## Abstract

Using three-dimensional classical ensembles, we have investigated the internuclear distance dependence of nonsequential double ionization (NSDI) of *H*
_{2} molecules by an 800 nm, 1×10^{14} W/cm^{2} laser pulse. For the internuclear distance R ranging from 2 to 12 a.u., the NSDI of *H*
_{2} provides rich correlation patterns in the two-electron momentum distributions. These correlation patterns essentially reveal different microscopic dynamics in NSDI process. Moreover, our calculations show that R ≈ 4 a.u. is the critical distance for double ionization yield of *H*
_{2}. These results are qualitatively explained based on the classical barrier expression model and back analysis.

©2010 Optical Society of America

Nonsequential double ionization (NSDI) is introduced since the observation of unexpectedly high double ionization probabilities of atoms which were inconsistent with sequential ionization [1]. Nowadays, it is widely accepted that rescattering model is the dominant mechanism for both atomic [2, 3] and molecular [4, 5, 6] NSDI processes. Compared with the case of atoms, molecules possess an additional vibrational coordinate (internuclear distance R), this additional coordinate should also exert influence on NSDI of molecules and make molecular NSDI process more complex to be investigated.

As the simplest molecules, double ionization (DI) of *H*
_{2} (*D*
_{2}) has been intensively investigated. It has been found that the mechanism for molecular ionization is not only dependent on laser intensity but also related to R. When R is stretched to several times of the equilibrium internuclear distances, the ionization rate of the molecular fragments is significantly enhanced. This anomalously high ionization of the molecular fragments was referred to as charge resonance enhanced ionization [7], which belongs to the sequential ionization regime [8]. Both experimental [9, 10] and theoretical studies [7, 11, 12] have revealed the above described enhanced ionization. On the other hand, short range double ionization of molecules is much less probable and includes ionization paths related to NSDI of molecules [5, 14].

In this paper, with the three-dimensional (3D) classical ensemble model, we investigate the R-dependence of electron correlation in NSDI of *H*
_{2} driven in a linearly polarized laser field. The two-electron momentum distributions, strongly dependent on the internuclear distance, exhibit rich correlated patterns. In addition, DI yield is also strongly dependent on R. When R ≈ 4 a.u., DI yield reaches the maximum. Qualitative analysis based on the classical barrier expression model [15] and back analysis reveals that this R-dependence of electron correlation and double ionization yield originates from the R-dependent potential well, the R-dependent ionization potential of the second electron and the R-dependent rescattering probability of the first ionized electron.

The 3D classical ensemble model has been described in detail in [16]. This model has been employed to study NSDI extensively [17, 18, 19, 20, 21]. For *H*
_{2}, the electron-nuclear interaction and the electron-electron interaction are represented by a two-center 3D soft-Coulomb potential (in atomic units) *V*(*r⃗*
_{1},*r⃗*
_{2}) = −1/((*r⃗*
_{1} − *R⃗*/2)^{2} + *a*
^{2})^{1/2} − 1/((*r⃗*
_{1} + *R⃗*/2)^{2} + *a*
^{2})^{1/2} − 1/((*r⃗*
_{2}−*R⃗*/2)^{2} +*a*
^{2})^{1/2}−1/((*r⃗*
_{2}+*R⃗*/2)^{2}+*a*
^{2})^{1/2}+1/((*r⃗*
_{1}−*r⃗*
_{2})2+*b*
^{2})^{1/2},*r⃗*
_{1},*r⃗*
_{2} represent the electronic coordinates, and *R⃗* is the internuclear vector. *a*, *b* are shielding parameters for the attraction of *H*
^{+} and the electron-electron interaction respectively. The evolution of the two electrons are governed by Newton’s equations of motion: $\frac{{d}^{2}{\overrightarrow{r}}_{i}}{d{t}^{2}}$ = −**E**(*t*)−Δ_{r⃗i}
*V*(*r⃗*
_{1},*r⃗*
_{1}), with *i* = 1,2 denoting different electrons. **E**(*t*) is a linearly polarized laser field. Without loss of generality the molecule is assumed to lie in the *x*−*z* plane. In present calculations, the two nuclei are fixed at (-Rsin*ϕ*, 0, -Rcos*ϕ*) and (Rsin*ϕ*, 0, Rcos*ϕ*), respectively, where *ϕ* is the angle between the molecular axis and the *z* axis. The electric field **E**(*t*) is a 800 nm linearly polarized laser pulse with a polarization direction along the z axis and a total duration of 10 optical cycles, two-cycle turn on, six cycles at full strength, and two-cycle turn off.

