Non-linear single and double ionisation of molecules ...

LETTER TO THE EDITOR

Non-linear single and double ionisation of molecules by strong laser pulses Alexander I. Pegarkov1,2 , Eric Charron1,4 and Annick Suzor-Weiner1,3,4 (1) Laboratoire de Photophysique Mol´eculaire du CNRS, Bˆatiment 213, Universit´e Paris XI, 91405 Orsay Cedex, France. (2) Physics Faculty, Voronezh State University, 1 Universitetskaya Ploschad, Voronezh 394693, Russia. (3) Laboratoire de Chimie Physique, 11 rue Pierre et Marie Curie, Universit´e Paris VI, 75231 Paris Cedex 05, France. (4) Laboratoire de Recherche correspondant du CEA (DSM-9702). Abstract. We present a theoretical treatment for the single and double ionisation of molecules in strong laser pulses, including the effect of electron correlations. Applied to a model N2 molecule, our simulations show that double ionisation occurs via a direct ✲ N2+ two-electron mechanism at moderate laser intensities (N2 ¯hω 2 +2e) while a step✲ N+ by-step sequential ionisation process dominates at higher intensity (N2 ¯hω 2 +e hω✲ N2+ +2e). At intermediate intensity, these two mechanisms have a comparable ¯ 2 contribution to the total double ionisation yield. This phenomenon is directly reflected 2+ on the spatial distributions of N+ 2 and N2 in the focal volume of the laser pulse. In addition we show that at high intensity the singly and doubly charged molecular ions are created in well-separated regions of space.

PACS numbers: 33.80.Rv,82.50.Fv

Letter to the Editor

2

Early theoretical models studying the interaction of molecules with intense laser pulses where based on the so-called single active electron (SAE) approximation [1]. These simple descriptions have been able to explain surprising experimental features such as the enhancement of ionisation at critical internuclear distances [2, 3, 4, 5]. The measurement, a few years ago, of a shoulder in the double ionisation yield of the He atom at moderate intensity [6] has shed a new light on the influence of electron correlations on intense field phenomena. This shoulder was clearly associated with the domination of a direct two-electron ionisation process in the He2+ signal at moderate intensity. It is only recently that some theoretical studies including two correlated electrons have been able to reproduce this non-sequential effect [7, 8, 9, 10]. More recently, two experimental groups reported the first evidence of non-sequential double ionisation in molecular systems [11, 12]. These observations were made on two diatomics (NO and N2 ), as well as a small linear polyatomic (C2 H2 ). In the present paper we use an ab-initio two-electron model similar to the atomic models [8, 9] to calculate the impact of electronic correlations on the direct and sequential double ionisation probabilities of the diatomic nitrogen molecule N2 . Our results are in good qualitative agreement with the experiment of Cornaggia and Hering [12] and confirm the predominance of the non-sequential mechanism at low intensity and of the sequential process at high intensity. In addition, we present here the spatial 2+ distributions of the N+ 2 and N2 ions formed in the focal volume of the pulse. As demonstrated experimentally [12], N2 mainly ejects its two first electrons at short internuclear distance, following the Franck-Condon principle. We have thus decided to fix the internuclear distance R in our calculation at the value of the equilibrium distance of the ground electronic state of the neutral nitrogen molecule Re = 2.075 au. Additionally, due to the linear polarisation of the laser pulse, we restrict the motion of the electrons along the electric field axis. Based on the experimental evidence of molecular alignment [13, 14], we also assume that the internuclear axis is aligned along the same vector. This two-electron molecular model has already been used by Wiedemann and Mostowski [15] and Yu et al [16]. The electronic wave function of this model molecule in a pulsed laser field evolves according to the time - dependent Schr¨odinger equation (in atomic units) ∂ (1) i Ψ(x1 , x2 , t) = [Hmol (x1 , x2 ) + Vint (x1 , x2 , t)]Ψ(x1 , x2 , t), ∂t where Hmol (x1 , x2 ) is the Hamiltonian of the field-free diatomic molecule Hmol (x1 , x2 ) = T (x1 ) + VeN (x1 ) + T (x2 ) + VeN (x2 ) + Vee (|x1 − x2 |).

