## Electromagnetic energy transport in finite photonic structures |

Optics Express, Vol. 22, Issue 11, pp. 12760-12772 (2014)

http://dx.doi.org/10.1364/OE.22.012760

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### Abstract

We have derived, for oblique propagation, an equation relating the averaged energy flux density to energy fluxes arising in the process of scattering by a lossless finite photonic structure. The latter fluxes include those associated with the dispersion relation of the structure, reflection, and interference between the incident and reflected waves. We have also derived an explicit relation between the energy flux density and the group velocity, which provides a simple and systematical procedure for studying theoretically and experimentally the properties of the energy transport through a wide variety of finite photonic structures. Such a relation may be regarded as a generalization of the corresponding one for infinite periodic systems to finite photonic structures. A finite, N-period, photonic crystal was used to illustrate the usefulness of our results.

© 2014 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987). [CrossRef] [PubMed]

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486–2489 (1987). [CrossRef] [PubMed]

3. K. Sakoda, *Optical Properties of Photonic Crystals* (Springer, Berlin, 2001). [CrossRef]

**S**to time-averaged energy density

*U*[5]. According to this definition, this velocity is in general a local quantity that provides an appropriate measure of the energy transport velocity inside the medium. Now, when describing the global propagation properties of these structures, the quantity of interest is the averaged energy transport velocity, defined as

**v**

*= 〈*

_{E}**S**〉/〈

*U*〉 [3

3. K. Sakoda, *Optical Properties of Photonic Crystals* (Springer, Berlin, 2001). [CrossRef]

**S**〉 and 〈

*U*〉 are the space-averaged energy flux density and space-averaged energy density, respectively. For infinite, higher dimensional photonic crystals, the average is taken over the unit cell [3

3. K. Sakoda, *Optical Properties of Photonic Crystals* (Springer, Berlin, 2001). [CrossRef]

7. S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B **72**,165112 (2005). [CrossRef]

**S**and

*U*at each point of the system [8

8. G. Torrese, J. Taylor, H. P. Schriemer, and M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A: Pure Appl. Opt. **8**, 973–980 (2006). [CrossRef]

*to represent the latter average.*

_{L}17. H. G. Winful, “Group delay, stored energy, and the tunneling of evanescent electromagnetic waves,” Phys. Rev. E **68**,016615 (2003). [CrossRef]

**S**〉

*and those of the energy fluxes arising in the process of scattering by the photonic structure. These Eqs. are used in Sec. 3 to express the components of both*

_{L}**v**

*and 〈*

_{E}**S**〉

*in terms of those of the group velocity*

_{L}**v**

*. The usefulness of the correlation between 〈*

_{g}**S**〉

*and*

_{L}**v**

*in the description and understanding of the energy transport through the considered structures is illustrated in Sec. 4 by applying it to a specific photonic structure. Finally, our conclusions are given in Sec. 5.*

_{g}## 2. Energy flux density

*z*= 0 and

*z*=

*L*planes, as shown schematically in Fig. 1. For simplicity, we assume the structure is sandwiched between two semi-infinite layers made of the same optical materials. We focus our attention on a monochromatic electromagnetic field propagating in the (

*x*,

*z*) plane with wave vector component

*K*along the

_{x}*x*-axis. For TE modes, the spatial part of the electric and magnetic fields can be written as [19

19. M. de Dios-Leyva and O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B **77**,125102 (2008). [CrossRef]

**x**,

**y**and

**z**are the unit vectors along the

*x*,

*y*and

*z*axes, respectively,

*φ*(

*z*) is the phase of

*E*(

*z*), is the total phase of the electric field

**E**(

**r**), and

*g*(

*z*,

*ω*) =

*ωμ*(

*z*), where

*μ*(

*z*) represents the magnetic permeability of the structure.

*μ*(

*z*) and the dielectric permittivity

*ε*(

*z*) are real quantities and may be frequency dependent.

