## A modified rate equation for the propagation of a femtosecond laser pulse in field-ionizing medium |

Optics Express, Vol. 21, Issue 5, pp. 5413-5423 (2013)

http://dx.doi.org/10.1364/OE.21.005413

Acrobat PDF (2945 KB)

### Abstract

A recombination rate of electron-ion in the strong-field atomic process is phenomenologically introduced into the ionization rate equation, and therefrom an ionization and recombination rates equation (IRRE) is obtained. By using the extended IRRE, the propagation equation of an intense femtosecond laser pulse in the gaseous medium is re-derived. Some new physical behaviors and characteristics caused by the introduced recombination rate are discussed in detail.

© 2013 OSA

## 1. Introduction

1. A. Rundquist, C. G. Durfee, Z. H. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft X-rays,” Science **280**(5368), 1412–1415 (1998) [CrossRef] [PubMed] .

2. M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. **60**, 389–486 (1997) [CrossRef] .

1. A. Rundquist, C. G. Durfee, Z. H. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft X-rays,” Science **280**(5368), 1412–1415 (1998) [CrossRef] [PubMed] .

3. S. C. Rae, K. Burnett, and J. Cooper, “Generation and propagation of high-order harmonics in a rapidly ionizing medium,” Phys. Rev. A **50**(4), 3438–3446 (1994) [CrossRef] [PubMed] .

4. E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, “Coherent soft X-ray generation in the water window with quasi-phase matching,” Science **302**(5642), 95–98 (2003) [CrossRef] [PubMed] .

3. S. C. Rae, K. Burnett, and J. Cooper, “Generation and propagation of high-order harmonics in a rapidly ionizing medium,” Phys. Rev. A **50**(4), 3438–3446 (1994) [CrossRef] [PubMed] .

5. S. C. Rae and K. Burnett, “Detailed simulations of plasma-induced spectral blueshifting,” Phys. Rev. A **46**(2), 1084–1090 (1992) [CrossRef] [PubMed] .

6. M. B. Gaarde, M. Murakami, and R. Kienberger, “Spatial separation of large dynamical blueshift and harmonic generation,” Phys. Rev. A **74**(5), 053401 (2006) [CrossRef] .

7. F. Brunel, “Not-so-resonant, resonant absorption,” Phys. Rev. Lett. **59** (1), 52–55 (1987) [CrossRef] [PubMed] .

9. P. B. Corkum, “Plasma perspective on strong-field multiphoton ionization,” Phys. Rev. Lett. **71**(13), 1994–1997 (1993) [CrossRef] [PubMed] .

10. M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A **49**(3), 2117–2132 (1994) [CrossRef] [PubMed] .

6. M. B. Gaarde, M. Murakami, and R. Kienberger, “Spatial separation of large dynamical blueshift and harmonic generation,” Phys. Rev. A **74**(5), 053401 (2006) [CrossRef] .

*P*(

*τ*) =

*e*∑

*(*

_{i}n_{i}*τ*)

*s*(

_{i}*τ*), where

*e*is the electron charge,

*n*(

_{i}*τ*) the number of electrons with the

*i*-th type of displacement

*s*(

_{i}*τ*) relative to their parent ions at time

*τ*.

*c*,

*τ*=

*t*−

*z/c*,

*ξ*=

*z*, and the international system of units (SI unit) is adopted unless otherwise specified.

## 2. Theoretical model

*E*=

*E*(

*r*,

*ξ*,

*τ*) through a gaseous medium can be described by where

*p*is the pressure of the gaseous medium and

*μ*

_{0}the permeability, and we denote

*P*(

*τ*) =

*P*(

*r*,

*ξ*,

*τ*).

*N*

_{e}(

*r*,

*ξ*,

*τ*) by

*N*

_{e}(

*τ*) with no ambiguity. Considering the recombination of electrons with their parent ions, the electron density satisfies where

*R*(

*τ*) is the rate of the electronic density. It consists of two parts: the production rate and the recombination rate, where

*N*

_{a}(

*τ*) =

*N*

_{0}−

*N*

_{e}(

*τ*) is the density of neutral atoms with

*N*

_{0}the initial density of neutral atoms depending on the applied temperature and pressure. In above equations, the time

*τ*

_{pro}is the releasing time of an electron from its parent ion under a strong femtosecond laser field, and

*α*(

*τ*,

*τ*

_{pro}) characterizes the recombination condition at

*τ*=

*τ*

_{rec}. In the quantum frame, the value of

*α*(

*τ*,

*τ*

_{pro}) ranges from 0 to 1 depending on overlap of the electronic wavepacket with the nuclear-dominated interaction region, which requires some further investigations via the transitions between bound and continuum states. More details about the capture of ionized electrons by ionic cores can be found in [23

23. J. Eichler and T. Stohlker, “Radiative electron capture in relativistic ion-atom collisions and the photoelectric effect in hydrogen-like high-Z systems,” Phys. Rep. **439**, 1–99 (2007) [CrossRef] .

