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PHYSICS B: ATO

MIC,

MOLECULAR

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OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 40 (2007) 2163-2177

doi:10.1088/0953-4075/40/11/017

Theoretical description of atomic photoionization by an attosecond XUV pulse in a strong laser field: effects of rescattering and orbital polarization
A K Kazansky1,2 and N M Kabachnik1,3
ЕЕ Е Fakultat fur Physik, Universitat Bielefeld, D-33615 Bielefeld, Germany Fock Institute of Physics, State University of Sankt Petersburg, Sankt Petersburg 198504, Russia 3 Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia
2 1

Received 8 March 2007, in final form 3 April 2007 Published 24 May 2007 Online at stacks.iop.org/JPhysB/40/2163 Abstract A theoretical description of atomic photoionization by attosecond pulses in the presence of an intense laser pulse is presented. It is based on the numerical Е solving of the non-stationary Schrodinger equation which includes on an equal footing the realistic atomic potential and the electric fields of both pulses. The calculated energy spectra and angular distributions of photoelectrons are compared with those obtained using a simple approximate model based on the strong-field approximation. The agreement is excellent for large energy of photoelectrons. When the energy is small, the rescattering of electrons by the ionic core affects the cross section considerably making the strong-field approximation inadequate. Influence of the electron orbital polarization on the ionization cross section is investigated. (Some figures in this article are in colour only in the electronic version)

1. Introduction The physical problem treated here is the photoexcitation of a bound atomic electron to a continuum state by a weak ultrashort extreme ultraviolet (XUV) pulse in the presence of a strong infrared (IR) laser field. This process forms the basis of the recent experiments related to the metrology of the attosecond pulses [1-4], as well as to the first applications of such pulses in the study of the time development of atomic processes [5, 6]. Therefore, accurate description of the laser-assisted ultrashort-pulse photoionization of atoms is necessary for a correct interpretation of those experiments. The considered process is closely related to the much more broadly discussed phenomenon of the strong-field ionization of atoms. Here reliable theoretical methods have been developed such as, for example, the strong-field approximation (SFA) [7-9], which allows one to understand the physics of the phenomenon, as well as to describe quantitatively the experimental data in a wide range of the parameters
0953-4075/07/112163+15$30.00 ї 2007 IOP Publishing Ltd Printed in the UK 2163


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A K Kazansky and N M Kabachnik

of the laser radiation from the regime of multiphoton ionization to the tunnelling ionization domain. Therefore, it is natural that the same methods were used for the description of the laser-assisted XUV photoionization. Theoretically, the process was treated quasi-classically [10] and quantum-mechanically within the SFA [11]. In the latter paper the validity of the SFA has been checked by comparison with a more exact method based on a numerical solution of Е the time-dependent Schrodinger equation for a hydrogen atom [12]. The agreement of the two calculations is shown to be only qualitative. In recent paper [13] the influence of the Coulomb atomic potential (which is ignored in the SFA) on the electron motion after XUV ionization was analysed for a one-dimensional model system. Strong difference between the predictions of the SFA and more accurate calculations involving Coulomb atomic potential is revealed both in the spectrum and in the left-right asymmetry of laser-driven photoelectrons. A modification of the SFA which includes the Coulomb asymptotics of the continuum electron wavefunction has been recently suggested [14]. This approach partly solves the problem of the influence of the core potential, at least of its Coulomb part. However, in our opinion the question of whether the SFA can be used as a solid basis for the attosecond-pulse metrology and for the description of other attosecond experiments is still open. Very recently we have suggested a simple model for the description of the laser-assisted attosecond-pulse photoionization which is based on the SFA [15]. Excellent agreement between the model calculations and the numerical solution Е of the non-stationary Schrodinger equation has been demonstrated. It confirms the accuracy of the SFA at least for high energy of the emitted photoelectrons and for moderate intensity of the IR field. In the experiments cited above, the electric field of the XUV pulse is weak, much weaker than the atomic field. Therefore, the first order perturbation theory is well applicable to the interaction of the XUV pulse with the atomic electrons. On the other hand, the IR laser field is rather strong. The laser intensity is typically 1013 -1014 W cm-2 . At such intensity the laser electric field is E L (0.1-0.3) Ч 109 V cm-1 . It is still smaller than the atomic field acting on the bound atomic electrons in outer shells (for comparison, the electric field at the 1s orbit in a hydrogen atom is 1 au = 5.14 Ч 109 V cm-1 ) and it is certainly much smaller than the atomic field felt by the subvalence and inner-shell electrons which are of primary interest in this work. When an electron leaves an atom as a result of photoionization, it moves in the atomic field which quickly decreases outside the atom and becomes much smaller than the laser field. This justifies the application of the SFA in which the atomic field is ignored outside some distance from the nucleus and the continuum electron is considered as moving in the field of the laser pulse. The situation is more complicated for the ionization of subvalence or inner-shell electrons as in the Auger-type experiments [5, 6]. The initial state of the electron to be ionized is concentrated inside the atom. Therefore, the part of the wavefunction of the outgoing electron which is essential for evaluating the transition probability, is in the region where atomic field dominates. In this case the applicability of the SFA is at least questionable. The typical carrier frequency of the XUV pulse in the cited experiments corresponds to the photon energy of 90 eV which determines the kinetic energy of the emitted electron. (A very recent experiment [4] have been done at lower energy of 40 eV.) Subsequently the photoelectron acquires additional energy from the IR laser field. Obviously, in a not extremely strong laser field, the total kinetic energy of electrons is still far from the relativistic domain, therefore one can apply a non-relativistic description of the process. The duration of the XUV pulse is several hundreds of attoseconds [1-4]. The laser pulse has a typical time duration of 5-7 fs with the carrier wavelength of about 800 nm (1.6 eV). With these parameters, the laser pulse has only few cycles, and the duration of the XUV pulse X is smaller than or comparable with the period of the laser pulse TL : X TL /2.


