Abstract

We investigate from a theoretical point of view the photonic properties of a two dimensional photonic aperiodic crystal. These structures are obtained by removing the lattice points from a square arrangement, following the inflation rules emerging from the Thue-Morse sequence. The photonic bandgap analysis is performed by means of the density of states calculation. The mechanism of bandgap formation is investigated adopting the single scattering model, and the Mie scattering. The electromagnetic field distribution can be represented as quasi-localized states. Finally, a generalized method to obtain aperiodic photonic structures has been proposed.

© 2007 Optical Society of America

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  7. M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740 (2000).
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  30. H. Altug and J. Vučkovic, "Two-dimensional coupled photonic crystal resonator arrays," Appl. Phys. Lett. 84, 161 (2004).
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  31. K. Nozaki and T. Baba, "Quasiperiodic photonic crystal microcavity lasers," Appl. Phys. Lett. 84, 4875 (2004).
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  32. Y. Wang, X. Hu, X. Hu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
    [CrossRef]
  33. S. Even-Dar Mandel, R. Lifshitz, "Electronic energy spectra and wave functions on the square Fibonacci tiling," Philos. Mag. 86, 759 (2006)
    [CrossRef]
  34. A. Bovier and J. M. Ghez, "Remarks on the spectral properties of tight-binding and Kronig-Penney models with substitution sequences," J. Phys A 28, 2313 (1995).
    [CrossRef]

2007 (2)

Discussions on "What is a crystal?," Z. Kristallog. 222, 308-319 (2007).

W. Steurer and D. Sutter-Widmer, "Photonic and phononic quasicrystals," J. Phys. D 40, R229 (2007).
[CrossRef]

2006 (4)

2005 (4)

R. Lifshitz, A. Arie, and A. Bahabad, "Photonic Quasicrystals for Nonlinear Optical Frequency Conversion," Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef] [PubMed]

A. Della Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band Gap Formation and Multiple Scattering in Photonic Quasicrystals with a Penrose-Type Lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, "Photonic band gaps and localization in the Thue-Morse structures," Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, "Spectrally enhanced light emission from aperiodic photonic structures," Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

2004 (3)

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc and J. Haavisto, "Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

H. Altug and J. Vučkovic, "Two-dimensional coupled photonic crystal resonator arrays," Appl. Phys. Lett. 84, 161 (2004).
[CrossRef]

K. Nozaki and T. Baba, "Quasiperiodic photonic crystal microcavity lasers," Appl. Phys. Lett. 84, 4875 (2004).
[CrossRef]

2003 (3)

Y. Wang, X. Hu, X. Hu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz, "Photonic band gaps in quasicrystal-related approximant structures," J. Mod. Opt. 50, 2095 (2003).

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, "Light Transport through the Band-Edge States of Fibonacci Quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef] [PubMed]

2002 (1)

R. Lifshitz, "The square Fibonacci tiling," J. All. Comp. 342, 186 (2002).
[CrossRef]

2000 (2)

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, 13458 (2000).
[CrossRef]

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740 (2000).
[CrossRef] [PubMed]

1998 (1)

Y. S. Chan, C. T. Chan and Z. Y. Liu, "Photonic Band Gaps in Two Dimensional Photonic Quasicrystals," Phys. Rev. Lett. 80, 956 (1998).
[CrossRef]

1997 (1)

N. Liu, "Propagation of light waves in Thue-Morse dielectric multilayers," Phys. Rev. B 55, 3543 (1997).
[CrossRef]

1995 (2)

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635 (1995).
[CrossRef]

A. Bovier and J. M. Ghez, "Remarks on the spectral properties of tight-binding and Kronig-Penney models with substitution sequences," J. Phys A 28, 2313 (1995).
[CrossRef]

1990 (1)

C. Godrèche and J. M. Luck, "Multifractal analysis in reciprocal space and the nature of the Fourier transform of self-similar structures," J. Phys A 23, 3769 (1990).
[CrossRef]

1989 (1)

T. Fujiwara, M. Kohmoto, and T. Tokihiro, "Multifractal wave functions on a Fibonacci lattice," Phys. Rev. B 40, 7413 (1989).
[CrossRef]

1988 (1)

Z. Cheng, R. Savit, and R. Merlin, "Structure and electronic properties of Thue-Morse lattices," Phys. Rev. B 37, 4375 (1988).
[CrossRef]

1987 (2)

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

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

1984 (1)

D. Shechtman, I. Blech, and D. Gratias, "Metallic phase with long-range orientational order and no translational symmetry," Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Abbate, G.

