Effect of tunneling ionization on dynamic alignment and post-ionization alignment of nitrogen molecules irradiated by different laser intensities

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Optik

Optics

Optik 121 (2010) 1219–1225 www.elsevier.de/ijleo

Effect of tunneling ionization on dynamic alignment and post-ionization alignment of nitrogen molecules irradiated by different laser intensities Xingshan Jiang, Hongchun Wu, Bomin Huang, Jianxin Chen, Shuangmu Zhuo Institute of Laser and Optoelectronics Technology, Fujian Provincial Key Laboratory for Photonics Technology, Key Laboratory of OptoElectronic Science and Technology for Medicine of Ministry of Education, Fujian Normal University, Fuzhou 350007, P. R. China Received 9 September 2008; accepted 12 January 2009

Abstract The dynamic alignment and post-ionization alignment of nitrogen molecules are investigated while considering the effect of tunneling ionization. The effects of tunneling ionization on the angular distribution are calculated when the molecules are irradiated by different laser intensities. The results show that laser intensity directly affects the time and extent of dynamic alignment. Furthermore, the extent of post-ionization alignment is not only determined by laser intensity but also affected by the final extent of dynamic alignment. The post-ionization alignment will dominate during the process of molecular (or molecular ion) rotational alignment for femtosecond laser pulse. The time of tunneling ionization is a significant factor to the final ensemble angular distribution of molecular ions when laser intensity is low. r 2009 Elsevier GmbH. All rights reserved. Keywords: Tunneling ionization; Post-ionization alignment; Dynamic alignment

1. Introduction In recent decades molecular alignment and orientation induced by intense field lasers are receiving increasing interest. Potential applications range from study and manipulation of chemical reactions [1] and trapping of molecules [2] through generation of highorder harmonic [3,4] to control photo-dissociation [5,6]. Dynamic and geometric alignment mechanisms have been proposed to interpret anisotropic angular distributions of ionic fragments under differently experimental conditions [7–10]. And the process of molecular alignCorresponding author. Fax: +86 591 8345 6462.

E-mail address: [email protected] (J. Chen). 0030-4026/$ - see front matter r 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.01.003

ment is tightly correlated with that of molecular multielectron dissociation ionization [11]. From the theoretical view, the time-dependent Schro¨dinger equation [12], a filed-ionization Coulomb explosion model [10,13,14], the two-dimensional Thomas–Fermi approach [15], the time-dependent unitary perturbation theory [16], strong-field S-matrix approach [17,18] and laser-driven rigid-rotor analytical model [19] are established to discuss molecular dynamic alignment. On the other hand, the theory of molecular tunneling ionization [20] was developed within the framework of tunneling ionization theory by Ammosov et al. [21]. Based on the MO-ADK theory, post-ionization alignment is studied [22]. And the mechanism of alignment-dependent ionization is becoming clear [23,24].

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In this paper, the effects of tunneling ionization to two alignment mechanisms are studied. One is the familiar dynamic alignment (DA) before the molecule’s breakup [25]. The other is the post-ionization alignment (PIA) effect, which occurs after the molecules are ionized. The case of O2 molecules has been studied by Tong et al. [22]. Here we extend their model by (1) considering the effect of tunneling ionization in calculating the DA and PIA processes, according to the MO-ADK theory, with different laser intensities and (2) calculating the ensemble final angle distribution when the PIA ended, including the molecules being tunneling ionized in the process of DA.

intensity variational field, part of or all of the N2 molecules will be ionized as N2+ 2 . And if two nuclei mutually repel due to the Coulomb force, then the internuclear distance R elongates. In the meanwhile, the dipole polarizability changes with R. The moment of increases quadratically inertia of the ‘rigid rotor’ N2+ 2 with R. The time dependence of B and the dipole polarizability Da can be entered into the time-dependent equations B(t) and Da(t), through their variations with R(t). The detail deductions of R(t) and Da(t) are demonstrated in Section 2.3. The solution of Eq. (2) can be expanded in a series of field-free wave function |J, MS ¼ YJ, M: X CðDoðtÞÞ ¼ C J ðDoðtÞÞjJ; Mi. (6) J

