Abstract

We present a numerical study of the structural properties, photonic density of states and bandedge modes of Vogel spiral arrays of dielectric cylinders in air. Specifically, we systematically investigate different types of Vogel spirals obtained by the modulation of the divergence angle parameter above and below the golden angle value (≈137.507°). We found that these arrays exhibit large fluctuations in the distribution of neighboring particles characterized by multifractal singularity spectra and pair correlation functions that can be tuned between amorphous and random structures. We also show that the rich structural complexity of Vogel spirals results in a multifractal photonic mode density and isotropic bandedge modes with distinctive spatial localization character. Vogel spiral structures offer the opportunity to create novel photonic devices that leverage radially localized and isotropic bandedge modes to enhance light-matter coupling, such as optical sensors, light sources, concentrators, and broadband optical couplers.

© 2012 OSA

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2011 (3)

J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett. 11(5), 2008–2016 (2011).
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S. F. Liew, H. Noh, J. Trevino, L. D. Negro, and H. Cao, “Localized photonic band edge modes and orbital angular momenta of light in a golden-angle spiral,” Opt. Express 19(24), 23631–23642 (2011).
[CrossRef] [PubMed]

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[CrossRef]

2009 (3)

M. E. Pollard and G. J. Parker, “Low-contrast bandgaps of a planar parabolic spiral lattice,” Opt. Lett. 34(18), 2805–2807 (2009).
[CrossRef] [PubMed]

C. Forestiere, G. Miano, G. Rubinacci, and L. Dal Negro, “Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticles arrays,” Phys. Rev. B 79(8), 085404 (2009).
[CrossRef]

C. Forestiere, G. F. Walsh, G. Miano, and L. Dal Negro, “Nanoplasmonics of prime number arrays,” Opt. Express 17(26), 24288–24303 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (1)

Y. Lai, Z. Zhang, C. Chan, and L. Tsang, “Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals,” Phys. Rev. B 76(16), 165132 (2007).
[CrossRef]

2005 (2)

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic bandgaps and localization in the Thue-Morse structures,” Appl. Phys. Lett. 86(20), 201110 (2005).
[CrossRef]

A. Baddeley and R. Turner, “Spatstat: an R package for analyzing spatial point patterns,” J. Stat. Softw. 12(6), 1–42 (2005).

2004 (1)

M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, “Lasing action due to the two-dimensional quasiperiodicity of photonic quasicrystals with a Penrose lattice,” Phys. Rev. Lett. 92(12), 123906 (2004).
[CrossRef] [PubMed]

2003 (2)

S. Torquato and F. H. Stillinger, “Local density fluctuations, hyperuniformity, and order metrics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(4), 041113 (2003).
[CrossRef] [PubMed]

E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376(4-5), 225–337 (2003).
[CrossRef]

2002 (1)

M. Naylor, “Golden, √ 2, and π Flowers: A Spiral Story,” Math. Mag. 75, 163–172 (2002).
[CrossRef]

2001 (1)

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64(6), 063808 (2001).
[CrossRef]

1999 (1)

P. K. Thakur and P. Biswas, “Multifractal scaling of electronic transmission resonances in perfect and imperfect Fibonacci δ-function potentials,” Physica A 265(1–2), 1–18 (1999).
[CrossRef]

1998 (2)

T. F. Krauss, D. Labilloy, A. Scherer, and R. M. De La Rue, “Photonic Crystals for Light-Emitting Devices,” Proc. SPIE 3278, 306–313 (1998).
[CrossRef]

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic Band Gaps in Two Dimensional Photonic Quasicrystals,” Phys. Rev. Lett. 80(5), 956–959 (1998).
[CrossRef]

1996 (1)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

1994 (2)

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low‐loss bends and high Q cavities,” J. Appl. Phys. 75(9), 4753–4755 (1994).
[CrossRef]

J. F. Muzy, E. Bacry, and A. Arneodo, “The multifractal formalism revisited with wavelets,” Int. J. Bifurcat. Chaos 4(2), 245–302 (1994).
[CrossRef]