(2)

T (x) is the electronic kinetic energy operator 1 ∂2 , x = {x1 , x2 }. (3) T (x) = − 2 ∂x2 VeN (x1 ), VeN (x2 ) and Vee (|x1 − x2 |) are the softened electron - nuclei and electron electron Coulomb interactions q q −q , (4) VeN (x) = − q (x + R/2)2 + a2 (x − R/2)2 + a2

3

Letter to the Editor 1 , Vee (|x1 − x2 |) = q (x1 − x2 )2 + a2

(5)

where q = 1 is the charge placed on each nucleus for the neutral nitrogen. The softening parameter a in the Coulomb potentials VeN (x) and Vee (|x1 − x2 |) of Eq. (4) and (5) has been chosen (a = 1.2928 au) so that the energy of the ground electronic state of the molecule equals the double ionisation potential of N2 , i.e. 42.68 eV. Vint (x1 , x2 , t) is the laser - molecule dipolar electric interaction energy Vint (x1 , x2 ) = −(x1 + x2 )E0 sin2 (πt/2τ ) cos(ωt).

(6)

E0 and ω are the amplitude and angular frequency of the electric wave, τ is the pulse duration. Starting from the ground electronic state of the molecule at time t = 0, the wavepacket Ψ(x1 , x2 , t) at time t is calculated by means of the split operator method [17]. Absorbing boundary conditions are imposed at the end of the grid. The outgoing ionisation flux is summed to calculate the probabilities P1e and P2e of single electron ionisation (for {|x1 | ≫ R, |x2 |} or {|x2 | ≫ R, |x1 |}) and of direct two-electron ionisation (for |x1 |, |x2| ≫ R), respectively. P1e and P2e are shown in Figure 1 as a function of the peak intensity of the pulse. The laser wavelength (λ = 800 nm) and pulse duration (τ = 50 fs) are comparable with those used experimentally by Cornaggia and Hering [12]. As the intensity increases, P1e increases rapidly and reaches saturation at the intensity I = 2 · 1014 W/cm2 . P1e stands here for the probability of finding a single detached photoelectron far from the nuclei. The probability of measuring a singly ionised molecule N+ 2 differs from P1e because the part of the ionised wavepacket associated with this probability is absorbed at the end of the numerical spatial grid whereas the real outgoing ionisation flux could undergo a second step of photoionisation before the pulse ends. The probability of this step-by-step (or sequential) double ionisation process must be subtracted from P1e to obtain the probability of formation of the singly ionised molecule N+ 2. is shown as a dotted line The sequential ionisation probability Pseq = P1e × PNSAE + 2

in Figure 1. PNSAE stands here for the ionisation probability of the molecular ion N+ + 2 2 considered as a single active electron system [3, 18]. In this one-electron model, the interaction potential between the electron and the two nuclei has the same form as in Eq. (4) with q = 1 and the parameter a = 1.3787 au chosen to get the correct single ionisation potential of N+ 2 , i.e. 27.21 eV. The sequential process corresponds to a stepby-step two-electron ionisation where the electrons are detached from the molecule at different times, and thus have non-correlated phases. The direct two-electron ionisation probability P2e (dashed line in Figure 1) has a more complicated dependence, with a few local maxima and minima for I < 1014 W/cm2 , before saturation is reached at I = 2 · 1014 W/cm2 . Due to the difference between the double and single ionisation potentials, the probability for the direct two-electron process at very high intensity is two orders of magnitude smaller

4

Letter to the Editor

than the probability of single electron ionisation. Our results demonstrate that the double ionisation of N2 is dominated by the direct mechanism at moderate intensity (I < 1.6 · 1014 W/cm2 ) while the sequential process is predominant at high intensity (I > 2.3 · 1014 W/cm2 ). For intermediate intensities, the direct and step-by-step mechanisms present comparable contributions to the total yield.

0

10

−1

10

−2

Probability

10

−3

10

P1e

−4

10

P2e

−5

10

Pseq

−6

10

PN + 2

−7

10

PN2+ 2

−8

10

13

10

14

10

15

2

10

16

10

I (W/cm ) Figure 1. Single and double ionisation probabilities of N2 as a function of the peak laser intensity, with wavelength λ = 800 nm and pulse duration τ = 50 fs. The single electron ionisation probability P1e (see text) is shown as a solid line while the dashed line represents the probability P2e of direct two-electron ionisation. The dotted line is the sequential two-electron ionisation probability Pseq . Finally, the probabilities of formation of the singly and doubly charged ions PN+ and PN2+ are shown as stars and 2 2 diamonds respectively.

