# Structural and Spectral Properties of Deterministic Aperiodic Optical Structures

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## Abstract

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## 1. Introduction

## 2. Spectral Measures of Aperiodic Arrays

## 3. Classification of Aperiodic Structures

## 4. Structural Properties of Aperiodic Arrays

## 5. Aperiodic Arrays and Spectral Graph Theory

## 6. The Green’s Matrix Method

## 7. Spectral Properties of Aperiodic Arrays

## 8. Level Statistics

## 9. Critical Modes and Resonances of Aperiodic Systems

## 10. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) $N=1024$ square array, and its diffraction pattern in (

**b**); (

**c**) $N=1108$ Penrose array, and its diffraction pattern in (

**d**); (

**e**) $N=1000$ pseudo-random 2D array, and its diffraction pattern in (

**f**). The diffraction patterns are created with $S{\lambda}^{2}=0.126$, in an angular range of 80 degrees. The third root of the diffraction pattern is plotted for better visualization.

**Figure 2.**$N=1000$ μ-spiral array (

**a**), and its diffraction pattern (

**b**); $N=1000$ π-spiral array (

**c**), and its diffraction pattern (

**d**); $N=1000$ τ-spiral array (

**e**), and its diffraction pattern (

**f**). The diffraction patterns are created with $S{\lambda}^{2}=0.314$, in an angular range of 80 degrees.

**Figure 3.**The radial distribution functions (RDFs), $g\left(r\right)$ of point patterns, for (

**a**) $N=1024$ square array; (

**b**) $N=1108$ Penrose array; (

**c**) $N=1000$ pseudo-random array; (

**d**) $N=1000$ μ-spiral array; (

**e**) $N=1000$ π-spiral array; and (

**f**) $N=1000$ τ-spiral array. The RDFs are calculated using the ImageJ software package, and the horizontal axis is in units of pixels. The dashed horizontal line $g\left(r\right)=1$ indicates the limit of an uncorrelated random gas model.

**Figure 4.**Probability distribution of Delaunay triangulation edge lengths. (

**a**) $N=1024$ square array; (

**b**) $N=1108$ Penrose array; (

**c**) $N=1000$ pseudo-random array; (

**d**) $N=1000$ μ-spiral array; (

**e**) $N=1000$ π-spiral array; and (

**f**) $N=1000$ τ-spiral array. The normalization edge length ${d}_{0}$ is defined as the edge length with the greatest probability in each case.

**Figure 5.**Delaunay triangulation with color-coded edge according to their lengths. (

**a**) $N=1024$ square array; (

**b**) $N=1108$ Penrose array; (

**c**) $N=1000$ pseudo-random array; (

**d**) $N=1000$ μ-spiral array; (

**e**) $N=1000$ π-spiral array; and (

**f**) $N=1000$ τ-spiral array. The edge lengths increase from blue (shortest edge of Delaunay triangulation of each array) to red (longest edge of Delaunay triangulation of each array).

**Figure 6.**Representative eigenvectors of the edge-length weighted graph Laplacian matrix for different aperiodic structures. (

**a**–

**c**) for an N = 1108 Penrose array with (eigenvalues equal to 0.3398, 5.4721, 8.2152, respectively); (

**d**–

**f**) for a μ-spiral array (eigenvalues equal to 1.2338, 20.0862 and 21.6709, respectively); (

**g**–

**i**) for a τ-spiral array (eigenvalues equal to 0.7915, 4.8394 and 20.0284).

**Figure 7.**Eigenvalue distributions of the dyadic Green’s matrix in the complex plane for $N=1024$ square array with (

**a**) $S{\lambda}^{2}=4.19$; (

**b**) $S{\lambda}^{2}=1.26$; (

**c**) $S{\lambda}^{2}=0.628$; (

**d**) $S{\lambda}^{2}=0.0628$; (

**e**) $S{\lambda}^{2}=0.00628$; (

**f**) $S{\lambda}^{2}=0.000628$. The color bar shows the inverse participation ratio (IPR) of the eigenmode associated to each eigenvalue in log scale.

**Figure 8.**Eigenvalue distributions of the dyadic Green’s matrix in the complex plane for $N=1108$ Penrose array with (

**a**) $S{\lambda}^{2}=4.19$; (

**b**) $S{\lambda}^{2}=1.26$; (

**c**) $S{\lambda}^{2}=0.628$; (

**d**) $S{\lambda}^{2}=0.0628$; (

**e**) $S{\lambda}^{2}=0.00628$; (

**f**) $S{\lambda}^{2}=0.000628$. The color bar shows the inverse participation ratio (IPR) of the eigenmode associated to each eigenvalue in log scale.

