Документ взят из кэша поисковой машины. Адрес оригинального документа : http://danp.sinp.msu.ru/LIIWM/JPhysB_40_3413_07KabachKazanski.pdf Дата изменения: Sun Oct 21 21:21:24 2007 Дата индексирования: Mon Oct 1 22:33:51 2012 Кодировка: Windows-1251

IOP PUBLISHING

JOURNA

L OF

PHYSICS B: ATO

MIC,

MOLECULAR

AND

OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 40 (2007) 3413-3424

doi:10.1088/0953-4075/40/17/009

Theoretical description of atomic photoionization by attosecond XUV pulses in a strong laser field: the case of p-shell ionization
A K Kazansky1,2 and N M Kabachnik1,3
1 2 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 Institute of Nuclear Physics, Moscow State University, Moscow 119991, Russia

Received 28 May 2007, in final form 23 July 2007 Published 21 August 2007 Online at stacks.iop.org/JPhysB/40/3413 Abstract A theoretical description of attosecond photoionization in the presence of Е a strong laser field, based on the numerical solution of the Schrodinger equation, is extended to the case of p-shell ionization. In particular, Ar(3p) photoionization is considered. The main difference between this case and the previously considered case of s-shell ionization stems from the interference of the two dipole allowed channels of p-shell photoionization, which determines the angular distribution of photoelectrons in the absence of the laser field. The latter additionally distorts the angular distributions. We also extend to the initial p-shell case the model based on the strong-field approximation (SFA), which has been suggested earlier. At high photoelectron energy and low laser intensity both calculations give similar results. However, at low electron energy the SFA is inadequate. The dependence of the angular distribution on the carrier-envelope phase and the effects of orbital polarization are considered. (Some figures in this article are in colour only in the electronic version)

1. Introduction The process of atomic photoionization by an ultrashort (attosecond) extreme ultraviolet (XUV) electromagnetic pulse in the presence of a strong infrared (IR) laser field forms a basis of the modern methods of detection, characterization and application of the attosecond pulses [1-5]. Therefore, its adequate theoretical description is an important task. Until quite recently the main theoretical method used for this purpose was the strong field approximation (SFA) [6, 7], which was successfully applied for the analysis of experimental data in the above-cited papers. The validity of this method and the range of its applicability were tested in several recent works [8-10]. In our previous paper [11], we have suggested a rigorous method for the description of the attosecond photoionization assisted by the strong laser field which is based on the
0953-4075/07/173413+12$30.00 ї 2007 IOP Publishing Ltd Printed in the UK 3413


3414

A K Kazansky and N M Kabachnik

Е numerical solution of the non-stationary Schrodinger equation. From one side, the method may be used to obtain the results necessary for the interpretation of experiments. From the other side, it provides a solid test-ground for checking various approximate approaches, such as the SFA, in order to find the borders of their applicability and to study the effects of the processes which are ignored in the approximate consideration. To the present time all calculations for the considered process have been performed for the initial s-state of the active electron, which is certainly easier for computations. However, all recent atomic experiments have been done for the p-shells of noble gases. Therefore, from the practical point of view, it is necessary to extend the calculations to the case of photoionization of 0 = 0 initial electronic states. Moreover, from the theoretical point of view, the latter case is more interesting, since at least two (in the non-relativistic approach) electronic partial waves = 0 + 1 are emitted in dipole photoionization. Interference of these partial waves should be taken into account in order to explain the angular distribution of the emitted electrons even without the IR field. This interference competes with another type of interference of electrons emitted coherently at different moments of the laser pulse which form the final complicated angular distribution pattern. 2. Basic equations and approximations For a description of the atomic photoionization by an attosecond pulse in the presence of a Е strong field of an IR few-cycle laser pulse we solve the time-dependent Schrodinger equation within a single active electron approximation. Using the perturbation approach with respect to the weak field of the attosecond pulse and the dipole approximation, one can derive the following system of time-dependent equations for partial wavefunctions (it is a straightforward generalization of the similar system derived in our previous paper [11] for an initial s-state): i ^ w m (r, t ) = [h( ) (r ) - t
L 0
max

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

C ( , ,m)w

m

(r, t ),

(1)

^ i u m (r, t ) = h( ) (r )u m (r, t ) + r EL (t ) t 1? + r EX (t ) exp[-i(X + 2

C ( , ,m)u
=0 L 0
max

m

(r, t )

)t ]
=0

C ( , ,m)w

m

(r, t ).