Finally, Figure 1 also presents the probabilities PN+2 and PN2+ of creating the singly 2 and doubly ionised molecular ions. As stated previously, the first probability is extracted from the difference PN+ = P1e − Pseq . On the other hand, N2+ 2 can be created either 2 directly or sequentially, thus implying the relation PN2+ = P2e + Pseq . One can see in 2 Figure 1 that an intensity I < 3 · 1014 W/cm2 mainly creates the N+ 2 ion, but for higher 14 2 intensities, I > 5 · 10 W/cm , this ion is further ionised to form predominantly the 14 doubly charged N2+ W/cm2 , a saturation of the N2+ 2 . For the intensity Is = 6 · 10 2 probability is observed, and the probability of formation of N+ becomes negligible. Note 2

5

Letter to the Editor

that this saturation intensity agrees with the experimental value [12]. These features 2+ have important consequences in the spatial distributions of the N+ 2 and N2 ions in the focal volume of the pulse. Although Figure 1 presents the quantum-mechanical probabilities calculated for a fixed peak intensity I, it is necessary to take into account the focalisation of the laser beam, which creates a continuous distribution of intensities inside a well-defined focal volume. The cylindrical symmetry of this three dimensional object with respect to the laser beam propagation axis imposes the usual coordinate system {r, z}. Since the experiments record the ions created inside this total volume, it is necessary to average the calculated probabilities over the intensity distribution. We have used the standard electrodynamic formulation for the intensity I(r, z) of a Gaussian shape in the r direction and a Lorentzian profile in the z direction, with realistic parameters for the laser waist [19].

10

Ion formation probability per laser shot

10 10 10 10

10 10 10

10 10 10

2

(a2)

I = 10 W/cm

(b1)

I = 9 x 10 W/cm

(c1)

I = 4 x 10 W/cm

−4

−6

−8

0

14

2

(b2)

15

2

(c2)

−2

−4

−6

−8

10 10

14

(a1)

−2

10 10

0

0

−2

−4

−6

−8

−2

−1

0 z (mm)

1

2 0

0.01 r (mm)

0.02

2+ Figure 2. Spatial distributions of the N+ 2 (solid line) and N2 (dashed line) ions inside the focal volume of the pulse for different peak laser intensities I. The rows (a), (b) and (c) refer to I = 1014 W/cm2 , 9 · 1014 W/cm2 and 4 · 1015 W/cm2 respectively. The first column shows the distribution along the laser beam propagation (z-) axis, while the second column shows the distribution with the orthogonal radius r.

6

Letter to the Editor

2+ Figure 2 shows the spatial distributions of N+ created along the laser 2 and N2 beam direction z (first column) and its orthogonal radius r (second column). For a peak intensity of 1014 W/cm2 (row (a)) the two kinds of ion are concentrated around the focal point {z = 0, r = 0} and their densities decrease sharply as one leaves the centre. For the higher intensity, I = 9 · 1014 W/cm2 (row (b) - note that in this case I > Is ), we see mainly the doubly charged ions N2+ 2 close to the centre of the focal volume. It is + only at larger distances that N2 may be seen. Similar but more pronounced features are obtained at higher intensities, as can be seen for I = 4 · 1015 W/cm2 (row (c)). As a consequence, this highest intensity pulse is able to separate spatially the singly from the doubly charged ions. In this example we can see the three predicted possible repartitions of the created ions inside the focal volume : the so-called potato distribution [20] corresponds to the lowest intensity (row (a)), the pyramid distribution [14, 21] is seen at intermediate intensity (row (b)), while the highest intensity distribution (row (c)) is well described by the onion model [22].

+

N2

1



10

2+

N2

0

10

−1

10

−2

10

−3

10

−4

10

−5

10

−6

SAE

10

−7

10

13

10

14

10

15

2

10

16

10

I (W/cm ) 2+ Figure 3. Number of ions N+ (dashed line) created per laser 2 (solid line) and N2 shot. The dotted line shows the results obtained using a single active electron (SAE) model. The number of ions is averaged over the intensity distribution of the focal volume.

Finally, Figure 3 shows the ion formation probabilities averaged over the intensity profile experienced by the molecules in the focal volume. The number of singly charged 2+ ions N+ 2 (solid line) always exceeds the number of doubly charged ions N2 (dashed line).

