# Near-Field Coupling and Mode Competition in Multiple Anapole Systems

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Ab-Initio Analysis of Multiple Anapole Systems

#### 2.2. Fano–Feshbach Analysis of the Internal Modes

## 3. Discussion and Conclusions

## 4. Materials and Methods

#### 4.1. FDTD Simulations

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

FDTD | Finite-Differences Time-Domain |

DOS | Density Of States |

UPML | Uniaxial Perfectly Matched Layer |

TFSF | Total-Field Scattered-field |

TM | Transverse Magnetic |

TE | Transverse Electric |

PEC | Perfect Electric Conductor |

PMC | Perfect Magnetic Conductor |

NSOM | Near-field scanning optical microscopy |

## References

- Wang, K.X.; Yu, Z.; Sandhu, S.; Liu, V.; Fan, S. Condition for perfect antireflection by optical resonance at material interface. Optica
**2014**, 1, 388–395. [Google Scholar] [CrossRef] - Liu, W.; Miroshnichenko, A.E.; Neshev, D.N.; Kivshar, Y.S. Broadband Unidirectional Scattering by Magneto-Electric Core-Shell Nanoparticles. ACS Nano
**2012**, 6, 5489–5497. [Google Scholar] [CrossRef] [PubMed] - Miroshnichenko, A.E.; Evlyukhin, A.B.; Yu, Y.F.; Bakker, R.M.; Chipouline, A.; Kuznetsov, A.I.; Luk’yanchuk, B.; Chichkov, B.N.; Kivshar, Y.S. Nonradiating anapole modes in dielectric nanoparticles. Nat. Commun.
**2015**, 6, 8069. [Google Scholar] [CrossRef] [PubMed] - Liu, W.; Zhang, J.; Lei, B.; Hu, H.; Miroshnichenko, A.E. Invisible nanowires with interfering electric and toroidal dipoles. Optics Lett.
**2015**, 40, 2293–2296. [Google Scholar] [CrossRef] [PubMed] - Luk’yanchuk, B.; Paniagua-Domínguez, R.; Kuznetsov, A.I.; Miroshnichenko, A.E.; Kivshar, Y.S. Suppression of scattering for small dielectric particles: anapole mode and invisibility. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci.
**2017**, 375, 20160069. [Google Scholar] [CrossRef] [PubMed] - Wei, L.; Xi, Z.; Bhattacharya, N.; Urbach, H.P. Excitation of the radiationless anapole mode. Optica
**2016**, 3, 799–802. [Google Scholar] [CrossRef] - Brongersma, M.L.; Hartman, J.W.; Atwater, H.A. Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit. Phys. Rev. B
**2000**, 62, R16356–R16359. [Google Scholar] [CrossRef] - Totero Gongora, J.S.; Fratalocchi, A. Harnessing Disorder at the Nanoscale. In Encyclopedia of Nanotechnology; Bhushan, B., Ed.; Springer Netherlands: Heidelberg, Germany, 2015; pp. 1–13. [Google Scholar]
- Bakker, R.M.; Yu, Y.F.; Paniagua-Domínguez, R.; Luk’yanchuk, B.; Kuznetsov, A. Silicon Nanoparticles for Waveguiding. Frontiers in Optics; Optical Society of America: Washington, DC, USA, 2015; p. FM1B.2. [Google Scholar]
- Evlyukhin, A.B.; Reinhardt, C.; Seidel, A.; Luk’yanchuk, B.S.; Chichkov, B.N. Optical response features of Si-nanoparticle arrays. Phys. Rev. B
**2010**, 82, 045404. [Google Scholar] [CrossRef] - Gongora, J.S.T.; Favraud, G.; Fratalocchi, A. Fundamental and high-order anapoles in all-dielectric metamaterials via Fano–Feshbach modes competition. Nanotechnology
**2017**, 28, 104001. [Google Scholar] [CrossRef] [PubMed] - Grinblat, G.; Li, Y.; Nielsen, M.P.; Oulton, R.F.; Maier, S.A. Efficient Third Harmonic Generation and Nonlinear Subwavelength Imaging at a Higher-Order Anapole Mode in a Single Germanium Nanodisk. ACS Nano
**2017**, 11, 953–960. [Google Scholar] [CrossRef] [PubMed] - Totero Gongora, J.S.; Miroshnichenko, A.E.; Kivshar, Y.S.; Fratalocchi, A. Anapole nanolasers for mode-locking and ultrafast pulse generation. Nat. Commun.
**2017**, in press. [Google Scholar] - Taflove, A.; Oskooi, A.; Johnson, S.G. Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology; Artech House Antennas and Propagation Series; Artech House: Norwood, MA, USA, 2013. [Google Scholar]
- Yariv, A.; Yeh, P. Photonics: Optical Electronics in Modern Communications (The Oxford Series in Electrical and Computer Engineering); Oxford University Press, Inc.: New York, NY, USA, 2006. [Google Scholar]
