## Role of group velocity in tracking field energy in linear dielectrics

Optics Express, Vol. 9, Issue 10, pp. 506-518 (2001)

http://dx.doi.org/10.1364/OE.9.000506

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### 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.

© Optical Society of America

## 1 Introduction

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]

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]

**P**and

**B**give the time/frequency connection for the polarization and magnetic field. In our convention, we take all fields to be real in the time domain, so the following symmetry holds in the frequency domain:

**B**and

**P**).

**E**(

**r**

_{0},

*ω*) is chosen at a point

**r**

_{0}. The solution renders the pulse form

**E**(

**r**

_{0}+Δ

**r**,

*ω*) (in terms of the initial pulse form) at any other point shifted by the displacement Δ

**r**. (Implicit in (6) is the restricting assumption that each frequency is associated with a single wave vector

**k**(

*ω*), although the wave vectors need not be all in the same direction.) The magnetic field is obtained in terms of the electric field through

*χ*(

*ω*) defines the (temporally non-local, but frequency local) constitutive relation between the electric field and the polarization:

*ω*to be real, whereas the wave vector

**k**(

*ω*) can be complex when the susceptibility

*χ*(

*ω*) is complex.

**r**. The function

*∂*Re

**k**/

*∂ω*·Δ

**r**is known as the

*group delay function*, and in the traditional context is evaluated at a single ‘carrier’ frequency

*. Group velocity is obtained by dividing the displacement Δ*ω ¯

**r**by the group delay function evaluated at

*.*ω ¯

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]

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]

**r**is described by a

*linear superposition of group delays, weighted by the surviving spectral content of the pulse*. Mathematically, this delay is given by

*ρ*(

**r**,

*ω*) is a normalized spectral distribution of Poynting flux (at the final point) defined by

*is a unit vector normal to the detector surface.) Another delay term due to spectral reshaping also plays a role in determining the total delay time. This additional delay term, which is evaluated without propagation, becomes important when an initially chirped pulse undergoes an alteration in its spectral profile. For an unchirped pulse, the reshaping effect is generally negligable. No approximations or expansions of the type (10) are used in the new formulation, making the results general.*η ^

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. R. 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]

## 2 Average arrival time of poynting flux to a point

13. R. L. Smith, “The Velocities of Light,” Am. J. Phys. **38**, 978–983 (1970). [CrossRef]

*ρ*(

**r**,

*t*) is a normalized temporal distribution of the Poynting flux at

**r**:

*refers to the direction in which the energy flow is detected (normal to a detector surface). It has importance for angularly dispersive systems such as grating pairs used for pulse compression where the result of the integral in the numerator is not necessarily parallel to that in the denominator. We have addressed the angularly dispersive case elsewhere [14*η ^

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]

*t*, the expression (18) can be recast as

*ω′*to be performed:

*T*depends only on the electric field since the magnetic field can be written in terms of the electric field through (7).

## 3 Delay time between two points

**r**

_{0}and

**r**=

**r**

_{0}+Δ

**r**. The difference in the arrival times of the two points is defined as

*t*and the group delay function

*∂*Re

**k**/

*∂ω*·Δ

**r**.

*T*[

**E**(

**r**

_{0}+Δ

**r**,

*ω*)]. With the solution (6) we can evaluate the integrand of the numerator of (22) at

**r**

_{0}+Δ

**r**as follows:

**r**

_{0}+Δ

**r**, giving

*net group delay*, is a spectral average of the group delay function over the spectral intensity that is experienced at the final point

**r**(as already mentioned in connection with):

*∂*Re

**k**/

*∂ω*·Δ

**r**. Clearly, the group delay function evaluated at every frequency present in the pulse influences the result. The net group delay depends only on the spectral content of the pulse, independent of its temporal organization (i.e. the phases of

**E**(

**r**,

*ω*) and

**B**(

**r**,

*ω*) do not contribute). Only the real part of

**k**plays a direct role in (28).

**r**

_{0},

*before*propagation takes place. The reshaping delay is the difference between the pulse arrival time at the initial point

**r**

_{0}evaluated

*without*and

*with*the spectral amplitude that will be lost (or gained) during propagation. Both terms in (30) utilize the phase of the fields at

**r**

_{0}. Thus, in contrast to the net group delay, the reshaping delay is sensitive to how the pulse is organized. The reshaping delay is zero if the spectral amplitude of the pulse is unaltered during propagation (i.e. if the imaginary part of

**k**can be neglected). The reshaping delay is also relatively unimportant if the pulse is unchirped before propagation and in the narrowband limit even if pulses experience strong absorption (or amplification).

