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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 21 — Oct. 13, 2008
  • pp: 16600–16608
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Optimal higher-lying band gaps for photonic crystals with large dielectric contrast

Ruey-Lin Chern and Sheng D. Chao  »View Author Affiliations


Optics Express, Vol. 16, Issue 21, pp. 16600-16608 (2008)
http://dx.doi.org/10.1364/OE.16.016600


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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

Photonic crystals have been the subject of intensive research in the past two decades [1

1. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

, 2

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

, 3

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

]. One of the most distinguished features is the existence of photonic band gap. Formation of band gap is considered as a result of hybridization of individual Mie resonance due to single scatterers and the Bragg-like multiple scattering due to periodicity [4

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

]. The former corresponds to a strongly localized photon state, and the latter to a nearly free photon state [5

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]

]. Nature of photonic band gap was elucidated with the variation of electrical energy inside the dielectric regions, through the use of a fill factor [6

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 10, 328–332 (1993).

]. As two consecutive bands exhibit markedly different fill factors, the discrepancy in frequency will be large and the band gap would be significant.

Due to the vector nature of electromagnetic fields, band gaps for two-dimensional structures behave in different manners for different polarizations. The common band gap for TM and TE modes (with respect to the normal of lattice plane) is an important issue for confining the light from arbitrary orientation. A general rule of thumb was proposed to characterize the band gap features for two-dimensional crystals: TM band gaps are favoured in a lattice of isolated high dielectric region, and TE gaps are favoured in a connected lattice [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]

]. This rule is useful for describing the basic features of a fundamental band gap. A compromise between an isolated and a connected lattice then leads to a common band gap for both polarizations. One typical example is the triangular lattice of air columns [8

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

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]

]. By incorporating two geometric parameters, the band gap size can be enlarged. The connected hexagonal lattice has an optimal common band gap (over 24%) for the silicon-air structure (with the dielectric contrast 13) [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]

, 11

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]

]. As the optimal condition is reached, the corresponding TM and TE modes exhibit 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

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]

].

The higher-lying band gap (which occurs at relatively higher frequency branches) shows a different characteristic. It is not necessary for the structure to be connected [13

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

, 14

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

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]

]. As the mid-gap frequency is higher, the minimum feature size is larger. This can be a very important issue in fabrication [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]

]. However, higher-lying band gaps are more difficult to appear [16

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]

] and receive much less attention in the past [17

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]

]. This may be attributed to a more complicated behavior at higher-order Bragg scattering. Although a genetic algorithm based on the selection of dielectric pixels can be utilized to obtain a large higher-lying band gap [18

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]

], the rule or mechanism for opening optimal gaps at higher bands still demands a further study.

2. Basic equations

Consider a periodic lattice of dielectric cylinders whose geometry is constant along the cylinder axis. For propagation of waves parallel to the lattice plane, the time-harmonic electromagnetic modes (with time dependence e -iωt) are described as

2E=ε(ωc)2E,
(1)
·(1εH)=(ωc)2H,
(2)

for transverse magnetic (TM) and transverse electric (TE) polarizations, respectively, where 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(r+ai)=eik·aiϕ(r),
(3)

applying at the unit cell boundary, where 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,√3a/2), where a is the lattice period.

The eigensystems (1) and (2) are solved by the inverse iteration method [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]

, 21

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]

], in which the 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

QE=E2dτεE2dτ,QH=1εH2dτH2dτ,
(4)

are employed to calculate the eigenfrequencies for TM and TE modes, respectively. By repeatedly solving a matrix inversion, the solution is refined through iterations until it is converged. The Rayleigh quotients (4) are utilized not only for obtaining the solutions, but also in the analysis of band gap features for different polarizations [22

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]

]. Details of the inverse iteration method can be found in Ref. [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]

].

Fig. 1. Contours of the higher-lying common gap ratio for a hexagonal lattice of circular cylinders with varying the dielectric constant ε d and cylinder radius r/a.

3. Results and discussion

3.1. Band gap features and localized modes

The hexagonal lattice of circular cylinders, where there are two dielectric 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]

]. In order to seek for an optimal condition, two parameters have been incorporated to arrange the structures [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]

]. In the present study, the optimal structure is attained by increasing the dielectric contrast on one hand, and varying the cylinder radius on the other. Let the dielectric constant of the cylinder be 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(2πc/a) and 0.575(2πc/a), respectively, with a band gap width 0.129(2πc/a) and the gap to mid-gap ratio 25.3%.

Fig. 2. Band structure for a hexagonal lattice of circular cylinders with radius r/a=0.19 and dielectric constant ε d=26. Shaded area is the common band gap for both polarizations; the band gap width is 0.129(2πc/a) and the gap to mid-gap ratio is 25.3%. The unit cell and geometric parameters are shown on the right.

It is noticed that the corresponding TM and TE modes exhibit simultaneous band edges, as has been observed in the fundamental (low-order) optimal band gap for a connected hexagonal structure [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]

]. The common gap width is therefore not trimmed off for either polarization. Around the band edges, the frequency branches becomes flattened over the whole wave vector space, which means that the resonant modes are dispersionless or insensitive to the change of Bloch wave vector. This is a typical feature of strong resonance and has been termed as the heavy photon state [23

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]

]. In the present study, the band flattening is a consequence of large dielectric contrast, as the energy has been strongly confined within the dielectric material and the frequency becomes insensitive to the change of wave vector. This phenomenon appears in other structures as well, such as the textured metallic microcavities [24

24. M. G. Salt and W. L. Barnes, “Flat photonic bands in guided modes of textured metallic microcavities,” Phys. Rev. B 61(16), 11,125–11,135 (2000).

