## Localized photonic band edge modes and orbital angular momenta of light in a golden-angle spiral |

Optics Express, Vol. 19, Issue 24, pp. 23631-23642 (2011)

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### 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.

© 2011 OSA

## 1. Introduction

2. M. Naylor, “Golden, *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]

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]

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]

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]

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]

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]

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

## 2. Structural analysis of the golden-angle spiral

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

*r*,

*θ*) as where

*q*= 0,1,2,... is an integer,

*b*is a constant scaling factor,

*α*= 360°/

*ϕ*

^{2}≈ 137.508° is an irrational number known as the “golden angle”,

*ϕ*= [1+ 5

^{1/2}]/2 = 1.6180339... is the golden ratio. The value of

*ϕ*is approached by the ratio of two consecutive numbers in the Fibonacci series (1,2,3,5,8,13,21,34,55,89,144,...). With this generation rule, the

*q*th circle is rotated azimuthally by the angle

*α*from the location of the (

*q*– 1)th one, and also pushed radially away from the origin by a distance

*N*= 1000 circles. Visually there are multiple families of spiral arms formed by the circles. Within each family, the spiral arms, also called parastichies, are regularly spaced. Some of the families have parastichies all twisting in the CW direction, and the others in the CCW direction. The families are all intertwined. The number of parastichies in every family is a Fibonacci number [2

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

**11**, 2008–2016 (2010). [CrossRef]

*d*by performing the Delaunay triangulation on the spiral array. In Fig. 1(c), each line segment connects two neighboring circles, and its length

*d*is color coded. The statistical distribution of

*d*in Fig. 1(d) is broad and non-Gaussian.

*d*is normalized by

*d*

_{0}, the most probable value of

*d*where the distribution is peaked.

*d*

_{0}scales linearly with

*b*, as shown in [4

**11**, 2008–2016 (2010). [CrossRef]

*d*is consistent with the rich Fourier spectrum. The brightest ring in the Fourier space, which is also the smallest, has a radius close to 2

*π/d*. The non-uniform color distribution in Fig. 1(c) reveals the spatial variation of neighboring particles spacing in the spiral structure. This special type of spatial inhomogeneity is a distinctive feature of the golden-angle spiral, and it has a significant impact on its optical resonances as will be shown later.

_{o}*ρ*(

*r*,

*θ*) is shown in Fig. 1(a), the azimuthal number

*m*is an integer, and

*k*represents a spatial frequency in the radial direction. The 2D plot of |

_{r}*f*(

*m*,

*k*)|

_{r}^{2}shown in Fig. 2(a) illustrates that there are multiple and well-defined azimuthal components

*m*in the golden-angle spiral. After integrating over the radial frequency

*k*, we obtain

_{r}*F*(

*m*) = ∫|

*f*(

*m*,

*k*)|

_{r}^{2}

*k*which is plotted in Fig. 2(b). The frequency range of integration is [

_{r}dk_{r}*π/d*, 3

_{o}*π/d*], centered around the dominant spatial frequency 2

_{o}*π/d*. We notice interestingly the dominant

_{o}*m*values are 5,8,13,21,34,55,89, which are Fibonacci numbers and represent the number of parastichies in each family. Later we will demonstrate that the parastichies encode discrete angular momenta, quantized in the Fibonacci numbers, onto the optical resonances.

## 3. Photonic bandgap and band edge modes

*N*= 1000 air cylinders in a dielectric medium with refractive index

*n*= 2.65. This structure, inverse of that in Ref. [5

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]

*E*,

_{r}*E*,

_{θ}*H*). We calculate the local density of optical states (LDOS) at the center of the spiral structure,

_{z}*g*(

**r**,

*ω*) = (2

*ω/πc*

^{2})

*Im*[

*G*(

**r**,

**r**,

*ω*)], where

*G*(

**r**,

**r’**,

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

*H*from point

_{z}**r**to

**r’**. The numerical calculation is implemented with a commercial program COMSOL (version 3.5) [13]. Since the golden-angle spiral has a finite dimension, light may leak through the outer boundary. In our simulation, the spiral structure is surrounded by a perfectly matched layer that absorbs the escaped light. From the calculated LDOS in Fig. 3, we clearly see a PBG, and its width is about 11% of the gap center frequency.

*d*= 0.323 and 0.331. They represent defect modes localized at the center of the spiral array where a small dielectric region free of air holes acts as a structural defect. At both edges of the gap, there are many more peaks which correspond to the band edge modes. Those on the higher (lower) frequency edge of the gap are denoted as upper (lower) band edge modes. Due to light leakage through the open boundary of the spiral structure, the band edge modes have complex frequencies, and the imaginary parts of the frequencies represent the leakage rates. We calculate the complex frequencies

_{o}/λ*ω*=

*ω*+

_{r}*iω*and spatial field distributions of the band edge modes using the eigensolver of COMSOL. The quality factor

_{i}*Q*=

*ω*/2

_{r}*ω*is obtained for every mode. From their frequencies and field patterns, we identify several classes of the band edge modes. Within each class the modes have similar field patterns and display monotonic variation of

_{i}*Q*. Two classes of the lower band edge modes are labeled in a plot of

*Q*vs.

*d*=

_{o}/λ*ω*/2

_{r}d_{o}*πc*in Fig. 4(a), another two classes of the upper band edge modes in Fig. 4(b). Within each class, the modes are ordered numerically following their spectral distances from the edge of the PBG. As the modes in each class move further away from the PBG, the frequency spacing of adjacent modes increases and the

*Q*decreases.