2. Description of the model

The expanded coefficient CJ(Do(t)) can be found from the differential equations:

2.1. Rotation dynamics

1 dC J;M ðDoðtÞÞ B dt ¼ C J;M ðDoðtÞÞ½JðJ þ 1Þ  o? ðtÞ

A quantum mechanical analysis of DA and PIA effect can be incorporated based on the standard treatment of the rotational motion of a homonuclear molecule in a laser field. Here, the diatomic molecule is approximately considered as a ‘rigid’ rotor [26–29]. The Hamiltonian for the diatomic molecules in intense laser field is

i

H ¼ BJ 2  122 ðtÞ½ðak  a? Þcos2 y þ a? ,

The time evolution of the expanded coefficient CJ, M(t) in Eq. (7) can be solved by using the finitedifference method combined with the fourth-order Runge–Kutta algorithm. From the calculated rotational wavepacket, the angular distribution of the molecules as a function of time can be obtained. The degree of alignment is often characterized by the alignment parameter defined by / cos2 yS, which is evaluated from X C J;M ðtÞC J 0 ;M ðtÞð2J þ 1Þ1=2 ð2J 0 þ 1Þ1=2 hcos2 yðtÞiJ;M ¼

(1)

where J2 is the squared angular momentum operator, B the rotational constant, and y the polar angle between the molecular axis and the direction of the field. a|| and a? are the components of the static polarizability parallel and perpendicular to the molecular axis, respectively. In what follows, a plane wave radiation of 800 nm wavelength and Gaussian time profile such that e(t) ¼ eo exp((t2t)2/2t2] are considered. The time-dependent Schro¨dinger equation corresponding to Hamiltonian Eq. (1) can be cast in a dimensionless form (in atomic units m ¼ _ ¼ e ¼ 1) @CðtÞ i ¼ HðtÞCðtÞ. @t

 DoðtÞC Jþ2;M ðDoðtÞÞhJ; Mjcos2 yjJ þ 2; Mi.

J;J 0

J 

M

J0

2 0

!

2

J0

0 0

0

J

M

(7)

! . (8)

And the ensemble average of the expectation value of the alignment cosine is given by hhcos2 yiiðtÞ ¼ Q1 r

2

ak ðtÞ ðtÞ , ok ðtÞ ¼ 2BðtÞ

(3)

a? ðtÞ2 ðtÞ , 2BðtÞ

DoðtÞ ¼ ok ðtÞ  o? ðtÞ ¼

 DoðtÞC J;M ðDoðtÞÞhJ; Mjcos2 yjJ; Mi

(2)

Solution of this depends on dimensionless interaction parameters:

o? ðtÞ ¼

 DoðtÞC J2;M ðDoðtÞÞhJ; Mjcos2 yjJ  2; Mi

(4) DaðtÞ2 ðtÞ . 2BðtÞ

(5)

With the effect of the molecular tunneling ionization (TI) and above-threshold ionization (ATI) in the

X J

eJðJþ1ÞB=kT r

M¼J X

hcos2 yðtÞi,

(9)

M¼J

where Tr is the rotational temperature and Qr ¼ Qr(Tr) is the rotation partition function. In calculations, Tr ¼ 30 K is set and nuclear spin statistics has to be accounted for in the partition function.

2.2. Time-dependent ionization rate The time-dependent ionization rate of N2 molecule within the framework of molecular tunneling ionization

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theory (MO-ADK theory) [22] can be theoretically deduced. When the direction of the molecule axis is parallel to the polarization direction of laser field, the field-dependent ionization rate w(e, 01) is given by Tong et al. [22] X B2 ðm; 0 Þwm ðÞ (10) wð; 0 Þ ¼ m

with the static ionization rate pffiffiffiffiffi " #ð2Z=pffiffiffiffiffi 2I p Þ1jmj ð2I p Þ1=2ðð2Z= 2I p Þ1Þ 2ð2I p Þ3=2 wm ðÞ ¼  2jmj jmj! " # 2ð2I p Þ3=2 (11) exp  3 and Bðm; 0 Þ ¼

X l

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l þ 1Þðl þ jmjÞ! : C l ð1Þm 2ðl  jmjÞ!

