## Optimal higher-lying band gaps for photonic crystals with large dielectric contrast

Optics Express, Vol. 16, Issue 21, pp. 16600-16608 (2008)

http://dx.doi.org/10.1364/OE.16.016600

Acrobat PDF (491 KB)

### Abstract

We investigate the characteristics of higher-lying band gaps for two-dimensional photonic crystals with large dielectric contrast. An optimal common band gap is attained on a hexagonal lattice of circular dielectric cylinders at relatively higher bands. The corresponding TM and TE modes exhibit simultaneous band edges, around which the frequency branches tend to be dispersionless. Unlike the fundamental band gap which usually appears between the dielectric and air bands, the optimal higher-lying gap in the present study occurs between two consecutive dielectric-like bands with high energy fill factors. The underlying mechanism is illustrated with the apparent change of eigenmode patterns inside the dielectric regions for both polarizations. In particular, the common gap region is bounded by two successive orders of Mie resonance frequencies on a single dielectric cylinder with the same geometry and material, where the Mie resonance modes show similar internal fields with the respective eigenmodes for the photonic crystal.

© 2008 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]

4. A. Moroz and A. Tip, “Resonance-induced effects in photonic crystals,” J. Phys. Condens.Matter **11**, 2503–2512 (1999). [CrossRef]

5. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B **61**, 458–13,464 (2000). [CrossRef]

7. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature **386**, 143–149 (1997). [CrossRef]

8. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. **61**, 495–497 (1992). [CrossRef]

9. P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B **46**, 4969–4972 (1992). [CrossRef]

10. R. L. Chern, C. C. Chang, C. C. Chang, and R. Hwang, “Large full band gaps for photonic crystals in two dimensions computed by an inverse method with multigrid acceleration,” Phys. Rev. E **68**, 26,704 (2003). [CrossRef]

11. H. K. Fu, Y. F. Chen, R. L. Chern, and C. C. Chang, “Connected hexagonal photonic crystals with largest full band gap,” Opt. Express **13**, 7854–7860 (2005). [CrossRef] [PubMed]

*simultaneous*band edges. Accordingly, the common gap width is not trimmed off for either polarization [10

10. R. L. Chern, C. C. Chang, C. C. Chang, and R. Hwang, “Large full band gaps for photonic crystals in two dimensions computed by an inverse method with multigrid acceleration,” Phys. Rev. E **68**, 26,704 (2003). [CrossRef]

12. R. L. Chern, C. C. Chang, C. C. Chang, and R. R. Hwang, “Two Classes of Photonic Crystals with Simultaneous Band Gaps,” Jpn. J. Appl. Phys. **43**, 3484–3490 (2004). [CrossRef]

13. D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B **53**, 7134–7142 (1996). [CrossRef]

14. C. S. Kee, J. E. Kim, and H. Y. Park, “Absolute photonic band gap in a two-dimensional square lattice of square dielectric rods in air,” Phys, Rev, E **56**, 6291–6293 (1997). [CrossRef]

15. L. Shen, S. He, and S. Xiao, “Large absolute band gaps in two-dimensional photonic crystals formed by large dielectric pixels,” Phys. Rev. B **66**, 165,315 (2002). [CrossRef]

7. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature **386**, 143–149 (1997). [CrossRef]

16. M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and D. Turner, “Localization of electromagnetic waves in two-dimensional disordered systems,” Phys. Rev. B **53**, 8340–8348 (1996). [CrossRef]

17. M. Straub, M. Ventura, and M. Gu, “Multiple Higher-Order Stop Gaps in Infrared Polymer Photonic Crystals,” Phys. Rev. Lett. **91**, 43,901 (2003). [CrossRef]

18. L. Shen, Z. Ye, and S. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B **68**, 35,109 (2003). [CrossRef]

## 2. Basic equations

*e*

^{-iωt)}are described as

*E*and

*H*are field components along the cylinder axis. For periodic structures with infinite extent, it is sufficient to solve the problem in one unit cell, along with the Bloch condition

*ϕ*is either

*E*or

*H*field,

**k**is the Bloch wave vector, and

**a**

_{i}(

*i*=1,2) is the lattice translation vector. For convenience in computation, the primitive unit cell (a hexagon) is replaced by a rectangle with the same area (cf. the right of Fig. 2). Accordingly, the lattice vectors are changed to

**a**

_{1}=(

*a*,0) and

**a**

_{2}=(

*a*/2,√3

*a*/2), where

*a*is the lattice period.

