Abstract

The evolution of the pulse centroid velocity of the Poynting vector for both ultrawideband rectangular and ultrashort Gaussian envelope pulses is presented as a function of the propagation distance in a dispersive, absorptive dielectric material. The index of refraction of the material is described by the Lorentz–Lorenz formula in which a single-resonance Lorentz model is used to describe the mean molecular polarizability. The results show that, as the propagation distance increases above a value that is on the order of an absorption depth at the pulse carrier frequency, the centroid velocity of an ultrawideband/ultrashort pulse tends toward the rate at which the Brillouin precursor travels through the medium. For small propagation distances when the carrier frequency of the optical pulse lies in the absorption band of the material, the centroid velocity can take on superluminal and negative values.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Lord Rayleigh, “On progressive waves,” Proc. London Math. Soc. IX, 21–26 (1877).
    [CrossRef]
  2. E. Wolf, “Significance and measurability of the phase of a spatially coherent optical field,” Opt. Lett. 28, 5–6 (2003).
    [CrossRef] [PubMed]
  3. W. R. Hamilton, “Researches respecting vibration, connected with the theory of light,” Proc. R. Ir. Acad. 1, 341–349 (1839).
  4. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
  5. A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. (Leipzig) 44, 177–202 (1914).
    [CrossRef]
  6. H. A. Lorentz, The Theory of Electrons (Tuebner, Leipzig, 1909), Chap. IV; reprinted (Dover, New York, 1952).
  7. L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. (Leipzig) 44, 204–240 (1914).
  8. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
  9. P. Debye, “Näherungsformeln für die zylinderfunktionen für grosse werte des arguments und unbeschränkt verander liche werte des index,” Math. Ann. 67, 535–558 (1909).
    [CrossRef]
  10. H. Baerwald, “Über die fortpflanzung von signalen in disperdierenden medien,” Ann. Phys. 7, 731–760 (1930).
    [CrossRef]
  11. K. E. Oughstun, G. C. Sherman, “Optical pulse propagation in temporally dispersive Lorentz media,” J. Opt. Soc. Am. 65, 1224A (1975).
  12. F. W. J. Olver, “Why steepest descents?” SIAM (Soc. Ind. Appl. Math.) Rev. 12, 228–247 (1970).
  13. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
    [CrossRef]
  14. K. E. Oughstun, S. Shen, “Velocity of energy transport for a time-harmonic field in a multiple-resonance Lorentz medium,” J. Opt. Soc. Am. B 5, 2395–2398 (1988).
    [CrossRef]
  15. K. E. Oughstun, 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]
  16. S. Shen, K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B 6, 948–963 (1989).
    [CrossRef]
  17. K. E. Oughstun, 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]
  18. K. Oughstun, G. Sherman, Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).
  19. G. Sherman, 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]
  20. G. Sherman, K. E. Oughstun, “Energy velocity description of pulse propagation in absorbing, dispersive dielectrics,” J. Opt. Soc. Am. B 12, 229–247 (1995).
    [CrossRef]
  21. R. Smith, “The velocities of light,” Am. J. Phys. 38, 978–984 (1970).
    [CrossRef]
  22. M. Lisak, “Energy expressions and energy velocity for wave packets in an absorptive and dispersive medium,” J. Phys. A 9, 1145–1158 (1976).
    [CrossRef]
  23. J. Peatross, S. A. Glasgow, M. Ware, “Average energy flow of optical pulses in dispersive media,” Phys. Rev. Lett. 84, 2370–2373 (2000).
    [CrossRef] [PubMed]
  24. M. Ware, S. A. Glasgow, J. Peatross, “Role of group velocity in tracking field energy in linear dielectrics,” Opt. Express 9, 506–518 (2001).
    [CrossRef] [PubMed]
  25. K. E. Oughstun, J. E. Laurens, “Asymptotic descriptionof ultrashort electromagnetic pulse propagation in a linear, causally dispersive medium,” Radio Sci. 26, 245–258 (1991).
    [CrossRef]
  26. K. E. Oughstun, 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]
  27. K. E. Oughstun, “Pulse propagation in a linear, causally dispersive medium,” Proc. IEEE 79, 1394–1420 (1991).
    [CrossRef]
  28. E. T. Copson, Asymptotic Expansions (Cambridge U. Press, London, 1965), Chap. 2.
  29. H. Xiao, K. E. Oughstun, “Failure of the group-velocity description for ultrawideband pulse propagation in a causally dispersive, absorptive dielectric,” J. Opt. Soc. Am. B 16, 1773–1785 (1999).
    [CrossRef]
  30. K. E. Oughstun, N. A. Cartwright, “On the Lorentz–Lorenz formula and the Lorentz model of dielectric dispersion,” Opt. Express 11, 1541–1546 (2003).
    [CrossRef] [PubMed]
  31. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sect. 5.12.
  32. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Macmillan, New York, 1943), Sect. 5.2.
  33. Here, “dominates the field” refers to the amplitude of the Brillouin precursor being larger than the other contributions to the field when the entire propagated field is considered, as it is in calculating the centroid velocity. However, there are θ domains within the evolved pulse in which the Sommerfeld precursor or the pole contribution is the dominant contribution to the field.
  34. C. M. Balictsis, K. E. Oughstun, “Uniform asymptotic description of ultrashort Gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E 47, 3645–3669 (1993).
    [CrossRef]
  35. K. E. Oughstun, C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210–2213 (1996).
    [CrossRef] [PubMed]
  36. C. M. Balictsis, 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]

