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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 15 — Jul. 23, 2007
  • pp: 9681–9691
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The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction

Marcelo Davanço, Yaroslav Urzhumov, and Gennady Shvets  »View Author Affiliations


Optics Express, Vol. 15, Issue 15, pp. 9681-9691 (2007)
http://dx.doi.org/10.1364/OE.15.009681


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Abstract

The propagation characteristics of a subwavelength plasmonic crystal are studied based on its complex Bloch band structure. Photonic crystal bands are generated with an alternative 2D Finite Element Method formulation in which the Bloch wave problem is reduced to a quadratic eigenvalue system for the Bloch wavevector amplitude k. This method constitutes an efficient and convenient alternative to nonlinear search methods normally employed in the calculation of photonic bands when dispersive materials are involved. The method yields complex wavevector Bloch modes that determine the wave-scattering characteristics of finite crystals. This is evidenced in a comparison between the band structure of the square-lattice plasmonic crystal and scattering transfer-functions from a corresponding finite crystal slab. We report on a wave interference effect that leads to transmission resonances similar to Fano resonances, as well as on the isotropy of the crystal’s negative index band. Our results indicate that effective propagation constants obtained from scattering simulations may not always be directly related to individual crystal Bloch bands.

© 2007 Optical Society of America

1. Introduction

The study of the optical properties of Photonic Crystals (PCs) has relied on the generation of Bloch-mode photonic band structures. Information such as the existence of forbidden band gaps, phase and group velocity, group velocity dispersion, photonic densities of states, propagation anisotropy, etc. may be conveniently obtained by inspection or manipulation of the computed propagation bands [1

1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. (Princteon University Press, 1995).

, 2

2. K. Sakoda, Optical Properties of Photonic Crystal, ser. Optical Sciences. (New York: Springer, 2001).

]. A commonly adopted approach for the calculation of PC band structures using frequency-domain methods consists in obtaining the allowed propagation frequencies ω for specific Bloch k-vectors as eigenvalues of an electromagnetic wave equation defined in a unit-cell of the periodic medium. If the PC incorporates only dispersionless materials (i.e., the dielectric constant ε(ω) constant for all frequencies), the problem is reduced to a linear generalized eigenvalue system of the form Akxk=ω 2 Bkxk [1

1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. (Princteon University Press, 1995).

4

4. B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, “Aplication of finite element methods to photonic crystal modelling,” IEE Proc - Sci. Meas. Technol. 149, 293–296 (2002). [CrossRef]

].

In Sec. 2, we derive the FEM formulation for the plasmonic crystal eigenvalue problem following similar steps as [4

4. B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, “Aplication of finite element methods to photonic crystal modelling,” IEE Proc - Sci. Meas. Technol. 149, 293–296 (2002). [CrossRef]

]. This formulation is implemented in a simple way with the COMSOL Finite Elements package [11] and used to produce the photonic bands presented in Sec. 3, for a square-lattice SPC displaying a negative refraction band [5

5. G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. 93, 243902 (2004). [CrossRef]

,6

6. G. Shvets and Y. Urzhumov, “Electric and magnetic properties of sub-wavelength plasmonic crystals,” J. Opt. A: Pure Appl. Opt. 7, S23–S31 (2005). [CrossRef]

]. In these references, the negative refraction band was calculated only along one of the high-symmetry crystal directions; in the present article, the photonic bands are obtained over the entire irreducible first Brillouin zone, such that the isotropy of the negative refraction band near k=0 is revealed. A comparison between scattering data from a finite SPC and its band diagram is also performed, in which the role of the multiple existing Bloch modes (including those with imaginary and complex wavevectors) is determined. In particular, we report on the existence of resonances in transmission through a finite crystal slab that stem from the coexistence of two SPC bands in the same frequency range, being similar in origin and character to Fano resonances [12

12. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961). [CrossRef]

]. In Sec. 4 we present conclusions.