1989 (1)

A. Chhabra and R. V. Jensen, “Direct determination of the f( α ) singularity spectrum,” Phys. Rev. Lett. 62(12), 1327–1330 (1989).
[CrossRef] [PubMed]

1988 (1)

H. E. Stanley and P. Meakin, “Multifractal phenomena in physics and chemistry,” Nature 335(6189), 405–409 (1988) (Review).
[CrossRef]

1986 (1)

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, “Fractal measures and their singularities: The characterization of strange sets,” Phys. Rev. A 33(2), 1141–1151 (1986).
[CrossRef] [PubMed]

1977 (2)

G. J. Mitchison, “Phyllotaxis and the fibonacci series,” Science 196(4287), 270–275 (1977).
[CrossRef] [PubMed]

B. D. Ripley, “Modelling spatial patterns,” J. R. Stat. Soc., B 39, 172–212 (1977).

Agrawal, A.

Albuquerque, E. L.

E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376(4-5), 225–337 (2003).
[CrossRef]

Alerhand, O. L.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low‐loss bends and high Q cavities,” J. Appl. Phys. 75(9), 4753–4755 (1994).
[CrossRef]

Arneodo, A.

J. F. Muzy, E. Bacry, and A. Arneodo, “The multifractal formalism revisited with wavelets,” Int. J. Bifurcat. Chaos 4(2), 245–302 (1994).
[CrossRef]

Bacry, E.

J. F. Muzy, E. Bacry, and A. Arneodo, “The multifractal formalism revisited with wavelets,” Int. J. Bifurcat. Chaos 4(2), 245–302 (1994).
[CrossRef]

Baddeley, A.

A. Baddeley and R. Turner, “Spatstat: an R package for analyzing spatial point patterns,” J. Stat. Softw. 12(6), 1–42 (2005).

Biswas, P.

P. K. Thakur and P. Biswas, “Multifractal scaling of electronic transmission resonances in perfect and imperfect Fibonacci δ-function potentials,” Physica A 265(1–2), 1–18 (1999).
[CrossRef]

Burin, A. L.

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64(6), 063808 (2001).
[CrossRef]

Cao, H.

J. K. Yang, H. Noh, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, “Lasing modes in polycrystalline and amorphous photonic structures,” Phys. Rev. A 84(3), 033820 (2011).
[CrossRef]

S. F. Liew, H. Noh, J. Trevino, L. D. Negro, and H. Cao, “Localized photonic band edge modes and orbital angular momenta of light in a golden-angle spiral,” Opt. Express 19(24), 23631–23642 (2011).
[CrossRef] [PubMed]

J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett. 11(5), 2008–2016 (2011).
[CrossRef] [PubMed]

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64(6), 063808 (2001).
[CrossRef]

Chan, C.

Y. Lai, Z. Zhang, C. Chan, and L. Tsang, “Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals,” Phys. Rev. B 76(16), 165132 (2007).
[CrossRef]

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(5), 956–959 (1998).
[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(5), 956–959 (1998).
[CrossRef]

Chang, R. P. H.

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64(6), 063808 (2001).
[CrossRef]

Chen, J.

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

Chhabra, A.

A. Chhabra and R. V. Jensen, “Direct determination of the f( α ) singularity spectrum,” Phys. Rev. Lett. 62(12), 1327–1330 (1989).
[CrossRef] [PubMed]

Cottam, M. G.

E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376(4-5), 225–337 (2003).
[CrossRef]

Dal Negro, L.

J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett. 11(5), 2008–2016 (2011).
[CrossRef] [PubMed]

C. Forestiere, G. Miano, G. Rubinacci, and L. Dal Negro, “Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticles arrays,” Phys. Rev. B 79(8), 085404 (2009).
[CrossRef]

C. Forestiere, G. F. Walsh, G. Miano, and L. Dal Negro, “Nanoplasmonics of prime number arrays,” Opt. Express 17(26), 24288–24303 (2009).
[CrossRef] [PubMed]

De La Rue, R. M.