], bi-dimensional metallic mesa gratings [25

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]

], and periodic arrays of plasmonic cylinders [26

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]

].

Fig. 3. Magnetic field contours of the TE eigenmodes at the point G for the photonic crystal in Fig. 2. (a) eigenmode near the lower edge with ω=0.429(2πc/a), (b) eigenmode near the upper edge with ω=0.593(2πc/a).

Figure 3 shows the TE eigenmode patterns near the band edges at the point Γ. The fields are strongly localized within the dielectric regions, where a lump of 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=εε0 E 20 H 2) for measuring the energy concentration within the dielectric regions [6

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 10, 328–332 (1993).

] does not exhibit a marked difference. In Figs. 3(a) and 3(b), f=0.9 and 0.88, respectively.

Fig. 4. Electric field contours of the TM eigenmodes at the point G for the photonic crystal in Fig. 2. (a) eigenmode near the lower edge with ω=0.425(2πc/a), (b) eigenmode near the upper edge with ω=0.618(2πc/a).

Similar localized field patterns of the respective eigenmodes are observed for TM polarization as well, as shown in Fig. 4. The fill factors are still large for both modes; 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 En=0. In contrast, the tangential H field is constrained in a different manner 1εHn=0. Consequently, the 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

Fig. 5. Amplitude coefficients of the scattered and internal fields for a dielectric circular cylinder of radius r=0.19a and dielectric constant ε d=26. (a) TE polarization, (b) TM polarization. Shaded areas correspond to the common band gap region for the photonic crystal in Fig. 2. Vertical solid and dashed lines indicate the waveguide mode frequencies related to Mie resonances.

Mie resonance is the scattering excitation of a single scatterer [27

27. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (JohnWiley & Sons, New York, 1983).

], which occurs when the wavelength matches the size of scatterer. At resonance, the scattered and internal fields may be strongly enhanced. For a dielectric circular cylinder of radius r and dielectric constant ε d, the scattered and internal fields subject to an incident plane wave are given as [28

28. C. A. Balanis, Advanced Enginerring Electromechanics (John Wiley & Sons, New York, 1989).

]

Hscat=n=anHn(kρ)einϕ,an=inJn(x)Jn'(x1)εdJn'(x)Jn(x1)εdHn'(x)Jn(x1)Hn(x)Jn'(x1),
(5)
Hint=n=bnJn(εdkρ)einϕ,bn=in2i/πxεdHn'(x)Jn(x1)Hn(x)Jn'(x1),
(6)

for TE polarization, where ρ and ϕ are cylindrical coordinates with the origin at the cylinder center, xkr=ωr/c, x1εdx, J n(x) and H n(x) are the nth 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]

]. Let r=0.19a 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(2πc/a) and ω=0.607(2πc/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 Hnmint=Jn(εdxnmρr)cos(nϕ), where x nm is the mth 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.

For TMpolarization, similar expressions are given for the scattered and internal fields as [28

28. C. A. Balanis, Advanced Enginerring Electromechanics (John Wiley & Sons, New York, 1989).

]

Escat=n=anHn(kρ)einϕ,an=inεdJn(x)Jn'(x1)Jn'(x)Jn(x1)Hn'(x)Jn(x1)εdHn(x)Jn'(x1),
(7)
Eint=n=bnJn(εdkρ)einϕ,bn=in2i/πxHn'(x)Jn(x1)εdHn(x)Jn'(x1),
(8)

where the amplitude coefficients are plotted in Fig. 5(b) for the same r and ε d. The peaks of the second and third orders of a n and b n occur at ω=0.382(2πc/a) and 0.617(2πc/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 Enmint=Jn(εdxnmρr)cos(nϕ), as shown in Figs. 6(c) and 6(d) for 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)].

Fig. 6. Internal field patterns of Mie resonances for a dielectric circular cylinder of radius r=0.19a and dielectric constant ε d=26. (a) H int 01 field with ω=0.382(2πc/a), (b) H int 11 field with ω=0.607(2πc/a), (c) E int 11 field with ω=0.382(2πc/a), (d) E int 21 field with ω=0.617(2πc/a).

3.3. Related waveguide modes

Likewise, Mie resonance condition for TM polarization is approximated (but not as good as the TE case) to Jn'(εdx)0 [the vanishing denominator of Eq. (7) or (8)] for large ε d. This is the boundary condition for TE waveguides [30

30. D. M. Pozar, Microwave engineering, 3rd ed. (Wiley, New York, 2005).

], and the frequency of TEnm waveguide mode is given as ωnmTE=xnm'crεd,, where x nm is the mth zero of J n (x). Accordingly, the second the third modes occur at ω TE 11=0.302(2πc/a) and ω TE 21=0.502(2πc/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(2πc/a) and 0.617(2πc/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

This work was supported in part by National Science Council of the Republic of China under Contract No. NSC 96-2221-E-002-190-MY3 and NSC 97-2120-M-002-013.

References and links

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.

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

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]

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 10, 328–332 (1993).

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]

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]

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]

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. 211, 205–213 (2002). [CrossRef]

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. 96, 6934 (2004). [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]

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]

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]

24.

M. G. Salt and W. L. Barnes, “Flat photonic bands in guided modes of textured metallic microcavities,” Phys. Rev. B 61(16), 11,125–11,135 (2000).

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]

27.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (JohnWiley & Sons, New York, 1983).

28.

C. A. Balanis, Advanced Enginerring Electromechanics (John Wiley & Sons, New York, 1989).

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]

30.

D. M. Pozar, Microwave engineering, 3rd ed. (Wiley, New York, 2005).

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. 56, 352–354 (1973). [CrossRef]

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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