. Q. Fang, Z. N. Guo, X. Y. Ma, X. Y. Yu, M. Han, C. Y. Wu, Q. H. Gong, and Y. Q. Liu, “Probing the spin-orbit time delay of multiphoton ionization of Kr by bicircular fields,” Phys. Rev. Lett. 126, 223001 (2021). [CrossRef]
7. I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. J. Cho, and C. H. Nam, “Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field,” Phys. Rev. Lett. 94, 243901 (2005). [CrossRef]
8. G. Lambert, B. Vodungbo, J. Gautier, B. Mahieu, V. Malka, S. Sebban, P. Zeitoun, J. Luning, J. Perron, A. Andreev, S. Stremoukhov, F. Ardana-Lamas, A. Dax, C. P. Hauri, A. Sardinha, and M. Fajardo, “Towards enabling femtosecond helicity-dependent spectroscopy with high-harmonic sources,” Nat. Commun. 6, 6167 (2015). [CrossRef]
9. O. Kfir, P. Grychtol, E. Turgut, R. Knut, D. Zusin, D. Popmintchev, T. Popmintchev, H. Nembach, J. M. Shaw, A. Fleischer, H. Kapteyn, M. Murnane, and O. Cohen, “Generation of bright phase-matched circularly-polarized extreme ultraviolet high harmonics,” Nat. Photonics 9, 99–105 (2014). [CrossRef]
10. T. T. Fan, P. Grychtol, R. Knut, C. Hernández-García, M. M. Murnane, and H. C. Kapteyn, “Bright circularly polarized soft x-ray high harmonics for x-ray magnetic circular dichroism,” Proc. Natl. Acad. Sci. USA 112, 14206–14211 (2015). [CrossRef]
11. K. M. Dorney, J. L. Ellis, C. Hernández-García, D. D. Hickstein, C. A. Mancuso, N. Brooks, T. T. Fan, G. Y. Fan, D. Zusin, C. Gentry, P. Grychtol, H. C. Kapteyn, and M. M. Murnane, “Helicity-selective enhancement and polarization control of attosecond high harmonic waveforms driven by bichromatic circularly polarized laser fields,” Phys. Rev. Lett. 119, 063201 (2017). [CrossRef]
12. S. Y. Ovchinnikov, J. Sternberg, J. Macek, T.-G. Lee, and D. R. Schultz, “Creating and manipulating vortices in atomic wave functions with short electric field pulses,” Phys. Rev. Lett. 105, 203005 (2010). [CrossRef]
13. J. M. Ngoko Djiokap, S. X. Hu, L. B. Madsen, N. L. Manakov, A. V. Meremianin, and A. F. Starace, “Electron vortices in photoionization by circularly polarized attosecond pulses,” Phys. Rev. Lett. 115, 113004 (2015). [CrossRef]
14. J. M. Ngoko Djiokap, A. V. Meremianin, N. L. Manakov, S. X. Hu, L. B. Madsen, and A. F. Starace, “Multistart spiral electron vortices in ionization by circularly polarized UV pulses,” Phys. Rev. A 94, 013408 (2016). [CrossRef]
15. D. Pengel, S. Kerbstadt, D. Johannmeyer, L. Englert, T. Bayer, and M. Wollenhaupt, “Electron vortices in femtosecond multiphoton ionization,” Phys. Rev. Lett. 118, 053003 (2017). [CrossRef]
16. J. M. Ngoko Djiokap, A. V. Meremianin, N. L. Manakov, S. X. Hu, L. B. Madsen, and A. F. Starace, “Kinematical vortices in double photoionization of helium by attosecond pulses,” Phys. Rev. A 96, 013405 (2017). [CrossRef]
17. L. Geng, F. C. Vélez, J. Z. Kamiński, L. Y. Peng, and K. Krajewska, “Vortex structures in photodetachment by few-cycle circularly polarized pulses,” Phys. Rev. A 102, 043117 (2020). [CrossRef]
18. L. Geng, F. C. Vélez, J. Z. Kamiński, L. Y. Peng, and K. Krajewska, “Structured photoelectron distributions in photodetachment induced by trains of laser pulses: vortices versus spirals,” Phys. Rev. A 104, 033111 (2021). [CrossRef]
19. M. Li, G. Z. Zhang, X. L. Kong, T. Q. Wang, X. Ding, and J. Q. Yao, “Dynamic stark induced vortex momentum of hydrogen in circular fields,” Opt. Express 26, 878–886 (2018). [CrossRef]
20. M. Li, G. Z. Zhang, X. Ding, and J. Q. Yao, “Carrier envelope phase description for an isolated attosecond pulse by momentum vortices,” Chin. Phys. Lett. 36, 063201 (2019). [CrossRef]
21. Q. Zhen, H. D. Zhang, S. Q. Zhang, L. Ji, T. Han, and X. S. Liu, “Generation of electron vortices in photoionization by counter-rotating circularly polarized attosecond pulses,” Chem. Phys. Lett. 738, 136885 (2020). [CrossRef]