For *H*
_{2} driven in an intense laser pulse, NSDI is dominated at laser intensities below 2×10^{14} W/cm^{2}, and sequential double ionization (SDI) is dominated at higher intensities [13, 14]. Based on our concerns of the R dependence of electron correlation in NSDI of *H*
_{2}, the laser
intensity employed in our simulation is 1×10^{14} W/cm^{2}. For this intensity, NSDI dominates double ionization of *H*
_{2} and the NSDI yield is significant. While for lower intensities, double ionization yield is much low. For each internuclear distance R, the ground-state energy of *H*
_{2} is obtained from Ref. [22]. Similar to [18], *b* is set to 0.05. In order to avoiding autoionization and guaranteeing a stable initial ensemble, a is set to be 1.25. The starting position of the electron pairs is distributed along z axis using a random probability distribution. We firstly select out the electron pairs whose available kinetic energy is positive, then the position and momentum distributions of the initial ensemble are obtained by the same method as that described in Ref. [21]. Figures 1(a) and 1(b) show the position distributions of the initial ensemble of *H*
_{2} along the x and z axes respectively when R=6 a.u. for parallel alignment (*ϕ* = 0).

Once the initial ensemble is obtained, the laser is turned on and all trajectories are evolved in the combined Coulomb and laser fields. At the end of the laser pulse, we examine the NSDI yield. We define that the molecule is doubly ionized if the energies of both electrons are positive and singly ionized (SI) if either electron firstly possesses positive energy. In the semiclassical model by Chen and Nam [24], the first electron is ionized through tunneling and the probability is calculated by ADK theory. While for the completely classical model in our paper, the first electron is ionized above the suppressed potential barrier, and there is no tunneling ionization. We define recollision if the distance between the two electrons is less than d=5 a.u. after the ionization of the first electron. The statistical results change very little when d ranges from 4 to 7 a.u. If recollision occurs, DI is classified into NSDI. Otherwise DI is classified into SDI.

Figure 2 shows the correlated electron-electron momentum spectra in the direction parallel to the laser field for different R ranging from 2 to 12 a.u. when the molecular axis is parallel to the laser polarization (*ϕ* = 0). *k*
^{∥}
_{1}, *k*
^{∥}
_{2} represent the momentum components of the two electrons parallel to the laser field direction, respectively. The NSDI of *H*
_{2} for different internuclear distances provides rich correlation patterns, which are strongly dependent on R. When R=2 a.u. [see Fig. 2(a)], the correlated pattern exhibits a fingerlike-structure. The fingerlike-structure corresponds to the recoil collision in a (e, 2e) recollision process [25]. When R=4 a.u. [see Fig. 2(c)], the correlated pattern exhibits a clear maximum at about *k*
^{∥}
_{1} = *k*
^{∥}
_{2} = 0.5 and -0.5 a.u.,
indicating both electrons emitted on the same side with similar momentum at almost the same time [26]. For large R [e.g., see Fig. 2(h)], the correlated pattern corresponds to the recollision-excitation with subsequent field-ionization (RESI) mechanism [27], for which both electrons are ejected independently after recollision and can be found in all quadrants. When R is even larger, a *H*
_{2} molecule is appreciated to two independent H atoms, the correlation between two electrons is expected to be more weak, and the DI events should be more uniformly distributed in all the quadrants.