2003

2001

2000

J. Peatross, S. A. Glasgow, M. Ware, “Average energy flow of optical pulses in dispersive media,” Phys. Rev. Lett. 84, 2370–2373 (2000).
[CrossRef] [PubMed]

1999

1997

C. M. Balictsis, 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]

1996

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

1995

1993

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

1991

K. E. Oughstun, J. E. Laurens, “Asymptotic descriptionof ultrashort electromagnetic pulse propagation in a linear, causally dispersive medium,” Radio Sci. 26, 245–258 (1991).
[CrossRef]

K. E. Oughstun, “Pulse propagation in a linear, causally dispersive medium,” Proc. IEEE 79, 1394–1420 (1991).
[CrossRef]

1990

K. E. Oughstun, 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. Sherman, 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]

1976

M. Lisak, “Energy expressions and energy velocity for wave packets in an absorptive and dispersive medium,” J. Phys. A 9, 1145–1158 (1976).
[CrossRef]

1975

K. E. Oughstun, G. C. Sherman, “Optical pulse propagation in temporally dispersive Lorentz media,” J. Opt. Soc. Am. 65, 1224A (1975).

1970

F. W. J. Olver, “Why steepest descents?” SIAM (Soc. Ind. Appl. Math.) Rev. 12, 228–247 (1970).

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

R. Smith, “The velocities of light,” Am. J. Phys. 38, 978–984 (1970).
[CrossRef]

1930

H. Baerwald, “Über die fortpflanzung von signalen in disperdierenden medien,” Ann. Phys. 7, 731–760 (1930).
[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, 204–240 (1914).

1909

P. Debye, “Näherungsformeln für die zylinderfunktionen für grosse werte des arguments und unbeschränkt verander liche werte des index,” Math. Ann. 67, 535–558 (1909).
[CrossRef]

1877

Lord Rayleigh, “On progressive waves,” Proc. London Math. Soc. IX, 21–26 (1877).
[CrossRef]

1839

W. R. Hamilton, “Researches respecting vibration, connected with the theory of light,” Proc. R. Ir. Acad. 1, 341–349 (1839).

Baerwald, H.

H. Baerwald, “Über die fortpflanzung von signalen in disperdierenden medien,” Ann. Phys. 7, 731–760 (1930).
[CrossRef]

Balictsis, C. M.

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

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).

Brillouin, L.

L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. (Leipzig) 44, 204–240 (1914).

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

Cartwright, N. A.

Copson, E. T.

E. T. Copson, Asymptotic Expansions (Cambridge U. Press, London, 1965), Chap. 2.

Debye, P.

P. Debye, “Näherungsformeln für die zylinderfunktionen für grosse werte des arguments und unbeschränkt verander liche werte des index,” Math. Ann. 67, 535–558 (1909).
[CrossRef]

Glasgow, S. A.

M. Ware, S. A. Glasgow, J. Peatross, “Role of group velocity in tracking field energy in linear dielectrics,” Opt. Express 9, 506–518 (2001).
[CrossRef] [PubMed]

J. Peatross, S. A. Glasgow, M. Ware, “Average energy flow of optical pulses in dispersive media,” Phys. Rev. Lett. 84, 2370–2373 (2000).
[CrossRef] [PubMed]

Hamilton, W. R.

