# Aperiodic Photonics of Elliptic Curves

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## Abstract

**:**

## 1. Introduction

## 2. Elliptic Curves and Discrete Logarithm Structures

## 3. Structural and Spectral Properties of Elliptic Curves and Discrete Logarithm Arrays

## 4. Light Scattering Properties and the Extended Green’s Matrix Method

## 5. Scattering Properties of Elliptic Curves and Discrete Logarithm Structures

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$EC$ | Elliptic curve structure |

$UR$ | Uniform random structure |

${\langle UR\rangle}_{e}$ | Ensemble averaged uniform random structures |

$EC$$DL$ | Elliptic curve discrete logarithm structure |

## References

- Anderson, P.W. Absence of Diffusion in Certain Random Lattices. Phys. Rev.
**1958**, 109, 1492–1505. [Google Scholar] [CrossRef] - Wiersma, D.S. Disordered photonics. Nat. Photonics
**2013**, 7, 188–196. [Google Scholar] [CrossRef] - Lagendijk, A.; Van Tiggelen, B.; Wiersma, D.S. Fifty years of Anderson localization. Phys. Today
**2009**, 62, 24–29. [Google Scholar] [CrossRef][Green Version] - Sheng, P. Introduction to Wave Scattering, Localization and Mesoscopic Phenomena; Taylor & Francis: London, UK, 2007. [Google Scholar]
- Bertolotti, J.; Gottardo, S.; Wiersma, D.S.; Ghulinyan, M.; Pavesi, L. Optical Necklace States in Anderson Localized 1D Systems. Phys. Rev. Lett.
**2005**, 94, 113903–113907. [Google Scholar] [CrossRef] [PubMed] - Cao, H.; Zhao, Y.G.; Ho, S.T.; Seelig, E.W.; Wang, Q.H.; Chang, R.P.H. Random Laser Action in Semiconductor Powder. Phys. Rev. Lett.
**1999**, 82, 2278–2281. [Google Scholar] [CrossRef][Green Version] - Cao, H. Review on latest developments in random lasers with coherent feedback. J. Phys. A Math. Gen.
**2005**, 38, 10497–10535. [Google Scholar] [CrossRef] - Leonetti, M.; Conti, C.; Lopez, C. The mode-locking transition of random lasers. Nat. Photonics
**2011**, 5, 615–617. [Google Scholar] [CrossRef][Green Version] - Lawandy, N.M. ‘Paint-On Lasers’ Light the Way for New Technologies. Photonics Spectra
**1994**, 28, 119–124. [Google Scholar] - Lawandy, N.M.; Balachandran, R.; Gomes, A.; Sauvain, E. Laser action in strongly scattering media. Nature
**1994**, 368, 436–438. [Google Scholar] [CrossRef] - Chen, Y.; Fiorentino, A.; Dal Negro, L. A fractional diffusion random laser. Sci. Rep.
**2019**, 9, 8686. [Google Scholar] [CrossRef] - Bertolotti, J.; Van Putten, E.G.; Blum, C.; Lagendijk, A.; Vos, W.L.; Mosk, A.P. Non-invasive imaging through opaque scattering layers. Nature
**2012**, 491, 232–234. [Google Scholar] [CrossRef] [PubMed] - Mosk, A.P.; Lagendijk, A.; Lerosey, G.; Fink, M. Controlling waves in space and time for imaging and focusing in complex media. Nat. Photonics
**2012**, 6, 283–292. [Google Scholar] [CrossRef][Green Version] - Sebbah, P. Waves and Imaging through Complex Media; Springer Science & Business Media: Berlin, Germany, 2001. [Google Scholar]
- Katz, O.; Heidmann, P.; Fink, M.; Gigan, S. Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations. Nat. Photonics
**2014**, 8, 784–790. [Google Scholar] [CrossRef] - Vellekoop, I.M.; Mosk, A. Focusing coherent light through opaque strongly scattering media. Opt. Lett.
**2007**, 32, 2309–2311. [Google Scholar] [CrossRef] [PubMed] - Redding, B.; Liew, S.F.; Sarma, R.; Cao, H. Compact spectrometer based on a disordered photonic chip. Nat. Photonics
**2013**, 7, 746–751. [Google Scholar] [CrossRef][Green Version] - Redding, B.; Popoff, S.M.; Cao, H. All-fiber spectrometer based on speckle pattern reconstruction. Opt. Express
**2013**, 21, 6584–6600. [Google Scholar] [CrossRef][Green Version] - Skipetrov, S.; Page, J.H. Red light for Anderson localization. New J. Phys.
**2016**, 18, 021001. [Google Scholar] [CrossRef] - Skipetrov, S.E.; Sokolov, I.M. Absence of Anderson localization of light in a random ensemble of point scatterers. Phys. Rev. Lett.
**2014**, 112, 023905. [Google Scholar] [CrossRef] - Bellando, L.; Gero, A.; Akkermans, E.; Kaiser, R. Cooperative effects and disorder: A scaling analysis of the spectrum of the effective atomic Hamiltonian. Phys. Rev. A
**2014**, 90, 063822. [Google Scholar] [CrossRef] - Maciá, E. Aperiodic Structures in Condensed Matter: Fundamentals and Applications; CRC Press: Boca Raton, FL, USA, 2008. [Google Scholar]
- Dal Negro, L. Optics of Aperiodic Structures: Fundamentals and Device Applications; Pan Stanford Publishing: Singapore, 2014. [Google Scholar]