2. Finite-element method

We consider a periodic 2D medium with electromagnetic waves propagating in the xy plane. Solutions toMaxwell’s equations are classified as either TE (H=Hz , where H is the magnetic field) or TM (E=Ez , where E is the electric field), with the respective z-components obeying the wave equation

·pϕ+ω2c2qϕ=0.
(1)

For TE waves, ϕ=Hz, p=1/ε, q=1 and ε is the electric permittivity; for TM waves, ϕ=Hz, p=1, q=ε. In Eq.(1), ω/c is the vacuum angular wavenumber.

The medium is periodic, and hence ε(r+T)=ε(r), where T is a lattice translation vector. From Bloch’s theorem we consider solutions ϕ(r)=u(r)exp(-i k·r), with u(r+T)=u(r) and k in the first Brillouin zone. Along with Eq.(1), this leads to:

·(pu)i·(pku)ik·puk2pu=ω2c2qu
(2)

Following the Galerkin procedure to reduce Eq.(1) to a weighted-residuals expression [13

13. J. Jin, The Finite Element Method in Electromagnetics, (2nd ed.Wiley, 2002).

], we multiply that equation on both sides by a weight function w(r), and note that

·(pu)=·[p(u)w]pu·w
(3)

and

·(pku)w=·[pkuw]k·(puw)
(4)

to arrive at

·[p(u)w]pu·wi·(pkuw)+
ik·{puw[p(u)w]}k2puw=
ω2c2quw
(5)

This expression is integrated over a closed domain Γ with boundary δ Γ that covers one unit cell of the periodic medium (the integration domain may actually extend over an integer number of unit cells). Making use of the Divergence theorem, the following weak-form expression is obtained:

Γpu·wdΓ+k2ΓpuwdΓ+
ik{Γpuk̂·wdΓΓpk̂·(u)wdΓ}=
=ω2c2ΓquwdΓ+δΓpw(uiku)·n̂dδΓ
(6)

In Eq.(6), n̂ is the outward normal unit vector to the boundary δ Γ. Since u is periodic, the line integral vanishes. The resulting integrodifferential equation may be transformed into matrix format by following the usual FEM discretization procedure [4

4. B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, “Aplication of finite element methods to photonic crystal modelling,” IEE Proc - Sci. Meas. Technol. 149, 293–296 (2002). [CrossRef]

, 13

13. J. Jin, The Finite Element Method in Electromagnetics, (2nd ed.Wiley, 2002).

]: the domain Γ is divided into several triangular subdomains (elements) in which locally supported expansion functions are defined; u is expanded in terms of such functions within each element; w is taken to be each one of the local expansion functions inside each element; and the material parameters p and q are allowed to be constant inside each element. Then the following quadratic matrix eigenvalue equation in k results:

[Aω2c2D]u=ik(CB)u+k2Cu.
(7)

3. Rectangular array plasmonic crystal

For comparison, the transmission (t) and reflection (r) coefficients for plane-waves incident on a finite (i.e., with N of unit cells) crystal slab were obtained using the same technique as described in [15

15. G. Shvets and Y. Urzhumov, “Negative index meta-materials based on two-dimensional metallic structures,” J. Opt. A: Pure Appl. Opt. 8, S122–S130 (2006). [CrossRef]

]. The incident plane wavew is in the Γ-X direction of the square-lattice Brillouin zone. Furthermone, an effective index n eff is calculated using expressions from the scattering-parameter technique of Smith et al. [15

15. G. Shvets and Y. Urzhumov, “Negative index meta-materials based on two-dimensional metallic structures,” J. Opt. A: Pure Appl. Opt. 8, S122–S130 (2006). [CrossRef]

, 16

16. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005). [CrossRef]

], i.e.:

neff=lnY(ω)iωΔc
(8)

with

X(1r2+t2)2t
(9)

and

YX±1X2
(10)

These expressions model a 1D Fabry-Pérot cavity for waves propagating with a single wavevector at each frequency. This in principle precludes their direct application in situations when multiple waves associated with different k-vectors exist simultaneously - for instance in systems displaying Fano-type resonances as shown below.

