Abstract

The accuracy of the group-velocity description of dispersive pulse propagation in a double-resonance Lorentz model dielectric is shown to decrease monotonically as the propagation distance increases, whereas the accuracy of the asymptotic description increases monotonically as the propagation distance increases above a single absorption depth in the medium at the pulse carrier frequency.

© 1999 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Sec. 10.4.
  2. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1990), Chap. 2.
  3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 3.
  4. C. Eckart, “The approximate solution of one-dimensional wave equations,” Rev. Mod. Phys. 20, 399–417 (1948).
    [CrossRef]
  5. G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1974), Chaps. 11 and 15.
  6. J. Jones, “On the propagation of a pulse through a dispersive medium,” Am. J. Phys. 42, 43–46 (1974).
    [CrossRef]
  7. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Secs. 7.8–7.9.
  8. R. B. Lindsay, Mechanical Radiation (McGraw-Hill, New York, 1960), Sec. 3.8.
  9. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), Chap. 3.
  10. E. G. Sauter, Nonlinear Optics (Wiley, New York, 1997), Chap. 9.
  11. G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Chaps. 2–5.
  12. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, New York, 1992), Chaps. 1 and 2.
  13. A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. (Leipzig) 44, 177–202 (1914).
    [CrossRef]
  14. L. Brillouin, “Über die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. (Leipzig) 44, 203–240 (1914).
    [CrossRef]
  15. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
  16. P. F. Curley, Ch. Spielmann, T. Brabec, F. Krausz, E. Wintner, and A. J. Schmidt, “Operation of a femtosecond Ti:sapphire solitary laser in the vicinity of zero group-delay dispersion,” Opt. Lett. 18, 54–56 (1993).
    [CrossRef] [PubMed]
  17. M. T. Asaki, C. P. Huang, D. Garvey, J. Zhou, H. C. Kapteyn, and M. M. Murname, “Generation of 11-fs pulses from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 18, 977–979 (1993).
    [CrossRef] [PubMed]
  18. J. Zhou, G. Taft, C. P. Huang, M. M. Murnane, H. C. Kapteyn, and I. Christov, “Pulse evolution in a broad-bandwidth Ti:sapphire laser,” Opt. Lett. 19, 1149–1151 (1994).
    [CrossRef] [PubMed]
  19. A. Stingl, M. Lenzner, Ch. Spielmann, and F. Krausz, “Sub-10 fs mirror-dispersion-controlled Ti:sapphire laser,” Opt. Lett. 20, 602–604 (1995).
    [CrossRef] [PubMed]
  20. M. Lenzner, Ch. Spielmann, E. Wintner, F. Krausz, and A. J. Schmidt, “Sub-20 fs, kilohertz-repetition-rate Ti:sapphire amplifier,” Opt. Lett. 20, 1397–1399 (1995).
    [CrossRef] [PubMed]
  21. G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. 20, 1562–1564 (1995).
    [CrossRef] [PubMed]
  22. S. Backus, J. Peatross, C. P. Huang, M. M. Murnane, and H. C. Kapteyn, “Ti:sapphire amplifier producing millijoule-level, 21-fs pulses at 1 kHz,” Opt. Lett. 20, 2000–2002 (1995).
    [CrossRef] [PubMed]
  23. S. H. Ashworth, M. Joschko, M. Woerner, E. Riedle, and T. Elsaesser, “Generation of 16-fs pulses at 425 nm by extracavity frequency doubling of a mode-locked Ti:sapphire laser,” Opt. Lett. 20, 2120–2122 (1995).
    [CrossRef] [PubMed]
  24. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
    [CrossRef]
  25. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).
  26. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 5.12.
  27. A. B. Shvartsburg, Time Domain Optics of Ultrashort Waveforms (Oxford U. Press, Oxford, 1996).
  28. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
    [CrossRef]
  29. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1394–1420 (1989).
    [CrossRef]
  30. J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” J. Eur. Opt. Soc. A 7, 575–602 (1998).
  31. S. Shen and K. E. Oughstun, “Dispersive pulse propagation in a double resonance Lorentz medium,” J. Opt. Soc. Am. B 6, 948–963 (1989).
    [CrossRef]
  32. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium,” Phys. Rev. A 41, 6090–6113 (1990).
    [CrossRef] [PubMed]
  33. C. M. Balictsis and K. E. Oughstun, “Uniform asymptotic description of ultrashort gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E 47, 3645–3669 (1993).
    [CrossRef]
  34. K. E. Oughstun, “Noninstantaneous, finite rise-time effects on the precursor field formation in linear dispersive pulse propagation,” J. Opt. Soc. Am. A 12, 1715–1729 (1995).
    [CrossRef]
  35. K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210–2213 (1996).
    [CrossRef] [PubMed]
  36. C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55, 1910–1921 (1997).
    [CrossRef]
  37. D. Anderson, J. Askne, and M. Lisak, “Wave packets in an absorptive and strongly dispersive medium,” Phys. Rev. A 12, 1546–1552 (1975).
    [CrossRef]
  38. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), Chap. 1.
  39. K. E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion relations and phase retrieval in optical spectroscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1997), Vol. XXXVII, pp. 57–94.
  40. E. T. Copson, Asymptotic Expansions (Cambridge U. Press, Cambridge, 1971), Chap. 2.
  41. R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).
    [CrossRef] [PubMed]
  42. J. Van Bladel, Singular Electromagnetic Fields and Sources (Oxford U. Press, Oxford, 1991), p. 3.
  43. D. C. Champeney, A Handbook of Fourier Theorems (Cambridge U. Press, Cambridge, 1990), theorem 10.7.
  44. K. E. Oughstun, “Dynamical structure of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin and L.B. Felsen, eds. (Plenum, New York, 1995), pp. 257–272.
  45. G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
    [CrossRef]
  46. G. C. Sherman and K. E. Oughstun, “Energy velocity description of pulse propagation in absorbing, dispersive dielectrics,” J. Opt. Soc. Am. B 12, 229–247 (1995).
    [CrossRef]
  47. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
    [CrossRef]
  48. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 6.
  49. Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998); see pp. 92–93.
    [CrossRef]