- Dal Negro, L.; Oton, C.J.; Gaburro, Z.; Pavesi, L.; Johnson, P.; Lagendijk, A.; Righini, R.; Colocci, M.; Wiersma, D.S. Light Transport through the Band-Edge States of Fibonacci Quasicrystals. Phys. Rev. Lett.
**2003**, 90, 055501. [Google Scholar] [CrossRef] - Dal Negro, L.; Wang, R.; Pinheiro, F.A. Structural and spectral properties of deterministic aperiodic optical structures. Crystals
**2016**, 6, 161. [Google Scholar] [CrossRef] - Dal Negro, L.; Boriskina, S.V. Deterministic aperiodic nanostructures for photonics and plasmonics applications. Laser Photonics Rev.
**2012**, 6, 178–218. [Google Scholar] [CrossRef] - Maciá, E. Physical nature of critical modes in Fibonacci quasicrystals. Phys. Rev. B
**1999**, 60, 10032. [Google Scholar] [CrossRef] - Ryu, C.; Oh, G.; Lee, M. Extended and critical wave functions in a Thue-Morse chain. Phys. Rev. B
**1992**, 46, 5162. [Google Scholar] [CrossRef] [PubMed] - Dal Negro, L.; Inampudi, S. Fractional transport of photons in deterministic aperiodic structures. Sci. Rep.
**2017**, 7, 2259. [Google Scholar] [CrossRef] [PubMed] - Sokolov, I.M.; Klafter, J.; Blumen, A. Fractional kinetics. Phys. Today
**2002**, 55, 48–54. [Google Scholar] [CrossRef] - Sgrignuoli, F.; Wang, R.; Pinheiro, F.; Dal Negro, L. Localization of scattering resonances in aperiodic Vogel spirals. Phys. Rev. B
**2019**, 99, 104202. [Google Scholar] [CrossRef][Green Version] - Froufe-Pérez, L.S.; Engel, M.; Sáenz, J.J.; Scheffold, F. Band gap formation and Anderson localization in disordered photonic materials with structural correlations. Proc. Natl. Acad. Sci. USA
**2017**, 114, 9570–9574. [Google Scholar] [CrossRef][Green Version] - Gellermann, W.; Kohmoto, M.; Sutherland, B.; Taylor, P. Localization of light waves in Fibonacci dielectric multilayers. Phys. Rev. Lett.
**1994**, 72, 633. [Google Scholar] [CrossRef] - Vardeny, Z.V.; Nahata, A.; Agrawal, A. Optics of photonic quasicrystals. Nat. Photonics
**2013**, 7, 177–187. [Google Scholar] [CrossRef] - Razi, M.; Wang, R.; He, Y.; Kirby, R.M.; Dal Negro, L. Optimization of Large-Scale Vogel Spiral Arrays of Plasmonic Nanoparticles. Plasmonics
**2019**, 14, 253–261. [Google Scholar] [CrossRef] - Trevino, J.; Cao, H.; Dal Negro, L. Circularly symmetric light scattering from nanoplasmonic spirals. Nano Lett.
**2011**, 11, 2008–2016. [Google Scholar] [CrossRef] [PubMed] - Lifshitz, R.; Arie, A.; Bahabad, A. Photonic quasicrystals for nonlinear optical frequency conversion. Phys. Rev. Lett.
**2005**, 95, 133901. [Google Scholar] [CrossRef] [PubMed] - Shalaev, V.M. Optical Properties of Nanostructured Random Media; Springer Science & Business Media: Berlin, Germany, 2002. [Google Scholar]
- Mahler, L.; Tredicucci, A.; Beltram, F.; Walther, C.; Faist, J.; Beere, H.E.; Ritchie, D.A.; Wiersma, D.S. Quasi-periodic distributed feedback laser. Nat. Photonics
**2010**, 4, 165–169. [Google Scholar] [CrossRef] - Capretti, A.; Walsh, G.F.; Minissale, S.; Trevino, J.; Forestiere, C.; Miano, G.; Dal Negro, L. Multipolar second harmonic generation from planar arrays of Au nanoparticles. Opt. Lett.
**2012**, 20, 15797–15806. [Google Scholar] [CrossRef] [PubMed] - Pecora, E.F.; Lawrence, N.; Gregg, P.; Trevino, J.; Artoni, P.; Irrera, A.; Priolo, F.; Dal Negro, L. Nanopatterning of silicon nanowires for enhancing visible photoluminescence. Nanoscale
**2012**, 4, 2863–2866. [Google Scholar] [CrossRef] [PubMed] - Schroeder, M. Number Theory in Science and Communication: With Applications In Cryptography, Physics, Digital Information, Computing, and Self-Similarity; Springer: Berlin, Germany, 2009. [Google Scholar]
- Wang, R.; Pinheiro, F.A.; Dal Negro, L. Spectral statistics and scattering resonances of complex primes arrays. Phys. Rev. B