T. F. Krauss, D. Labilloy, A. Scherer, and R. M. De La Rue, “Photonic Crystals for Light-Emitting Devices,” Proc. SPIE 3278, 306–313 (1998).
[CrossRef]

Devenyi, A.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low‐loss bends and high Q cavities,” J. Appl. Phys. 75(9), 4753–4755 (1994).
[CrossRef]

Edagawa, K.

M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, “Lasing action due to the two-dimensional quasiperiodicity of photonic quasicrystals with a Penrose lattice,” Phys. Rev. Lett. 92(12), 123906 (2004).
[CrossRef] [PubMed]

Fan, S.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

Feng, S.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic bandgaps and localization in the Thue-Morse structures,” Appl. Phys. Lett. 86(20), 201110 (2005).
[CrossRef]

Forestiere, C.

C. Forestiere, G. F. Walsh, G. Miano, and L. Dal Negro, “Nanoplasmonics of prime number arrays,” Opt. Express 17(26), 24288–24303 (2009).
[CrossRef] [PubMed]

C. Forestiere, G. Miano, G. Rubinacci, and L. Dal Negro, “Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticles arrays,” Phys. Rev. B 79(8), 085404 (2009).
[CrossRef]

Grattan, K. T. V.

Halsey, T. C.

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, “Fractal measures and their singularities: The characterization of strange sets,” Phys. Rev. A 33(2), 1141–1151 (1986).
[CrossRef] [PubMed]

Huang, K. C.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic bandgaps and localization in the Thue-Morse structures,” Appl. Phys. Lett. 86(20), 201110 (2005).
[CrossRef]

Jensen, M. H.

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, “Fractal measures and their singularities: The characterization of strange sets,” Phys. Rev. A 33(2), 1141–1151 (1986).
[CrossRef] [PubMed]

Jensen, R. V.

A. Chhabra and R. V. Jensen, “Direct determination of the f( α ) singularity spectrum,” Phys. Rev. Lett. 62(12), 1327–1330 (1989).
[CrossRef] [PubMed]

Jiang, X.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic bandgaps and localization in the Thue-Morse structures,” Appl. Phys. Lett. 86(20), 201110 (2005).
[CrossRef]

Joannopoulos, J. D.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic bandgaps and localization in the Thue-Morse structures,” Appl. Phys. Lett. 86(20), 201110 (2005).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low‐loss bends and high Q cavities,” J. Appl. Phys. 75(9), 4753–4755 (1994).
[CrossRef]

Kadanoff, L. P.

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, “Fractal measures and their singularities: The characterization of strange sets,” Phys. Rev. A 33(2), 1141–1151 (1986).
[CrossRef] [PubMed]

Kash, K.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low‐loss bends and high Q cavities,” J. Appl. Phys. 75(9), 4753–4755 (1994).
[CrossRef]

Kejalakshmy, N.

Krauss, T. F.

T. F. Krauss, D. Labilloy, A. Scherer, and R. M. De La Rue, “Photonic Crystals for Light-Emitting Devices,” Proc. SPIE 3278, 306–313 (1998).
[CrossRef]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

Labilloy, D.

T. F. Krauss, D. Labilloy, A. Scherer, and R. M. De La Rue, “Photonic Crystals for Light-Emitting Devices,” Proc. SPIE 3278, 306–313 (1998).
[CrossRef]

Lai, Y.

Y. Lai, Z. Zhang, C. Chan, and L. Tsang, “Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals,” Phys. Rev. B 76(16), 165132 (2007).
[CrossRef]

Liew, S. F.

J. K. Yang, H. Noh, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, “Lasing modes in polycrystalline and amorphous photonic structures,” Phys. Rev. A 84(3), 033820 (2011).
[CrossRef]

S. F. Liew, H. Noh, J. Trevino, L. D. Negro, and H. Cao, “Localized photonic band edge modes and orbital angular momenta of light in a golden-angle spiral,” Opt. Express 19(24), 23631–23642 (2011).
[CrossRef] [PubMed]

Ling, Y.

