Abstract

A model-independent theorem demonstrates how a causal linear dielectric medium responds to the instantaneous spectrum, that is, the spectrum of the electric field pulse that is truncated at each new instant (as a given locale in the medium experiences the pulse). This process leads the medium to exchange energy with the front of a pulse differently than with the back as the instantaneous spectrum laps onto or off of nearby resonances. So-called superluminal pulse propagation in either absorbing or amplifying media as well as highly subluminal pulse propagation are understood qualitatively and quantitatively within this context.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34–R37 (1993).
    [CrossRef] [PubMed]
  2. E. L. Bolda, J. C. Garrison, R. Y. Chiao, “Optical pulse propagation at negative group velocities due to a nearby gain line,” Phys. Rev. A 49, 2938–2947 (1994).
    [CrossRef] [PubMed]
  3. R. Y. Chiao, A. M. Steinberg, “Tunneling times and superluminality,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1997), Vol. 37, pp. 347–406.
  4. L. J. Wang, A. Kuzmmich, A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
    [CrossRef] [PubMed]
  5. C. G. B. Garrett, D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
    [CrossRef]
  6. S. Chu, S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
    [CrossRef]
  7. L. V. Hau, S. E. Haris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
    [CrossRef]
  8. C. H. Page, “Instantaneous power spectra,” J. Appl. Phys. 23, 103–106 (1952).
    [CrossRef]
  9. M. B. Priestley, “Power spectral analysis of nonstationary random processes,” J. Sound Vib. 6, 86–97 (1967).
    [CrossRef]
  10. J. H. Eberly, K. Wodkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am. 67, 1252–1260 (1977).
    [CrossRef]
  11. The authors are preparing the following paper for publication: “Poynting’s theorem and luminal energy transport in causal dielectrics.”
  12. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998), pp. 323, 325, 334.
  13. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
  14. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), pp. 667–670.
  15. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
    [CrossRef]
  16. G. C. Sherman, K. E. Oughstun, “Energy-velocity description of pulse propagation in absorbing, dispersive dielectrics,” J. Opt. Soc. Am. B 12, 229–247 (1995).
    [CrossRef]
  17. K. E. Oughstun, C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210–2213 (1996).
    [CrossRef] [PubMed]
  18. G. C. 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]
  19. 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]
  20. G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), pp. 425–426.
  21. L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, UK, 1982), p. 274.

2000 (2)

L. J. Wang, A. Kuzmmich, A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[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]

1999 (1)

L. V. Hau, S. E. Haris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

1996 (1)

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

1995 (1)

1994 (1)

E. L. Bolda, J. C. Garrison, R. Y. Chiao, “Optical pulse propagation at negative group velocities due to a nearby gain line,” Phys. Rev. A 49, 2938–2947 (1994).
[CrossRef] [PubMed]

1993 (1)

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34–R37 (1993).
[CrossRef] [PubMed]

1982 (1)

S. Chu, S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[CrossRef]

1981 (1)

G. C. 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]

1977 (1)

1970 (2)

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

C. G. B. Garrett, D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[CrossRef]

1967 (1)

M. B. Priestley, “Power spectral analysis of nonstationary random processes,” J. Sound Vib. 6, 86–97 (1967).
[CrossRef]

1952 (1)

C. H. Page, “Instantaneous power spectra,” J. Appl. Phys. 23, 103–106 (1952).
[CrossRef]

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), pp. 425–426.

Balictsis, C. M.

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

Behroozi, C. H.

L. V. Hau, S. E. Haris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Bolda, E. L.

E. L. Bolda, J. C. Garrison, R. Y. Chiao, “Optical pulse propagation at negative group velocities due to a nearby gain line,” Phys. Rev. A 49, 2938–2947 (1994).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), pp. 667–670.

Brillouin, L.

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

Chiao, R. Y.

E. L. Bolda, J. C. Garrison, R. Y. Chiao, “Optical pulse propagation at negative group velocities due to a nearby gain line,” Phys. Rev. A 49, 2938–2947 (1994).
[CrossRef] [PubMed]

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34–R37 (1993).
[CrossRef] [PubMed]

R. Y. Chiao, A. M. Steinberg, “Tunneling times and superluminality,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1997), Vol. 37, pp. 347–406.

Chu, S.

S. Chu, S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[CrossRef]

Dogariu, A.

L. J. Wang, A. Kuzmmich, A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Dutton, Z.

L. V. Hau, S. E. Haris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Eberly, J. H.

Garrett, C. G. B.

C. G. B. Garrett, D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[CrossRef]

Garrison, J. C.

E. L. Bolda, J. C. Garrison, R. Y. Chiao, “Optical pulse propagation at negative group velocities due to a nearby gain line,” Phys. Rev. A 49, 2938–2947 (1994).
[CrossRef] [PubMed]

Glasgow, S. A.

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]

Haris, S. E.

L. V. Hau, S. E. Haris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Hau, L. V.

L. V. Hau, S. E. Haris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998), pp. 323, 325, 334.

Kuzmmich, A.

L. J. Wang, A. Kuzmmich, A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, UK, 1982), p. 274.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, UK, 1982), p. 274.

Loudon, R.

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

McCumber, D. E.

C. G. B. Garrett, D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[CrossRef]

Oughstun, K. E.

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

G. C. 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. C. 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]

Page, C. H.

