Strong-field approximation for high-order above-threshold ionization of randomly oriented diatomic molecules

Chemical Physics 366 (2009) 85–90

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Strong-field approximation for high-order above-threshold ionization of randomly oriented diatomic molecules D.B. Miloševic´ a,c, M. Busuladzˇic´ b, A. Gazibegovic´-Busuladzˇic´ a, W. Becker c,* a

Faculty of Science, University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Herzegovina Medical Faculty, University of Sarajevo, Cˇekaluša 90, 71000 Sarajevo, Bosnia and Herzegovina c Max-Born-Institut, Max-Born-Strasse 2a, 12489 Berlin, Germany b

a r t i c l e

i n f o

Article history: Received 27 May 2009 Accepted 1 September 2009 Available online 6 September 2009 Keywords: Molecules Attosecond Above-threshold ionization Electron spectra Alignment Double-slit interference

a b s t r a c t High-order above-threshold ionization of diatomic molecules by a strong linearly polarized field is considered using the molecular strong-field approximation. Tunneling–rescattering as the mechanism of this ionization gives rise to a novel two-source double-slit interference, which involves the four geometrical paths that are available to an electron, which can be ionized from and rescattered off either of the two centers of the diatomic molecule. For a comparison of this theory with experiments in the absence of molecular alignment, it is necessary to average the theoretical results over the molecular orientation. This paper presents technical details of this averaging procedure. It is shown that, depending on the molecular symmetry, the destructive two-source double-slit interference minima can survive the orientation averaging and can be observed in the angle-resolved electron spectra. This is illustrated on the examples of N2 and O2 molecules. It is also shown that two- and three-dimensional versions of orientation averaging lead to qualitatively similar results. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction The strong-field approximation (SFA) is at the core of the semianalytical theory of laser-induced strong-field ionization of atoms [1–4]. When it is extended to allow for rescattering and atom-specific electron-atom scattering potentials are introduced, its predictions are rather reliable, especially for the angle-resolved energy spectra (see [5] and references therein). In contrast, accurate results for total ionization rates are harder to come by because the latter are dominated by the emission of electrons of very low energy, which are only poorly represented by the SFA because it does not properly account for the atomic Coulomb potential [6]. The extension of the SFA to molecules has only recently come under closer scrutiny [7–11]. It is much more demanding not only because of the structure of the molecule that needs be taken into account but also because other problems that already hamper the atomic SFA, such as its neglect of all bound states except for the initial ground state and its notorious dependence [12] on the gauge employed, become much more severe [11,13]. Yet, many advances have recently been made and the molecular SFA has progressed to the point [14,15] – and sufficiently detailed experimental data have become available [16,17] – where a comparison with the latter has become meaningful [18]. * Corresponding author. Tel.: +49 3063921372; fax: +49 3063921309. E-mail address: [email protected] (W. Becker). 0301-0104/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2009.09.002

The SFA provides the T-matrix element into a fully specified final continuum state. All parameters that are not observed have to be integrated over. This is one of the reasons of why a T-matrix computation of the total ionization rate is particularly cumbersome. In experiments and also in approaches based on the solution of the time-dependent Schrödinger equation the situation is the exact opposite. Inclusive results are easiest to obtain, and more differential ones require harder work [19] (see also [20] and references therein). For observations on molecules, one parameter that is difficult to fix in experiments with high precision is the orientation of the molecule. Interesting features that exist for well-defined orientation may be washed out in the average. This paper considers technical details of the orientation average in SFA calculations, especially the extent to which a planar geometry is sufficient, and applications to high-order above-threshold ionization of molecules. A novel interference phenomenon that goes beyond the phenomenology of the standard double slit and the question of whether its footprint in the data survives the orientation average provide a case in point [21]. The paper is organized as follows. In Section 2 we briefly summarize our earlier work in so much as it is indispensable for the present paper. In the next Section 3, we discuss the dependence of the T matrix on the various angles and define the problem of averaging over the orientation of the molecule. In Section 4, we simplify the problem by considering the case where all vectors are coplanar. Next, in Section 5 we present the results of full

