## S-matrix theory of two-electron momentum distribution produced by double ionization in intense laser fields

Optics Express, Vol. 8, Issue 7, pp. 383-394 (2001)

http://dx.doi.org/10.1364/OE.8.000383

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### Abstract

Recently observed momentum distribution of doubly charged recoil-ions of atoms produced by femtosecond infrared laser pulses is analyzed using the so-called intense-field many-body S-matrix theory. Observed characteristics of the momentum distributions, parallel and perpendicular to the polarization axis, are reproduced by the theory. It is shown that correlated energy-sharing between the two electrons in the intermediate state and their 'Volkov-dressing' in the final state, can explain the origin of these characteristics.

© Optical Society of America

## 1 Introduction

1. Th. Weber, M. Weckenbrock, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, F. Afaneh, G. Urbasch, M. Vollmer, H. Giessen, and R. Dörner, “Recoil-ion momentum distribution for single and double ionization of helium in strong laser fields,” Phys. Rev. Lett. **84**, 443–446 (2000). [CrossRef] [PubMed]

2. R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C.D. Schröder, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner “Momentum distribution of Ne^{n+} ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. **84**, 447–450 (2000). [CrossRef] [PubMed]

3. B. Walker, B. Sheehy, L.F. DiMauro, P. Agostini, K.J. Schafer, and K.C. Kulander “Precision measurement od strong field double ionization of helium,” Phys. Rev. Lett. **73**, 1227–1230 (1994). [CrossRef] [PubMed]

4. S. Larochelle, A. Talebpour, and S.L. Chin, “Non-sequential multiple ionization or rare gas atoms in a Ti:sapphire laser field,” J. Phys. B **31**, 1201–1214 (1998). [CrossRef]

_{i≠j}1/

*r*

_{ij}.

*N*,

*N*=2, 3…, dimensional partial differential equations over realistically large space-time grids. Much progress (especially at higher frequencies) towards numerical solutions of the six-dimensional Schrödinger problem for the double ionization of He have recently been made using state-of-the-art computations (e.g. [5

5. J.S. Parker, K.T. Taylor, C.W. Clark, and S. Blodgett-Ford, “Intense-field mutliphoton ionisation of a two-electron atom,” J. Phys. B **29**, L33–L42 (1996). [CrossRef]

7. J.S. Parker, L.R. Moore, K.J. Mehring, D. Dundas, and K.T. Taylor, “Double-electron above threshold ionization of helium,” J. Phys. B **34**, L69–L78 (2001). [CrossRef]

*S*-matrix theory’ (IMST) is designed to this end [8

8. F.H.M. Faisal and A. Becker, “‘Intense-Field Many-Body S-Matrix Theory’ and mechanism of laser induced double ionization of Helium,” in *Selected Topics on Electron Physics*, D.H. Campbell and H. Kleinpoppen, eds. (Plenum Press, New York, 1996) pp. 397–410. [CrossRef]

8. F.H.M. Faisal and A. Becker, “‘Intense-Field Many-Body S-Matrix Theory’ and mechanism of laser induced double ionization of Helium,” in *Selected Topics on Electron Physics*, D.H. Campbell and H. Kleinpoppen, eds. (Plenum Press, New York, 1996) pp. 397–410. [CrossRef]

16. A. Becker and F.H.M. Faisal, “Interpretation of momentum distribution of recoil ions from laser induced nonsequential double ionization,” Phys. Rev. Lett. **84**, 3546–3549 (2000). [CrossRef] [PubMed]

16. A. Becker and F.H.M. Faisal, “Interpretation of momentum distribution of recoil ions from laser induced nonsequential double ionization,” Phys. Rev. Lett. **84**, 3546–3549 (2000). [CrossRef] [PubMed]

## 2 TheIMST

*S*-matrix theory (IMST), the

*S*-matrix (or the probability amplitude for any process) is rearranged in such a way that the dominant features of the process can appear in the leading terms of the

*S*-matrix series. It is interesting to note that, besides providing an ab-initio systematic approximation method for calculating the amplitude of interest, IMST can help in identifying possible mechanisms involved in the process of interest. This can be done by analyzing the Feynman diagrams that are generated by the leading terms of the theory. It is also interesting to note that, in addition to the above use of the theory, IMST can also help in constructing simple models by suggesting appropriate places for introduction of

*physical*hypotheses. The latter possibility is particularly interesting for gaining insights into complex situations that may be otherwise unaccessible to (or extremly impractical for) ab-initio analyses. In this section we briefly present the IMST and point out the motivations behind its construction. We also point out the structural flexibilities of the theory, that lead to the methodical advantages mentioned above.

