Abstract

We present a S-matrix description of the process of high order harmonic generation during the interaction of atoms with strong electromagnetic fields. In contrast with the state-of-the-art approaches, our model does not employ the stationary phase approximation and accounts as well for the continuum-continuum transitions. Therefore we are able to reproduce quantitatively the higher frequency part of the spectrum for arbitrary pulse shapes, and for intensities corresponding to multiphoton, tunnel and soft over-the barrier ionization regimes. In addition this model can be implemented very efficiently in a Personal Computer to calculate the harmonic generation for the atom interacting with an eight-cycle pulse at λ =800 nm in, roughly, ten minutes (a reduction of two orders of magnitude from the typical time requirements of the exact integration).

© 2007 Optical Society of America

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  1. Th. Brabec, F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545 (2000)
    [CrossRef]
  2. P. B. Corkum, "Plasma Perspective on Strong Field Multiphoton Ionization," Phys. Rev. Lett. 71, 1994 (1993)
    [CrossRef] [PubMed]
  3. M. Lewenstein,  et al, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117 (1994)
    [CrossRef] [PubMed]
  4. M. Yu. Ivanov,  et al, "Coulomb corrections and polarization effects in high-intensity high-harmonic emission," Phys. Rev. A 54, 742 (1996)
    [CrossRef] [PubMed]
  5. A. Gordon, F. Kärtner, "Quantitative Modelling of Single Atom High Harmonic Generation," Phys. Rev. Lett. 95, 223901 (2005)
    [CrossRef] [PubMed]
  6. N. Milosevic,  et al, "Numerical Characterization of High Harmonic Attosecond Pulses," Phys. Rev. Lett. 88, 093905 (2002)
    [CrossRef] [PubMed]
  7. W. Becker,  et al, "A unified theory of high-harmonic generation: Application to polarization properties of the harmonics," Phys. Rev. A 56645 (1997)
    [CrossRef]
  8. A. Becker, F. H. M. Faisal, "Intense-field many-body S-matrix theory," J. Phys. B: At. Mol. Opt. Phys. 38R1 (2005)
    [CrossRef]
  9. Other models beyond the stationary phase approximation are restricted to the case of a constant field envelope [8], not including quantitative results.
  10. M. Yu. Kuchiev, V. N. Ostrovsky, "Quantum theory of high harmonic generation as a three-step process," Phys. Rev. A 603111 (1999)
    [CrossRef]
  11. L. V. Keldysh, "Ionization in the field of strong electromagnetic wave," Zh. Eksp. Teor. Fiz. 47, 1945 (1964) [Sov. Phys. JETP 20, 1307 (1965)]
  12. F. H. M. Faisal, Multiple absorption of laser photons by atoms, J. Phys. B 6L89 (1973)
    [CrossRef]
  13. H. R. Reiss, "Effect of an intense electromagnetic field on a weakly bound system," Phys. Rev. A 22, 1786 (1980).
    [CrossRef]
  14. 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,  14425 (1997)
    [CrossRef]
  15. A. Becker et al, "Total ionization rates and ion yields of atoms at nonperturbative laser intensities," Phys. Rev. A 64, 023408 (2001)
    [CrossRef]
  16. P. Moreno,  et al, "Influence of barrier suppression in high-order harmonic generation," Phys. Rev. A 51, 4746 (1995)
    [CrossRef] [PubMed]
  17. P. Moreno,  et al, "High-order harmonic generation in a partially ionized medium," J. Opt. Soc. Am. B 13430 (1996)
    [CrossRef]

2005 (2)

A. Gordon, F. Kärtner, "Quantitative Modelling of Single Atom High Harmonic Generation," Phys. Rev. Lett. 95, 223901 (2005)
[CrossRef] [PubMed]

A. Becker, F. H. M. Faisal, "Intense-field many-body S-matrix theory," J. Phys. B: At. Mol. Opt. Phys. 38R1 (2005)
[CrossRef]

2002 (1)

