## Energy transport in linear dielectrics

Optics Express, Vol. 9, Issue 10, pp. 519-532 (2001)

http://dx.doi.org/10.1364/OE.9.000519

Acrobat PDF (491 KB)

### Abstract

We examine the energy exchanged between an electromagnetic pulse and a linear dielectric medium in which it propagates. While group velocity indicates the presence of field energy (the locus of which can move with arbitrary speed), the velocity of energy transport maintains strict luminality. This indicates that the medium treats the leading and trailing portions of the pulse differently. The principle of causality requires the medium to respond to the instantaneous spectrum, the spectrum of the pulse truncated at each new instant as a given locale in the medium experiences the pulse.

© Optical Society of America

## 1 Introduction

1. 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]

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

3. 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]

4. 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]

5. R. Y. Chiao and A. M. Steinberg, “Tunneling Times and Superluminality,” Progress in Optics37, pp. 347–406 (Emil Wolf ed., Elsevier, Amsterdam, 1997). [CrossRef]

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

*presence*of field energy in dielectric media (irrespective of whether the field energy is transported from point to point or converted to or from energy stored locally in the medium). In this article, we examine the actual transport of energy and how energy is exchanged between the pulse field and the medium. In section 2 we briefly review Poyntings theorem and the concept of energy transport velocity. Section 3 demonstrates that the global energy transport velocity is strictly bounded by

*c*. We also show that there is no such limit on the velocity at which the centroid of field energy (i.e., the average position of the field energy density) moves, even though the velocity at which field energy is transported from point to point is strictly bounded by

*c*. This effect is a result of the medium exchanging energy asymmetrically with the leading and trailing portions of the pulse.

## 2 Poynting’s theorem and the energy transport velocity

*u*(

**r**,-∞), which corresponds to energy stored in the medium before the arrival of any pulse. The electromagnetic field energy density is

*u*

_{exchange}increases, the energy in the medium increases. Conversely, as

*u*

_{exchange}decreases, the medium surrenders energy to the electromagnetic field. While it is possible for

*u*

_{exchange}to become negative, the combination

*u*

_{exchange}+

*u*(-∞) (i.e., the net energy in the medium) cannot go negative since a material cannot surrender more energy than it has to begin with. Both

*u*

_{field}and

*u*

_{exchange}are zero before the arrival of the pulse (i.e. at

*t*=-∞). In addition, the field energy density returns to zero after the pulse has passed (i.e. at

*t*=+∞).

*V*, yields

*V*be small enough to take

**S**to be uniform throughout. The energy transport velocity (directed along

**S**) is then defined to be the effective speed at which the energy contained in the volume (i.e. the result of the 3-D integral) would need to travel in order to achieve the power transmitted through one side of the volume (e.g., the power transmitted through one end of a tiny cylinder aligned with

**S**). The energy transport velocity as traditionally written [15] is then

**S**and

*u*over rapid oscillations, although this average is often made [11

11. R. Loudon, “The Propagation of Electromagnetic Energy through an Absorbing Dielectric,” J. Phys. A **3**, 233–245 (1970). [CrossRef]

**S**. However, this possibility should not be injected into (7) since it cannot contribute to the integral in (6).)

*u*is used in computing (7), the energy transport velocity is

*fictitious*in its nature; it is not the actual velocity of the total energy (since part is stationary), but rather the effective velocity necessary to achieve the same energy transport that the electromagnetic flux alone delivers. There is no behind the scenes flow of mechanical energy. Moreover, if only

*u*

_{field}is used in evaluating (7), the Cauchy-Schwartz inequality (i.e.,

*α*

^{2}+

*β*

^{2}≥2

*αβ*) ensures an energy transport velocity that is strictly bounded by the speed of light in vacuum

*c*. We insist that the total energy density

*u*at a minimum should be at least as great as the field energy density so that this strict luminality is maintained. In this we differ from previous usage of the energy transport velocity in connection with amplifying media [3

3. 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]

4. 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]

5. R. Y. Chiao and A. M. Steinberg, “Tunneling Times and Superluminality,” Progress in Optics37, pp. 347–406 (Emil Wolf ed., Elsevier, Amsterdam, 1997). [CrossRef]