(2)

Here and below atomic units are used unless otherwise indicated. In equations (1) and (2) 0 is the single electron energy in the initial state and C ( , ,m) is the angular part of the dipole matrix element, C ( , ,m) Y
m

| cos |Y

m

=

(2 +1)(2 +1)( 0, 0|10)( - m, m|10),

(3)

^ with ( m, m |1M) being the Clebsch-Gordan coefficient. The operator h( ) (r ) is the radial part of the single-electron Hamiltonian 1 2 ( +1) ^ + U (r ) + , (4) h( ) (r ) = - 2 r 2 2r 2 where U (r ) is a single-electron atomic potential. We note here that we consider the ^ Hamiltonians h( ) (r ) as independent of time, i.e., we ignore the polarization of the tightly bound atomic core by the laser field. On the other hand, the polarization of the active electron


Photoionization of p-shells by attosecond XUV pulses in a strong laser field

3415

orbital is fully taken into account (see below). The strong laser field is presented as ? ? EL (t ) = EL (t ) cos (L t + 0 ) = E0L L (t ) cos (L t + 0 ), (5) ? where L is the carrier frequency, EL (t ) is the pulse envelope with the shape described by ? the function L (t ) and with the amplitude E0L , and 0 is the carrier-envelope (CE) phase. Similarly, the XUV attosecond pulse is presented as ? ? EX (t ) = EX (t - td ) cos [X (t - td )] = E0X X (t - td ) cos [X (t - td )], (6) where td is the time delay between both pulses. We suppose that both pulses are linearly polarized along the z direction in the laboratory frame. Equation (1) describes the evolution of the active electron orbital in the strong IR laser field, i.e., the polarization of the orbital by the strong field. Equation (2) describes the evolution of the emitted electron wave packet, created by the XUV pulse and steered by the laser field. Since the laser field is strong, it mixes the states with different orbital angular momenta, which leads to forming a superposition of the Volkov states at large distance from the atom. The number of essential partial waves (the values of Lmax and L max ) is selected from the condition of accuracy of the calculated results. We remind that, as in the previous work [11], equations (1) and (2) are derived within the rotating wave approximation (RWA) which is used for the description of ionization by the XUV field. This approximation works very well in the considered frequency domain. Its application 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. (The factor 1/2 in equation (2)is a consequence of the RWA.) Since the interactions considered in the problem are axially symmetrical with respect to the z axis, the projection of the angular momentum of the electron is conserved during the evolution. This means that the solutions of the equations for different m can be considered independently. The system of equations (1) and (2) should be solved with the initial condition that at t -, i.e., before the atom is illuminated by the radiation pulses, the active electron is in the initial orbital l0 m (r) in the ground atomic state. Therefore w m (r, t ) = u m (r, t ) = 0,
,
0

l0 (r ) exp(-i 0 t), (7)

where l0 (r ) is the radial part of the initial-state wavefunction and , 0 is the Kronecker symbol. If the laser field is not strong enough, the polarization of the electron orbital by the laser field can be neglected. In this case the system (1) and (2) reduces to the equation i ^ u m (r, t ) = h( ) (r )u m (r, t ) + r EL (t ) t
L
max

C ( , ,m)u
=0

m

(r, t )

1? (8) + r EX (t ) exp[-i(X + 0 )t ]C ( , 0 ,m) 0 (r ). 2 In the following we shall use the solution of this equation in order to demonstrate the influence of the orbital polarization. Solving numerically the system (1) and (2), we then calculate the partial amplitudes of photoionization by projecting the solution u m (r, t ) at very large time onto the continuum ( eigenfunctions of the ionic Hamiltonian with the necessary asymptotics - ) (E ; r) (for details see [12]):
A m (E ) = exp[i (E )]
0

dru m (r, t +)

() -

(E ; r),

(9)


3416

A K Kazansky and N M Kabachnik

where (E ) is the photoelectron phase in the absence of the IR field. The complex amplitudes A m (E ) determine the double differential cross section (DDCS): d2 (E , ) = 2X K dE d = 2X K
-1

|A(E , )|

2 2

-1 m0

(2 - m0 )

A m (E )Y m (, 0) .

