Ion momentum analysis of double ionization of stretched moleculesby circularly polarized laser pulses

Aihong Tong n

Department of Physics and Mechanical & Electrical Engineering, Hubei University of Education, Wuhan 430205, China

a r t i c l e i n f o

Article history:Received 11 July 2013Received in revised form4 September 2013Accepted 16 September 2013Available online 1 October 2013

Keywords:Double ionizationCollisionStretched moleculeIon momentum distribution

a b s t r a c t

Using classical ensemble method, we study double ionization (DI) of stretched molecules driven bycircularly polarized laser pulses. Classical DI trajectories show that DI is dominated by nonsequentialdouble ionization (NSDI). We find that as R increases, the ion sum-momentum distribution translatesfrom a single-band structure to a three-band structure, and its projection in the direction parallel to themolecular axis changes from a single central peak, to a central-peak with two side shoulders, then toa double-peak. From the ion sum-momentum distributions, the dominant ionization mechanism isidentified that translates from collision excitation with subsequent field-ionization (CESI) to collisionionization (CI) as R increases.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Double ionization (DI) is a fundamental and important processin intense laser fields. It is roughly divided into two categories:sequential double ionization (SDI) [1–4] and non-sequential dou-ble ionization (NSDI) [5–12]. SDI is usually described by a single-active-electron approximation [13], in which two electrons aretunnel-ionized successively and independently. In NSDI, strongelectron correlation exists between two ionized electrons and twoelectrons cannot be treated independently. The most widelyaccepted mechanism to interpret NSDI is recollision model [14],which has been confirmed by a series of experimental [15–19] andtheoretical [20–24] studies. In recollision model, an electron thattunnels out from the atom or the molecule is then driven back tothe parent ion and knocks out the second electron.

According to the above recollision model, the ellipticity of thepolarized field will has significant effect on NSDI process since theperpendicular component of the laser field will drive the firstionized electron away from the nucleus, and then recollisionprobability will be reduced. This picture has been confirmed byprevious experiment performed by Dietrich et al. [25]. However,it is contrast to some recent experiments. Characteristic NSDI eventshave been observed for some molecules such as NO and O2 [26] andatomic magnesium [27] driven by circular laser pulses. Theseunexpected observations have drawn great attention from theoreticalresearchers. Theoretical investigations of atomic double ionization in

elliptical laser fields have confirmed the existence of NSDI eventsand shown that recollision is also the underlying mechanism forNSDI of atoms in elliptical laser fields, and it is via ellipticaltrajectories [28,29]. However, it is long quantum orbits but notshort quantum orbits that make dominant contribution to NSDI[30,31], in contrast to linear polarization.

For diatomic molecules driven by elliptical fields with largeellipticity or circular fields, recollision seems more possible formolecules with large internuclear distance, since the electronemitted from one nucleus is possible to return to the othernucleus. Motivated by this supposition, we investigate DI ofstretched molecules with R¼8 a.u., 10 a.u., 12 a.u. by circularlypolarized laser pulses in this paper. We find that NSDI is trulydominated. As R increases, the 2D ion sum-momentum distribu-tions change from a single-band structure to a three-band struc-ture. When the 2D ion sum-momentum distribution is projectedto the direction parallel to the molecular axis, the parallel ion sum-momentum distribution changes from a single central peak, to acentral-peak with two side shoulders, then to a double-peak. Fromthe change of the ion sum-momentum distribution with R and thedominance of NSDI in total DI events, the dominant DI mechanismis identified that changes from collision-excitation with subsequentfield-ionization (CESI) to collision ionization (CI) as R increases.

2. The classical ensemble model

We employ the classical model that proposed by Haan andEberly et al. and it has been widely used to study DI of atoms andmolecules in intense laser fields [32–36]. In this study we restrict

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Optics Communications

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n Corresponding author. Tel.: þ86 15327157279.E-mail address: tah1979@126.com

Optics Communications 312 (2014) 252–257

the motions of electrons in the x–y plane for simplicity since theout-of-plain effects are negligible [37]. The evolution of the two-electron system is determined by Newton's classical motionequations (atomic units are used throughout this paper unlessotherwise stated): d2r