Photoionization by an attosecond XUV pulse in a strong laser field

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In what follows we treat the XUV photoionization in the presence of the strong laser field Е by solving numerically the non-stationary Schrodinger equation which describes the motion of the active electron in the field of the residual ion and in the fields of both XUV and IR electromagnetic pulses. We suggest an algorithm which permits one to solve this problem with quite moderate computer facilities and at the same time to overcome the difficulty connected with the strong Coulomb attractive potential which is present in any realistic atomic potential. Our main goal is to develop a robust and accurate theoretical description of the process. Although the numerical solution of the non-stationary problem is usually time consuming, it is necessary to have such results as a test ground for the development of approximate methods. In particular, we compare the numerical results with a simple approximation based on the SFA which is suggested in [15]. It is computationally fast and can be used, for example, in fitting the experimental data. In paper [15] we have shown that the SFA gives very good agreement with the exact calculations for sufficiently fast emitted electrons. In the present work we extend the comparison to smaller energies. If the energy of the photoelectron is small, and, on the other side, the laser field is strong enough to drive the ejected electron back to the atom, the rescattering process can play a noticeable role and the SFA may fail to describe the angular and energy distribution of the rebound electrons. The developed non-stationary approach permits us also to study the influence of the polarization of the atomic orbital on the cross section and angular distributions for photoionization in the strong laser field. This problem was considered in [13] on the basis of a one-dimensional model. In the present paper we discuss the polarization effects on the basis of our realistic three-dimensional calculations. 2. Basic equations and approximations We assume that the atom can be described by the independent electron model. The evolution of the single-electron wavefunction for an active electron can be described by the non-stationary Е Schrodinger equation (atomic units are used throughout unless otherwise indicated) i (r, t ) ^ = [H 1 (r) + zEL (t ) + zEX (t )](r, t ). t (1)

^ Here H 1 (r) is a single-electron Hamiltonian describing the interaction of the electron with the atomic core, and the last two terms in the square brackets describe (in the dipole approximation) the interaction of the electron with the electric fields of the two pulses: EX (t ) is the weak electric field of the ultrashort (attosecond) XUV pulse, and EL (t ) is the strong field of the longer (femtosecond) IR laser pulse. Both the pulses are supposed to be linearly polarized along the same direction which we choose as the z axis of the laboratory frame. Both can be presented as ? ? Ei (t ) = Ei (t ) cos i t = E0i i (t ) cos i t (2) ? (i = L, X) with the carrier frequency i and the envelope Ei (t ) which in turn is presented as ? a product of the amplitude E0i and the pulse shape function i (t ) with a unit amplitude. Initially, for t -, i.e. before the atom is illuminated by the radiation pulses, the atom is in the ground state, therefore the initial boundary condition for the function (r, t ) is (r, t -) = 0 (r) exp(-i 0 t), where 0 (r) is a single-electron wavefunction of the active electron in the ground atomic state with the corresponding single-electron energy 0 . For simplicity, we shall assume that the initial state is an s-state. The single-electron ^ Hamiltonian H 1 (r) contains an effective interaction of the active electron with the core. This interaction can depend on the electron angular momentum. Importantly, it will be described