- Türeci, H.E.; Ge, L.; Rotter, S.; Stone, A.D. Strong Interactions in Multimode Random Lasers. Science
**2008**, 320, 643–646. [Google Scholar] [CrossRef] [PubMed] - Kristensen, P.T.; Hughes, S. Modes and Mode Volumes of Leaky Optical Cavities and Plasmonic Nanoresonators. ACS Photonics
**2014**, 1, 2–10. [Google Scholar] [CrossRef] - Viviescas, C.; Hackenbroich, G. Field quantization for open optical cavities. Phys. Rev. A
**2003**, 67, 013805. [Google Scholar] [CrossRef] - Jin, J.M. Theory and Computation of Electromagnetic Fields; John Wiley & Sons: Hoboken, NJ, USA, 2011; Google-Books-ID: D6SqmxJVV5wC. [Google Scholar]
- Viviescas, C.; Hackenbroich, G. Quantum theory of multimode fields: Applications to optical resonators. J. Opt. B Quantum Semiclass. Opt.
**2004**, 6, 211. [Google Scholar] [CrossRef] - Van Bladel, J. Radiation in Free Space. In Electromagnetic Fields; Wiley-IEEE Press: Hoboken, NJ, USA, 2007; pp. 277–356. [Google Scholar]
- Tribelsky, M.I.; Miroshnichenko, A.E. Giant in-particle field concentration and Fano resonances at light scattering by high-refractive-index particles. Phys. Rev. A
**2016**, 93, 053837. [Google Scholar] [CrossRef] - Cherchi, M.; Ylinen, S.; Harjanne, M.; Kapulainen, M.; Aalto, T. Dramatic size reduction of waveguide bends on a micron-scale silicon photonic platform. Opt. Express
**2013**, 21, 17814. [Google Scholar] [CrossRef] [PubMed] - Gentilini, S.; Fratalocchi, A.; Angelani, L.; Ruocco, G.; Conti, C. Ultrashort pulse propagation and the Anderson localization. Opt. Lett.
**2009**, 34, 130–132. [Google Scholar] [CrossRef] [PubMed] - Crosta, M.; Fratalocchi, A.; Trillo, S. Bistability and instability of dark-antidark solitons in the cubic-quintic nonlinear Schrödinger equation. Phys. Rev. A
**2011**, 84, 063809. [Google Scholar] [CrossRef] - Huang, J.; Liu, C.; Zhu, Y.; Masala, S.; Alarousu, E.; Han, Y.; Fratalocchi, A. Harnessing structural darkness in the visible and infrared wavelengths for a new source of light. Nat. Nanotechnol.
**2016**, 11, 60–66. [Google Scholar] [CrossRef] [PubMed] - Totero Gongora, J.S.; Miroshnichenko, A.E.; Kivshar, Y.S.; Fratalocchi, A. Energy equipartition and unidirectional emission in a spaser nanolaser. Laser Photonics Rev.
**2016**, 10, 432–440. [Google Scholar] [CrossRef] - Labelle, A.J.; Bonifazi, M.; Tian, Y.; Wong, C.; Hoogland, S.; Favraud, G.; Walters, G.; Sutherland, B.; Liu, M.; Li, J.; et al. Broadband Epsilon-near-Zero Reflectors Enhance the Quantum Efficiency of Thin Solar Cells at Visible and Infrared Wavelengths. ACS Appl. Mater. Interfaces
**2017**, 9, 5556–5565. [Google Scholar] [CrossRef] [PubMed] - Galinski, H.; Favraud, G.; Dong, H.; Totero Gongora, J.S.; Favaro, G.; Döbeli, M.; Spolenak, R.; Fratalocchi, A.; Capasso, F. Scalable, ultra-resistant structural colors based on network metamaterials. Light Sci. Appl.
**2017**, 6, e16233. [Google Scholar] [CrossRef] - Berenger, J.P. A Perfectly Matched Layer for the Absorption of Electromagnetic-Waves. J. Comput. Phys.
**1994**, 114, 185–200. [Google Scholar] [CrossRef] - Taflove, A.; Hagness, S.C. Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed.; Artech House Antennas and Propagation Library, Artech House: Boston, MA, USA, 2005. [Google Scholar]