,i=dt2 ¼ �E

,ðtÞ�∇Vneðr,iÞ�∇Veeðr,1; r

,2Þ; where

the subscript i is the label of two electrons. The electric field E,ðtÞ is

a 800 nm circularly polarized electric field and written as: E,ðtÞ¼ f

(t)E0 [sin(ωt)_xþcos(ωt)_y]. _x and _y are the polarization vectors.f(t) is the laser envelope that turns on and turns off linearly during2 T0 and keeps full strength for 6 T0 (T0 is the cycle of the circularlaser pulse). E0 is the amplitude of the circularly polarized electricfield, ω is the laser frequency. The intensity of the laser field is1.4�1014 W/cm2. For simplicity, we employ H2 as our model

molecule. The potential Vneðr,iÞ ¼ �1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxi�R=2Þ2þy2i þa2

q�1=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxiþR=2Þ2þy2i þa2

qis the ion-electron interaction, andVeeðr,1; r

,2Þ

¼ 1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx1�x2Þ2þðy1�y2Þ2þb2

qrepresents the electron-electron

interaction. R is the internuclear distance that is aligned along xaxis. Two nuclei are fixed at (�R/2, 0) and (R/2, 0), respectively. Foreach R, the ground-state energy of the model molecule is obtainedfrom Ref. [38]. In order to avoid autoionization and guarantee astable initial ensemble, the softing parameters a and b are set to be1.25 and 0.1, respectively. The initial ensemble is obtained byallowing the two-electron system to evolve a sufficiently long timewithout the laser field [32]. When the initial ensemble is obtained,the circular electric field is turned on and the entire process isdetermined by Newton's motion equation presented above.

We mention that the model employed in this paper treats theentire process classically. Though this classical model does nottake into account tunneling, it can catch most of the features instrong field double ionization because of the high laser intensity[32–34]. Recently the classical model has also been extended tostudy double ionization in elliptical laser fields [4,28] and goodagreement with experiment [2] has been achieved. In addition, forstrong field double ionization in the circularly polarized laserfields, the most accurate method, i.e., numerically solving thetime-dependent Schrödinger equation for the two-electron sys-tem, is a task of great challenge. Thus, we employ the classicalmodel to investigate the complex electron dynamics in doubleionization by circular laser fields. We use this model to investigateNSDI of molecules with large internuclear distance by circularlypolarized laser fields.

In this paper, we calculate DI of molecules with R¼8 a.u., 10 a.u.and 12 a.u. In our calculations, H2 is just as a simple model fordiatomic molecules. Experimentally, the internuclear distance of

H2 could hardly reach such large values. We call our modelmolecule as H2 only based on the fact that the ionization potentialof the model molecule is fitted to be that of H2. Qualitatively, theresults of our calculations is general for other diatomic moleculesand they do not depend on the choice of the ionization potential.In the experimental side, one could study employ the dimers ofrare gases (such as KrXe, whose equilibrium internuclear distanceis larger than 8 a.u. [39]) to study the phenomena shown in this paper.

3. Results and discussion

Fig. 1 displays the ion sum-momentum distributions of doublyionized molecules created by a 800 nm circularly polarized laserpulses at intensity of 1.4�1014 W/cm2. The ion sum-momentumof doubly ionized molecules is obtained by the negative sum of thetwo electron momentum vector, i.e. pion;sum ¼ �ðpe1þpe2Þ, becausethe momentum of the absorbed photons is negligibly small. Theinternuclear distances of the stretched molecules are (a) 8 a.u.,(b) 10 a.u. and (c) 12 a.u., respectively. Here, the horizontal axisshows the ion sum-momentum in the direction parallel to themolecular axis whereas the vertical axis corresponds to the ionsum-momentum component perpendicular to the molecular axis.For R¼8 a.u., the ion sum-momentum distribution is mainlydistributed in a continuous region along the diagonal px;sumþpy;sum ¼ 0. As R increases to 10 a.u., except for the dominance in thesame region as R¼8 a.u., other two regions in the first and thirdquadrants becomes obvious. When R further increases to 12 a.u.,the two regions in the first and third quadrants are denselypopulated as the center continuous region, and the ion sum-momentum distribution displays a clear three-band structure.Thus, the ion sum-momentum distributions show an importantfeature: the population of the ion sum-momentum moves fromthe origin to the first and third quadrants as R increases.