1998

J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” J. Eur. Opt. Soc. A 7, 575–602 (1998).

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998); see pp. 92–93.
[CrossRef]

1997

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55, 1910–1921 (1997).
[CrossRef]

K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[CrossRef]

1996

K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210–2213 (1996).
[CrossRef] [PubMed]

1995

1994

1993

1991

1990

K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium,” Phys. Rev. A 41, 6090–6113 (1990).
[CrossRef] [PubMed]

1989

1988

1981

G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

1975

D. Anderson, J. Askne, and M. Lisak, “Wave packets in an absorptive and strongly dispersive medium,” Phys. Rev. A 12, 1546–1552 (1975).
[CrossRef]

1974

J. Jones, “On the propagation of a pulse through a dispersive medium,” Am. J. Phys. 42, 43–46 (1974).
[CrossRef]

1948

C. Eckart, “The approximate solution of one-dimensional wave equations,” Rev. Mod. Phys. 20, 399–417 (1948).
[CrossRef]

1914

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. (Leipzig) 44, 177–202 (1914).
[CrossRef]

L. Brillouin, “Über die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. (Leipzig) 44, 203–240 (1914).
[CrossRef]

Anderson, D.

D. Anderson, J. Askne, and M. Lisak, “Wave packets in an absorptive and strongly dispersive medium,” Phys. Rev. A 12, 1546–1552 (1975).
[CrossRef]

Asaki, M. T.

Ashworth, S. H.

Askne, J.

D. Anderson, J. Askne, and M. Lisak, “Wave packets in an absorptive and strongly dispersive medium,” Phys. Rev. A 12, 1546–1552 (1975).
[CrossRef]

Backus, S.

Balictsis, C. M.

C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55, 1910–1921 (1997).
[CrossRef]

K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210–2213 (1996).
[CrossRef] [PubMed]

C. M. Balictsis and K. E. Oughstun, “Uniform asymptotic description of ultrashort gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E 47, 3645–3669 (1993).
[CrossRef]

Brabec, T.

Brillouin, L.

L. Brillouin, “Über die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. (Leipzig) 44, 203–240 (1914).
[CrossRef]

Cavallari, M.

Christov, I.

Curley, P. F.

Driscoll, T. J.

Eckart, C.

C. Eckart, “The approximate solution of one-dimensional wave equations,” Rev. Mod. Phys. 20, 399–417 (1948).
[CrossRef]

Elsaesser, T.

