## Non-local energy transport in tunneling and plasmonic structures |

Optics Express, Vol. 19, Issue 16, pp. 15281-15296 (2011)

http://dx.doi.org/10.1364/OE.19.015281

Acrobat PDF (1788 KB)

### Abstract

Various definitions of the velocity of propagation of the electromagnetic field have been adopted in experimental and theoretical studies of tunneling and plasmonic systems. Tunneling problems are often analyzed by invoking the group delay (or dwell time) velocities. On the other hand, slow light and plasmonic systems are considered by using the wave packet group velocity. This paper discusses various definitions for the velocity of the electromagnetic wave propagation and compares them in applications to the problems of slow light and superluminality in resonant and tunneling structures. Energy propagation is, in general, a nonlocal quantity and depends on the global properties of the system, rather than being simply a local quantity. The energy propagation velocity takes into account the non-local characteristics of the wave propagation and offers a natural generalization for those situations when the group velocity is ill defined or gives unphysical results. It is shown that the group delay velocity, which may be superluminal away from the resonance, becomes equal to the energy velocity at the resonant point.

© 2011 OSA

## 1. Introduction

2. A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A **49**, 3283–3295 (1994). [CrossRef] [PubMed]

3. A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E **78**, 016601 (2008). [CrossRef]

6. A. Enders and G. Nimtz, “Evanescent-mode propagation and quantum tunneling,” Phys. Rev. E **48**, 632–634 (1993). [CrossRef]

7. T. Baba, “Slow light in photonic crystals,” Nat. Photonics **2**, 465–463 (2008). [CrossRef]

11. C. M. de Sterke, K. B. Dossou, T. P. White, L. C. Botten, and R. C. McPhedran, “Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes,” Opt. Express **17**, 17338–17343 (2009). [CrossRef]

12. M. Mojahedi, K. Malloy, G. Eleftheriades, J. Woodley, and R. Chiao, “Abnormal wave propagation in passive media,” IEEE J. Sel. Top. Quantum Electron. **9**, 30–39 (2003). [CrossRef]

14. C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. **12**, 104003 (2010). [CrossRef]

*v*, the group velocity

_{d}**v**

*=*

_{g}*∂ω*/

*∂*

**k**, and the energy velocity

**v**

*[15]. The group delay velocity*

_{E}*v*is introduced by using the group delay [16

_{d}16. E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. **98**, 145–147 (1955). [CrossRef]

*v*=

_{d}*∂ω*/

*∂κ*, where

*κ*=

*Lφ*and

_{t}*φ*is the phase of the transmission coefficient. The energy velocity

_{t}**v**

*=*

_{E}**S**/

*U*is introduced as the ratio of the Poynting vector to the electromagnetic energy density inside the medium [15]. One can easily show that for a finite wave packet in an unbounded non-dissipative medium, the energy velocity is equal to the group velocity [17]. This work aims to study the differences and mutual relations between various measures of the propagation speed of electromagnetic radiation. One important point to note is that the various definitions of the velocity do not necessarily apply to the velocity of energy (or information) propagation.

18. A. I. Smolyakov, E. A. Fourkal, S. I. Krasheninnikov, and N. Sternberg, “Resonant modes and resonant transmission in multi-layer structures,” Prog. Electromagn. Res. **107**, 293–314 (2010). [CrossRef]

19. E. Fourkal, I. Velchev, C.-M. Ma, and A. Smolyakov, “Resonant transparency of materials with negative permittivity,” Phys. Lett. A **361**, 277–282 (2007). [CrossRef]

*ɛ*(

*ω*) > 0 and

*∂ω*/

*∂*

**k**≠ 0, which nevertheless does not transport any energy. The group velocity,

*∂ω*/

*∂*

**k**, is a local quantity and is defined for wavepackets in an infinite medium. It does not take into account wave reflections, and, as such, it does not depend on the geometry of the medium, e.g. on the boundary conditions. As a matter of fact, the energy flux through the medium, and, consequently the energy flow velocity, is in general a non-local quantity. It is not defined solely by the local properties of the dielectric medium via the local function