**2018**, 97, 024202. [Google Scholar] [CrossRef][Green Version] - Miller, S.J.; Takloo-Bighash, R. An Invitation to Modern Number Theory; Princeton University Press: Princeton, NJ, USA, 2006. [Google Scholar]
- Schwarz, W.; Spilker, J. Arithmetical Functions. An Introduction to Elementary and Analytic Properties of Arithmetic Functions and to Some of Their Almost-Periodic Properties; Cambridge University Press: Cambridge, UK, 1994. [Google Scholar]
- Zharekeshev, I.K.; Kramer, B. Asymptotics of universal probability of neighboring level spacings at the Anderson transition. Phys. Rev. Lett.
**1997**, 79, 717. [Google Scholar] [CrossRef] - Hoffstein, J.; Pipher, J.; Silverman, J.H.; Silverman, J.H. An Introduction to Mathematical Cryptography; Springer: Berlin, Germany, 2008. [Google Scholar]
- Silverman, J.H. The Arithmetic of Elliptic Curves; Springer Science & Business Media: Berlin, Germany, 2009. [Google Scholar]
- Stewart, I.; Tall, D. Algebraic Number Theory and Fermat’s Last Theorem, 4th ed.; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- Washington, L.C. Elliptic Curves Number Theory and Cryptography; Chapman and Hall/CRC: Boca Raton, FL, USA, 2008. [Google Scholar]
- Millennium Problems. Available online: http://www.claymath.org/millennium-problems (accessed on 9 September 2019).
- Birch, B. Conjectures concerning elliptic curves. In Proceedings of Symposia in Pure Mathematics; American Mathematical Society: Providence, RI, USA, 1965; pp. 106–112. [Google Scholar]
- Hasse, H. Beweis des Analogons der Riemannschen Vermutung fur die Artinschen und F.K. Schmidtschen Kongruenzzetafunktionen in gewissen elliptischen Fällen. Nachr. Gesell. Wiss. Göttingen
**1933**, 42, 253–262. [Google Scholar] - Taylor, R. Automorphy for some ℓ-adic lifts of automorphic mod ℓ Galois representations. II. Publ. Math. Inst. Hautes Études Sci.
**2008**, 108, 183–239. [Google Scholar] [CrossRef] - Illian, J.; Penttinen, A.; Stoyan, H.; Stoyan, D. Statistical Analysis and Modelling of Spatial Point Patterns; John Wiley & Sons: New York, NY, USA, 2008. [Google Scholar]
- Baake, M.; Grimm, U. Aperiodic Order; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- Senechal, M. Quasicrystals and Geometry; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Oğuz, E.C.; Socolar, J.E.; Steinhardt, P.J.; Torquato, S. Hyperuniformity of quasicrystals. Phys. Rev. B
**2017**, 95, 054119. [Google Scholar] [CrossRef][Green Version] - Torquato, S. Hyperuniform states of matter. Phys. Rep.
**2018**, 745, 1–95. [Google Scholar] [CrossRef][Green Version] - Queffélec, M. Substitution Dynamical Systems-Spectral Analysis; Springer: New York, NY, USA, 2010. [Google Scholar]
- Sgrignuoli, F.; Röntgen, M.; Morfonios, C.V.; Schmelcher, P.; Dal Negro, L. Compact localized states of open scattering media: A graph decomposition approach for an ab initio design. Opt. Lett.
**2019**, 44, 375–378. [Google Scholar] [CrossRef] [PubMed] - Lagendijk, A.; Van Tiggelen, B.A. Resonant multiple scattering of light. Phys. Rep
**1996**, 270, 143–215. [Google Scholar] [CrossRef] - Pinheiro, F.A.; Rusek, M.; Orlowski, A.; Van Tiggelen, B.A. Probing Anderson localization of light via decay rate statistics. Phys. Rev. E
**2004**, 69, 026605. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pinheiro, F.A. Statistics of quality factors in three-dimensional disordered magneto-optical systems and its applications to random lasers. Phys. Rev. A
**2008**, 78, 023812. [Google Scholar] [CrossRef] - Skipetrov, S.E. Finite-size scaling analysis of localization transition for scalar waves in a three-dimensional ensemble of resonant point scatterers. Phys. Rev. B
**2016**, 94, 064202. [Google Scholar] [CrossRef][Green Version] - Skipetrov, S.E.; Sokolov, I.M. Magnetic-field-driven localization of light in a cold-atom gas. Phys. Rev. Lett.