Gale, G. M.

Garvey, D.

Hache, F.

Hagness, S. C.

Huang, C. P.

Jones, J.

J. Jones, “On the propagation of a pulse through a dispersive medium,” Am. J. Phys. 42, 43–46 (1974).
[CrossRef]

Joschko, M.

Joseph, R. M.

Kapteyn, H. C.

Kivshar, Y. S.

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998); see pp. 92–93.
[CrossRef]

Krausz, F.

Lenzner, M.

Lisak, M.

D. Anderson, J. Askne, and M. Lisak, “Wave packets in an absorptive and strongly dispersive medium,” Phys. Rev. A 12, 1546–1552 (1975).
[CrossRef]

Luther-Davies, B.

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998); see pp. 92–93.
[CrossRef]

Murname, M. M.

Murnane, M. M.

Oughstun, K. E.

J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” J. Eur. Opt. Soc. A 7, 575–602 (1998).

K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[CrossRef]

C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55, 1910–1921 (1997).
[CrossRef]

K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210–2213 (1996).
[CrossRef] [PubMed]

K. E. Oughstun, “Noninstantaneous, finite rise-time effects on the precursor field formation in linear dispersive pulse propagation,” J. Opt. Soc. Am. A 12, 1715–1729 (1995).
[CrossRef]

G. C. Sherman and K. E. Oughstun, “Energy velocity description of pulse propagation in absorbing, dispersive dielectrics,” J. Opt. Soc. Am. B 12, 229–247 (1995).
[CrossRef]

C. M. Balictsis and K. E. Oughstun, “Uniform asymptotic description of ultrashort gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E 47, 3645–3669 (1993).
[CrossRef]

K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium,” Phys. Rev. A 41, 6090–6113 (1990).
[CrossRef] [PubMed]

S. Shen and K. E. Oughstun, “Dispersive pulse propagation in a double resonance Lorentz medium,” J. Opt. Soc. Am. B 6, 948–963 (1989).
[CrossRef]

K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1394–1420 (1989).
[CrossRef]

K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
[CrossRef]

G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

Peatross, J.

Riedle, E.

Schmidt, A. J.

Shen, S.

Sherman, G. C.

Smith, P. D.

J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” J. Eur. Opt. Soc. A 7, 575–602 (1998).

Solhaug, J. A.

J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” J. Eur. Opt. Soc. A 7, 575–602 (1998).

Sommerfeld, A.

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. (Leipzig) 44, 177–202 (1914).
[CrossRef]

Spielmann, Ch.

Stamnes, J. J.

J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” J. Eur. Opt. Soc. A 7, 575–602 (1998).

Stingl, A.

Taflove, A.

Taft, G.

Wintner, E.

Woerner, M.

Xiao, H.

K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[CrossRef]

Zhou, J.

Am. J. Phys.

J. Jones, “On the propagation of a pulse through a dispersive medium,” Am. J. Phys. 42, 43–46 (1974).
[CrossRef]

Ann. Phys. (Leipzig)

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. (Leipzig) 44, 177–202 (1914).
[CrossRef]

L. Brillouin, “Über die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. (Leipzig) 44, 203–240 (1914).
[CrossRef]

J. Eur. Opt. Soc. A

J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” J. Eur. Opt. Soc. A 7, 575–602 (1998).

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).
[CrossRef] [PubMed]

P. F. Curley, Ch. Spielmann, T. Brabec, F. Krausz, E. Wintner, and A. J. Schmidt, “Operation of a femtosecond Ti:sapphire solitary laser in the vicinity of zero group-delay dispersion,” Opt. Lett. 18, 54–56 (1993).
[CrossRef] [PubMed]

M. T. Asaki, C. P. Huang, D. Garvey, J. Zhou, H. C. Kapteyn, and M. M. Murname, “Generation of 11-fs pulses from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 18, 977–979 (1993).
[CrossRef] [PubMed]

J. Zhou, G. Taft, C. P. Huang, M. M. Murnane, H. C. Kapteyn, and I. Christov, “Pulse evolution in a broad-bandwidth Ti:sapphire laser,” Opt. Lett. 19, 1149–1151 (1994).
[CrossRef] [PubMed]

