Abstract

We present a numerical study on photonic bandgap and band edge modes in the golden-angle spiral array of air cylinders in dielectric media. Despite the lack of long-range translational and rotational order, there is a large PBG for the TE polarized light. Due to spatial inhomogeneity in the air hole spacing, the band edge modes are spatially localized by Bragg scattering from the parastichies in the spiral structure. They have discrete angular momenta that originate from different families of the parastichies whose numbers correspond to the Fibonacci numbers. The unique structural characteristics of the golden-angle spiral lead to distinctive features of the band edge modes that are absent in both photonic crystals and quasicrystals.

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  1. P. Stevens, Patterns in Nature (Little, Brown and Co., New York, 1974).
  2. M. Naylor, “Golden, 2, and π Flowers : A Spiral Story,” Math. Mag. 75, 163 (2002).
    [CrossRef]
  3. A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, “Golden-angle spiral photonic crystal fiber : polarization and dispersion properties,” Opt. Lett. 22, 2716–2718 (2008).
    [CrossRef]
  4. J. Trevino, H. Cao, and L. D. Negro, “Circularly Symmetric Light Scattering from Nanoplasmonic Spirals,” Nano Lett. 11, 2008–2016 (2010).
    [CrossRef]
  5. M. E. Pollard and G. J. Parker, “Low-contrast bandgaps of a planar parabolic spiral lattice,” Opt. Lett. 34, 2805–2807 (2009).
    [CrossRef] [PubMed]
  6. 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).
  7. 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]
  8. M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,” Phys. Rev. B 80, 155112 (2009).
    [CrossRef]
  9. D. Lusk, I. Abdulhalim, and F. Placido, “Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,” Opt. Commun. 198, 273 (2001).
    [CrossRef]
  10. W. Steurer and D. Sutter-Widmer, “Photonic and phononic quasicrystals,” J. Phys. D: Appl. Phys. 40, R229 (2007).
    [CrossRef]
  11. 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]
  12. H. Vogel, “A better way to construct the sunflower head,” Math. Biosci.44, 179 (1979).
  13. http://www.comsol.com

2010 (1)

J. Trevino, H. Cao, and L. D. Negro, “Circularly Symmetric Light Scattering from Nanoplasmonic Spirals,” Nano Lett. 11, 2008–2016 (2010).
[CrossRef]

2009 (2)

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

M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,” Phys. Rev. B 80, 155112 (2009).
[CrossRef]

2008 (1)

A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, “Golden-angle spiral photonic crystal fiber : polarization and dispersion properties,” Opt. Lett. 22, 2716–2718 (2008).
[CrossRef]

2007 (1)

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

2003 (1)

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)

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

2001 (1)

D. Lusk, I. Abdulhalim, and F. Placido, “Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,” Opt. Commun. 198, 273 (2001).
[CrossRef]

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]

Abdulhalim, I.

D. Lusk, I. Abdulhalim, and F. Placido, “Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,” Opt. Commun. 198, 273 (2001).
[CrossRef]

Agrawal, A.

A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, “Golden-angle spiral photonic crystal fiber : polarization and dispersion properties,” Opt. Lett. 22, 2716–2718 (2008).
[CrossRef]

Cao, H.

J. Trevino, H. Cao, and L. D. Negro, “Circularly Symmetric Light Scattering from Nanoplasmonic Spirals,” Nano Lett. 11, 2008–2016 (2010).
[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, 956 (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, 956 (1998).
[CrossRef]

Chen, J.

A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, “Golden-angle spiral photonic crystal fiber : polarization and dispersion properties,” Opt. Lett. 22, 2716–2718 (2008).
[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, 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]

Florescu, M.

M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,” Phys. Rev. B 80, 155112 (2009).
[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]

Grattan, K. T. V.

A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, “Golden-angle spiral photonic crystal fiber : polarization and dispersion properties,” Opt. Lett. 22, 2716–2718 (2008).
[CrossRef]

Joannopoulos, J. D.

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).

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]

Johnson, S. G.

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).

Kejalakshmy, N.

A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, “Golden-angle spiral photonic crystal fiber : polarization and dispersion properties,” Opt. Lett. 22, 2716–2718 (2008).
[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]

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]

Lusk, D.

D. Lusk, I. Abdulhalim, and F. Placido, “Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,” Opt. Commun. 198, 273 (2001).
[CrossRef]

Meade, R. D.

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).

Naylor, M.

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

Negro, L. D.

J. Trevino, H. Cao, and L. D. Negro, “Circularly Symmetric Light Scattering from Nanoplasmonic Spirals,” Nano Lett. 11, 2008–2016 (2010).
[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.

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]

Placido, F.

D. Lusk, I. Abdulhalim, and F. Placido, “Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,” Opt. Commun. 198, 273 (2001).
[CrossRef]

Pollard, M. E.

Rahman, B. M. A.

A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, “Golden-angle spiral photonic crystal fiber : polarization and dispersion properties,” Opt. Lett. 22, 2716–2718 (2008).
[CrossRef]

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).
[CrossRef] [PubMed]

Steinhardt, P. J.

M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,” Phys. Rev. B 80, 155112 (2009).
[CrossRef]

Steurer, W.

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

Stevens, P.

P. Stevens, Patterns in Nature (Little, Brown and Co., New York, 1974).

Sutter-Widmer, D.

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

Torquato, S.

M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,” Phys. Rev. B 80, 155112 (2009).
[CrossRef]

Trevino, J.

J. Trevino, H. Cao, and L. D. Negro, “Circularly Symmetric Light Scattering from Nanoplasmonic Spirals,” Nano Lett. 11, 2008–2016 (2010).
[CrossRef]

Vogel, H.

H. Vogel, “A better way to construct the sunflower head,” Math. Biosci.44, 179 (1979).

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).
[CrossRef] [PubMed]

Winn, J. N.

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).

J. Phys. D: Appl. Phys. (1)

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

Math. Mag. (1)

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

Nano Lett. (1)

J. Trevino, H. Cao, and L. D. Negro, “Circularly Symmetric Light Scattering from Nanoplasmonic Spirals,” Nano Lett. 11, 2008–2016 (2010).
[CrossRef]

Opt. Commun. (1)

D. Lusk, I. Abdulhalim, and F. Placido, “Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,” Opt. Commun. 198, 273 (2001).
[CrossRef]

Opt. Lett. (2)

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

A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, “Golden-angle spiral photonic crystal fiber : polarization and dispersion properties,” Opt. Lett. 22, 2716–2718 (2008).
[CrossRef]

Phys. Rev. B (1)

M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,” Phys. Rev. B 80, 155112 (2009).
[CrossRef]

Phys. Rev. Lett. (2)

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).
[CrossRef] [PubMed]

Other (4)

H. Vogel, “A better way to construct the sunflower head,” Math. Biosci.44, 179 (1979).

http://www.comsol.com

P. Stevens, Patterns in Nature (Little, Brown and Co., New York, 1974).

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).

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