**2015**, 114, 053902. [Google Scholar] [CrossRef] - Rusek, M.; Mostowski, J.; Orłowski, A. Random Green matrices: From proximity resonances to Anderson localization. Phys. Rev. A
**2000**, 61, 022704. [Google Scholar] [CrossRef] - Rusek, M.; Orłowski, A.; Mostowski, J. Localization of light in three-dimensional random dielectric media. Phys. Rev. E
**1996**, 53, 4122. [Google Scholar] [CrossRef] - Sgrignuoli, F.; Mazzamuto, G.; Caselli, N.; Intonti, F.; Cataliotti, F.S.; Gurioli, M.; Toninelli, C. Necklace state hallmark in disordered 2D photonic systems. ACS Photonics
**2015**, 2, 1636–1643. [Google Scholar] [CrossRef] - Goetschy, A.; Skipetrov, S. Non-Hermitian Euclidean random matrix theory. Phys. Rev. E
**2011**, 84, 011150. [Google Scholar] [CrossRef][Green Version] - Noh, H.; Yang, J.K.; Boriskina, S.V.; Rooks, M.J.; Solomon, G.S.; Dal Negro, L.; Cao, H. Lasing in Thue–Morse structures with optimized aperiodicity. Appl. Phys. Lett.
**2011**, 98, 201109. [Google Scholar] [CrossRef] - Efimov, V. Energy levels arising from resonant two-body forces in a three-body system. Phys. Lett. B
**1970**, 33, 563–564. [Google Scholar] [CrossRef] - Wang, J.; Genack, A.Z. Transport through modes in random media. Nature
**2011**, 471, 345–348. [Google Scholar] [CrossRef] [PubMed] - Skipetrov, S.; Sokolov, I. Ioffe-Regel criterion for Anderson localization in the model of resonant point scatterers. Phys. Rev. B
**2018**, 98, 064207. [Google Scholar] [CrossRef][Green Version] - Skipetrov, S.; Sokolov, I. Search for Anderson localization of light by cold atoms in a static electric field. Phys. Rev. B
**2019**, 99, 134201. [Google Scholar] [CrossRef][Green Version] - Haake, F. Quantum Signatures of Chaos; Springer Science & Business Media: Berlin, Germany, 2013. [Google Scholar]
- Mehta, M.L. Random Matrices; Elsevier: Amsterdam, The Netherlands, 2004. [Google Scholar]
- Mulholland, G.W.; Bohren, C.F.; Fuller, K.A. Light scattering by agglomerates: Coupled electric and magnetic dipole method. Langmuir
**1994**, 10, 2533–2546. [Google Scholar] [CrossRef] - Nieto-Vesperinas, M.; Gomez-Medina, R.; Saenz, J. Angle-suppressed scattering and optical forces on submicrometer dielectric particles. JOSA A
**2011**, 28, 54–60. [Google Scholar] [CrossRef] - GarcOpt; Expressía-Cámara, B.; Moreno, F.; González, F.; Martin, O.J. Light scattering by an array of electric and magnetic nanoparticles. Opt. Express
**2010**, 18, 10001–10015. [Google Scholar] [CrossRef][Green Version] - Yurkin, M.A.; Hoekstra, A.G. The discrete dipole approximation: An overview and recent developments. J. Quant. Spectrosc. Radiat. Transf.
**2007**, 106, 558–589. [Google Scholar] [CrossRef][Green Version] - Eyges, L. The Classical Electromagnetic Field; Dover: New York, NY, USA, 2012. [Google Scholar]
- Chaumet, P.C.; Rahmani, A. Coupled-dipole method for magnetic and negative-refraction materials. J. Quant. Spectrosc. Radiat. Transf.
**2009**, 110, 22–29. [Google Scholar] [CrossRef] - Doyle, W.T. Optical properties of a suspension of metal spheres. Phys. Rev. B
**1989**, 39, 9852. [Google Scholar] [CrossRef] [PubMed] - Draine, B.T. The discrete-dipole approximation and its application to interstellar graphite grains. Astrophys. J.
**1988**, 333, 848–872. [Google Scholar] [CrossRef] - Jackson, J.D. Classical Electrodynamics; John Wiley & Sons: New York, NY, USA, 1999. [Google Scholar]
- Wiscombe, W.J. Improved Mie scattering algorithms. Appl. Opt.
**1980**, 19, 1505–1509. [Google Scholar] [CrossRef] [PubMed] - Chen, Y.; Dal Negro, L. Pole-zero analysis of scattering resonances of multilayered nanospheres. Phys. Rev. B
**2018**, 98, 235413. [Google Scholar] [CrossRef][Green Version] - Bohren, C.F.; Huffman, D.R. Absorption and Scattering of Light by Small Particles; John Wiley & Sons: New York, NY, USA, 2008. [Google Scholar]
- Person, S.; Jain, M.; Lapin, Z.; Sáenz, J.J.; Wicks, G.; Novotny, L. Demonstration of zero optical backscattering from single nanoparticles. Nano Lett.
**2013**, 13, 1806–1809. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) Continuous elliptic curve generated by Equation (1) when the coefficients A and B are equal to 27 and 4, respectively. The sum operation on elliptic curve ${R}^{\prime}=P\oplus Q$ is also shown. (