W. R. Hamilton, “Researches respecting vibration, connected with the theory of light,” Proc. R. Ir. Acad. 1, 341–349 (1839).

Laurens, J. E.

K. E. Oughstun, J. E. Laurens, “Asymptotic descriptionof ultrashort electromagnetic pulse propagation in a linear, causally dispersive medium,” Radio Sci. 26, 245–258 (1991).
[CrossRef]

Lisak, M.

M. Lisak, “Energy expressions and energy velocity for wave packets in an absorptive and dispersive medium,” J. Phys. A 9, 1145–1158 (1976).
[CrossRef]

Lorentz, H. A.

H. A. Lorentz, The Theory of Electrons (Tuebner, Leipzig, 1909), Chap. IV; reprinted (Dover, New York, 1952).

Loudon, R.

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

Olver, F. W. J.

F. W. J. Olver, “Why steepest descents?” SIAM (Soc. Ind. Appl. Math.) Rev. 12, 228–247 (1970).

Oughstun, K.

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

Oughstun, K. E.

K. E. Oughstun, N. A. Cartwright, “On the Lorentz–Lorenz formula and the Lorentz model of dielectric dispersion,” Opt. Express 11, 1541–1546 (2003).
[CrossRef] [PubMed]

H. Xiao, K. E. Oughstun, “Failure of the group-velocity description for ultrawideband pulse propagation in a causally dispersive, absorptive dielectric,” J. Opt. Soc. Am. B 16, 1773–1785 (1999).
[CrossRef]

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

G. Sherman, 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, 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, J. E. Laurens, “Asymptotic descriptionof ultrashort electromagnetic pulse propagation in a linear, causally dispersive medium,” Radio Sci. 26, 245–258 (1991).
[CrossRef]

K. E. Oughstun, “Pulse propagation in a linear, causally dispersive medium,” Proc. IEEE 79, 1394–1420 (1991).
[CrossRef]

K. E. Oughstun, 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, 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, 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, S. Shen, “Velocity of energy transport for a time-harmonic field in a multiple-resonance Lorentz medium,” J. Opt. Soc. Am. B 5, 2395–2398 (1988).
[CrossRef]

K. E. Oughstun, 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. Sherman, 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, G. C. Sherman, “Optical pulse propagation in temporally dispersive Lorentz media,” J. Opt. Soc. Am. 65, 1224A (1975).

Peatross, J.

M. Ware, S. A. Glasgow, J. Peatross, “Role of group velocity in tracking field energy in linear dielectrics,” Opt. Express 9, 506–518 (2001).
[CrossRef] [PubMed]

J. Peatross, S. A. Glasgow, M. Ware, “Average energy flow of optical pulses in dispersive media,” Phys. Rev. Lett. 84, 2370–2373 (2000).
[CrossRef] [PubMed]

Rayleigh, Lord

Lord Rayleigh, “On progressive waves,” Proc. London Math. Soc. IX, 21–26 (1877).
[CrossRef]

Shen, S.

Sherman, G.

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

G. Sherman, 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. Oughstun, G. Sherman, Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).

Sherman, G. C.

K. E. Oughstun, 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]

K. E. Oughstun, 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, 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]

K. E. Oughstun, G. C. Sherman, “Optical pulse propagation in temporally dispersive Lorentz media,” J. Opt. Soc. Am. 65, 1224A (1975).

Smith, R.

R. Smith, “The velocities of light,” Am. J. Phys. 38, 978–984 (1970).
[CrossRef]

Sommerfeld, A.

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

Stratton, J. A.

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

Ware, M.

M. Ware, S. A. Glasgow, J. Peatross, “Role of group velocity in tracking field energy in linear dielectrics,” Opt. Express 9, 506–518 (2001).
[CrossRef] [PubMed]

J. Peatross, S. A. Glasgow, M. Ware, “Average energy flow of optical pulses in dispersive media,” Phys. Rev. Lett. 84, 2370–2373 (2000).
[CrossRef] [PubMed]

Watson, G. N.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Macmillan, New York, 1943), Sect. 5.2.

Whittaker, E. T.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Macmillan, New York, 1943), Sect. 5.2.

Wolf, E.

Xiao, H.

Am. J. Phys.

R. Smith, “The velocities of light,” Am. J. Phys. 38, 978–984 (1970).
[CrossRef]

Ann. Phys.

H. Baerwald, “Über die fortpflanzung von signalen in disperdierenden medien,” Ann. Phys. 7, 731–760 (1930).
[CrossRef]

Ann. Phys. (Leipzig)

L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. (Leipzig) 44, 204–240 (1914).