*ω*=

*ω*(

**k**), but also depends on the overall geometry of the system, in particular, on boundary conditions.

**v**

*=*

_{E}**S**/

*U*is a better measure of the energy propagation velocity. It fully takes into account non-local effects (finite system dimensions and presence of boundaries). It is also remains well defined in those cases, when the group velocity becomes ill-determined (as in tunneling problems) and/or produces unphysical results. There exists noticeable disconnect between the approaches used to study slow light in various optical and electromagnetic structures and the methods used to study pulse propagation in tunneling problems. The slow light phenomenon is mainly interpreted as an effect of very small group velocity

*v*≡

_{g}*∂ω*/

*∂k*≪

*c*, resulting from large dispersion near the resonances (the nature of resonance varies for different systems). The concept of group velocity does not exist for evanescent waves in tunneling. In the latter, the concept of the group delay and dwell time are often used as a measure of the propagation time. The interpretation of the group delay as the tunneling (traversal) time leads to apparent paradoxes such as superluminality [20

20. J. C. Garrison, M. W. Mitchell, R. Y. Chiao, and E. L. Bolda, “Superluminal signals: causal loop paradoxes revisited,” Phys. Lett. A **245**, 19–25 (1998). [CrossRef]

22. A. M. Steinberg, “How much time does a tunneling particle spend in the barrier region?” Phys. Rev. Lett. **74**, 2405–2409 (1995). [CrossRef] [PubMed]

6. A. Enders and G. Nimtz, “Evanescent-mode propagation and quantum tunneling,” Phys. Rev. E **48**, 632–634 (1993). [CrossRef]

23. A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. **71**, 708–711 (1993). [CrossRef] [PubMed]

26. D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. **84**, 4830–4833 (2000). [CrossRef] [PubMed]

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

23. A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. **71**, 708–711 (1993). [CrossRef] [PubMed]

26. D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. **84**, 4830–4833 (2000). [CrossRef] [PubMed]

**v**

*, the group velocity*

_{E}**v**

*=*

_{g}*∂ω*/

*∂*

**k**, and the group delay velocity

*v*in several configurations involving tunneling and evanescent wave energy transport. Our particular interest is in resonant regimes, in particular, those involving surface mode and standing wave resonances [30

_{d}30. A. Dogariu, A. Kuzmich, H. Cao, and L. Wang, “Superluminal light pulse propagation via rephasing in a transparent anomalously dispersive medium,” Opt. Express **8**, 344–350 (2001). [CrossRef] [PubMed]

32. N. Malkova, G. W. Bryant, S. Polyakov, and A. Migdall, “Effect of surface modes on photon traversal through stop bands of dielectric stacks,” Phys. Rev. B **80**, 165127 (2009). [CrossRef]

18. A. I. Smolyakov, E. A. Fourkal, S. I. Krasheninnikov, and N. Sternberg, “Resonant modes and resonant transmission in multi-layer structures,” Prog. Electromagn. Res. **107**, 293–314 (2010). [CrossRef]

## 2. Group delay time and energy velocity

16. E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. **98**, 145–147 (1955). [CrossRef]

29. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. **436**, 1–69 (2006). [CrossRef]

*ω*

_{0}, can be presented in the form This packet propagates in vacuum with a group velocity

*∂*

*ω*

*/∂k*. Upon collision with a generic barrier, the reflected and transmitted wavepackets are generated Assuming that the functions

*f*(

*ω*–

*ω*

_{0}) |

*R*| and

*f*(

*ω*–

*ω*

_{0}) |

*T*| are slowly varying functions of

*ω*and that the phase changes rapidly with

*ω*, using the stationary phase method [29

29. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. **436**, 1–69 (2006). [CrossRef]