**b**) Point pattern generated from the continuous curve of panel (

**a**) defined over the finite field ${\mathbb{F}}_{2111}$ rescaled to have an average interparticles separation equal to 450 nm. The red and blue point marker identifies two representative points W and M, respectively. Panels (

**c**,

**d**) show the point patterns generated by solving the discrete log problem $W=kM$. Specifically, panels (

**c**,

**d**) are characterized by the coordinates $({M}_{x};k)$ and $({M}_{y};k)$, respectively. Also for these geometries we rescaled the generated point patterns to have an average interparticles separation equal to 450 nm.

**Figure 2.**(

**a**) Radial distribution function $g\left(r\right)$ of the $EC$ of Figure 1b (blue line) as compared to the averaged two-point correlation function of 200 different disorder realizations of Poissonian point patterns (red curve). The black line identifies the averaged $g\left(r\right)$ of 900 different elliptic curves generated by all the possible combinations of the coefficients A and B in the range $(1,30)$. (

**b**) First (blue bars) and second (pastel green bars) neighbor probability density function of the $EC$ point pattern generated by the equation ${y}^{2}={x}^{3}+27x+4$ as compared to the Poissonian first and second neighbor distribution defined by Equation (3) [31,55]. The two dotted curves are the averaged $P\left({d}_{1}\right)$ and $P\left({d}_{2}\right)$ of the 900 different elliptic curves generated as explained above. Panels (

**c**,

**d**) display, respectively, the behavior of the integrated intensity function defined by Equation (5) of the elliptic curve of Figure 1b over the finite fields ${\mathbb{F}}_{2111}$ and of a representative Poissonian point pattern. Insets report their structure factors $S\left(k\right)$ evaluated by using Equation (4).