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three-dimensional (3D) averaging. In Section 6 we consider for the coplanar case the partial rates for specific molecular orientation angles in order to explain the differences between the molecules N2 and O2 , and in Section 7 we compare two-dimensional (2D) and 3D averaging. 2. General theory In this paper, we will only be concerned with the part of the T matrix that is due to rescattering (in earlier work, this part was denoted by T ð1Þ ; here we drop the superscript ‘‘(1)”). For all details of the molecular SFA, which underlies this paper, we refer to Refs. [11,15], for direct and rescattering ionization, respectively. The total rescattering T-matrix element can be decomposed into four terms that are distinguished by the geometrical pathway followed by the electron on its way from the molecule to the detector as illustrated in Fig. 1:

T ¼ T þþ þ T  þ T þ þ T þ :

ð1Þ

The electron can tunnel out of either atom and after its excursion in the continuum rescatter at either atom, which gives rise to four terms. Note that the terms T þ and T þ , where the electron is ionized and rescatters off different atoms, describe topologically different pathways: in the lower left panel, the electron passes the second center, where it will subsequently rescatter, shortly after it was ionized, at the beginning of its excursion away from the molecule. In contrast, in the lower right panel, it encounters the second center for the first time when it returns to rescatter. In consequence, these two terms are different and exhibit, for example, slightly different cutoffs. However, for the comparatively small internuclear separations that we shall consider the differences are not very pronounced, and we shall assume that the four contributions depicted in Fig. 1 are identical up to respective phases, which are related to the length of the respective orbits. For the two ‘‘nontransfer” terms where the electron is born at and rescatters off the same atom this path-length difference is R0 cosðhfe  hL Þ so that

  T þþ þ T  ¼ eia þ sak eia A

ð2Þ

with a ¼ pf  R0 =2. The quantity sak depends on the symmetry of the molecular orbital which is a linear combination of the atomic orbi-

k st

+

k st

–

R0

L

L

pf

fe

pf T

T

L

k st pf T

tals a. Hence, the right-hand side of Eq. (2) is proportional to cos a or sin a, respectively, for sak ¼ þ1 and sak ¼ 1. This gives rise to the well-known two-center destructive interference, which for ‘‘direct” processes (not involving rescattering) was predicted in Ref. [7] and first observed in Ref. [16]. With our assumption that all four terms have about the same magnitude, the above symmetry will not be observed owing to the contributions of the ‘‘transfer” terms T þ and T þ . The corresponding two paths differ from each other by twice the projection of the internuclear axis on the direction of the field polarization minus the projection of the internuclear axis on the direction of the final electron momentum:

  T þ þ T þ ¼ eib þ sak eib A

ð3Þ

with b ¼ kst  R0  pf  R0 =2. We note that the factor sak is associated with the center where the electron starts, not where it rescatters. The interference pattern (3) of the transfer terms is different from that of the nontransfer terms, Eq. (2). Because they overlap neither one can be observed separately. The sum of all four terms can be factorized as follows:

h i ih i i i i T þþ þ T  þ T þ þ T þ ¼ e2ðabÞ þ e2ðabÞ e2ðaþbÞ þ sak e2ðaþbÞ A  cos d ðsak ¼ 1Þ ; ð4Þ ¼ 4A cos c i sin d ðsak ¼ 1Þ where

2c ¼ a  b ¼ ðpf  kst Þ  R0  2ðc cos hfR  dÞ; 2d ¼ a þ b ¼ kst  R0  2b cos hL

ð5Þ

and we have introduced the parameters b ¼ kst R0 =2 and c ¼ pf R0 =2 for later use. Remarkably, when c ¼ np þ p=2 with integer n, Eq. (4) predicts destructive interference regardless of the molecular symmetry. No double slit can create this pattern. It requires four paths to contribute, such as they are generated by two sources plus two slits. To be more specific, the total rescattering T-matrix element T þþ þ T  þ T þ þ T þ for the ath constituent of the molecular orbital having the symmetry r (for N2 ) or p (for O2 ) is found to be proportional to [14,15]