Letter to the Editor

7

Our calculated ion yields behave as I 3/2 for high intensity : I > 6 · 1014 W/cm2 for N+ 2 and I > 1015 W/cm2 for N2+ , which is well known from experiments. In this domain 2 2+ of intensity, the branching ratio between the formation of N+ 2 and N2 agrees very well with the experimental value [12]. A comparison of Figures 1 and 3 shows the substantial difference between the quantum probabilities calculated at a fixed spatial point (i.e. for a single intensity) and the total number of ionised molecules. Note that the result of a complete sequential SAE calculation (dotted line in Figure 3) underestimates the N2+ signal by several orders of magnitude at moderate intensities. This is a direct 2 indication that non-sequential double ionisation dominates the N2+ 2 signal in this range 14 2 of peak intensities (I < 3 · 10 W/cm ). The characteristic hollow of the N2+ yield 2 14 2 14 2 in the interval 1.6 · 10 W/cm < I < 6 · 10 W/cm occurs due to the saturation of direct two-electron detachment in conjunction with a fast increase of sequential double ionisation. In conclusion, the present model of a diatomic molecule with two Coulombic centres and two active one dimensional electrons gives a correct qualitative picture of the non-linear single and double molecular ionisation in short and intense laser pulses. This numerical simulation highlights the role played by electron correlations in the competition between the direct two-electron ionisation mechanism and the step-by-step sequential process. The spatial intensity distribution created by focalisation of the laser has a large effect on the ion yields. Different focalisations can change the ion densities inside the focal volume and can modify the ion yields significantly. Our model calculations show that for each ion charge state, different spatial distributions can be obtained depending on the peak intensity of the pulse. A further development of the present approach is now in progress in order to analyse the non-linear dynamics of excitation of a two electron molecule and the impact of high excited states upon it. We would like to thank Christian Cornaggia for fruitful discussions. Tamar Seideman is gratefully acknowledged for stimulating discussions at the preliminary stage of this work and for having provided a subroutine used in our SAE programs. AIP thanks the French Ministry for Education and Research and the Russian Ministry for Basic and Professional Education (programme ”Universities of Russia - Fundamental Research”) for support. References [1] [2] [3] [4]

Codling K., Frasinsky L. J. and Hatherly P. A. 1989 J. Phys. B: At. Mol. Opt. Phys. 22 4321 Schmidt M., Normand D. and Cornaggia C. 1994 Phys. Rev. A50 5037 Seideman T., Ivanov M. and Corkum P. B. 1995 Phys. Rev. Lett. 75 2819 Posthumus J. H., Frasinsky L. J., Giles A. J. and Codling K. 1995 J. Phys. B: At. Mol. Opt. Phys. 28 L349 [5] Chelkowski S. and Bandrauk A. D. 1995 J. Phys. B: At. Mol. Opt. Phys. 28 L723 Yu H., Zuo T. and Bandrauk A. D. 1998 J. Phys. B: At. Mol. Opt. Phys. 31 1533 [6] Fittinghoff D. N., Bolton P. R., Chang B. and Kulander K. C. 1992 Phys. Rev. Lett. 69 2642 Walker B., Sheehy B., DiMauro L. F., Agostini P., Schafer K. J. and Kulander K. C. 1994 Phys. Rev. Lett. 73 1227

Letter to the Editor

8

[7] Watson J. B., Sanpera A., Lappas D. G., Knight P. L. and Burnett K. 1997 Phys. Rev. Lett. 78 1884 [8] Bauer D. 1997 Phys. Rev. A56 3028 [9] Lappas D. G. and Leeuven van R. 1998 J. Phys. B: At. Mol. Opt. Phys. 31 L249 [10] Lambropoulos P., Maragakis P. and Zhang J. 1998 Phys. Rep. 305 203 [11] Talebpour A., Larochelle S. and Chin S. L. 1997 J. Phys. B: At. Mol. Opt. Phys. 30 L245 [12] Cornaggia C. and Hering Ph. 1999 J. Phys. B: At. Mol. Opt. Phys. 31 L503 [13] Normand D., Lompr´e L. A. and Cornaggia C. 1992 J. Phys. B: At. Mol. Opt. Phys. 25 L497 Strickland D.T., Beaudouin Y., Dietrich P. and Corkum P. B. 1992 Phys. Rev. Lett. 68 2755 [14] Schmidt M., Dobosz S., Meynadier P., D’Oliveira P., Normand D., Charron E. and Suzor-Weiner A. Phys. Rev. A submitted [15] Wiedemann H. and Mostowski J. 1994 Phys. Rev. A49 2719 [16] Yu H., Zuo T. and Bandrauk A.D. 1996 Phys. Rev. A54 3290 Yu H., Zuo T. and Bandrauk A.D. 1997 Phys. Rev. A56 685 [17] Feit M.J., Fleck J.A. and Steiger A. 1982 J. Comput. Phys. 47 412 [18] Ivanov M., Seideman T., Corkum P., Ilkov F. and Dietrich P. 1996 Phys. Rev. A54 1541 [19] Cornaggia C. Private Communication [20] Vrijen R. B., Hoogenraad J. H., Muller H. G. and Noordam L. D. 1993 Phys. Rev. Lett. 70 3016 Story J. G., Duncan D. I. and Gallagher T. F. 1993 Phys. Rev. Lett. 71 3431 [21] Dobosz S., Lewenstein M., Lezius M., Normand D. and Schmidt M. 1997 J. Phys. B: At. Mol. Opt. Phys. 30 L757 [22] Freeman R. R., Bucksbaum P. H., Milchberg H., Darack S., Schumacher D. and Geusic M. E. 1987 Phys. Rev. Lett. 59 1092