(12)

Here, Ip is the ionization potential, Z the ion charge, and m the magnetic quantum number of the orbital the electron is ejected from. Intensity of laser field e in the Gaussian beam is the function of time t: ðtÞ ¼ o exp½ðt  2tÞ2 =2t2 . So, the N2 molecular ionization rate in the field polarization direction at the moment of t can be written as wðt; 0 Þ ¼ wððtÞ; 0 Þ.

(13)

According to Ref. [22], when the laser field is 1013–1015 W/cm2, the ratio of ionization rates for N2 molecules aligned along the laser field direction to the randomly distributed ones remains at 3.15–3.0. So, in the following calculation, the ensemble average ionization rate can be approximated by w(t) ¼ w(t, 01)/ cos2 y(t)S. And the ionization probability is expressed as [22] R (14) pðtÞ ¼ 1  e½ wðtÞ dt .

2.3. Ensemble average of B(t) and Da(t) The occurrence of the Coulomb explosion of molecules at a critical internuclear distance Rc, which is larger than the equilibrium internuclear distance Ro, has been proved in many experiments and several theoretical calculations [30–32]. The model indicates that there are two steps in which the molecular Coulomb explosion occurs [33]. The first step is that in which the neutral molecules are ionized. Once the molecules are ionized beyond the first stage, the two nuclei mutually repel due to the Coulomb force. Thereafter additional electrons lose rapidly when the internuclear distance increases to the critical distance Rc ¼ 4.07/Ip (in atomic unites), where Ip is the ionization potential of the molecule. As

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the internuclear distance increase to the critical distance, repulsion between the two nuclei vanishes, which means the dynamic rotation of molecules (or molecular ions) is over at this moment. In 2002, Tong et al. [22] published the molecular tunneling ionization MO-ADK theory, pointing out that still part molecules are being ionized under the threshold intensity of laser field due to the quantum tunneling effect. Meanwhile the alignment dependence of the ionization rates of molecules depends critically on the symmetry of the outmost orbital of the molecule. On the other hand, tunneling ionization affects the dynamic of alignment as well, including the dynamical alignment (DA) and the post-ionization alignment (PIA) because the variation of R(t) will have some effect on the ensemble average of rotational constant B and dipole polarizability Da. The motion equation for internuclear distance R(t) can be written as € ¼ Q1 Q2 RðtÞ M½RðtÞ2

(15)

Q1 and Q2 are, respectively, the charges of the two ions. In the following calculation, the situation of Q1 ¼ Q2 ¼ 1 is considered. And tf is the time that R(t) elongates from the equilibrium internuclear distance Ro to the critical distance Rc. Since the diatomic molecule (or molecular ion) is treated as a linear rotor, the moment of inertia quadratically depends on R and thus the rotation constant B is inversely proportional to the square of R. For the dipole polarizability, the component perpendicular to the internuclear axis is not changed when R is increased. The parallel component is increased almost linearly with R. We use the quantum chemistry code GAMESS to obtain the dipole polarizat a number of R’s. The perpendicular ability for N2+ 2 component is nearly constant and a value of 10.2351 was used for the parallel component. The R-dependence was found to be well fitted by a|| ¼ 14.8425+ ( is the equilibrium 3.7189(RRo), where Ro ¼ 1.098 A distance of N2. In our calculation of ensemble average of B(t) and Da(t), the following three cases arise: (a) totf: At the moment of t, the ensemble average of rotational constant B(t) is given by  Z t  Z t BðtÞ ¼ Bo exp  wðt0 Þ dt0 þ Bo 0

0

2 Ro wðt0 Þ dt0 . Rðt  t0 Þ

(16a)

The first term of Eq. (16a) is the rotational constant to which the molecules contribute without being ionized, which is induced from Eq. (14). The second is contributed by the molecules being ionized at different time t0 . w(t) being the ionization rate and R(tt0 ) is the internuclear distance of molecules that have been ionized at the moment of t0 , derived from Eq. (15).