33. T. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. Rep. **33**, 3427–3433 (1962). [CrossRef]

*x*=

*d*(for the transmitted packet) and

*x*= 0 (for the reflected packet) with delays given by [29

29. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. **436**, 1–69 (2006). [CrossRef]

33. T. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. Rep. **33**, 3427–3433 (1962). [CrossRef]

34. H. G. Winful, “Delay time and the Hartman effect in quantum tunneling,” Phys. Rev. Lett. **91**, 260401 (2003). [CrossRef]

34. H. G. Winful, “Delay time and the Hartman effect in quantum tunneling,” Phys. Rev. Lett. **91**, 260401 (2003). [CrossRef]

*U*is the electromagnetic energy density,

**S**is the Poynting vector (electromagnetic energy flux) and

*Q*represents the Joule losses. The flux of any quantity can be represented as the density of this quantity multiplied by its flux velocity, that is,

**S**=

*U*

**v**

*. This last representation of the Poynting vector is known as the Umov form [37]. For a wave packet in an infinite homogeneous medium, the energy flux and the energy density are constant with*

_{E}**v**

*=*

_{E}*∂ω*/

*∂*

**k**, that is, the energy velocity equals the group velocity. In general,

*U*and

**S**can be position dependent and the local energy velocity can be non-uniform in space. In general, this energy velocity is always less than

*c*even for gain media [38

38. E. Schulz-DuBois, “Energy transport velocity of electromagnetic propagation in dispersive media,” Proc. IEEE **57**, 1748–1757 (1969). [CrossRef]

*E*

_{0},

*H*

_{0}) and the transmission coefficient where

*Z*

_{0}=

*E*

_{0}/

*H*

_{0}is the impedance of free space.

*H*(

_{x}*z*) is given by with

*κ*a real number but

*C*and

*D*possibly complex, being the amplitudes of two waves in the region. By Maxwell’s equations, the electric field associated with these waves is given by or For the Poynting flux we have As can be seen from Eq. (15) the z-component of the Poynting flux becomes finite when the product

*CD*

^{*}has an imaginary part other than zero; in other words, there has to be a finite phase shift between the amplitudes

*C*and

*D*. Therefore, a finite energy flux occurs as a result of the superposition of two evanescent modes with a finite phase shift [18

18. A. I. Smolyakov, E. A. Fourkal, S. I. Krasheninnikov, and N. Sternberg, “Resonant modes and resonant transmission in multi-layer structures,” Prog. Electromagn. Res. **107**, 293–314 (2010). [CrossRef]

**107**, 293–314 (2010). [CrossRef]

**107**, 293–314 (2010). [CrossRef]

38. E. Schulz-DuBois, “Energy transport velocity of electromagnetic propagation in dispersive media,” Proc. IEEE **57**, 1748–1757 (1969). [CrossRef]

## 3. Velocities in dispersive slab

*A, B, R*and

*T*are obtained by using the boundary conditions for the fields (e.g continuity of

*E*and

_{y}*H*at the boundaries

_{x}*z*= 0 and

*z*=

*d*). The resulting reflection and transmission coefficients for this structure are given by where and The above formulas also remain valid in the evanescent case, where

*k*is replaced with

*iκ*.

*ɛ*(

*ω*).

*U*is given by For the case of a dispersive slab, Eq. (30) reduces to For a medium with dielectric permittivity given by Eq. (22), such that the energy velocity divided by

*c*(normalized to the speed of light), reduces to which, as can be seen clearly, remains always subluminal and will oscillate due to the term

*v*=

_{g}*∂ω*/

*∂k*, which is valid only for regions where

*ɛ*> 0. The energy, group and group delay velocities are plotted in Fig. 2(a). A close-up look of the region where

*ɛ*→ 0 is given in Fig. 2(b).