*u*(-∞) was left at zero, resulting in the viewpoint of superluminal and negative (opposite to the direction of

**S**) energy transport velocities.

## 3 Average energy transport velocity

*all relevant energy*) is also bounded by

*c*. This has been discussed for pulses propagating in vacuum [16

16. K. R. Bownstein, “Some Time Evolution Properties of an Electromagnetic Wave,” Am. J. Phys. **65**, 510–515 (1997). [CrossRef]

*u*. If, for example, only the field energy density

*u*

_{field}is used in defining the energy transport velocity, the time derivative cannot be brought out in front of the entire expression as in (10) since the integral in the denominator would retain time dependence. Although (10) guarantees that the centroid of the

*total*energy moves strictly luminally (since v

_{E}is pointwise luminal), there is no such guarantee on the centroid of field energy alone. Mathematically, we have

4. 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]

5. R. Y. Chiao and A. M. Steinberg, “Tunneling Times and Superluminality,” Progress in Optics37, pp. 347–406 (Emil Wolf ed., Elsevier, Amsterdam, 1997). [CrossRef]

*t*

_{0}to time

*t*

_{0}+Δ

*t*, the difference in the average position of the field energy is given by

**r**

_{G}(typically the dominant contributor to the total displacement Δ

**r**) is a linear superposition of the group velocity given by

*ρ*(

**k**,

*t*) is a normalized k-space distribution of field energy density (see Eq. (40)) at the final time

*t*≡

*t*

_{0}+Δ

*t*:

_{k}Re

*ω*(

**k**) is connected to the presence of field energy. The velocity of the pulse is predicted by an average of the group velocity function weighted by the k-space distribution of field energy in the

*final*pulse (i.e. the pulse at

*t*=

*t*

_{0}+Δ

*t*). To the extent that this k-space distribution of the field energy is modified due to amplification or absorption, the displacement of the centroid changes accordingly. Since, as is well known, the group velocity function can be superluminal or negative, the displacement per time Δ

**r**

_{G}/Δ

*t*can take on virtually any value.

*ω*(

**k**) is single-valued. For details, see the appendix

**r**

_{R}in (15) represents a displacement which arises solely from a reshaping of the pulse through absorption or amplification (without considering the dispersion introduced by propagation). This reshaping displacement is the difference between the pulse position at the

*initial time t*

_{0}evaluated without and with the spatial frequency amplitude that is lost during propagation (speaking as though the medium is absorptive). The reshaping displacement is zero if the amplitudes of the spatial frequency components are unaltered during propagation (i.e. if the imaginary part of

*ω*(

**k**) is tiny). The reshaping displacement is also relatively modest (negligible) if the pulse is unchirped before propagation. In addition, it goes to zero in the narrowband limit even if pulses experience strong absorption or amplification. (In the narrowband limit, Δ

**r**

_{G}reduces to ∇

_{k}Re

*ω*(

**k̄**), where k̄ is the central wave-vector in the pulse. This recovers the standard group velocity obtained using expansion techniques.)

**r**

_{R}is ususally small, the presence of field energy is generally tracked by group velocity as shown in Eq. (16). Thus, while the velocity of the centroid of total energy is strictly bounded by

*c*(as demonstrated in Eq. (10)), the centroid of field energy can move with any speed. This is not very mysterious when one recalls that in our discussion of field energy we have made no mention of where this energy comes from. Since a dielectric medium continually exchanges energy with the field of a pulse, the rapid movement of the centroid of field energy requires only that the medium exchange energy differently with various portions of the pulse. For example, the centroid of the field can be made to move extra fast if the medium gives energy to the leading portion and takes energy from the trailing portion (very slow propagation requires the converse).

## 4 Energy exchange and the instantaneous spectrum

*ω*.) We assume a linear, isotropic medium so that the polarization is connected to the electric field in the frequency domain via

*t*goes to +∞. In this case, the middle integral can also be performed. Therefore, after the point

**r**has experienced the entire pulse, the total amount of energy density that the medium has exchanged with the field is

**r**. We can modify this formula in a simple and intuitive way so that it describes

*u*

_{exchange}for any time during the pulse. This requires no approximations; the slowly-varying envelope approximation need not be made. The principle of causality guides us in considering how the medium perceives the electric field for any time.