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Similarly, the ensemble average of Da(t) is evaluated from  Z t  Z t wðt0 Þ dt0 þ Daðt  t0 Þwðt0 Þ dt0 : DaðtÞ ¼ Dao exp  0

the moment of tion is contributed and the second term is associated with that have been ionized before tion.

0

(16b)

3. Results and discussion

0

Here Dao is the polarizability tensor of N2. Da(tt ) is being ionized at the moment of t0 and the that of N2+ 2 internuclear distance is R(tt0 ). (b) tfptotion: Here, tion is the time of laser field reaching the ionization threshold intensity of the N2 molecule. At time tion, the molecules will be quickly ionized. tion is also considered as the moment that separate dynamical alignment and post-ionization alignment. In this stage, the ensemble average of B(t) and Da(t) is BðtÞ ¼

 Z t  Z Bo exp  wðt0 Þ dt0 þ

ttf

0

Z



t

ttf

 exp

 wðt Þ dt , 0

Ro Rðt  t0 Þ

2

The ensemble averages of the alignment cosine under different laser intensities (5  1014 W/cm2, 1  1015 W/ cm2, 5  1015 W/cm2, and 1  1016 W/cm2) are calculated 0.7 0.65 0.6

#

0.55

wðt0 Þ dt0

0



"

3.1. Effects on DA and PIA

(17a)

0.5 0.45

0

"

DaðtÞ ¼

 Z t  Z Dao exp  wðt0 Þ dt0 þ 0

Z

ttf

 exp

 wðt0 Þ dt0 ,

0.4

#

t

Daðt  t0 Þwðt0 Þ dt0

0.35

ttf 0

(17b)

50

Time (unite:fs)

100

150

0

ttf

wðt0 Þ dt0

 ¼

  Z exp 

0

ttf

wðt0 Þ dt0

1

0

is the probability of the molecules that internuclear distance R reaches the critical distance Rc, and these molecules are considered to vanish (c) tionpt: In this case, the intensity of laser field exceeds the threshold field intensity. The N2 molecules that did not undergo tunneling ionization are considered to ionize to N2+ 2 . At the moment tion, "  Z tion  2 R0 wðt0 Þ dt0 BðtÞ ¼ B0 exp  Rðt  tion Þ 0 # 2 Z tion  R0 0 0 wðt Þ dt þB0 Rðt  tion Þ ttf Z ttf  (18a) wðt0 Þ dt0 ,  exp 0

 Z DaðtÞ ¼ Daðt  tion Þexp  Z

tion 0

tion

ttf

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35



þ

Fig. 1. The red line and the blue line correspond to expected value of the alignment cosine with and without considering tunneling ionization, respectively. /cos2 yS is a function of time with 100 fs laser pulses of 5  1014 W/cm2. Dynamic alignment (solid line) and post-ionization alignment (dash line) are presented at point of the end of DA and PIA labeled as triangles and diamonds. For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.



where Z exp



wðt0 Þ dt0 # Z

Daðt  t0 Þwðt0 Þ dt0 exp

0

20

40

60

80

100

120

Time (unite:fs)

ttf

 wðt0 Þ dt0 .

0

(18b) To first term in the square brackets of Eqs. (18a) and (18b) the proportion of N2 molecules being ionized at

Fig. 2. The red line and the blue line correspond to expected value of the alignment cosine with and without considering tunneling ionization with 100fs laser pulses of 10  1014 W/ cm2, respectively. For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.

