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

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

http://dx.doi.org/10.1364/OE.20.003015

Acrobat PDF (2422 KB)

### 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

## 1. Introduction

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. SPIE **3278**, 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]

*et al.*have shown that a GA-spiral array of air cylinders in a dielectric medium supports a large PBG for TE polarized light and characteristic bandedge modes that are absent in both photonic crystals and quasicrystals [11

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. Express **19**(24), 23631–23642 (2011). [CrossRef] [PubMed]

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

## 2. Geometrical structure of aperiodic spirals

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

*n*= 0,1,2,…is an integer,

*a*is a constant scaling factor, and α is an irrational angle that gives the constant aperture between adjacent position vectors

*r(n)*and

*r(n + 1)*of particles. In the case of the “sunflower spiral”, also called golden-angle spiral (GA-spiral), α ≈137.508° is an irrational number known as the “golden angle” that can be expressed as α = 360/φ

^{2}, where φ = (1 + √5)/2 ≈1.618 is the golden number, which can be approximated by the ratio of consecutive Fibonacci numbers. Rational approximations to the golden angle can be obtained by the formula α = 360 × (1 +

*p/q*)

^{−1}where

*p*and

*q*<

*p*are consecutive Fibonacci numbers. The GA-spiral is shown in Fig. 1(e) for

*n*= 1000 particles.

_{1}-spiral) and 137.6° (i.e., β

_{4}-spiral), shown in Fig. 1(a) and Fig. 1(i), respectively [10

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]

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

_{1}- and β

_{4}-spirals are called “nearly golden spirals”, and their families of diverging arms, known as

*parastichies,*are considerably fewer.

_{1}-spiral and the GA-spiral, and also between the golden angle and β

_{4}, as summarized in Table 1 . These structures can be considered as one-parameter (α) structural perturbations of the GA-spiral, and possess fascinating geometrical features, which are responsible for unique mode localization properties and optical spectra, as it will be discussed from section 3.

_{1}and β

_{4}in Fig. 1), gaps appear in the center head of the spirals and the resulting point patterns mostly consist of either CW or CCW spiraling arms. Stronger structural perturbations (i.e., further increase in the diverge angle) eventually lead to less interesting spiral structures containing only radially diverging parastichies (not investigated here).

18. 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]

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. Express **19**(24), 23631–23642 (2011). [CrossRef] [PubMed]

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]

**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. Express **19**(24), 23631–23642 (2011). [CrossRef] [PubMed]

**19**(24), 23631–23642 (2011). [CrossRef] [PubMed]

*g(r)*, also known as the radial density distribution function, is used to characterize the probability of finding two particles separated by a distance

*r*, measuring the local (correlation) order in the structure. Figure 3(a) displays the calculated

*g(r)*for spiral arrays with divergence angles between α

_{1}and the golden angle (α series), while Fig. 3(b) shows the results of the analysis for arrays generated with divergence angles between the golden angle and β

_{4}. (β series). In order to better capture the geometrical features associated to the geometrical structure (i.e., array pattern) of Vogel spirals, the

*g(r)*was calculated directly from the array point patterns (i.e., no form factor associated to finite-size particles) using the library

*spatstat*[19] within the R statistical analysis package. The pair correlation function is calculated as:where r is the radius of the observation window and

*K'(r)*is the first derivative of the reduced second moment function (“Ripley’s K function”) [20].

*d*between first neighboring particles by performing a Delaunay triangulation of the spiral array [8

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]

**19**(24), 23631–23642 (2011). [CrossRef] [PubMed]

*d*, which measures the spatial uniformity of spatial point patterns [23]. In Fig. 4 , we show the calculated statistical distribution, obtained by the Delaunay triangulation, of the parameter

*d*normalized by

*d*, which is the most probable value (where the distribution is peaked). In all the investigated structures, the most probable value

_{0}*d*is generally found to be close in value to the average inter-particle separation.

_{0}*d*is color coded consistently with the scale of Fig. 4 (i.e., increasing numerical values from blue to red colors).