**Figure 3.**Eigenvalues of the Green’s matrix (7) are shown by points on the complex plane for 1630 electric point dipoles arranged in a Poissonian configuration (panel (

**a**,

**b**)) and elliptic curve geometry (panel (

**c**,

**d**)), respectively. The $EC$ point pattern is shown in Figure 1b. Specifically, panels (

**a**,

**c**) and panels (

**b**,

**d**) refer to low ($\rho {\lambda}^{2}$ = 0.01) and high ($\rho {\lambda}^{2}$ = 50) optical density, respectively. The data are colored according to the ${log}_{10}$ values of the MSE. The different markers identify representative scattering resonances displayed in Figure 4. For the traditional uniform random configuration a total of at least $5\times {10}^{4}$ eigenvalues for each optical densities are considered.

**Figure 4.**Representative spatial distributions of the Green’s matrix eigenvectors that belong to the class of scattering resonances identified in the complex plane of Figure 3b,d. Specifically, panels (

**a**–

**d**) and panels (

**e**–

**h**) refer to the $UR$ and $EC$ configurations, respectively.

**Figure 5.**Probability distribution of scattering resonances as a function of $\rho {\lambda}^{2}$ of representative $EC$ geometries as compared to $UR$ for various MSE intervals (

**a**–

**f**). Specifically, the $UR$ configuration is located in the first row of each panel, while the representative $EC$ structures, 30 planar arrays in total, are characterized by the parameters A and B defined by the relation $A=B=n$, where n is defined in the range (1, 30).

**Figure 6.**(

**a**) Averaged modal lifetime as a function of different optical densities of the 900 $EC$ geometries generated by all the possible combinations of the coefficients A and B in the range $(1,30)$ as compared to the uniform random configuration ensemble averaged over 10 different disordered realizations (red curve). The highest ($E{C}_{H}$), the middle ($E{C}_{M}$), and the lower ($E{C}_{L}$) $\widehat{\mathsf{\Gamma}}$ trend are highlighted in violet, pink and grey color, respectively. In particular, $E{C}_{H}$, $E{C}_{M}$, and $E{C}_{L}$ are the elliptic curves over the finite field ${\mathbb{F}}_{2111}$ generated by the parameters combination $(a=27;b=4;)$, $(a=11;b=8;)$, and $(a=28;b=19;)$ respectively. (

**b**) Thouless conductance as a function of the scattering strength $\rho {\lambda}^{2}$ averaged over the frequency stripe of width 2${\mathsf{\Gamma}}_{0}$ centered in ${\omega}_{0}$ for the $E{C}_{H}$ (circle violet markers) and $UR$ (circle red markers) configurations, respectively. The dashed-black lines identify the threshold of the diffusion-localization transition $g=1$.

**Figure 7.**Level spacing statistics of the Green’s matrix eigenvalues for two different regimes: $\rho {\lambda}^{2}$ = $0.05$ (panels (

**a**–

**c**)) and $\rho {\lambda}^{2}$ = 50 (panels (

**d**–

**f**)). Panels (

**a**–

**d**), (

**b**–

**e**), (

**c**–

**f**) refer to $E{C}_{H}$, $E{C}_{M}$, and $E{C}_{L}$ configurations, respectively. The fitting curves are performed by using the critical cumulative distribution [25,43,46] (dotted dashed lines in panels (

**a**–

**c**)) and the Poisson distribution (dotted dashed lines in panels (

**d**–

**f**)). The dotted dashed black lines in panels (

**a**–

**c**) indicates the level spacing distribution of a representative $UR$ structure defined by Equation (9).

**Figure 8.**Panels (

**a**,

**b**) and panels (

**c**,

**d**) report the radial distribution function $g\left(r\right)$ and the first and second neighbor probability density function of the $EC$$DL$ point patterns reported in Figure 1 panels (

**c**,

**d**), respectively. Moreover, their averaged values, with respect to 72 different $EC$$DL$ geometries (generated by randomly selecting the starting point W from the elliptic curve point patterns generated by the coefficients combination $(a=27;b=4)$ and $(a=28;b=19)$, named $E{C}_{H}$ and $E{C}_{L}$ respectively), are compared with respect the $UR$ scenario.