N. Milosevic,  et al, "Numerical Characterization of High Harmonic Attosecond Pulses," Phys. Rev. Lett. 88, 093905 (2002)
[CrossRef] [PubMed]

2001 (1)

A. Becker et al, "Total ionization rates and ion yields of atoms at nonperturbative laser intensities," Phys. Rev. A 64, 023408 (2001)
[CrossRef]

2000 (1)

Th. Brabec, F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545 (2000)
[CrossRef]

1999 (1)

M. Yu. Kuchiev, V. N. Ostrovsky, "Quantum theory of high harmonic generation as a three-step process," Phys. Rev. A 603111 (1999)
[CrossRef]

1997 (2)

W. Becker,  et al, "A unified theory of high-harmonic generation: Application to polarization properties of the harmonics," Phys. Rev. A 56645 (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. B,  14425 (1997)
[CrossRef]

1996 (2)

P. Moreno,  et al, "High-order harmonic generation in a partially ionized medium," J. Opt. Soc. Am. B 13430 (1996)
[CrossRef]

M. Yu. Ivanov,  et al, "Coulomb corrections and polarization effects in high-intensity high-harmonic emission," Phys. Rev. A 54, 742 (1996)
[CrossRef] [PubMed]

1995 (1)

P. Moreno,  et al, "Influence of barrier suppression in high-order harmonic generation," Phys. Rev. A 51, 4746 (1995)
[CrossRef] [PubMed]

1994 (1)

M. Lewenstein,  et al, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117 (1994)
[CrossRef] [PubMed]

1993 (1)

P. B. Corkum, "Plasma Perspective on Strong Field Multiphoton Ionization," Phys. Rev. Lett. 71, 1994 (1993)
[CrossRef] [PubMed]

1980 (1)

H. R. Reiss, "Effect of an intense electromagnetic field on a weakly bound system," Phys. Rev. A 22, 1786 (1980).
[CrossRef]

1973 (1)

F. H. M. Faisal, Multiple absorption of laser photons by atoms, J. Phys. B 6L89 (1973)
[CrossRef]

1964 (1)

L. V. Keldysh, "Ionization in the field of strong electromagnetic wave," Zh. Eksp. Teor. Fiz. 47, 1945 (1964) [Sov. Phys. JETP 20, 1307 (1965)]

Becker, A.

A. Becker, F. H. M. Faisal, "Intense-field many-body S-matrix theory," J. Phys. B: At. Mol. Opt. Phys. 38R1 (2005)
[CrossRef]

A. Becker et al, "Total ionization rates and ion yields of atoms at nonperturbative laser intensities," Phys. Rev. A 64, 023408 (2001)
[CrossRef]

Becker, W.

W. Becker,  et al, "A unified theory of high-harmonic generation: Application to polarization properties of the harmonics," Phys. Rev. A 56645 (1997)
[CrossRef]

Brabec, Th.

Th. Brabec, F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545 (2000)
[CrossRef]

Corkum, P. B.

P. B. Corkum, "Plasma Perspective on Strong Field Multiphoton Ionization," Phys. Rev. Lett. 71, 1994 (1993)
[CrossRef] [PubMed]

Faisal, F. H. M.

A. Becker, F. H. M. Faisal, "Intense-field many-body S-matrix theory," J. Phys. B: At. Mol. Opt. Phys. 38R1 (2005)
[CrossRef]

F. H. M. Faisal, Multiple absorption of laser photons by atoms, J. Phys. B 6L89 (1973)
[CrossRef]

Gordon, A.

A. Gordon, F. Kärtner, "Quantitative Modelling of Single Atom High Harmonic Generation," Phys. Rev. Lett. 95, 223901 (2005)
[CrossRef] [PubMed]

Ivanov, M. Yu.

M. Yu. Ivanov,  et al, "Coulomb corrections and polarization effects in high-intensity high-harmonic emission," Phys. Rev. A 54, 742 (1996)
[CrossRef] [PubMed]

Kärtner, F.