Figure 1 shows the calculated band diagram in the Γ-X direction of the square lattice irreducible Brillouin zone, with ωc=0, in terms of both kr and ki (k=kr+iki). Modes for which ki=0 are marked with crosses, while complex-k modes are marked with circles and triangles. Duplicates of such eigensolutions are also produced by the formulation, with kr=kr±2m·π/a (kr in the first B.Z., m=1,2, …) and identical ki. Such solutions only arise because they satisfy the established periodic boundary conditions at the edges of the unit-cell; they do not participate in the Bloch-mode expansion of the total field in the crystal. Notice as well that since the calculated bands are for a crystal with zero damping losses (wc=0), the existence of complex eigenvalues is completely unrelated to power dissipation.

Same as in the case of completely dielectric photonic crystals, real-k modes extend over the entire crystal, with a phase evolution given by the k(Ω) dispersion and with no amplitude variation apart from that of the periodic Bloch envelope. In Fig. 1, purely real eigenvalues are found in the ranges Ω≈0.2-0.36 and Ω≈0.54-0.6.

It has been suggested that imaginary k modes play an important role in representing the decaying field inside a finite or semi-infinite PC upon wave incidence at bandgap frequencies [10

10. E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005). [CrossRef]

]. The inference that the imaginary part of the Bloch wavevector relates to the field decay rate in this situation is supported by our analysis. In Fig. 1, non-real eigenvalues may be either purely imaginary (kr=0), or complex. Modes with purely imaginary k (found in the intervals Ω=0.43-0.52 and Ω=0.6-0.61) have no phase variation across a unit-cell, and are non-propagating. Complex k modes, on the other hand, present a phase variation concomitant with an exponential amplitude decay (or growth), which translate into interference effects in scattering from a finite crystal. In Fig. 1, complex k modes with |kr| <π/a are found between Ω=0.36 and 0.43, and between Ω=0.52 and 0.54. Bands with ki marked with triangles in Fig. 1(c) have real parts at the Brillouin zone boundaries, krπ/a, such that the phase varies in full cycles across a unit cell. These solutions will henceforth be referred to as zone-boundary modes.

Fig. 1. (a)Square-lattice plasmonic crystal band structure as function of kr. (b) Band structure as function of ki. Crosses indicate bands with ki=0; circles, ki≠0; and triangles, krπ/a, ki≠0
Fig. 2. (a)Amplitude-squared transmission and reflection coefficients, (b) Band structure as function of kr and (c) Band structure as function of ki for a.

Fig. 3. (a) Comparison between numerically calculated transmission coefficient amplitudes for 3- and 4-unit-cell crystal slabs (thin lines) and the analytical expression |t(Ω)|=A·| exp(-ki(Ω)·N·a)+i·β· t FP(ki(Ω)·N·a)| (thick lines), with ki(Ω) and kr(Ω) taken from the purely real and imaginary Bloch bands respectively. (b)Amplitude and phase of the transmission transfer function for the 4-unit-cell crystal with ωc/ωp=0.001 (thick line) and ωc=0 (thin line).

In the range Ω≈0.36-0.43, the transmitted power is extremely low, yet the transfer function presents a series of anti-resonances corresponding to the observed peaks in Im{k eff} with concomitant step-transitions in Re{k eff}. At the same time, disregarding the resonances, both real and imaginary parts of the effective wavevector closely follow the corresponding photonic bands. Hence, this behavior has an origin similar to the resonances discussed above, given the existence of two evanescent Bloch bands, as seen in Figs. 1(a) and 1(b).

Fig. 4. (a) Band structure for (a)kr and (b)ki. Dots correspond to ωc=0, crosses to ωc/ωp=0.001.

As mentioned above, in the lossless case, the matrices involved are Hermitian, such that eigenvalues are either purely real or come in complex conjugate pairs. The latter case is observed in Ω=0.36-0.43, where two degenerate modes at each kr exist. The degeneracy is evident from the opening of bifurcations in ki at Ω≈ 0.36 and splitting of ki bands in Ω=0.36-0.43 due to small damping losses. The singled-out band in Fig. 4(a) is a perturbed version of a zone-boundary band that exists at Ω⪷0.365 in the lossless case. Above Ω⪷0.365, the latter ceases to be a zone-boundary band, and becomes a degenerate complex-k band. This transition is evident in the departure of the perturbed band from kr=π/a and its merging with a second complex-k band around Ω=0.36.