*t*

_{i}). To this end, at first, we make the usual partition of the total Hamiltonian of the system

*H*(

*t*) (Hartree atomic units, a.u., are used below),

*V*

_{i}(

*t*) with the atomic electrons, and rewrite the Schrödinger equation

*G*(

*t*,

*t*

^{′}) by the definition

*ϕ*

_{i}(t) is the solution of the initial Schrödinger problem with the unperturbed Hamiltonian

_{i}(t) is given by the interaction Hamiltonian of the two electrons coupled to the vector potential of the laser field,

*t*) is the final reference Hamiltonian (e.g., the sum Hamiltonian of two electrons in the field [17

17. F.H.M. Faisal, “Exact Solution of the Schrödinger Equation of Two Electrons Interacting with an Intense Electromagnetic Field,” Phys. Lett. A **187**, 180–184 (1994). [CrossRef]

18. A. Becker and F.H.M. Faisal, “Correlated Keldysh-Faisal-Reiss theory of above-threshold double ionization of He in intense laser fields,” Phys. Rev. A **50**, 3256–3264 (1994). [CrossRef] [PubMed]

*V*

_{f}(

*t*) is the final state interaction (e.g., the electron-electron correlation, and also any residual interaction with the nucleus). The final Green’s function satisfies

*t*)> above, Eq. (3), to get

*not*identical to the initial reference Hamiltonian. This is because the projection is orthogonal and hence extracts only

*one*term from the final reference Green’s function (given in the proper state representation). Such a formulation is, therefore, useful not only in the present context but also for many other problems, e.g., the charge exchange reactions, chemical reactions and rearrangement processes in general (which invariably involve unequal reference Hamiltonians in the input and output channels). Thus, projecting onto the final state |

*ϕ*

_{f}(

*t*)>, belonging to

*t*), we write the transition amplitude for the

*i*→

*f*transition as:

*master equation*for the

*S*-matrix, whose two most important features are that (a) the total Green’s function appears

*between*the initial and the final interactions. This is unlike the appearence of the Green’s function at an end, as in the usual ‘prior (or direct time)’ and the ‘post (or time reversed)’ form (see e.g. [20, 21

21. H.R. Reiss, “Effect of an intense electromagnetic field on weakly bound system,” Phys. Rev. A **22**, 1786–1813 (1980). [CrossRef]

*or*unequal reference Hamiltonians. This structural flexibility of Eq. (11) permits the introduction of any

*virtual*fragments-propagator of interest, already in the leading terms of the IMST, which is responsible for its potential usefulness for the general many-body

*reaction*problems. Note that, inclusion of the effects of such virtual fragments-channels in the usual ‘prior’ and ‘post’ expansions may only be attemped indirectly, if at all, by summing them to very high orders, if not to infinite orders. We may now introduce the third intermediate partitioning of the total Hamiltonian:

*H*

_{0}(

*t*), together with the corresponding propagator

*G*

_{0}(

*t*,

*t*

^{′}). For the double ionization problem of present interest, the virtual intermediate fragments can consist of one electron in the Volkov states of all virtual momenta {

**k**}, and a singly charged residual ion in its virtual eigenstates; the corresponding fragments-Hamiltonian is, therefore,

8. F.H.M. Faisal and A. Becker, “‘Intense-Field Many-Body S-Matrix Theory’ and mechanism of laser induced double ionization of Helium,” in *Selected Topics on Electron Physics*, D.H. Campbell and H. Kleinpoppen, eds. (Plenum Press, New York, 1996) pp. 397–410. [CrossRef]