**E**

_{t}(

**r**,

*ω*)|

^{2}, the

*instantaneous power spectrum*, which has been used to describe the response of driven electronic circuits [12

12. C. H. Page, “Instantaneous Power Spectra,” J. Appl. Phys. **23**, 103–106 (1952). [CrossRef]

13. M. B. Priestley, “Power Spectral Analysis of Non-Stationary Random Processes,” J. Sound Vib. **6**, 86–97 (1967). [CrossRef]

14. J. H. Eberly and K. Wodkiewicz, “The Time-Dependent Physical Spectrum of Light,” J. Opt. Soc. Am. **67**, 1252–1260 (1977). [CrossRef]

*χ*(

**r**,

*ω*) plane, which leads to the Kramers-Kronig relations [18]. We have given formal proof starting from this more familiar context of causality in Ref. [8], while including the possibility of both material anisotropy and diamagnetism. A streamlined proof is given in Ref. [9

9. J. Peatross, M. Ware, and S. A. Glasgow, “The Role of the Instantaneous Spectrum in Pulse Propagation in Causal Linear Dielectrics,” J. Opt. Soc. Am. A **18**, 1719–1725 (2001). [CrossRef]

*χ*(

**r**,

*ω*)). Since the function

**E**

_{t}(

**r**,

*ω*) is the Fourier transform of the pulse truncated at the current time and set to zero thereafter, it can include frequency components that are not present in the pulse taken in its entirety. As a point in the medium experiences the pulse, the instantaneous spectrum can lap onto or off of resonances in the medium, causing

*u*

_{exchange}to change accordingly. As discussed in section 2, as

*u*

_{exchange}increases the medium absorbs energy from the pulse and as

*u*

_{exchange}decreases the medium surrenders energy to the pulse. Thus a point in the medium may amplify the pulse at one instant while absorbing at another. As noted at the end of section 3, this allows for the possibility of dramatic superluminal or highly subluminal effects when observing the field energy alone. In section 5 we discuss specific examples in which this exotic behavior occurs.

11. R. Loudon, “The Propagation of Electromagnetic Energy through an Absorbing Dielectric,” J. Phys. A **3**, 233–245 (1970). [CrossRef]

*c*. If a signal edge begins abruptly at time

*t*

_{0}, the instantaneous spectrum

**E**

_{t}(

*ω*) clearly remains identically zero until that time. In other words, no energy may be exchanged with a material until the field energy from the pulse arrives. Since, as was pointed out in connection with Eq. (7), the Cauchy-Schwartz inequality prevents the field energy from traveling faster than

*c*, at no point in the medium can a signal front exceed

*c*.

## 5 Discussion

3. 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]

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

*c*. The effect is not only consistent with the principle of causality (as has been previously demonstrated via the Lorentz model [3

**48**, R34–R37 (1993). [CrossRef] [PubMed]

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

*ω*

_{0}and a damping frequency

*γ*. (Note that the results derived above are independent of any specific model.) In this model, the linear susceptibility is

*χ*(

*ω*)=

*ω*

^{2}-

*iγω*), where

*ω*

_{p}is the plasma frequency and

*f*is the oscillator strength, which is negative for an inverted medium. We have chosen the medium parameter values as follows:

*ω*

_{0}=1×10

^{5}

*γ*,

*γ*, and consider propagation through a thickness of 1.9 (

*c*/

*γ*). Figure 1(a) shows the imaginary parts of

*χ*(

*ω*) obtained using these parameters. The electric field of the initial pulse is chosen to be Gaussian,

**E**(0,

*t*)=

**E**

_{0}exp(-

*t*

^{2}/

*τ*

^{2})cos(

*), with the following parameters:*ω ¯
t

*τ*=2/

*γ*and

*-*ω ¯

*ω*

_{0}=10

*γ*. Thus, the resonance structure is centered a modest distance above the pulse carrier frequency, and there is only minor spectral overlap between the pulse and the resonance structure. Figure 1(b) shows the total spectrum of the initial pulse.

