Abstract

We present a study of the effect of time-dependent ionization on harmonic generation by bound–bound transitions. The introduction of tunneling and above-barrier ionization rates in a simple bound system, a two-level atom, induces an unexpected richness in the harmonic spectra, leading to the production of radiation at high frequencies. Such frequencies are not reached when no ionization or steady ionization is considered. Time-dependent ionization is, therefore, a key process in the understanding of generation of middle plateau frequency harmonics from atoms in the tunneling ionization regime and shows that bound–free transitions are not uniquely responsible for the higher-harmonic frequencies in realistic atoms. Following previous authors’ studies, we use an adiabatic approach to understand the presence of these higher frequencies.

© 1996 Optical Society of America

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  1. Ph. Balcou, C. Cornaggia, A. S. L. Gomes, L. A. Lompré, and A. L’Huillier, “Optimizing high-order harmonic generation in strong fields,” J. Phys. B 25, 4467 (1992).
    [CrossRef]
  2. L. Plaja and L. Roso, “Contribution of bound-bound transitions to high-order harmonic generation,” in SILAP III Proceedings, B. Pireaux, A. L’Huiller, and K. Rzazewski, eds. (Plenum, New York, 1993), pp. 53–61.
  3. P. B. Corkum, “Plasma perspective on strong-field multiphoton ionization,” Phys. Rev. Lett. 71, 1994 (1993).
    [CrossRef] [PubMed]
  4. 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, 2117 (1994).
    [CrossRef] [PubMed]
  5. J. L. Krause, K. J. Schafer, and K. C. Kulander, “Calculation of photoemission from atoms subject to intense laser fields,” Phys. Rev. A 45, 4998 (1992).
    [CrossRef] [PubMed]
  6. B. Sundaram and P. W. Milonni, “High-order harmonic generation: simplified model and relevance of single-atom theories to experiment,” Phys. Rev. A 41, 6571 (1990).
    [CrossRef] [PubMed]
  7. L. Plaja and L. Roso-Franco, “Adiabatic theory for high-order harmonic generation in a two-level atom,” J. Opt. Soc. Am. B 9, 2210 (1992).
    [CrossRef]
  8. F. Brunel, “Harmonic generation due to plasma effects in a gas undergoing multiphoton ionization in the high-intensity limit,” J. Opt. Soc. Am. B 7, 521 (1990).
    [CrossRef]
  9. V. Malyshev, E. Conejero Jarque, and L. Roso, “Self-reflection of an intense ultrashort laser pulse by tunnel ionization on a solid surface,” J. Opt. Soc. Am. B (to be published).
  10. N. B. Delone and V. P. Krainov, Multiphoton Processes in Atoms (Springer-Verlag, Berlin, 1994).
    [CrossRef]
  11. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two Electron Atoms (Plenum, New York, 1954).
  12. V. P. Krainov and B. Shokri, “Energy and angular distributions of electrons resulting from barrier-suppression ionization of atoms by strong low-frequency radiation,” JETP 80, 657 (1995).
  13. V. P. Krainov, Department of Theoretical Physics, Moscow Institute of Physics and Technology, 141700 Dolgopzudny, Moscow, Russia (personal communication).
  14. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory (Pergamon, Oxford, 1977).
  15. V. P. Krainov and Z. S. Mulyukov, “A plateau in high-order harmonic generation for a two-level atom,” Laser Phys. 4, 544 (1994).
  16. L. Plaja, “Theoretical study on optical high-order harmonic generation in three matter models,” Ph.D. dissertation (Universitat Autònoma de Barcelona, Barcelona, 1993).

1995 (1)

V. P. Krainov and B. Shokri, “Energy and angular distributions of electrons resulting from barrier-suppression ionization of atoms by strong low-frequency radiation,” JETP 80, 657 (1995).

1994 (2)

V. P. Krainov and Z. S. Mulyukov, “A plateau in high-order harmonic generation for a two-level atom,” Laser Phys. 4, 544 (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. 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]

1992 (3)

Ph. Balcou, C. Cornaggia, A. S. L. Gomes, L. A. Lompré, and A. L’Huillier, “Optimizing high-order harmonic generation in strong fields,” J. Phys. B 25, 4467 (1992).
[CrossRef]

J. L. Krause, K. J. Schafer, and K. C. Kulander, “Calculation of photoemission from atoms subject to intense laser fields,” Phys. Rev. A 45, 4998 (1992).
[CrossRef] [PubMed]

L. Plaja and L. Roso-Franco, “Adiabatic theory for high-order harmonic generation in a two-level atom,” J. Opt. Soc. Am. B 9, 2210 (1992).
[CrossRef]

1990 (2)

F. Brunel, “Harmonic generation due to plasma effects in a gas undergoing multiphoton ionization in the high-intensity limit,” J. Opt. Soc. Am. B 7, 521 (1990).
[CrossRef]

B. Sundaram and P. W. Milonni, “High-order harmonic generation: simplified model and relevance of single-atom theories to experiment,” Phys. Rev. A 41, 6571 (1990).
[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. A 49, 2117 (1994).
[CrossRef] [PubMed]

Ph. Balcou, C. Cornaggia, A. S. L. Gomes, L. A. Lompré, and A. L’Huillier, “Optimizing high-order harmonic generation in strong fields,” J. Phys. B 25, 4467 (1992).
[CrossRef]

Bethe, H. A.

