Abstract

The effects of coherent propagation of an ultrashort pulse through an extended collection of multilevel systems is examined. It is found that the transmitted field shows significant deviation from both Beer’s law absorption and the conventional area theorem. In addition, when the system is in a nonstationary state it is found that additional frequencies can appear in the transmitted pulse spectrum. Such effects may be important in several areas of current interest, such as ultrafast pump–probe, coherent control, and inversionless amplification experiments in extended, multilevel media as well as in ultrafast-laser development.

© 1996 Optical Society of America

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  1. S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
    [Crossref]
  2. D. C. Burnham and R. Y. Chiao, “Coherent resonance fluorescence excited by short light pulses,” Phys. Rev. 188, 667–675 (1969).
    [Crossref]
  3. M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
    [Crossref]
  4. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).
  5. J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
    [Crossref]
  6. S. M. Hamadani, J. Goldhar, N. A. Kurnit, and A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
    [Crossref]
  7. H.-J. Hartmann and A. Laubereau, “Transient infrared spectroscopy on the picosecond time-scale by coherent pulse propagation,” J. Chem. Phys. 80, 4663–4670 (1984).
    [Crossref]
  8. J. N. Sweetser, “Wave packet-modulated coherent emission and amplification of femtosecond optical pulses,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1994).
  9. H. Harde, N. Katzenellenbogen, and D. Grischkowsky, “Terahertz coherent transients from methyl chloride vapor,” J. Opt. Soc. Am B 11, 1018–1030 (1994).
    [Crossref]
  10. C. Lin, M. G. Littman, L. Shen, H. Rabitz, D. Reitze, A. M. Weiner, and D. E. Leaird, “Optically induced birefringence in lithium dimer as the result of resonant ultrafast pumping,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 28.
  11. T. J. Dunn, J. N. Sweetser, I. A. Walmsley, and C. Radzewicz, “Experimental determination of the dynamics of a molecular nuclear wave packet via the spectra of spontaneous emission,” Phys. Rev. Lett. 70, 3388–3391 (1993).
    [Crossref] [PubMed]
  12. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, New York, 1977).
  13. M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  14. T. J. Bridges, H. A. Haus, and P. W. Hoff, “Small-signal step response of laser amplifiers and measurement of CO2 laser linewidth,” IEEE J. Quantum Electron. QE-4, 777–782 (1968).
    [Crossref]
  15. Note that the optical density defined here has units of inverse time and is different from the conventional optical density, which is unitless. The conventional on-resonance optical density is defined as μz/γ or μzT2, where T2 is the polarization decay time. However, because of the transient or weakly damped nature of the system, T2 is typically much larger than the incident pulse width. As γ can be made to be arbitrarily small (i.e., an undamped system) without affecting the basic form of the transmitted field for ultrashort pulses, this could lead to a deceivingly large value of the conventional optical density. For this reason the optical density used here does not contain the decay time and thus has units of inverse time. Furthermore, it is natural to use μz to characterize the medium because it is this quantity that appears explicitly in the expressions for the transmitted field.
  16. J. N. Sweetser and I. A. Walmsley, “Quantum interference and propagation effects in the free-induction decay from nonstationary multi-level systems,” Laser Phys. 5, 461–465 (1995).
  17. A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5, 1563–1572 (1988).
    [Crossref]
  18. P. M. Felker and A. H. Zewail, “Purely rotational coherence effect and time-resolved sub-Doppler spectroscopy of large molecules. I. Theoretical,” J. Chem. Phys. 86, 2460–2482 (1987); D. M. Willberg, J. J. Breen, M. Guttman, and A. H. Zewail, “Rotational constants of vibrationally excited iodine from purely rotational coherence observed in pump-probe experiments,” J. Phys. Chem. 95, 7136–7138 (1991).
    [Crossref]
  19. G. Hertzberg, Spectra of Diatomic Molecules (Van Nostrand Reinhold, New York, 1950).
  20. O. Kinrot and Y. Prior, “Nonlinear interaction of propagating short pulses in optically dense media,” Phys. Rev. A 51, 4996–5007 (1995).
    [Crossref] [PubMed]
  21. M. Woerner, A. Seilmeier, and W. Kaiser, “Reshaping of infrared picosecond pulses after passage through atmospheric CO2,” Opt. Lett. 14, 636–638 (1989).
    [Crossref] [PubMed]
  22. W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
    [Crossref] [PubMed]
  23. O. Kocharovskaya, “Amplification and lasing without inversion,” Phys. Rep. 219, 175–190 (1992), and references therein.
    [Crossref]
  24. A. Nottlemann, C. Peters, and W. Lange, “Inversionless amplification of picosecond pulses due to Zeeman coherence,” Phys. Rev. Lett. 70, 1783–1786 (1993).
    [Crossref]
  25. J. D. Harvey, J. M. Dudley, P. F. Curley, C. Spielmann, and F. Krausz, “Coherent effects in a self-mode-locked Ti:sapphire laser,” Opt. Lett. 19, 972–974 (1994).
    [Crossref] [PubMed]
  26. J. Zhou, G. Taft, C.-P. Huang, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, “Pulse evolution in a broad-bandwidth Ti:sapphire laser,” Opt. Lett. 19, 1149–1151 (1994).
    [Crossref] [PubMed]