**r**and

**r**

_{0}in (27) can be interchanged which gives

*t*is unaffected by the ordering. The group delay computed over the initial spectrum is

*ρ*(

*ω*)→

*δ*(

*ω*-

*)) so that the total delay reduces to Δ*ω ¯

*t*→Δ

*t*

_{G}=

*∂*Re

**k**/

*∂ω*|

*·Δ*ω ¯

**r**. This is in agreement with the well verified observation [7

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. R. 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]

*in the narrowband limit.*ω ¯

## 4 The effect of pulse bandwidth on delay time

**k**=

*kẑ*. We will consider the delay time of a pulse as it propagates through a displacement Δ

**r**=0.1 (

*c*/

*γ*)

*ẑ*. Figure 1(b) depicts the group delay function

*∂k*/

*∂ω*·Δ

**r**for these parameters.

**r**

_{0}is chosen to be Gaussian,

**E**(

**r**

_{0},

*t*)=

*x̂*

**E**

_{0}exp(-

*t*

^{2}/

*τ*

^{2})cos (

*), with initial duration*ω ¯
t

*τ*=10/

*γ*. In all of the following examples the pulses are initially unchirped, so that the reshaping delay Δ

*t*

_{R}is negligable. Figure 2(a) depicts the normalized spectral distribution function (Eq. (29)) of the narrowband pulse before (dotted line) and after (solid line) propagation in the medium, with the initial spectrum centered on resonance (i.e.

*=*ω ¯

*ω*

_{0}). Because the initial pulse spectrum is narrow compared to the resonance,

*ρ*changes little during propagation. Since the reshaping delay is negligable, the total delay time is given by the expectation of the group delay function (shown in Fig. 1(b)) weighted by the spectral distribution (shown in Fig. 2(a)) as in Eq. (28). Figure 2(b) plots the delay time as the pulse’s central frequency

*is varied relative to the resonance frequency*ω ¯

*ω*

_{0}. Since

*ρ*(

**r**,

*ω*) is narrow compared to the width of the resonance, the total delay closely resembles the group delay function in Fig. 2 (b). Figure 2 (c) plots the ratio of the pulse energy after propagation to that before propagation as the pulse’s central frequency is varied.

*-*ω ¯

*ω*

_{0}=

*γ*where the pulse in Fig. 2 exhibits the most dramatic superluminal behavior. The dotted line represents the pulse as it would have travelled in vacuum and the solid line indicates the actual pulse profile. In the animation it is clear that the centroid of the narrowband pulse is shifted forward relative to the vacuum pulse upon traversing the medium. In contrast, the broadband pulse is delayed relative to the vacuum pulse.

*-*ω ¯

*ω*

_{0}=

*γ*. Figure 5 plots the delay time as the pulse duration

*τ*(and hence the bandwidth) is varied. As expected, the pulses with narrow bandwidths (large

*τ*) exhibit superluminal effects. The behavior becomes subluminal as the bandwidth increases (

*τ*decreases). This is easily understood in terms of (28). As the bandwidth increases,

*ρ*(

**r**,

*ω*) includes more and more of the ‘slow’ spectral components near

*ω*

_{0}. When the expectation (28) is taken, these ‘slow’ spectral components outweigh the effect of the frequencies with a negative group delay function resulting in a subluminal delay time.

*ρ*(

*ω*)→constant). By comparison, any resonance structure encompasses a finite bandwidth, and the reshaping delay tends to zero. If, for simplicity, we consider all wave vectors to be parallel (i.e.

**k**(

*ω*)=

*k*(

*ω*)

*k̂*) we have from (28) or (32) that Δ

*t*→Δ

*t*

_{G}≈Re

**k**/

*ω*|

_{∞}·Δ

**r**=

*k̂*·Δ

**r**/

*c*, assuming a refractive index which approaches unity at high frequency. This demonstrates (without lengthy analyses or numerical simulations) the well-known fact that the velocity for a sharply defined signal is exactly c. Thus, the Sommerfeld-Brillouin result [2] of luminality for pulses of definite support is consistent with and even implied by the present context.

**1**, 305–313 (1970). [CrossRef]

11. S. Chu and S. Wong, “Linear Pulse Propagation in an Absorbing Medium,” Phys. Rev. Lett. **48**, 738–741 (1982). [CrossRef]

*ρ*(

**r**,

*ω*) will outweigh the on resonance components resulting in luminal delay times. Figure 6 is an animation illustrating the difference between narrowband and broadband pulses in an absorbing medium. The pulse parameters are the same as in fig. 4, and the medium parameters are the same except

*f*=+1 in order to make an absorbing medium. (These parameters are identical to those used in [1

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]

## 5 Propagation distance and delay time

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]

*t*

_{R}(

**r**

_{0}) is negligable, so that the total delay is given by Δ

*t*

_{G}(

**r**). When the pulse spectrum is centered on-resonance,

*ρ*(

**r**,

*ω*) will primarily include the ‘fast’ spectral components near

*ω*

_{0}(remember that in an absorbing medium, frequencies near

*ω*

_{0}have the shortest group delay function). However, as it travels, the fast spectral components near

*ω*

_{0}are absorbed from the pulse spectrum. Eventually only spectral components away from the resonance (i.e. frequencies where the group delay function gives a luminal delay) are left in the spectrum. The delay time, given by the net group delay (28), then becomes subluminal.