*K*and the effective wavevector

_{x}*K*, which is defined in terms of the phase Φ of the complex transmission amplitude

_{z}*t*as Φ =

*LK*[20

_{z}20. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E **53**, 4107–4121 (1996). [CrossRef]

21. M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,” Phys. Rev. E **60**, 4891–4898 (1999). [CrossRef]

*K*and

_{x}*K*are independent quantities and the dispersion relation of the structure is characterized by the dependence of the frequency

_{z}*ω*on the wavevector

**K**=

**x**

*K*+

_{x}**z**

*K*, i. e.

_{z}*ω*=

*ω*(

*K*,

_{x}*K*). Further, since Eqs. (8) and (9) depend explicitly on

_{z}*ω*and

*K*, the vectors

_{x}**u**and

**v**also depend on

**K**. Note, however, that the dependence of

**u**and

**v**on

*K*is only through the dispersion relation. That is, these vectors are composite functions of

_{z}*K*. This difference between

_{z}*K*and

_{x}*K*will be taken into account in our calculations.

_{z}*K*-derivative in Eqs. (8) and (9) and combining both results, we obtain the Poynting theorem [16

_{α}16. M. de Dios-Leyva and J. C. Drake-Pérez, “Group velocity and nonlocal energy transport velocity in finite photonic structures,” J. Opt. Soc. Am. B **29**, 2275–2281 (2012). [CrossRef]

**S**and

*U*are the time-averaged Poynting vector and energy density, respectively, and

*v*=

_{g}_{α}*∂ω*(

**K**)/

*∂K*is the

_{α}*α*-component of the group velocity. To obtain these expressions we followed the procedure used in [3

3. K. Sakoda, *Optical Properties of Photonic Crystals* (Springer, Berlin, 2001). [CrossRef]

11. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. **69**, 742–756 (1979). [CrossRef]

**u**is a real quantity. It is important to note that since Φ =

*LK*, the

_{z}*z*-component of

**v**

*is given by*

_{g}*v*=

_{gz}*∂ω*(

**K**)/

*∂K*=

_{z}*L/τ*, where

_{d}*τ*=

_{d}*∂*Φ/

*∂ω*. That is,

*τ*represents the group delay or Wigner delay time [22

_{d}22. E. P. Wigner, “Lower Limit for the Energy Derivative of the Scattering Phase Shift,” Phys. Rev. **98**, 145–147 (1955). [CrossRef]

*corresponds to the group delay velocity.*

_{gz}*S*and lateral

_{z}*S*components and determines the local energy flux density inside the structure. The lateral component disappears for normal propagation.

_{x}*z*-component of the Poynting vector

**z**·

**S**is conserved throughout the structure in the absence of losses, the spatial average of Eq. (11) over the entire sample along the

*z*-direction leads to the formulas for

*α*=

*z*,

*x*, respectively, where

*φ*(

*L*) and

*φ*(0) are the phases of the electric field

*E*(

*z*) at the right (

*z*=

*L*) and left (

*z*= 0) interfaces of the photonic structure, and

*θ*is the phase of

*E*(0) = 1 +

*r*= 1 +

*r*

_{1}+

*ir*

_{2}and satisfies the relation

*∂K*=

_{i}/∂K_{j}*δ*, with

_{i,j}*i*,

*j*=

*x*,

*z*, and the fact that the normal energy flow is given by Equations (19) and (20) can be written as for

*α*=

*x*,

*z*, where

*T*= |

*t*|

^{2}is the transmission coefficient and

*g*=

_{L}*ωμ*.

_{L}*G*(

_{α}*z*,

**K**) at the right (

*z*=

*L*) and left (

*z*= 0) interfaces of the structure and Eqs. (23) and (24), it is straightforward to show that

*G*(

_{α}*L*,

**K**) = 0 and for

*α*=

*x*,

*z*.

*Q*(

_{L}*r*−

*r*

^{*}) = 2

*ir*

_{2}

*Q*in the latter Eq. arises from the interference between incident and reflected waves.

_{L}*G*(

_{α}**K**) and

*∂θ/∂K*calculated from (26) into Eq. (28), we get where

_{α}*R*= |

*r*|

^{2}is the reflection coefficient and Φ

*is the phase of the complex reflection amplitude*

_{R}*r*.