10. R. L. Chern, C. C. Chang, C. C. Chang, and R. Hwang, “Large full band gaps for photonic crystals in two dimensions computed by an inverse method with multigrid acceleration,” Phys. Rev. E **68**, 26,704 (2003). [CrossRef]

21. R. L. Chern, C. C. Chang, C. C. Chang, and R. R. Hwang, “Numerical Study of Three-Dimensional Photonic Crystals with Large Band Gaps,” J. Phys. Soc. Jpn. **73**, 727–737 (2004). [CrossRef]

*Hermitian*property of the differential operators is of full use. An arbitrary distribution of fields over the unit cell is given as the initial guess of the eigenfunction, and the Rayleigh quotients

22. C. C. Chang, J. Y. Chi, R. L. Chern, C. C. Chang, C. H. Lin, and C. O. Chang, “Effect of the inclusion of small metallic components in a two-dimensional dielectric photonic crystal with large full band gap,” Phys. Rev. B **70**, 75,108 (2004). [CrossRef]

**68**, 26,704 (2003). [CrossRef]

## 3. Results and discussion

### 3.1. Band gap features and localized modes

*atoms*in one unit cell, is useful for studying higher-lying band gaps [7

7. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature **386**, 143–149 (1997). [CrossRef]

**68**, 26,704 (2003). [CrossRef]

*ε*

_{d}, and that of the surrounding medium be unity. Figure 1 shows the contours of higher-lying common gap ratio with respect to

*ε*

_{d}and the cylinder radius

*r/a*. The optimal condition is reached at

*r/a*=0.19 and

*ε*

_{d}=26, and the corresponding band structure is plotted in Fig. 2. A common band gap is opened between the third and fourth bands for TE polarization, and between the sixth and seventh bands for TMpolarization. The lower and upper band edges occur at

*ω*=0.446(2

*πc/a*) and 0.575(2

*πc/a*), respectively, with a band gap width 0.129(2

*πc/a*) and the gap to mid-gap ratio 25.3%.

*simultaneous*band edges, as has been observed in the fundamental (low-order) optimal band gap for a connected hexagonal structure [10

**68**, 26,704 (2003). [CrossRef]

23. K. Ohtaka and Y. Tanabe, “Photonic Band Using Vector Spherical Waves. I. Various Properties of Bloch Electric Fields and Heavy Photons,” J. Phys. Soc. Jpn. **65**, 2265–2275 (1996). [CrossRef]

25. J. Plouin, E. Richalot, O. Picon, M. Carras, and A. de Rossi, “Photonic band structures for bi-dimensional metallic mesa gratings,” Opt. Express **14**, 9982–9987 (2006). [CrossRef] [PubMed]

26. R. L. Chern, C. C. Chang, and C. C. Chang, “Analysis of surface plasmon modes and band structures for plasmonic crystals in one and two dimensions,” Phys. Rev. E **73**, 36,605 (2006). [CrossRef]

*inphase*oscillation is associated with the lower band edge [Fig. 3(a)], and a pair of antiphase oscillations is with the upper band edge [Fig. 3(b)]. Unlike the fundamental band gap which are likely to appear between the dielectric and air bands, the higher-lying gap in the present problem occurs between two consecutive dielectric-like bands with high energy fill factors. The fill factor

*f*≡∫

_{ε=εd}

*uda*/∫

*uda*(

*u*=εε

_{0}

*E*

^{2}+µ

_{0}

*H*

^{2}) for measuring the energy concentration within the dielectric regions [6] does not exhibit a marked difference. In Figs. 3(a) and 3(b),

*f*=0.9 and 0.88, respectively.

*f*=0.79 and 0.87 for Figs. 4(a) and 4(b), respectively. However, the field localization within the dielectric regions are not so strong. This is due to the boundary condition for tangential

*E*field, which is required to be smooth across the boundary

*H*field is constrained in a different manner

*H*field could be more localized within the high dielectric region than the

*E*field. In addition, the respective eigenmodes display one higher order of oscillations than those in TE polarization; one pair of oscillations is associated with the lower edge [Fig. 4(a)] and two pairs are with the upper edge [Fig. 4(b)].