*et al.*[24

24. A. Scrinzi, M. Geissler, and T. Brabec, “Ionization above the Coulomb barrier,” Phys. Rev. Lett. **83**(4), 706–709 (1999) [CrossRef] .

*α*(

*τ*,

*τ*

_{pro}) = 1 at the first return time of an ionized electron. Although it will overestimate the density of recombined electrons, it does not change the mathematics expressions of the ionization rate equation and the below propagation equation.

*m*

_{e}

*ẍ*(

*τ*,

*τ*

_{pro}) =

*eE*(

*τ*) with two

*τ*

_{pro}-dependent initial conditions

*x*

_{0}and

*v*

_{0}for

*τ*<

*τ*

_{rec}, where

*τ*

_{rec}is the recombination time of the electron with its parent ion. Generally, the ionization position is

*x*

_{0}=

*I*

_{p}/ (

*eE*(

*τ*

_{pro})), and

*v*

_{0}= 0 is always assumed. The explicit solution of electronic motion equation can be obtained by performing integration over time [

*τ*

_{pro},

*τ*] where

*v*(

*τ*,

*τ*

_{pro}) is the corresponding velocity which is given by Here the vector potential

*A*(

*τ*) of the electric field is defined as

*α*(

*τ*,

*τ*

_{pro}) = 1, we investigate the condition

*x*(

*τ*,

*τ*

_{pro}) → 0, which determines the recombination of an electron with its parent ion. At this moment, Coulomb effects can not be neglected. The extreme of distorted Coulomb potential by external strong field,

*V*= −

*e*

^{2}/(4

*πε*

_{0}

*x*)+

*exE*, is given as

*x*(

*τ*,

*τ*

_{pro}) <

*x*

_{c}, Coulomb potential will be in dominant status, so that the free electron can be captured by its parent ion. In this paper, we neglect the two or more collisions between an electron and its parent ion before they are recombined. The return time

*τ*

_{rec}is determined by where

*α*(

*τ*,

*τ*

_{pro}) = 1 for

*τ*=

*τ*

_{rec}, and

*α*(

*τ*,

*τ*

_{pro}) = 0 otherwise. In Fig. 1, we show the releasing time

*τ*

_{pro}and the corresponding return time

*τ*

_{rec}. Because of the non-zero initial position and the non-planar electric field, some of the electrons will take several optical cycles to return to their parent ions. The ansatz that

*α*(

*t*,

*t*

_{b}) = 1 at the first return time of an ionized electron leads to the truncation of the subsequent excursions which are marked out with red pentagrams.

*τ*

_{pro}[

*τ*] as the releasing time

*τ*

_{pro}corresponding to the return time

*τ*, and

*τ*

_{rec}[

*τ*] as the return time

*τ*

_{rec}corresponding to the releasing time

*τ*, which is determined by Eq. (5). In this sense, the recombination parameter

*α*(

*τ*,

*τ*

_{pro}[

*τ*]) = 1, and the recombination rate can be rewritten as

*R*

_{rec}(

*τ*) = ∑

_{τpro}

*R*

_{pro}(

*τ*

_{pro}[

*τ*]).

*N*

_{e}(

*τ*)

*s*(

*τ*)〉 as where

*x*(

*τ*,

*τ*′) is the electronic displacement relative to its parent ion at time

*τ*, and

*τ*′ is the releasing time of the electron. In order to involve the recombination effects, we introduce the Heaviside function

*θ*(

*τ*) defined as

*θ*(

*τ*) = 1 for

*τ*≥ 0 and

*θ*(

*τ*) = 0 for

*τ*< 0. The derivative of the heaviside function is and the integral of an arbitrary given

*τ*-dependent function

*F*(

*τ*) over (−∞,

*τ*] is where the right-hand side sums over the releasing time

*τ*

_{pro}of electrons that return to the ions at time

*τ*. Moreover, the density of free electron can be re-expressed by

*θ*(

*τ*)-function

*J*(

_{P}*τ*) =

*∂P*(

*τ*)/

*∂τ*, that is where the releasing position

*x*(

*τ*,

*τ*) =

*I*

_{p}/(

*eE*(

*τ*)) and

*θ*(

*τ*−

*τ*

_{rec}[

*τ*]) = 1 are used. Notably, the second term on the right-hand side actually vanishes due to the zero return position