*u*

_{exchange}+

*u*(-∞) (energy density in the medium). We have assigned

*u*(-∞) to be the same value at each point in the medium, chosen such that the energy density in the medium never becomes negative at any point. For reference, the dashed line represents the field energy density of a pulse that propagates exactly at

*c*(i.e. as if the medium were not there). The actual pulse exiting the medium is ahead of this pulse, indicating that the centroid of field energy moved superluminally through the medium. Figure 2(b) shows the instantaneous spectrum for the first point in the medium. Notice that as this point experiences the leading portion of the pulse, the amount of overlap of the instantaneous spectrum with the resonance (at

*ω*

_{0}) increases and the medium surrenders energy to the leading portion of the pulse. As this point experiences the entire pulse, the instantaneous spectrum withdraws from the resonance, and energy is returned to the medium from the trailing portion of the pulse (notice that the energy in the medium rebounds slightly at the end of the pulse).

*t*the instantaneous spectrum withdrew from the resonance. However, at points farther in the medium (after the pulse has experienced modification), the spectrum of the pulse taken in its entirety acquires significant on-resonance spectral components. Therefore, as the pulse propagates farther into the medium the instantaneous spectrum does not withdraw entirely from the resonance during the trailing portion of the pulse. Because the instantaneous spectrum has a greater overlap with the resonance in the trailing portion than the leading portion, the trailing portion of the pulse tends to be amplified to a greater extent than the leading portion. This explains why superluminal propagation in an amplifying medium does not occur over indefinite lengths. (For the pulse shown in Fig. 2, the transition from superluminal to subluminal transit times occurs when the medium thickness is increased from 1.9 (

*c*/

*γ*) to 2(

*c*/

*γ*).)

*µ*s pulse to traverse a Δ

*r*=6cm amplifying medium was Δ

*t*≈-63ns, meaning the pulse moved forward in time by about 1% of its width. The strength of the wings in the instantaneous spectrum can be approximated as

*E*

_{t}(

*ω*)~

*E*(

*t*)/(

*ω*-

*), where*ω ¯

*represents a carrier frequency and*ω ¯

*E*(

*t*) is the strength of the field at the moment the pulse is truncated. The imaginary part of the susceptibility in a low-density vapor is approximately Im

*χ*(

*ω*)≈

*cg*/

*ω*, where

*g*is the frequency dependent gain coefficient (in the Wang experiment,

*g*≈0.1cm

^{-1}at a spectral shift of

*δω*≡

*ω*-

*≈2*ω ¯

*π*(2MHz)). A crude approximation to the integral (28) renders

*u*

_{exchange}=

*∊*

_{0}

*E*

^{2}(

*t*)

*cg*/

*δω*. This suggests that in the case of the Wang experiment the front of the pulse extracts about 250×

*∊*

_{0}

*E*

^{2}(

*t*) in energy density from the medium (i.e. 250 times the energy density in the electromagnetic field of the pulse). This energy density (extracted from the 6cm vapor cell) is distributed over about a kilometer, corresponding to the duration of the front half of the pulse. Thus, the the electromagnetic field energy on the forward part of the pulse is enhanced by several percent and similarly the field energy diminishes on the trailing edge. This is consistent with the data presented in the paper. (The traditional group velocity analysis used by Wang is perhaps a more convenient way to predict the transit time of the pulse. The utility of (28) lies primarily in its interpretation of how the pulse and the medium interact. Neither analysis substitutes for the full solution to Maxwell’s equations, but rather indicates some features of the solution.)