**Figure 9.**Panels (

**a**,

**b**) display the complex eigenvalues distributions of a representative $EC$$DL$ configuration (integer starting point equal to $W=(379;1735)$ on the elliptic curve defined by the equation ${y}^{2}={x}^{3}+27x+4$ defined over the finite field ${\mathbb{F}}_{2111}$) of the Green’s matrix (7) for two different optical densities, respectively. The data are colored according to the ${log}_{10}$ values of the MSE. The different markers in panel (

**b**) identify representative scattering resonances displayed in panels (

**c**–

**f**). In particular, panels (

**c**,

**d**) display, respectively, a proximity and a clustered scattering resonance on 4 particles, while panels (

**e**,

**f**) show two representative modes with the lowest decay rates.

**Figure 10.**Probability distribution of scattering resonances as a function of $\rho {\lambda}^{2}$ of 36 representative $EC$$DL$ geometries as compared to $UR$ structures for various MSE intervals (

**a**–

**f**). Specifically, the $UR$ configuration is reported in the first row of each panels. The reported $EC$$DL$ structures are generated by randomly selecting 9 integers starting points W on the two elliptic curves defined by the parameter combination $(a=27;b=4;)$, and $(a=28;b=19;)$ over the finite field ${\mathbb{F}}_{2111}$, respectively. These two $EC$ point patterns are the $E{C}_{H}$ and $E{C}_{L}$ structures introduced in Figure 6. Each selected integer W generates two different $EC$$DL$ aperiodic point patterns by solving the two discrete logarithmic problems $W=k{M}_{x}$ and $W=k{M}_{y}$, where ${M}_{x}$ and ${M}_{y}$ are the components of a each point M of $EC$ that satisfies these two relations (see Section 2 and in the caption of Figure 1 for more details). In particular, the 9 integer points selected on the $E{C}_{H}$ are: (1893;1826), (114;1753), (375;1739), (340;936), (1124;999), (1881;1246), (1902;389), (1129;395), and (305;329). On the other hand, the 9 integer points selected on the $E{C}_{L}$ are: (1719;1909), (1122;1836), (382;1761), (212;889), (1021;1138), (1841;1105), (1768;330), (1066;243), and (295;235).

**Figure 11.**(

**a**) Averaged modal lifetime as a function of different optical densities of 36 $EC$$DL$ geometries generated by randomly selecting the point W from the $E{C}_{H}$ and $E{C}_{L}$ point patterns as compared to the uniform random configuration ensemble averaged over 10 different disordered realizations (red line). The highest ($EC$$D{L}_{H}$), the middle ($EC$$D{L}_{M}$), and the lower ($EC$$D{L}_{L}$) $\widehat{\mathsf{\Gamma}}$ trends are highlighted in orange, cyan, and green colors, respectively. Specifically, $EC$$D{L}_{H}$ and $EC$$D{L}_{M}$ are the point patterns characterized by the coordinates $({M}_{x};k)$ and $({M}_{y};k)$ generated, respectively, by solving the discrete logarithmic problem $W=kM$ associated to the $E{C}_{H}$ when W is equal to (375;1739) and (1902;389). $EC$$D{L}_{L}$ is, instead, generated by solving the discrete logarithmic problem $W=k{M}_{x}$ associated to the elliptic curve ${y}^{2}={x}^{3}+28x+19$ defined over the finite field ${\mathbb{F}}_{2111}$ when the integer W is equal to (295;235). (

**b**) Thouless conductance as a function of the scattering strength $\rho {\lambda}^{2}$ averaged over the frequency stripe of width of 2${\mathsf{\Gamma}}_{0}$ centered in ${\omega}_{0}$ for the $EC$$D{L}_{H}$ (circle carrot orange markers) and $E{C}_{H}$ (circle violet markers) configurations, respectively. The dashed-black lines identify the threshold of the diffusion-localization transition $g=1$.

**Figure 12.**Level spacing statistics of the Green’s matrix eigenvalues for two different regimes: $\rho {\lambda}^{2}$ = $0.05$ (panels (

**a**–

**c**)) and $\rho {\lambda}^{2}$ = 50 (panels (

**d**–

**f**)). Panels (

**a**,

**d**), (

**b**,

**e**), (

**c**,

**f**) refer to $EC$$D{L}_{H}$, $EC$$D{L}_{M}$, and $EC$$D{L}_{L}$ configurations, respectively. The fitting curves are performed by using the critical cumulative distribution [25,43,46] (dotted dashed lines in panels (