1995 (2)

J. N. Sweetser and I. A. Walmsley, “Quantum interference and propagation effects in the free-induction decay from nonstationary multi-level systems,” Laser Phys. 5, 461–465 (1995).

O. Kinrot and Y. Prior, “Nonlinear interaction of propagating short pulses in optically dense media,” Phys. Rev. A 51, 4996–5007 (1995).
[Crossref] [PubMed]

1994 (3)

1993 (3)

T. J. Dunn, J. N. Sweetser, I. A. Walmsley, and C. Radzewicz, “Experimental determination of the dynamics of a molecular nuclear wave packet via the spectra of spontaneous emission,” Phys. Rev. Lett. 70, 3388–3391 (1993).
[Crossref] [PubMed]

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
[Crossref] [PubMed]

A. Nottlemann, C. Peters, and W. Lange, “Inversionless amplification of picosecond pulses due to Zeeman coherence,” Phys. Rev. Lett. 70, 1783–1786 (1993).
[Crossref]

1992 (1)

O. Kocharovskaya, “Amplification and lasing without inversion,” Phys. Rep. 219, 175–190 (1992), and references therein.
[Crossref]

1989 (1)

1988 (1)

1987 (1)

P. M. Felker and A. H. Zewail, “Purely rotational coherence effect and time-resolved sub-Doppler spectroscopy of large molecules. I. Theoretical,” J. Chem. Phys. 86, 2460–2482 (1987); D. M. Willberg, J. J. Breen, M. Guttman, and A. H. Zewail, “Rotational constants of vibrationally excited iodine from purely rotational coherence observed in pump-probe experiments,” J. Phys. Chem. 95, 7136–7138 (1991).
[Crossref]

1984 (2)

H.-J. Hartmann and A. Laubereau, “Transient infrared spectroscopy on the picosecond time-scale by coherent pulse propagation,” J. Chem. Phys. 80, 4663–4670 (1984).
[Crossref]

J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[Crossref]

1974 (1)

S. M. Hamadani, J. Goldhar, N. A. Kurnit, and A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
[Crossref]

1970 (1)

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
[Crossref]

1969 (2)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[Crossref]

D. C. Burnham and R. Y. Chiao, “Coherent resonance fluorescence excited by short light pulses,” Phys. Rev. 188, 667–675 (1969).
[Crossref]

1968 (1)

T. J. Bridges, H. A. Haus, and P. W. Hoff, “Small-signal step response of laser amplifiers and measurement of CO2 laser linewidth,” IEEE J. Quantum Electron. QE-4, 777–782 (1968).
[Crossref]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).