In Fig. 2, the 2D ion sum-momentum distributions are pro-jected to the horizontal (in the direction parallel to the molecularaxis, i.e., x axis) [the left column] and vertical (in the directionperpendicular to the molecular axis, i.e., y axis) [the right column]axes. The internuclear distances are 8 a.u. [(a) and (d)], 10 a.u.[(b) and (e)], and 12 a.u. [(c) and (f)]. In the left column of Fig. 2,the distribution of the ion sum-momentum parallel to the mole-cular axis peaks at zero for R¼8 a.u. When R increases to 10 a.u.,except for the high central peak at zero, two side shouldersemerges. In below discussion, this structure can be decomposedinto a central peak and a lower double-peak. When R is 12 a.u., thetwo side shoulders become higher and exceed the central peak,

−2 −1 0 1 2

−2

−1

0

1

2

px sum

(a.u.)

p y sum

(a.u

.)

−2 −1 0 1 2

−2

−1

0

1

2

px sum

(a.u.)−2 −1 0 1 2

−2

−1

0

1

2

px sum

(a.u.)

Fig. 1. The ion sum-momentum distributions of doubly ionized model molecules created by a 800 nm circularly polarized laser pulses at intensity of 1.4�1014 W/cm2.The internuclear distance of model molecules is (a) R¼8 a.u., (b) R¼10 a.u. and (c) R¼12 a.u., respectively. The horizontal axis shows the ion sum-momentum in thedirection parallel to the molecular axis whereas the vertical axis corresponds to the ion sum-momentum component perpendicular to the molecular axis.

A. Tong / Optics Communications 312 (2014) 252–257 253

showing a double-peak structure with a central minimum in Fig. 2(c).It's should be noted that the value of the central minimum in Fig. 2(c)is not small and this central minimum is resulted from the higherpopulation of the ion sum-momentum in the first and thirdquadrants than in the central continuous region. So there is nocontradiction between the three-band structure in Fig. 1(c) andtwo-peak structure in Fig. 2(c). In the right column of Fig. 2, alldistributions of the ion sum-momentum perpendicular to themolecular axis peak at zero and the width of the distributionincreases with the increasing internuclear distance. It is easy toobtain the correspondence between Fig. 1 and Fig. 2: the centralpeak is related to the distribution of ion sum-momentum in thecenter continuous region, whereas the double-peak structurecorresponds to the ion sum-momentum distribution in the firstand third quadrants.

The results above indicate that the ion sum-momentum dis-tribution depends on the internuclear distance of moleculessensitively, implying a sensitive dependence of ionization mechan-ism on internuclear distance. As is well known, the identificationof ionization mechanism via analysis of ion momentum distribu-tion is based on momentum conservation, and on the angularstreaking principle, which provides a clear relation between thetime when an electron is set free and the drift momentum that theelectron acquired from the field. An electron ionized to an oscillatoryfield at time ti with initial momentum p0 will achieve final momen-tum pf ¼ p0�AðtiÞ, AðtiÞ is the vector potential at the instant ofionization. Neglecting the initial momentum, the final momentaof two electrons are determined by the vector potential at instantof ionization. In the circular laser field: Ex¼E0sin(ωt), Ey¼E0cos(ωt),the final momentum of the electron ionized at instant ti will be:pxf ¼ E0

ω cos ðωtiÞ; pyf ¼ �E0ω sin ðωtiÞ. In order to explain the origin

of the above ion sum-momentum distributions and identify theionization mechanism for different internuclear distances via theion sum-momentum distributions, we back analyze the doubleionization trajectories. We find that most DI events occur throughelectron-electron collision and DI is dominated by nonsequentialprocess. Our statistic shows that the ratios of nonsequential double

ionization probability to the total double ionization probability are70%, 87%, 91% for R¼8 a.u., 10 a.u., 12 a.u., respectively. The dominanceof NSDI in our calculations is because that the laser intensity weemployed is not strong enough to induce sequential double ionization.We find out the ionization time and collision time of the DI events. Foreach DI trajectory, We judge collision if the closest distance betweenthe two electrons is less than 3 a.u. and the collision time is defined asthe instant of the closest approach of the two electrons. The doubleionization time is defined as the instant when both electrons achievepositive energy, where the energy contains the kinetic energy, the ion-electron interaction and half electron-electron interaction. We men-tion that the results below did not depend on the details of ourdefinitions above.