Figure 2 also gives information about the total DI, NSDI yields for different R. Based on the above definitions of DI, NSDI and SI, we calculate the total DI, NSDI, SI probabilities and the ratio of the NSDI probability to the DI probability as functions of R, which are shown in Figs. 3(a–d). When R=4 a.u., the total DI, NSDI, SI probabilities reaches the maximum at the same time, while the ratio of the NSDI probability to the DI probability is minimum. As shown in Figs. 3 (a–c), the DI, NSDI and SI probabilities increase rapidly at first and then decrease slowly with R increasing from 2 to 12 a.u. While the ratio of the NSDI probability to the DI probability decreases quickly at first and then increases slowly and finally decreases with R increasing. The dependence of correlated electron momentum distributions on R can be understood in the ratio of the NSDI probability to the DI probability.

Finally, we qualitatively discuss the origin of above results about correlated momentum distributions and the NSDI probability. As is known, SDI is a DI process in which two electrons are independently emitted via tunnelling ionization while NSDI is resulted from the recollision between the returning electron and the parent ion. Therefore, for a fixed R, the SDI probability is determined by the SI probability, the ionization potential of the second electron (*I*
_{p2}) and the double potential well distorted by the external laser field. In addition, the NSDI probability is also determined by the rescattering probability and the returning energy of the first ionized electron.

The R-dependent SI probability has been plotted in Fig. 3(c), which is consistent with previous theoretical results [11] and explained by the occurrence of an avoided crossing between the covalent ground state and the ionic excited state [12]. Figure 4(a) shows the R-dependent *I*
_{p2} [22]. As R increasing, *I*
_{p2} decreases monotonically. In Figs. 5(a-c), we show the field distorted double potential wells (the laser electric field reaches the maximum) for R=2, 4, 9 a.u. respectively. The energy of the second electron is denoted by the black dashed line. Based on the same statistical criterion as that for NSDI, the rescattering probability of the first ionized electron as a function of R is plotted in Fig. 4 (b), which decreases with R increasing. The returning energies obtained by back analysis are similar for different R and thus make negligible influence on recollisions when R varies. Thus, the NSDI probability is mainly determined by *I*
_{p2} and the rescattering probability.

When R=2 a.u., the potential well resembles that of helium. For ionization to occur, the second electron must tunnel through the large outer barrier, as shown in Fig. 5(a), resulting the low ionization probability. Moreover, the ionization of the second electron should be aided by the recollision between the first electron and the parent ion and thus most DI paths are related to NSDI. Thus, the probability ratio of NSDI to DI is very high. However, the NSDI probability is low because of the low SI yield and the high *I*
_{p2}.

As R increases to 4 a.u., the outer barrier is more effectively suppressed [see Fig. 5(b)], resulting in the increase of the tunnelling probability of the second electron. Additionally, the SI probability increases to its maximum and *I*
_{p2} is lower. As a result the SDI yield is largely enhanced. For the NSDI yield, it also increases largely for two reasons. Firstly, though the rescattering probability is reduced, the high SI yield still leads to a large number of the rescattering electrons. Secondly, *I*
_{p2} is lower. By comparison, the enhancement in SDI yield is more remarkable, resulting the probability ratio of NSDI to DI decreasing.

When R increases further, the inner barrier becomes wider [see Fig. 5(c)], impeding the tunnelling of the second electron and thus resulting in the decrease of the SDI yield. Besides, the NSDI yield is also reduced as a result of the decrease of the SI yield and the rescattering probability of the first ionized electron. When R ranges from 4 to 7 a.u., the decrease of the SDI yield is more remarkable than that in the NSDI yield, therefore, the probability ratio of NSDI to DI increases. When R is larger than 7 a.u., the rapidly decreasing rescattering probability leads to a more prominent decrease of the NSDI yield compared to that of the SDI yield, thus the probability ratio of NSDI to DI starts to decrease.