(10)

Here is the fine-structure constant, K = dt |X (t )|2 ,Ylm (, ) are the spherical functions and is the polar angle of the photoelectron emission. In the examples presented and discussed below we show the calculated values of the amplitude square |A(E , )|2 2 A m (E )Y m ( ) which is proportional to the DDCS. By integrating m 0 (2 - m0 ) equation (10) over the emission angle, one can obtain the photoelectron spectrum: d(E ) |A m (E )|2 . (11) = 2X K -1 dE m Non-stationary equations (1) and (2) have been solved numerically using the splitpropagation method with the Crank-Nicolson propagator. The details of the numerical procedure are published elsewhere [11, 12]. The calculations have been done for the ionization of the 3p subshell in Ar. The single electron potential has been calculated within the Hartree- Slater (HS) approach [13]. The initial wavefunction of the active electron has been calculated as a proper eigenfunction in this HS potential. It is well known that for large energies of photoelectrons the HS approach gives rather good results for the differential cross section of photoionization which are close to those obtained by more accurate methods [14]. For smaller energies the HS results may be considered as only qualitatively correct. However, for our purposes of the first estimation of the DDCS for p-electron ionization in a strong IR field, this accuracy is sufficient. In what follows we compare the results of the calculations made by solving the Е non-stationary Schrodinger equation with the calculations made within the strong field approximation (SFA). For this purpose we have generalized our model based on the SFA [10] in order to apply it to the case of p-electron ionization. We remind that in the model the XUV pulse is presented as a sequence of very short pulses (pulselets):
N

? EX (t ) =
j =1

Cj (t - tj ).

(12)

Here (t ) describes a pulselet with a duration, characterized by the parameter . The number of pulselets N, their duration, positions tj and the coefficients Cj are determined by fitting the expansion to the XUV pulse. For each of the pulselets the amplitude of atomic ionization in the laser field can be easily written within the SFA (see [10]). The total amplitude is presented as a coherent sum of such contributions. A straightforward extension of the model to the case of the initial p-electron gives the following equation for the partial amplitude of photoionization: ? A m (E , ) = 1
N

C
j =1

j

4

E exp[i (Ej )]R Ej

0

(Ej )C ( 0 , ,m)Y m (j , 0)

Ч ~

Ej + | 0 |- X i 2
t tj

exp[-i(X -| 0 |)tj ] , (13)

Ч exp -

{[p cos - A(t )]2 + p2 sin2 }dt


Photoionization of p-shells by attosecond XUV pulses in a strong laser field

3417

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 = 1fs; (b) t = 1.5fs; (c) t = 2 fs. The solid curve shows the laser electric field at 0 = 0. The dashed and dot-dashed curves show the same field at 0 = -/4 and /4, respectively.

where p = 2E, () is a Fourier transform of (t ), and R 0 (Ej ) is the radial integral ~ for the dipole operator, calculated within the same HS model as above. Ej and j are the energy and the emission angle of the photoelectron at the moment of its production by the j th pulselet. They are related to the final (asymptotic) energy E and the detection angle by the relations E/Ej sin , Ej = E + 2EA(tj ) cos + A2 (tj )/2,
t

sin j =

(14) (15)

where A(tj ) = tj EL (t ) dt is the z component of the vector potential of the laser field in the Coulomb gauge. The last exponential factor in equation (13) contains the Volkov phase [15] which describes the propagation of the free electron with momentum p in the laser field. The square of the total amplitude |A(E , )|2 , which is related to the DDCS by equation (10), is expressed in terms of partial amplitudes (13) as follows:
2

|A(E , )|2 =
m0

(2 - m0 )
= 0 +1

? A m (E , ) .