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by a realistic model potential, which includes, in particular, a very strong Coulomb interaction ^ between the electron and the nucleus at small distances. Note that the Hamiltonian H 1 (r) is assumed to be independent of time. Thus we ignore the polarization of the core by the laser field (see discussion below). The great difference in magnitudes of the XUV and IR fields and in frequencies of the pulses allows one to use the first order perturbation treatment and the rotating wave approximation (RWA) for the description of ionization by the XUV field. The application of the RWA is advantageous since in this case the calculated probabilities or the cross sections are simply proportional to the squared electric field amplitude which is often not known in the experiment. Keeping this in mind, it is convenient to present the active electron wavefunction as the following sum (r, t ) = exp(-i 0 t)(r, t ) + (r, t ), (3)

where (r, t ) describes a perturbation of the active electron wavefunction due to interaction with the XUV field. Separation of the fast oscillating exponential factor in equation (3) leaves us with the functions (r, t ) and (r, t ) rather slowly changing in time what will save considerably the computation time. Substituting equation (3) into equation (1) one obtains the set of non-stationary Е Schrodinger equations for the functions (r, t ) and (r, t ) (r, t ) = ( t i (r, t ) = t i ^ H 1 (r) -
0

) (r, t ) + zEL (t ) (r, t ),
0

(4) + X )t ](r, t ), (5)

1? ^ H 1 (r) (r, t ) + zEL (t ) (r, t ) + zEX (t ) exp[-i( 2

which is the basic equation set for our study. (The factor 1/2 in the last term is due to the RWA.) The obvious boundary conditions are (r, -) = 0 (r) and (r, -) = 0. Equation (4) describes the evolution of the initial electron state under the influence of a rather strong IR field or, in other words, the polarization of the initial orbital by the IR field. It is important to keep in mind that the electron wavefunctions which were orthogonal before switching on the IR field, remain to be orthogonal during all the evolution. (This statement is a consequence of Hermiticity of the Hamiltonian in equation (4).) Therefore, the initially filled atomic orbitals remain to be filled and do not interfere with each other. We can safely consider the evolution of only the orbital involved in the ionization process without taking care of possible admixture of other filled orbitals during the evolution. It is worth noting that we treat the case when the IR field is not very strong. The considered laser intensities are in the range 1013 -1014 Wcm-2 as in the recent attosecond experiments [1-3]. Then the laser electric field is much weaker than the atomic field especially in the region of the atomic core. The relative weakness of ^ the IR field allows us to neglect the time variation in the single-electron Hamiltonian H 1 (r) in equations (4) and (5), caused by the core electron wavefunction polarization by the IR laser field. In principle, the time variation of the electronic potential could be accounted for within the time-dependent density functional method. In contrast, the polarization of the active electron orbital is fully taken into account by equation (4). Equation (5) describes the excitation of the polarized initial state in the presence of the strong laser field. The excited state is populated by the XUV pulse, but its depopulation is neglected as an effect of higher order in the XUV field strength. The electric IR field substantially influences the electron in continuum forming the well-known Volkov states [16]. To proceed further, we expand the active electron wavefunction in spherical harmonics (only harmonics with m = 0 are involved due to axial symmetry of the considered process)


Photoionization by an attosecond XUV pulse in a strong laser field
L
max

2167

(r, t ) = (r, t ) =

w (r, t )Y 0 ( , ),
=0 Lmax

(6)

u (r, t )Y 0 ( , ).
=0

(7)

Е Substituting the expansions (6) and (7) into the time-dependent Schrodinger equations (4) and (5) we obtain the following system of coupled non-stationary equations for the coefficient functions w (r, t ) and u (r, t ): i ^ w (r, t ) = [h( ) (r ) - t
L 0
max

]w (r, t ) + r EL (t )
=0 L
max

C ( , )

, +1

w (r, t ),

(8)

i

^ u (r, t ) = h( ) (r )u (r, t ) + r EL (t ) t

C ( , )
=0 L 0
max

, +1

u (r, t )

1? + r EX (t ) exp[-i(X + 2

)t ]
=0

C ( , )

, +1

w (r, t ).

(9)

Here C ( , ) is the angular part of the dipole matrix element, C ( , ) = (2 +1)(2 +1)( 0, 0|10)2 , with ( 0, 0|10) being the Clebsch-Gordan coefficient, , +1 is the Kronecker symbol reflecting the dipole selection rule (in the following the Kronecker symbol is omitted since this selection rule is guaranteed by the Clebsch-Gordan ^ coefficient in C ( , )). The operator h( ) (r ) is the radial part of the single-electron Hamiltonian: 1 2 ( +1) ^ h( ) (r ) = - + U (r ) + , (10) 2 2 r 2r 2 where U (r ) is a single-electron potential. Since the laser IR field is strong, it mixes the states with different orbital angular momenta. The numbers of partial waves (the values of Lmax and Lmax ) are selected from the condition of accuracy of the calculated results. If the IR laser field is only moderately strong and the initial electronic state is deeply bound, the effect of the laser field on the initial orbital is negligible. Then one can ignore Е the last term in equation (4) and the equation becomes the usual Schrodinger equation for the bound electron with the solution (r, t ) = 0 (r). In this case of `frozen' initial orbital, the first equation (8) gives w (r, t ) = , 0 0 (r ) where 0 (r ) is the radial part of the initial wavefunction, and the second equation (9) becomes i ^ u (r, t ) = h( ) (r )u (r, t ) + r EL (t ) t
L
max