We also calculated NSDI of *H*
_{2} for two cases of nonparallel alignment: *ϕ* = *π*/4 and *π*/2.
We found that the correlated electron spectra also exhibit rich similar patterns as the case of parallel alignment. For nonparallel alignment [see Fig. 6], the dependence of double ionization yield on the internuclear distance is similar to that of the parallel alignment case. However, the double ionization yield decreases quickly with the alignment angle increasing. The yield for perpendicular alignment is much less than that for parallel alignment. When the molecular axis is aligned perpendicular to the laser polarization, the electron is more tightly bound than when the laser field is applied along the molecular axis, and the laser field is now acting over a shorter distance and is less effective in causing a suppression of the potential barrier [15]. As a result, the ionization is suppressed greatly.

In summary, we have theoretically investigated the internuclear distance-dependence of electron correlation in NSDI of *H*
_{2} molecules by an intense linearly polarized laser field with classical 3D ensemble model. The correlated electron momentum spectra strongly depends on the internuclear distance R. The rich correlated patterns essentially reflect the different microscopic dynamics in double ionization process for different internuclear distance. Our simulations show a critical internuclear distance of about 4 a.u. for NSDI of *H*
_{2}, where NSDI probability reaches a maximum. These results provide an insight into the important role of the structure of diatomic molecules in NSDI.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant No. 10774054, National Science Fund for Distinguished Young Scholars under Grant No.60925021, and the 973 Program of China under Grant No. 2006CB806006.

## References and links

**1. **D. N. Fittinghoff, P. R. Bolton, B. Chang, and K. C. Kulander, “Observation of nonsequential double ionization of helium with optical tunneling,” Phys. Rev. Lett. **69**, 2642–2645 (1992). [CrossRef] [PubMed]

**2. **P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. **71**, 1994–1997 (1993). [CrossRef] [PubMed]

**3. **M. Lein, E. K. U. Gross, and V. Engel, “Intense-Field Double Ionization of Helium: identifying the Mechanism,” Phys. Rev. Lett. **85**, 4707–4710 (2000). [CrossRef] [PubMed]

**4. **A. S. Alnaser, T. Osipov, E. P. Benis, A. Wech, B. Shan, C. L. Cocke, X. M. Tong, and C. D. Lin, “Rescattering double ionization of *D*_{2} and *H*_{2} by intense laser pulses,” Phys. Rev. Lett. **91**, 163002 (2003). [CrossRef] [PubMed]

**5. **S. Saugout and C. Cornaggia, “Temporal separation of *H*_{2} double-ionization channels using intense ultrashort 10-fs laser pulses,” Phys. Rev. A. **73**, 041406(R) (2006). [CrossRef]

**6. **Q. Liao, P. Lu, Q. Zhang, Z. Yang, and X. Wang, “Double ionization of HeH+ molecules in intense laser fields,” Opt. Express **16**, 17070–17075 (2008). [CrossRef] [PubMed]

**7. **T. Zuo and A. D. Bandrauk, “Charge-resonance-enhanced ionization of diatomic molecular ions by intense lasers,” Phys. Rev. A. **52**, R2511–R2514 (1996). [CrossRef]

**8. **S. Saugout, “*H*_{2} double ionization with few-cycle laser pulses,” Phys. Rev. A. **77**, 023404 (2008). [CrossRef]

**9. **G. N. Gibson, M. Li, C. Guo, and J. Neira, “Strong-field dissociation and ionization of *H*^{+}_{2} using ultrashort laser pulses,” Phys. Rev. Lett. **79**, 2022–2025 (1997). [CrossRef]

**10. **C. Trump, H. Rottke, M. Wittmann, G. Korn, W. Sandner, M. Lein 3,4, and V. Engel, “Pulse-width and isotope effects in femtosecond-pulse strong-field dissociation of *H*^{+}_{2} and *D*^{+}_{2},” Phys. Rev. A. **62**, 063402 (2000). [CrossRef]

**11. **H. Yu, T. Zuo, and A. D. Bandrauk, “Molecules in intense laser fields: enhanced ionization in a one-dimensional model of *H*_{2},” Phys. Rev. A. **54**, 3290–3298 (1996). [CrossRef] [PubMed]

**12. **A. Saenz, “Enhanced ionization of molecular hydrogen in very strong fields,” Phys. Rev. A. **61**, 051402 (2000). [CrossRef]