Finally, to show that the method may be conveniently applied to the generation of bands in arbitrary directions in the k-vector space, Figs. 5 and 6 display, respectively, the low-frequency and two high-frequency purely real-k bands. These diagrams were obtained simply by changing the direction of the unit k-vector in Eq. (6).

Fig. 5. Low-frequency, purely real bands of the square-lattice plasmonic crystal over the first Brillouin zone.

Figure 5 shows that the lowest purely real band is highly isotropic at lower frequencies, however becomes anisotropic as the photonic bandgap is approached. Figure 6 shows slices of the upper real bands, which in reality form a closed surface. The negative group velocity band Ω=0.57-0.6 is highly isotropic near its top, thus making the square array of circular rods a perfectly isotropic negative-index metamaterial for the range of frequencies Ω=0.58-0.6.

Fig. 6. High-frequency, purely real bands of the square-lattice plasmonic crystal over the first Brillouin zone.

On the other hand, the longitudinal-wave, or Bulk Plasmon, band near Ω=0.0.61 predicted in [5

5. G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. 93, 243902 (2004). [CrossRef]

] displays very strong anisotropy near the Γ-point. This anisotropy is apparent in Fig. 7, where the Bulk Plasmon band was plotted for the Γ-M (0°) and Γ-X (45°) directions.

Fig. 7. Bulk Plasmon band along Γ-M (0°) and Γ-X (45°) directions.

4. Conclusions

The Finite-Element formulation presented in Section 2 is convenient for the calculation o photonic band structures of periodic metamaterials composed of dispersive materials. The formulation yields a quadratic eigenvalue equation system that may be easily implemented with commercial, generic Finite-Element packages such as COMSOLMultiphysics, and that may be more efficiently solved than the general nonlinear eigenvalue problem that is normally solved in the metamaterials community. In addition, calculation of bands in arbitrary directions in k-space simply involves determination of the components of the unit k-vector, which is convenient for the analysis of medium anisotropy.

An important advantage associated with solving the quadratic eigenvalue problem is that it yields Bloch eigenmodes with not only real but also complex wavevectors, which, as suggested in [10

10. E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005). [CrossRef]

], may be associated with many effects observed in the scattering of waves from finite crystals, especially at, but not limited to, bandgap frequency ranges. A clear indication of this in fact is the existence of the Fano-type resonances in transmission as shown in Section 3.

It is clear from the analysis in Section 3 that the effective index n eff determined from Eq.(8) via scattering from finite crystals will not completely coincide with effective indices of particular Bloch modes when multiple such modes exist that influence the scattering process at similar levels. At the same time, just the Bloch band structure by itself does not give an immediately clear picture of wave scattering and therefore should be used with some attention in predicting the scattering response of finite structures. At any rate, both methods may be used in conjunction to form a more complete understanding of the propagation characteristics of a metamaterial.

Finally, it must be pointed out that alternative formulations to the present finite element-based one presented in this article may also lead to quadratic eigenvalue equations, see for instance the plane-wave expansion formulation used in [10

10. E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005). [CrossRef]

].

Acknowledgements

The authors thank Dr. Stephen Forrest for support, encouragement and useful comments. This work was supported by the Air Force Office of Scientific Research under contract FA 9550-06-01-0279 through the Multidisciplinary University Research Initiative Program, as well as ARO MURI W911NF-04-01-0203, and DARPA contract HR0011-05-C-0068.

References and links

1.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. (Princteon University Press, 1995).

2.

K. Sakoda, Optical Properties of Photonic Crystal, ser. Optical Sciences. (New York: Springer, 2001).

3.

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]

4.

B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, “Aplication of finite element methods to photonic crystal modelling,” IEE Proc - Sci. Meas. Technol. 149, 293–296 (2002). [CrossRef]

5.