Fig. 3 shows the laser phase (in cycles) at the time of DI versusthe laser phase at collision. In Fig. 3, the two concentrations forR¼8 a.u. are indicated by red circles [Fig. 3(a)], while the twoconcentrations for R¼12 a.u. are indicated by black circles [Fig. 3(c)].For R¼10 a.u., four concentrations appear in Fig. 3(b). Since thepositions of two concentrations in Fig. 3(b) are similar to that inFig. 3(a), we indicate them by red circles. Accordingly, the othertwo concentrations in Fig. 3(b) are indicated by black circles.Before an explanation of the ion sum-momentum distributionsin Fig. 1 and Fig. 2, we firstly give an intuitive correspondencebetween Figs. 1–3 as following. The distributions in the red circlescorrespond to the center part in Fig. 1 and the central single-peakin Fig. 2, and the distributions in the black circles correspond tothe population in the first and third quadrants in Fig. 1 and thedouble-peak in Fig. 2. Seen from Fig. 3, for R¼8 a.u. the centers ofthe red circles are located at (0.33 T0, 0.78 T0) and (0.83 T0, 0.28 T0)while for R¼12 a.u. the centers of the black circles are located at(0.34 T0, 0.57 T0) and (0.87 T0, 0.08 T0).

In order to get an easy understanding of the ion sum-momentum distributions in Fig. 1 and Fig. 2, in Fig. 4 we plotthe x component (Ex) and y component (Ey) of the circular electricfield, where the signs “þ” and “�” denote the directions of thecorresponding vector potential. The black solid and black dashedlines indicate the dominant collision time (tc) and DI time (ti2) for

0.5

1

Cou

nts(

arb.

u.)

0.5

1

Cou

nts(

arb.

u.)

0.5

1

Cou

nts(

arb.

u.)

0.5

1

Cou

nts(

arb.

u.)

−4 −2 0 2 40

0.5

1

py sum (a.u.)C

ount

s(ar

b.u.

)−4 −2 0 2 40

0.5

1

px sum (a.u.)

Cou

nts(

arb.

u.)

Fig. 2. Projections of Fig. 1 onto the horizontal (a)–(c) and vertical (d)–(f) axes. The internuclear distances are R¼8 a.u. [(a) and (d)], R¼10 a.u. [(b) and (e)], and R¼8 a.u.[(c) and (f)].

A. Tong / Optics Communications 312 (2014) 252–257254

R¼8 a.u., and the dominant collision time (tc) and DI time (ti2) forR¼12 a.u. are indicated by the red solid and red dashed lines.

With the approximation that the final ionization time of thefirst electron is equal to the collision time (i.e., one electron is stillfree after the collision), one can easily obtain the direction ofthe electrons’ final momentum and understand the ion sum-momentum distributions from Fig. 4. For R¼8 a.u., the concentra-tions of the first ionization time and the final DI time are just afterthe peak times in x direction, so the x momentum components oftwo electrons are nearly zero, and the ion sum-momentumdistribution in the parallel direction is singly peaked at zero [seeFig. 2(a)]. In the perpendicular direction, two electrons are emittedinto opposite direction, so the ion sum-momentum distribution inthe perpendicular direction is also singly peaked at zero [see Fig. 2(d)].For R¼12 a.u., two electrons are emitted into the same hemi-sphere along x direction but opposite directions along y direction.

As a result, the ion sum-momentum distribution in the paralleldirection displays a double-peak structure [see Fig. 2(c)], while theion sum-momentum distribution in the perpendicular directionalso shows a single central peak [see Fig. 2(f)]. For R¼10 a.u., twoconcentrations locate as that of R¼8 a.u. and the other two locateas that of R¼12 a.u. As a result, the ion sum-momentum in theparallel direction is a mixture of a single central peak and a doublepeak, showing a triple-peak structure in Fig. 2(b), and the ion sum-momentum in the perpendicular direction is also peaked at zero,with its width ranging between that of R¼8 a.u. and 12 a.u.

Finally we identify the ionization mechanism for differentinternuclear distance via ion sum-momentum distributions. Sincecollision is commonly occurred for the internuclear distances inour investigations, the main ionization mechanism includes colli-sion ionization (CI) and collision excitation with subsequent fieldionization (CESI). Since the ion sum-momentum distributions inthe direction perpendicular to the molecular axis exhibit thesimilar structureless single peak structure for all internucleardistances in the right column of Fig. 2, our analysis is concentratedon the ion sum-momentum distribution in the direction parallel tothe molecular axis. For R¼8 a.u., the simple peak at zero impliesthat the dominant ionization mechanism is CESI. For R¼12 a.u.,the clear double-peak with a minimum at zero demonstrates thedominance of CI. For R¼10 a.u., the tripe-peak structure can bedecomposed into a single-peak and a double-peak, implying amixture of CI and CESI.