r ðsak ¼ þ1Þ; r ðsak ¼ 1Þ; sin hL sin d cos c for a ¼ 2p and p ðsak ¼ 1Þ:

cos d cos c for a ¼ s and

cos hL sin d cos c for a ¼ p and

ð6Þ

As mentioned before, the magnitudes of the four terms T þþ ; T  ; T þ , and T þ are not exactly equal and, hence, the interference pattern predicted by Eq. (6) will not be exactly observed. However, for the moderate values of R0 that we consider the destructive interference is very pronounced and, moreover, its position does not depend on the molecular symmetry, as predicted by Eq. (4). 3. Definition of the problem and 3D averaging

k st

Theoretically, in the spherical coordinate system with the z axis ^L (see Fig. 2) the vectors of the internualong the polarization axis e

pf T

Fig. 1. Schematic diagram of the four rescattering T-matrix contributions to strongfield ionization of a diatomic molecule represented by two atomic centers denoted by ‘‘+” and ‘‘”. The electron drift momentum kst between ionization and rescattering is along the laser polarization axis. The final momentum pf and the relative nuclear coordinate R0 include the angles hfe and hL with respect to the former. In the upper panels the contributions T þþ and T  are depicted, where the electron rescatters at the same center at which it was born, and the corresponding electron path-length difference D ¼ R0 cosðhfe  hL Þ is indicated. The two lower panels show the contributions of the processes where ionization and rescattering occur at different centers. The path-length difference of these two contributions, before rescattering, is 2DL ¼ 2R0 cos hL .

z

eˆ L pf

fe

R0

L

L

y

x Fig. 2. Definition of the 3D coordinate system used in our theoretical considerations.

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87

a

clear separation R0 and the electron momentum pf are represented by ðR0 ; hL ; uL Þ and ðpf ; hfe ; ufe Þ, respectively, while the vector pf in the spherical coordinate system with the z axis along the internub 0 is ðp ; hfR ; u Þ. The following connection is valid clear axis R f fR

cos hfR ¼ cos hL cos hfe þ sin hL sin hfe cosðufe  uL Þ:

ð8Þ

so that the dependence on the angle ufe is eliminated. Since the molecular axis orientation is unknown, in order to compare our numerical results with the experimental results, we b 0 . Therefore, for a fixed angle have to average over all directions R he we have to calculate the orientation-averaged differential ionization rate

 e Þ ¼ 2ppf wðh

Z 2p Z duL p sin hL dhL jTðhe ; hL ; uL Þj2 : 2p 0 2 0

eˆ L R0

ð9Þ

With this formula all molecular orientations with respect to the ^L and pf , in which the experimental plane defined by the vectors e spectrum is recorded, are taken into account with equal statistical weight. Since the uL -dependence of the rescattering T-matrix element is only via cos uL ¼ cosð2p  uL Þ, in Eq. (9) we can replace Rp R 2p duL with 2 0 duL . 0

b

eˆ L

pf

fR

fR

x L

L

z

0

L

e

L

0,

L

e

pf x

eˆ L

L

L

L

R0

fe

L

2

L

,2

Fig. 3. (a) Definition of the 2D coordinate system. (b) and (c) Connection between the angles in 2D and 3D systems.

5. Analysis of 3D averaging for r and p orbitals Introducing Eq. (6) into (9) and calculating numerically the 2D integral over uL and hL , we can investigate how the ionization yield depends on the electron emission angle he and the laser parameters. We are interested in the high-energy electrons. Therefore, we will fix the electron energy to be equal to the cutoff energy for he ¼ 0, which is given by [22]: Epf ;max ðhe ¼ 0Þ ¼ 10:007U p þ0:538Ip , where Ip is the ionization potential and U p ¼ A20 =4 is the electron’s ponderomotive energy in the laser field with the vector-potential amplitude A0 . The corresponding stationary momentum is kst ¼ 2A0 =ð3pÞ [15]. In Fig. 4 we present these ionization yields as functions of the electron emission angle he for the laser parameters relevant to the experiment [18]. The yield for the O2 molecule shows a minimum for he ¼ 0, while for N2 , both for the s and p orbitals, such minima do not appear. For he ¼ 0 the rescattering T-matrix element does not depend R 2p on uL so that 0 duL ! 2p. Introducing the new variable x ¼ cos hL , we obtain