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64(6), 063808 (2001).
[CrossRef]

Liu, X.

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64(6), 063808 (2001).
[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(5), 956–959 (1998).
[CrossRef]

Meade, R. D.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low‐loss bends and high Q cavities,” J. Appl. Phys. 75(9), 4753–4755 (1994).
[CrossRef]

Meakin, P.

H. E. Stanley and P. Meakin, “Multifractal phenomena in physics and chemistry,” Nature 335(6189), 405–409 (1988) (Review).
[CrossRef]

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

Miano, G.

C. Forestiere, G. Miano, G. Rubinacci, and L. Dal Negro, “Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticles arrays,” Phys. Rev. B 79(8), 085404 (2009).
[CrossRef]

C. Forestiere, G. F. Walsh, G. Miano, and L. Dal Negro, “Nanoplasmonics of prime number arrays,” Opt. Express 17(26), 24288–24303 (2009).
[CrossRef] [PubMed]

Mitchison, G. J.

G. J. Mitchison, “Phyllotaxis and the fibonacci series,” Science 196(4287), 270–275 (1977).
[CrossRef] [PubMed]

Muzy, J. F.

J. F. Muzy, E. Bacry, and A. Arneodo, “The multifractal formalism revisited with wavelets,” Int. J. Bifurcat. Chaos 4(2), 245–302 (1994).
[CrossRef]

Naylor, M.

M. Naylor, “Golden, √ 2, and π Flowers: A Spiral Story,” Math. Mag. 75, 163–172 (2002).
[CrossRef]

Negro, L. D.

Noh, H.

J. K. Yang, H. Noh, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, “Lasing modes in polycrystalline and amorphous photonic structures,” Phys. Rev. A 84(3), 033820 (2011).
[CrossRef]

S. F. Liew, H. Noh, J. Trevino, L. D. Negro, and H. Cao, “Localized photonic band edge modes and orbital angular momenta of light in a golden-angle spiral,” Opt. Express 19(24), 23631–23642 (2011).
[CrossRef] [PubMed]

Notomi, M.

M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, “Lasing action due to the two-dimensional quasiperiodicity of photonic quasicrystals with a Penrose lattice,” Phys. Rev. Lett. 92(12), 123906 (2004).
[CrossRef] [PubMed]

Parker, G. J.

Pollard, M. E.

Procaccia, I.

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, “Fractal measures and their singularities: The characterization of strange sets,” Phys. Rev. A 33(2), 1141–1151 (1986).
[CrossRef] [PubMed]

Rahman, B. M. A.

Ratner, M. A.

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64(6), 063808 (2001).
[CrossRef]

Ripley, B. D.

B. D. Ripley, “Modelling spatial patterns,” J. R. Stat. Soc., B 39, 172–212 (1977).

Rooks, M. J.

J. K. Yang, H. Noh, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, “Lasing modes in polycrystalline and amorphous photonic structures,” Phys. Rev. A 84(3), 033820 (2011).
[CrossRef]

Rubinacci, G.

C. Forestiere, G. Miano, G. Rubinacci, and L. Dal Negro, “Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticles arrays,” Phys. Rev. B 79(8), 085404 (2009).
[CrossRef]

Scherer, A.

T. F. Krauss, D. Labilloy, A. Scherer, and R. M. De La Rue, “Photonic Crystals for Light-Emitting Devices,” Proc. SPIE 3278, 306–313 (1998).
[CrossRef]

Shraiman, B. I.

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, “Fractal measures and their singularities: The characterization of strange sets,” Phys. Rev. A 33(2), 1141–1151 (1986).
[CrossRef] [PubMed]

Smith, D. A.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low‐loss bends and high Q cavities,” J. Appl. Phys. 75(9), 4753–4755 (1994).
[CrossRef]

Solomon, G. S.

