# Fast and Accurate Prediction of Light Scattering from Plasmonic Nanoarrays in Multiple Directions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Method of Moments

_{1}, and the magnetic-field-integral equation, MFIE

_{1}, outside the medium as follows [22]:

_{2}and the MFIE

_{2}inside the medium can also be obtained as follows:

#### 2.2. H-Matrix Method

_{I}is generated by a recursive subdivision of I. One index set is subdivided into two subsets recursively until the number of basis functions in the subset (denoted as “#”) is smaller than a threshold n

_{leaf}. The resulting cluster tree is called a binary tree, as shown in Figure 3.

- $\#t\le {n}_{\mathrm{leaf}}$ or $\#s\le {n}_{\mathrm{leaf}}$
- Clusters t and s satisfy the admissibility condition of$$\mathrm{min}\left\{diam\left({\Omega}_{t}\right),diam\left({\Omega}_{s}\right)\right\}\le \eta dist\left({\Omega}_{t},{\Omega}_{s}\right)$$$$G=X{Y}^{T}(G\in {\mathbb{R}}^{m\times n},X\in {\mathbb{R}}^{m\times k},Y\in {\mathbb{R}}^{n\times k},k\ll m,n)$$

_{H}, all the non-zero matrix entries in Z are filled in inadmissible leaves while admissible leaves remain empty because the partial differential operator is local. Hence, the representation of Z

_{H}is exact without approximation.