C ( , )u (r, t )
=0

1? (11) + r EX (t ) exp[-i(X + 0 )t ]C ( , 0 ) 0 (r ). 2 The system of equations (8), (9) has been solved numerically using the split-propagation method [17, 18]. The details of the numerical procedure are presented and discussed in the appendix. Calculating the set of functions u (r, t ) at t , we have evaluated the partial amplitudes of photoionization by projecting the functions onto the appropriate ( continuum functions - ) (E ; r) which are eigenfunctions of the final ionic state Hamiltonian, corresponding to the electron energy E:
A (E ) = exp[i (E )]
0

dru (r, t +)

() -

(E ; r),

(12)


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( where (E ) is the photoionization phase, the eigenfunctions - ) (E ; r) are normalized on the () energy scale and obey the boundary condition - (E ; 0) = 0 (for further details see [19]). ? As follows from equations (9) and (11), the solution u (r, t ) is proportional to the factor 1 E0X . 2 In the calculations of the amplitudes A (E ) we have always put this factor equal to unity. It is later taken into account by the general normalization factor for the cross section. Knowing the amplitudes, A (E ), one can evaluate the double differential cross section (DDCS) defined as an electron yield per one photon of carrier frequency in the pulse (see [19]):

d2 (E , ) = 2X K dE d

2 -1

|A(E , )| = 2X K
2

-1

A (E )Y 0 ( ) .

(13)

Here is the fine-structure constant, K = dt |X (t )|2 , and is the polar angle of the photoelectron emission. In the examples presented and discussed below we show the calculated 2 values of the amplitude square |A(E , )|2 A (E )Y 0 ( ) which is proportional to the DDCS. By integrating equation (13) over the emission angle one can obtain the photoelectron spectrum: d(E ) |A (E )|2 . = 2X K -1 (14) dE In the following section 3 we compare the results obtained by numerically solving the Е non-stationary Schrodinger equation with the model calculations based on the SFA. A detailed description of the model has been published recently [15] therefore we will not repeat it here. The comparison of the two calculations permits one to explore the accuracy of the SFA and the region of its applicability. Since the atomic field and the orbital polarization effects are ignored in the SFA, the comparison of the two approaches reveals the importance of the effects for particular parameters of the experiment, namely, the laser intensity and the photoelectron energy. 3. Results and discussion 3.1. Particular example: Ar (3s) ionization As an example, we consider the photoionization of the 3s electron in Ar by the XUV pulse of a few hundred attosecond duration in the field of the laser with the intensity of (1 - 4) Ч 1013 W cm-2 . The laser carrier wavelength is taken 800 nm (photon energy 1.6 eV) and the laser pulse duration (full width at half maximum of the field envelope) FWHM = 5 fs. Such a choice of parameters corresponds to the existing experimental facilities [1-3]. At the chosen laser intensity its electric field is weaker than the atomic field at the orbit of the Ar 3s electron, therefore the polarization of the electron orbital by the laser field should not be strong. This question will be specially investigated in section 3.4. We have chosen the laser pulse of the cosine shape: L (t ) = 0.5{cos[(t /L - 1)]+1}. (15) For such a shape the pulse duration is FWHM = L . The onset of the IR pulse corresponds to t = 0. The carrier-envelope phase is supposed to be zero (cosine-type pulse) in the present study. For the XUV pulse we have taken the hyperbolic secant shape with some time delay td relative to the IR pulse: X (t - td ) = 1/ cosh[(t - td )/X ]. (16) For such a pulse FWHM = 2.634X . In the examples below the XUV pulse duration is 330 as (X = 125 as). The delay td = 0 corresponds to the maximum of the XUV pulse at the onset


Photoionization by an attosecond XUV pulse in a strong laser field

2169

Figure 1. Schematics of the XUV and IR pulses. The duration of the IR pulse is 5 fs, that of the XUV pulse is 330 as. Arrows show several particular delays which are discussed in the text: (a) t = -0.5fs; (b) t = 0;(c) t = 0.5fs; (d) t = 1.2fs.

