# 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

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**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 |

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**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