J. K. Yang, H. Noh, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, “Lasing modes in polycrystalline and amorphous photonic structures,” Phys. Rev. A 84(3), 033820 (2011).
[CrossRef]

Stanley, H. E.

H. E. Stanley and P. Meakin, “Multifractal phenomena in physics and chemistry,” Nature 335(6189), 405–409 (1988) (Review).
[CrossRef]

Stillinger, F. H.

S. Torquato and F. H. Stillinger, “Local density fluctuations, hyperuniformity, and order metrics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(4), 041113 (2003).
[CrossRef] [PubMed]

Suzuki, H.

M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, “Lasing action due to the two-dimensional quasiperiodicity of photonic quasicrystals with a Penrose lattice,” Phys. Rev. Lett. 92(12), 123906 (2004).
[CrossRef] [PubMed]

Tamamura, T.

M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, “Lasing action due to the two-dimensional quasiperiodicity of photonic quasicrystals with a Penrose lattice,” Phys. Rev. Lett. 92(12), 123906 (2004).
[CrossRef] [PubMed]

Thakur, P. K.

P. K. Thakur and P. Biswas, “Multifractal scaling of electronic transmission resonances in perfect and imperfect Fibonacci δ-function potentials,” Physica A 265(1–2), 1–18 (1999).
[CrossRef]

Torquato, S.

S. Torquato and F. H. Stillinger, “Local density fluctuations, hyperuniformity, and order metrics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(4), 041113 (2003).
[CrossRef] [PubMed]

Trevino, J.

Tsang, L.

Y. Lai, Z. Zhang, C. Chan, and L. Tsang, “Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals,” Phys. Rev. B 76(16), 165132 (2007).
[CrossRef]

Turner, R.

A. Baddeley and R. Turner, “Spatstat: an R package for analyzing spatial point patterns,” J. Stat. Softw. 12(6), 1–42 (2005).

Villeneuve, P. R.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

Walsh, G. F.

Yang, J. K.

J. K. Yang, H. Noh, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, “Lasing modes in polycrystalline and amorphous photonic structures,” Phys. Rev. A 84(3), 033820 (2011).
[CrossRef]

Yi, Y.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic bandgaps and localization in the Thue-Morse structures,” Appl. Phys. Lett. 86(20), 201110 (2005).
[CrossRef]

Zhang, Y.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic bandgaps and localization in the Thue-Morse structures,” Appl. Phys. Lett. 86(20), 201110 (2005).
[CrossRef]

Zhang, Z.

Y. Lai, Z. Zhang, C. Chan, and L. Tsang, “Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals,” Phys. Rev. B 76(16), 165132 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic bandgaps and localization in the Thue-Morse structures,” Appl. Phys. Lett. 86(20), 201110 (2005).
[CrossRef]

Int. J. Bifurcat. Chaos (1)

J. F. Muzy, E. Bacry, and A. Arneodo, “The multifractal formalism revisited with wavelets,” Int. J. Bifurcat. Chaos 4(2), 245–302 (1994).
[CrossRef]

J. Appl. Phys. (1)

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low‐loss bends and high Q cavities,” J. Appl. Phys. 75(9), 4753–4755 (1994).
[CrossRef]

J. R. Stat. Soc., B (1)

B. D. Ripley, “Modelling spatial patterns,” J. R. Stat. Soc., B 39, 172–212 (1977).

J. Stat. Softw. (1)

A. Baddeley and R. Turner, “Spatstat: an R package for analyzing spatial point patterns,” J. Stat. Softw. 12(6), 1–42 (2005).