1. 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]

9. J. Peatross, M. Ware, and S. A. Glasgow, “The Role of the Instantaneous Spectrum in Pulse Propagation in Causal Linear Dielectrics,” J. Opt. Soc. Am. A **18**, 1719–1725 (2001). [CrossRef]

20. M. D. Crisp, “Concept of Group Velocity in Resonant Pulse Propagation,” Phys. Rev. A4, (1971). [CrossRef]

*χ*(

*ω*)=

*f*

_{1}

*ω*

^{2}-

*iγ*

_{1}

*ω*)+

*f*

_{2}

*ω*

^{2}-

*iγ*

_{2}

*ω*). For this example we choose the medium parameter values as follows:

*ω*

_{0}=1×10

^{5}

*γ*

_{1},

*ω*

_{p}=10

*γ*

_{1},

*f*

_{1}=0.02,

*f*

_{2}=-0.1, and

*γ*

_{2}=5

*γ*

_{1}. Figure 3(a) illustrates the imaginary part of

*χ*(

*ω*) for these parameters. The pulse is Gaussian as before, with parameters as follows:

*τ*=70/

*γ*

_{1}and

*=*ω ¯

*ω*

_{0}. Figure 3(b) shows the spectrum of the initial pulse.

*c*/

*γ*

_{1}). Again, the purple line in the medium represents the combination

*u*

_{exchange}+

*u*(-∞). Figure 4(b) shows the instantaneous spectrum for the first point in the medium as it experiences the pulse. In this case, the enhancement of the leading portion and the absorption of the trailing portion causes the exiting pulse to emerge from the medium before the incoming pulse enters.

21. G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A223, (1996). [CrossRef]

## 6 Summary

*transport*is strictly luminal. We also pointed out that the velocity at which field energy transported from one point to another is strictly bound by

*c*. The centroid of only field energy density can move at any speed, as predicted by group velocity. The overly rapid motion of the centroid of field energy can occur when the medium exchanges energy asymmetrically with the leading and trailing portions of the pulse. The principle of causality requires this asymmetric energy exchange as governed by the instantaneous power spectrum used in Eq. (28).

## A Appendix

**P**enters into Maxwell’s equations through a time derivative as opposed to a spatial derivative) we take a moment to review how the solutions are obtained.

**B**and

**P**give the k-space representation for the magnetic and polarization fields. We take

**E**(

**r**,

*t*),

**B**(

**r**,

*t*), and

**P**(

**r**,

*t*) to be real functions, so that the following symmetry holds for their k-space representations:

**B**(

**k**,

*t*), and

**P**(

**k**,

*t*). In a homogeneous, isotropic medium, Maxwell’s equations have as a solution

**E**(

**k**,

*t*

_{0}) is chosen at the instant

*t*

_{0}for each frequency associated with the wave number

**k**. The solution renders the pulse form

**E**(

**k**,

*t*

_{0}+Δ

*t*) (in terms of the initial pulse form) after an arbitrary time interval Δ

*t*. The magnetic field is connected to the electric field via

*ω*

_{m}and wave number

*k*is:

*k*and solve Eq. (35) for

*ω*

_{m}. The subscript

*m*and the summations in (33) and (34) reflect the fact that the solution to (35) is in general multi-valued. We take this degeneracy to be countable and therefore use a summation rather than an integral. (For example, a single Lorentz oscillator is four-fold degenerate with two distinct frequencies for a given

**k**which can each propagate forwards or backwards.) This degeneracy reflects the physical reality that in the presence of a complex linear susceptibility

*χ*(

*ω*), different frequencies can correspond to the same wavelength. As mentioned in the text, we make the simplifying assumption that only a single frequency ω is associated with each k, so that we can write the solution to Maxwell’s equations as:

**k**leads to the use of complex frequencies

*ω*. The meaning of complex frequencies is clear. The susceptibility of a complex frequency is determined by the medium’s response to an oscillatory field whose amplitude either decays or builds exponentially in time. If the susceptibility

*χ*(

*ω*) is known (measured) only for real values of

*ω*, its behavior for complex frequencies can be inferred through a Fourier transform followed by an inverse Fourier transform with complex frequency arguments. Given the real k vectors, the complex frequencies correspond to uniform plane waves that decay or build everywhere in space as a function of time. (This is in contrast with the time picture where the pulse is comprised of waves that are steady in time but which decayed or build as a function of position.)