Altug, H.

H. Altug and J. Vučkovic, "Two-dimensional coupled photonic crystal resonator arrays," Appl. Phys. Lett. 84, 161 (2004).
[CrossRef]

Arie, A.

R. Lifshitz, A. Arie, and A. Bahabad, "Photonic Quasicrystals for Nonlinear Optical Frequency Conversion," Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef] [PubMed]

Baba, T.

K. Nozaki and T. Baba, "Quasiperiodic photonic crystal microcavity lasers," Appl. Phys. Lett. 84, 4875 (2004).
[CrossRef]

Bahabad, A.

R. Lifshitz, A. Arie, and A. Bahabad, "Photonic Quasicrystals for Nonlinear Optical Frequency Conversion," Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef] [PubMed]

Baumberg, J. J.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740 (2000).
[CrossRef] [PubMed]

Blech, I.

D. Shechtman, I. Blech, and D. Gratias, "Metallic phase with long-range orientational order and no translational symmetry," Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Bovier, A.

A. Bovier and J. M. Ghez, "Remarks on the spectral properties of tight-binding and Kronig-Penney models with substitution sequences," J. Phys A 28, 2313 (1995).
[CrossRef]

Capolino, F.

A. Della Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band Gap Formation and Multiple Scattering in Photonic Quasicrystals with a Penrose-Type Lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Chan, C. T.

Y. S. Chan, C. T. Chan and Z. Y. Liu, "Photonic Band Gaps in Two Dimensional Photonic Quasicrystals," Phys. Rev. Lett. 80, 956 (1998).
[CrossRef]

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635 (1995).
[CrossRef]

Chan, Y. S.

Y. S. Chan, C. T. Chan and Z. Y. Liu, "Photonic Band Gaps in Two Dimensional Photonic Quasicrystals," Phys. Rev. Lett. 80, 956 (1998).
[CrossRef]

Charlton, M. D. B.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740 (2000).
[CrossRef] [PubMed]

Chelnokov, A.

K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz, "Photonic band gaps in quasicrystal-related approximant structures," J. Mod. Opt. 50, 2095 (2003).

Cheng, B.

Y. Wang, X. Hu, X. Hu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Cheng, Z.

Z. Cheng, R. Savit, and R. Merlin, "Structure and electronic properties of Thue-Morse lattices," Phys. Rev. B 37, 4375 (1988).
[CrossRef]

Colocci, M.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, "Light Transport through the Band-Edge States of Fibonacci Quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef] [PubMed]

Dal Negro, L.

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, "Spectrally enhanced light emission from aperiodic photonic structures," Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc and J. Haavisto, "Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, "Light Transport through the Band-Edge States of Fibonacci Quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef] [PubMed]

David, S.

K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz, "Photonic band gaps in quasicrystal-related approximant structures," J. Mod. Opt. 50, 2095 (2003).

De Stefano, L.

Della Villa, A.

A. Della Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band Gap Formation and Multiple Scattering in Photonic Quasicrystals with a Penrose-Type Lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Duan, X.

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc and J. Haavisto, "Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

Economou, E. N.

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, 13458 (2000).
[CrossRef]

Enoch, S.

A. Della Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band Gap Formation and Multiple Scattering in Photonic Quasicrystals with a Penrose-Type Lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Even-Dar Mandel, S.

S. Even-Dar Mandel, R. Lifshitz, "Electronic energy spectra and wave functions on the square Fibonacci tiling," Philos. Mag. 86, 759 (2006)
[CrossRef]

Feng, S.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, "Photonic band gaps and localization in the Thue-Morse structures," Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

Fujiwara, T.