### 3.2. Connection with Mie resonances

*r*and dielectric constant

*ε*

_{d}, the scattered and internal fields subject to an incident plane wave are given as [28]

*ρ*and

*ϕ*are cylindrical coordinates with the origin at the cylinder center,

*x*≡

*kr*=

*ωr*/

*c*,

*J*

_{n}(

*x*) and

*H*

_{n}(

*x*) are the

*n*th order Bessel function and Hankel function of the first kind, respectively, the prime denotes derivative with respect to the argument, and the dielectric constant of surrounding medium is assumed to be unity. Mie resonance corresponds to divergence of the amplitude coefficients for the scattered and internal fields; that is, the vanishing denominator of

*a*

_{n}or

*b*

_{n}[29

29. C. Rockstuhl, U. Peschel, and F. Lederer, “Correlation between single-cylinder properties and bandgap formation in photonic structures,” Opt. Lett. **31**, 1741–1743 (2006). [CrossRef] [PubMed]

*r*=0.19

*a*and

*ε*

_{d}=26 be the same as the photonic structure in Fig. 2. The first two orders of

*a*

_{n}and

*b*

_{n}for TE polarization are plotted in Fig. 5(a), where the Mie resonances (located at the peak positions) occur at

*ω*=0.382(2

*πc/a*) and

*ω*=0.607(2

*πc/a*). For comparison, the common gap region for the photonic crystal in Fig. 2 is overlaid in the same plot (shaded area), showing that the common gap region is approximately bounded by the two Mie resonance frequencies. The corresponding internal field is given by

*x*

_{nm}is the

*m*th zero of the denominator of

*a*

_{n}or

*b*

_{n}. The field patterns for

*H*

^{int}

_{01}and

*H*

^{int}

_{11}are shown in Figs. 6(a) and 6(b), respectively. A notable similarity is observed with the respective eigenmodes (in the dielectric region) for the photonic crystal [cf. Figs. 3(a) and 3(b)]. This feature further confirms the correlation of Mie resonances to the higher-lying band gaps in the present problem.

*r*and

*ε*

_{d}. The peaks of the second and third orders of

*a*

_{n}and

*b*

_{n}occur at

*ω*=0.382(2

*πc/a*) and 0.617(2

*πc/a*), respectively. As in the case of TE polarization, the common gap region is approximately bounded by the two Mie resonance frequencies. Note that the peak of

*a*

_{1}(

*b*

_{1}) in TM modes coincides with that of

*a*

_{0}(

*b*

_{0}) in TE modes, which can be realized by the relations:

*J*

^{′}

_{0}=-

*J*

_{1}and

*H*

^{′}

_{0}=-

*H*

_{1}. In addition, the connection with the zeroth order coefficients

*a*

_{0}and

*b*

_{0}is not obvious. This is in accordance with the observation that the band gap locates at relatively higher bands for TM polarization (cf. Fig. 2). The internal field is given by

*E*

^{int}

_{11}and

*E*

^{int}

_{21}, respectively. As expected, the Mie resonance modes display similar field patterns with the respective eigenmodes (in the dielectric region) for the photonic crystal [cf. Figs. 4(a) and 4(b)].

### 3.3. Related waveguide modes

*ε*

_{d}. This is the boundary condition for TE waveguides [30], and the frequency of TE

_{nm}waveguide mode is given as

*x*

^{′}

_{nm}is the

*m*th zero of

*J*

^{′}

_{n}(

*x*). Accordingly, the second the third modes occur at

*ω*

^{TE}

_{11}=0.302(2

*πc/a*) and

*ω*

^{TE}

_{21}=0.502(2

*πc/a*) (

*x*

^{′}

_{11}≈1.841 and

*x*

^{′}

_{21}≈3.054), respectively. Note that both

*ω*

^{TE}

_{11}and

*ω*

^{TE}

_{21}are not so close to Mie resonances at 0.382(2

*πc/a*) and 0.617(2

*πc/a*), respectively [cf. Figs. 6(c) and 6(d)]. This is due to the

*weaker*localization of fields (inside the dielectric) in TM polarization (compare Figs. 3 and 4), so that TM waveguide modes are not indeed very accurate approximations to Mie resonances. Nonetheless, they serve as a convenient way for roughly estimating the Mie resonances and describing the band gap features of the underlying problem.

## 4. Concluding remarks

## 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. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

4. | A. Moroz and A. Tip, “Resonance-induced effects in photonic crystals,” J. Phys. Condens.Matter |

5. | E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B |

6. | R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Nature of the photonic band gap: some insights from a field analysis,” J. Opt. Soc. Am. B |

7. | J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature |

8. | R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. |

9. | P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B |

10. | R. L. Chern, C. C. Chang, C. C. Chang, and R. Hwang, “Large full band gaps for photonic crystals in two dimensions computed by an inverse method with multigrid acceleration,” Phys. Rev. E |

11. | H. K. Fu, Y. F. Chen, R. L. Chern, and C. C. Chang, “Connected hexagonal photonic crystals with largest full band gap,” Opt. Express |