*ɛ*→ 1. It can be seen that the group delay velocity becomes superluminal in the tunneling regime (

*ɛ*< 0). It is worth noting here that while the group velocity approaches zero at

*ɛ*= 0, the energy flow velocity is never zero. In tunneling, for the non resonant case it can be very small. It is interesting to notice that the energy velocity remains small for tunneling even in the resonant case, contrary to the resonant cases with surface wave resonances as in Section 4.

*ω*

_{0}is the resonant frequency of the electron motion [17], the situation is similar. The energy velocity will oscillate and it will be equal to the group delay velocity when the transmission is total. The group velocity is not equal to the energy velocity. Similarly to the previous model, all the velocities tend to the same value as

*ɛ*→ 1 as can be seen from Fig. 2(c).

*ɛ*< 1 implies dispersion and therefore Eq. (36) is no longer valid. For 0 <

*ɛ*< 1, the group velocity will be superluminal.

*et al*[39

39. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E **63**, 036610 (2001). [CrossRef]

## 4. Velocities for a double layer and a double barrier

*ɛ*

_{1}=

*ɛ*

_{1}(

*ω*) and

*ɛ*

_{2}=

*ɛ*

_{2}(

*ω*) and widths

*L*

_{1}and

*L*

_{2}, for a total width of

*L*=

*L*

_{1}+

*L*

_{2}. The electromagnetic wave that will be considered is TM-polarized (p-polarized), with the electric field in the incidence plane (

*y,z*) and the magnetic field in the x-direction. Denoting: the fields in the structure are given by

_{1}, Δ

_{2}and Δ

_{3}are defined as The resonant conditions of the structure are given by the conditions [18

**107**, 293–314 (2010). [CrossRef]

*c*.At the resonance, where the transmission coefficient is 1, the group delay velocity is equal to the energy velocity. The group delay velocity becomes superluminal away from the resonance.

*a*are bounded by vacuum and separated by another vacuum region of width

*L*as shown in Fig. 4(b). As in the double layer case, a TM-polarized electromagnetic wave will be considered and the same notations will be used.

36. V. S. Olkhovsky, E. Recami, and A. K. Zaichenko, “Resonant and non-resonant tunneling through a double barrier,” Europhys. Lett. **70**, 712–718 (2005). [CrossRef]

40. M. Razavy, *Quantum Theory of Tunneling* (WorldScientific, 2003). [CrossRef]

**107**, 293–314 (2010). [CrossRef]

**107**, 293–314 (2010). [CrossRef]

*c*. The group delay velocity is equal to the energy velocity at the resonance and becomes superluminal away from the resonance.

## 5. Discussion and summary

**v**

*=*

_{g}*∂ω*/

*∂*

**k**can be used as a measure of the energy velocity only for transparent (non-dissipative) weakly dispersive and infinite medium when the wave packet width is shorter than the characteristic width of the propagation region and no multiple reflections occur. In the latter case, the conventional group velocity is equal to the energy flow velocity given by the ratio of the Poynting flux to the energy density,

**v**

*=*

_{g}**v**

*≡*

_{E}**S**/

*U*. This equality is broken however in many instances. Even in the simplest case of a transparent (non-dissipative) dispersive medium the energy transport velocity in a finite width region differs from the group velocity due to nonlocal (boundary) effects as it was discussed in Section 3.

**v**

*offers a natural generalization of the conventional group velocity*

_{E}**v**

*=*

_{g}*∂ω*/

*∂*

**k**. It is worth noting here that the actual energy transport (and the energy velocity

**v**

*) remains finite even when the conventional group velocity becomes zero [41*

_{E}41. Y. Chen and S. Blair, “Nonlinearity enhancement in finite coupled-resonator slow-light waveguides,” Opt. Express **12**, 3353–3366 (2004). [CrossRef] [PubMed]

**v**

*describes the energy transport in other situations, such as in dissipative media, where the group velocity may become superluminal (and thus has not the physical meaning of the energy/information propagation velocity) and when the group velocity is not well defined (as in tunneling problems).*