16. A. Becker and F.H.M. Faisal, “Interpretation of momentum distribution of recoil ions from laser induced nonsequential double ionization,” Phys. Rev. Lett. **84**, 3546–3549 (2000). [CrossRef] [PubMed]

*θ*(

*t*-

*t*

^{′}) is the Heaviside theta function, and {|

*G*

_{0}(

*t*,

*t*

^{′}) is available, we can expand

*G*(

*t*,

*t*

^{′}) as

*S*-matrix series,

*S*

^{(1)}

*f*

_{i}(

*t*), are generally

*equivalent*for long interaction times ([10], footnote 11), but, as discussed above, IMST differs term by term

*qualitatively*from the next term onward. We further note that the first term of the IMST series is the same as that of the so-called KFR theory [22, 23

23. F.H.M. Faisal, “Multiple Absorption of Laser Photons by Atoms,” J. Phys. B **6**, L89–92 (1973). [CrossRef]

21. H.R. Reiss, “Effect of an intense electromagnetic field on weakly bound system,” Phys. Rev. A **22**, 1786–1813 (1980). [CrossRef]

## 3 Two-Electron Sum-Momentum Distributions

**P**

_{par.}=−[(

**k**

_{a})

_{par.}+(

**k**

_{b})

_{par.}]

**P**

_{perp.}≈−[(

**k**

_{a})

_{perp.}+(

**k**

_{b})

_{perp.}]

**P**is the momentum of the doubly charged ion, and

**k**

_{a}and

**k**

_{b}are the momenta of the two outgoing electrons.

1. Th. Weber, M. Weckenbrock, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, F. Afaneh, G. Urbasch, M. Vollmer, H. Giessen, and R. Dörner, “Recoil-ion momentum distribution for single and double ionization of helium in strong laser fields,” Phys. Rev. Lett. **84**, 443–446 (2000). [CrossRef] [PubMed]

2. R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C.D. Schröder, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner “Momentum distribution of Ne^{n+} ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. **84**, 447–450 (2000). [CrossRef] [PubMed]

*parallel*to the laser polarization direction shows a prominent double-hump distribution with a central

*minimum*at peak intensities near the saturation point,

*single*-hump distribution,

1. Th. Weber, M. Weckenbrock, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, F. Afaneh, G. Urbasch, M. Vollmer, H. Giessen, and R. Dörner, “Recoil-ion momentum distribution for single and double ionization of helium in strong laser fields,” Phys. Rev. Lett. **84**, 443–446 (2000). [CrossRef] [PubMed]

2. R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C.D. Schröder, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner “Momentum distribution of Ne^{n+} ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. **84**, 447–450 (2000). [CrossRef] [PubMed]

*γ*, 2

*e*) reaction by (weak field) synchrotron photons (that favors the Warnier back-to-back mechanism near zero momentum).

*total*double ionization yields using the present theory [8

*Selected Topics on Electron Physics*, D.H. Campbell and H. Kleinpoppen, eds. (Plenum Press, New York, 1996) pp. 397–410. [CrossRef]

11. A. Becker and F.H.M. Faisal, “Mechanism of laser-induced double ionization of helium,” J. Phys. B **29**, L197–L202 (1996). [CrossRef]

15. A. Becker and F.H.M. Faisal, “*S*-matrix analysis of ionization yields of noble gas atoms at the focus of Ti:sapphire laser pulses,” J. Phys. B **32**, L335–L343 (1999). [CrossRef]

24. P. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. **71**, 1994–1997 (1993). [CrossRef] [PubMed]

13. A. Becker and F.H.M. Faisal, “Interplay of electron correlation and intense field dynamics in double ionization of helium,” Phys. Rev. A **59**, R1742–R1745 (1999). [CrossRef]

**84**, 3546–3549 (2000). [CrossRef] [PubMed]

*Selected Topics on Electron Physics*, D.H. Campbell and H. Kleinpoppen, eds. (Plenum Press, New York, 1996) pp. 397–410. [CrossRef]

*I*=6.6×10

^{14}W/cm

^{2}used for the He experiment [1

**84**, 443–446 (2000). [CrossRef] [PubMed]

*relative*importance of the eight leading diagrams are in general quite different for different