*x*(

*τ*,

*τ*

_{pro}[

*τ*]) = 0. However, we should maintain this term for the second derivative of polarization because the return velocity of the electron

*ẋ*(

*τ*,

*τ*

_{pro}[

*τ*])≠ 0, where the single dot refers to the first derivative respect to time

*τ*. The last term means the average current generated by motion of the

*N*

_{e}(

*τ*) electrons, i.e., 〈

*N*

_{e}(

*τ*)

*v*(

*τ*,

*τ*′)〉. Using the electronic motion equation

*m*

_{e}

*ẍ*(

*τ*,

*τ*′) =

*eE*(

*τ*) with

*τ*<

*τ*

_{rec}[

*τ*′], the second derivative of polarization

*P*(

*τ*) is where the return velocity

*v*(

*τ*,

*τ*

_{pro}[

*τ*]) of the electron at time

*τ*is calculated by Eq. (4). Substituting Eq. (7) into Eq. (1) and integrating the two sides over time (−∞,

*τ*], one can obtain the wave equation of the intense laser pulse propagating through gaseous field-ionizing medium,

## 3. Discussions

*I*

_{p}in the form of high-order harmonics if they can return to their own parent ions. Contrast with the total energy conversion, the rate of energy transfer can better reveal the details of the temporal conversion process, which directly affects the laser profile.

*The ionization loss*

*I*

_{p}to excite its bound electrons to the continuum states. Therefore, the analysis above on the laser-atom interaction shows that there are

*N*

_{pro}electrons released throughout the laser pulse, in which the laser field loses

*I*

_{p}per ionized electron. Then the energy loss rate

*T*

_{pro}is proportional to the ionization rate

*R*

_{pro}, where

*R*

_{nonRecom}(

*τ*) is the ionization rate without recombination involved, and

*T*

_{ioni}(

*τ*) is the corresponding energy transfer rate from laser field to the ionized atoms. The comparison up to a factor

*I*

_{p}is depicted in Fig. 4. During the front edge of laser pulse, the two curves almost coincide, indicating that there is almost no electron-ion recombination. While the electron-ion recombination results in the increase of the neutral density during the right side of pulse peak, and then more laser energies are needed to excite the atoms, which is just the multiple ionization of helium atoms. This kind of laser loss process can be embodied by the third term on the right-hand side in Eq. (8).

*The Joule heating process of ionized electrons*

*τ*′ is moving in the laser field, its velocity is denoted by

*v*(

*τ*,

*τ*′). Then the electric currents in the laser pulse can be evaluated by where the introduction of

*θ*-function accounts for decrease of the ionized electrons due to the effects of the electron-ion recombination. Applying the Poynting theorem

*T*(

*τ*) =

*J*(

*τ*) ·

*E*(

*τ*), we have the absorption energy rate of ionized electrons from the laser field where we have used

*E*(

*τ*) = −

*∂A*(

*τ*)/

*∂τ*,

*v*(

*τ*,

*τ*′) = (

*e/m*

_{e})[

*A*(

*τ*′)−

*A*(

*τ*)], and the expression of

*N*

_{e}(

*τ*) in terms of

*θ*-function. While the absorption rate of the laser energy without electron-ion recombination can be calculated as [25

25. M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. **83**(15), 2930–2933 (1999) [CrossRef] .

*N*

_{nonRecom}(

*τ*) is the time-dependent density of ionized electrons without recombination involved. As is shown in Fig. 5, the introduction of the electron-ion recombination results in the decline of the ionized electrons, so do the laser energies which are needed to accelerate the electrons. Moreover, the negative value of the energy transfer rate means that the electrons give their kinetics back to the laser field.

*T*

_{drift}(

*τ*) can be built from the kinetic energy obtained by an ionized electron from the laser field,

*ρ*(

*τ*,

*τ*′) =

*m*

_{e}[

*v*(

*τ*,

*τ*′)]

^{2}/2, so the rate of the total kinetic energy changes at time

*τ*can be calculated as

*τ*′ is given by

*v*

_{d}=

*v*(+∞,

*τ*) =

*eA*(

*τ*)/

*m*

_{e}at the end of the laser pulse. Therefore, the first term in the laser line in Eq. (10) refers to the total kinetic energy gained by the

*N*

_{e}(+∞) ionized electrons, while the number of free electrons

*N*

_{e}(

*τ*) will be decreased due to the electron-ion recombination.