_{E}## Acknowledgments

## References and links

1. | R. Chiao and A. Steinberg, |

2. | A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A |

3. | A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E |

4. | L. Ragni, “Group delay of evanescent signals in a waveguide with barrier,” Phys. Rev. E |

5. | M. T. Reiten, D. Grischkowsky, and R. A. Cheville, “Optical tunneling of single-cycle terahertz bandwidth pulses,” Phys. Rev. E |

6. | A. Enders and G. Nimtz, “Evanescent-mode propagation and quantum tunneling,” Phys. Rev. E |

7. | T. Baba, “Slow light in photonic crystals,” Nat. Photonics |

8. | J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. |

9. | R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science |

10. | J. F. Galisteo-López, M. Galli, A. Balestreri, M. Patrini, L. C. Andreani, and C. López, “Slow to superluminal light waves in thin 3D photonic crystals,” Opt. Express |

11. | C. M. de Sterke, K. B. Dossou, T. P. White, L. C. Botten, and R. C. McPhedran, “Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes,” Opt. Express |

12. | M. Mojahedi, K. Malloy, G. Eleftheriades, J. Woodley, and R. Chiao, “Abnormal wave propagation in passive media,” IEEE J. Sel. Top. Quantum Electron. |

13. | C. Ling, M. Zheng, and K. Yu, “Slowing light in diatomic nanoshelled chains,” Opt. Commun. |

14. | C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. |

15. | L. Brillouin and A. Sommerfeld, |

16. | E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. |

17. | P. Yeh and A. Yariv, |

18. | A. I. Smolyakov, E. A. Fourkal, S. I. Krasheninnikov, and N. Sternberg, “Resonant modes and resonant transmission in multi-layer structures,” Prog. Electromagn. Res. |

19. | E. Fourkal, I. Velchev, C.-M. Ma, and A. Smolyakov, “Resonant transparency of materials with negative permittivity,” Phys. Lett. A |

20. | J. C. Garrison, M. W. Mitchell, R. Y. Chiao, and E. L. Bolda, “Superluminal signals: causal loop paradoxes revisited,” Phys. Lett. A |

21. | T. Sauter and F. Paschke, “Can Bessel beams carry superluminal signals?” Phys. Lett. A |

22. | A. M. Steinberg, “How much time does a tunneling particle spend in the barrier region?” Phys. Rev. Lett. |

23. | A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. |

24. | N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. |

25. | N. Borjemscaia, S. V. Polyakov, P. D. Lett, and A. Migdall, “Single-photon propagation through dielectric bandgaps,” Opt. Express |

26. | D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. |

27. | L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature |

28. | H. G. Winful, “Apparent superluminality and the generalized hartman effect in double-barrier tunneling,” Phys. Rev. E |

29. | H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. |

30. | A. Dogariu, A. Kuzmich, H. Cao, and L. Wang, “Superluminal light pulse propagation via rephasing in a transparent anomalously dispersive medium,” Opt. Express |

31. | J. Wang, Y. Zhang, J. Zhang, Y. Cai, X. Zhang, and P. Yuan, “Simultaneous observation of superluminal and slow light propagation in a nested fiber ring resonator,” Opt. Express |

32. | N. Malkova, G. W. Bryant, S. Polyakov, and A. Migdall, “Effect of surface modes on photon traversal through stop bands of dielectric stacks,” Phys. Rev. B |

33. | T. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. Rep. |

34. | H. G. Winful, “Delay time and the Hartman effect in quantum tunneling,” Phys. Rev. Lett. |

35. | E. H. Hauge and J. A. Støvneng, “Tunneling times: a critical review,” Rev. Mod. Phys. |

36. | V. S. Olkhovsky, E. Recami, and A. K. Zaichenko, “Resonant and non-resonant tunneling through a double barrier,” Europhys. Lett. |

37. | L. Landau and E. Lifshitz, |

38. | E. Schulz-DuBois, “Energy transport velocity of electromagnetic propagation in dispersive media,” Proc. IEEE |

