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.

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  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,” Science280(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]
  3. 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]
  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,” Science302(5642), 95–98 (2003).
    [CrossRef] [PubMed]
  5. S. C. Rae and K. Burnett, “Detailed simulations of plasma-induced spectral blueshifting,” Phys. Rev. A46(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. A74(5), 053401 (2006).
    [CrossRef]
  7. F. Brunel, “Not-so-resonant, resonant absorption,” Phys. Rev. Lett.59 (1), 52–55 (1987).
    [CrossRef] [PubMed]
  8. I. P. Christov, “Enhanced generation of attosecond pulses in dispersion-controlled hollow-core fiber,” Phys. Rev. A60(4), 3244–3250 (1999).
    [CrossRef]
  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. A49(3), 2117–2132 (1994).
    [CrossRef] [PubMed]
  11. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).
  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. A57(6), 4553–4560 (1998).
    [CrossRef]
  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. A83 (1), 013419 (2011).
    [CrossRef]
  14. 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]
  15. 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]
  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. JETP64, 1191–1194 (1986).
  17. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory (Pergamon, 1977).
  18. I. P. Christov, “Propagation of ultrashort pulses in gaseous medium: breakdown of the quasistatic approximation,” Opt. Express.6(2), 34–39 (1999).
    [CrossRef]
  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. B14(2), 425–431 (1997).
    [CrossRef]
  20. 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]
  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.38, 2593–2600 (2005).
    [CrossRef]
  22. 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]
  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]
  24. A. Scrinzi, M. Geissler, and T. Brabec, “Ionization above the Coulomb barrier,” Phys. Rev. Lett.83(4), 706–709 (1999).
    [CrossRef]
  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]

2012

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]

2011

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]

2008

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]

2007

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]

2006

M. B. Gaarde, M. Murakami, and R. Kienberger, “Spatial separation of large dynamical blueshift and harmonic generation,” Phys. Rev. A74(5), 053401 (2006).
[CrossRef]

2005

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]

2003

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]

1999

I. P. Christov, “Enhanced generation of attosecond pulses in dispersion-controlled hollow-core fiber,” Phys. Rev. A60(4), 3244–3250 (1999).
[CrossRef]

I. P. Christov, “Propagation of ultrashort pulses in gaseous medium: breakdown of the quasistatic approximation,” Opt. Express.6(2), 34–39 (1999).
[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]

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]

1998

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]

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]

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]

1997

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

1994

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]

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]

1993

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

1992

S. C. Rae and K. Burnett, “Detailed simulations of plasma-induced spectral blueshifting,” Phys. Rev. A46(2), 1084–1090 (1992).
[CrossRef] [PubMed]

1987

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

1986

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).

Ammosov, M. V.

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).

Antoine, Ph.

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]

Aquila, A.

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]

Attwood, D. T.

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]

Backus, S.

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]

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]

Balcou, Ph.

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]

Bauer, D.

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]

Brabec, T.

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]

Brunel, F.

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

Burnett, K.

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]

S. C. Rae and K. Burnett, “Detailed simulations of plasma-induced spectral blueshifting,” Phys. Rev. A46(2), 1084–1090 (1992).
[CrossRef] [PubMed]

Buth, C.

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]

Chang, Z. H.

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]

Christov, I. P.

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]

I. P. Christov, “Enhanced generation of attosecond pulses in dispersion-controlled hollow-core fiber,” Phys. Rev. A60(4), 3244–3250 (1999).
[CrossRef]

I. P. Christov, “Propagation of ultrashort pulses in gaseous medium: breakdown of the quasistatic approximation,” Opt. Express.6(2), 34–39 (1999).
[CrossRef]

Cooper, J.

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]

Corkum, P. B.

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]

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

Delone, N. B.

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]

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).

Durfee, C. G.

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]

Eichler, J.

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]

Etches, A.

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]

Gaarde, M. B.

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. 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]

M. B. Gaarde, M. Murakami, and R. Kienberger, “Spatial separation of large dynamical blueshift and harmonic generation,” Phys. Rev. A74(5), 053401 (2006).
[CrossRef]

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]

Gaudiosi, D.

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]

Geissler, M.

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]

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

Gibson, E. A.

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]

Gullikson, E. M.

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]

Herne, C.

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]

Ivanov, M. Yu.

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]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).

Kapteyn, H. C.

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]

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]

Keitel, C. H.

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

Kienberger, R.

M. B. Gaarde, M. Murakami, and R. Kienberger, “Spatial separation of large dynamical blueshift and harmonic generation,” Phys. Rev. A74(5), 053401 (2006).
[CrossRef]

Knight, P. L.

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

Krainov, V. P.

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]

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]

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).

Krausz, F.

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]

Kulander, K. C.

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]

L’Huillier, A.

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. 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]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory (Pergamon, 1977).

Lewenstein, M.

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]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory (Pergamon, 1977).

Lin, C. D.

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]

Madsen, L. B.

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]

Mulser, P.

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]

Murakami, M.

M. B. Gaarde, M. Murakami, and R. Kienberger, “Spatial separation of large dynamical blueshift and harmonic generation,” Phys. Rev. A74(5), 053401 (2006).
[CrossRef]

Murnane, M. M.

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]

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]

Paul, A.

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]

Protopapas, M.

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

Rae, S. C.

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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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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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).
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[CrossRef]

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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]

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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]

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[CrossRef]

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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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[CrossRef] [PubMed]

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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]

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[CrossRef]

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[CrossRef]

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[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]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[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).
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[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).
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Figures (6)

Fig. 1
Fig. 1

Return time v.s. releasing time of the freed electrons. The subsequent excursions (red pentagrams) of an ionized electron is truncated due to the recombination coefficient α(t, tb) = 1 at the first return time.