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two Electron Atoms (Plenum, New York, 1954).

Brunel, F.

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. A 49, 2117 (1994).
[CrossRef] [PubMed]

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

Cornaggia, C.

Ph. Balcou, C. Cornaggia, A. S. L. Gomes, L. A. Lompré, and A. L’Huillier, “Optimizing high-order harmonic generation in strong fields,” J. Phys. B 25, 4467 (1992).
[CrossRef]

Delone, N. B.

N. B. Delone and V. P. Krainov, Multiphoton Processes in Atoms (Springer-Verlag, Berlin, 1994).
[CrossRef]

Gomes, A. S. L.

Ph. Balcou, C. Cornaggia, A. S. L. Gomes, L. A. Lompré, and A. L’Huillier, “Optimizing high-order harmonic generation in strong fields,” J. Phys. B 25, 4467 (1992).
[CrossRef]

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. A 49, 2117 (1994).
[CrossRef] [PubMed]

Jarque, E. Conejero

V. Malyshev, E. Conejero Jarque, and L. Roso, “Self-reflection of an intense ultrashort laser pulse by tunnel ionization on a solid surface,” J. Opt. Soc. Am. B (to be published).

Krainov, V. P.

V. P. Krainov and B. Shokri, “Energy and angular distributions of electrons resulting from barrier-suppression ionization of atoms by strong low-frequency radiation,” JETP 80, 657 (1995).

V. P. Krainov and Z. S. Mulyukov, “A plateau in high-order harmonic generation for a two-level atom,” Laser Phys. 4, 544 (1994).

N. B. Delone and V. P. Krainov, Multiphoton Processes in Atoms (Springer-Verlag, Berlin, 1994).
[CrossRef]

V. P. Krainov, Department of Theoretical Physics, Moscow Institute of Physics and Technology, 141700 Dolgopzudny, Moscow, Russia (personal communication).

Krause, J. L.

J. L. Krause, K. J. Schafer, and K. C. Kulander, “Calculation of photoemission from atoms subject to intense laser fields,” Phys. Rev. A 45, 4998 (1992).
[CrossRef] [PubMed]

Kulander, K. C.

J. L. Krause, K. J. Schafer, and K. C. Kulander, “Calculation of photoemission from atoms subject to intense laser fields,” Phys. Rev. A 45, 4998 (1992).
[CrossRef] [PubMed]

L’Huillier, A.

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, 2117 (1994).
[CrossRef] [PubMed]

Ph. Balcou, C. Cornaggia, A. S. L. Gomes, L. A. Lompré, and A. L’Huillier, “Optimizing high-order harmonic generation in strong fields,” J. Phys. B 25, 4467 (1992).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory (Pergamon, Oxford, 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. A 49, 2117 (1994).
[CrossRef] [PubMed]

Lifshitz, E. M.

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

Lompré, L. A.

Ph. Balcou, C. Cornaggia, A. S. L. Gomes, L. A. Lompré, and A. L’Huillier, “Optimizing high-order harmonic generation in strong fields,” J. Phys. B 25, 4467 (1992).
[CrossRef]

Malyshev, V.

V. Malyshev, E. Conejero Jarque, and L. Roso, “Self-reflection of an intense ultrashort laser pulse by tunnel ionization on a solid surface,” J. Opt. Soc. Am. B (to be published).

Milonni, P. W.

B. Sundaram and P. W. Milonni, “High-order harmonic generation: simplified model and relevance of single-atom theories to experiment,” Phys. Rev. A 41, 6571 (1990).
[CrossRef] [PubMed]

Mulyukov, Z. S.

V. P. Krainov and Z. S. Mulyukov, “A plateau in high-order harmonic generation for a two-level atom,” Laser Phys. 4, 544 (1994).

Plaja, L.

L. Plaja and L. Roso-Franco, “Adiabatic theory for high-order harmonic generation in a two-level atom,” J. Opt. Soc. Am. B 9, 2210 (1992).
[CrossRef]

L. Plaja, “Theoretical study on optical high-order harmonic generation in three matter models,” Ph.D. dissertation (Universitat Autònoma de Barcelona, Barcelona, 1993).

L. Plaja and L. Roso, “Contribution of bound-bound transitions to high-order harmonic generation,” in SILAP III Proceedings, B. Pireaux, A. L’Huiller, and K. Rzazewski, eds. (Plenum, New York, 1993), pp. 53–61.

Roso, L.

L. Plaja and L. Roso, “Contribution of bound-bound transitions to high-order harmonic generation,” in SILAP III Proceedings, B. Pireaux, A. L’Huiller, and K. Rzazewski, eds. (Plenum, New York, 1993), pp. 53–61.

