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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  1. A. L’Huillier and P. Balcou, Phys. Rev. Lett. 70, 774 (1993); J. J. Macklin, J. D. Kmetec, and C. L. Gordon, 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, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
    [Crossref] [PubMed]

1998 (5)

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]

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]

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, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
[Crossref] [PubMed]

1997 (3)

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

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]

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]

1993 (1)

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

1991 (1)

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]

1986 (1)

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]

Ammosov, M. V.

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]

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, 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]

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

Brabec, T.

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

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

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

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

Burnett, N.

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

Burnett, N. H.

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]

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, 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]

Delone, N. B.

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]

Durfee, C. G.

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

Ferenc, K.

Geissler, M.

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

Hänsch, T.

Herne, C.

A. Rundquist, C. G. Durfee, 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.

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]

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, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
[Crossref] [PubMed]

Koppitsch, R.

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]

Krainov, V. P.

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]

Krausz, F.

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

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

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]

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

Kulander, K. C.

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]

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]

L’Huillier, A.

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

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]

Lenzner, M.

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]

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, 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, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
[Crossref] [PubMed]

Sartania, S.

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]

Schafer, K. J.

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]

Schnürer, M.

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

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]

Scrinzi, A.

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

Spielmann, C.

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

Spielmann, Ch.

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

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]

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

Stingl, A.

Takano, S.

Tempea, G.

Wobrauschek, P.

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]

Xu, L.

Opt. Lett. (3)

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. (5)

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]

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

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

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

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

Science (2)

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]

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

Sov. Phys. JETP (1)

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]

Other (1)

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)

Equations on this page are rendered with MathJax. Learn more.

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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