**R**since the magnetic field can be obtained through (34).

*t*

_{0}to time

*t*

_{0}+Δ

*t*. The difference in the average position at these two times is

*at the initial time t*

_{0}evaluated without and with the spatial frequency amplitude that is lost during propagation. Dispersion effects due to propagation are not included since

**E**(

**k**,

*t*

_{0}) is used in both terms of Eq. (43).

## References and links

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

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

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

4. | 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 |

5. | R. Y. Chiao and A. M. Steinberg, “Tunneling Times and Superluminality,” Progress in Optics37, pp. 347–406 (Emil Wolf ed., Elsevier, Amsterdam, 1997). [CrossRef] |

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

7. | M. Ware, S. A. Glasgow, and J. Peatross “The Role of Group Velocity in Tracking Field Energy in Linear Dielectrics,” Opt. Express (Submitted). |

8. | S. A. Glasgow, M. Ware, and J. Peatross, “Poynting’s Theorem and Luminal Energy Transport Velocity in Causal Dielectrics,” Phys. Rev. E, (to be published 2001). |

9. | J. Peatross, M. Ware, and S. A. Glasgow, “The Role of the Instantaneous Spectrum in Pulse Propagation in Causal Linear Dielectrics,” J. Opt. Soc. Am. A |

10. | L. V. Hau, S. E. Haris, Z. Dutton, and C. H. Behroozi, “Light Speed Reduction to 17 Metres per Second in an Ultracold Atomic Gas,” Nature |

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

12. | C. H. Page, “Instantaneous Power Spectra,” J. Appl. Phys. |

13. | M. B. Priestley, “Power Spectral Analysis of Non-Stationary Random Processes,” J. Sound Vib. |

14. | J. H. Eberly and K. Wodkiewicz, “The Time-Dependent Physical Spectrum of Light,” J. Opt. Soc. Am. |

15. | L. Brillouin, |

16. | K. R. Bownstein, “Some Time Evolution Properties of an Electromagnetic Wave,” Am. J. Phys. |

17. | L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, |

18. | J. D. Jackson, |

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

20. | M. D. Crisp, “Concept of Group Velocity in Resonant Pulse Propagation,” Phys. Rev. A4, (1971). [CrossRef] |

21. | G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A223, (1996). [CrossRef] |

**OCIS Codes**

(260.2030) Physical optics : Dispersion

(260.2110) Physical optics : Electromagnetic optics

(260.2160) Physical optics : Energy transfer

(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, "Energy transport in linear dielectrics," Opt. Express **9**, 519-532 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-10-519

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### References

- 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]
- S. Chu and S. Wong, "Linear Pulse Propagation in an Absorbing Medium," Phys. Rev. Lett. 48, 738-741 (1982). [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]
- 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]
- L. J. Wang, A. Kuzmmich, and A. Dogariu, "Gain-Assisted Superluminal Light Propagation," Nature 406, 277-279 (2000). [CrossRef] [PubMed]
- M. Ware, S. A. Glasgow, and J. Peatross "Role of Group Velocity in Tracking Field Energy in Linear Dielectrics," Opt. Express 9, 506-518 (2001), http://www.opticsexpress.org/oearchive/source/35288.htm
- S. A. Glasgow, M. Ware, and J. Peatross, "Poynting's Theorem and Luminal Energy Transport Velocity in Causal Dielectrics," Phys. Rev. E, (to be published 2001).
- J. Peatross, M. Ware, and S. A. Glasgow, "The Role of the Instantaneous Spectrum in Pulse Propagation in Causal Linear Dielectrics," J. Opt. Soc. Am. A 18, 1719-1725 (2001). [CrossRef]
- L. V. Hau, S. E. Haris, Z. Dutton, and C. H. Behroozi, "Light Speed Reduction to 17 Metres per Second in an Ultracold Atomic Gas," Nature 397, 594-598 (1999). [CrossRef]
- R. Loudon, "The Propagation of Electromagnetic Energy through an Absorbing Dielectric," J. Phys. A 3, 233-245 (1970). [CrossRef]
- C. H. Page, "Instantaneous Power Spectra," J. Appl. Phys. 23, 103-106 (1952). [CrossRef]
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