*ρ*(

**r**,

*ω*) changes to include more near-resonance frequencies.

*τ*=10/

*γ*and

*-*ω ¯

*ω*

_{0}=1

*γ*. For ease of comparison, the delay time is scaled by

*c*/Δ

**r**so that propagation at the speed of light has a scaled delay time of 1. Figure 7 (b) shows the group delay function (also scaled by

*c*/Δ

**r**). For small displacements, the delay time matches the group delay function evaluated at

*. As the pulse propagates and acquires more on-resonance spectral intensity, the delay time shifts to the on-resonance value of the group delay function.*ω ¯

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]

*c*(i.e. as if the absorber and amplifier were not there). The Lorentz model is used for the two media with

*ω*

_{0}=2×10

^{7}γ,

*ω*

_{p}=100

*γ*, and Δ

*r*=0.5

*c*/

*γ*. The oscillator strength is

*f*=1 for the absorbing medium and

*f*=-1 for the amplifier. The pulse is chosen to be gaussian as above with

*τ*=0.264 √2

*γ*and

*-*ω ¯

*ω*

_{0}=

*ω*

_{p}/3.

*t*

_{R}also plays a role since the pulse was chirped in the absorber. The reshaping delay is still small, however, amounting to about a 5% correction to Δ

*t*

_{G}.) Because the near-resonance spectral components were removed, the large values of the group delay function near resonance do not play a role in the expectation. Thus, the pulse passing through the amplifier can exhibit superluminal behavior over a relatively long distance. If the full gaussian is sent through the amplifier without this preparation, superluminal behavior persists over a displacement of only Δ

*r*=7×10

^{-3}

*c*/

*γ*before becoming subluminal (and experiencing extreme amplification). This transition occurs when the amplitude of the on-resonance peak in

*ρ*(

**r**,

*ω*) has roughly the same amplitude as the peak at the carrier frequency

**49**, 2938–2947 (1994). [CrossRef] [PubMed]

## 6 Conclusion

*electromagnetic field energy*only. It is important to realize that a superluminal delay time for the field energy center of mass does not require superluminal energy transport. For example, energy stored in the medium downstream from a pulse’s centroid may convert to field energy. The appearance of downstream field energy can make the pulse’s center of mass move superluminally while no actual energy transport occurs faster than c. In the absorbing case, the medium can preferentially absorb the trailing portion of a pulse, resulting in an advance of the pulse’s center of mass. In both of these cases the medium treats the leading and trailing portions of the pulse differently. In a companion article [17], we discuss why dielectric media behave in this manner as well as addressing the issue of energy transport velocity.

## References and links

1. | J. Peatross, S. A. Glasgow, and M. Ware, “Average Energy Flow of Optical Pulses in Dispersive Media,” Phys. Rev. Lett. |

2. | L. Brillouin, |

3. | M. Born and E. Wolf, |

4. | J. D. Jackson, |

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. |

6. | K. E. Oughstun and H. Xiao, “Failure of the Quasimonochromatic Approximation for Ultrashort Pulse Propagation in a Dispersive, Attenuative Medium,” Phys. Rev. Lett. |

7. | C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium,” Phys. Rev. A |

8. | R. Y. Chiao, “Superluminal (but Causal) Propagation of Wave Packets in Transparent Media with Inverted Atomic Populations,” Phys. Rev. A. |

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 |

10. | R. Y. Chiao and A. M. Steinberg, “Tunneling Times and Superluminality,” Progress in Optics |

11. | S. Chu and S. Wong, “Linear Pulse Propagation in an Absorbing Medium,” Phys. Rev. Lett. |

12. | L. J. Wang, A. Kuzmmich, and A. Dogariu, “Gain-Assisted Superluminal Light Propagation,” Nature |

13. | R. L. Smith, “The Velocities of Light,” Am. J. Phys. |

14. | M. Ware, W. E. Dibble, S. A. Glasgow, and J. Peatross, “Energy Flow in Angularly Dispersive Optical Systems,” J. Opt. Soc. Am. B |

15. | R. Loudon, “The Propagation of Electromagnetic Energy through an Absorbing Dielectric,” J. Phys. A |

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. |

17. | M. Ware, S. A. Glasgow, and J. Peatross “Energy Transport in Linear Dielectrics,” Opt. Express (Submitted). |

**OCIS Codes**

(260.2030) Physical optics : Dispersion

(260.2110) Physical optics : Electromagnetic optics

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 19, 2001

Published: November 5, 2001

**Citation**

Michael Ware, S. Glasgow, and Justin Peatross, "Role of group velocity in tracking field energy in linear dielectrics," Opt. Express **9**, 506-518 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-10-506

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### References

- 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

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