(16)

3. Results and discussion In the calculations discussed below the attosecond pulse was taken to be of hyperbolic secant form with a typical duration (FWHM of the field envelope) of 330 as, which is quite realistic for the modern experiment. For the IR few-cycle laser pulse we have chosen the cosine-shape envelope with a duration of 5 fs, a carrier wavelength of 800 nm, and intensity in the range (1 - 5) Ч 1013 Wcm-2 . More details can be found in [11]. It is convenient to define the time delay t as a time difference between the maximum of the XUV pulse and the middle point of the IR pulse envelope. For our choice of pulse envelopes t = td - L , where L is the FWHM of the laser pulse envelope. For the laser pulse of a cosine type, t = 0 corresponds to the situation where the maxima of both pulses coincide. In figure 1 we show schematically both pulses at the time delay t = 1 fs. The arrows show other delays which are considered in the examples below. The solid curve corresponds to the cosine-type pulse (The CE phase is zero). The dashed and dash-dotted lines show the laser pulses for the CE phase 0 = +/4.


3418

A K Kazansky and N M Kabachnik

Figure 2. 2D plots of the calculated |A(E , )|2 in atomic units for Ar(3p) photoionization at X = 90 eV. (a) Calculation without an IR laser field. (b) Calculation with the laser field of intensity 1 Ч 1013 Wcm-2 and the time delay t = 1.5 fs. (c) The same as in (b) but for the time delay of 2 fs. Colour (grey) scale plots show the results obtained by solving the non-stationary Е Schrodinger equation; contour plots show the results of the model calculation within the SFA.

3.1. Angular and energy distributions of photoelectrons First we discuss the results of calculation for the Ar(3p) photoionization at high photon energies. Figure 2 shows the calculated values of |A(E , )|2 , proportional to the DDCS, as a function of the photoelectron energy and emission angle. The results are shown as a 2D colour-scale plot. The calculations have been done for the XUV carrier frequency of 90 eV and the pulse duration of 330 as. The intensity of the laser is 1 Ч 1013 W cm-2 . The left panel, figure 2(a), shows the DDCS calculated by the solution of the non-stationary ? Е Schrodinger equation but without the IR laser field (E0L = 0). The angular distributions for each energy are clearly symmetric with respect to 90 and are described by the standard distribution 1 + (E )P2 (cos ). The energy distribution at any angle is determined by the shape of the Fourier transform of the pulse. Since the calculated binding energy of the Ar(3p) electron is 14.5 eV, the spectral maximum is at the energy of 75.5 eV. Integrating over the spectrum, one obtains the averaged angular distribution. In the present case, the average angular distribution is characterized by the asymmetry parameter = 1.22 which practically coincides with the value, calculated within the same HS model for monochromatic light [14]. The other two panels show the angular distributions for two different time delays between XUV and IR pulses. The distributions are strongly asymmetric, as it was already observed in our previous calculations for the s-state. Such strong forward-backward asymmetry was discussed for atomic photoionization by a strong field of few-cycle lasers [16, 17]. It is related to the fact that the electric field of the incomplete linearly polarized few-cycle pulse, which is felt by the electron, pushes electrons more to one of the two directions. In contrast to the angular distributions for the initial s-state which has zero at 90 , the p-state distributions show a more complicated structure due to the interference of the s and d waves at m = 0. Similar to the case of the spectra for the Ar(3s) ionization, the maxima of the 3p spectra change their position (streaking effect) and width with time delay. In the same two panels of figures 2(b) and (c) we show as contour plots the results of the calculations using SFA (expressions (13)-(16)) . As in the case of the 3s ionization, the results of the two calculations are in good agreement. This, of course, was expected, since the energy of the photoelectrons is large (even larger than for the 3s ionization). In this case the distortion of the ejected electron wavefunction due to the atomic field is relatively small, and the SFA is a very good approximation. Note that, in fact in our version of the SFA, the atomic field is partly taken


Photoionization of p-shells by attosecond XUV pulses in a strong laser field

3419

Figure 3. Contribution of different projections of the angular momentum of the initial state to the DDCS. (a) (m = 0) component; (b) (m = 1) component. The parameters of the pulses and the notations are the same as in figure 2(b).