Balant, A. C.

J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[Crossref]

Bridges, T. J.

T. J. Bridges, H. A. Haus, and P. W. Hoff, “Small-signal step response of laser amplifiers and measurement of CO2 laser linewidth,” IEEE J. Quantum Electron. QE-4, 777–782 (1968).
[Crossref]

Burnham, D. C.

D. C. Burnham and R. Y. Chiao, “Coherent resonance fluorescence excited by short light pulses,” Phys. Rev. 188, 667–675 (1969).
[Crossref]

Chiao, R. Y.

D. C. Burnham and R. Y. Chiao, “Coherent resonance fluorescence excited by short light pulses,” Phys. Rev. 188, 667–675 (1969).
[Crossref]

Christov, I. P.

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, New York, 1977).

Crisp, M. D.

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
[Crossref]

Curley, P. F.

Dahleh, M.

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
[Crossref] [PubMed]

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, New York, 1977).

Dudley, J. M.

Dunn, T. J.

T. J. Dunn, J. N. Sweetser, I. A. Walmsley, and C. Radzewicz, “Experimental determination of the dynamics of a molecular nuclear wave packet via the spectra of spontaneous emission,” Phys. Rev. Lett. 70, 3388–3391 (1993).
[Crossref] [PubMed]

Eberly, J. H.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).

Felker, P. M.

P. M. Felker and A. H. Zewail, “Purely rotational coherence effect and time-resolved sub-Doppler spectroscopy of large molecules. I. Theoretical,” J. Chem. Phys. 86, 2460–2482 (1987); D. M. Willberg, J. J. Breen, M. Guttman, and A. H. Zewail, “Rotational constants of vibrationally excited iodine from purely rotational coherence observed in pump-probe experiments,” J. Phys. Chem. 95, 7136–7138 (1991).
[Crossref]

Goldhar, J.

S. M. Hamadani, J. Goldhar, N. A. Kurnit, and A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
[Crossref]

Grischkowsky, D.

H. Harde, N. Katzenellenbogen, and D. Grischkowsky, “Terahertz coherent transients from methyl chloride vapor,” J. Opt. Soc. Am B 11, 1018–1030 (1994).
[Crossref]

J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[Crossref]

Hahn, E. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[Crossref]

Hamadani, S. M.

S. M. Hamadani, J. Goldhar, N. A. Kurnit, and A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
[Crossref]

Harde, H.

H. Harde, N. Katzenellenbogen, and D. Grischkowsky, “Terahertz coherent transients from methyl chloride vapor,” J. Opt. Soc. Am B 11, 1018–1030 (1994).
[Crossref]

Hartmann, H.-J.

H.-J. Hartmann and A. Laubereau, “Transient infrared spectroscopy on the picosecond time-scale by coherent pulse propagation,” J. Chem. Phys. 80, 4663–4670 (1984).
[Crossref]

Harvey, J. D.

Haus, H. A.

T. J. Bridges, H. A. Haus, and P. W. Hoff, “Small-signal step response of laser amplifiers and measurement of CO2 laser linewidth,” IEEE J. Quantum Electron. QE-4, 777–782 (1968).
[Crossref]

Heritage, J. P.

Hertzberg, G.

G. Hertzberg, Spectra of Diatomic Molecules (Van Nostrand Reinhold, New York, 1950).

Hoff, P. W.

T. J. Bridges, H. A. Haus, and P. W. Hoff, “Small-signal step response of laser amplifiers and measurement of CO2 laser linewidth,” IEEE J. Quantum Electron. QE-4, 777–782 (1968).
[Crossref]

Huang, C.-P.

Javan, A.

S. M. Hamadani, J. Goldhar, N. A. Kurnit, and A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
[Crossref]

Kaiser, W.

Kapteyn, H. C.

Katzenellenbogen, N.

H. Harde, N. Katzenellenbogen, and D. Grischkowsky, “Terahertz coherent transients from methyl chloride vapor,” J. Opt. Soc. Am B 11, 1018–1030 (1994).
[Crossref]

Kinrot, O.