180

10 15 5 20 25

p, σ

10 3 15 20 25

2p, π

5 10 15 20

4 2 3 160

ð10Þ

2

In the 2D case the factor sin hL , which appears in Eq. (9), is absent. This factor expresses the fact that in 3D perpendicular alignment is much more likely than parallel alignment. The connection between the angles in 2D and 3D geometries is given in Fig. 3b and c where the case hfe ¼ he P 0 is considered. The same formulas are valid for he < 0. According to Fig. 3b and c the connection between the 2D rescattering matrix element, which appears in Eq. (10), and its 3D counterpart (9) is given by

T 2D ðhe ; 0 6 /L 6 pÞ ¼ Tðhe ; /L ; 0ÞjhfR ¼jhe hL j ; T 2D ðhe ; p 6 /L 6 2pÞ ¼ Tðhe ; 2p  /L ; pÞjhfR ¼jhe þhL j :

fe

fR

Yield (arb. units)

Z 2p d/L jT 2D ðhe ; /L Þj2 : 2p 0

c

z

R0

s, σ

Eq. (9) corresponds to full 3D orientation averaging. However, since such calculations are very time-consuming, in Ref. [18] we have used 2D averaging, restricting ourselves to the case where ^L , R0 , and pf lie in the same plane. In this case, the anall vectors e gles hL and uL do not appear in the problem and only the angle /L ^L and R0 is relevant [see Fig. 3a]. It assumes between the vectors e values from 0 to 2p, and the 2D averaged differential ionization rate is defined by

0, 2

L

x

4. 2D averaging and its connection with 3D averaging

 2D ðhe Þ ¼ 2ppf w

L

ð7Þ

It can be shown analytically that for our diatomic molecules the rescattering T-matrix element depends on the above-defined angles only via sin hL ; cos hL ; cos hfe , and cos hfR [15]. Therefore, the only dependence on the angles ufe and uL is given by Eq. (7). In the experiment [18] the ionization yields of N2 and O2 molecules have been measured and the results were presented in false colors in the electron emission angle-energy ðhe ; Epf Þ plane (the subscript ‘‘e” stands for experimental). The spectrum is recorded ^L and pf and with the z axis in the plane defined by the vectors e along the laser polarization axis. Since the molecules are randomly oriented, the experimental results do not depend on the angle ufe , i.e., we are free to take any value ufe ¼ const: For he P 0 we can fix ufe ¼ 0; he ¼ hfe 2 ½0; p (this case is presented in Fig. 2), while for he < 0 we can choose ufe ¼ p; he ¼ hfe 2 ½p; 0Þ. Only the angles p=2 6 he 6 p=2 are considered since the other angles correspond ^L . With this ^L ! e to the flip of the laser polarization direction: e definition, Eq. (7) can be rewritten as

cos hfR ¼ cos he cos hL þ sin he sin hL cos uL ;

z

ð11Þ

-45 0 45 θe (deg)

1

1

-45 0 45 θe (deg)

-45 0 45 θe (deg)

Fig. 4. Estimate of the orientation-averaged differential ionization rates of N2 (s; r orbital and p; r orbital) and O2 (2p; p orbital) in arbitrary units as functions of the electron emission angle. The wavelength of the linearly polarized laser field is 800 nm, while its intensity is j  1013 W=cm2 ; j ¼ 5; 10; 15; 20; 25, with the value of j given in each panel. The electron energy corresponds to the respective cutoff, while the stationary momentum is kst ¼ 2A0 =ð3pÞ, as explained in the text.