T. Fujiwara, M. Kohmoto, and T. Tokihiro, "Multifractal wave functions on a Fibonacci lattice," Phys. Rev. B 40, 7413 (1989).
[CrossRef]

Gaburro, Z.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, "Light Transport through the Band-Edge States of Fibonacci Quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef] [PubMed]

Galdi, V.

A. Della Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band Gap Formation and Multiple Scattering in Photonic Quasicrystals with a Penrose-Type Lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Ghez, J. M.

A. Bovier and J. M. Ghez, "Remarks on the spectral properties of tight-binding and Kronig-Penney models with substitution sequences," J. Phys A 28, 2313 (1995).
[CrossRef]

Godrèche, C.

C. Godrèche and J. M. Luck, "Multifractal analysis in reciprocal space and the nature of the Fourier transform of self-similar structures," J. Phys A 23, 3769 (1990).
[CrossRef]

Gratias, D.

D. Shechtman, I. Blech, and D. Gratias, "Metallic phase with long-range orientational order and no translational symmetry," Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Haavisto, J.

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc and J. Haavisto, "Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

Ho, K. M.

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635 (1995).
[CrossRef]

Hu, X.

Y. Wang, X. Hu, X. Hu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Y. Wang, X. Hu, X. Hu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Huang, K. C.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, "Photonic band gaps and localization in the Thue-Morse structures," Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

Jiang, X.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, "Photonic band gaps and localization in the Thue-Morse structures," Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

Joannopoulos, J. D.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, "Photonic band gaps and localization in the Thue-Morse structures," Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

John, S.

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

Johnson, P.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, "Light Transport through the Band-Edge States of Fibonacci Quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef] [PubMed]

Kimerling, L. C.

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, "Spectrally enhanced light emission from aperiodic photonic structures," Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc and J. Haavisto, "Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

Kohmoto, M.

T. Fujiwara, M. Kohmoto, and T. Tokihiro, "Multifractal wave functions on a Fibonacci lattice," Phys. Rev. B 40, 7413 (1989).
[CrossRef]

Lagendijk, A.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, "Light Transport through the Band-Edge States of Fibonacci Quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef] [PubMed]

LeBlanc, J.

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc and J. Haavisto, "Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

Lederer, F.

Lidorikis, E.

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, 13458 (2000).
[CrossRef]

Lifshitz, R.

S. Even-Dar Mandel, R. Lifshitz, "Electronic energy spectra and wave functions on the square Fibonacci tiling," Philos. Mag. 86, 759 (2006)
[CrossRef]

R. Lifshitz, A. Arie, and A. Bahabad, "Photonic Quasicrystals for Nonlinear Optical Frequency Conversion," Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef] [PubMed]

R. Lifshitz, "The square Fibonacci tiling," J. All. Comp. 342, 186 (2002).
[CrossRef]

Liu, N.

N. Liu, "Propagation of light waves in Thue-Morse dielectric multilayers," Phys. Rev. B 55, 3543 (1997).
[CrossRef]

Liu, Z. Y.

Y. S. Chan, C. T. Chan and Z. Y. Liu, "Photonic Band Gaps in Two Dimensional Photonic Quasicrystals," Phys. Rev. Lett. 80, 956 (1998).
[CrossRef]

Lourtioz, J. M.

K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz, "Photonic band gaps in quasicrystal-related approximant structures," J. Mod. Opt. 50, 2095 (2003).

Luck, J. M.

C. Godrèche and J. M. Luck, "Multifractal analysis in reciprocal space and the nature of the Fourier transform of self-similar structures," J. Phys A 23, 3769 (1990).
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E. Maciá, "The role of aperiodic order in science and technology," Rep. Prog. Phys. 69, 397 (2006).
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Z. Cheng, R. Savit, and R. Merlin, "Structure and electronic properties of Thue-Morse lattices," Phys. Rev. B 37, 4375 (1988).
[CrossRef]

Michel, J.