O. Kinrot and Y. Prior, “Nonlinear interaction of propagating short pulses in optically dense media,” Phys. Rev. A 51, 4996–5007 (1995).
[Crossref] [PubMed]

Kirschner, E. M.

Kocharovskaya, O.

O. Kocharovskaya, “Amplification and lasing without inversion,” Phys. Rep. 219, 175–190 (1992), and references therein.
[Crossref]

Krausz, F.

Kurnit, N. A.

S. M. Hamadani, J. Goldhar, N. A. Kurnit, and A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
[Crossref]

Laloe, F.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, New York, 1977).

Lange, W.

A. Nottlemann, C. Peters, and W. Lange, “Inversionless amplification of picosecond pulses due to Zeeman coherence,” Phys. Rev. Lett. 70, 1783–1786 (1993).
[Crossref]

Laubereau, A.

H.-J. Hartmann and A. Laubereau, “Transient infrared spectroscopy on the picosecond time-scale by coherent pulse propagation,” J. Chem. Phys. 80, 4663–4670 (1984).
[Crossref]

Leaird, D. E.

C. Lin, M. G. Littman, L. Shen, H. Rabitz, D. Reitze, A. M. Weiner, and D. E. Leaird, “Optically induced birefringence in lithium dimer as the result of resonant ultrafast pumping,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 28.

Lin, C.

C. Lin, M. G. Littman, L. Shen, H. Rabitz, D. Reitze, A. M. Weiner, and D. E. Leaird, “Optically induced birefringence in lithium dimer as the result of resonant ultrafast pumping,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 28.

Littman, M. G.

C. Lin, M. G. Littman, L. Shen, H. Rabitz, D. Reitze, A. M. Weiner, and D. E. Leaird, “Optically induced birefringence in lithium dimer as the result of resonant ultrafast pumping,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 28.

McCall, S. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[Crossref]

Murnane, M. M.

Nottlemann, A.

A. Nottlemann, C. Peters, and W. Lange, “Inversionless amplification of picosecond pulses due to Zeeman coherence,” Phys. Rev. Lett. 70, 1783–1786 (1993).
[Crossref]

Peters, C.

A. Nottlemann, C. Peters, and W. Lange, “Inversionless amplification of picosecond pulses due to Zeeman coherence,” Phys. Rev. Lett. 70, 1783–1786 (1993).
[Crossref]

Prior, Y.

O. Kinrot and Y. Prior, “Nonlinear interaction of propagating short pulses in optically dense media,” Phys. Rev. A 51, 4996–5007 (1995).
[Crossref] [PubMed]

Rabitz, H.

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
[Crossref] [PubMed]

C. Lin, M. G. Littman, L. Shen, H. Rabitz, D. Reitze, A. M. Weiner, and D. E. Leaird, “Optically induced birefringence in lithium dimer as the result of resonant ultrafast pumping,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 28.

Radzewicz, C.

T. J. Dunn, J. N. Sweetser, I. A. Walmsley, and C. Radzewicz, “Experimental determination of the dynamics of a molecular nuclear wave packet via the spectra of spontaneous emission,” Phys. Rev. Lett. 70, 3388–3391 (1993).
[Crossref] [PubMed]

Reitze, D.

C. Lin, M. G. Littman, L. Shen, H. Rabitz, D. Reitze, A. M. Weiner, and D. E. Leaird, “Optically induced birefringence in lithium dimer as the result of resonant ultrafast pumping,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 28.

Rothenberg, J. E.

J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[Crossref]

Seilmeier, A.

Shen, L.

C. Lin, M. G. Littman, L. Shen, H. Rabitz, D. Reitze, A. M. Weiner, and D. E. Leaird, “Optically induced birefringence in lithium dimer as the result of resonant ultrafast pumping,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 28.

Spielmann, C.

Sweetser, J. N.

