Abstract

This study investigates the dispersion relation of two dimensional photonic crystals conformed in a hybrid triangular-graphite configuration. This lattice includes, as limiting cases, two major well-known structures, the triangular and the graphite lattices. The analysis has been carried out by using preconditioned block-iterative algorithms for computing eigenstates of Maxwell’s equations for periodic dielectric systems, using a plane-wave basis. We present the evolution of the so-called gap maps as a function of the radii of the structures. We conclude that a number of gaps exist for both TM and TE polarizations. We also predict the appearance of sizeable complete band gaps for structures the can be achieved using present fabrication capabilities.

© 2004 Optical Society of America

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References

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  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. S.Y. Lin et al. �??A three dimensional photonic crystal operating at infrared wavelenghts�??, Nature 394, 251 (1998); A. Blanco, et al. �??Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres�??, Nature 405, 437 (2000); Y. Vlasov, et al., �??On-chip natural assembly of silicon photonic band gap crystals�??, 414, 289 (2001).
    [CrossRef]
  4. R. D. Meade, K. L. Brommer, A. M. Rappe, and Joannopoulos, �??Existence of a photonic band gap in two dimensions,�?? Appl. Phys. Lett. 61, 495-497 (1992).
    [CrossRef]
  5. D. Cassagne, C. Jouanin, and D. Bertho, �??Photonic band gaps in a two-dimensional graphite structure,�?? Phys. Rev. B 52, R2217-R2220 (1995).
    [CrossRef]
  6. D. Cassagne, C. Jouanin, and D. Bertho, �??Hexagonal photonic-band-gap structures,�?? Phys. Rev. B 53, 7134-7142 (1995).
    [CrossRef]
  7. R. Padjen, J. M. Gérad, and J. Y. Marzin, �??Analysis of the filling pattern dependence of the photonic bandgap for two-dimensionl systems,�?? J. Mod. Opt. 41, 295-310 (1994).
    [CrossRef]
  8. S. G. Johnson and J. D. Joannopoulos, �??Block-iterative frequency-domain methods for Maxwell�??s equations in planewave basis,�?? Opt. Express 8, no. 3, 173-190 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEN-8-3-173.">http://www.opticsexpress.org/abstract.cfm?URI=OPEN-8-3-173.</a>
    [CrossRef] [PubMed]
  9. S. G. Johnson and J. D. Joannopoulos, The MIT Photonic-Bands Package home page <a href="http://ab-initio.mit.edu/mpb/.">http://ab-initio.mit.edu/mpb/.</a>

Appl. Phys. Lett.

R. D. Meade, K. L. Brommer, A. M. Rappe, and Joannopoulos, �??Existence of a photonic band gap in two dimensions,�?? Appl. Phys. Lett. 61, 495-497 (1992).
[CrossRef]

J. Mod. Opt.

R. Padjen, J. M. Gérad, and J. Y. Marzin, �??Analysis of the filling pattern dependence of the photonic bandgap for two-dimensionl systems,�?? J. Mod. Opt. 41, 295-310 (1994).
[CrossRef]

Nature

S.Y. Lin et al. �??A three dimensional photonic crystal operating at infrared wavelenghts�??, Nature 394, 251 (1998); A. Blanco, et al. �??Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres�??, Nature 405, 437 (2000); Y. Vlasov, et al., �??On-chip natural assembly of silicon photonic band gap crystals�??, 414, 289 (2001).
[CrossRef]

Opt. Express

Phys. Rev. B

D. Cassagne, C. Jouanin, and D. Bertho, �??Photonic band gaps in a two-dimensional graphite structure,�?? Phys. Rev. B 52, R2217-R2220 (1995).
[CrossRef]

D. Cassagne, C. Jouanin, and D. Bertho, �??Hexagonal photonic-band-gap structures,�?? Phys. Rev. B 53, 7134-7142 (1995).
[CrossRef]

Phys. Rev. Lett.

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]

Other

S. G. Johnson and J. D. Joannopoulos, The MIT Photonic-Bands Package home page <a href="http://ab-initio.mit.edu/mpb/.">http://ab-initio.mit.edu/mpb/.</a>

Supplementary Material (4)

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Figures (4)

Fig. 1.
Fig. 1.

Structure of the hybrid lattice under consideration. Small dots are at the position of the triangular lattice and have radii R/a, whereas big dots are located at the positions of a graphite structure and have radii Rg/a. The lattice parameter is a and the vectors defining the primitive cell are a 1 and a 2. The unit cell is delimited by the thick lines and contains three elements in its interior. These elements are located at u 0=0, u 1=(a 1+a 2)/3 and u 0=-(a 1+a 2)/3 with respect to the center of the unit cell.

Fig. 2.
Fig. 2.

Photonic band structure for TE (red long-dashed lines) and TM (blue dashed lines) polarization of the hybrid triangular-graphite structure of air holes in a dielectric matrix of index ε=10.24 for radius of the holes R/a=0.1 and Rg/a=0.24. A complete band-gap is observed for a normalized frequency around 0.84 (shadowed area).

Fig. 3.
Fig. 3.

Photonic band maps for the TE (red) polarization and TM (blue) polarization of a hybrid triangular-graphite structure of air cavities in a dielectric matrix of index ε=10.24. Left (275Kb animation): Graphite-lattice maps, varying the radius of the cylinders of the triangular lattice. Right (238Kb animation): Triangular lattice maps as the radius of the cylinders of the graphite sublattice grows.

Fig. 4.
Fig. 4.

Photonic band maps for the TE (red) polarization and TM (blue) polarization of a hybrid triangular-graphite structure of dielectric rods in air. Left (245Kb animation): GGMs, varying the radius of the cylinders of the triangular lattice. Right (228Kb animation): TGMs as the radius of the cylinders of the graphite sublattice increases.

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