Math. Mag. (1)

M. Naylor, “Golden, √ 2, and π Flowers: A Spiral Story,” Math. Mag. 75, 163–172 (2002).
[CrossRef]

Nano Lett. (1)

J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett. 11(5), 2008–2016 (2011).
[CrossRef] [PubMed]

Nature (1)

H. E. Stanley and P. Meakin, “Multifractal phenomena in physics and chemistry,” Nature 335(6189), 405–409 (1988) (Review).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rep. (1)

E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376(4-5), 225–337 (2003).
[CrossRef]

Phys. Rev. A (3)

J. K. Yang, H. Noh, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, “Lasing modes in polycrystalline and amorphous photonic structures,” Phys. Rev. A 84(3), 033820 (2011).
[CrossRef]

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, “Fractal measures and their singularities: The characterization of strange sets,” Phys. Rev. A 33(2), 1141–1151 (1986).
[CrossRef] [PubMed]

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64(6), 063808 (2001).
[CrossRef]

Phys. Rev. B (2)

Y. Lai, Z. Zhang, C. Chan, and L. Tsang, “Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals,” Phys. Rev. B 76(16), 165132 (2007).
[CrossRef]

C. Forestiere, G. Miano, G. Rubinacci, and L. Dal Negro, “Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticles arrays,” Phys. Rev. B 79(8), 085404 (2009).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

S. Torquato and F. H. Stillinger, “Local density fluctuations, hyperuniformity, and order metrics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(4), 041113 (2003).
[CrossRef] [PubMed]

Phys. Rev. Lett. (4)

A. Chhabra and R. V. Jensen, “Direct determination of the f( α ) singularity spectrum,” Phys. Rev. Lett. 62(12), 1327–1330 (1989).
[CrossRef] [PubMed]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, “Lasing action due to the two-dimensional quasiperiodicity of photonic quasicrystals with a Penrose lattice,” Phys. Rev. Lett. 92(12), 123906 (2004).
[CrossRef] [PubMed]

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic Band Gaps in Two Dimensional Photonic Quasicrystals,” Phys. Rev. Lett. 80(5), 956–959 (1998).
[CrossRef]

Physica A (1)

P. K. Thakur and P. Biswas, “Multifractal scaling of electronic transmission resonances in perfect and imperfect Fibonacci δ-function potentials,” Physica A 265(1–2), 1–18 (1999).
[CrossRef]

Proc. SPIE (1)

T. F. Krauss, D. Labilloy, A. Scherer, and R. M. De La Rue, “Photonic Crystals for Light-Emitting Devices,” Proc. SPIE 3278, 306–313 (1998).
[CrossRef]

Science (1)

G. J. Mitchison, “Phyllotaxis and the fibonacci series,” Science 196(4287), 270–275 (1977).
[CrossRef] [PubMed]

Other (17)

C. Janot, Quasicrystals: A Primer (Clarendon Press, 1992).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, Princeton, 2008).

L. Dal Negro and S. V. Boriskina, “Deterministic Aperiodic Nanostructures for Photonics and Plasmonics Applications,” Laser Photon. Rev. (2011), doi: 10.1002/lpor.201000046.
[CrossRef]

J. A. Adam, A Mathematical Nature Walk (Princeton University Press, 2009).

E. Macia, Aperiodic Structures in Condensed Matter: Fundamentals and Applications (CRC Press Taylor & Francis, Boca Raton, 2009).

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J. Gouyet, Physics and Fractal Structures (Springer, 1996).

B. B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman and Co., 1982).

J. Illian, A. Penttinen, H. Stoyan, and D. Stoyan, Statistical Analysis and Modeling of Spatial Point Patterns, S. Senn, M. Scott, and V. Barnett, ed. (John Wiley, 2008).

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U. Frisch and G. Parisi, “Fully developed turbulence and intermittency,” Turbulence and Predictability in Geophysical Fluid Dynamic and Climate Dynamics, M. Ghil, R. Benzi, and G. Parisi, eds. (North Holland, 1985).

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Figures (12)

Fig. 1
Fig. 1

Vogel spiral array consisting of 1000 particles, created with a divergence angle of (a) 137.3° (α1), (b) 137.3692546° (α2), (c) 137.4038819° (α3), (d) 137.4731367° (α4), (e) 137.5077641° (GA), (f) 137.5231367° (β1), (g) 137.553882° (β2), (h) 137.5692547° (β3), (i) 137.6° (β1).