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0.7

0.65

0.65

0.6

0.6

0.55

0.55





X. Jiang et al. / Optik 121 (2010) 1219–1225

0.5

0.5

0.45

0.45

0.4

0.4

0.35

0.35

0

10

20

30

40

50

60

0

Time (unite:fs)

using the method described in Section 2. The pulse durations are all set to t ¼ 100 fs for different intensities. Figs. 1–3 present the angular distributions for N2 with and without consideration of tunneling ionization for different laser intensities (5  1014 W/cm2, 10  1014 W/cm2, and 5  1015 W/cm2). Solid lines and dashed lines represent DA and PIA, respectively. Triangles are the end of DA and diamonds are the end of PIA. The red dashed line represents the PIA considering tunneling ionization effect and the blue dashed line represents PIA without considering tunneling ionization effect. It can be seen that the effect of tunneling ionization on DA and PIA decreases when the laser intensity increased for N2. The decreasing alignment time reduces the ionization probability of molecules and has less effect to the alignment when the laser intensity increases. Tunneling ionization can promote the extent of DA and PIA, where B(t) and Da(t) increase. For different intensities and the same duration laser pulse, the time taken for intensity to increase to ionization threshold is different, for which tunneling ionization takes place at this period. So if the longer tunneling time directly induces larger tunneling ionization probability, then effect of tunneling ionization is more clear. For dynamic alignment, it can be noted that the majority of dynamic alignment for N2 molecule takes place before the molecule ionizes and begins to dissociate. The increasing torque and decrease in alignment time compete during the course of molecular dynamic alignment. Fig. 4 shows that the aligning parameter /cos2 yS(tion) at the end of the DA process decreases from 0.4167 to 0.3492 when the laser intensity increases from 5  1014 W/cm2 to 1  1016 W/cm2. It indicates that the reducing extent of molecular dynamic alignment with increase in laser intensity mainly is due

20

40

60

80

100

120

140

160

180

Time (unite:fs)

Fig. 4. Expected value of the alignment cosine, /cos2 yS as a function of time, with 100 fs laser pulses of different intensities. The different colours correspond to different laser intensities. Dynamic alignment (solid line) and post-ionization alignment (dash line) are presented at point of the end of DA and PIA labeled by triangles and diamonds, respectively. 0.8

DA PIA Ensemble alignment

0.7 0.6 0.5



Fig. 3. The red line and the blue line correspond to expected value of the alignment cosine with and without considering tunneling ionization with 100 fs laser pulses of 5  1015 W/cm2, respectively. For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.

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0.4 0.3 0.2 0.1 0

5x1014 W/cm2

1x1015 W/cm2

5x1015 W/cm2

1x1016 W/cm2

Fig. 5. Comparison of DA and PIA final values and ensemble final value.

to decrease in the alignment time. For the postionization alignment, laser intensity is the single factor to the rate of alignment because the interaction time of laser field and molecular ion is constant as tf which is the time that molecule take to ionize and disappear. The start point of the PIA process is a factor that cannot be neglected. As Figs. 4 and 5 show final extent of PIA does not constantly increase when laser intensity increases. The final value of /cos2 yS at the laser intensity 1  1016 W/cm2 is less than 5  1015 W/cm2. It is also clear that /cos2 yS value increases more quickly with increase in laser intensity.

3.2. Ensemble angular distribution When the laser pulse and molecules interacted, many molecules undergo tunneling ionization. The final

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angular distribution of PIA is evaluated as /cos2 yS(t0 +tf), in which t0 is the moment of molecule being first ionized and breakup starts. The ensemble final angle distribution of all the molecules is given by  Z tion  wðt0 Þ dt0 hhcos2 yii ¼ hcos2 yiðtion þ tf Þ exp  0 Z tion 2 0 0 hcos yiðt þ tf Þwðt Þ dt0 . (19) þ 0

The first term is the final alignment extent of those molecules being ionized at tion. The second term that being tunneling ionized before tion. Fig. 5 presents a comparison of ensemble final angular distribution and final angular distribution of DA and PIA. It can be seen that the difference of final angular distribution of ensemble and PIA decreases with increase in laser intensity due to decrease in DA time. It means that the proportion of molecules being tunneling ionized is less when the laser intensity increases.

4. Conclusions In summary, the roles of laser intensity and tunneling ionization in dynamic alignment and post-ionization alignment of N2 molecule are presented. The time and the extent of dynamic alignment are directly affected by laser intensity. Furthermore, the extent of post-ionization alignment is not only determined by laser intensity but also affected by the final extent of dynamic alignment. The post-ionization alignment will dominate during the process of molecular (or molecular ion) rotational alignment for femtosecond laser pulses. The time of tunneling ionization is a significant factor for the final ensemble angular distribution of N2 molecular ions for low laser intensity due to the large proportion of tunneling ionization.