of the laser pulse. Maxima of both pulses coincide at the delay td = L . It is convenient to characterize the delay by t = td - L . Then the positive t corresponds to the situation when the IR pulse reaches maximum before the XUV pulse, negative t corresponds to the opposite situation (i.e., the XUV pulse comes before the IR field reaches its maximum). In figure 1 we show schematically the two pulses, 5 fs IR pulse and 330 as XUV pulse, at t = 0 as a function of time (in atomic units, 1 au = 24.2 as). The arrows show the particular delays which will be discussed below. Since the IR pulse has only few cycles, it is possible to uniquely define the `forward' direction (positive z) as a direction of the electric field in the main maximum. Hereafter we use this definition. The single-electron potential U (r ) in equation (10) has been calculated in the Hartree- Slater (HS) approximation [20]. This potential is local and independent of the orbital angular momentum of the electron. The choice of the potential is determined by its simplicity although it is known that it does not give accurately the photoemission cross section at small energies [21]. We should mention that at small energies also the single-channel approximation becomes inaccurate, and the channel coupling can play an important role (for our particular example of Ar(3s) photoionization the channel coupling effects were investigated in [22]). With these reservations we consider the quality of the used approximations as sufficient for our purposes. The initial state wavefunction u3s (r ) and the corresponding binding energy (| 0 | = 28.6 eV) for the 3s subvalence electron in Ar have been calculated with the same potential. 3.2. High electron energies, X = 90 eV First we consider a comparatively large carrier frequency of the XUV pulse, corresponding to 90 eV. This case has been treated in our paper [15] within the frozen orbital approximation. Now we include the initial wavefunction polarization. The corresponding photoelectron energy at the maximum of the spectral line is 61.4 eV (without influence of the IR field), i.e. larger than the typical streaking energy shift in the chosen laser field (about 30 eV, as will be seen below). For such fast electrons one can expect that the SFA description works well. In figure 2 the averaged spectra of photoelectrons emitted in the backward direction are shown in a 3D plot as a function of the electron energy and the delay time between the Е XUV and IR pulses. The spectra are calculated by solving the non-stationary Schrodinger equation for the XUV pulse duration of 330 as and the 5 fs laser pulse with the intensity of 1 Ч


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.

..

.



(a.u.)

Figure 2. 3D plot of the Ar (3s) photoelectron spectra (in arbitrary units) at X = 90 eV as a function of the time delay (in atomic units) between IR and XUV pulses. Dots on the plane show the positions of the spectrum maxima calculated using the classical expression (see text).

1013 W cm-2 . To imitate the final acceptance angle of a detector, for each time delay the DDCS was convoluted with the Gaussian, representing the `instrumental' function, with the dispersion = 15 . The results show a typical streaking effect similar to that observed in experiment at similar conditions [1, 3]. In the figure the dotted line on the plane traces the `trajectory' of the spectral maximum calculated by the classical expression (see expression (2) in [10]) as a function of the delay. The curve perfectly corresponds to the positions of the spectral maxima calculated within the time-dependent theory. The maximal shift of the peak occurs at the time delays corresponding to the maximal values of the vector potential (close to zeros of the electric field). Note the strong variation of the spectral width of the electron pulse which was used in papers [1-3] for determination of the XUV pulse duration. The maximal width is achieved at the time delays corresponding to the extrema of the laser field (zeros of the vector potential). In figure 3 the calculated spectra of electrons emitted in the forward and backward directions ( = 15 ) are shown for two particular delays, t = 0.5 fs and t = 1.2 fs, corresponding to the positions `c' and `d' in figure 1. In this figure we compare the results of the three types of calculations: the solution of equations (8) and (9) which includes the orbital polarization effect, the frozen orbital results (equation (11)) and the SFA model results (equation (11) of [15]). All three calculations give very similar results. It shows that for the used parameters of the pulses the polarization effect is not large, and the SFA agrees very well with the non-stationary approaches. Especially good is the agreement with the frozen orbital approach which was already demonstrated earlier [15] in more detail. Figure 4 shows the 2D plots of the complete energy and angular distributions of photoelectrons for the case of time delays t = 0.5 fs and t = 1.2 fs. The amplitude square |A(E , )|2 which is proportional to the DDCS is presented. The colour (grey) Е scale plot shows the results of the calculations by solving the non-stationary Schrodinger equations (8) and (9) which include the polarization effect. The contour plot shows the results of the SFA model calculations [15]. The agreement between the two calculations is very good. Therefore, also for the angular distributions the SFA works very well and the polarization effect included in the first calculation is practically negligible in this case. Interesting that in

E (eV)