39. | G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E |

40. | M. Razavy, |

41. | Y. Chen and S. Blair, “Nonlinearity enhancement in finite coupled-resonator slow-light waveguides,” Opt. Express |

**OCIS Codes**

(240.0310) Optics at surfaces : Thin films

(240.6690) Optics at surfaces : Surface waves

(240.7040) Optics at surfaces : Tunneling

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: March 4, 2011

Revised Manuscript: June 22, 2011

Manuscript Accepted: July 5, 2011

Published: July 26, 2011

**Citation**

Winston Frias, Andrei Smolyakov, and Akira Hirose, "Non-local energy transport in tunneling and plasmonic structures," Opt. Express **19**, 15281-15296 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-16-15281

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### References

- R. Chiao and A. Steinberg, Progress in Optics , E. Wolf, ed. (Elsevier, 1997).
- A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994). [CrossRef] [PubMed]
- A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008). [CrossRef]
- L. Ragni, “Group delay of evanescent signals in a waveguide with barrier,” Phys. Rev. E 79, 046609 (2009). [CrossRef]
- M. T. Reiten, D. Grischkowsky, and R. A. Cheville, “Optical tunneling of single-cycle terahertz bandwidth pulses,” Phys. Rev. E 64, 036604 (2001). [CrossRef]
- A. Enders and G. Nimtz, “Evanescent-mode propagation and quantum tunneling,” Phys. Rev. E 48, 632–634 (1993). [CrossRef]
- T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–463 (2008). [CrossRef]
- J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. 2, 287–318 (2010). [CrossRef]
- R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009). [CrossRef] [PubMed]
- J. F. Galisteo-López, M. Galli, A. Balestreri, M. Patrini, L. C. Andreani, and C. López, “Slow to superluminal light waves in thin 3D photonic crystals,” Opt. Express 15, 15342–15350 (2007). [CrossRef] [PubMed]
- C. M. de Sterke, K. B. Dossou, T. P. White, L. C. Botten, and R. C. McPhedran, “Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes,” Opt. Express 17, 17338–17343 (2009). [CrossRef]
- M. Mojahedi, K. Malloy, G. Eleftheriades, J. Woodley, and R. Chiao, “Abnormal wave propagation in passive media,” IEEE J. Sel. Top. Quantum Electron. 9, 30–39 (2003). [CrossRef]
- C. Ling, M. Zheng, and K. Yu, “Slowing light in diatomic nanoshelled chains,” Opt. Commun. 283, 1945–1949 (2010). [CrossRef]
- C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12, 104003 (2010). [CrossRef]
- L. Brillouin and A. Sommerfeld, Wave Propagation and Group Velocity (Academic Press, 1960).
- E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955). [CrossRef]
- P. Yeh and A. Yariv, Optical Waves in Crystals (Wiley-Interscience, 1984).
- A. I. Smolyakov, E. A. Fourkal, S. I. Krasheninnikov, and N. Sternberg, “Resonant modes and resonant transmission in multi-layer structures,” Prog. Electromagn. Res. 107, 293–314 (2010). [CrossRef]
- E. Fourkal, I. Velchev, C.-M. Ma, and A. Smolyakov, “Resonant transparency of materials with negative permittivity,” Phys. Lett. A 361, 277–282 (2007). [CrossRef]
- J. C. Garrison, M. W. Mitchell, R. Y. Chiao, and E. L. Bolda, “Superluminal signals: causal loop paradoxes revisited,” Phys. Lett. A 245, 19–25 (1998). [CrossRef]
- T. Sauter and F. Paschke, “Can Bessel beams carry superluminal signals?” Phys. Lett. A 285, 1–6 (2001). [CrossRef]
- A. M. Steinberg, “How much time does a tunneling particle spend in the barrier region?” Phys. Rev. Lett. 74, 2405–2409 (1995). [CrossRef] [PubMed]
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