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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 3015–3033

Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals

Jacob Trevino, Seng Fatt Liew, Heeso Noh, Hui Cao, and Luca Dal Negro  »View Author Affiliations

Optics Express, Vol. 20, Issue 3, pp. 3015-3033 (2012)

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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

OCIS Codes
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(160.5293) Materials : Photonic bandgap materials

ToC Category:
Photonic Crystals

Original Manuscript: November 22, 2011
Revised Manuscript: January 15, 2012
Manuscript Accepted: January 19, 2012
Published: January 25, 2012

Jacob Trevino, Seng Fatt Liew, Heeso Noh, Hui Cao, and Luca Dal Negro, "Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals," Opt. Express 20, 3015-3033 (2012)

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  1. 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).
  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]
  3. 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]
  4. T. F. Krauss, D. Labilloy, A. Scherer, and R. M. De La Rue, “Photonic Crystals for Light-Emitting Devices,” Proc. SPIE3278, 306–313 (1998). [CrossRef]
  5. 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]
  6. 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]
  7. 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]
  8. 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]
  9. A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, “Golden spiral photonic crystal fiber: polarization and dispersion properties,” Opt. Lett.33, 2716–2718 (2008). [CrossRef]
  10. J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett.11(5), 2008–2016 (2011). [CrossRef] [PubMed]
  11. 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. Express19(24), 23631–23642 (2011). [CrossRef] [PubMed]
  12. J. A. Adam, A Mathematical Nature Walk (Princeton University Press, 2009).
  13. E. Macia, Aperiodic Structures in Condensed Matter: Fundamentals and Applications (CRC Press Taylor & Francis, Boca Raton, 2009).
  14. M. Naylor, “Golden, √ 2, and π Flowers: A Spiral Story,” Math. Mag.75, 163–172 (2002). [CrossRef]
  15. G. J. Mitchison, “Phyllotaxis and the fibonacci series,” Science196(4287), 270–275 (1977). [CrossRef] [PubMed]
  16. C. Janot, Quasicrystals: A Primer (Clarendon Press, 1992).
  17. 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. B79(8), 085404 (2009). [CrossRef]
  18. C. Forestiere, G. F. Walsh, G. Miano, and L. Dal Negro, “Nanoplasmonics of prime number arrays,” Opt. Express17(26), 24288–24303 (2009). [CrossRef] [PubMed]
  19. A. Baddeley and R. Turner, “Spatstat: an R package for analyzing spatial point patterns,” J. Stat. Softw.12(6), 1–42 (2005).
  20. B. D. Ripley, “Modelling spatial patterns,” J. R. Stat. Soc., B39, 172–212 (1977).
  21. 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. A84(3), 033820 (2011). [CrossRef]
  22. 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]
  23. 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).
  24. http://www.comsol.com .
  25. Y. Lai, Z. Zhang, C. Chan, and L. Tsang, “Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals,” Phys. Rev. B76(16), 165132 (2007). [CrossRef]
  26. E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep.376(4-5), 225–337 (2003). [CrossRef]
  27. P. K. Thakur and P. Biswas, “Multifractal scaling of electronic transmission resonances in perfect and imperfect Fibonacci δ-function potentials,” Physica A265(1–2), 1–18 (1999). [CrossRef]
  28. J. Feder, Fractals (Plenum Press, 1988).
  29. J. Gouyet, Physics and Fractal Structures (Springer, 1996).
  30. B. B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman and Co., 1982).
  31. H. E. Stanley and P. Meakin, “Multifractal phenomena in physics and chemistry,” Nature335(6189), 405–409 (1988) (Review). [CrossRef]
  32. B. B. Mandelbrot, “An introduction to multifractal distribution functions,” Fluctuations and Pattern Formation, H.E. Stanley, N. Ostrowsky, eds., (Kluwer, 1988).
  33. 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).
  34. J. F. Muzy, E. Bacry, and A. Arneodo, “The multifractal formalism revisited with wavelets,” Int. J. Bifurcat. Chaos4(2), 245–302 (1994). [CrossRef]
  35. 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. A33(2), 1141–1151 (1986). [CrossRef] [PubMed]
  36. A. Chhabra and R. V. Jensen, “Direct determination of the f( α ) singularity spectrum,” Phys. Rev. Lett.62(12), 1327–1330 (1989). [CrossRef] [PubMed]
  37. A. Karperien, FracLac for ImageJ, version 2.5. http://rsb.info.nih.gov/ij/plugins/fraclac/FLHelp/Introduction.htm . (1999–2007).
  38. W. S. Rasband and J. Image, U. S. National Institutes of Health, Bethesda, Maryland, USA, http://imagej.nih.gov/ij/ , (1997–2011).
  39. 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]
  40. S. Mallat, A Wavelet Tour of Signal Processing, 3rd ed., (Elsevier, 2009).
  41. J. C. van den Berg, ed., Wavelets in Physics (Cambridge University Press, 2004).
  42. J. Buckheit, S. Chen, D. Donoho, I. Johnstone, and J. Scargle, WaveLab850 http://www.stat.stanford.edu/~wavelab Stanford University & NASA-Ames Research Center (2005).
  43. 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. A64(6), 063808 (2001). [CrossRef]

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