D.B. Miloševic´ et al. / Chemical Physics 366 (2009) 85–90

 p;r ð0Þ / pf w

Z

1

dx cos2 ðbxÞ cos2 ððc  bÞxÞ; 0

Z

 2p;p ð0Þ / pf w

ð12Þ

1

2

2

dxx sin ðbxÞ cos2 ððc  bÞxÞ;

ð13Þ

0

Z

1

2

dxð1  x2 Þ sin ðbxÞ cos2 ððc  bÞxÞ:

ð14Þ

0

All these integrals can be calculated analytically. For high-energy electrons we have pf > kst , so that c > b. In this case, for 2b > 1 we obtain that the integrals (12) and (13) are equal to a constant plus a term proportional to 1=ð2bÞ, while the integral (14) is equal 2 to a constant plus a term proportional to 1=ð2b Þ. Therefore, for 2b > 1 the integral that corresponds to the p-orbital is b times smaller. This is valid only for he ¼ 0 since in this case the terms pro2 portional to 1=b cancel and only the terms proportional to 1=b remain in Eq. (14). For a laser field whose electric-field vector changes sign upon a translation in time by one half of the optical period, which is the case for a monochromatic elliptically polarized laser field, the differential ionization rate for emission of an electron having the momentum pf at the detector satisfies the twofold symmetry ([23]; see also Refs. [24,25] for atoms and negative atomic ions): wðpf Þ ¼ wðpf Þ, i.e., in the spherical coordinates, wðpf ; p  hfe ; ufe þ pÞ ¼ wðpf ; hfe ; ufe Þ. As we have explained in Section 3, the ^L and spectrum is recorded in the plane defined by the vectors e pf so that the above inversion-symmetry relation reduces to wðhe þ pÞ ¼ wðhe Þ. Also, a homonuclear diatomic molecule is symmetric with respect to the substitution R0 ! R0 , i.e., ðR0 ; hL ; uL Þ ! ðR0 ; p  hL ; uL þ pÞ. It can be shown that the orientation-averaged differential ionization rate (9) is invariant with respect to the transformation he ! he , i.e.

 e Þ:  wðh e Þ ¼ wðh

to he ¼ 0, as it is shown for /L ¼ 45 in the right-hand panel of Fig. 5. The sum of these two contributions (/L ¼ 45 and /L ¼ 135 in Fig. 5) gives the summed partial rate. Summing over the orientation angles /L produces curves as shown in the righthand panel of Fig. 4, which display a prominent depression centered at he ¼ 0 in agreement with the experimental observations [18,21]. In Fig. 6 we show analogous results for the N2 molecule. Comparing with the previous case of O2 , we notice that for orientation angles /L below 60° or 70° the results are not dramatically different; especially, the deep suppressions whose positions in view of Eq. (6) are given by the zeroes of cos c are almost identical. The crucial difference between N2 and O2 comes in for angles /L greater than about 70°. For these angles, the position of the suppression

-10

1 10 20 30 40

-11

-12

-14 -45

Namely, using relations (8) and (5), (6) it can be shown that

T a;p ðhe ; p  hL ; uL Þ ¼ T a;p ðhe ; hL ; uL Þ;

ð16Þ

so that sin hL jTj2 is invariant with respect to the substitution ðhe ; hL ; uL Þ ! ðhe ; p  hL ; uL Þ and relation (15) is valid. For the 2D rescattering T-matrix element (11) the symmetry relation (16) takes the form

T 2D ðhe ; 0 6 /L 6 pÞ ¼ Tðhe ; p  /L ; 0Þ ¼ T 2D ðhe ; p  /L Þ;

-10

so that

More precise numerical results for the T-matrix element can be obtained by numerical integration as explained in Refs. [14,15,18]. We will calculate the partial differential ionization rates, defined by the quantity 2ppf jT 2D ðhe ; /L Þj2 , as functions of the electron emission angle he , for different values of the molecular orientation angle /L . We will also use the symmetry relation (18). Let us first analyze the case of the O2 molecule. From the two left panels of Fig. 5 we see that the dominant contribution to the rate integrated over /L comes from the interval 30 6 /L 6 70 . All these partial rates have a minimum not too far from he ¼ 0, which moves from positive to negative he for increasing molecular-orientation angle /L . For each /L -contribution there is a ð180  /L Þ-contribution having the mirror symmetry with respect

2

log10 [2πpf |T| (a.u.)]