#### 2.3. Extraction of Light Scattering Characteristics

## 3. Results and Discussion

#### 3.1. Silver Nanosphere Array

#### 3.2. Silver Nanocylinder Array

#### 3.3. Gold-Nano-Truncated Cone Array

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Stewart, M.E.; Anderson, C.R.; Thompson, L.B.; Maria, J.; Gray, S.K.; Rogers, J.A.; Nuzzo, R.G. Nano-Structured Plasmonic Sensors. Chem. Rev.
**2008**, 108, 494–521. [Google Scholar] [CrossRef] [PubMed] - Kelly, K.L.; Coronado, E.; Zhao, L.L.; Schatz, G.C. The Optical Properties of Metal Nanoparticles: The Influence of Size, Shape, and Dielectric Environment. ChemInform
**2003**, 34, 668–677. [Google Scholar] [CrossRef] - Mock, J.J.; Barbic, M.; Smith, D.R.; A Schultz, D.A.; Schultz, S. Shape effects in plasmon resonance of individual colloidal silver nanoparticles. J. Chem. Phys.
**2002**, 116, 6755–6759. [Google Scholar] [CrossRef] - Sun, Y.; Xia, Y. Gold and silver nanoparticles: A class of chromophores with colors tunable in the range from 400 to 750 nm. Analyst
**2003**, 128, 686–691. [Google Scholar] [CrossRef] - Rechberger, W.; Hohenau, A.; Leitner, A.; Krenn, J.; Lamprecht, B.; Aussenegg, F. Optical properties of two interacting gold nanoparticles. Opt. Commun.
**2003**, 220, 137–141. [Google Scholar] [CrossRef] - Kedem, O.; Tesler, A.B.; Vaskevich, A.; Rubinstein, I. Sensitivity and optimization of Localized Surface Plasmon Resonance Transducers. ACS Nano
**2011**, 5, 748–760. [Google Scholar] [CrossRef] - Wan, T.; Guo, Y.X.; Tang, B.L. Photothermal Modeling and Characterization of Graphene Plasmonic Waveguides for Optical Interconnect. Opt. Express
**2019**, 27, 33268–33281. [Google Scholar] [CrossRef] - Hirsch, L.R.; Stafford, R.J.; Bankson, J.A.; Sershen, S.R.; Rivera, B.; Price, R.E.; Hazle, J.D.; Halas, N.J.; West, J.L. Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance. Proc. Natl. Acad. Sci. USA
**2003**, 100, 13549–13554. [Google Scholar] [CrossRef] [Green Version] - Bernardi Ronald, J.; Lowery, A.R.; Thompson, P.A.; Blaney, S.M.; West, J.L. Immunonanoshells for targeted photothermal ablation in medulloblastoma and glioma: An in vitro evaluation using human cell lines. J. Neuro-Oncol.
**2008**, 86, 165–172. [Google Scholar] [CrossRef] - Zeman, M.; Isabella, O.; Jaeger, K.; Santbergen, R.; Liang, R.; Solntsev, S.; Krc, J. Advanced Light Trapping in Thin-film Silicon Solar Cells. MRS Online Proc. Libr. OPL
**2010**, 1245. [Google Scholar] [CrossRef] [Green Version] - Derkacs, D.; Lim, S.H.; Matheu, P.; Mar, W.; Yu, E.T. Improved performance of amorphous silicon solar cells via scattering from surface plasmon polaritons in nearby metallic nanoparticles. Appl. Phys. Lett.
**2006**, 89, 514–518. [Google Scholar] [CrossRef] [Green Version] - Liu, W.; Wang, X.D.; Li, Y.; Geng, Z.; Yang, F.; Li, J. Surface plasmon enhanced GaAs thin film solar cells. Sol. Energy Mater. Sol. Cells
**2011**, 95, 693–698. [Google Scholar] [CrossRef] - Notarianni, M.; Vernon, K.; Chou, A.; Aljada, M.; Liu, J.; Motta, N. Plasmonic effect of gold nanoparticles in organic solar cells. Sol. Energy
**2014**, 106, 23–37. [Google Scholar] [CrossRef] [Green Version] - Lynch, D.R.; Paulsen, K.D. Time-Domain integration of the Maxwell Equations on finite Elements. IEEE Trans. Antennas Propag.
**1991**, 38, 1933–1942. [Google Scholar] [CrossRef] - Harrington, R.; Mautz, J. Theory of characteristic modes for conducting bodies. IEEE Trans. Antennas Propag.
**1971**, 19, 622–628. [Google Scholar] [CrossRef] - Gan, H.H.; Xia, T.; Dai, Q.I.; Li, Y.; Chew, W. Augmented electric field integral equation for inhomogeneous media. IEEE Antennas Wirel. Propag. Lett.
**2017**, 16, 2967–2970. [Google Scholar] [CrossRef] - de Abajo, F.J.G.; Howie, A. Retarded field calculation of electron energy loss in inhomogeneous dielectrics. Phys. Rev. B
**2002**, 65, 115418. [Google Scholar] [CrossRef] - Song, J.M.; Lu, C.C.; Chew, W.C. Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects. IEEE Trans. Antennas Propag.
**1997**, 45, 1488–1493. [Google Scholar] [CrossRef] [Green Version] - Feng, X.P.; Chen, J.Q.; Mo, L.; Wang, D.X.; Chen, R.S.; Yung, E.K.N.; Chan, C.H. Fast analysis of microwave integrated circuits using preconditioned SMCG method. In Proceedings of the 2005 IEEE Antennas and Propagation Society International Symposium, Washington, DC, USA, 3–8 July 2005. [Google Scholar]
- Ling, F.; Wang, C.F.; Jin, J.M. An efficient algorithm for analyzing large-scale microstrip structures using adaptive integral method combined with discrete complex-image method. IEEE Trans. Microw. Theory Tech.
**2000**, 48, 832–839. [Google Scholar] [CrossRef] - Stewart, G.W. Matrix Algorithms: Basic Decompositions; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1998. [Google Scholar]
- Yla-Oijala, P.; Taskinen, M. Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects. IEEE Trans. Antennas Propag.
**2005**, 53, 1168–1173. [Google Scholar] [CrossRef] - Bao, Y.; Wan, T.; Liu, Z.W.; Bowler, J.R.; Song, J.M. Integral equation fast solver with truncated and degenerated kernel for computing flaw signals in eddy current non-destructive testing. NDT E Int.
**2021**, 124, 102544. [Google Scholar] [CrossRef] - Wan, T.; Tang, B.L.; Li, M.Z. An Iteration-Free Domain Decomposition Method for the Fast Finite Element Analysis of Electromagnetic Problems. IEEE Trans. Antennas Propag.
**2020**, 68, 400–410. [Google Scholar] [CrossRef] - Wan, T.; Li, M.Z.; Li, L.F. Direct Solution of Finite Element- Boundary Integral System for Electromagnetic Analysis in Half-Space. IEEE Trans. Antennas Propag.
**2020**, 68, 6461–6466. [Google Scholar] [CrossRef] - Hohenester, U. Making simulations with the MNPBEM toolbox big: Hierarchical matrices and iterative solvers. Comput. Phys. Commun.
**2018**, 222, 209–228. [Google Scholar] [CrossRef] - Johnson, P.B.; Christy, R.W. Optical Constants of the Noble Metals. Phys. Rev. B
**1972**, 6, 4370–4379. [Google Scholar] [CrossRef]