J. N. Sweetser and I. A. Walmsley, “Quantum interference and propagation effects in the free-induction decay from nonstationary multi-level systems,” Laser Phys. 5, 461–465 (1995).

T. J. Dunn, J. N. Sweetser, I. A. Walmsley, and C. Radzewicz, “Experimental determination of the dynamics of a molecular nuclear wave packet via the spectra of spontaneous emission,” Phys. Rev. Lett. 70, 3388–3391 (1993).
[Crossref] [PubMed]

J. N. Sweetser, “Wave packet-modulated coherent emission and amplification of femtosecond optical pulses,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1994).

Taft, G.

Walmsley, I. A.

J. N. Sweetser and I. A. Walmsley, “Quantum interference and propagation effects in the free-induction decay from nonstationary multi-level systems,” Laser Phys. 5, 461–465 (1995).

T. J. Dunn, J. N. Sweetser, I. A. Walmsley, and C. Radzewicz, “Experimental determination of the dynamics of a molecular nuclear wave packet via the spectra of spontaneous emission,” Phys. Rev. Lett. 70, 3388–3391 (1993).
[Crossref] [PubMed]

Warren, W. S.

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
[Crossref] [PubMed]

Weiner, A. M.

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5, 1563–1572 (1988).
[Crossref]

C. Lin, M. G. Littman, L. Shen, H. Rabitz, D. Reitze, A. M. Weiner, and D. E. Leaird, “Optically induced birefringence in lithium dimer as the result of resonant ultrafast pumping,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 28.

Woerner, M.

Zewail, A. H.

P. M. Felker and A. H. Zewail, “Purely rotational coherence effect and time-resolved sub-Doppler spectroscopy of large molecules. I. Theoretical,” J. Chem. Phys. 86, 2460–2482 (1987); D. M. Willberg, J. J. Breen, M. Guttman, and A. H. Zewail, “Rotational constants of vibrationally excited iodine from purely rotational coherence observed in pump-probe experiments,” J. Phys. Chem. 95, 7136–7138 (1991).
[Crossref]

Zhou, J.

Appl. Phys. Lett. (1)

S. M. Hamadani, J. Goldhar, N. A. Kurnit, and A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
[Crossref]

IEEE J. Quantum Electron. (1)

T. J. Bridges, H. A. Haus, and P. W. Hoff, “Small-signal step response of laser amplifiers and measurement of CO2 laser linewidth,” IEEE J. Quantum Electron. QE-4, 777–782 (1968).
[Crossref]

J. Chem. Phys. (2)

P. M. Felker and A. H. Zewail, “Purely rotational coherence effect and time-resolved sub-Doppler spectroscopy of large molecules. I. Theoretical,” J. Chem. Phys. 86, 2460–2482 (1987); D. M. Willberg, J. J. Breen, M. Guttman, and A. H. Zewail, “Rotational constants of vibrationally excited iodine from purely rotational coherence observed in pump-probe experiments,” J. Phys. Chem. 95, 7136–7138 (1991).
[Crossref]

H.-J. Hartmann and A. Laubereau, “Transient infrared spectroscopy on the picosecond time-scale by coherent pulse propagation,” J. Chem. Phys. 80, 4663–4670 (1984).
[Crossref]

J. Opt. Soc. Am B (1)

H. Harde, N. Katzenellenbogen, and D. Grischkowsky, “Terahertz coherent transients from methyl chloride vapor,” J. Opt. Soc. Am B 11, 1018–1030 (1994).
[Crossref]

J. Opt. Soc. Am. B (1)

Laser Phys. (1)

J. N. Sweetser and I. A. Walmsley, “Quantum interference and propagation effects in the free-induction decay from nonstationary multi-level systems,” Laser Phys. 5, 461–465 (1995).