*domains*of the laser parameters, in particular, in different wavelength domains. Thus, for example, one of the two first rank diagrams in fact was found to be dominant at λ=248 nm, in the UV wavelength domain, over the entire range of the intensity in an experiment of interest (see, experiment [25

25. D. Charalambidis, D. Xenakis, C.J.G.J. Uiterwaal, P. Maragakis, Jian Zhang, H. Schröder, O. Faucher, and P. Lambropoulos “Multiphoton ionisation saturation intensities and generalised cross sections from ATI spectra,” J. Phys. B **30**, 1467–1480 (1997). [CrossRef]

13. A. Becker and F.H.M. Faisal, “Interplay of electron correlation and intense field dynamics in double ionization of helium,” Phys. Rev. A **59**, R1742–R1745 (1999). [CrossRef]

*t*=

*t*

_{2}-

*t*

_{1}=

*τ*(or the phase difference, Δ

*ϕ*=

*ϕ*

_{2}-

*ϕ*

_{1}), as it naturally occurs in the time integrations of the amplitude of the process (see below). Under appropriate limiting conditions, however, the time integrations over the interval

*τ*may be estimated semiclassically. Under the latter conditions (e.g.,

*U*

_{p}≫

*ω*,

*ω*≪

*E*

_{B}) a significant contribution can arise from time intervals of the order of the classical return time(s) of the first emitted electron to the core region, with a corresponding return energy of the order of 3

*U*

_{p}. This provides the theoretical connection between the present quantum theory and the classical ‘rescattering model’ of Corkum [24

24. P. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. **71**, 1994–1997 (1993). [CrossRef] [PubMed]

*exactly*after expanding all the Volkov waves in the amplitude in terms of the generalized Bessel functions (see Eqs. (22),(23) below).

_{a}, k

_{b};

*t*

_{2}) is the two-electron product Volkov final state in the field (c.f. [17

17. F.H.M. Faisal, “Exact Solution of the Schrödinger Equation of Two Electrons Interacting with an Intense Electromagnetic Field,” Phys. Lett. A **187**, 180–184 (1994). [CrossRef]

*V*

_{corr}(

*t*

_{2}) stands for correlation operator and Ψ

_{i}(

*t*

_{2}) is the total wavefunction of the system, at time

*t*

_{2}. In the present approximation, Ψ

_{i}(

*t*

_{2}) is given by (c.f. Fig. 1),

*V*

_{ATI}(

*t*

_{1}) is the interaction operator for the virtual ATI-like process at the time

*t*

_{1},

*ϕ*

^{V}(

**k**;

**r**,

*t*) is the one-electron Volkov wave function, and

*ϕ*

_{1S}(

**r**

_{1},

**r**

_{2};

*t*

_{1}) is the ground state wavefunction of the He atom with binding energy

*E*

_{B}=2.904 (a.u.). An exact evaluation of this amplitude, including all orders of correlation and ATI interaction, is practically an impossible task. We have, therefore, restricted ourselves to the lowest significant terms of the theory by replacing

*V*

_{corr}(

*t*

_{2}) by 1/

*r*

_{12}and

*VAT*

_{I}(

*t*

_{1}) by

*ϕ*

^{V}(

**k**,

**r**;

*t*), in terms of the corresponding Fourier components defined by the generalized Bessel functions of two arguments (e.g., [19]). This allows us to evaluate the two-fold time integrations, over the instants

*t*

_{1}and

*t*

_{2}of the two interactions, exactly; for pulses much longer than a laser period (as in the experiments of present interest) this gives,

*ϕ*

^{0}(

**k**,

**r**) is a plane wave state.

*k*is performed by pole approximation and the integrals over the angles of

**k**and the sum over

*n*are performed numerically. Contribution of the lowest term of the sum over

*j*(corresponding to the ground state of the ionic state) is retained only since it is found to dominate over the contribution from any excited state (c.f. [11

11. A. Becker and F.H.M. Faisal, “Mechanism of laser-induced double ionization of helium,” J. Phys. B **29**, L197–L202 (1996). [CrossRef]