A. Stingl, M. Lenzner, Ch. Spielmann, and F. Krausz, “Sub-10 fs mirror-dispersion-controlled Ti:sapphire laser,” Opt. Lett. 20, 602–604 (1995).
[CrossRef] [PubMed]

M. Lenzner, Ch. Spielmann, E. Wintner, F. Krausz, and A. J. Schmidt, “Sub-20 fs, kilohertz-repetition-rate Ti:sapphire amplifier,” Opt. Lett. 20, 1397–1399 (1995).
[CrossRef] [PubMed]

G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. 20, 1562–1564 (1995).
[CrossRef] [PubMed]

S. Backus, J. Peatross, C. P. Huang, M. M. Murnane, and H. C. Kapteyn, “Ti:sapphire amplifier producing millijoule-level, 21-fs pulses at 1 kHz,” Opt. Lett. 20, 2000–2002 (1995).
[CrossRef] [PubMed]

S. H. Ashworth, M. Joschko, M. Woerner, E. Riedle, and T. Elsaesser, “Generation of 16-fs pulses at 425 nm by extracavity frequency doubling of a mode-locked Ti:sapphire laser,” Opt. Lett. 20, 2120–2122 (1995).
[CrossRef] [PubMed]

Phys. Rep.

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998); see pp. 92–93.
[CrossRef]

Phys. Rev. A

D. Anderson, J. Askne, and M. Lisak, “Wave packets in an absorptive and strongly dispersive medium,” Phys. Rev. A 12, 1546–1552 (1975).
[CrossRef]

K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium,” Phys. Rev. A 41, 6090–6113 (1990).
[CrossRef] [PubMed]

Phys. Rev. E

C. M. Balictsis and K. E. Oughstun, “Uniform asymptotic description of ultrashort gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E 47, 3645–3669 (1993).
[CrossRef]

C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55, 1910–1921 (1997).
[CrossRef]

Phys. Rev. Lett.

G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210–2213 (1996).
[CrossRef] [PubMed]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[CrossRef]

Rev. Mod. Phys.

C. Eckart, “The approximate solution of one-dimensional wave equations,” Rev. Mod. Phys. 20, 399–417 (1948).
[CrossRef]

Other

G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1974), Chaps. 11 and 15.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Sec. 10.4.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1990), Chap. 2.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 3.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Secs. 7.8–7.9.

R. B. Lindsay, Mechanical Radiation (McGraw-Hill, New York, 1960), Sec. 3.8.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), Chap. 3.

E. G. Sauter, Nonlinear Optics (Wiley, New York, 1997), Chap. 9.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Chaps. 2–5.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, New York, 1992), Chaps. 1 and 2.

K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 5.12.

A. B. Shvartsburg, Time Domain Optics of Ultrashort Waveforms (Oxford U. Press, Oxford, 1996).

L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), Chap. 1.

K. E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion relations and phase retrieval in optical spectroscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1997), Vol. XXXVII, pp. 57–94.

E. T. Copson, Asymptotic Expansions (Cambridge U. Press, Cambridge, 1971), Chap. 2.

J. Van Bladel, Singular Electromagnetic Fields and Sources (Oxford U. Press, Oxford, 1991), p. 3.

D. C. Champeney, A Handbook of Fourier Theorems (Cambridge U. Press, Cambridge, 1990), theorem 10.7.

K. E. Oughstun, “Dynamical structure of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin and L.B. Felsen, eds. (Plenum, New York, 1995), pp. 257–272.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 6.

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Figures (15)

Fig. 1
Fig. 1

Frequency dispersion of (a) the real and (b) the imaginary parts of the double-resonance Lorentz model of the complex index of refraction of a fluoride glass with infrared (ω0, b0, δ0) and near-ultraviolet (ω2, b2, δ2) resonance lines (solid curves). The three-, four-, and ten-term Taylor series approximations about the minimum dispersion point ωc=ωmin between the two resonance lines are depicted.

Fig. 2
Fig. 2

rms error over the frequency interval [ωc-Δω, ωc+Δω] of the Taylor series approximation of the real part of the complex index of refraction for the double-resonance Lorentz model of the complex refractive index of a fluoride glass with infrared (ω0, b0, δ0) and near-ultraviolet (ω2, b2, δ2) resonance lines about the minimum dispersion point ωc=ωmin between the two resonance lines as a function of the number of terms. Notice that the ordinate values for the data in (a) have been magnified by a factor of 10.

