Abstract

We have theoretically investigated the phase dependence of high-harmonic generation with few-cycle laser pulses. We performed the analysis by solving the coupled evolution equations for the laser pulse and for the harmonic field in one space dimension. Our calculations reveal phase-dependent features in the cutoff region, such as a phase-dependent shift of the cutoff, that are preserved during propagation.

© 1999 Optical Society of America

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References

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  1. A. L’Huillier and P. Balcou, Phys. Rev. Lett. 70, 774 (1993); J. J. Macklin, J. D. Kmetec, and C. L. Gordon III, Phys. Rev. Lett. 70, 766 (1993).
    [CrossRef] [PubMed]
  2. C. Kan, N. H. Burnett, C. E. Capjack, and R. Rankin, Phys. Rev. Lett. 79, 2971 (1997).
    [CrossRef]
  3. Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, Science 278, 661 (1997); Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Capteyn, Phys. Rev. Lett. 79, 2967 (1997).
    [CrossRef]
  4. I. P. Christov, M. M. Murnane, and H. C. Capteyn, Phys. Rev. Lett. 78, 1251 (1997).
    [CrossRef]
  5. F. Krausz, T. Brabec, M. Schnürer, and C. Spielmann, Opt. Photonics News 9(7), 46 (1998).
    [CrossRef]
  6. A. de Bohan, P. Antoine, D. B. Milośevič, and B. Piraux, Phys. Rev. Lett. 81, 1837 (1998).
    [CrossRef]
  7. L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. Hänsch, Opt. Lett. 21, 2008 (1996).
    [CrossRef] [PubMed]
  8. G. Tempea and T. Brabec, Opt. Lett. 23, 1286 (1998); M. Geissler, G. Tempea, A. Schrinzi, F. Krausz, and T. Brabec, “Polarization response of tunnel ionizing media,” submitted to Phys. Rev. Lett.
    [CrossRef]
  9. M. V. Ammosov, N. B. Delone, and V. P. Krainov, Sov. Phys. JETP 64, 1191 (1986); Ch. Spielmann, C. Kan, N. H. Burnett, T. Brabec, M. Geissler, A. Scrinzi, M. Schnürer, and F. Krausz, IEEE J. Select. Top. Quantum Electronics 4, 249 (1998).
    [CrossRef]
  10. A. Scrinzi, M. Geissler, and T. Brabec, “Above barrier ionization,” submitted to Phys. Rev. Lett.
  11. M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
    [CrossRef] [PubMed]
  12. M. Y. Ivanov, T. Brabec, and N. Burnett, Phys. Rev. A 54, 742 (1996).
    [CrossRef] [PubMed]
  13. A. L’Huillier, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 66, 2200 (1991); P. L. Shkolnikov, A. E. Kaplan, and A. Lago, J. Opt. Soc. Am. B 13, 412 (1996).
    [CrossRef]
  14. L. Xu, G. Tempea, Ch. Spielmann, F. Krausz, A. Stingl, K. Ferenc, and S. Takano, Opt. Lett. 23, 789 (1998).
    [CrossRef]
  15. A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
    [CrossRef] [PubMed]

1998 (4)

F. Krausz, T. Brabec, M. Schnürer, and C. Spielmann, Opt. Photonics News 9(7), 46 (1998).
[CrossRef]

A. de Bohan, P. Antoine, D. B. Milośevič, and B. Piraux, Phys. Rev. Lett. 81, 1837 (1998).
[CrossRef]

L. Xu, G. Tempea, Ch. Spielmann, F. Krausz, A. Stingl, K. Ferenc, and S. Takano, Opt. Lett. 23, 789 (1998).
[CrossRef]

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
[CrossRef] [PubMed]

1997 (2)

C. Kan, N. H. Burnett, C. E. Capjack, and R. Rankin, Phys. Rev. Lett. 79, 2971 (1997).
[CrossRef]

I. P. Christov, M. M. Murnane, and H. C. Capteyn, Phys. Rev. Lett. 78, 1251 (1997).
[CrossRef]

1996 (2)

1994 (1)

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

Antoine, P.