A. Gordon, F. Kärtner, "Quantitative Modelling of Single Atom High Harmonic Generation," Phys. Rev. Lett. 95, 223901 (2005)
[CrossRef] [PubMed]

Keldysh, L. V.

L. V. Keldysh, "Ionization in the field of strong electromagnetic wave," Zh. Eksp. Teor. Fiz. 47, 1945 (1964) [Sov. Phys. JETP 20, 1307 (1965)]

Krainov, V. P.

Krausz, F.

Th. Brabec, F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545 (2000)
[CrossRef]

Kuchiev, M. Yu.

M. Yu. Kuchiev, V. N. Ostrovsky, "Quantum theory of high harmonic generation as a three-step process," Phys. Rev. A 603111 (1999)
[CrossRef]

Lewenstein, M.

M. Lewenstein,  et al, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117 (1994)
[CrossRef] [PubMed]

Milosevic, N.

N. Milosevic,  et al, "Numerical Characterization of High Harmonic Attosecond Pulses," Phys. Rev. Lett. 88, 093905 (2002)
[CrossRef] [PubMed]

Moreno, P.

P. Moreno,  et al, "High-order harmonic generation in a partially ionized medium," J. Opt. Soc. Am. B 13430 (1996)
[CrossRef]

P. Moreno,  et al, "Influence of barrier suppression in high-order harmonic generation," Phys. Rev. A 51, 4746 (1995)
[CrossRef] [PubMed]

Ostrovsky, V. N.

M. Yu. Kuchiev, V. N. Ostrovsky, "Quantum theory of high harmonic generation as a three-step process," Phys. Rev. A 603111 (1999)
[CrossRef]

Reiss, H. R.

H. R. Reiss, "Effect of an intense electromagnetic field on a weakly bound system," Phys. Rev. A 22, 1786 (1980).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. B (1)

F. H. M. Faisal, Multiple absorption of laser photons by atoms, J. Phys. B 6L89 (1973)
[CrossRef]

J. Phys. B: At. Mol. Opt. Phys. (1)

A. Becker, F. H. M. Faisal, "Intense-field many-body S-matrix theory," J. Phys. B: At. Mol. Opt. Phys. 38R1 (2005)
[CrossRef]

Phys. Rev. A (7)

M. Lewenstein,  et al, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117 (1994)
[CrossRef] [PubMed]

M. Yu. Ivanov,  et al, "Coulomb corrections and polarization effects in high-intensity high-harmonic emission," Phys. Rev. A 54, 742 (1996)
[CrossRef] [PubMed]

H. R. Reiss, "Effect of an intense electromagnetic field on a weakly bound system," Phys. Rev. A 22, 1786 (1980).
[CrossRef]

A. Becker et al, "Total ionization rates and ion yields of atoms at nonperturbative laser intensities," Phys. Rev. A 64, 023408 (2001)
[CrossRef]

P. Moreno,  et al, "Influence of barrier suppression in high-order harmonic generation," Phys. Rev. A 51, 4746 (1995)
[CrossRef] [PubMed]

M. Yu. Kuchiev, V. N. Ostrovsky, "Quantum theory of high harmonic generation as a three-step process," Phys. Rev. A 603111 (1999)
[CrossRef]

W. Becker,  et al, "A unified theory of high-harmonic generation: Application to polarization properties of the harmonics," Phys. Rev. A 56645 (1997)
[CrossRef]

Phys. Rev. Lett. (3)

A. Gordon, F. Kärtner, "Quantitative Modelling of Single Atom High Harmonic Generation," Phys. Rev. Lett. 95, 223901 (2005)
[CrossRef] [PubMed]

N. Milosevic,  et al, "Numerical Characterization of High Harmonic Attosecond Pulses," Phys. Rev. Lett. 88, 093905 (2002)
[CrossRef] [PubMed]

P. B. Corkum, "Plasma Perspective on Strong Field Multiphoton Ionization," Phys. Rev. Lett. 71, 1994 (1993)
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

Th. Brabec, F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545 (2000)
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

L. V. Keldysh, "Ionization in the field of strong electromagnetic wave," Zh. Eksp. Teor. Fiz. 47, 1945 (1964) [Sov. Phys. JETP 20, 1307 (1965)]

Other (1)

Other models beyond the stationary phase approximation are restricted to the case of a constant field envelope [8], not including quantitative results.