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, "Spectrally enhanced light emission from aperiodic photonic structures," Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc and J. Haavisto, "Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

Moretti, L.

Netti, M. C.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740 (2000).
[CrossRef] [PubMed]

Nguyen, V.

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, "Spectrally enhanced light emission from aperiodic photonic structures," Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

Nozaki, K.

K. Nozaki and T. Baba, "Quasiperiodic photonic crystal microcavity lasers," Appl. Phys. Lett. 84, 4875 (2004).
[CrossRef]

Oton, C. J.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, "Light Transport through the Band-Edge States of Fibonacci Quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef] [PubMed]

Parker, G. J.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740 (2000).
[CrossRef] [PubMed]

Pavesi, L.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, "Light Transport through the Band-Edge States of Fibonacci Quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef] [PubMed]

Peschel, U.

Pierro, V.

A. Della Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band Gap Formation and Multiple Scattering in Photonic Quasicrystals with a Penrose-Type Lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Rea, I.

Rendina, I.

Righini, R.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, "Light Transport through the Band-Edge States of Fibonacci Quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
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Rotiroti, L.

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[CrossRef]

Soukoulis, C. M.

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, 13458 (2000).
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W. Steurer and D. Sutter-Widmer, "Photonic and phononic quasicrystals," J. Phys. D 40, R229 (2007).
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L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc and J. Haavisto, "Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

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W. Steurer and D. Sutter-Widmer, "Photonic and phononic quasicrystals," J. Phys. D 40, R229 (2007).
[CrossRef]

Tayeb, G.

A. Della Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band Gap Formation and Multiple Scattering in Photonic Quasicrystals with a Penrose-Type Lattice," Phys. Rev. Lett. 94, 183903 (2005).
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K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz, "Photonic band gaps in quasicrystal-related approximant structures," J. Mod. Opt. 50, 2095 (2003).

Wang, Y.

Y. Wang, X. Hu, X. Hu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Wiersma, D. S.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, "Light Transport through the Band-Edge States of Fibonacci Quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
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L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, "Spectrally enhanced light emission from aperiodic photonic structures," Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

Yi, Y.

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, "Spectrally enhanced light emission from aperiodic photonic structures," Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, "Photonic band gaps and localization in the Thue-Morse structures," Appl. Phys. Lett. 86, 201110 (2005).
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L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc and J. Haavisto, "Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
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C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635 (1995).
[CrossRef]

Zhang, D.

Y. Wang, X. Hu, X. Hu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Zhang, Y.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, "Photonic band gaps and localization in the Thue-Morse structures," Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

Zoorob, M. E.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740 (2000).
[CrossRef] [PubMed]

Appl. Phys. Lett. (5)

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, "Photonic band gaps and localization in the Thue-Morse structures," Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc and J. Haavisto, "Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, "Spectrally enhanced light emission from aperiodic photonic structures," Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

H. Altug and J. Vučkovic, "Two-dimensional coupled photonic crystal resonator arrays," Appl. Phys. Lett. 84, 161 (2004).
[CrossRef]

K. Nozaki and T. Baba, "Quasiperiodic photonic crystal microcavity lasers," Appl. Phys. Lett. 84, 4875 (2004).
[CrossRef]

J. All. Comp. (1)

R. Lifshitz, "The square Fibonacci tiling," J. All. Comp. 342, 186 (2002).
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J. Mod. Opt. (1)

K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz, "Photonic band gaps in quasicrystal-related approximant structures," J. Mod. Opt. 50, 2095 (2003).