*d*, thus defining “radial heterostructures” that can trap radiation in regions of different lattice constants, similar in nature to the concentric rings of Omniguide Bragg fibers. The sharp contrast between adjacent rings radially traps radiation by Bragg scattering along different circular loops. The circular regions discovered in the spatial map of local particle coordination in Fig. 5 well correspond to the scattering rings observed in the Fourier spectra (Fig. 2), and are at the origin of the recently discovered circular scattering resonances carrying orbital angular momentum in Vogel spirals [10

**11**(5), 2008–2016 (2011). [CrossRef] [PubMed]

**19**(24), 23631–23642 (2011). [CrossRef] [PubMed]

## 3. Density of states and optical modes

^{−7}. The LDOS is calculated at the center of the spiral structure using the well-known relation g(

**r**, ω) = (2ω/π

*c*

^{2})

*Im*[G(

**r**,

**r’**, ω)], where G(

**r**,

**r’**, ω) is the Green’s function for the propagation of the E

_{z}component from point

**r**to

**r’**. The numerical calculations are implemented using the Finite Element method within COMSOL Multiphysics (version 3.5a) [11

**19**(24), 23631–23642 (2011). [CrossRef] [PubMed]

_{0}= 13.2x10

^{14}Hz), respectively. For comparison, the LDOS of the GA-spiral is also reported in both panels.

*et al*for the GA-spiral [8

**34**(18), 2805–2807 (2009). [CrossRef] [PubMed]

_{0}= 1.122 (1.273 µm) represents a defect mode localized at the center of the spiral array where a small air region free of dielectric cylinders acts as a structural defect. Several peaks corresponding to localized modes appear both along the band edges and within the gap. These dense series of photonic bandedge modes have been observed for all types of Vogel spirals and correspond to spatially localized modes due to the inhomogeneous distribution of neighboring particles, as previously demonstrated by Seng

*et al*. for the GA-spiral [11

**19**(24), 23631–23642 (2011). [CrossRef] [PubMed]

*d*in Fig. 5 are sandwiched between two other regions of distinctively different values of

*d*, thus creating a photonic heterostructure that can efficiently localize optical modes. In this picture, the outer regions of the spirals act as “effective barriers” that confine different classes of modes within the middle spirals regions. According to this mode localization mechanism, the reduced number of bandedge modes calculated for spirals α

_{4}and β

_{4}is attributed to the monotonic decrease (i.e., gradual fading) of interparticle separations when moving away from the central regions of the spirals, consistent with the corresponding Delaunay triangulation maps in Figs. 5(i) and 5(d). In particular, since these strongly perturbed spiral structures do not display clearly contrasted (i.e., sandwiched) areas of differing interparticle separations, their bandedge LDOS is strongly reduced and circularly symmetric bandedge modes cannot be formed. These observations will be validated by the detailed optical mode analysis presented in section 5 for all the different spirals.

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. B **76**(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]

## 4. Multifractal and scaling properties of aperiodic Vogel spirals

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

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

*singularity strength*α(

*x*) of the multifractal measure μ obeys the local scaling law:where

*x*and of size ε. The smaller the exponent α(

*x*), the more singular will be the measure around

*x*(i.e., local singularity). The

*multifractal spectrum*f ( α ) , also known as

*singularity spectrum*, characterizes the statistical distribution of the singularity exponent α(

*x*) of a multifractal measure. If we cover the support of the measure μ with balls of size ε, the number of balls

*x*with scaling index α [29]. The spectrum

34. J. F. Muzy, E. Bacry, and A. Arneodo, “The multifractal formalism revisited with wavelets,” Int. J. Bifurcat. Chaos **4**(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. A **33**(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).