*S*whose velocity is determined by the dispersion relation of the structure. The second one is the energy flow along the

_{g}_{α}*α*-axis arising from the interference between the incident and reflected waves, as noted above. In order to interpret the third term we consider, for simplicity, a symmetric photonic structure. In this case, Φ

*= Φ ±*

_{R}*π*/2 =

*LK*±

_{z}*π*/2 and (1/

*L*)

*∂*Φ

*= 0 and 1 for*

_{R}/∂K_{α}*α*=

*x*and

*z*, respectively. Thus, the energy flow associated with the third term vanishes along the lateral direction, whereas it is exactly equal to the reflected energy flow along the normal direction.

## 3. Relation between group velocity and energy flux density

*∂/∂K*in (30) by for

_{α}*α*=

*z*and

*x*, respectively, where the latter term on the right-hand side of (32) represents the derivative with respect to

*K*keeping the frequency

_{x}*ω*constant.

*α*=

*z*and

*x*, respectively, and use Eq. (27) and the relation 〈

*S*〉

_{α}*=*

_{L}*v*〈

_{Eα}*U*〉

*, we obtain, after some algebraic manipulation, the expressions where*

_{L}*τ*=

_{d}*L/v*is the group delay,

_{gz}*v*,

_{Ez}*v*and

_{gz}*K*, that is to say, the effects of oblique propagation do not modify it.

_{x}20. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E **53**, 4107–4121 (1996). [CrossRef]

23. M. de Dios-Leyva and J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. **109**,103526 (2011). [CrossRef]

*X*and

*Y*are the real and imaginary parts of

*t*,

*F*(

*ω*,

*K*) depends only explicitly on

_{x}*ω*and

*K*, as discussed above, and Φ =

_{x}*LK*is the phase of

_{z}*t*=

*X*+

*iY*.

*τ*and the ratio between the components of the group velocity: where

_{d}*T*=

*X*

^{2}+

*Y*

^{2}is the transmission coefficient.

## 4. Application to finite, N-period, photonic crystals

13. W. Frias, A. Smolyakov, and A. Hirose, “Non-local energy transport in tunneling and plasmonic structures,” Opt. Express **19**, 15281–15296 (2011). [CrossRef] [PubMed]

19. M. de Dios-Leyva and O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B **77**,125102 (2008). [CrossRef]

24. H. Daninthe, S. Foteinopoulou, and C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications **4**, 123–131 (2006). [CrossRef]

25. A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, and Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A **79**,013820 (2009). [CrossRef]

*A*[

*BABA...BA*]

*A*sandwiched between two semi-infinite layers made of the same optical materials

*A*, characterized by positive and frequency independent optical parameters

*ε*

_{1}and

*μ*

_{1}. For now we will leave the nature of layer

*B*unspecified, beyond requiring that their optical parameters

*ε*

_{2}(

*ω*) and

*μ*

_{2}(

*ω*) be real quantities, as assumed above.

*E*(

*z*) for

*z*< 0 and

*z*>

*L*are giving by Eqs. (23) and (24), respectively, and using the transfer-matrix technique [23

23. M. de Dios-Leyva and J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. **109**,103526 (2011). [CrossRef]

*T*

_{22}and

*T*

_{21}: where

*i*= 1, 2,

*η*=

*μ*

_{2}

*Q*

_{1}/

*μ*

_{1}

*Q*

_{2},

*N*is the number of unit cells,

*a*and

*b*are the widths of layers

*A*and

*B*, respectively, and

*β*is the Bloch phase associated with the corresponding infinite photonic crystal which satisfies the dispersion relation:

*Q*

_{1}is real,

*Q*

_{2}may be real or purely imaginary, and the Bloch phase

*β*is real inside the allowed bands and equal to

*iψ*or to

*π*+

*iψ*in the energy gap regions, where

*ψ*is a real angle, the function

*g*and both cos

*Nβ*and sin

*Nβ*/sin

*β*are always real quantities. These properties and the first relation in Eq. (46) lead to the following formulas for the transmission amplitude and the dispersion relation of the finite photonic crystal: where is the transmission coefficient and is the real part of

*t*.