22. Q. Zhen, J. H. Chen, S. Q. Zhang, Z. J. Yang, and X. S. Liu, “Effects of initial electronic state on vortex patterns in counter-rotating circularly polarized attosecond pulses,” Chin. Phys. B 30, 024203 (2021). [CrossRef]
23. L. Ji, W. W. Yu, S. Y. Han, Y. F. Han, Q. Zhen, S. Q. Zhang, and X. S. Liu, “Trajectories of electron vortices in photoionization by bichromatic co-rotating circularly polarized laser fields,” Chem. Phys. Lett. 754, 137634 (2020). [CrossRef]
24. J. C. Jia, H. F. Cui, C. P. Zhang, J. Shao, J. Ma, and X. Y. Miao, “Investigation of the photoionization process of helium ion in bichromatic circularly XUV fields with different time delays,” Chem. Phys. Lett. 725, 119–123 (2019). [CrossRef]
25. H. F. Cui and X. Y. Miao, “Photoelectron momentum distributions of single-photon ionization under a pair of elliptically polarized attosecond laser pulses,” Chin. Phys. B 29, 074203 (2020). [CrossRef]
26. M. Y. Ma, J. P. Wang, W. Q. Jing, Z. Guan, Z. H. Jiao, G. L. Wang, J. H. Chen, and S. F. Zhao, “Controlling the atomic-orbital-resolved photoionization for neon atoms by counter-rotating circularly polarized attosecond pulses,” Opt. Express 29, 33245–33256 (2021). [CrossRef]
27. I. Barth and M. Lein, “Numerical verification of the theory of nonadiabatic tunnel ionization in strong circularly polarized laser fields,” J. Phys. B 47, 204016 (2014). [CrossRef]
28. I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields. II. Derivation of formulas,” Phys. Rev. A 87, 013433 (2013). [CrossRef]
29. R. R. Wang, M. Y. Ma, J. P. Wang, Z. Guan, and S. F. Zhao, “Orbital-resolved photoelectron momentum distributions of neon atoms in bichromatic elliptically polarized attosecond pulses,” Eur. Phys. J. D 76, 145 (2022). [CrossRef]
30. J. M. N. Djiokap and A. F. Starace, “Doubly-excited state effects on two-photon double ionization of helium by time-delayed, oppositely circularly-polarized attosecond pulses,” J. Opt. 19, 124003 (2017). [CrossRef]
31. K. J. Yuan, S. Chelkowski, and A. D. Bandrauk, “Photoelectron momentum distributions of molecules in bichromatic circularly polarized attosecond UV laser fields,” Phys. Rev. A 93, 053425 (2016). [CrossRef]
32. K. J. Yuan, H. Z. Lu, and A. D. Bandrauk, “Photoionization of triatomic molecular ions $H_3^{2 +}$ by intense bichromatic circularly polarized attosecond UV laser pulses,” J. Phys. B 50, 124004 (2017). [CrossRef]
33. J. Guo, S. Q. Zhang, J. Zhang, S. P. Zhou, and P. F. Guan, “Exploration of electron vortices in the photoionization of diatomic molecules in intense laser fields,” Laser Phys. 31, 065301 (2021). [CrossRef]
34. T. Bayer and M. Wollenhaupt, “Molecular free electron vortices in photoionization by polarization-tailored ultrashort laser pulses,” Front. Chem. 10, 899461 (2022). [CrossRef]
35. J. M. N. Djiokap, A. V. Meremianin, N. L. Manakov, L. B. Madsen, S. X. Hu, and A. F. Starace, “Dynamical electron vortices in attosecond double photoionization of H2,” Phys. Rev. A 98, 063407 (2018). [CrossRef]
36. J. M. N. Djiokap and A. F. Starace, “Temporal coherent control of resonant two-photon double ionization of the hydrogen molecule via doubly excited states,” Phys. Rev. A 103, 053110 (2021). [CrossRef]
37. X. M. Tong, Z. X. Zhao, and C. D. Lin, “Theory of molecular tunneling ionization,” Phys. Rev. A 66, 033402 (2002). [CrossRef]
38. E. P. Benis, J. F. Xia, X. M. Tong, M. Faheem, M. Zamkov, B. Shan, P. Richard, and Z. Chang, “Ionization suppression of Cl2 molecules in intense laser fields,” Phys. Rev. A 70, 025401 (2004). [CrossRef]
39. Z. Y. Lin, X. Y. Jia, C. L. Wang, Z. L. Hu, H. P. Kang, W. Quan, X. Y. Lai, X. J. Liu, J. Chen, B. Zeng, W. Chu, J. P. Yao, Y. Cheng, and Z. Z. Xu, “Ionization suppression of diatomic molecules in an intense midinfrared laser field,” Phys. Rev. Lett. 108, 223001 (2012). [CrossRef]