*The electron-ion recombination process*

*I*

_{p}will be also released, which is a kind of atomic processes. During this process, the change of energy conversion can be calculated by where

*v*(

*τ*,

*τ*

_{pro}[

*τ*]) is the return velocity of an electron born at

*τ*

_{pro}[

*τ*], and the term enclosed in the curly brace means the energy of an high-energy photon. Ostensibly, the ionization potential

*I*

_{p}comes from the atomic process and is independent of the laser field, but actually the atomic ionization is just caused by the laser field, and the energy conversion of

*I*

_{p}equivalent is carried by an ionized electron. So the emitted energy of electromagnetic normalized to incident single-photon energy

*h*̄

*ω*at

*τ*is

*W*

_{EM}(

*τ*) = {·}/(

*h*̄

*ω*), where the {·} means the portion in the curly brace in Eq. 12,

*h*̄ is the reduced Planck constant and

*ω*the angular frequency of applied laser pulse. For the incident laser pulse at the entrance, the time-dependent radiant energy

*W*

_{EM}(

*τ*) is shown in Fig. 6.

## 4. Conclusions

*τ*of the ionized electrons the recombination rate is directly relating to the production rate at ionization time

*τ*

_{pro}, and the electron’s releasing and return times,

*τ*=

*τ*

_{rec}[

*τ*

_{pro}], are determined by the saddle-point-like equation with nonzero ionization position. Furthermore, this field-bounded electron-ion recombination retards the ionization saturation of medium and some atoms have to experience the multiple ionizations, which results in more energy losing of the incident laser in the atomic ionization process.

## Acknowledgments

## References and links

1. | A. Rundquist, C. G. Durfee, Z. H. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft X-rays,” Science |

2. | M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. |

3. | S. C. Rae, K. Burnett, and J. Cooper, “Generation and propagation of high-order harmonics in a rapidly ionizing medium,” Phys. Rev. A |

4. | E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, “Coherent soft X-ray generation in the water window with quasi-phase matching,” Science |

5. | S. C. Rae and K. Burnett, “Detailed simulations of plasma-induced spectral blueshifting,” Phys. Rev. A |

6. | M. B. Gaarde, M. Murakami, and R. Kienberger, “Spatial separation of large dynamical blueshift and harmonic generation,” Phys. Rev. A |

7. | F. Brunel, “Not-so-resonant, resonant absorption,” Phys. Rev. Lett. |

8. | I. P. Christov, “Enhanced generation of attosecond pulses in dispersion-controlled hollow-core fiber,” Phys. Rev. A |

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

10. | M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A |

11. | J. D. Jackson, |

12. | M. B. Gaarde, Ph. Antoine, A. L’Huillier, K. J. Schafer, and K. C. Kulander, “Macroscopic studies of short-pulse high-order harmonic generation using the time-dependent Schrödinger equation,” Phys. Rev. A |

13. | M. B. Gaarde, C. Buth, J. L. Tate, and K. J. Schafer, “Transient absorption and reshaping of ultrafast XUV light by laser-dressed helium,” Phys. Rev. A |

14. | M. B. Gaarde, J. L. Tate, and K. J. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B: At. Mol. Opt. Phys. |

15. | A. Etches, M. B. Gaarde, and L. B. Madsen, “Laser-induced bound-state phases in high-order-harmonic generation,” Phys. Rev. A |

16. | M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP |

17. | L. D. Landau and E. M. Lifshitz, |

18. | I. P. Christov, “Propagation of ultrashort pulses in gaseous medium: breakdown of the quasistatic approximation,” Opt. Express. |

19. | V. P. Krainov, “Ionization rates and energy and angular distributions at the barrier-suppression ionization of complex atoms and atomic ions,” J. Opt. Soc. Am. B |

20. | D. Bauer and P. Mulser, “Exact field ionization rates in the barrier-suppression regime from numerical time-dependent Schrödinger-equation calculations,” Phys. Rev. A |

21. | X. M. Tong and C. D. Lin, “Empirical formula for static field ionization rates of atoms and molecules by lasers in the barrier-suppression regime,” J. Phys. B: At. Mol. Opt. Phys. |

22. | N. B. Delone and V. P. Krainov, “Tunneling and barrier-suppression ionization of atoms and ions in a laser radiation field,” Phys. Usp. |

23. | J. Eichler and T. Stohlker, “Radiative electron capture in relativistic ion-atom collisions and the photoelectric effect in hydrogen-like high-Z systems,” Phys. Rep. |

24. | A. Scrinzi, M. Geissler, and T. Brabec, “Ionization above the Coulomb barrier,” Phys. Rev. Lett. |