Photoionization by an attosecond XUV pulse in a strong laser field
1.2

2171

Electron spectrum (arb. units)

Electron spectrum (arb. units)

(a)
1.5

1 0.8 0.6 0.4 0.2 30

(b)

1

0.5

40

60 Energy (eV)

80

100

40

50

60 70 Energy (eV)

80

90

100

Figure 3. The spectra of Ar (3s) photoelectrons emitted in the forward (black lines and symbols) and the backward (red (grey) lines and symbols) directions at X = 90 eV. The time delays are (a) 0.5 fs and (b) 1.2 fs. Three types of the calculations are presented: with solving equations (8), (9) (solid lines), with solving equation (11) with frozen orbitals (dashed line) and the SFA model results (dots). The two curves for the low-energy peak in (a) cannot be distinguished in the figure.

(a)

(b)

Figure 4. 2D plot of the amplitude square |A(E , )|2 in atomic units for Ar (3s) photoionization at X = 90 eV for two different time delays (a) t = 0.5 fs, (b) t = 1.2 fs. The colour (grey) Е scale plot presents the results obtained by solving of the Schrodinger equations (8) and (9), the contour plot shows the SFA model results. The maximum value of the DDCS in both calculations is almost the same.

both cases the angular distributions of electrons in the strong laser field have zero at 90 which reflects the zero in the `original' angular distribution produced by the XUV pulse (without the laser field) which is described by the cos2 ( ) distribution. 3.3. Low electron energies, X = 40 eV Now we consider the case of the XUV pulse with lower carrier frequency, corresponding to 40 eV. Very recently the isolated attosecond pulses of such energy have been reported [4]. In this case for the Ar(3s) ionization the mean energy of the ejected electrons is about 11.4 eV, while the streaking shift of the electron spectrum due to the interaction with the IR field remains in the 30 eV range. This means that under some conditions the emitted electrons can


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A K Kazansky and N M Kabachnik

300

z (a.u.)

200

100

0

0

100

200

300

t (a.u.)
Figure 5. Trajectories of electrons emitted along the electric field vector of the IR field at different times (delays).

be returned to the ion and be scattered by it. This phenomenon is well known: it explains the high harmonic generation in strong laser fields [23, 24]. Here we investigate how it affects the energy and angular distributions of photoelectrons. The origin of the phenomenon can be clarified with the classical picture. In figure 5 we give a set of trajectories of electrons ejected along the IR laser field direction with different delays. It is clear that most of the trajectories are weakly perturbed by the interaction with the Coulomb core, but at few delays the character of the electron motion is changed drastically by that field. We see that for the delays around 180 au ( t = -0.5 fs) the electrons ejected close to the z axis are turned backward to the core by the IR field and undergo the secondary scattering, including rebounding and even possibly the capture into the Rydberg states. The classical picture should be reflected in the full quantum mechanical computations which are universal and reproduce the physical reality. One should realize, however, that only a narrow bunch of the ejected electron trajectories concentrated close to the z axis suffers a substantial deviation in the vicinity of the nucleus. The majority of the ejected electrons are only slightly deviated by the ionic field, and their contribution to the DDCS should qualitatively agree with the prediction of the SFA. In figure 6 the squared amplitude |A(E , )|2 proportional to the DDCS is shown for the low XUV pulse energy (40 eV) for two delays t = -0.5 fs and t = 0 (positions `a' and `b' in figure 1). For the first delay (figure 6(a)) the XUV pulse is close to the zero of the IR field, the angular distribution is strongly asymmetric. The zero at 90 is washed out, but there is a minimum shifted towards larger angles. The backward peak at large energies is similar in shape to the corresponding peaks at high XUV energy (cf figure 4). However, the low-energy forward peak is strongly distorted. The prediction of the SFA model (contour plot) agrees very well with the exact solution for the backward peak, but the agreement is only qualitative for the forward peak. In figure 6(b) similar results are shown for the delay t = 0. Here both forward and backward maxima are at rather small electron energies. The zero at 90 is almost filled. The agreement with the SFA calculation is only qualitative at the low electron energy. When the intensity of the IR laser field is increasing, the angular and energy distributions of the emitted electrons are more and more distorted. Practically, all emitted electrons


Photoionization by an attosecond XUV pulse in a strong laser field

2173

(a)

(b)

Figure 6. 2D plot of the amplitude square |A(E , )|2 in atomic units for Ar (3s) photoionization at X = 40 eV for two different time delays (a) t = -0.5 fs, (b) t = 0. The notations are the same as in figure 4.