Fig. 2
Fig. 2

Calculated spatial Fourier spectrum of the spiral structures show in Fig. 1. The reciprocal space structure of a (a) α1-spiral, (b) α2-spiral, (c) α3-spiral, (d) α4-spiral, (e) g.a-spiral, (f) β1-spiral (g) β2-spiral, (h) β3-spiral, and (i) β4-spiral are plotted where Δ represents the average edge-to-edge minimum inter-particle separation.

Fig. 3
Fig. 3

Pair correlation function g(r) for spiral arrays with divergence angles between (a) α1 and the golden angle and (b) between the golden angle and β 4.

Fig. 4
Fig. 4

Statistical distribution of spiral structures shown in Fig. 1. Values represent the distance between neighboring particles d normalized to the most probable value do, obtained by Delaunay triangulation (increasing numerical values from blue to red colors). The Y-axis displays the fraction of d in the total distribution.

Fig. 5
Fig. 5

Delaunay triangulation of spiral structures shown in Fig. 1. The line segments that connect neighboring circles are color-coded by their lengths d. The colors are consistent to those in Fig. 4.

Fig. 6
Fig. 6

LDOS calculated at the center of the each spiral array as a function of normalized frequency (as described in Section 3) for spiral arrays with divergence angles between (a) α1 and the golden angle and (b) between the golden angle and β4.

Fig. 7
Fig. 7

Multifractal singularity spectra f(α) of direct space spiral arrays (N = 1000) with divergence angles between (a) α1 and the golden angle and (b) between the golden angle and β4. Multifractal spectra for spiral LDOS with divergence angles between (c) α1 and the golden angle and (d) between the golden angle and β4.

Fig. 8
Fig. 8

Quality factors of the air band edge modes for (a) α1-spiral and (b) g-spiral and β4-spiral versus normalized frequency (as described in Section 3).

Fig. 9
Fig. 9

Spatial distributions of electric field Ez for the first three band edge modes of (a-c) class B in a α1-spiral, (d-f) class A in a g-spiral and (g-i) class A in a β4-spiral. Spectrally located at ω/ω0 = (a) 0.9248, (b) 0.9290, (c) 0.9376, (d) 1.1629, (e) 1.1638, (f) 1.1657, (g) 1.1781, (h) 1.1900, and (i) 1.2152 (normalized as described in Section 3).

Fig. 10
Fig. 10

Spatial distributions of electric field Ez class D1 band edge mode (ω/ω0 = 1.053 normalized as described in Section 3) in a α1-spiral with (a) 1000 particles, (b) 750 particles and (c) 500 particles. (d) LDOS calculated at the center of α1-spirals with varying number of particles between n = 150 and n = 1000.

Fig. 12
Fig. 12

Spatial distributions of electric field Ez class A band edge mode (ω/ω0 = 1.190 normalized as described in Section 3) in a β4-spiral with (a) 1000 particles, (b) 750 particles and (c) 500 particles. (d) LDOS calculated at the center of β4-spirals with varying number of particles between n = 150 and n = 1000.

Fig. 11
Fig. 11

Spatial distributions of electric field Ez class B band edge mode (ω/ω0 = 1.175 normalized as described in Section 3) in a GA-spiral with (a) 1000 particles, (b) 750 particles and (c) 500 particles. (d) LDOS calculated at the center of g.a-spirals with varying number of particles between n = 150 and n = 1000.

Tables (1)

Tables Icon

Table 1 Divergence Angle Structural Perturbations of GA-spiral

Equations (12)

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

r=a n
θ=nα
g(r)= K'(r) 2πr
N(ε)= ε D f
μ( B x (ε)) ε α(x)
N α (ε) ε f(α)
μ i = P (i) q P (i) q
α= μ i ×lnP(i)/lnε
f(a)= [ μ i × ln μ i ]/lnε
W ψ [f](b,a)= 1 a + ψ ¯ ( xb a )f(x)dx
Ζ(q,a)= p | W ψ [f](x,a) | q
Ζ(q,a) a τ(a)

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