*f*(α) of a digitized spiral image, FracLac generates a partition of the image into a group of covering boxes of size ε labeled by the index

*i*= 1,2,…,

*N*(ε). The fraction of the mass of the object (i.e., number of pixels) that falls within box

*i*of radius ε is indicated by

*P*(

*i*), and it is used to define the generalized measure:where

*q*is an integer and the sum runs over all the ε-boxes. The quantity in Eq. (7) represents the (

*q*-1)-th order moment of the “probability” (i.e., pixel fraction)

*P*(

*i*)/

*N,*where

*N*is the total number of pixels of the image. The singularity exponent and singularity spectrum can then be directly obtained as [36

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]

*f*(α) spectrum generally displays a single humped shape (i.e., downward concavity) which extends over a compact interval [α

_{min}, α

_{max}], where α

_{min}(respectively α

_{max}) correspond to the strongest (respectively the weakest) singularities. The maximum value of

*f*(α) corresponds to the (average) box-counting dimension of the multifractal object, while the difference Δα = α

_{max}-α

_{min}can be used as a parameter reflecting the fluctuations in the length scales of the intensity measure [26

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

*, = 1.873), which is consistent with its more regular structure. We also notice that the Δα for the GA-spiral is the largest, consistently with the diffuse nature (absolutely continuous) of its Fourier spectrum (Fig. 2(e)). On the other hand, the less homogeneous α*

_{f}_{1}-spiral structure features the lowest fractal dimensionality (D

*, = 1.706), consistent with a larger degree of structural disorder. All other spirals in the α series were found to vary in between these two extremes. On the other hand, the results shown in Figs. 7(c) and 7(d) demonstrate significantly reduced differences in the singularity spectra of the spirals in the β series, due to the much smaller variation of the perturbing divergence angle α (137.5-137.6) reported in Table 1. These results demonstrate that multifractal analysis is suitable to detect the small local structural differences among Vogel spirals obtained by very small variations in the divergence angle α.*

_{f}26. E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. **376**(4-5), 225–337 (2003). [CrossRef]

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]

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

*f*is a decomposition into elementary

*space-scale contributions,*associated to so-called wavelets that are constructed from one single function ψ by means of translations and dilation operations. The WT of the function

*f*is defined as:where

*a*is the real scale parameter,

*b*is the real translation parameter, and

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

*q*is a real number and the sum runs over the local maxima of

*x*. For each

*q,*from the scaling behavior of the partition function at fine scales one can obtain the scaling exponent τ(

*q*):

*f*(α) is derived from τ(

*q*) by a Legendre transform [40, 41]. In order to analyze the LDOS of photonic Vogel’s spirals we have implemented the aforementioned WTMM method within the free library of Matlab wavelet routines WaveLab850 [42

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).

*D*≈0.6-0.74, with the two extremes belonging to the α-series (i.e., α

_{f}_{1}and α

_{2}, respectively). The strength of the LDOS singularity is measured by the value of α

_{0}= α

_{max}, which is the singularity exponent corresponding to the peak of the

*f*(α) spectrum. In Fig. 7(c), we notice that the singular character of the LDOS spectra steadily increases from spiral α

_{1}to the GA-spiral across the α-series. On the other hand, a more complex behavior is observed across the β series, where the strength of the LDOS singularity increases from β

_{2}to β

_{4}spirals.

## 5. Optical mode analysis of Vogel spirals

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. A **64**(6), 063808 (2001). [CrossRef]

_{r}+

*i*ω

_{i}were calculated using eigenmode analysis within COMSOL. Complex mode frequencies naturally arise from radiation leakage through the open boundary of the arrays. The imaginary components of the complex mode frequencies give the leakage rates of the mode, from which the quality factor can be defined as

*Q =*ω

_{r}/2ω

_{i}. The calculated quality factors of the modes are plotted in Fig. 8(a) for the α

_{1}-spiral and in Fig. 8(b) for the GA-spiral and the β

_{4}-spiral as a function of normalized frequency. We limit our analysis to only these three structures since they cover the full perturbation spectrum and are representative of the general behavior of the localized bandedge modes in Vogel spirals. By examining the spatial electric field patterns of the modes across the air bandedge of Vogel spirals we discovered that it is possible to group them into several different classes. The Q factors of modes in the same class depend linearly on frequency, as shown in Fig. 8. In particular, their quality factors are found to linearly decrease as the modes in each class move further away from the central PBG region. The frequency range spanned by each class of modes depends on the class and the spiral type. For example, the modes in classes A and B (Fig. 8(a)) of the α

_{1}-spiral span the entire air-bandedge, while modes in classes C-F are confined within a narrower region of the bandedge.