ð18Þ

6. Numerical results for 2D averaging

0 45 θe(deg)

-45

0 45 θe(deg)

-45

0 45 θe(deg)

Fig. 5. 2D partial differential ionization rates of O2 as functions of the electron emission angle for different values of the molecular orientation angle /L whose values (in degrees) are denoted in each panel. The intensity of the linearly polarized laser field is 0:7  1014 W=cm2 and the wavelength is 800 nm. The electron energy is Epf ¼ 9:1 U p ¼ 38:03 eV. The results for the T-matrix element are obtained by numerical integration as explained in [14,15,18].

T 2D ðhe ; p 6 /L 6 2pÞ ¼ Tðhe ; /L  p; pÞ ¼ T 2D ðhe ; p  /L Þ; ð17Þ

jT 2D ðhe ; /L Þj2 ¼ jT 2D ðhe ; p  /L Þj2 :

45 135 sum

-13

ð15Þ

T a;r ðhe ; p  hL ; uL Þ ¼ T a;r ðhe ; hL ; uL Þ;

89 80 70 60 50

2

 s;r ð0Þ / pf w

log10[2πpf |T| (a.u.)]

88

0 10 20 30 40

45 135 sum

-11

-12

90 80 70 60 50 -45 0 45 θe (deg)

-45 0 45 θe (deg)

-45 0 45 θe (deg)

Fig. 6. Same as in Fig. 5 but for N2 , the laser intensity is 0:9  1014 W=cm2 , and the electron energy Epf ¼ 9:35 U p ¼ 50:3 eV.

D.B. Miloševic´ et al. / Chemical Physics 366 (2009) 85–90

7. Comparison of 2D and 3D averaging It is obvious that the 2D averaging (10) is an approximation to the 3D averaging (9). We will now investigate how good this approximation is by comparing the corresponding partial rates for fixed values of the angle hL . In this case, Eq. (9) for the 3D case gives

Z 2p  eÞ dwðh duL 2 ¼ 2ppf jTðhe ; hL ; uL Þj2 : sin hL dhL 2p 0

L

ð20Þ

log10 [Averaged rate (arb. units)]

-10

θL=30

o

3D 2D

θL=45

o

3D 2D

θL=60

o

-11

-12

0 -45 45 θe (deg)

3D 2D

o

3D 2D

o

θL=45

0 -45 45 θe (deg)

θL=60

o

-11

-12

0 -45 45 θe (deg)

0 -45 45 θe (deg)

Fig. 8. Same as Fig. 7 but for N2 and for the electron energy and the laser parameters as in Fig. 6.

-10

3D 2D o θL=1

O2

3D 2D o θL=89

O2

3D 2D o θL=0

N2

0 -45 45 θe (deg)

Fig. 7. Comparison of the 3D (integrated over uL ) and the 2D (sum of /L ¼ hL and /L ¼ 2p  hL contributions) partial differential ionization rates of O2 for fixed hL whose value is denoted in each panel. The electron energy and the laser parameters are as in Fig. 5.

N2

-11

-12

3D 2D o θL=90

-13

/L ¼2phL

These 3D and 2D averaged rates as functions of the angle he are compared in Figs. 7–9. One can see that the 3D and 2D results are in qualitative agreement. Especially for the case of perpendicular alignment (Fig. 9) the agreement is very good. (For parallel alignment, we have a genuinely coplanar situation so that 2D averaging is exact.) We conclude that 2D averaging, which was utilized in Ref. [18], is sufficient for our analysis of the two-source two-slit interference.