25. | M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. |

**OCIS Codes**

(320.0320) Ultrafast optics : Ultrafast optics

(020.2649) Atomic and molecular physics : Strong field laser physics

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: December 13, 2012

Revised Manuscript: February 15, 2013

Manuscript Accepted: February 21, 2013

Published: February 26, 2013

**Citation**

Cheng-Xin Yu, Shi-Bing Liu, Xiao-Fang Shu, Hai-Ying Song, and Zhi Yang, "A modified rate equation for the propagation of a femtosecond laser pulse in field-ionizing medium," Opt. Express **21**, 5413-5423 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5413

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### References

- A. Rundquist, C. G. Durfee, Z. H. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft X-rays,” Science280(5368), 1412–1415 (1998). [CrossRef] [PubMed]
- M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys.60, 389–486 (1997). [CrossRef]
- S. C. Rae, K. Burnett, and J. Cooper, “Generation and propagation of high-order harmonics in a rapidly ionizing medium,” Phys. Rev. A50(4), 3438–3446 (1994). [CrossRef] [PubMed]
- E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, “Coherent soft X-ray generation in the water window with quasi-phase matching,” Science302(5642), 95–98 (2003). [CrossRef] [PubMed]
- S. C. Rae and K. Burnett, “Detailed simulations of plasma-induced spectral blueshifting,” Phys. Rev. A46(2), 1084–1090 (1992). [CrossRef] [PubMed]
- M. B. Gaarde, M. Murakami, and R. Kienberger, “Spatial separation of large dynamical blueshift and harmonic generation,” Phys. Rev. A74(5), 053401 (2006). [CrossRef]
- F. Brunel, “Not-so-resonant, resonant absorption,” Phys. Rev. Lett.59 (1), 52–55 (1987). [CrossRef] [PubMed]
- I. P. Christov, “Enhanced generation of attosecond pulses in dispersion-controlled hollow-core fiber,” Phys. Rev. A60(4), 3244–3250 (1999). [CrossRef]
- P. B. Corkum, “Plasma perspective on strong-field multiphoton ionization,” Phys. Rev. Lett.71(13), 1994–1997 (1993). [CrossRef] [PubMed]
- M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A49(3), 2117–2132 (1994). [CrossRef] [PubMed]
- J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).
- M. B. Gaarde, Ph. Antoine, A. L’Huillier, K. J. Schafer, and K. C. Kulander, “Macroscopic studies of short-pulse high-order harmonic generation using the time-dependent Schrödinger equation,” Phys. Rev. A57(6), 4553–4560 (1998). [CrossRef]
- M. B. Gaarde, C. Buth, J. L. Tate, and K. J. Schafer, “Transient absorption and reshaping of ultrafast XUV light by laser-dressed helium,” Phys. Rev. A83 (1), 013419 (2011). [CrossRef]
- M. B. Gaarde, J. L. Tate, and K. J. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B: At. Mol. Opt. Phys.41, 132001 (2008). [CrossRef]
- A. Etches, M. B. Gaarde, and L. B. Madsen, “Laser-induced bound-state phases in high-order-harmonic generation,” Phys. Rev. A86(2), 023818 (2012). [CrossRef]
- M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP64, 1191–1194 (1986).
- L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory (Pergamon, 1977).
- I. P. Christov, “Propagation of ultrashort pulses in gaseous medium: breakdown of the quasistatic approximation,” Opt. Express.6(2), 34–39 (1999). [CrossRef]
- V. P. Krainov, “Ionization rates and energy and angular distributions at the barrier-suppression ionization of complex atoms and atomic ions,” J. Opt. Soc. Am. B14(2), 425–431 (1997). [CrossRef]
- D. Bauer and P. Mulser, “Exact field ionization rates in the barrier-suppression regime from numerical time-dependent Schrödinger-equation calculations,” Phys. Rev. A59(1), 569–577 (1999). [CrossRef]
- X. M. Tong and C. D. Lin, “Empirical formula for static field ionization rates of atoms and molecules by lasers in the barrier-suppression regime,” J. Phys. B: At. Mol. Opt. Phys.38, 2593–2600 (2005). [CrossRef]
- N. B. Delone and V. P. Krainov, “Tunneling and barrier-suppression ionization of atoms and ions in a laser radiation field,” Phys. Usp.41, 469–485 (1998). [CrossRef]
- J. Eichler and T. Stohlker, “Radiative electron capture in relativistic ion-atom collisions and the photoelectric effect in hydrogen-like high-Z systems,” Phys. Rep.439, 1–99 (2007). [CrossRef]
- A. Scrinzi, M. Geissler, and T. Brabec, “Ionization above the Coulomb barrier,” Phys. Rev. Lett.83(4), 706–709 (1999). [CrossRef]
- M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett.83(15), 2930–2933 (1999). [CrossRef]

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