_{1}spiral, and we discover that its bandedge modal classes contain modes spatially confined to the red region in Fig. 5(a), bounded on either side by areas of higher particle density (i.e., shorter interparticle separations). The spatial profiles of the representative class-B modes of the α

_{1}spiral, shown in Fig. 9(a), are centered around this low density circular region (i.e., central red ring in Fig. 5(a)) and have the same number of oscillations in the radial field (i.e., radial number 2) while displaying increasing azimuthal oscillations (i.e., increasing azimuthal numbers). On the other hand, for the GA-spiral the modal patterns in class A, shown in Figs. 9(d)-9(f), are also confined to this spatial region, but have radial number equal to one along the series of increasing azimuthal numbers. Modes in class C also occupy the same spatial region of the spiral, but with a radial number of three (not shown here). This characteristic cascade process of “radial splitting” of the modes continues for classes D, E and F in each spiral. However, as the radial numbers increase, the less confined outer portions of the modes result in a reduced quality factor. It is also relevant to note here that the slopes of the linear trends in Q-factors with frequency are all approximately the same for modes that are confined approximately within the same spatial region.

_{1}, and β

_{4}). The LDOS is again computed at the center of the array utilizing the same methodology described previously. In Figs. 10 -12 we show the calculated LDOS for the three spiral types with progressively decreasing number of cylinders from 1000 to 150. Also included in Figs. 10-12 are representative air-bandedge modes calculated for a spirals with decreasing size (from panels (a) to (c)).

_{1}-spiral scaling shown in Fig. 10, which we found to possess the lowest fractal dimension (i.e., or the highest degree of structural inhomogeneity). Below 750 cylinders, the air bandedge region is almost completely depopulated of bandedge modes, which become strongly leaky as shown in Figs. 10(a) and 10(b). This behavior can directly be attributed to the loss of the outermost boundary region (outer blue region in Fig. 5(a)) when the number of cylinder is decreased, eliminating the radial heterostructure confinement scheme needed to support localized bandedge modes. On the other hand, the modes in the LDOS whose confinement regions remain intact upon size scaling, such as the ones shown in Figs. 11(a) -11(c) and Figs. 12(a)-12(c), exist even when scaling the size of the spiral down to only a few hundred cylinders. These results demonstrate the localized nature of the air bandedge modes that densely populate the multifractal LDOS spectra of Vogel spirals.

## 6. Conclusions

## Acknowledgments

## References and links

1. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: |

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. |

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. |

4. | T. F. Krauss, D. Labilloy, A. Scherer, and R. M. De La Rue, “Photonic Crystals for Light-Emitting Devices,” Proc. SPIE |

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. |

6. | Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic Band Gaps in Two Dimensional Photonic Quasicrystals,” Phys. Rev. Lett. |

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. |

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. |

10. | J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett. |

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. Express |

12. | J. A. Adam, |

13. | E. Macia, |

14. | M. Naylor, “Golden, √ 2, and π Flowers: A Spiral Story,” Math. Mag. |

15. | G. J. Mitchison, “Phyllotaxis and the fibonacci series,” Science |

16. | C. Janot, |

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. B |

18. | C. Forestiere, G. F. Walsh, G. Miano, and L. Dal Negro, “Nanoplasmonics of prime number arrays,” Opt. Express |

19. | A. Baddeley and R. Turner, “Spatstat: an R package for analyzing spatial point patterns,” J. Stat. Softw. |

20. | B. D. Ripley, “Modelling spatial patterns,” J. R. Stat. Soc., B |

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. A |

22. | S. Torquato and F. H. Stillinger, “Local density fluctuations, hyperuniformity, and order metrics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