3D 2D

θL=30

0 -45 45 θe (deg)

log10[Averaged rate (arb. units)]

i 1h 2ppf jTðhe ; hL ; 0Þj2 þ 2ppf jTðhe ; hL ; pÞj2 2 ¼ ppf jT 2D ðhe ; /L ¼ hL Þj2 þ ppf jT 2D ðhe ; /L ¼ 2p  hL Þj2    2D ðhe Þ  2D ðhe Þ dw dw   ¼p þ p :   d/ d/ /L ¼hL

3D 2D

ð19Þ

According to Fig. 3b and c, for fixed hL 2 ½0; p, from all values of uL in Eq. (19), in the 2D approximation only the following two configurations contribute: the angle uL ¼ 0 for which /L ¼ hL and the angle uL ¼ p for which /L ¼ 2p  hL , i.e., we have that (19) should be connected with

L

-10

log10 [Averaged rate (arb. units)]

has moved far away from he ¼ 0. Adding the contribution from 180  /L now leads to a maximum at he . For O2 , these contributions are small—indeed, the rescattering plateau is completely absent for /L ¼ 0 and 90°—hence, the yield averaged over /L still exhibits the minimum for he ¼ 0. In contrast, for N2 the contributions from large /L are dominant and cover the minima that are due to the smaller values of /L . This explains why the minima are not seen in the orientation-averaged results and the experimental data for N2 .

89

-45 0 45 θe (deg)

-45 0 45 θe (deg)

-45 0 45 θe (deg)

-45 0 45 θe (deg)

Fig. 9. Same as Figs. 7 and 8 but for parallel (almost parallel for O2 ) and perpendicular (almost perpendicular for O2 ) orientation. The parameters are those of Fig. 7 for O2 and Fig. 8 for N2 .

8. Conclusions For fixed molecular orientation, the interference of the four geometrical orbits in the rescattering regime that an electron can choose for its path from the initial molecular ground state into the final continuum state with specified asymptotic momentum can be almost completely destructive, giving rise to a pronounced minimum in the electron yield for emission in a certain direction. Experimental data have been mostly collected for random molecular orientation so that the theoretical results have to be averaged over the former before they can be compared to the data. For specified molecular orientation, the electron momentum for which the four-orbit (two-source double-slit) interference is most destructive is almost the same for the molecules O2 (p symmetry) and N2 (r symmetry). However, only for O2 does a manifestation of the destructive interference survive in the orientation-averaged spectrum in the form of a marked dip in the direction of the applied field. This is due to the fact that for N2 the averaged yield is dominated by the contribution of perpendicular alignment, which is not subject to the destructive interference. For O2 , owing to its p symmetry, there is no contribution from perpendicular alignment, and the interference survives. We have also confirmed that for qualitative results it is sufficient to perform the average for the coplanar case where the three vectors of the problem (the molec-

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D.B. Miloševic´ et al. / Chemical Physics 366 (2009) 85–90

ular orientation, the electron momentum, and the field polarization) lie in the same plane. Finally, it should be mentioned that the destructive-interference minima can also be seen in the process of high-order harmonic generation, which is analogous to high-order abovethreshold ionization in the sense that the tunneling–rescattering scenario is replaced by the tunneling–recombination scenario. The interference minima in the harmonic spectrum are caused by the destructive interference of the corresponding partial recombination amplitudes [26,27]. In this case, there is no final electron momentum but, nevertheless, it will be interesting to explore how the random orientation of the molecules will influence this process. Acknowledgments We are grateful to K. Ueda for illuminating discussions. This work was supported in part by the Federal Ministry of Education and Science, Bosnia and Herzegovina. References [1] L.F. DiMauro, P. Agostini, Adv. At. Mol. Opt. Phys. 35 (1995) 79. [2] W. Becker, F. Grasbon, R. Kopold, D.B. Miloševic´, G.G. Paulus, H. Walther, Adv. At. Mol. Opt. Phys. 48 (2002) 35. [3] D.B. Miloševic´, F. Ehlotzky, Adv. At. Mol. Opt. Phys. 49 (2003) 373. [4] A. Becker, F.H.M. Faisal, J. Phys. B 38 (2005) R1. [5] A. Cˇerkic´, E. Hasovic´, D.B. Miloševic´, W. Becker, Phys. Rev. A 79 (2009) 053410. [6] A. Becker, L. Plaja, P. Moreno, M. Nurhuda, F.H.M. Faisal, Phys. Rev. A 64 (2001) 023408. [7] J. Muth-Böhm, A. Becker, F.H.M. Faisal, Phys. Rev. Lett. 85 (2000) 2280; 96 (2006) 039902.

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