A. de Bohan, P. Antoine, D. B. Milośevič, and B. Piraux, Phys. Rev. Lett. 81, 1837 (1998).
[CrossRef]

Backus, S.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
[CrossRef] [PubMed]

Balcou, P.

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

Brabec, T.

F. Krausz, T. Brabec, M. Schnürer, and C. Spielmann, Opt. Photonics News 9(7), 46 (1998).
[CrossRef]

M. Y. Ivanov, T. Brabec, and N. Burnett, Phys. Rev. A 54, 742 (1996).
[CrossRef] [PubMed]

L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. Hänsch, Opt. Lett. 21, 2008 (1996).
[CrossRef] [PubMed]

Burnett, N.

M. Y. Ivanov, T. Brabec, and N. Burnett, Phys. Rev. A 54, 742 (1996).
[CrossRef] [PubMed]

Burnett, N. H.

C. Kan, N. H. Burnett, C. E. Capjack, and R. Rankin, Phys. Rev. Lett. 79, 2971 (1997).
[CrossRef]

Capjack, C. E.

C. Kan, N. H. Burnett, C. E. Capjack, and R. Rankin, Phys. Rev. Lett. 79, 2971 (1997).
[CrossRef]

Capteyn, H. C.

I. P. Christov, M. M. Murnane, and H. C. Capteyn, Phys. Rev. Lett. 78, 1251 (1997).
[CrossRef]

Chang, Z.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
[CrossRef] [PubMed]

Christov, I. P.

I. P. Christov, M. M. Murnane, and H. C. Capteyn, Phys. Rev. Lett. 78, 1251 (1997).
[CrossRef]

Corkum, P. B.

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

de Bohan, A.

A. de Bohan, P. Antoine, D. B. Milośevič, and B. Piraux, Phys. Rev. Lett. 81, 1837 (1998).
[CrossRef]

Durfee III, C. G.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
[CrossRef] [PubMed]

Ferenc, K.

Hänsch, T.

Herne, C.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
[CrossRef] [PubMed]

Ivanov, M. Y.

M. Y. Ivanov, T. Brabec, and N. Burnett, Phys. Rev. A 54, 742 (1996).
[CrossRef] [PubMed]

Ivanov, M. Yu.

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

Kan, C.

C. Kan, N. H. Burnett, C. E. Capjack, and R. Rankin, Phys. Rev. Lett. 79, 2971 (1997).
[CrossRef]

Kapteyn, H. C.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
[CrossRef] [PubMed]

Krausz, F.

L’Huiller, A.

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

Lewenstein, M.

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

Milosevic, D. B.

A. de Bohan, P. Antoine, D. B. Milośevič, and B. Piraux, Phys. Rev. Lett. 81, 1837 (1998).
[CrossRef]

Murnane, M. M.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
[CrossRef] [PubMed]

I. P. Christov, M. M. Murnane, and H. C. Capteyn, Phys. Rev. Lett. 78, 1251 (1997).
[CrossRef]

Piraux, B.

A. de Bohan, P. Antoine, D. B. Milośevič, and B. Piraux, Phys. Rev. Lett. 81, 1837 (1998).
[CrossRef]

Poppe, A.

Rankin, R.

C. Kan, N. H. Burnett, C. E. Capjack, and R. Rankin, Phys. Rev. Lett. 79, 2971 (1997).
[CrossRef]

Rundquist, A.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
[CrossRef] [PubMed]

Schnürer, M.

F. Krausz, T. Brabec, M. Schnürer, and C. Spielmann, Opt. Photonics News 9(7), 46 (1998).
[CrossRef]

Spielmann, C.

F. Krausz, T. Brabec, M. Schnürer, and C. Spielmann, Opt. Photonics News 9(7), 46 (1998).
[CrossRef]

Spielmann, Ch.

Stingl, A.

Takano, S.

Tempea, G.

Xu, L.