V. Malyshev, E. Conejero Jarque, and L. Roso, “Self-reflection of an intense ultrashort laser pulse by tunnel ionization on a solid surface,” J. Opt. Soc. Am. B (to be published).

Roso-Franco, L.

Salpeter, E. E.

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two Electron Atoms (Plenum, New York, 1954).

Schafer, K. J.

J. L. Krause, K. J. Schafer, and K. C. Kulander, “Calculation of photoemission from atoms subject to intense laser fields,” Phys. Rev. A 45, 4998 (1992).
[CrossRef] [PubMed]

Shokri, B.

V. P. Krainov and B. Shokri, “Energy and angular distributions of electrons resulting from barrier-suppression ionization of atoms by strong low-frequency radiation,” JETP 80, 657 (1995).

Sundaram, B.

B. Sundaram and P. W. Milonni, “High-order harmonic generation: simplified model and relevance of single-atom theories to experiment,” Phys. Rev. A 41, 6571 (1990).
[CrossRef] [PubMed]

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

J. Phys. B (1)

Ph. Balcou, C. Cornaggia, A. S. L. Gomes, L. A. Lompré, and A. L’Huillier, “Optimizing high-order harmonic generation in strong fields,” J. Phys. B 25, 4467 (1992).
[CrossRef]

JETP (1)

V. P. Krainov and B. Shokri, “Energy and angular distributions of electrons resulting from barrier-suppression ionization of atoms by strong low-frequency radiation,” JETP 80, 657 (1995).

Laser Phys. (1)

V. P. Krainov and Z. S. Mulyukov, “A plateau in high-order harmonic generation for a two-level atom,” Laser Phys. 4, 544 (1994).

Phys. Rev. A (3)

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, 2117 (1994).
[CrossRef] [PubMed]

J. L. Krause, K. J. Schafer, and K. C. Kulander, “Calculation of photoemission from atoms subject to intense laser fields,” Phys. Rev. A 45, 4998 (1992).
[CrossRef] [PubMed]

B. Sundaram and P. W. Milonni, “High-order harmonic generation: simplified model and relevance of single-atom theories to experiment,” Phys. Rev. A 41, 6571 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

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

Other (7)

L. Plaja, “Theoretical study on optical high-order harmonic generation in three matter models,” Ph.D. dissertation (Universitat Autònoma de Barcelona, Barcelona, 1993).

L. Plaja and L. Roso, “Contribution of bound-bound transitions to high-order harmonic generation,” in SILAP III Proceedings, B. Pireaux, A. L’Huiller, and K. Rzazewski, eds. (Plenum, New York, 1993), pp. 53–61.

V. P. Krainov, Department of Theoretical Physics, Moscow Institute of Physics and Technology, 141700 Dolgopzudny, Moscow, Russia (personal communication).

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

V. Malyshev, E. Conejero Jarque, and L. Roso, “Self-reflection of an intense ultrashort laser pulse by tunnel ionization on a solid surface,” J. Opt. Soc. Am. B (to be published).

N. B. Delone and V. P. Krainov, Multiphoton Processes in Atoms (Springer-Verlag, Berlin, 1994).
[CrossRef]

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two Electron Atoms (Plenum, New York, 1954).

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Figures (3)

Fig. 1
Fig. 1

Dipole spectra of a two-level atom representing the hydrogen 1s–2p transition for a field of frequency ω0=0.04 a.u. and amplitudes (a) E0=0.05 a.u. and (b) E0=0.4 a.u. The pulse has a square profile of 10 optical cycles.

Fig. 2
Fig. 2

Harmonic spectra calculated (a) for our open two-level atom and (b) for a conventional two-level atom without ionization. The incident pulse has an amplitude E0=0.05 a.u., a frequency ω0=0.04 a.u., and a duration of 10 optical cycles, three of them of linear ramp turn-on and the rest of constant amplitude.

Fig. 3
Fig. 3

Time evolution of our system during the interaction with the pulse. (a), (b) Evolution of the population (solid curves) and the corresponding ionization rate (dotted curves) of (a) the ground and (b) the excited levels, respectively. (c) Dipole acceleration whose Fourier transform is the harmonic depicted in Fig. 2(a).

Equations (7)

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dρ11(t)dt=[idE(t)ρ21(t)+c.c.]-γ1(t)ρ11(t),
dρ22(t)dt=-[idE(t)ρ21(t)+c.c.]-γ2(t)ρ22(t),
dρ21(t)dt=idE(t)[ρ22(t)-ρ11(t)]-iωT+γ1(t)+γ2(t)2ρ21(t),
γ1(t)=4|E(t)|exp-23|E(t)|.
γ2(t)={2[-|ω2|+2|E(t)|1/2]}1/2Θ(|E|>Ean=2),
H=ω2dE(t)dE(t)ω1
λ±=ω1+ω22±½[(ω2-ω1)2+4d2E(t)2]1/2.

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