C. H. Page, “Instantaneous power spectra,” J. Appl. Phys. 23, 103–106 (1952).
[CrossRef]

Peatross, J.

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]

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, UK, 1982), p. 274.

Priestley, M. B.

M. B. Priestley, “Power spectral analysis of nonstationary random processes,” J. Sound Vib. 6, 86–97 (1967).
[CrossRef]

Sherman, G. C.

G. C. 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. C. 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]

Steinberg, A. M.

R. Y. Chiao, A. M. Steinberg, “Tunneling times and superluminality,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1997), Vol. 37, pp. 347–406.

Wang, L. J.

L. J. Wang, A. Kuzmmich, A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Ware, M.

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]

Weber, H. J.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), pp. 425–426.

Wodkiewicz, K.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), pp. 667–670.

Wong, S.

S. Chu, S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[CrossRef]

J. Appl. Phys. (1)

C. H. Page, “Instantaneous power spectra,” J. Appl. Phys. 23, 103–106 (1952).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. A (1)

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

J. Sound Vib. (1)

M. B. Priestley, “Power spectral analysis of nonstationary random processes,” J. Sound Vib. 6, 86–97 (1967).
[CrossRef]

Nature (2)

L. V. Hau, S. E. Haris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

L. J. Wang, A. Kuzmmich, A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Phys. Rev. A (3)

C. G. B. Garrett, D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[CrossRef]

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34–R37 (1993).
[CrossRef] [PubMed]

E. L. Bolda, J. C. Garrison, R. Y. Chiao, “Optical pulse propagation at negative group velocities due to a nearby gain line,” Phys. Rev. A 49, 2938–2947 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (4)

S. Chu, S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[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. C. 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]

Other (7)

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), pp. 425–426.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, UK, 1982), p. 274.

R. Y. Chiao, A. M. Steinberg, “Tunneling times and superluminality,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1997), Vol. 37, pp. 347–406.

The authors are preparing the following paper for publication: “Poynting’s theorem and luminal energy transport in causal dielectrics.”

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998), pp. 323, 325, 334.

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

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), pp. 667–670.

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 (4)

Fig. 1
Fig. 1

(a) Electric field envelope in units of E0. The vertical lines indicate times for the assessment of the instantaneous spectrum. (b) Refractive index associated with an amplifying resonance. (c) Exchange energy density in units of 0E02/2. (d) Instantaneous spectra of the field pulse in units of E02/γ2. The spectra are assessed at the times indicated by the vertical lines in (a) and (c).

Fig. 2
Fig. 2

(a) Electric field envelope in units of E0. The vertical lines indicate times for the assessment of the instantaneous spectrum. (b) Refractive index associated with an absorbing resonance. (c) Exchange energy density in units of 0E02/2. (d) Instantaneous spectra of the field pulse in units of E02/γ2. The spectra are assessed at the times indicated by the vertical lines in (a) and (c). Note that the pulse duration is longer than that shown in Fig. 1.

Fig. 3
Fig. 3

Same as is shown in Fig. 1 with an absorbing resonance.

Fig. 4
Fig. 4

Similar to Fig. 2 but with an amplifying resonance. In addition, a wider absorptive resonance is superimposed on the amplifying resonance.

Equations (25)

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

×E+Bt=0,×Bμ0-0Et=Pt.
  S+ut=0,
u(t)=ufield+uexchange+u(-).
ufieldB22μ0+0E22.
uexchange=-tE  Ptdt.
vES/u.
E(ω)12π-exp(iωt)E(t)dt.
P(ω)=0χ(ω)E(ω).
P(t)=12π-P(ω)exp[-(iωt)]dω.
uexchange=0-|Et(ω)|2ω Im χ(ω)dω,
Et(ω)12π-tdtE(t)exp(iωt).
P(t)=-dtE(t)G(t-t),
G(t)02π-χ(ω)exp[-(iωt)]dω.
G(t)=GRe(t)+GIm(t),
GRe(t)02π-Re[χ(ω)]exp[-(iωt)]dω,
GIm(t)i 02π-Im[χ(ω)]exp[-(iωt)]dω.
Re χ(ω)=1π P-Im χ(ω)ω-ωdω.
GRe(t)=02π2-dωIm χ(ω)P-exp[-(iωt)]dωω-ω,
P-exp[-(iωt)]ω-ωdω=iπ exp[-(iω)]t>0-iπ exp[-(iωt)]t<0.
GRe(t)=GIm(t)t>0-GIm(t)t<0.
P(t)=-tdtE(t)2GIm(t-t)=i0π-dω Im[χ(ω)]exp[-(iωt)]×-tdtE(t)exp(iωt),
P(t)t=0π-dωω Im[χ(ω)]exp[-(iωt)]×-tdtE(t)exp(iωt)+0E(t)π-dω Im χ(ω).
uexchange(t)=20-dωω Im χ(ω)-t12πexp[-(iωt)]×E(t)  Et(ω)dt,
uexchange(t)=20-dωω Im χ(ω)×-Et*(ω)t Et(ω)dt.
uexchange(t)=0-dωω Im χ(ω)-tEt*(ω)t  Et(ω)+Et*(ω)  Et(ω)tdt=0-tdωω Im χ(ω)-|Et(ω)|2tdt=0-dωω Im χ(ω)|Et(ω)|2,

Metrics