J. Phys A (2)

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C. Godrèche and J. M. Luck, "Multifractal analysis in reciprocal space and the nature of the Fourier transform of self-similar structures," J. Phys A 23, 3769 (1990).
[CrossRef]

J. Phys. D (1)

W. Steurer and D. Sutter-Widmer, "Photonic and phononic quasicrystals," J. Phys. D 40, R229 (2007).
[CrossRef]

Nature (1)

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740 (2000).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Philos. Mag. (1)

S. Even-Dar Mandel, R. Lifshitz, "Electronic energy spectra and wave functions on the square Fibonacci tiling," Philos. Mag. 86, 759 (2006)
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Phys. Rev. B (6)

Y. Wang, X. Hu, X. Hu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[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, 13458 (2000).
[CrossRef]

T. Fujiwara, M. Kohmoto, and T. Tokihiro, "Multifractal wave functions on a Fibonacci lattice," Phys. Rev. B 40, 7413 (1989).
[CrossRef]

N. Liu, "Propagation of light waves in Thue-Morse dielectric multilayers," Phys. Rev. B 55, 3543 (1997).
[CrossRef]

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635 (1995).
[CrossRef]

Z. Cheng, R. Savit, and R. Merlin, "Structure and electronic properties of Thue-Morse lattices," Phys. Rev. B 37, 4375 (1988).
[CrossRef]

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Y. S. Chan, C. T. Chan and Z. Y. Liu, "Photonic Band Gaps in Two Dimensional Photonic Quasicrystals," Phys. Rev. Lett. 80, 956 (1998).
[CrossRef]

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, "Light Transport through the Band-Edge States of Fibonacci Quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
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R. Lifshitz, A. Arie, and A. Bahabad, "Photonic Quasicrystals for Nonlinear Optical Frequency Conversion," Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

D. Shechtman, I. Blech, and D. Gratias, "Metallic phase with long-range orientational order and no translational symmetry," Phys. Rev. Lett. 53, 1951 (1984).
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E. Maciá, "The role of aperiodic order in science and technology," Rep. Prog. Phys. 69, 397 (2006).
[CrossRef]

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Discussions on "What is a crystal?," Z. Kristallog. 222, 308-319 (2007).

Other (5)

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M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge England 1999) 7th ed.

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[CrossRef]

International Union of Crystallography," Report of the Executive Committee for 1991," Acta Crystallographica A 48, 922.

We underline the fact that x and y coordinates of RTM are independent of each other, therefore the 2D calculation is performed by using independent 1D computations.

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Figures (6)

Fig. 1.
Fig. 1.

First orders and their complements of square Thue-Morse photonic aperiodic crystal made of dielectric cylinders lying in the vacuum: ε=12.25, r=0.35 a. The cylinders are located at the vertices of a two dimensional square Thue-Morse lattice. St1 St2 and St3 represent the local structures replied aperiodically in the infinite Thue-Morse structures.

Fig. 2.
Fig. 2.

Fourier spectrum (magnitude) of dielectric function for M 5. The solid-line circles correspond to the Bragg condition in Eq. (7) for the ω16 bandgaps frequencies. The dot-line circle corresponds to the Bragg condition for the highest-frequency bandgap of PhC. The dash-line represents the path of reciprocal space investigated in the FDTD calculations.

Fig. 3.
Fig. 3.

Density of States of TM modes, for a perfect PhC (black line) and for several 2D ThMo structures: M 3 (red line), M 4 (green line), M 5 (blue line), M 6 (magenta line).

Fig. 4.
Fig. 4.

Mie Scattering Cross Section (a) and the amplitude of the first three Mie coefficients (b) for the cylinder with r=0.35 a, and ε=12.25 for TM polarization. The gray zone represents the first band gap region of ThMo structure.

Fig. 5.
Fig. 5.

The band diagram (a) of the M3 approximant of ThMo infinite structure for the TM modes compared with DOS (b).

Fig. 6.
Fig. 6.

Electric-field pattern along the z direction at Γ point for bands I (a), II (b), III (c), and IV (d).

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

ρ N ( 1 ) ( k ) = 2 2 N j = 0 N 1 sin 2 ( 2 j πk )
k l , m = 2 l + 1 3 × 2 m
S ij = T N ( i ) T N ( j ) with i , j = 1 , 2 N
S N + 1 = S ̅ N S N S N S ̅ N and S ̅ N = S N
R TM i j = s ij + 1 2 R SQ i j i , j = 1 , 2 N
R ( π 2 ) M i = { M i for i even M ̅ i for i odd
k = ω πc

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