Fig. 3
Fig. 3

Frequency dispersion of (a) the real and (b) the imaginary parts of the complex wave number (solid curves) for the double-resonance Lorentz model dielectric of Fig. 1. The three-, four-, and ten-term Taylor series approximations about the minimum dispersion point ωc=ωmin between the two resonance lines are depicted.

Fig. 4
Fig. 4

Detail of the frequency dispersion of (a) the real and (b) the imaginary parts of the complex wave number of Fig. 3.

Fig. 5
Fig. 5

Frequency dispersion of (a) the real and (b) the imaginary parts of the double-resonance Lorentz model of the complex index of refraction of the reduced-loss fluoride glass with infrared (ω0, b0, δ0/10) and near-ultraviolet (ω2, b2, δ2/10) resonance lines (solid curves). The Taylor series approximations about the minimum dispersion point ωc=ωmin between the two resonance lines are depicted.

Fig. 6
Fig. 6

Frequency dispersion of the real and the imaginary parts of the double-resonance Lorentz model of the complex index of refraction of a fluoride glass with an infrared and a near-ultraviolet resonance line (solid curves). For comparison, the relative magnitude (drawn to an arbitrary scale) of the input pulse spectrum is illustrated for single-cycle, five-cycle, and ten-cycle Van Bladel envelope pulses.

Fig. 7
Fig. 7

Impulse response of the double-resonance Lorentz model of a fluoride glass at the fixed propagation distance z=3.24zd, where zd=1/α(ωmin) is the e-1 absorption depth at the minimum dispersion point ωmin=1.615×1015 r/s indicated in Figs. 1 and 3. The field evolution is described as a function of the dimensionless space–time parameter θ=ct/z, which, at fixed z, is a dimensionless time parameter.

Fig. 8
Fig. 8

Dynamic field evolution of the propagated plane-wave field that results from an input Heaviside unit step-function–modulated signal with carrier frequency ωc=1.615×1015 r/s at the fixed propagation distance z=3α-1(ωc) in the double-resonance Lorentz model dielectric whose frequency dispersion is depicted in Figs. 1 and 3. Solid curves, the exact propagated field behavior for the full dispersion relation; dotted curves, the numerically determined field behavior with (a) the quadratic dispersion approximation and (b) the cubic dispersion approximation.

Fig. 9
Fig. 9

Dynamic field evolution of the propagated plane-wave field that results from an input rectangular envelope modulated pulse with initial pulse width T=38.9 fs and carrier frequency ωc=1.615×1015 r/s at the fixed propagation distance z=3α-1(ωc) in the double-resonance Lorentz model dielectric whose frequency dispersion is depicted in Figs. 1 and 3. Solid curves, the exact propagated field behavior with the full dispersion relation; dotted curves, the numerically determined field behavior with (a) the quadratic dispersion approximation and (b) the cubic dispersion approximation.

Fig. 10
Fig. 10

Numerically determined propagated field evolution that results from an input unit amplitude single-cycle pulse (τ=3.89 fs) with the exact dispersion model (solid curves) and the cubic dispersion approximation (dotted curves) of the double-resonance Lorentz medium as a function of dimensionless time parameter θ=ct/z at several fixed propagation distances z<zd into the dispersive, lossy medium.

Fig. 11
Fig. 11

Numerically determined propagated field evolution that results from an input unit amplitude single-cycle pulse (τ=3.89 fs) with the exact dispersion model (solid curves) and the cubic dispersion approximation (dotted curves) of the double-resonance Lorentz medium as a function of the dimensionless time parameter θ=ct/z at several fixed propagation distances zzd into the dispersive, lossy medium.

Fig. 12
Fig. 12

Same as Fig. 11 but for an input unit amplitude five-cycle pulse (τ=19.45 fs).

Fig. 13
Fig. 13

Same as Fig. 11 but for an input unit amplitude ten-cycle pulse (τ=38.9 fs).

Fig. 14
Fig. 14

Same as Fig. 11 but for an input unit amplitude 20-cycle pulse (τ=77.8 fs).