(a)

(b)

Figure 7. 2D plot of the amplitude square |A(E , )|2 in atomic units for Ar (3s) photoionization for the time delay t = -0.5 fs and two different intensities (a) 2 Ч 1013 W cm-2 (b) 4 Ч 1013 Wcm-2 . The notations are the same as in figure 4.

independent of the original emission direction move now in one direction which is determined by the laser field. As illustration, in figure 7 we show the results of calculations for the intensities of 2 Ч 1013 W cm-2 and 4 Ч 1013 W cm-2 . The delay is t = -0.5 fs, so that these results may be compared with figure 6(a) where the corresponding results for the intensity 1 Ч 1013 Wcm-2 are shown. One sees that at larger intensities the main (backward) peak is shifted to higher energies (streaking effect) but the second (forward) peak practically disappears. The backward peak at large energy is described rather well by the SFA model (contour plot). However, at lower energies the SFA model does not work well. Note the points of closeness of contours (dark spots at the upper right part of the figures). In this area the SFA gives unphysical results due to zero in the denominator of equation determining the SFA amplitude (see equation (11)of[15]). The origin of the small maxima in the forward direction is not clear.


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A K Kazansky and N M Kabachnik

Electron spectrum (arb. units)

6 5 4 3 2 1

Electron spectrum (arb. units)

6 5 4 3 2 1 20

15

20

30 25 Energy (eV)

35

40

30

40 50 Energy (eV)

60

Figure 8. The spectra of Ar (3s) photoelectrons emitted in the backward directions at the time delays of t = -0.5 fs for two intensities of the IR laser, (a) 1 Ч 1013 W cm-2 , (b) 4 Ч 1013 W cm-2 . Solid curves present the results obtained by solving equations (8) and (9) which include the orbital polarization effect. Dashed curves present the results obtained by solving equation (11) with frozen orbitals.

Very interesting interference patterns (horizontal stripes) are seen at low energies as well as at the low-energy side of the main peak. This structure has a period of about 1.6 eV, close to the energy of the IR photon. Since the freely moving electron cannot absorb photons, the appearance of such a structure clearly indicates the substantial influence of the ion field. The electrons emitted in the direction of the ion are scattered by the core and then accelerated by the laser field to the final velocity. During the scattering they can absorb several photons what is reflected in the structure of the spectrum. This phenomenon is quite similar to the above-threshold ionization but occurs with electrons ionized by the XUV pulse. Using a semiclassical picture, one can say that the discussed structure appears due to the interference of the electron wave packets which have the same energy but achieve it by moving along different trajectories including one or several rotations around the ionic core. Certainly, such a phenomenon cannot be described within the SFA. 3.4. Polarization of electron orbital by the laser field Е The time-dependent Schrodinger equations (8) and (9) which we solve in this work include the polarization of the orbital of the active electron by the strong laser field. We have already mentioned in section 3.2 that at comparatively small laser intensity of 1 Ч 1013 Wcm-2 and for the considered rather tightly bound subvalence electron, the influence of the orbital polarization is small. Roughly, the distortion of the orbital by the strong field can be estimated by the ratio of the ponderomotive energy (Up ) of the electron in the field and its binding energy. | 0 |, which is equivalent to the Keldysh parameter [7] 1, the polarization If Up effect should be negligible. In the above considered cases at the laser intensity of 1 Ч 1013 W cm-2 , 5 and the polarization effect is indeed small. However, with the increase of the laser intensity it may become considerable. In figure 8 we show the spectra of electrons emitted in the backward direction by the 40 eV XUV pulse in the laser field at the delay t = -0.5 fs. The first spectrum (a) is obtained for the laser intensity 1 Ч 1013 W cm-2 , it corresponds to the DDCS in figure 6(a). The second spectrum (b) is obtained for the laser intensity 4 Ч 1013 W cm-2 , it corresponds to the DDCS in figure 7(b). In the second case 2.5. The calculations have been made with (solid lines) and without (dashed lines) accounting for the polarization of the active-electron orbital. One can clearly see that