**Figure 1.**

**Mutual coupling of non-radiating anapole states**. (

**a**) Near-field coupling between two silicon nanodisks ($n=3.5$) excited at the anapole wavelength ${\lambda}_{\mathrm{an}}=568$ nm. (inset) The dielectric resonators are mutually displaced by a centre-to-centre distance d and by an angle $\alpha $; (

**b**) Scattering cross-section ${C}_{\mathrm{sca}}$ (blue line) and internal electric energy (orange dotted line) as a function of the incident wavelength $\lambda $. The anapole state (green-dashed line) is characterized by the simultaneous suppression of the scattering cross-section, and by a strong enhancement of the internal field intensity; (

**c**) Coupled electric energy as a function of the rotation angle $\alpha $ ($d=450$ nm). The mutual coupling is maximum at $\alpha =0,\pi $ and negligible at $\alpha =\pi /2,3\pi /2$; (

**d**) Coupled electric energy as a function of the mutual distance d. The results correspond to the angular condition of maximum scattering $\alpha =2\pi $.

**Figure 2.**

**Local density of states and interacting resonant modes**. (

**a**,

**b**) Local density of state for the (

**a**) ${H}_{z}$ and (

**b**) ${E}_{z}$ field components, corresponding to the transverse electric (TE) and transverse magnetic (TM) modes of the silicon nanostructure, respectively. The resonant wavelengths (vertical dashed lines) in both configurations are computed by means of Equation (4).

**Figure 3.**

**Fano–Feshbach partitioning of the anapole state**. The characteristic mode-profile of the anapole state is originated by the superposition of a cylindrically symmetric ${\mathrm{TM}}_{020}$ and a quadrupolar ${\mathrm{TM}}_{210}$.

**Figure 4.**

**Anapole nanochain**. Steady-state electromagnetic energy distribution along a chain of anapole nanoparticles. The centre-to-centre distance is $d=400$ nm. The external excitation is restricted to the first anapole of the chain (not included in the panel).

**Figure 5.**

**Robust sub-wavelength guiding via near-field transfer of anapole states**. Due to the near-field confinement produced by the anapole state, the anapole nanochain is robust against bending and splitting of the integrated wave-guide. This opens to the realization of (

**a**) integrated splitters and (

**b**) 90-degree bends without introducing radiation losses.

© 2017 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**

Mazzone, V.; Totero Gongora, J.S.; Fratalocchi, A. Near-Field Coupling and Mode Competition in Multiple Anapole Systems. *Appl. Sci.* **2017**, *7*, 542.
https://doi.org/10.3390/app7060542

**AMA Style**

Mazzone V, Totero Gongora JS, Fratalocchi A. Near-Field Coupling and Mode Competition in Multiple Anapole Systems. *Applied Sciences*. 2017; 7(6):542.
https://doi.org/10.3390/app7060542

**Chicago/Turabian Style**

Mazzone, Valerio, Juan Sebastian Totero Gongora, and Andrea Fratalocchi. 2017. "Near-Field Coupling and Mode Competition in Multiple Anapole Systems" *Applied Sciences* 7, no. 6: 542.
https://doi.org/10.3390/app7060542