Fig. 15
Fig. 15

Comparison of the numerically determined propagated field evolution that results from an input unit amplitude single-cycle pulse (τ=3.89 fs) with the exact dispersion model (solid curves) and the nonuniform asymptotic approximation of that pulse evolution with numerically determined saddle point locations (dotted curves) at several absorption depths into the double-resonance Lorentz medium.

Equations (26)

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A(z, t)=12πRi exp(-iψ)ia-ia+u˜(ω-ωc)×exp{i[k˜(ω)z-ωt]}dω
[2+k˜2(ω)]A˜(z, ω)=0,
k˜(ω)=β (ω)+iα(ω)=ωn(ω)/c
k˜(ω)=j=01j!k˜(j)(ωc)(ω-ωc)j,
k˜(ω)k˜(ωc)+k˜(1)(ωc)(ω-ωc)+12!k˜(2)(ωc)(ω-ωc)2.
A(z, t)Rexp{i[k˜(ωc)z-ωct+3π/4-ψ]}[2πk˜(2)(ωc)z]1/2×-u(t)exp-i[k˜(1)(ωc)z+t-t]22k˜(2)(ωc)zdtR1[2πk˜(2)(ωc)z]1/2u˜t-k˜(1)(ωc)zk˜(2)(ωc)z×expik˜(ωc)z-ωct+3π/4-ψ-[k˜(1)(ωc)z-t]22k˜(2)(ωc)z,
vg(ω)=[β(ω)/ω]-1,
k˜(ω)k˜(ωc)+k˜(1)(ωc)(ω-ωc)+12!k˜(2)(ωc)(ω-ωc)2+13!k˜(3)(ωc)(ω-ωc)3
n(ω)=1-b02ω2-ω02+2iδ0ω-b22ω2-ω22+2iδ2ω1/2,
A(z, t)=12πRia-ia+ f˜(ω)expzcϕ(ω, θ)dω,
ϕ(ω, θ)=i(c/z)[k˜(ω)z-ωt]=iω[n(ω)-θ]
A(z, t)AS(z, t)+Am(z, t)+AB(z, t)
Aj(z, t)=ajc2πz1/2Ri[-ϕ(2)(ωSPj, θ)]-1/2×expzcϕ(ωSPj, θ)
A(z, t)AS(z, t)+Am(z, t)+AB(z, t)+Ac(z, t)
Aj(z, t)=ajc2πz1/2Riu˜(ωSPj-ωc)[-ϕ(2)(ωSPj, θ)]1/2×expzcϕ(ωSPj, θ)
Ac(z, t)R(iu(t-z/v)exp{i[k˜(ωc)z-ωct]}),
u(t)=exp1+τ24t(t-τ),0tτ
A(z, t)=12πRi exp(-iψ)Cu˜(ω-ωc)×expzcϕ(ω, θ)dω
A(z, t)=AS(z, t)+Am(z, t)+AB(z, t)
AS(z, t)c2πz1/2Rexp(-iψ)×j=±[-ϕ(2)(ωdj, θ)]-1/2×u˜(ωdj-ωc)expzcϕ(ωdj, θ),
AB(z, t)c2πz1/2Rexp(-iψ)[-ϕ(2)(ωn+, θ)]-1/2×u˜(ωn+-ωc)expzcϕ(ωn+, θ)
AB(z, t1)Γ(1/3)2π31/62cz|ϕ(3)(ωn, θ1)|1/3×Ri exp(-iψ)u˜(ωn-ωc)× expzcϕ(ωn, θ1)
AB(z, t)c2πz1/2Rexp(-iψ)j=±[-ϕ(2)(ωnj, θ)]-1/2×u˜(ωnj-ωc)expzcϕ(ωnj, θ)
Am(z, t)c2πz1/2Rexp(-iψ)j[-ϕ(2)(ωmj, θ)]-1/2×u˜(ωmj-ωc)expzcϕ(ωmj, θ)
Am(z, t)R([2πk˜(2)(ωm+)]-1/2u˜(ωm+-ωc)×exp{i[k˜(ωm+)z-ωm+t-ψ-π/4]}+[2πk˜(2)(ωm-)]-1/2 u˜(ωm--ωc)×exp{i[k˜(ωm-)z-ωm-t-ψ-π/4]}).
Am(z, t)R 2πk˜(2)(ωm+)1/2u˜(ωm+-ωc)×exp{i[k˜(ωm+)z-ωm+t-ψ-π/4]},

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