**a**–

**c**)) and the Poisson distribution (dotted dashed lines in panels (

**d**–

**f**)). The dotted dashed black lines in panels (

**a**–

**c**) indicates the level spacing distribution of a representative $UR$ structure defined by Equation (9).

**Figure 13.**(

**a**) Scattering efficiency of a single TiO${}_{2}$ nanoparticle (R = 70 nm) evaluated by using the Mie–Lorentz theory by truncating $\widehat{l}$ up to the convergence order provided by $\widehat{l}=x+4.05{x}^{1/3}+2$ [87] (x is the size parameter) as compared to both the analytical result with only the dipolar contribution ($\widehat{l}=1$) and the numerical electric and magnetic coupled dipole approximation (EMCDA) calculation (red circle markers). The electric dipole (ED) and the magnetic dipole (MD) contributions are also shown. The grey-left y-axis indicates the phase difference $\Delta \varphi =\varphi \left({b}_{1}\right)-\varphi \left({a}_{1}\right)$, normalized with respect to $\pi $, between the magnetic and electric dipole. Here, ${a}_{1}$ and ${b}_{1}$ are the Mie–Lorentz coefficients evaluated by using Equation (20) with $\nu =1$. Panels (

**b**,

**c**) show the same benchmark for the differential scattering efficiency in the forward and backward direction, respectively. Panel (

**d**–

**f**) displays the benchmark between the EMCDA and the finite element method (FEM) technique applied to different dimer nanoparticle ($R=$ 70 nm) configurations characterized by an interparticle separation of 10 nm, 50 nm, and 450 nm, respectively. Panels (

**g**–

**i**) show, respectively, the scattering efficiency and the differential scattering efficiency in the forward and backward directions of $E{C}_{H}$ (violet line), $E{C}_{L}$ (grey line), $EC$$D{L}_{H}$ (circle orange markers), $EC$$D{L}_{L}$ (green diamond markers), and $UR$ (red line) arrays rescaled to avoid touching scatterers. The error bars of the $UR$ case are evaluated as the standard deviation over 20 different disorder realizations.

**Figure 14.**(

**a**) Normalized backscattering cone of different representative $EC$ (blue lines) and $EC$$DL$ (black lines) structures as compared to 20 different disorder realizations of traditional uniform random arrays (red lines). Table 1 summarizes the averaged structural parameters of the different analyzed devices. Specifically, the $EC$ arrays were selected equidistantly from the 900 different $EC$ structures generated by all the possible combination of the coefficients A and B in the range (1, 30) ordered by following the $\widehat{\mathsf{\Gamma}}$ trend of Figure 6. In the same way, the 26 different $EC$$DL$ arrays were selected equidistantly from the 32 $EC$$DL$ point patterns, generated as discussed in Section 3, order by following the $\widehat{\mathsf{\Gamma}}$ trend of Figure 11. (

**b**) Intensity peak of the differential scattering efficiency evaluated in the backward direction by using Equation (36). Panels (

**c**,

**d**) report, respectively, the full width half maximum of the backscattering cone of the $EC$ and $EC$$DL$ aperiodic arrays.

**Table 1.**Averaged structural parameters of the different analyzed devices. Specifically, ${d}_{min}$, ${d}_{1}$ and $\rho $ indicates, respectively, the minimum particle separation, the averaged first-neighbor particle separation, and the particle density evaluated as $N/{L}_{x}{L}_{y}$ (${L}_{x,y}$ are the lateral dimension along the x and y direction).

Structural Parameters | EC | EC DL | UR |
---|---|---|---|

$\langle {d}_{min}\rangle $$\left[nm\right]$ | $152.04\pm 1.72$ | $189.79\pm 19.35$ | $151.05\pm 1.05$ |

$\langle {d}_{1}\rangle $$\left[nm\right]$ | $482.99\pm 5.03$ | $499.43\pm 12.69$ | $484.11\pm 3.62$ |

$\rho $$\left[\mu {m}^{-2}\right]$ | $1.22\pm 0.03$ | $1.20\pm 0.11$ | $1.23\pm 0.04$ |

© 2019 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/).

## Share and Cite

**MDPI and ACS Style**

Dal Negro, L.; Chen, Y.; Sgrignuoli, F. Aperiodic Photonics of Elliptic Curves. *Crystals* **2019**, *9*, 482.
https://doi.org/10.3390/cryst9090482

**AMA Style**

Dal Negro L, Chen Y, Sgrignuoli F. Aperiodic Photonics of Elliptic Curves. *Crystals*. 2019; 9(9):482.
https://doi.org/10.3390/cryst9090482

**Chicago/Turabian Style**

Dal Negro, Luca, Yuyao Chen, and Fabrizio Sgrignuoli. 2019. "Aperiodic Photonics of Elliptic Curves" *Crystals* 9, no. 9: 482.
https://doi.org/10.3390/cryst9090482