12. | R. L. Chern, C. C. Chang, C. C. Chang, and R. R. Hwang, “Two Classes of Photonic Crystals with Simultaneous Band Gaps,” Jpn. J. Appl. Phys. |

13. | D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B |

14. | C. S. Kee, J. E. Kim, and H. Y. Park, “Absolute photonic band gap in a two-dimensional square lattice of square dielectric rods in air,” Phys, Rev, E |

15. | L. Shen, S. He, and S. Xiao, “Large absolute band gaps in two-dimensional photonic crystals formed by large dielectric pixels,” Phys. Rev. B |

16. | M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and D. Turner, “Localization of electromagnetic waves in two-dimensional disordered systems,” Phys. Rev. B |

17. | M. Straub, M. Ventura, and M. Gu, “Multiple Higher-Order Stop Gaps in Infrared Polymer Photonic Crystals,” Phys. Rev. Lett. |

18. | L. Shen, Z. Ye, and S. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B |

19. | J. Y. Ye, V. Mizeikis, Y. Xu, S. Matsuo, and H. Misawa, “Fabrication and optical characteristics of silicon-based two-dimensional photonic crystals with honeycomb lattice,” Opt. Commun. |

20. | J. Y. Ye, S. Matsuo, V. Mizeikis, and H. Misawa, “Silicon-based honeycomb photonic crystal structures with complete photonic band gap at 1.5 mm wavelength,” J. Appl. Phys. |

21. | R. L. Chern, C. C. Chang, C. C. Chang, and R. R. Hwang, “Numerical Study of Three-Dimensional Photonic Crystals with Large Band Gaps,” J. Phys. Soc. Jpn. |

22. | C. C. Chang, J. Y. Chi, R. L. Chern, C. C. Chang, C. H. Lin, and C. O. Chang, “Effect of the inclusion of small metallic components in a two-dimensional dielectric photonic crystal with large full band gap,” Phys. Rev. B |

23. | K. Ohtaka and Y. Tanabe, “Photonic Band Using Vector Spherical Waves. I. Various Properties of Bloch Electric Fields and Heavy Photons,” J. Phys. Soc. Jpn. |

24. | M. G. Salt and W. L. Barnes, “Flat photonic bands in guided modes of textured metallic microcavities,” Phys. Rev. B |

25. | J. Plouin, E. Richalot, O. Picon, M. Carras, and A. de Rossi, “Photonic band structures for bi-dimensional metallic mesa gratings,” Opt. Express |

26. | R. L. Chern, C. C. Chang, and C. C. Chang, “Analysis of surface plasmon modes and band structures for plasmonic crystals in one and two dimensions,” Phys. Rev. E |

27. | C. F. Bohren and D. R. Huffman, |

28. | C. A. Balanis, |

29. | C. Rockstuhl, U. Peschel, and F. Lederer, “Correlation between single-cylinder properties and bandgap formation in photonic structures,” Opt. Lett. |

30. | D. M. Pozar, |

31. | R. C. Kell, A. C. Greenham, and G. C. E. Olds, “High-Permittivity Temperature-Stable Ceramic Dielectrics with Low Microwave Loss,” J. Am. Ceram. Soc. |

**OCIS Codes**

(260.5740) Physical optics : Resonance

(290.4020) Scattering : Mie theory

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: August 5, 2008

Revised Manuscript: September 23, 2008

Manuscript Accepted: September 24, 2008

Published: October 2, 2008

**Citation**

Ruey-Lin Chern and Sheng D. Chao, "Optimal higher-lying band gaps for photonic crystals with large dielectric contrast," Opt. Express **16**, 16600-16608 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16600

