Abstract

A new context for the group delay function (valid for pulses of arbitrary bandwidth) is presented for electromagnetic pulses propagating in a uniform linear dielectric medium. The traditional formulation of group velocity is recovered by taking a narrowband limit of this generalized context. The arrival time of a light pulse at a point in space is defined using a time expectation integral over the Poynting vector. The delay between pulse arrival times at two distinct points consists of two parts: a spectral superposition of group delays and a delay due to spectral reshaping via absorption or amplification. The use of the new context is illustrated for pulses propagating both superluminally and subluminally. The inevitable transition to subluminal behavior for any initially superluminal pulse is also demonstrated.

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References

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  1. J. Peatross, S. A. Glasgow, and M. Ware, "Average Energy Flow of Optical Pulses in Dispersive Media," Phys. Rev. Lett. 84, 2370-2373 (2000).
    [CrossRef] [PubMed]
  2. L. Brillouin, Wave Propagation and Group Velocity (Academic Press, New York, 1960).
  3. M. Born and E. Wolf, Principles of Optics, 7th Ed. (Cambridge, 1999), pp. 19-24.
  4. J. D. Jackson, Classical Electrodynamics, 3rd Ed. (Wiley, New York, 1998), pp. 323, 330-335.
  5. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, and B. A. Richman, "Measuring Ultrashort Laser Pulses in the Time-Frequency Domain Using Frequency-Resolved Optical Gating," Rev. Sci. Instrum. 68, 3277-3295 (1997).
    [CrossRef]
  6. 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]
  7. C. G. B. Garrett and D. E. McCumber, "Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium," Phys. Rev. A 1, 305-313 (1970).
    [CrossRef]
  8. 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]
  9. E. L. Bolda, J. C. Garrison, and 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]
  10. Y. Chiao and A. M. Steinberg, "Tunneling Times and Superluminality," Progress in Optics 37, pp. 347-406 (Emil Wolf ed., Elsevier, Amsterdam, 1997).
    [CrossRef]
  11. S. Chu and S. Wong, "Linear Pulse Propagation in an Absorbing Medium," Phys. Rev. Lett. 48, 738-741 (1982).
    [CrossRef]
  12. L. J. Wang, A. Kuzmmich, and A. Dogariu, "Gain-Assisted Superluminal Light Propagation," Nature 406, 277-279 (2000).
    [CrossRef] [PubMed]
  13. R. L. Smith, "The Velocities of Light," Am. J. Phys. 38, 978-983 (1970).
    [CrossRef]
  14. M. Ware, W. E. Dibble, S. A. Glasgow, and J. Peatross, "Energy Flow in Angularly Dispersive Optical Systems," J. Opt. Soc. Am. B 18 839-845 (2001) .
    [CrossRef]
  15. R. Loudon, "The Propagation of Electromagnetic Energy through an Absorbing Dielectric," J. Phys. A 3, 233-245 (1970).
    [CrossRef]
  16. Md. Aminul Islam Talukder, Yoshimitsu Amagishi, and Makoto Tomita, "Superluminal to Subluminal Transition in the Pulse Propagation in a Resonantly Absorbing Medium," Phys. Rev. Lett. 86, 3546-3549 (2001).
    [CrossRef]
  17. M. Ware, S. A. Glasgow, and J. Peatross "Energy Transport in Linear Dielectrics," Opt. Express 9, 519-532 (2001), http://www.opticsexpress.org/oearchive/source/35289.htm

Other (17)

J. Peatross, S. A. Glasgow, and M. Ware, "Average Energy Flow of Optical Pulses in Dispersive Media," Phys. Rev. Lett. 84, 2370-2373 (2000).
[CrossRef] [PubMed]

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

M. Born and E. Wolf, Principles of Optics, 7th Ed. (Cambridge, 1999), pp. 19-24.

J. D. Jackson, Classical Electrodynamics, 3rd Ed. (Wiley, New York, 1998), pp. 323, 330-335.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, and B. A. Richman, "Measuring Ultrashort Laser Pulses in the Time-Frequency Domain Using Frequency-Resolved Optical Gating," Rev. Sci. Instrum. 68, 3277-3295 (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]