into account through the dipole matrix element and the photoelectron phase (Ej ). But we neglect the corrections to the ejected electron wavefunction arising from the joint action of the IR laser and atomic field in the vicinity of the atom. The corrections to the SFA due to the Coulomb distortion of the Volkov continuum wavefunctions have been discussed recently in papers [8, 9]. However, in our opinion the uncorrected version of the SFA works reasonably well in the range of parameters (electron energy, laser intensity) where the rescattering effect and initial state polarization are negligible. The corrections become essential when it is not already possible to ignore rescattering and polarization effects. In this case the SFA is strictly speaking inapplicable. We should remark that the agreement between the exact calculation and the SFA for the p-electron ionization is not as good as for the s-electron [10]. The fine details of the distributions are not reproduced by the SFA. Probably it is connected with the fact that in the case of p-electron two interfering partial waves contribute, which makes the results more sensitive to the simultaneous action of the atomic and the IR laser fields on the ejected electron. It is confirmed by the comparison of the contributions of (m = 0) and (m = 1) projections of the initial angular momentum to the DDCS, as calculated within the two models. As is clear from figure 3, the results of both calculations practically coincide for the component where only the d wave contributes. In contrast, the results for the component, where both s and d waves contribute, are slightly shifted with respect to each other. Note that and projections give principal contributions at different emission angles, and the component contributes about twice less than the one. In figure 4 we show the angular distributions of photoelectrons for several electron energies for a particular time delay ( t = 1.5 fs). The curves represent the cuts of the surface shown in figure 2(b) along the constant energy lines. One can see that the angular distribution strongly depends on the electron energy. Almost always the distribution is forward-backward asymmetric. This asymmetry is introduced by the IR pulse, as has been explained above. Figure 5 presents the averaged electron spectra in the forward and backward directions, calculated for the time delay t = 1.5 fs. For averaging we have folded the DDCS with the Gaussian curve with a dispersion of = 15 , representing an exceptance angle of a detector. In the figure the results of three calculations are compared. The solid lines show the results obtained by the solution of equations (1) and (2), where the orbital polarization effect is taken into account. The dashed lines show the results obtained by the solution of equation (8) for the frozen initial orbital. The dash-dotted lines show the results of the SFA


3420
0.5 0.4
84.2 65.8

A K Kazansky and N M Kabachnik

DDCS (arb. units)

0.3
71.9

90.3 78.9

0.2
59.7 96.4

0.1 0 0

45

90 Angle (grad)

135

180

Figure 4. Angular distributions of photoelectrons calculated by solving the non-stationary Е Schrodinger equation (solid lines) and using the SFA (dashed lines). Numbers near the curves indicate the energy of photoelectrons in eV. The parameters of the pulses are the same as in figure 2(b).

0.5

Electron spectrum (arb. units)

0.4 0.3 0.2 0.1 0 40

60

80 Electron energy (eV)

100

Figure 5. The spectra of Ar (3p) photoelectrons emitted in the forward (black lines) and the backward (red lines) directions at X = 90 eV. The time delay is 1.5 fs. The IR field intensity is 1 Ч 1013 W cm-2 . Three types of calculations are presented: with solving equations (1) and (2) (solid lines), with solving equation (8) for the frozen initial orbital (dashed line) and the SFA model results (dash-dotted line).

calculation. All three calculations give close results, showing that at such large electron energies and for not very strong laser field the polarization effect is small and the SFA is a very good approximation. 3.2. Dependence on the carrier-envelope phase All the above calculations have been done with the IR laser pulse of a cosine type, i.e., with setting 0 = 0 in expression (5). It is clear that the angular distribution of the emitted electrons should be strongly dependent on the CE phase. In figure 6 we present the calculated DDCSs for one and the same time delay ( t = 1 fs) but for different CE phases. The left panel (figure 6(a)) shows the DDCS for the cosine-type IR pulse (0 = 0). The middle panel (figure 6(b)) shows the DDCS for the CE phase 0 = -/4. With this phase the laser pulse is shifted in such a way that the XUV pulse occurs close to the zero of the IR field (see dashed curve in figure 1). In this case the vector potential is maximal and the IR laser field strongly shifts the spectral maximum to higher (lower) energy at the forward (backward) emission angle. The angular


Photoionization of p-shells by attosecond XUV pulses in a strong laser field

3421

Figure 6. 2D plots of the calculated |A(E , )|2 in atomic units for Ar(3p) photoionization at X = 90 eV, for the time delay td = 1 fs, an IR laser intensity of 1 Ч 1013 Wcm-2 , and different carrier-envelope phases of the laser field: (a) 0 = 0, (b) 0 = -/4and (c) 0 = /4.