Fig. 2
Fig. 2

Density of ionized electrons. The red dashed line represents the recombination-corrected density, and as a comparison, the non-corrected one is shown by the magenta connected asterisks.

Fig. 3
Fig. 3

The contours of the laser pulse at (a) the entrance and (b) the exit. The laser pulse is selected with a Gaussian profile, E ( r , τ ) = E 0 exp ( r 2 / r f 2 ) exp ( τ 2 / τ f 2 ) cos ( ω τ ), where the angular frequency ω = 2πc/λ with the wavelength λ = 800 nm, the laser intensity is I0 = 3.0×1015W/cm2, and the full width at half maximum (FWHM) τf = 5 fs, rf = 40μm.

Fig. 4
Fig. 4

Comparison of the recombination-corrected production rate with the non-corrected ionization rate of electrons in helium gas. Clearly, the electron production rates depend strongly on the electric field strength. The result of the corrected model almost coincides with that of the non-corrected model during the front edge of laser pulse, while during the back edge of laser pulse the result of the corrected model is significantly larger than that of the non-corrected one.

Fig. 5
Fig. 5

Comparison of the energy transfer rate to accelerate the ionized electrons between with/without the electron-ion recombination incorporated. Obviously, the electron-ion recombination causes the decrease of the ionized electrons, thus reduces the energy transfer rate.

Fig. 6
Fig. 6

The emission of the high-energy photon. The maximal single-photon energy is about 270 times that of the fundamental laser, which appears at the right side of the main peak of the pulse. Furthermore, there is a periodic harmonic emission with the period of a half cycle of the driving laser field.

Equations (22)

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2 c 2 E τ ξ + 2 E = n 2 ( p ) 1 c 2 2 E τ 2 + μ 0 2 P ( τ ) τ 2
d N e ( τ ) d τ = R ( τ ) = R pro ( τ ) R rec ( τ )
R pro ( τ ) = w ( τ ) N a ( τ ) , R rec ( τ ) = τ pro α ( τ , τ pro ) R pro ( τ pro )
x ( τ , τ pro ) = x 0 + τ pro τ v ( τ , τ pro ) d τ
v ( τ , τ pro ) = e m e τ pro τ E ( τ ) d τ = e m e [ A ( τ pro ) A ( τ ) ]
x ( τ , τ pro ) = x 0 + e m e [ A ( τ pro ) ( τ τ pro ) τ pro τ A ( τ ) d τ ]
p st ( τ rec , τ pro ) e A ( τ pro ) = m x 0 τ rec τ pro
N e ( τ ) = N 0 { 1 exp ( τ w ( τ ) d τ ) } N 1 ( τ ) exp ( τ w ( τ ) d τ ) τ d τ R rec ( τ ) exp ( τ w ( τ ) d τ ) N 2 ( τ ) N rec ( τ )
N pro ( τ ) = τ R pro ( τ ) d τ , N rec ( τ ) = τ R rec ( τ ) d τ
N e ( τ ) s ( τ ) = τ [ 1 θ ( τ τ rec [ τ ] ) ] R pro ( τ ) x ( τ , τ ) d τ
τ θ ( τ τ rec [ τ ] ) = δ ( τ τ rec [ τ ] )
τ δ ( τ τ rec [ τ ] ) F ( τ ) d τ = τ pro F ( τ pro [ τ ] )
N e ( τ ) = τ [ 1 θ ( τ τ rec [ τ ] ] R pro ( τ ) d τ
P ( τ ) τ = I p R pro ( τ ) E ( τ ) e τ pro R pro ( τ pro [ τ ] ) x ( τ , τ pro [ τ ] ] + e τ [ 1 θ ( τ τ rec [ τ ] ] R pro ( τ ) x ( τ , τ ) τ d τ
2 P ( τ ) τ 2 = I p τ [ R pro ( τ ) E ( τ ) ] + e 2 m e N e ( τ ) E ( τ ) 2 e τ pro R pro ( τ pro [ τ ] ) v ( τ , τ pro [ τ ] )
E ( τ ) d ξ = c 2 τ 2 E ( τ ) d τ 1 2 c τ [ n 2 ( p ) 1 ] 2 E ( τ ) τ 2 d τ c μ 0 I p 2 [ R pro ( τ ) E ( τ ) ] c μ 0 e 2 2 m e τ N e ( τ ) E ( τ ) d τ + c μ 0 e 2 m e τ τ pro [ R pro ( τ pro [ τ ] ) τ pro [ τ ] τ E ( τ ) d τ ] d τ
T pro ( τ ) = I p R pro ( τ ) , T ioni ( τ ) = I p R nonRecom ( τ )
J drift = e τ ( 1 θ ( τ τ rec [ τ ] ) ) R pro ( τ ) ν ( τ , τ ) d τ
T drift ( τ ) = e 2 2 m e N e ( τ ) [ A ( τ ) ] 2 τ e 2 m e A ( τ ) τ τ ( 1 θ ( τ τ rec [ τ ] ) ) R pro ( τ ) A ( τ ) d τ
T drift ( τ ) = e 2 2 m e [ A ( τ ) ] 2 τ N nonRecom ( τ )
T drift ( τ ) = τ ( 1 θ ( τ τ rec [ τ ] ) ) ρ ˙ ( τ , τ ) R pro ( τ ) d τ
T recom ( τ ) = τ pro R pro ( τ pro [ τ ] ) { 1 2 m e [ v ( τ , τ pro [ τ ] ) ] 2 + I p }

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