23. | J. Illian, A. Penttinen, H. Stoyan, and D. Stoyan, |

24. | |

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. B |

26. | E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. |

27. | P. K. Thakur and P. Biswas, “Multifractal scaling of electronic transmission resonances in perfect and imperfect Fibonacci δ-function potentials,” Physica A |

28. | J. Feder, |

29. | J. Gouyet, |

30. | B. B. Mandelbrot, |

31. | H. E. Stanley and P. Meakin, “Multifractal phenomena in physics and chemistry,” Nature |

32. | B. B. Mandelbrot, “An introduction to multifractal distribution functions,” |

33. | U. Frisch and G. Parisi, “Fully developed turbulence and intermittency,” |

34. | J. F. Muzy, E. Bacry, and A. Arneodo, “The multifractal formalism revisited with wavelets,” Int. J. Bifurcat. Chaos |

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. A |

36. | A. Chhabra and R. V. Jensen, “Direct determination of the |

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. |

40. | S. Mallat, |

41. | J. C. van den Berg, ed., |

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. A |

**OCIS Codes**

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

(160.5293) Materials : Photonic bandgap materials

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: November 22, 2011

Revised Manuscript: January 15, 2012

Manuscript Accepted: January 19, 2012

Published: January 25, 2012

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-3015

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### References

- 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).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett.11(5), 2008–2016 (2011). [CrossRef] [PubMed]
- 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]
- 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).
- M. Naylor, “Golden, √ 2, and π Flowers: A Spiral Story,” Math. Mag.75, 163–172 (2002). [CrossRef]
- G. J. Mitchison, “Phyllotaxis and the fibonacci series,” Science196(4287), 270–275 (1977). [CrossRef] [PubMed]
- C. Janot, Quasicrystals: A Primer (Clarendon Press, 1992).
- 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]
- C. Forestiere, G. F. Walsh, G. Miano, and L. Dal Negro, “Nanoplasmonics of prime number arrays,” Opt. Express17(26), 24288–24303 (2009). [CrossRef] [PubMed]
- A. Baddeley and R. Turner, “Spatstat: an R package for analyzing spatial point patterns,” J. Stat. Softw.12(6), 1–42 (2005).
- B. D. Ripley, “Modelling spatial patterns,” J. R. Stat. Soc., B39, 172–212 (1977).
- 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]
- 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]
- 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).
- http://www.comsol.com .
- 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]
- E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep.376(4-5), 225–337 (2003). [CrossRef]
- 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]
- J. Feder, Fractals (Plenum Press, 1988).
- J. Gouyet, Physics and Fractal Structures (Springer, 1996).
- B. B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman and Co., 1982).
- H. E. Stanley and P. Meakin, “Multifractal phenomena in physics and chemistry,” Nature335(6189), 405–409 (1988) (Review). [CrossRef]
- B. B. Mandelbrot, “An introduction to multifractal distribution functions,” Fluctuations and Pattern Formation, H.E. Stanley, N. Ostrowsky, eds., (Kluwer, 1988).
- 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).
- J. F. Muzy, E. Bacry, and A. Arneodo, “The multifractal formalism revisited with wavelets,” Int. J. Bifurcat. Chaos4(2), 245–302 (1994). [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. A33(2), 1141–1151 (1986). [CrossRef] [PubMed]
- 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. Karperien, FracLac for ImageJ, version 2.5. http://rsb.info.nih.gov/ij/plugins/fraclac/FLHelp/Introduction.htm . (1999–2007).
- W. S. Rasband and J. Image, U. S. National Institutes of Health, Bethesda, Maryland, USA, http://imagej.nih.gov/ij/ , (1997–2011).
- 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]
- S. Mallat, A Wavelet Tour of Signal Processing, 3rd ed., (Elsevier, 2009).
- J. C. van den Berg, ed., Wavelets in Physics (Cambridge University Press, 2004).
- 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).
- 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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