Acknowledgements The project was supported by the National Natural Science Foundation of China (No. 60508017), the Natural Science Foundation of Fujian Province of China (2007J0007, C0720001), the Program for New Century Excellent Talents in University (NCET-070191).

References [1] H.J. Loesch, A. Remscheid, Brute force in molecular reaction dynamics: a novel technique for measuring steric effects, J. Chem. Phys. 93 (1990) 4779–4790.

[2] B. Friedrich, D. Herschbach, Alignment and trapping of molecules in intense laser fields, Phys. Rev. Lett. 74 (1995) 4623–4626. [3] N. Hay, R. Verlotta, B. Mason M, M. Castillejo, J.P. Marangos, High-order harmonic generation efficiency increased by controlled dissociation of molecular iodine, J. Phys. B: At. Mol. Opt. Phys. 35 (2002) 1051–1060. [4] V. Kalosha, M. Spanner, J. Herrmann, M. Ivanov, Generation of single dispersion precompensated 1-fs pulses by shaped-pulse optimized high-order stimulated Raman scattering, Phys. Rev. Lett. 88 (2002) 1039011–103901-4. [5] M.D. Poulsen, E. Skovsen, H.J. Stapefeldt, Photodissociation of laser aligned iodobenzene: towards selective photoexcitation, J. Chem. Phys. 117 (2002) 2097–2102. [6] N.H. Nahler, R. Baumfalk, U. Buck, Z. Bihary, R.B. Gerber, B. Friedrich, Photodissociation of oriented HXeI molecules generated from HI–Xen clusters, J. Chem. Phys. 119 (2003) 224–231. [7] D. Normand, L.A. Lompre, C. Cornaggia, Laser-induced molecular alignment probed by a double-pulse experiment, J. Phys. B: At. Mol. Opt. Phys. 25 (1992) L497–L503. [8] P. Dietrich, D.T. Strickland, M. Laberge, P.B. Corkum, Molecular reorientation during dissociative multiphoton ionization, Phys. Rev. A 47 (1993) 2305–2311. [9] L.J. Frasinski, K. Codling, P.A. Hatherly, Femtosecond dynamics of multielectron dissociative ionization by use of a picosecond laser, Phys. Rev. Lett. 58 (1987) 2424–2427. [10] J.H. Posthumus, J. Plumridge, M.K. Thomas, K. Codling, L.J. Frsinski, A.J. Langley, P.F. Taday, Dynamic and geometric laser-induced alignment of molecules in intense laser fields, J. Phys. B: At. Mol. Opt. Phys. 31 (1998) L553–L562. [11] J.X. Chen, X.F. Li, Characteristics of dynamic alignment for diatomic and linear triatomic molecules in intense femtosecond laser fields, Int. J. Mass Spectrometry 243 (2005) 155–161. [12] C.M. Dion, A. Keller, O. Atabek, A.D. Bandrauk, Laserinduced alignment dynamics of HCN: roles of the permanent dipole moment and the polarizability, Phys. Rev. A 59 (1999) 1382–1391. [13] E. Springate, F. Rosca-Pruna, H.L. Offerhaus, M. Krishnamurthy, M.J.J. Vrakking, Spatial alignment of diatomic molecules in intense laser fields: II. Numerical modeling, J. Phys. B: At. Mol. Opt. Phys. 34 (2001) 4939–4956. [14] J.H. Posthumus, A.J. Giles, M.R. Thompson, K. Codling, Field-ionization, Coulomb explosion of diatomic molecules in intense laser fields, J. Phys. B: At. Mol. Opt. Phys. 29 (1996) 5811–5829. [15] L. Quaglia, M. Brewczyk, C. Cornaggia, Molecular reorientation in intense femtosecond laser fields, Phys. Rev. A 65 (2002) 031404-1–031404-4. [16] D. Sugny, A. Keller, O. Atabek, Time-dependent unitary perturbation theory for intense laser-driven molecular orientation, Phys. Rev. A 69 (2004) 043407-1–043407-11. [17] A. Jaron´-Becker, A. Becker, F.H.M. Faisal, Ionization of N2, O2, and linear carbon clusters in a strong laser pulse, Phys. Rev. A 69 (2004) 023410-1–023410-9.