*of the complex reflection amplitude*

_{R}*r*, which leads immediately to

*∂*Φ

*=*

_{R}/∂ω*τ*−

_{d}*τ*and (

_{a}*∂*Φ

*)*

_{R}/∂K_{x}*=*

_{ω}*aK*

_{x}/Q_{1}, where

*τ*=

_{a}*a∂Q*

_{1}

*/∂ω*is the time the electromagnetic wave spends in layer

*A*. If the latter relations are used in combination with Eq. (36) and the fact that

*T*+

*R*= 1, Eq. (44) becomes:

*v*should be calculated by combining Eqs. (49), (53) and (55). As a result, we obtain: where

_{gx}/v_{gz}*g*and

*f*are the functions shown in Eqs. (52)–(53) and

*g′*= (

*∂g/∂K*)

_{x}*and*

_{ω}*f′*= (

*∂f/∂K*)

_{x}*.*

_{ω}*x*-component of the space-averaged energy flux density.

*θ*, the structure of 〈

_{i}*S*〉

_{z}*as a function of*

_{L}*ω*is the same as that of the transmission coefficient. In consequence, the

*z*-component of the energy flux density exhibits maxima at transmission resonances which, according to Eq. (56), correspond to the conditions

*Nβ*=

*mπ*, with

*m*= ±1, ±2,...,±(

*N*− 1);

*bQ*

_{2}=

*nπ*, with

*n*= 1, 2, 3,...; and

*η*= ±1. Note that only the former condition depends on the number

*N*of unit cells. Thus, when the frequency

*ω*varies within an allowed miniband of the corresponding infinite photonic crystal, 〈

*S*〉

_{z}*is an oscillating function of*

_{L}*ω*and exhibits a resonant structure. For frequencies inside the bandgaps of the infinite crystal, the Bloch phase

*β*is a complex quantity and, according to Eq. (56), the resonant structure of 〈

*S*〉

_{z}*should disappear. These properties of the 〈*

_{L}*S*〉

_{z}*-spectra are illustrated in Fig. 2 for a finite,*

_{L}*N*-period, quarter-wave-stack (

*λ*

_{0}/4 =

*πc*/2

*ω*

_{0}structure)[20

20. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E **53**, 4107–4121 (1996). [CrossRef]

26. G Torrese, J. Taylor, T. J. Hall, and P. Mégret, ”Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E **73**,066616 (2006). [CrossRef]

*μ*

_{1}=

*μ*

_{2}= 1,

*N*= 5, 10 and various values of the angle of incidence

*θ*. One sees in Fig. 2 that the main effects of increasing

_{i}*θ*, for a given value of

_{i}*N*, are to shift the 〈

*S*〉

_{z}*-spectra to higher frequencies and to reduce the corresponding resonant-peak values.*

_{L}*R*=

*r*

_{2}= 0 at transmission resonances, it follows from Eqs.(49), (57) and (62) that, for a fixed value of

*θ*≠ 0 and ignoring quantitative differences, the lateral energy flux density 〈

_{i}*S*〉

_{x}*and the ratio*

_{L}*V*=

_{fin}*v*for the finite photonic crystals should exhibit resonant structures similar to those of 〈

_{gx}/v_{gz}*S*〉

_{z}*. It is clearly seen in Eq. (62) that the peak values of 〈*

_{L}*S*〉

_{x}*and*

_{L}*V*are the same at each transmission resonance if the factor cos

_{fin}*θ*is ignored. These theoretical results are illustrated in Fig. 3 for the

_{i}*λ*

_{0}/4 photonic structure with the same parameters used in Fig. 2.