**k**

_{a}and

**k**

_{b}, from the formula,

*sum*of the two momenta, (

**k**

_{a}+

**k**

_{b}), are determined, in each case by integrating over the remaining variables and summing the contributions from all significant

*N*s. We recall that under the condition of the experiment the He

^{2+}recoil momentum

**P**≈-(

**k**

_{a}+

**k**

_{b}) [1

**84**, 443–446 (2000). [CrossRef] [PubMed]

^{n+} ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. **84**, 447–450 (2000). [CrossRef] [PubMed]

^{++},

*parallel*to the (linear) polarization axis by Weber et al. [1

**84**, 443–446 (2000). [CrossRef] [PubMed]

^{14}W/cm

^{2}and λ=800 nm. They are compared with the corresponding results of the present theory [16

**84**, 3546–3549 (2000). [CrossRef] [PubMed]

*one*point, namely the maximum of the distribution (at -(

**k**

*+*

**a****k**

_{b})=-2 a.u.), with the experimental data. This one-point fit determines the

*relative*scale for the entire theoretical set in the Figure. As it can be seen from the comparison, all the essential features of the experimental distribution of the parallel component of the recoil momentum are reproduced by the theoretical results. Thus, both the distributions show a double-hump structure with a central minimum. The positions and the heights of the two maxima are also well reproduced. We observe that the maximum size of the recoil ion momentum (cut-off momentum) of the experiment and the calculation of the sum electron-momenta also agree well with each other and in fact in both cases it is as large as ≈5 a.u. Finally, we note that there is a quantitative difference at the minimum of the distribution. This suggests that either the uncertainty in the intensity measurement and momentum resolution [27] and/or the higher order contributions in the theory might be involved here.

*and*its absence in the perpendicular case? To gain a greater insight into their origin we calculated the corresponding distributions by deliberately neglecting the

*final*-state interaction of the two electrons with the laser pulse, i.e. dropping the ‘Volkov dressing’ of the two electron in the final state. This is easily done in the present theory by replacing the final state two-electron Volkov wavefunction, by two free (plane) waves of momenta

**k**

_{a}and

**k**

_{b}, and keeping everything else the same.

*with*(Fig. 2b) and

*without*(Fig. 4a) the Volkov dressing in the final state, clearly shows that the double-hump character of the parallel component of the distribution collapses into a single-hump structure in the

*absence*of the final state Volkov dressing. This unequivocally suggests that the final-state field interaction of the two electrons is primarily responsible for the double-hump structure of the observed distribution in this case. In Fig. 4b we show the corresponding result of calculation for the perpendicular component in the absence of the final state laser interaction. Comparison of this distribution with that of Fig. 3b shows that in this (perpendicular) case the presence of the final state laser interaction does not play a significant role. This is as might be expected in the present case, since the force due to the electric field is negligible in the direction perpendicular to the polarization direction. This is also consistent with the observed narrow width of the perpendicular distribution due to the absence of significant laser coupling in the final state. Therefore, in this case we have the interesting situation of observing the double ionization in the laboratory as if the laser field is switched off as soon as the two electrons are freed from the bound atomic system.

**84**, 3546–3549 (2000). [CrossRef] [PubMed]

_{B}is the binding energy. For the case of the distribution shown in Fig. 2 use of this formula predicts the cut-off momentum ≈5 a.u., in good agreement with both the experimental value, and the numerical result. Satisfactory agreement between the above cut-off formula and the available experimental values at other intensities for He [1

**84**, 443–446 (2000). [CrossRef] [PubMed]

^{n+} ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. **84**, 447–450 (2000). [CrossRef] [PubMed]

**84**, 3546–3549 (2000). [CrossRef] [PubMed]

## 4 Models of Total Ionization Yields

*Selected Topics on Electron Physics*, D.H. Campbell and H. Kleinpoppen, eds. (Plenum Press, New York, 1996) pp. 397–410. [CrossRef]

11. A. Becker and F.H.M. Faisal, “Mechanism of laser-induced double ionization of helium,” J. Phys. B **29**, L197–L202 (1996). [CrossRef]