ARTICLE IN PRESS X. Jiang et al. / Optik 121 (2010) 1219–1225

[18] A. Jaron´-Becker, A. Becker, F.H.M. Faisal, Signatures of molecular orientation and orbital symmetry in strongfield photoelectron angular and energy distributions of diatomic molecules and small carbon clusters, Laser Phys. 14 (2004) 179–185. [19] S. Banerjee, D. Mathur, G.R. Kumar, Propensity of molecules to spatially align in intense light fields, Phys. Rev. A 63 (2001) 045401-1–045401-4. [20] X.M. Tong, Z.X. Zhao, C.D. Lin, Theory of molecular tunneling ionization, Phys. Rev. A 66 (2002) 0334021–033402-11. [21] M.V. Ammosov, N.B. Delone, V.P. Krainov, Eksp. Teor. Fiz. 91 (1986) 2008; M.V. Ammosov, N.B. Delone, V.P. Krainov, Sov. Phys. JETP 64 (1986) 1191. [22] X.M. Tong, Z.X. Zhao, A.S. Alnaser, S. Voss, C.L. Cocke, C.D. Lin, Post ionization alignment of the fragmentation of molecules in an ultrashort intense laser field, J. Phys. B: At. Mol. Opt. Phys. 38 (2005) 333–341. [23] D. Zeidler, B. Bardon A, A. Staudte, D.M. Villeneuve, R. Do¨rner, P.B. Corkum, Alignment independence of the instantaneous ionization rate for nitrogen molecules, Phys. B: At. Mol. Opt. Phys. 39 (2006) L159–L166. [24] Z.X. Zhao, X.M. Tong, C.D. Lin, Alignment-dependent ionization probability of molecules in a double-pulse laser field, Phys. Rev. A 67 (2003) 043404-1–043404-5. [25] H. Stapelfelt, T. Seideman, Colloquium: aligning molecules with strong laser pulses, Rev. Mod. Phys. 75 (2003) 543–557.

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[26] T. Seideman, Rotational excitation and molecular alignment in intense laser fields, J. Chem. Phys. 103 (1995) 7887–7896. [27] J. Otigoso, M. Rodriguez, M. Gupta, B. Friedrich, Time evolution of pendular states created by the interaction of molecular polarizability with a pulsed nonresonant laser field, J. Chem. Phys. 110 (1999) 3870–3875. [28] L. Cai, J. Marango, B. Friedrich, Time-dependent alignment and orientation of molecules in combined electrostatic and pulsed nonresonant laser fields, Phys. Rev. Lett. 86 (2001) 775–778. [29] C.M. Dion, A.B. Haj-Yedder, E. Cances, C.L. Bris, A. Keller, O. Atabek, Optimal laser control of orientation: the kicked molecule, Phys. Rev. A 47 (2002) 0634081–063408-7. [30] M. Schmidt, D. Normand, C. Cornaggia, Laser-induced trapping of chlorine molecules with pico- and femtosecond pulse, Phys. Rev. A 50 (1994) 5037–5045. [31] P.A. Hatherly, M. Stankiewicz, K. Coding, L.J. Fransinski, G.M. Cross, The multielectron dissociative ionization of molecular iodine in intense laser field, J. Phys. B: At. Mol. Opt. Phys. 27 (1994) 2993–3003. [32] H.Z. Ren, R. Ma, J.X. Chen, X. Li, H. Yang, Q.H. Gong, Field ionization and Coulomb explosion of CO in an intense femtosecond laser field, J. Phys. B: At. Mol. Opt. Phys. 36 (2003) 2179–2188. [33] S. Chelkowski, A.D. Bandrauk, Two-step Coulomb explosions of diatoms in intense laser fieds, J. Phys. B: At. Mol. Opt. Phys. 28 (1995) L723–L731.