**Figure 9.**Eigenvalue distributions of the dyadic Green’s matrix in the complex plane for $N=1000$ random array with (

**a**) $S{\lambda}^{2}=4.19$; (

**b**) $S{\lambda}^{2}=1.26$; (

**c**) $S{\lambda}^{2}=0.628$; (

**d**) $S{\lambda}^{2}=0.0628$; (

**e**) $S{\lambda}^{2}=0.00628$; (

**f**) $S{\lambda}^{2}=0.000628$. The color bar shows the inverse participation ratio (IPR) of the eigenmode associated to each eigenvalue in log scale. The insets in (

**a**–

**d**) highlight the central part of the eigenvalue cluster.

**Figure 10.**Eigenvalue distributions of the dyadic Green’s matrix in the complex plane for $N=1000$ μ-spiral array with (

**a**) $S{\lambda}^{2}=4.19$; (

**b**) $S{\lambda}^{2}=1.26$; (

**c**) $S{\lambda}^{2}=0.628$; (

**d**) $S{\lambda}^{2}=0.0628$; (

**e**) $S{\lambda}^{2}=0.00628$; (

**f**) $S{\lambda}^{2}=0.000628$. The color bar shows the inverse participation ratio (IPR) of the eigenmode associated to each eigenvalue in log scale.

**Figure 11.**Eigenvalue distributions of the dyadic Green’s matrix in the complex plane for $N=1000$ π-spiral array with (

**a**) $S{\lambda}^{2}=4.19$; (

**b**) $S{\lambda}^{2}=1.26$; (

**c**) $S{\lambda}^{2}=0.628$; (

**d**) $S{\lambda}^{2}=0.0628$; (

**e**) $S{\lambda}^{2}=0.00628$; (

**f**) $S{\lambda}^{2}=0.000628$. The color bar shows the inverse participation ratio (IPR) of the eigenmode associated to each eigenvalue in log scale.

**Figure 12.**Eigenvalue distributions of the dyadic Green’s matrix in the complex plane for $N=1000$ τ-spiral array with (

**a**) $S{\lambda}^{2}=4.19$; (

**b**) $S{\lambda}^{2}=1.26$; (

**c**) $S{\lambda}^{2}=0.628$; (

**d**) $S{\lambda}^{2}=0.0628$; (

**e**) $S{\lambda}^{2}=0.00628$; (

**f**) $S{\lambda}^{2}=0.000628$. The color bar shows the inverse participation ratio (IPR) of the eigenmode associated to each eigenvalue in log scale.

**Figure 13.**Level statistics of first-neighbor eigenvalues in the complex plane for an $N=1108$ Penrose array with $S{\lambda}^{2}$ values as indicated in the panels. The value of first level spacing, ${S}_{1}$, is computed at the nearest neighbor distance between eigenvalues in the complex plane for each eigenvalue, and is normalized by the average ${S}_{1}$ for each case. The probability density is normalized such that the total probability equals to one.

**Figure 14.**Level statistics of first-neighbor eigenvalues in the complex plane for an $N=1000$ random array with $S{\lambda}^{2}$ values as indicated in the panels. In the case of random systems we can write $k\ell $=2$\pi /\left(S{\lambda}^{2}\right)$ where ℓ provides is the scattering mean free path in the system. The value of first level spacing, ${S}_{1}$, is computed at the nearest neighbor distance between eigenvalues in the complex plane for each eigenvalue, and is normalized by the average ${S}_{1}$ for each case. The probability density is normalized such that the total probability equals to one.

**Figure 15.**Level statistics of first-neighbor eigenvalues in the complex plane for an $N=1000$ μ-spiral array with $S{\lambda}^{2}$ values as indicated in the panels. The value of first level spacing, ${S}_{1}$, is computed at the nearest neighbor distance between eigenvalues in the complex plane for each eigenvalue, and is normalized by the average ${S}_{1}$ for each case. The probability density is normalized such that the total probability equals to one.

**Figure 16.**Level statistics of first-neighbor eigenvalues in the complex plane for an $N=1000$ π-spiral array with $S{\lambda}^{2}$ values as indicated in the panels. The value of first level spacing, ${S}_{1}$, is computed at the nearest neighbor distance between eigenvalues in the complex plane for each eigenvalue, and is normalized by the average ${S}_{1}$ for each case. The probability density is normalized such that the total probability equals to one.

**Figure 17.**Level statistics of first-neighbor eigenvalues in the complex plane for an $N=1000$ τ-spiral array with $S{\lambda}^{2}$ values as indicated in the panels. The value of first level spacing, ${S}_{1}$, is computed at the nearest neighbor distance between eigenvalues in the complex plane for each eigenvalue, and is normalized by the average ${S}_{1}$ for each case. The probability density is normalized such that the total probability equals to one.