G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. 93, 243902 (2004). [CrossRef]

6.

G. Shvets and Y. Urzhumov, “Electric and magnetic properties of sub-wavelength plasmonic crystals,” J. Opt. A: Pure Appl. Opt. 7, S23–S31 (2005). [CrossRef]

7.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005), doi:10.1103/PhysRevLett.95.203901. [CrossRef] [PubMed]

8.

A. Ruhe, “Algorithms for the nonlinear eigenvalue problem,” SIAM J. Numer. Anal. 10, 674–689 (1973). [CrossRef]

9.

A. Spence and C. Poulton, “Photonic band structure calculations using nonlinear eigenvalue techniques,” J. Comput. Phys. 204, 65–81 (2005). [CrossRef]

10.

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005). [CrossRef]

11.

[Online]. Available: http://www.comsol.com

12.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961). [CrossRef]

13.

J. Jin, The Finite Element Method in Electromagnetics, (2nd ed.Wiley, 2002).

14.

F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. 43, 235–286 (2001). [CrossRef]

15.

G. Shvets and Y. Urzhumov, “Negative index meta-materials based on two-dimensional metallic structures,” J. Opt. A: Pure Appl. Opt. 8, S122–S130 (2006). [CrossRef]

16.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005). [CrossRef]

17.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, “Nature of lossy bloch states in polaritonic photonic crystals,” Phys. Rev. B 69, 195111 (2004). [CrossRef]

18.

H. G. Winful, “The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities,” New J. Phys., Phys. 8, 101 (2006). [CrossRef]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(160.4670) Materials : Optical materials
(260.2030) Physical optics : Dispersion
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Physical Optics

History
Original Manuscript: June 1, 2007
Revised Manuscript: July 15, 2007
Manuscript Accepted: July 16, 2007
Published: July 19, 2007

Citation
Marcelo Davanco, Yaroslav Urzhumov, and Gennady Shvets, "The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction," Opt. Express 15, 9681-9691 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9681


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References

  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. (Princteon University Press, 1995).
  2. K. Sakoda, Optical Properties of Photonic Crystal, ser. Optical Sciences. (New York: Springer, 2001).
  3. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for maxwell’s equations in a planewave basis," Opt. Express 8, 173-190 (2001). [CrossRef] [PubMed]
  4. B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, "Aplication of finite element methods to photonic crystal modelling," IEE Proc - Sci. Meas. Technol. 149, 293-296 (2002). [CrossRef]
  5. G. Shvets and Y. A. Urzhumov, "Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances," Phys. Rev. Lett. 93, 243902 (2004). [CrossRef]
  6. G. Shvets and Y. Urzhumov, "Electric and magnetic properties of sub-wavelength plasmonic crystals," J. Opt. A: Pure Appl. Opt. 7, S23-S31 (2005). [CrossRef]
  7. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005). [CrossRef] [PubMed]
  8. A. Ruhe, "Algorithms for the nonlinear eigenvalue problem," SIAM J. Numer. Anal. 10, 674-689 (1973). [CrossRef]
  9. A. Spence and C. Poulton, "Photonic band structure calculations using nonlinear eigenvalue techniques," J. Comput. Phys. 204, 65-81 (2005). [CrossRef]
  10. E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195122 (2005). [CrossRef]
  11. [Online]. Available: http://www.comsol.com
  12. U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866-1878 (1961). [CrossRef]
  13. J. Jin, The Finite Element Method in Electromagnetics, (2nd ed. Wiley, 2002).
  14. F. Tisseur and K. Meerbergen, "The quadratic eigenvalue problem," SIAM Rev. 43, 235-286 (2001). [CrossRef]
  15. G. Shvets and Y. Urzhumov, "Negative index meta-materials based on two-dimensional metallic structures," J. Opt. A: Pure Appl. Opt. 8, S122-S130 (2006). [CrossRef]
  16. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005). [CrossRef]
  17. K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004). [CrossRef]
  18. H. G. Winful, "The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities," New J. Phys., Phys. 8, 101 (2006). [CrossRef]

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