distribution of photoelectrons is strongly asymmetric. The right panel shows the DDCS for the CE phase 0 = /4. Here the laser pulse is shifted so that the XUV pulse falls onto the maximum magnitude of the IR field (see the dash-dotted curve in figure 1). In this case the vector potential is close to zero and the electron spectrum is practically not shifted, becoming, however, broader. The angular distribution is more symmetric with respect to 90 . The results presented in figure 6 clearly show that the angular distributions of photoelectrons are strongly affected by the CE phase in full correspondence with the basic idea of streaking effect. 3.3. Rescattering and polarization effects Until now we have presented the results of the calculations for the carrier frequency of the XUV photons corresponding to 90 eV; therefore for large electron energies (around 75 eV, before streaking by the IR field). At such energies and at the considered intensity of the laser field (1 Ч 1013 Wcm-2 ) the effects of polarization and rescattering are negligible. However, if the energy is not so high (for example, 40 eV, as in recent experiments by an Italian group [4], or lower) these effects should be significant. As we have shown earlier for the 3s ionization, rescattering and polarization lead to the breakdown of the SFA [11]. In our calculations we can switch on and off the polarization effect by solving either the system of equations (1) and (2) or equation (8), respectively. First, consider the effect of electron scattering by the atomic field, disregarding the polarization. In figure 7 we show the results of calculation of the Ar(3p) photoionization at the carrier photon energy of 40 eV. The calculations have been done for three different delay times, but for one and the same intensity of the IR field of 1 Ч 1013 W cm-2 . Already here we see that the SFA does not describe properly the angular and energy distributions of photoelectrons. Although the general shape of the distribution is similar in both calculations, the details differ considerably. The influence of the atomic field, not included in the SFA, becomes stronger for smaller photon energies. Comparing with our previous work, one can conclude that the disagreement of the SFA with the exact solution Е of the Schrodinger equation is stronger for the 3p ionization than for the 3s one. It may be connected with the fact that the angular distributions are determined additionally by the interference of the two partial waves, originally emitted by the atom in photoionization. Thus the p-shell photoionization is more sensitive to the simultaneous action of the atomic and the laser fields on the ejected electron.


3422

A K Kazansky and N M Kabachnik

(a)

(b)

(c)

Figure 7. 2D plots of the calculated |A(E , )|2 in atomic units for Ar(3p) photoionization for the XUV carrier frequency X = 40 eV, for the laser intensity of 1Ч1013 Wcm-2 and for different time delays: (a) t = 1fs, (b) t = 1.5fsand (c) t = 2 fs. The notations are the same as in figure 2.

(a)

(b)

(c)

Figure 8. 2D plots of the calculated |A(E , )|2 in atomic units for Ar(3p) photoionization for the XUV carrier frequency X = 30 eV, for the time delay of 1 fs and for different laser intensities: (a) 1 Ч 1013 Wcm-2 ,(b) 2 Ч 1013 Wcm-2 and (c) 4 Ч 1013 Wcm-2 .

We note that the calculations at 40 eV do not show the interference pattern, which was present in the results for the Ar(3s) ionization [11] and which was interpreted as an effect of the rescattering of electrons by the ionic core. Since the binding energy of the 3p electron is only 14.5 eV, the kinetic energy of photoelectrons is larger in this case than for the 3s ionization (the binding energy 28.6 eV). Thus the laser field is not strong enough to turn emitted electrons back to the core. At smaller XUV photon energies and at larger intensities of the IR field the interference pattern is clearly seen. In figure 8 we present the results of calculations for X = 30 eV for three different laser intensities at one and the same time delay of 1 fs. The horizontal stripes at small energies reveal the interference structure in the DDCS due to the rescattering of the emitted electrons by the ionic core. The calculations have been done taking into account the polarization of the initial orbital by the laser field, i.e. by solving the system (1) and (2). In order to demonstrate the effect of the orbital polarization clearer we have calculated the spectra of photoelectrons emitted in the forward direction (averaged over = 15 ) with and without the polarization. The results, shown in figure 9 for three different intensities, confirm that the polarization effect is not negligible. It considerably diminishes the cross section and becomes more important at higher intensities.