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2175

the photoelectron spectra become broader due to the polarization and the polarization effect increases as diminishes, it becomes essential at 1. Qualitatively this is in agreement with the conclusion in [13] where the effect of polarization has been studied within the onedimensional model. Note the structure at the low-energy side of the spectrum in figure 8(b). It reflects the interference structure in the DDCS which we have discussed above. 4. Conclusions We have presented the theoretical description of the ultrashort XUV pulse photoionization of atoms in the presence of the strong IR laser field which includes the effects of the core potential and the orbital polarization by the IR field. The method is based on the numerical Е solution of the non-stationary Schrodinger equation for an active electron. The interaction of the electron with the residual ion is included, as well as the interaction with the electric fields of both pulses. The DDCSs calculated in this way are compared with the results of the model calculations based on the SFA. We have demonstrated that for high-energy electrons and for moderately intense laser pulses the SFA results agree very well with the more exact non-stationary calculations. The polarization of the initial electron orbital by the laser field plays a minor role in these conditions and modifies the DDCSs only slightly. This confirms the conclusion of our previous paper [15] where the polarization was not included. However, when the photoelectron energy is small or/and the intensity of the IR laser is large, the rescattering of the emitted electron by the ionic core becomes important. For the larger IR field intensity, also the orbital polarization by the laser field becomes essential. In this case, the interference fringes associated with the photon absorption by the scattered electron are predicted. Naturally the SFA cannot accurately describe the DDCS in this limit. Acknowledgments We gratefully acknowledge numerous useful discussions with U Heinzmann. We are grateful to Bielefeld University for hospitality and financial support via SFB 613. We also acknowledge the financial support from Russian Foundation for Fundamental Researches via grant 06-0216289. Appendix In order to solve numerically the system of equations (8) and (9) we have used the following computation scheme (as an example, we discuss equation (9); equation (8) has been solved in parallel by the same method):
L
max

ul (r, t + t ) =
=0

t 1 r exp[-i(X + S (t + t , t )ul (r, t ) + 22
0

L 0

max

)t ]
=0

C( , ) (A.1)

? Ч{EX (t + t ) exp[-i(X +

? ) t ]w (r, t + t ) + EX (t )w (r, t )},

where Sll (t + t , t ) is a propagation operator over the radial variable, corresponding to equation (9). For this operator we have used the split-propagation approximation Sll (t + t , t ) = Kl (t + t , t + t /2)Fll (t + t , t )Kl (t + t /2,t ). (A.2) The operator Kl (t + t , t ) is the Crank-Nicolson propagator [17] over the radial variable and it is diagonal with respect to the angular momenta. Here we do not describe this operator in


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A K Kazansky and N M Kabachnik

detail since it was fully discussed in our recent paper (Appendix in [25]) and elsewhere [26]. The operator
Fll (t + t , t ) = exp -irC ( , )
t t +t

EL (t ) dt

(A.3)

is diagonal with respect to the radial variable and it is a matrix with respect to the angular momentum indices. To accelerate the computations, we have diagonalized the matrix C ( , ) only once, then the action of the operator (A.3) can be presented as a set of the projection operations. Indeed, if | (j ) is an eigenvector of the matrix C ( , ) with the eigenvalue чj , then the operator (A.3) acts on any vector |f as follows:
F |f =
j

exp -iчj r
t

t +t

EL (t ) dt |

(j )



(j )

|f .

(A.4)

This method of computation is quite effective in our case. The described algorithm is reasonably accurate, fast and presents moderate requirements to the computer resources. It has a number of advantages, but also contains some internal restrictions. The principal advantage is that it allows one to consider all the interactions on the same footing. With the quadratic substitution of the variables r = x 2 and the uniform mesh over the x variable, one can treat quite accurately the close vicinity of the nucleus, where the active electron is influenced by a very strong Coulomb field. This allows us to use a realistic model potential describing the electron-atom interaction. It is especially important for subvalence and inner-shell electrons, since the absorption of the XUV photon occurs at a rather small r. However, this approach contains also a drawback: we can consider the laser field only in the length gauge. It is rather restrictive, since the phase in the operator (A.3) increases with r and the requirement for this phase to be small leads to a small time step. The critical value of the radius rmax , which should be guaranteed in the calculation is determined 2 au, the duration of the IR laser pulse by the highest considered electron velocity kmax 200 au = 5fs , and the field strength EL 0.02 au (the IR pulse intensity about 2 Ч L 1013 W cm-2 ). The condition reads dt k
max L

EL

0.1,

(A.5)

i.e. dt 0.01 au and rmax = kmax L 400 au. So, with our method we cannot easily consider long IR pulses. The problem is somewhat trivial: it is difficult to reproduce an IR field-modified plane wave (Volkov state) at large r with a finite sum of the partial waves. We do not discuss the improvements of the method here, but should stress that with increase of intensity of the IR field or the IR pulse duration, the algorithm has to be corrected. The results presented in this paper have been calculated with a step in the x variable of 0.003 au, a step in time of 0.005 au, and the number of partial waves Lmax = 40-60. The number of partial waves included in the computation of the orbital polarization by the laser field is smaller, Lmax = 10. The gobbler (absorbing potential) has been smoothly switched on starting from the radius of about 1500 au. High accuracy of the algorithm implementation has been proven by a broad comparison of the results obtained with this algorithm, with the results obtained with the SFA (see section 3 and paper [15]). References
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