40. H. P. Kang, Z. Y. Lin, S. P. Xu, C. L. Wang, W. Quan, X. Y. Lai, X. J. Liu, X. Y. Jia, Y. Cheng, and Z. Z. Xu, “Wavelength-dependent ionization suppression of diatomic molecules in intense circularly polarized laser fields,” Phys. Rev. A 90, 063426 (2014). [CrossRef]
41. X. H. Ren, J. Yang, Y. F. Niu, and J. T. Zhang, “Ionization suppression of diatomic molecules in different wavelength laser fields,” Opt. Commun. 329, 140–144 (2014). [CrossRef]
42. X. Y. Jia, W. D. Li, J. Fan, J. Liu, and J. Chen, “Suppression effect in the nonsequential double ionization of molecules by an intense laser field,” Phys. Rev. A 77, 063407 (2008). [CrossRef]
43. K. Lin, X. Y. Jia, Z. Q. Yu, F. He, J. Y. Ma, H. Li, X. C. Gong, Q. Y. Song, Q. Y. Ji, W. B. Zhang, J. Chen, and J. Wu, “Comparison study of strong-field ionization of molecules and atoms by bicircular two-color femtosecond laser pulses,” Phys. Rev. Lett. 119, 203202 (2017). [CrossRef]
44. B. Shan, X. M. Tong, Z. X. Zhao, Z. H. Chang, and C. D. Lin, “High-order harmonic cutoff extension of the O2 molecule due to ionization suppression,” Phys. Rev. A 66, 061401 (2002). [CrossRef]
45. G. H. Li, J. P. Yao, H. S. Zhang, C. R. Jing, B. Zeng, W. Chu, J. L. Ni, H. Q. Xie, X. J. Liu, J. Chen, Y. Cheng, and Z. Z. Xu, “Influence of ionization suppression on high-harmonic generation in molecules: dependence of cutoff energy on driver wavelength,” Phys. Rev. A 88, 043401 (2013). [CrossRef]
46. B. Y. Wang, Y. Wu, and S. F. Zhao, “Influence of the alignment angle of molecules on the cutoff of the high-order harmonics,” Optik 180, 733–737 (2019). [CrossRef]
47. M. Peters, T. T. Nguyen-Dang, E. Charron, A. Keller, and O. Atabek, “Laser-induced electron diffraction: a tool for molecular orbital imaging,” Phys. Rev. A 85, 053417 (2012). [CrossRef]
48. H. D. Zhang, S. Q. Zhang, L. Ji, Q. Zhen, J. Guo, and X. S. Liu, “Dependence of photoelectron-momentum distribution of H+2 molecule on orientation angle and laser ellipticity,” Chin. Phys. B 28, 053201 (2019). [CrossRef]
49. M. D. Feit, J. A. Fleck Jr., and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982). [CrossRef]
50. T. Zuo, A. D. Bandrauk, and P. B. Corkum, “Laser-induced electron diffraction: a new tool for probing ultrafast molecular dynamics,” Chem. Phys. Lett. 259, 313–320 (1996). [CrossRef]
51. V. F. Hund, “Zur dentung der molekelspektren. IV,” Z. Phys. 51, 759 (1928). [CrossRef]
52. R. S. Mulliken, “The assignment of quantum numbers for electrons in molecules. I,” Phys. Rev. 32, 186–222 (1928). [CrossRef]
53. H. Akagi, T. Otobe, A. Staudte, A. Shiner, F. Turner, R. Dörner, D. M. Villeneuve, and P. B. Corkum, “Laser tunnel ionization from multiple orbitals in HCl,” Science 325, 1364–1367 (2009). [CrossRef]
54. H. Chen, V. Tagliamonti, and G. N. Gibson, “Enhanced ionization of an inner orbital of I2 by strong laser fields,” Phys. Rev. A 86, 051403 (2012). [CrossRef]
55. H. Liu, S. F. Zhao, M. Li, Y. K. Deng, C. Y. Wu, X. X. Zhou, Q. H. Gong, and Y. Q. Liu, “Molecular-frame photoelectron angular distributions of strong-field tunneling from inner orbitals,” Phys. Rev. A 88, 061401 (2013). [CrossRef]
56. S. F. Zhao, C. Jin, A. T. Le, T. F. Jiang, and C. D. Lin, “Determination of structure parameters in strong-field tunneling ionization theory of molecules,” Phys. Rev. A 81, 033423 (2010). [CrossRef]
57. J. P. Wang, S. F. Zhao, C. R. Zhang, W. Li, and X. X. Zhou, “Determination of structure parameters in molecular tunnelling ionisation model,” Mol. Phys. 112, 1102–1114 (2014). [CrossRef]
58. C. Jin, A. T. Le, S. F. Zhao, R. R. Lucchese, and C. D. Lin, “Theoretical study of photoelectron angular distributions in single-photon ionization of aligned N2 and CO2,” Phys. Rev. A 81, 033421 (2010). [CrossRef]