Opt. Lett. (2)

Opt. Photonics News (1)

F. Krausz, T. Brabec, M. Schnürer, and C. Spielmann, Opt. Photonics News 9(7), 46 (1998).
[CrossRef]

Phys. Rev. A (2)

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

M. Y. Ivanov, T. Brabec, and N. Burnett, Phys. Rev. A 54, 742 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

A. de Bohan, P. Antoine, D. B. Milośevič, and B. Piraux, Phys. Rev. Lett. 81, 1837 (1998).
[CrossRef]

C. Kan, N. H. Burnett, C. E. Capjack, and R. Rankin, Phys. Rev. Lett. 79, 2971 (1997).
[CrossRef]

I. P. Christov, M. M. Murnane, and H. C. Capteyn, Phys. Rev. Lett. 78, 1251 (1997).
[CrossRef]

Science (1)

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
[CrossRef] [PubMed]

Other (6)

A. L’Huillier, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 66, 2200 (1991); P. L. Shkolnikov, A. E. Kaplan, and A. Lago, J. Opt. Soc. Am. B 13, 412 (1996).
[CrossRef]

A. L’Huillier and P. Balcou, Phys. Rev. Lett. 70, 774 (1993); J. J. Macklin, J. D. Kmetec, and C. L. Gordon III, Phys. Rev. Lett. 70, 766 (1993).
[CrossRef] [PubMed]

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, Science 278, 661 (1997); Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Capteyn, Phys. Rev. Lett. 79, 2967 (1997).
[CrossRef]

G. Tempea and T. Brabec, Opt. Lett. 23, 1286 (1998); M. Geissler, G. Tempea, A. Schrinzi, F. Krausz, and T. Brabec, “Polarization response of tunnel ionizing media,” submitted to Phys. Rev. Lett.
[CrossRef]

M. V. Ammosov, N. B. Delone, and V. P. Krainov, Sov. Phys. JETP 64, 1191 (1986); Ch. Spielmann, C. Kan, N. H. Burnett, T. Brabec, M. Geissler, A. Scrinzi, M. Schnürer, and F. Krausz, IEEE J. Select. Top. Quantum Electronics 4, 249 (1998).
[CrossRef]

A. Scrinzi, M. Geissler, and T. Brabec, “Above barrier ionization,” submitted to Phys. Rev. Lett.

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Figures (4)

Fig. 1
Fig. 1

Fourier transform of single-atom dipole moment d for HHG in hydrogen as determined from Eqs. (4)–(9) (solid curve) and as determined by a numerical solution of the exact Schrödinger equation. The laser parameters are I0=5×1014 W/cm2, Ψ0=0, laser center wavelength λ0=750 nm, and τp=2.84 fs, which corresponds to a FWHM duration of 5 fs.

Fig. 2
Fig. 2

Spectral intensity of the harmonic radiation for the fundamental pulse phases Ψ0=0 (dotted curves) and Ψ0=π/2 (solid curves) after propagation distances of a, 0.2; b, 5; c, 9; and d, 18 µm in helium at a pressure of 500 Torr. The laser parameters are I0=2×1015 W/cm2, λ0=750 nm, and τp=2.84 fs.

Fig. 3
Fig. 3

Spectral intensity of the harmonic radiation for the fundamental pulse phases Ψ0=π/6 (dotted curves) and Ψ0=π/3 (solid curves) after propagation distances of a, 0.2; b, 5; c, 9; and d, 18 µm. For the other parameters, see Fig. 2.

Fig. 4
Fig. 4

Fourier transform of the harmonic amplitude spectrum between orders 210 and 230 for the parameters of Fig. 1c. Dotted and solid curves, fundamental pulse phases Ψ0=0 and Ψ0=π/2, respectively.

Equations (9)

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ξ El=-2πe2mc-τ ρ(El)El dτ-2πIpcτρ(El)El,
ρ(ξ, τ)=ρ01-exp--τdτw[El(ξ, τ)],
ξ Eh=2πcτ D(El),
d(τb, τ)=2meτb 1iaion(τb)apr (τb, τ)arec(τ)
aion(τb)={w(τb)[ρ0-ρ(τb)]/ρ0}1/2,
apr(τ, τb)=m1/4e22πτ-τb3/2(2Ip)1/4El(τb)×exp-iτbτdτ12m[p(τb, τ)-eAl(τ)]2+Ip,
arec(τ)=m2e102p(τb, τ)-eAl(τ)Ip+12m[p(τb, τ)-eAl(τ)]23,
p(τb, τ)=eτ-τbτbτ dτAl(τ)
p(τb, τ)-eAl(τb)=0.

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