Opt. Lett. (3)

Phys. Rep. (1)

O. Kocharovskaya, “Amplification and lasing without inversion,” Phys. Rep. 219, 175–190 (1992), and references therein.
[Crossref]

Phys. Rev. (2)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[Crossref]

D. C. Burnham and R. Y. Chiao, “Coherent resonance fluorescence excited by short light pulses,” Phys. Rev. 188, 667–675 (1969).
[Crossref]

Phys. Rev. A (2)

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
[Crossref]

O. Kinrot and Y. Prior, “Nonlinear interaction of propagating short pulses in optically dense media,” Phys. Rev. A 51, 4996–5007 (1995).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

A. Nottlemann, C. Peters, and W. Lange, “Inversionless amplification of picosecond pulses due to Zeeman coherence,” Phys. Rev. Lett. 70, 1783–1786 (1993).
[Crossref]

J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[Crossref]

T. J. Dunn, J. N. Sweetser, I. A. Walmsley, and C. Radzewicz, “Experimental determination of the dynamics of a molecular nuclear wave packet via the spectra of spontaneous emission,” Phys. Rev. Lett. 70, 3388–3391 (1993).
[Crossref] [PubMed]

Science (1)

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
[Crossref] [PubMed]

Other (7)

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, New York, 1977).

M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

Note that the optical density defined here has units of inverse time and is different from the conventional optical density, which is unitless. The conventional on-resonance optical density is defined as μz/γ or μzT2, where T2 is the polarization decay time. However, because of the transient or weakly damped nature of the system, T2 is typically much larger than the incident pulse width. As γ can be made to be arbitrarily small (i.e., an undamped system) without affecting the basic form of the transmitted field for ultrashort pulses, this could lead to a deceivingly large value of the conventional optical density. For this reason the optical density used here does not contain the decay time and thus has units of inverse time. Furthermore, it is natural to use μz to characterize the medium because it is this quantity that appears explicitly in the expressions for the transmitted field.

G. Hertzberg, Spectra of Diatomic Molecules (Van Nostrand Reinhold, New York, 1950).

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).

C. Lin, M. G. Littman, L. Shen, H. Rabitz, D. Reitze, A. M. Weiner, and D. E. Leaird, “Optically induced birefringence in lithium dimer as the result of resonant ultrafast pumping,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 28.

J. N. Sweetser, “Wave packet-modulated coherent emission and amplification of femtosecond optical pulses,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1994).

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the propagation of an ultrashort pulse through a resonant absorbing medium that is characterized by a coupling constant μ. The medium is composed of three-level systems whose excited-state level separation is spanned by the spectrum. Δn represents the detuning of the center of the pulse spectrum from level n.

Fig. 2
Fig. 2

Result of the linear model for the field transmitted through a sample of three-level absorbers (μz = 1.2 ps−1). The input pulse is a Gaussian of 100-fs duration (τp/T2 = 10−4) and unity amplitude centered at 0.5 ps. The pulse spectrum is centered between the frequencies of the two transitions. The rapid oscillations are due to polarization interference, and the slower modulation is a result of coherent propagation through the weakly damped medium.

Fig. 3
Fig. 3

Calculated transmitted field after propagation through a sample of two-level absorbers (μz = 15 ps−1). The peak is attenuated, and the pulse area has decayed to zero, as expected, but the total energy transmission is close to 100%.

Fig. 4
Fig. 4

Plot of the calculated transmitted pulse area (normalized to the incident area) versus optical depth for a TLS and a MLS (three levels) with the same nominal absorption coefficient. Whereas the TLS shows exponential decay as expected from the weak-field area theorem, the MLS displays a significant departure from this behavior.

Fig. 5
Fig. 5

Plots of the calculated transmitted intensity versus time in a three-level medium in which an initial sublevel coherence is present as measured by the magnitude of the coherence, B, in Eq. (31). W = −1 in all cases. In (a) the system is in a stationary state (i.e., no coherence). In (b) the system is in a nonstationary state and the magnitude of the coherence is given by B = 1; in (c) the coherence is increased to B = 2. The insets show the corresponding spectrum for each plot. Note that for increasing coherence there is a corresponding increase in the number of harmonic sidebands in the spectrum. The harmonics are labeled by their corresponding periods, which are approximately submultiples of the fundamental wave packet period of 314 fs.