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

J. Opt. Soc. Am.

K. E. Oughstun, G. C. Sherman, “Optical pulse propagation in temporally dispersive Lorentz media,” J. Opt. Soc. Am. 65, 1224A (1975).

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. A

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

M. Lisak, “Energy expressions and energy velocity for wave packets in an absorptive and dispersive medium,” J. Phys. A 9, 1145–1158 (1976).
[CrossRef]

Math. Ann.

P. Debye, “Näherungsformeln für die zylinderfunktionen für grosse werte des arguments und unbeschränkt verander liche werte des index,” Math. Ann. 67, 535–558 (1909).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

K. E. Oughstun, 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, 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, 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.

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

G. Sherman, 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]

J. Peatross, S. A. Glasgow, M. Ware, “Average energy flow of optical pulses in dispersive media,” Phys. Rev. Lett. 84, 2370–2373 (2000).
[CrossRef] [PubMed]

Proc. IEEE

K. E. Oughstun, “Pulse propagation in a linear, causally dispersive medium,” Proc. IEEE 79, 1394–1420 (1991).
[CrossRef]

Proc. London Math. Soc.

Lord Rayleigh, “On progressive waves,” Proc. London Math. Soc. IX, 21–26 (1877).
[CrossRef]

Proc. R. Ir. Acad.

W. R. Hamilton, “Researches respecting vibration, connected with the theory of light,” Proc. R. Ir. Acad. 1, 341–349 (1839).

Radio Sci.

K. E. Oughstun, J. E. Laurens, “Asymptotic descriptionof ultrashort electromagnetic pulse propagation in a linear, causally dispersive medium,” Radio Sci. 26, 245–258 (1991).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) Rev.

F. W. J. Olver, “Why steepest descents?” SIAM (Soc. Ind. Appl. Math.) Rev. 12, 228–247 (1970).

Other

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

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).

H. A. Lorentz, The Theory of Electrons (Tuebner, Leipzig, 1909), Chap. IV; reprinted (Dover, New York, 1952).

E. T. Copson, Asymptotic Expansions (Cambridge U. Press, London, 1965), Chap. 2.

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

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Macmillan, New York, 1943), Sect. 5.2.

Here, “dominates the field” refers to the amplitude of the Brillouin precursor being larger than the other contributions to the field when the entire propagated field is considered, as it is in calculating the centroid velocity. However, there are θ domains within the evolved pulse in which the Sommerfeld precursor or the pole contribution is the dominant contribution to the field.

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Real and imaginary parts of the complex index of refraction for the Lorentz–Lorenz modified single-resonance Lorentz model dielectric with plasma frequency b=20×1016 rad/s, phenomenological damping constant δ=0.28×1016 rad/s, and undamped resonance frequency ω0=(16×1032+b2/3)1/2 rad/s4.761×1016 rad/s. Note that the Lorentz–Lorenz formula shifts this undamped resonance frequency to the effective resonance frequency ω0eff4×1016 rad/s.

Fig. 2
Fig. 2

Real (solid curves) and imaginary (dashed curves) parts of the normal-incidence transmission coefficients for the vacuum/Lorenz–Lorentz medium interface for the electric (top) and magnetic (bottom) fields for the single-resonance Lorentz–Lorenz model dielectric whose frequency dispersion is presented in Fig. 1.

Fig. 3
Fig. 3

Relative centroid velocities in the Lorenz–Lorentz model dielectric for an input rectangular-envelope-modulated sine wave with carrier frequency ωc in the normal-dispersion region below the region of anomalous dispersion as a function of the relative absorption depth z/zd.

Fig. 4
Fig. 4

Relative centroid velocities in the Lorenz–Lorentz model dielectric for an input rectangular-envelope-modulated sine wave with carrier frequency ωc in the region of anomalous dispersion as a function of the relative absorption depth z/zd.

Fig. 5
Fig. 5

Top, initial (dashed curve) and propagated (solid curve) Poynting vectors due to the input rectangular-envelope-modulated pulse at 1.24 absorption depths with on-resonance carrier frequency ωc=4×1016 rad/s. Bottom, magnification of the region around the initial and propagated temporal centers of gravity, indicated by the circle and the asterisk, respectively.