C. G. B. Garrett and 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, and 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]

Y. Chiao and A. M. Steinberg, "Tunneling Times and Superluminality," Progress in Optics 37, pp. 347-406 (Emil Wolf ed., Elsevier, Amsterdam, 1997).
[CrossRef]

S. Chu and S. Wong, "Linear Pulse Propagation in an Absorbing Medium," Phys. Rev. Lett. 48, 738-741 (1982).
[CrossRef]

L. J. Wang, A. Kuzmmich, and A. Dogariu, "Gain-Assisted Superluminal Light Propagation," Nature 406, 277-279 (2000).
[CrossRef] [PubMed]

R. L. Smith, "The Velocities of Light," Am. J. Phys. 38, 978-983 (1970).
[CrossRef]

M. Ware, W. E. Dibble, S. A. Glasgow, and J. Peatross, "Energy Flow in Angularly Dispersive Optical Systems," J. Opt. Soc. Am. B 18 839-845 (2001) .
[CrossRef]

R. Loudon, "The Propagation of Electromagnetic Energy through an Absorbing Dielectric," J. Phys. A 3, 233-245 (1970).
[CrossRef]

Md. Aminul Islam Talukder, Yoshimitsu Amagishi, and Makoto Tomita, "Superluminal to Subluminal Transition in the Pulse Propagation in a Resonantly Absorbing Medium," Phys. Rev. Lett. 86, 3546-3549 (2001).
[CrossRef]

M. Ware, S. A. Glasgow, and J. Peatross "Energy Transport in Linear Dielectrics," Opt. Express 9, 519-532 (2001), http://www.opticsexpress.org/oearchive/source/35289.htm

Supplementary Material (3)

» Media 1: MOV (1424 KB)     
» Media 2: MOV (1746 KB)     
» Media 3: MOV (1742 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

(a) Real and imaginary parts of the refractive index. (b) Group delay function for the given index

Fig. 2.
Fig. 2.

(a) Normalized spectral distribution ρ for the narrowband pulse centered on resonance before (dotted) and after (solid) propagation. (b) Total delay as a function of ω ¯ for the narrowband pulse. (c) Overall pulse transmission as a function of ω ¯

Fig. 3.
Fig. 3.

(a) Normalized spectral distribution ρ for the broadband pulse centered on resonance before (dotted) and after(solid) propagation. (b) Total delay as a function of ω ¯ for the broadband pulse. (c) Overall pulse transmission as a function of ω ¯

Fig. 4.
Fig. 4.

Animation of narroband (top) and broadband (bottom) gaussian pulses traversing an amplifying medium (1.4 MB)

Fig. 5.
Fig. 5.

Total delay as the width of the pulse is changed from broadband to narrow band.

Fig. 6.
Fig. 6.

Animation of narroband (top) and broadband (bottom) gaussian pulses traversing an absorbing medium (1.7 MB)

Fig. 7.
Fig. 7.

(a) Total delay, scaled by cr, as a function of displacement. (b) The group delay function divided by displacement.

Fig. 8.
Fig. 8.

Animation of a gaussian pulse traversing an absorbing medium followed by an amplifying medium as proposed by Bolda [9] (1.7 MB)

Fig. 9.
Fig. 9.

(a) Spectral distribution before (dashed) and after (solid) traversing the absorber. The distribution after the amplifier is the same as the initial distribution. (b) Group delay function for the amplifying medium.

Equations (38)