**13. **X. M. Tong, Z. X. Zhao, and C. D. Lin, “Probing molecular dynamics at attosecond resolution with femtosecond laser pulses,” Phys. Rev. Lett. **91**, 233203 (2003). [CrossRef] [PubMed]

**14. **A. S. Alnaser, X. M. Tong, T. Osipov, S. Voss, C. M. Maharjan, P. Ranitovic, B. Ulrich, B. Shan, Z. Chang, C. D. Lin, and C. L. Cocke, “Routes to control of *H*_{2} coulomb explosion in few-cycle laser pulses,” Phys. Rev. Lett. **93**, 183202 (2004). [CrossRef] [PubMed]

**15. **K. Codling, L. J. Frasinski, and P. A. Hatherly, “On the field ionisation of diatomic molecules by intense laser fields,” J.Phys.B **22**, L321–L327 (1989). [CrossRef]

**16. **S. L. Haan, L. Breen, A. Karim, and J. H. Eberly, “Variable time lag and backward ejection in full-dimensional analysis of strong-field double ionization,” Phys. Rev. Lett. **97**, 103008 (2006). [CrossRef] [PubMed]

**17. **S. L. Haan, L. Breen, A. Karim, and J. H. Eberly, “Recollision dynamics and time delay in strong-field double ionization,” Opt. Express **15**, 767–778 (2007). [CrossRef] [PubMed]

**18. **S. L. Haan, J. S. Van Dyke, and Z. S. Smith, “Recollision excitation, electron correlation, and the production of high-momentum electrons in double ionization,” Phys. Rev. Lett. **101**, 113001 (2008). [CrossRef] [PubMed]

**19. **S. L. Haan, Z.S. Smith, K. N. Shomsky, and P. W. Plantinga, “Electron drift directions in strong-field double ionization of atoms,” J. Phys. B **42**, 134009 (2009). [CrossRef]

**20. **Q. Liao and P. Lu, “Manipulating nonsequential double ionization via alignment of asymmetric molecules,” Opt. Express **17**, 15550–15557 (2009). [CrossRef] [PubMed]

**21. **Y. Zhou, Q. Liao, and P. Lu, “Mechanism for high-energy electrons in nonsequential double ionization below the recollision-excitation threshold,” Phys. Rev. A. **80**, 023412 (2009). [CrossRef]

**22. **T.-T. Nguyen-Dang, F. Châteauneuf, and S. Manoli, “Tunnel ionization of *H*_{2} in a low-frequency laser field: a wave-packet approach,” Phys. Rev. A. **56**, 2142–2167 (1997). [CrossRef]

**23. **R. Panfili and J. H. Eberly, “Comparing classical and quantum dynamics of strong-field double ionization,” Opt. Express **8**, 431–435 (2001). [CrossRef] [PubMed]

**24. **J. Chen and C. H. Nam, “Ion momentum distributions for He single and double ionization in strong laser fields,” Phys. Rev. A. **66**, 053415 (2002). [CrossRef]

**25. **A. Staudte, C. Ruiz, M. Schöffler, S. Schössler, D. Zeidler, Th. Weber, M. Meckel, D. M. Villeneuve, P. B. Corkum, A. Becker, and R. Dörner, “Binary and recoil collisions in strong field double ionization of Helium,” Phys. Rev. Lett. **99**, 263002 (2007). [CrossRef]

**26. **Th. Weber, H. Giessen, M. Weckenbrock, G. Urbasch, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, M. Vollmer, and R. Dörner, “Correlated electron emission in multiphoton double ionization,” Nature **405**, 658–661 (2000). [CrossRef] [PubMed]

**27. **B. Feuerstein, R. Moshammer, D. Fischer, A. Dorn, C. D. Schröter, J. Deipenwisch, J. R. Crespo Lopez-Urrutia, C. Höhr, P. Neumayer, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, and W. Sandner, “Separation of recollision mechanisms in nonsequential strong field double ionization of Ar: the role of excitation tunneling,” Phys. Rev. Lett. **87**, 043003 (2001). [CrossRef] [PubMed]