*n*-sum), semi-empirical ‘

*e*- 2

*e*’ rates at collison energies given by the maximum classical energy [12] or the back-scattering energy [13

13. A. Becker and F.H.M. Faisal, “Interplay of electron correlation and intense field dynamics in double ionization of helium,” Phys. Rev. A **59**, R1742–R1745 (1999). [CrossRef]

15. A. Becker and F.H.M. Faisal, “*S*-matrix analysis of ionization yields of noble gas atoms at the focus of Ti:sapphire laser pulses,” J. Phys. B **32**, L335–L343 (1999). [CrossRef]

*and*multiple ionization yields when compared with the available experimental data, in a wide variety of cases [12–15

15. A. Becker and F.H.M. Faisal, “*S*-matrix analysis of ionization yields of noble gas atoms at the focus of Ti:sapphire laser pulses,” J. Phys. B **32**, L335–L343 (1999). [CrossRef]

*model*calculations for double ionization of He at λ=780 nm and a pulse duration of

*τ*=160 fs using the model of collision energy for the ‘

*e*- 2

*e*’ collision to be equal to the ‘rescattering energy’ (of the order of 3

*U*

_{p}, cf. [12]), or the back-scattering energy (of the order of 8

*U*

_{p}, cf. [13

**59**, R1742–R1745 (1999). [CrossRef]

*S*-matrix analysis of ionization yields of noble gas atoms at the focus of Ti:sapphire laser pulses,” J. Phys. B **32**, L335–L343 (1999). [CrossRef]

3. B. Walker, B. Sheehy, L.F. DiMauro, P. Agostini, K.J. Schafer, and K.C. Kulander “Precision measurement od strong field double ionization of helium,” Phys. Rev. Lett. **73**, 1227–1230 (1994). [CrossRef] [PubMed]

*U*

_{p}) may not be quite enough as a higher collision energy of the order of the back-scattering energy (≈8

*U*

_{p}) apparently fare better.

*e*- 2

*e*’

*rates*as a function of the collision energy, which is shown for He

^{+}ions in Fig. 6, as calculated from the well-known formula given by Lotz [28

28. W. Lotz, “Electron-Impact Ionization Cross Sections and Ionization Rate Coefficients for Atoms and Ions from Hydrogen to Calcium,” Zeit. f. Phys. **216**, 241–247 (1968). [CrossRef]

*e*- 2

*e*’ rate increases rapidly at first and remains virtually

*constant*at higher collision energies. This appears to be the reason for the relative insensitivity of the ion yields, specially at not too low intensities, with respect to the assumption of the two collision energies. Furthermore, it should be noted that the actual value of the near constant rate is Γ

^{(e-2e)}≈3×10

^{-3}a.u., is close to the magnitude of the

*ratio*between the double to single ionization yields of He of about 2×10

^{-3}, that has been observed [3

3. B. Walker, B. Sheehy, L.F. DiMauro, P. Agostini, K.J. Schafer, and K.C. Kulander “Precision measurement od strong field double ionization of helium,” Phys. Rev. Lett. **73**, 1227–1230 (1994). [CrossRef] [PubMed]

## 5 Conclusion

## Acknowledgments

## References and links

1. | Th. Weber, M. Weckenbrock, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, F. Afaneh, G. Urbasch, M. Vollmer, H. Giessen, and R. Dörner, “Recoil-ion momentum distribution for single and double ionization of helium in strong laser fields,” Phys. Rev. Lett. |

2. | R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C.D. Schröder, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner “Momentum distribution of Ne |

3. | B. Walker, B. Sheehy, L.F. DiMauro, P. Agostini, K.J. Schafer, and K.C. Kulander “Precision measurement od strong field double ionization of helium,” Phys. Rev. Lett. |

4. | S. Larochelle, A. Talebpour, and S.L. Chin, “Non-sequential multiple ionization or rare gas atoms in a Ti:sapphire laser field,” J. Phys. B |

5. | J.S. Parker, K.T. Taylor, C.W. Clark, and S. Blodgett-Ford, “Intense-field mutliphoton ionisation of a two-electron atom,” J. Phys. B |