Two sample DI trajectories are plotted in Fig. 5. The upper,middle, and bottom rows show the energy versus time, separationbetween two electrons versus time, and the electron trajectories inx–y plane, respectively. ree is the separation between two elec-trons. The left and right trajectories are chosen from DI events forR¼8 a.u. and 12 a.u., respectively. In the left trajectory, oneelectron ionizes immediately after collision while the other isexcited [see Fig. 5(a)]. The process corresponds to the CESImechanism. In the right trajectory, both electrons are emittedimmediately after collision almost at the same time [see Fig. 5(d)],which belongs to the typical CI process. Also seen from Fig. 5(b)and (e), the distance between the two electrons decreases untilthe collision occurs. This is the characteristic of the collisionprocess, which is different from the common recollision, wherethe distance of the two electrons first increases and thendecreases. In Fig. 5(c) and (f), the electron trajectories in x–y planeare plotted around the collision time, and the inset figures are theelectron trajectories from the beginning to the end of the laserpulse. The behavior in the collision process can be more wellunderstood from Fig. 5(c) and (f). It is clearly shown that the firstelectron ionizes through the inner barrier, moves directly towardsthe bound electron at the other nucleus, transfers energy to thesecond electron and excites or ionizes it.

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

tc(T0)

t i2 (T

0)

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

tc (T0)0 0.25 0.5 0.75 1

0

0.25

0.5

0.75

1

tc(T0)

Fig. 3. Double ionization phase ti2 versus collision phase tc (both in cycle) where the internuclear distance are (a) R¼8 a.u., (b) R¼10 a.u., and (c) R¼12 a.u., respectively.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2.0 2.25 2.5 2.75 3.0−0.05

0

0.05

t (T0)

Ey (a

.u.)

−0.05

0

0.05

E x (a.u

.)

R=8a.u.

R=12a.u.

tc

tc

_

t i2

_ +

tc

_

_

_t i2

+

+

+

+

_

_

_

tc +

t i2

t i2

+

+

Fig. 4. Sketch of the electron dynamics. The x component (Ex) and y component(Ey) of the circular electric field are shown in Fig. 4(a) and (b), respectively.The black solid and black dashed lines indicate the collision time (tc) and DI time (ti2)for R¼8 a.u., and the collision time (tc) and DI time (ti2) for R¼12 a.u. are indicatedby there solid and red dashed lines. The signs “þ” and “�” are used to denote thedirections of the corresponding vector potential during one cycle. (For interpreta-tion of the references to color in this figure legend, the reader is referred to the webversion of this article.)

A. Tong / Optics Communications 312 (2014) 252–257 255

4. Conclusion

In summary, using the classical ensemble model, we haveinvestigated DI of stretched molecules by circularly polarized laserpulses. We find that even in the circular laser field, DI also occursthrough a nonsequential process, where the second electron isionized by the collision of the first electron. The 2D ion sum-momentum distribution changes from a one-band structure to athree-band structure. More interesting, the projected ion sum-momentum in the direction parallel to the molecular axis displaysa single-peak for R¼8 a.u., then a central peak with two sideshoulders for R¼10 a.u., and a double-peak for R¼12 a.u. From thechange of the ion sum-momentum distributions, the dominantNSDI mechanism changes from CESI to CI as R increases.

Acknowledgment

This work was supported by the National Science Foundation ofChina under Grant No. 11004070, 11234004, and the fund forexcellent youths of Hubei Provincial Department of Educationunder Grant No. Q20133001.

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0 2 4 6 8 10

−10123

t (T0)

Ener

gy (a

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20

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r ee (a

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−15 −10 −5 0 5 10

0

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60

x (a.u.)

y (a

.u.)

0 400 800−500

0

500

0 2 4 6 8 10

−10123

t (T0)

Ener

gy (a

.u.)

0

20

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r ee (a

.u.)

−10 −5 0 5 10 15−20

0

20

40

60

x (a.u.)

y (a

.u.)

0 400 800

−5000

500

Fig. 5. Two sample trajectories for R¼8 a.u. [left] and R¼12 a.u. [right]. The upper, middle, and bottom rows show the energy versus time, separation between two electronsversus time, and the electron trajectories in x–y plane, respectively. The two circles in Fig. 5(c) and (f) indicate the positions of the two nuclei.

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