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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References

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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).
    [CrossRef]
  13. http://www.comsol.com

2010

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

2009

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

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

2008

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

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

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

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

2001

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

1998

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).
[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).
[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.

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

Math. Mag.

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

Nano Lett.

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

Opt. Commun.

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.

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

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.

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

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

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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Figures (12)

Fig. 1
Fig. 1

(a) Golden-angle spiral array consisting of 1000 circles. (b) Spatial Fourier spectrum of the spiral structure in (a). (c) Delaunay triangulation of (a). The line segments that connect neighboring circles are color-coded by their lengths d. (d) Statistical distribution of the distance between neighboring particles d normalized to the most probable value do. The colors are consistent to those in (c).

Fig. 2
Fig. 2

Fourier Bessel Transform (FBT) of the golden-angle spiral structure in Fig. 1(a) gives |f(m,k)|2 (a) and F(m) (b).

Fig. 3
Fig. 3

LDOS calculated at the center of the golden-angle spiral array as a function of the normalized frequency do. The regions at the lower and upper band edge where the band edge modes exist are highlighted.

Fig. 4
Fig. 4

Quality factors of the lower band edge modes (a) and upper band edge modes (b) versus the normalized frequency do/λ.

Fig. 5
Fig. 5

Spatial distributions of magnetic field Hz for the first three pairs of band edge modes of class A. The modes are localized within a ring of radius ∼ 12do.

Fig. 6
Fig. 6

Spatial distributions of magnetic field Hz for the first three pairs of band edge modes of class B. The modes are localized within a ring of radius ∼ 7do, close to the center of spiral than the modes of class A.

Fig. 7
Fig. 7

Spatial distributions of magnetic field Hz for the first three pairs of band edge modes of class C. The modes are located near the boundary of the spiral and have stronger light leakage through the boundary.

Fig. 8
Fig. 8

Spatial distributions of magnetic field Hz for the first three pairs of band edge modes of class D. The modes are localized closer to the center of the spiral and have small light leakage through the outer boundary. Mode D1” has the same field distribution as D1 after the air cylinders in the outer layer of the spiral are removed. The light leakage increases since the mode is closer to the boundary now. The insensitivity of mode D1 to the change at the boundary confirms it is a localized mode.

Fig. 9
Fig. 9

(a) Overlay of the region where class A modes are localized on the color map of the neighboring particles distance of air cylinders revealing class A modes stay mostly inside a ring labeled (ii) and sandwiched between two other rings (i) and (iii). (b) LDOS in the regions (i), (ii) and (iii).

Fig. 10
Fig. 10

(a) Magnetic field distribution of mode A1 revealing the field maxima follow a family of 21 parastichies twisting in the CCW direction and another family of 89 parastichies in the CW direction (both are marked by the dashed arrows). (b) FBT of the field distribution in (a) gives F(m′,kr). (c) Region of the spiral array that contains 90% energy of mode A1 is shown after removing the air cylinders outside. (d) FBT of the structure in (c) gives F(m,kr) of the local region where mode A1 stays.

Fig. 11
Fig. 11

(a) Magnetic field distribution of mode D1 revealing the field maxima follow a family of 34 parastichies twisting in the CW direction(marked by the dashed arrow). (b) FBT of the field distribution in (a) gives F(m′,kr). (c) Region of the spiral array that contains 90% energy of mode D1 is shown after removing the air cylinders outside. (d) FBT of the structure in (c) gives F(m,kr) of the local region where mode D1 stays.

Fig. 12
Fig. 12

F(m′) from the FBT of the field profiles of mode D1 (a), D2 (b), and D3 (c) illustrating the splitting of the peak due to azimuthal modulations of the envelop functions of D2 and D3.

Equations (3)

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r = b q ,
θ = q α ,
f ( m , k r ) = 1 2 π 0 0 2 π r dr d θ ρ ( r , θ ) J m ( k r r ) e i m θ ,

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