**Figure 18.**Example of long-lived ($Re\mathsf{\Lambda}\simeq -1$) eigenvectors with large IPR value in each case of (

**a**) $N=1024$ square array $(\mathsf{\Lambda}=-0.9930+0.8807i,IPR=0.0021)$; (

**b**) $N=1108$ Penrose array $(\mathsf{\Lambda}=-0.9980+1.3426i,IPR=0.0113)$; (

**c**) $N=1000$ random array $(\mathsf{\Lambda}=-0.9989-3877i,IPR=0.5)$; (

**d**) $N=1000$ μ-spiral array $(\mathsf{\Lambda}=-0.9596+1.0960i,IPR=0.1076)$; (

**e**) $N=1000$ π-spiral array $(\mathsf{\Lambda}=-1.0000+5.4620i,IPR=0.0244)$; and (

**f**) $N=1000$ τ-spiral array $(\mathsf{\Lambda}=-0.9846+0.4464i$, $IPR=0.0054)$. The $S{\lambda}^{2}$ values are all equal to 4.19 and color-coding is the vector norm of the eigenvector component $(x,y,z)$ associated with each particle.

**Figure 19.**Example of long-lived ($Re\simeq -1$) eigenvectors with small IPR value in each case of (

**a**) $N=1024$ square array $(\mathsf{\Lambda}=-0.9323-1.9290i,IPR=0.0008)$; (

**b**) $N=1108$ Penrose array $(\mathsf{\Lambda}=-0.9963-1.0690i$, $IPR=0.003)$; (

**c**) $N=1000$ random array $(\mathsf{\Lambda}=-0.9091-8518i,IPR=0.0052)$; (

**d**) $N=1000$ μ-spiral array $(\mathsf{\Lambda}=-0.9995+0.9646i,IPR=0.0024)$; (

**e**) $N=1000$ π-spiral array $(\mathsf{\Lambda}=-1.0000-0.5143i$, $IPR=0.0064)$; and (

**f**) $N=1000$ τ-spiral array $(\mathsf{\Lambda}=-0.9896-0.4762i,IPR=0.002)$. The $S{\lambda}^{2}$ values are all equal to 4.19 and color-coding is the vector norm of the eigenvector component $(x,y,z)$ associated with each particle.

**Figure 20.**Examples of long-lived critical eigenvectors with small IPR for $N=1108$ Penrose array with (

**a**) $(\mathsf{\Lambda}=-0.9896-1.0865i,IPR=0.0029)$; (

**b**) $(\mathsf{\Lambda}=-0.9897-1.1118i,IPR=0.0033)$; N = 1000 μ-spiral array with (

**c**) $(\mathsf{\Lambda}=-0.9904-0.6844i,IPR=0.0021)$ (

**d**) $(\mathsf{\Lambda}=-0.9911-0.7660i,IPR=0.002)$; $N=1000$ τ-spiral array with (

**e**) $(\mathsf{\Lambda}=-0.9882+0.9366i,IPR=0.0014)$; and (

**f**) $(\mathsf{\Lambda}=-0.9876+0.9359i$, $IPR=0.0019)$. The $S{\lambda}^{2}$ values are all equal to 4.19 and color-coding is the vector norm of the eigenvector component $(x,y,z)$ associated with each particle.

Structure | ${\mathit{d}}_{\mathbf{min}}\mathbf{/}\mathit{\lambda}$ | $\mathbf{\langle}\mathit{d}\mathbf{\rangle}\mathbf{/}\mathit{\lambda}$ | $\mathbf{\langle}\mathit{D}\mathbf{\rangle}$ | R | ${\mathit{C}}_{\mathit{N}}$ | ${\mathit{C}}_{\mathit{WS}}$ | $\mathit{AC}$ |
---|---|---|---|---|---|---|---|

Square | 0.50436 | 8.4195 | 5.752 | −0.15535 | 0.38633 | 0.44005 | 0.017206 |

Penrose | 0.34091 | 8.4625 | 5.917 | −0.29777 | 0.39162d | 0.41858 | 0.018345 |

Random | 0.011587 | 8.042 | 5.926 | −0.15252 | 0.38313 | 0.43558 | 0.022469 |

μ-spiral | 0.29777 | 7.9442 | 5.942 | −0.17738 | 0.40156 | 0.41145 | 0.01878 |

π-spiral | 0.17491 | 8.0401 | 5.838 | 0.17011 | 0.39641 | 0.41163 | 0.01591 |

τ-spiral | 0.40869 | 7.9687 | 5.936 | −0.15421 | 0.39988 | 0.40776 | 0.021046 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Dal Negro, L.; Wang, R.; Pinheiro, F.A. Structural and Spectral Properties of Deterministic Aperiodic Optical Structures. *Crystals* **2016**, *6*, 161.
https://doi.org/10.3390/cryst6120161

**AMA Style**

Dal Negro L, Wang R, Pinheiro FA. Structural and Spectral Properties of Deterministic Aperiodic Optical Structures. *Crystals*. 2016; 6(12):161.
https://doi.org/10.3390/cryst6120161

**Chicago/Turabian Style**

Dal Negro, Luca, Ren Wang, and Felipe A. Pinheiro. 2016. "Structural and Spectral Properties of Deterministic Aperiodic Optical Structures" *Crystals* 6, no. 12: 161.
https://doi.org/10.3390/cryst6120161