Fig. 6
Fig. 6

Schematic diagram of the experimental setup. The input pulse is split into the pump and the gate pulses at the beam splitter (BS). The dispersion of the input pump pulse is adjusted by a dispersive delay line (DDL) in the form of a prism pair, and the pulse delay is varied by a roof mirror mounted upon a programmable motion controller (PMC). It is then focused (FL) into the heat pipe (HP), which contains the Na2. The emission from the sample is collected and focused onto the BBO crystal by two parabolic mirrors (PM’s). The gate pulse is delayed (DL) and collimated (CL) before being focused onto the BBO by the second PM. The resulting upconverted light is imaged with a quartz lens (QL) into the monochromator (M/C) and detected with a photomultiplier tube (PMT).

Fig. 7
Fig. 7

Measured transmitted intensity of FID from Na2. The input is a 60-fs pulse whose spectrum overlaps approximately four vibrational levels in the excited state.

Fig. 8
Fig. 8

Experimental setup for adjusting and monitoring the input pulse spectrum. The pulse is spectrally filtered in the pulse sharper; its width and central frequency are controlled by the adjustable slit (SL). We measure the spectrum of the pulse by picking off the beam with a mirror mounted upon a kinematic mount (KM) and sending it to the monochromator (M/C). The signal is detected with a photodiode (PD), amplified in a transimpedance amplifier (A), and digitized (A/D). The signal transmitted through the sample (S) is chopped (C) to increase the signal-to-noise ratio. The signal detection is the same as in Fig. 6. PMC, programmable motion controller; BS, beam splitter; DL, delay line.

Fig. 9
Fig. 9

(a) Measured FID from Na2 using a spectrally narrow pump. Also shown are (b) the cross correlation of the input and the gate pulses, showing the >700-fs input pulse width, and (c) the ∼1.5-nm-wide input pulse spectrum. The vertical lines in (c) denote the wavelengths corresponding to transitions to two excited-state vibrational levels.

Fig. 10
Fig. 10

Calculated intensity transmitted through a 150-mm-long sample of two-level absorbers. The sample consists of TLS’s with three different absorption coefficients [α = 0.05, 0.03, 0.005 (mm ps)−1]. This leads to a smoothing effect on the signal.

Equations (36)