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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]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).
- A. Moroz and A. Tip, "Resonance-induced effects in photonic crystals," J. Phys. Condens Matter 11, 2503-2512 (1999). [CrossRef]
- E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, "Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials," Phys. Rev. B 61, 458-13,464 (2000). [CrossRef]
- R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, "Nature of the photonic band gap: some insights from a field analysis," J. Opt. Soc. Am. B 10, 328-332 (1993).
- J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997). [CrossRef]
- R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Existence of a photonic band gap in two dimensions," Appl. Phys. Lett. 61, 495-497 (1992). [CrossRef]
- P. R. Villeneuve and M. Piche´, "Photonic band gaps in two-dimensional square and hexagonal lattices," Phys. Rev. B 46, 4969-4972 (1992). [CrossRef]
- R. L. Chern, C. C. Chang, C. C. Chang, and R. Hwang, "Large full band gaps for photonic crystals in two dimensions computed by an inverse method with multigrid acceleration," Phys. Rev. E 68, 26,704 (2003). [CrossRef]
- H. K. Fu, Y. F. Chen, R. L. Chern, and C. C. Chang, "Connected hexagonal photonic crystals with largest full band gap," Opt. Express 13, 7854-7860 (2005). [CrossRef] [PubMed]
- R. L. Chern, C. C. Chang, C. C. Chang, and R. R. Hwang, "Two Classes of Photonic Crystals with Simultaneous Band Gaps," Jpn. J. Appl. Phys. 43, 3484-3490 (2004). [CrossRef]
- D. Cassagne, C. Jouanin, and D. Bertho, "Hexagonal photonic-band-gap structures," Phys. Rev. B 53, 7134-7142 (1996). [CrossRef]
- C. S. Kee, J. E. Kim, and H. Y. Park, "Absolute photonic band gap in a two-dimensional square lattice of square dielectric rods in air," Phys. Rev. E 56, 6291-6293 (1997). [CrossRef]
- L. Shen, S. He, and S. Xiao, "Large absolute band gaps in two-dimensional photonic crystals formed by large dielectric pixels," Phys. Rev. B 66, 165,315 (2002). [CrossRef]
- M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and D. Turner, "Localization of electromagnetic waves in twodimensional disordered systems," Phys. Rev. B 53, 8340-8348 (1996). [CrossRef]
- M. Straub, M. Ventura, and M. Gu, "Multiple Higher-Order Stop Gaps in Infrared Polymer Photonic Crystals," Phys. Rev. Lett. 91, 43,901 (2003). [CrossRef]
- L. Shen, Z. Ye, and S. He, "Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm," Phys. Rev. B 68, 35,109 (2003). [CrossRef]
- J. Y. Ye, V. Mizeikis, Y. Xu, S. Matsuo, and H. Misawa, "Fabrication and optical characteristics of silicon-based two-dimensional photonic crystals with honeycomb lattice," Opt. Commun. 211, 205-213 (2002). [CrossRef]
- J. Y. Ye, S. Matsuo, V. Mizeikis, and H. Misawa, "Silicon-based honeycomb photonic crystal structures with complete photonic band gap at 1.5 μm wavelength," J. Appl. Phys. 96, 6934 (2004). [CrossRef]
- R. L. Chern, C. C. Chang, C. C. Chang, and R. R. Hwang, "Numerical Study of Three-Dimensional Photonic Crystals with Large Band Gaps," J. Phys. Soc. Jpn. 73, 727-737 (2004). [CrossRef]
- C. C. Chang, J. Y. Chi, R. L. Chern, C. C. Chang, C. H. Lin, and C. O. Chang, "Effect of the inclusion of small metallic components in a two-dimensional dielectric photonic crystal with large full band gap," Phys. Rev. B 70, 75,108 (2004). [CrossRef]
- K. Ohtaka and Y. Tanabe, "Photonic Band using Vector Spherical Waves. I. Various Properties of Bloch Electric Fields and Heavy Photons," J. Phys. Soc. Jpn. 65, 2265-2275 (1996). [CrossRef]
- M. G. Salt and W. L. Barnes, "Flat photonic bands in guided modes of textured metallic microcavities," Phys. Rev. B 61, 11,125-11,135 (2000).
- J. Plouin, E. Richalot, O. Picon,M. Carras, and A. de Rossi, "Photonic band structures for bi-dimensional metallic mesa gratings," Opt. Express 14, 9982-9987 (2006). [CrossRef] [PubMed]
- R. L. Chern, C. C. Chang, and C. C. Chang, "Analysis of surface plasmon modes and band structures for plasmonic crystals in one and two dimensions," Phys. Rev. E 73, 36,605 (2006). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (JohnWiley & Sons, New York, 1983).
- C. A. Balanis, Advanced Enginerring Electromechanics (John Wiley & Sons, New York, 1989).
- C. Rockstuhl, U. Peschel, and F. Lederer, "Correlation between single-cylinder properties and bandgap formation in photonic structures," Opt. Lett. 31, 1741-1743 (2006). [CrossRef] [PubMed]
- D. M. Pozar, Microwave engineering, 3rd ed. (Wiley, New York, 2005).
- R. C. Kell, A. C. Greenham, and G. C. E. Olds, "High-Permittivity Temperature-Stable Ceramic Dielectrics with Low Microwave Loss," J. Am. Ceram. Soc. 56, 352-354 (1973). [CrossRef]

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