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× E ( r , t ) + B ( r , t ) t = 0 ,
× B ( r , t ) μ 0 0 E ( r , t ) t = P ( r , t ) t .
E ( r , ω ) = 1 2 π e i ω t E ( r , t ) d t ,
E ( r , t ) = 1 2 π e i ω t E ( r , ω ) d ω .
E ( r , ω ) = E * ( r , ω )
E ( r 0 + Δ r , ω ) = E ( r 0 , ω ) e i k · Δ r ,
B ( r , ω ) = k ( ω ) × E ( r , ω ) ω .
P ( r , ω ) = 0 χ ( ω ) E ( r , ω ) .
k 2 ( ω ) = ω 2 c 2 [ 1 + χ ( ω ) ] .
k ( ω ) · Δ r [ k ω - · Δ r ] + [ k ω ω ¯ · Δ r ] ( ω ω - ) +
Δ t G ( r ) = ρ ( r , ω ) ( Re k ω · Δ r ) d ω ,
ρ ( r , ω ) η ̂ · S ( r , ω ) η ̂ · S ( r , ω ) d ω .
t r t ρ ( r , t ) d t ,
ρ ( r , t ) η ̂ · S ( r , t ) η ̂ · S ( r , t ) d t .
S ( r , t ) E ( r , t ) × B ( r , t ) μ 0 .
S ( r , t ) d t = S ( r , ω ) d ω ,
S ( r , ω ) E ( r , ω ) × B * ( r , ω ) μ 0 ,
t S ( r , t ) d t = t [ 1 2 π E ( r , ω ) e i ω t d ω ] × [ 1 2 π B ( r , ω ' ) e i ω ' t d ω ' ] d t .
t S ( r , t ) d t = i d ω E ( r , ω ) × ω [ d ω B ( r , ω ) μ 0 1 2 π e i ( ω + ω ) t d t ] .
t S ( r , t ) d t = i E ( r , ω ) × ω B ( r , ω ) μ 0 d ω = i E ( r , ω ) ω × B * ( r , ω ) μ 0 d ω .
t r = T [ E ( r , ω ) ] ,
T [ E ( r , ω ) ] i η ̂ · E ( r , ω ) ω × B * ( r , ω ) μ 0 d ω η ̂ · S ( r , ω ) d ω .
Δ t t r t r 0 = T [ E ( r 0 + Δ r , ω ) ] T [ E ( r 0 , ω ) ] .
E ( r 0 , ω ) e i k · Δ r ω × B * ( r 0 , ω ) e i k * · Δ r μ 0 = i Re k · Δ r ω E ( r 0 , ω ) × B * ( r 0 , ω ) e 2 Im k · Δ r + E ( r 0 , ω ) e Im k · Δ r ω × B * ( r 0 , ω ) e Im k · Δ r μ 0 .
S ( r 0 + Δ r , ω ) = E ( r 0 , ω ) e i k · Δ r × B * ( r 0 , ω ) e i k * · Δ r μ 0
= E ( r 0 , ω ) e Im k · Δ r × B * ( r 0 , ω ) e Im k * · Δ r μ 0 .
T [ E ( r 0 + Δ r , ω ) ] = η ̂ · S ( r , ω ) ( Re k ω · Δ r ) d ω η ̂ · S ( r , ω ) d ω
i η ̂ · E ( r 0 , ω ) e Im k · Δ r ω × B * ( r 0 , ω ) e Im k * · Δ r μ 0 d ω η ̂ · e 2 Im k · Δ r S ( r 0 , ω ) d ω .
Δ t = Δ t G ( r ) + Δ t R ( r 0 ) .
Δ t G ( r ) = ρ ( r , ω ) ( Re k ω · Δ r ) d ω ,
ρ ( r , ω ) η ̂ · S ( r , ω ) η ̂ · S ( r , ω ) d ω .
Δ t R ( r 0 ) T [ e Im k · Δ r E ( r 0 , ω ) ] T [ E ( r 0 , ω ) ] .
Δ t G ( r ) + Δ t R ( r 0 ) = Δ t G ( r 0 ) + Δ t R ( r ) .
Δ t G ( r 0 ) = ρ ( r 0 , ω ) ( Re k ω · Δ r ) d ω .
Δ t R ( r ) = T [ E ( r 0 + Δ r , ω ) ] T [ e Im k · Δ r E ( r 0 + Δ r , ω ) ] .
T [ e Im k · r E ( r 0 + Δ r , ω ) ] = η ̂ · S ( r 0 , ω ) ( Re k ω · Δ r ) d ω η ̂ · S ( r 0 , ω ) d ω
i η ̂ · E ( r 0 , ω ) ω × B * ( r 0 , ω ) μ 0 d ω η ̂ · S ( r 0 , ω ) d ω .
[ Re n ( ω ) + i Im n ( ω ) ] 2 = 1 + χ ( ω )

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