Photoionization of p-shells by attosecond XUV pulses in a strong laser field
20

3423

Electron spectrum (arb. units)

15

a

b

c

10

5

0 0

10

20

30

40

50

60

70

Electron energy (eV)

Figure 9. The spectra of Ar (3p) photoelectrons emitted in the forward direction at X = 30 eV. The time delay is 1 fs. The solid lines are obtained by solving equations (1) and (2), the dashed lines by solving equation (8) with the frozen initial orbital. The laser intensities are (a) 1 Ч 1013 Wcm-2 ,(b) 2 Ч 1013 Wcm-2 and (c) 4 Ч 1013 Wcm-2 .

4. Conclusions In conclusion, we have extended the calculations of attosecond photoionization in the presence of a strong laser field to the case of atomic subshells with 0 = 0. Both methods, suggested earlier for the s-state ionization, namely the numerical solution of the non-stationary Е Schrodinger equation [11] and the method based on the SFA [10], are generalized to the 0 = 0 case. Particular calculations have been done for the Ar(3p) ionization. All characteristic features which were observed for the s-state ionization are present in the case of the initial p state. Besides, the angular distributions of photoelectrons are characterized by the interference of the different partial waves even without an IR field which mixes them additionally. We have demonstrated a strong dependence of the angular distribution on the carrier-envelope phase of the laser pulse, which may be important in the analysis of experiments. A comparison of the two models shows that the SFA works well for high electron energies but becomes inadequate for low electron energies. The angular distribution for p-electron emission is more sensitive to the approximations than in the s-electron case. Acknowledgments We are grateful to U Heinzmann for numerous useful discussions. We are indebted to Bielefeld University for hospitality and financial support via SFB 613. We also acknowledge the financial support from the Russian Foundation for Fundamental Researches via grant 06-02-16289. The research was supported in part through a visitor grant for AKK by the DFG cluster of excellence Munich-Centre for Advanced Photonics (www.munich-photonics.de). References
[1] Hentschel M, Kienberger R, Spielmann Ch, Reider G A, Milosevic N, Brabec T, Corkum P, Heinzmann U, Drescher M and Krausz F 2001 Nature 414 509 [2] Drescher M, Hentschel M, Kienberger R, Tempea G, Spielmann Ch, Reider G A, Corkum P B and Krausz F 2001 Science 291 1923 [3] Kienberger R et al 2004 Nature 427 817 [4] Sansone G et al 2006 Science 314 443


3424

A K Kazansky and N M Kabachnik

[5] Drescher M, Hentschel M, Kienberger R, Uiberacker M, Yakovlev V, Scrinzi A, Westerwalbesloh Th, Kleineberg U, Heinzmann U and Krausz F 2002 Nature 419 803 ?? [6] Itatani J, Quere F, Yudin G L, Ivanov M Yu, Krausz F and Corkum P B 2002 Phys. Rev. Lett. 88 173903 [7] Kitzler M, Milosevic N, Scrinzi A, Krausz F and Brabec T 2002 Phys. Rev. Lett. 88 173904 [8] Smirnova O, Spanner M and Ivanov M Yu 2006 J. Phys. B: At. Mol. Opt. Phys. 39 S323 [9] Yudin G L, Patchkovskii S, Corkum P B and Bandrauk A D 2007 J. Phys. B: At. Mol. Opt. Phys. 40 F93 [10] Kazansky A K and Kabachnik N M 2006 J. Phys. B: At. Mol. Opt. Phys. 39 5173 [11] Kazansky A K and Kabachnik N M 2007 J. Phys. B: At. Mol. Opt. Phys. 40 2163 [12] Kazansky A K and Kabachnik N M 2006 Phys. Rev. A 73 062712 [13] Herman F and Skillman S 1963 Atomic Structure Calculations (Englewood Cliffs, NJ: Prentice-Hall) [14] Kennedy D J and Manson S T 1972 Phys. Rev. A 5 227 [15] Volkov D V 1935 Z. Phys. 94 250 [16] Chelkowski S and Bandrauk A D 2002 Phys. Rev. A 65 061802(R) [17] Makris M G and Lambropoulos P 2004 J. Phys. B: At. Mol. Opt. Phys. 37 2247