Fig. 6
Fig. 6

Top, initial (dashed curve) and propagated (solid curve) Poynting vectors due to the input rectangular-envelope-modulated pulse at 1.25 absorption depths with on-resonance carrier frequency ωc=4×1016 rad/s. Bottom, a magnification of the region around the initial and propagated temporal centers of gravity, indicated by the circle and the asterisk, respectively.

Fig. 7
Fig. 7

Net group delay (solid curve) and reshaping delay (dashed curve) of a rectangular-envelope-modulated sine wave of ten oscillations with on-resonance carrier frequency ωc=4×1016 rad/s as a function of the relative propagation distance z/zd in the Lorenz–Lorentz model dielectric.

Fig. 8
Fig. 8

Relative centroid velocities in the Lorenz–Lorentz model dielectric for an input rectangular-envelope-modulated sine wave with carrier frequency ωc in the normal-dispersion region above the region of anomalous dispersion as a function of the relative propagation distance z/zd.

Fig. 9
Fig. 9

Relative centroid velocities in the Lorenz–Lorentz model dielectric for an input Gaussian-envelope-modulated cosine wave with carrier frequency ωc in the normal-dispersion region below the region of anomalous dispersion as a function of the relative propagation distance z/zd.

Fig. 10
Fig. 10

Relative centroid velocities in the Lorenz–Lorentz model dielectric for an input Gaussian-envelope-modulated cosine wave with carrier frequency ωc in the region of anomalous dispersion as a function of the relative propagation distance z/zd.

Fig. 11
Fig. 11

Relative centroid velocities in the Lorenz–Lorentz model dielectric for an input Gaussian-envelope-modulated cosine wave with carrier frequency ωc in the normal-dispersion region above the anomalous-dispersion region as a function of the relative propagation distance z/zd.

Fig. 12
Fig. 12

Relative instantaneous centroid velocities in the Lorenz–Lorentz model dielectric for an input rectangular-envelope-modulated sine wave with carrier frequency ωc in the normal-dispersion region below the anomalous-dispersion region as a function of the relative propagation distance z/zd.

Equations (26)

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

-tE2(r, t)dt-E2(r, t)dt-1,
S(r, t)=E(r, t)×H(r, t)
αm(ω)=-qe2/meω2-ω02+2iδω,
n(ω)=1+(2/30)Nαm(ω)1-(1/30)Nαm(ω)1/2,
τEE˜tE˜i=2n1(ω)n1(ω)+n2(ω),
τBB˜tB˜i=2n2(ω)n1(ω)+n2(ω),
vc=ztz-t0,
tz=zˆ·-tS(z, t)dtzˆ·-S(z, t)dt
tz-t0=Gz+R0,
Gz=-R(k˜)ωzS(z, ω)dω-S(z, ω)dω,
R0=i-ω[exp(I{k˜}z)E(0, ω)]×exp(I{k˜}z)H*(0, ω)dω-exp(2I{k˜}z)S(0, ω)dω-i-ω[E(0, ω)]×H*(0, ω)dω-S(0, ω)dω,
E(z, t)=-τE(ω)E˜(0, ω)exp[i(k˜(ω)z-ωt)]dω.
B(z, t)=1c-n(ω)τB(ω)E˜(0, ω)×exp[i(k˜(ω)z-ωt)]dω,
E(0, t)=u(t)sin(wct+ψ),
A(z, t)=12πRi exp(-iψ)ia-ia+τE(ω)u˜(ω-ωc)expzcϕ(ω, θ)dω
[2+k˜2(ω)]A˜(z, ω)=0,
ϕ(ω, θ)=icz[k˜(ω)z-ωt]=iω[n(ω)-θ],
A(z, t)=AS(z, t)+AB(z, t)+AC(z, t),
S(z, t)=(1/μ0)E(z, t)B(z, t)zˆ
A(z, t)=AS(z, t, 0)+AB(z, t, 0)+AC(z, t, 0)-[AS(z, t, T)+AB(z, t, T)+AC(z, t, T)].
A(z, t, T)=12πRexp(-iωcT)×ia-ia+τE(ω)u˜(ω-ωc)×expzcϕT(ω, θ)dω,
ϕT(ω, θ)=iω[n(ω)-θT]=iωn(ω)-c(t-T)z.
u(t)=exp-(t-t0)2T2
u˜(ω)=π1/2T exp(iωt0)exp-T2ω24,
A(z, t)=AS(z, t)+AB(z, t).
vic=limz2z1z2-z1t2-t1,

Metrics