**Figure 2.**A schematic diagram of a typical H-matrix structure. Black represents full matrix blocks and white represents low-rank matrix blocks.

**Figure 4.**Structure of the silver nanosphere array. (

**a**) 3D view; (

**b**) dimensions of the nanosphere array, where d = 5 nm, R = 20 nm.

**Figure 5.**Comparison of the ECS results obtained by the proposed method and the COMSOL software for the nanosphere array. (

**a**) 90° incident angle; (

**b**) 120° incident angle.

**Figure 6.**ECSs of the nanosphere array with the incident angle varying from 0° to 360° for different wavelengths.

**Figure 7.**Surface current distribution under different incident angles at 350 nm wavelength for the nanosphere array. (

**a**) 120° incident angle; (

**b**) 180° incident angle.

**Figure 8.**(

**a**) 3D view of the silver nanocylinder array with meshes; (

**b**) dimensions of the nanocylinder array, where R = 10 nm, d = 5 nm, h = 10 nnm.

**Figure 9.**Comparison of the ECS results obtained by the proposed method and the COMSOL software for the nanocylinder array. (

**a**) 90° incident angle; (

**b**) 120° incident angle.

**Figure 10.**ECSs of the nanocylinder array with the incident angle varying from 0° to 360° for different wavelengths.

**Figure 11.**Surface current distribution under different incident angles at 350 nm wavelength for the nanocylinder array. (

**a**) 120° incident angle; (

**b**) 180° incident angle.

**Figure 12.**Structure of the gold-nano-truncated cone array. (

**a**) 3D view; (

**b**) dimensions of the nanosphere array, where d = 10 nm, R2 = 20 nm, R1 = 10 nm, h = 20 nm.

**Figure 13.**Comparison of the ECS results obtained by the proposed method and the COMSOL software for the nano-truncated cone array. (

**a**) 90° incident angle; (

**b**) 120° incident angle.

**Figure 14.**Surface current distribution under different incident angles at 620 nm wavelength for the nano-truncated cone array. (

**a**) 120° incident angle; (

**b**) 180° incident angle.

**Table 1.**Comparison of the computational costs between the proposed method and the traditional MoM for the nanosphere array.

Number of Unknowns | Method | Solution Time (s) | Memory Requirement (MB) |
---|---|---|---|

3834 | Traditional MoM | 57,009.4 | 448.6 |

Proposed method | 1351.5 | 271.6 |

**Table 2.**Comparison of the computational costs between the proposed method and the traditional MoM for the nanocylinder array.

Number of Unknowns | Method | Solution Time (s) | Memory Requirement (MB) |
---|---|---|---|

12,750 | Traditional MoM | 570,695.6 | 4961.0 |

Proposed method | 14,267.4 | 2591.3 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wan, T.; Chen, T.; Bao, Y.; Wang, S.
Fast and Accurate Prediction of Light Scattering from Plasmonic Nanoarrays in Multiple Directions. *Micromachines* **2022**, *13*, 613.
https://doi.org/10.3390/mi13040613

**AMA Style**

Wan T, Chen T, Bao Y, Wang S.
Fast and Accurate Prediction of Light Scattering from Plasmonic Nanoarrays in Multiple Directions. *Micromachines*. 2022; 13(4):613.
https://doi.org/10.3390/mi13040613

**Chicago/Turabian Style**

Wan, Ting, Tianhao Chen, Yang Bao, and Shiyi Wang.
2022. "Fast and Accurate Prediction of Light Scattering from Plasmonic Nanoarrays in Multiple Directions" *Micromachines* 13, no. 4: 613.
https://doi.org/10.3390/mi13040613