6. | E.S. Smyth, J.S. Parker, and K.T. Taylor, “Numerical integration of the time-depedent Schrödinger equation for laser-driven helium,” Comp. Phys. Comm. |

7. | J.S. Parker, L.R. Moore, K.J. Mehring, D. Dundas, and K.T. Taylor, “Double-electron above threshold ionization of helium,” J. Phys. B |

8. | F.H.M. Faisal and A. Becker, “‘Intense-Field Many-Body S-Matrix Theory’ and mechanism of laser induced double ionization of Helium,” in |

9. | F.H.M. Faisal and A. Becker, “Effect of rescattering on ATI and e-e correlation on double ionization in intense laser fields” in |

10. | F.H.M. Faisal, A. Becker, and J. Muth-Böhm, “Intense-Field Many-Body |

11. | A. Becker and F.H.M. Faisal, “Mechanism of laser-induced double ionization of helium,” J. Phys. B |

12. | F.H.M. Faisal and A. Becker, “Non-sequential double ionization: Mechanism and model formula,” Laser Phys. |

13. | A. Becker and F.H.M. Faisal, “Interplay of electron correlation and intense field dynamics in double ionization of helium,” Phys. Rev. A |

14. | A. Becker and F.H.M. Faisal, “Production of high charge states of Xe in a femtosecond laser pulse,” Phys. Rev. A |

15. | A. Becker and F.H.M. Faisal, “ |

16. | A. Becker and F.H.M. Faisal, “Interpretation of momentum distribution of recoil ions from laser induced nonsequential double ionization,” Phys. Rev. Lett. |

17. | F.H.M. Faisal, “Exact Solution of the Schrödinger Equation of Two Electrons Interacting with an Intense Electromagnetic Field,” Phys. Lett. A |

18. | A. Becker and F.H.M. Faisal, “Correlated Keldysh-Faisal-Reiss theory of above-threshold double ionization of He in intense laser fields,” Phys. Rev. A |

19. | F.H.M. Faisal, |

20. | C. Joachain, |

21. | H.R. Reiss, “Effect of an intense electromagnetic field on weakly bound system,” Phys. Rev. A |

22. | L.V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP20, 1307–1314 (1965) [Zh. Eksp. Teor. Fiz.47, 1945–1957 (1964)] |

23. | F.H.M. Faisal, “Multiple Absorption of Laser Photons by Atoms,” J. Phys. B |

24. | P. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. |

25. | D. Charalambidis, D. Xenakis, C.J.G.J. Uiterwaal, P. Maragakis, Jian Zhang, H. Schröder, O. Faucher, and P. Lambropoulos “Multiphoton ionisation saturation intensities and generalised cross sections from ATI spectra,” J. Phys. B |

26. | A. Erdélyi (Ed.), |

27. | The uncertainty in the intensity measurement is ≈15–30%, and that of momentum resolution ≈0.2–0. 4 a.u. [R. Dörner, and H. Rottke (private communication)]. |

28. | W. Lotz, “Electron-Impact Ionization Cross Sections and Ionization Rate Coefficients for Atoms and Ions from Hydrogen to Calcium,” Zeit. f. Phys. |

**OCIS Codes**

(020.4180) Atomic and molecular physics : Multiphoton processes

(190.4180) Nonlinear optics : Multiphoton processes

**ToC Category:**

Focus Issue: Laser-induced multiple ionization

**History**

Original Manuscript: February 8, 2001

Published: March 26, 2001

**Citation**

Andreas Becker and F. Faisal, "S-matrix theory of two-electron momentum distribution produced by double ionization in intense laser fields," Opt. Express **8**, 383-394 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-7-383

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### References

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- A. Erd'elyi (Ed.), Higher Transcendental Functions, Vol. 2, (New York: McGraw-Hill, 1953).
- The uncertainty in the intensity measurement is $approx$ 15 - 30, and that of momentum resolution $approx$ 0.2 - 0.4 a.u. [R. Doerner, and H. Rottke (private communication)].
- W. Lotz, "Electron-Impact Ionization Cross Sections and Ionization Rate Coefficients for Atoms and Ions from Hydrogen to Calcium," Zeit. f. Phys. 216, 241-247 (1968).

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