*H*) for the first three modes in classes A, B, C and D are presented in Figs. 5–8. Every mode is accompanied by a degenerate mode, e.g., A1 and A1’ have the same frequency and complementary spatial profile. The lower band edge modes have magnetic (electric) field mostly concentrated in the air (dielectric) part of the structure, while the upper band edge modes are just the opposite. This behavior is similar to that of a photonic crystal, but there are also remarkable differences. The band edge modes in the golden-angle spiral are spatially localized, each class of modes is confined within a ring of different radius. As long as the ring is notably smaller than the system size, the modes are insensitive to the boundary, as for the localized states. For example, mode D1 remains unchanged when the air cylinders near the boundary are removed [D1” in Fig. 8].

_{z}## 4. Spatial inhomogeneity and localization

*d*. From the colors of line segments connecting neighboring circles in Fig. 1(c), we see alternating rings of green color [(i) and (iii) in Fig. 1(c), 1.1

*d*

_{0}<

*d*< 1.3

*d*

_{0}] and blue-reddish color [(ii) in Fig. 1 (c),

*d*

_{0}<

*d*< 1.1

*d*

_{0}(blue) and 1.3

*d*

_{0}<

*d*< 1.5

*d*

_{0}(red)]. Different classes of band edge modes are localized in the rings of distinct colors. For example, by overlaying the region that contains 90% energy of modes in class A on the color map of

*d*in Fig. 1(c), we find these modes are confined in region (ii), which is sandwiched by regions (i) and (iii) of different color. The distribution of

*d*in region (ii) is distinct from that in (i) or (iii), leading to a change of PBG. We compute the LDOS in regions (i), (ii) and (iii) by removing air cylinders outside that region. As highlighted in Fig. 9(b), the frequency range of class A modes is inside the PBG of region (i) and (iii) but outside the PBG of region (ii). Consequently, light within this frequency range is allowed to propagate in region (ii) but not in (i) or (iii). Hence, regions (i) and (iii) act like barriers that confine class A modes in region (ii).

*d*, the golden-angle spiral can support numerous modes at different frequencies. The spatial inhomogeneity of

*d*leads to mode confinement in different parts of the structure.

## 5. Discrete angular momentum

*H*oscillates between the positive maxima on one parastichy and the negative maxima on the next one of the same family. We perform the FBT on the field distribution by replacing

_{z}*ρ*(

*r*,

*θ*) in Eq. (3) with

*H*(

_{z}*r*,

*θ*). To compare with the FBT of the structure [

*ρ*(

*r*,

*θ*) > 0], we set

*m*′ = 2

*m*and

*k*′

*= 2*

_{r}*k*for the field FBT, which is equivalent to considering the FBT of the field intensity distribution.

_{r}*m*′ = 21 and

*m*′ = 89, both are Fibonacci numbers. To find their origin, we perform FBT of the structure in the region where mode A1 is localized [Fig. 10(c)]. The result is presented in Fig. 10(d), and show indeed

*m*= 21 and

*m*= 89 components with radial frequency

*k*similar to that in the field profile of mode A1. While there are also

_{r}*m*= 34 and

*m*= 55 components in the structure, they are at lower

*k*, thus corresponding to modes at lower frequencies and further away from the band edge. Hence, these analysis show that the angular momenta of the band edge modes are imparted by the underlying structure, more specifically, the parastichies in the golden-angle spiral. Similar analysis of mode B1 reveals that it supports angular momenta

_{r}*m*= 13 and

*m*= 55. They are also Fibonacci numbers, but smaller than those of mode A1, because mode B1 is localized in a smaller ring that has less number of parastichies.

*m*′ = 34. FBT of the corresponding region where D1 locates also reveals that there is a

*m*= 34 component at the similar value of

*k*. Other

_{r}*m*components in the structure have higher

*k*, thus corresponding to higher-frequency modes farther away from the band edge. Similar analysis of mode C1 reveals that it has angular momentum

_{r}*m*′ = 55, and the underlying structure contains a family of 55 parastichies.

*π*to the phase of the returning field, and does not affect the constructive interference at the starting point. We perform FBT on the field distributions of the higher-order modes, and find the additional nodes in the envelope function causes a splitting of the peaks in

*F*(

*m*′). For example, mode D1 has only a single peak at

*m*= 34 [Fig. 12(a)], while D2 has two peaks at

*m*= 32 and

*m*= 36 [Fig. 12(b)]. The change in

*m*, Δ

*m*= 2, is equal to the number of nodes in the envelope function. For D3 mode [Fig. 12(c)], Δ

*m*= 4 due to four nodes in the envelop function. Such splitting is observed in all higher-order modes of classes A, B and C. The azimuthal modulation of the envelop function introduces additional angular momenta to the band edge modes.

## 6. Conclusion

## Acknowledgments

## References and links

1. | P. Stevens, |

2. | M. Naylor, “Golden, |

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. |

4. | J. Trevino, H. Cao, and L. D. Negro, “Circularly Symmetric Light Scattering from Nanoplasmonic Spirals,” Nano Lett. |

5. | M. E. Pollard and G. J. Parker, “Low-contrast bandgaps of a planar parabolic spiral lattice,” Opt. Lett. |

6. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, |

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

8. | M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,” Phys. Rev. B |

9. | D. Lusk, I. Abdulhalim, and F. Placido, “Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,” Opt. Commun. |

10. | W. Steurer and D. Sutter-Widmer, “Photonic and phononic quasicrystals,” J. Phys. D: Appl. Phys. |

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. |

12. | H. Vogel, “A better way to construct the sunflower head,” |

13. |

**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: September 19, 2011

Revised Manuscript: October 10, 2011

Manuscript Accepted: October 11, 2011

Published: November 7, 2011

**Citation**

, "Localized photonic band edge modes and orbital angular momenta of light in a golden-angle spiral," Opt. Express **19**, 23631-23642 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-23631

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