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τ P n ( z , τ ) = Γ n P n ( z , τ ) i 2 χ ( z , τ ) × [ W n 1 + k n , 1 N σ n k ( τ ) ] ,
z χ ( z , τ ) i μ j = 2 N P j ( z , τ ) ,
μ = 2 π N | d | 2 ћ λ ,
τ P ˜ n ( s , τ ) = Γ n P ˜ n ( s , τ ) i κ n ( τ ) 2 s χ ( 0 , τ ) μ κ n ( τ ) 2 s j = 2 N P ˜ j ( s , τ ) ,
κ n ( τ ) = [ W n 1 + k n , 1 N σ n k ( τ ) ]
τ P ˜ ( s , τ ) = M ( τ ) · P ˜ ( s , τ ) + ξ ( τ ) ,
P ˜ = ( P ˜ 2 P ˜ n ) ,
ξ = i χ ( 0 , τ ) 2 s ( κ 2 κ n ) .
M ( τ ) = Γ + β ( τ ) B ,
Γ = [ Γ 2 0 0 0 0 0 0 Γ n ] ,
β ( τ ) = μ 2 s [ κ 2 0 0 0 0 0 0 κ n ] ,
B = [ 1 1 1 1 ] .
P ˜ ( s , τ ) = exp [ Λ ( τ ) ] P ˜ ( s , 0 ) + 0 τ d τ exp [ Λ ( τ ) Λ ( τ ) ] ξ ( τ ) ,
Λ ( τ ) = 0 τ d τ M ( τ ) .
χ ˜ ( s , τ ) = χ ( 0 , τ ) s + i μ s [ 1 T · P ˜ ( s , τ ) ] ,
1 T · P ˜ ( s , τ ) = j = 2 N P ˜ j ( s , τ )
χ ˜ ( s , τ ) = χ ( 0 , τ ) s + i μ s 1 T · { exp [ Λ ( τ ) ] · P ˜ ( s , 0 ) + 0 τ d τ exp [ Λ ( τ ) Λ ( τ ) ] · ξ ( τ ) } .
M ( τ ) = C ( τ ) + D ( τ ) ,
C ( τ ) = [ β ( τ ) γ ] · I ,
D ( τ ) = β ( τ ) [ i Δ 2 β ( τ ) 1 1 i Δ 2 β ( τ ) ] ,
β ( τ ) = μ κ 2 s
exp ( C + D ) = exp ( C ) exp ( D ) exp ( 1 2 [ C , D ] ) .
exp [ Λ ( τ ) ] = exp [ 0 τ d τ C ( τ ) ] exp [ 0 τ d τ D ( τ ) ] .
exp [ Λ ( z ) ] = n = 0 [ 0 z d z 1 0 z 1 d z 2 0 z n 1 d z n D ( z 1 ) × D ( z 2 ) D ( z n ) ] .
Δ | μ κ 2 s | ,
U ( ± ) T exp [ 0 τ d τ D ( τ ) ] U ( ± ) exp ( ± i Δ τ ) ,
χ ˜ ( s , τ ) = χ ( 0 , τ ) s + i μ j exp ( Γ j τ ) P ˜ j ( s , 0 ) × { exp [ μ I ( 0 , τ ) 2 s ] s } μ 2 0 τ d τ χ ( 0 , τ ) × j exp [ Γ j ( τ τ ) ] κ j ( τ ) { exp [ μ I ( 0 , τ ) 2 s ] s }
I ( τ , τ ) = τ τ d τ κ ( τ ) .
χ ( z , τ ) = χ ( 0 , τ ) + i μ 0 z d z J 0 { [ 2 μ ( z z ) I ( 0 , τ ) ] 1 / 2 } × j P j ( z , 0 ) exp ( Γ j τ ) μ z 0 τ d τ χ ( 0 , τ ) κ ( τ ) J 1 { [ 2 μ z I ( τ , τ ) ] 1 / 2 } 2 μ z I ( τ , τ ) × j exp [ Γ j ( τ τ ) ] ,
κ ( τ ) = [ W + B cos ( Ω τ ) ] ,
χ ( z , τ ) = χ ( 0 , τ ) + i μ 0 z d z J 0 ( 2 μ ( z z ) { [ W τ B sin ( Ω τ ) Ω ] } 1 / 2 ) × j P j ( z , 0 ) exp ( Γ j τ ) + μ z 0 z d τ χ ( 0 , τ )[ W + B cos ( Ω τ ) ] × J 1 A ( z ) A ( z ) j exp [ Γ j ( τ τ ) ] ,
A ( z ) = ( 2 μ z { W ( τ τ ) B [ sin ( Ω τ ) sin ( Ω τ ) Ω ] } ) 1 / 2 .
χ ˜ ( z , s ) = i μ 0 z d z n P n ( z , 0 ) s Γ n exp [ j a j ( z z ) s G j ] + χ ˜ ( 0 , s ) exp [ j α j z Γ j ] ,
α j = μ W j 1 2 .
χ ( z , τ ) = χ ( 0 , τ ) + i μ 0 z d z J 0 [ 2 μ W ( z z ) τ ] 1 / 2 × n P n ( z , 0 ) exp ( Γ n τ ) + μ W z 0 τ d τ χ ( 0 , τ ) J 1 { [ 2 μ W z ( τ τ ) ] 1 / 2 } [ 2 μ W z ( τ τ ) ] 1 / 2 × n exp [ Γ n ( τ τ ) ] ,
Δ | μ W z 2 , |

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