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Figures (3)

Fig. 1.
Fig. 1.

Schemes of the direct (a) and field-assisted (b) recombination processes leading to harmonic generation. The lower picture shows the Δt → 0 considered in the time integral of Eq. (4): for t - t 2 > Δt the recombination is negligible.

Fig. 2.
Fig. 2.

Acceleration spectra computed through the exact numerical integration of the Schrödinger equation (black lines), the theory described in the text (red lines), and the contribution of the direct recombination adr (dashed lines) to the total acceleration. We have considered a Hydrogen atom interacting with a 8-cycle sin2 linearly-polarized pulse at λ = 800nm and intensities (a) 5×1013, (b) 8.75×1013, (c) 3.5×1014 and (d) 5×1014 W/cm2.

Fig. 3.
Fig. 3.

Contributions of the different canonical momenta kz to the total harmonic spectrum, for the cases depicted in (a): Fig. 2(a) , and (b): 2(c). The dashed and dotted lines mark the maximum harmonic order corresponding to the electrons returning to the nucleus, Ip + 3.17Up , and to the non-returning electrons, Ip +8Up , respectively. The harmonic intensities are plotted in logarithmic scale.

Equations (10)

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a ( t ) = a dr ( t ) + a ar ( t ) + c . c . = h ̄ 1 t 0 t dt 1 ψ ( t 0 ) G a t t 0 â G + t t 1 V i ( t 1 ) G a + t 1 t 0 ψ ( t 0 ) + h ̄ 2 t 0 t dt 2 t 0 t 2 dt 1 ψ ( t 0 ) G a t 2 t 0 V i ( t 2 ) G t t 2 â G + t t 1 V i ( t 1 ) G a + t 1 t 0 ψ ( t 0 ) + c . c .
iG + t t ´ ϕ k ( t ´ ) e i h ̄ t ´ t ε k ( τ ) ϕ k ( t ´ )
a k t i h ̄ t 0 t dt 1 e 0 ( t t 1 ) h ̄ e i h ̄ t 1 t ε k ( τ ) α 0 , k V i , k ( t 1 ) ϕ ( k )
a ar ( t ) = d k d k ´ t Δt t dt 2 h ̄ e i ε 0 h ̄ t 2 e i h ̄ t 2 t ε k ' ( τ ) ϕ 0 V i ( t 2 ) k ´ k ´ × t 0 t 2 dt 1 h ̄ e i ε 0 h ̄ t 1 e i h ̄ t 1 t ε k ( τ ) â k V i , k ( t 1 ) ϕ ( k )
a ar ( t ) i h ̄ d k d k ´ e i ε 0 h ̄ t 1 ε k ' ( t ) ε 0 ϕ 0 [ ε k ´ ( t ) h ̄ 2 k ' 2 2 m ] k ´ k ´ × t 0 t dt 1 e i ε 0 h ̄ t 1 e i h ̄ t 1 t ε k ( τ ) â k V i , k ( t 1 ) ϕ ( k )
a ar ( t ) d k ( 1 + ε 0 h ̄ 2 k 2 2 m U p ) a k t + c . c .
h ̄ 2 2 2 m ϕ 0 â k = ( H a V c ) ϕ 0 â k = ϕ 0 â ( ε 0 V c ) k
= ϕ 0 â ( ε 0 H + H f ) k ϕ 0 â h ̄ 2 k 2 2 m k
a ( t ) C F d k ε 0 h ̄ 2 k 2 2 m U p a k t + c . c .
d dt a ( k , t ) = i h ̄ [ ε 0 ε k ( t ) ] a ( k , t ) i h ̄ α 0 , k V i , k ( t ) ϕ ( k )

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