*V*with those of the ratio

_{fin}*V*=

_{inf}*v*for the corresponding infinite crystals. This will provide an appropriate understanding of the behavior of

_{gx}/v_{gz}*V*. Such a comparison is shown in Fig. 4 for the

_{fin}*λ*

_{0}/4 structure with the same parameters used in Fig. 2. One clearly sees that, for a fixed value of N, the oscillations of

*V*in a pass band always occur around the curve associated with

_{fin}*V*. This means that the pass bands of the infinite crystals are similar to the corresponding ones of the finite crystal, specially for large values of N. The difference inside the bandgaps of the infinite crystals, which tends to disappear for large values of

_{inf}*N*, is due to the fact that the effect of finite crystal size is to create photon states inside these gaps.

## 5. Conclusion

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | K. Sakoda, |

4. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, |

5. | L. Brillouin, |

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

7. | S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B |

8. | G. Torrese, J. Taylor, H. P. Schriemer, and M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A: Pure Appl. Opt. |

9. | R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A |

10. | P. Y. Chen, R. C. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A |

11. | P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. |

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

13. | W. Frias, A. Smolyakov, and A. Hirose, “Non-local energy transport in tunneling and plasmonic structures,” Opt. Express |

14. | R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A |

15. | N. Le Thomas and R. Houdré, “Group velocity and energy transport velocity near the band edge of a disordered coupled cavity waveguide: an analytical approach,” J. Opt. Soc. Am. B |

16. | M. de Dios-Leyva and J. C. Drake-Pérez, “Group velocity and nonlocal energy transport velocity in finite photonic structures,” J. Opt. Soc. Am. B |

17. | H. G. Winful, “Group delay, stored energy, and the tunneling of evanescent electromagnetic waves,” Phys. Rev. E |

18. | R. E. Collin, |

19. | M. de Dios-Leyva and O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B |

20. | J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E |

21. | M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,” Phys. Rev. E |

22. | E. P. Wigner, “Lower Limit for the Energy Derivative of the Scattering Phase Shift,” Phys. Rev. |

23. | M. de Dios-Leyva and J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. |

24. | H. Daninthe, S. Foteinopoulou, and C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications |

25. | A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, and Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A |

26. | G Torrese, J. Taylor, T. J. Hall, and P. Mégret, ”Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E |

**OCIS Codes**

(260.2160) Physical optics : Energy transfer

(160.5293) Materials : Photonic bandgap materials

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: February 24, 2014

Revised Manuscript: April 21, 2014

Manuscript Accepted: April 22, 2014

Published: May 19, 2014

**Citation**

M. de Dios-Leyva, C. A. Duque, and J. C. Drake-Pérez, "Electromagnetic energy transport in finite photonic structures," Opt. Express **22**, 12760-12772 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-12760

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### References

- E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]
- S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]
- K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001). [CrossRef]
- J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 2008).
- L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).
- A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, 1984).
- S. Foteinopoulou, C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72,165112 (2005). [CrossRef]
- G. Torrese, J. Taylor, H. P. Schriemer, M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A: Pure Appl. Opt. 8, 973–980 (2006). [CrossRef]
- R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970). [CrossRef]
- P. Y. Chen, R. C. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82,053825 (2010). [CrossRef]
- P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979). [CrossRef]
- G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63,036610 (2001). [CrossRef]
- W. Frias, A. Smolyakov, A. Hirose, “Non-local energy transport in tunneling and plasmonic structures,” Opt. Express 19, 15281–15296 (2011). [CrossRef] [PubMed]
- R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002). [CrossRef]
- N. Le Thomas, R. Houdré, “Group velocity and energy transport velocity near the band edge of a disordered coupled cavity waveguide: an analytical approach,” J. Opt. Soc. Am. B 27, 2095–2101 (2010). [CrossRef]
- M. de Dios-Leyva, J. C. Drake-Pérez, “Group velocity and nonlocal energy transport velocity in finite photonic structures,” J. Opt. Soc. Am. B 29, 2275–2281 (2012). [CrossRef]
- H. G. Winful, “Group delay, stored energy, and the tunneling of evanescent electromagnetic waves,” Phys. Rev. E 68,016615 (2003). [CrossRef]
- R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1992).
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