## 2 Theoretical Modal of Coupled Nanoparticles  
When the feature size of the high refractive index dielectric nanoparticles is significantly smaller than the incident wavelength, the resulting optical resonance can be described by the effective electric dipole and magnetic dipole as [32]:  
\[\underset{P}{\rightarrow}=\varepsilon_{0}\alpha_{E}\underset{E_{0}}{ \rightarrow},\underset{m}{\rightarrow}=\ \chi_{H}\underset{H_{0}}{\rightarrow} \tag{1}\]  
where \(\underset{P}{\rightarrow}\) and \(\underset{m}{\rightarrow}\) denote the induced electric and magnetic dipoles, respectively, and \(\alpha_{E}\) and \(\chi_{H}\) are the electric and magnetic polarizabilities, respectively, and \(\overset{\rightarrow}{E}_{0}\) and \(\overset{\rightarrow}{H}_{0}\) are the incident fields.  
When considering the interaction between multiple nanoparticles, the electric and magnetic fields at the \(i\)th particle, as caused by the \(j\)th particle, can be represented by [32]  
\[\left\{\begin{array}{c}\underset{E_{i}}{\rightarrow}=\alpha_{ij}\alpha_{E} \underset{E_{j}}{\rightarrow}+b_{ij}\alpha_{E}\left(\underset{E_{j}}{ \rightarrow}\underset{n_{ji}}{\rightarrow}\right)\underset{n_{ji}}{ \rightarrow}-d_{ij}(\frac{\mu_{0}}{\kappa_{0}})^{1/2}\chi_{H}\left( \underset{n_{ji}}{\rightarrow}\times\underset{H_{ij}}{\rightarrow}\right)\\ \underset{H_{i}}{\rightarrow}=\alpha_{ij}\chi_{H}\underset{H_{ij}}{ \rightarrow}+b_{ij}\chi_{H}\left(\underset{H_{ij}}{\rightarrow}\underset{n_{ji} }{\rightarrow}\right)\underset{n_{ji}}{\rightarrow}+d_{ij}(\frac{\kappa_{0 }}{\mu_{0}})^{1/2}\alpha_{E}\left(\underset{n_{ji}}{\rightarrow}\times \underset{E_{j}}{\rightarrow}\right)\end{array}\right. \tag{2}\]  
where \(\overset{\rightarrow}{n_{ji}}\) denote the direction vector from the \(j\)th particle to \(i\)th particle. The coefficients \(a_{ij}\), \(b_{ij}\) and \(d_{ij}\) are expressed as:  
\[\left\{\begin{array}{c}\alpha_{ij}=\frac{1}{4\pi}\frac{\epsilon^{\text{i} \alpha_{ij}}}{r_{ij}}\left(k^{2}-\frac{1}{r_{ij}^{2}}+\frac{ik}{r_{ij}}\right) \\ b_{ij}=\frac{1}{4\pi}\frac{\epsilon^{\text{i}\alpha_{ij}}}{r_{ij}}\left(-k^{2 }+\frac{3}{r_{ij}^{2}}-\frac{3 ik}{r_{ij}}\right)\\ d_{ij}=\frac{1}{4\pi}\frac{\epsilon^{\text{i}\alpha_{ij}}}{r_{ij}}\left(k^{2 }+\frac{ik}{r_{ij}}\right)\end{array}\right. \tag{3}\]where \(r_{ij}\) denotes the distance between \(i\)th and \(j\)th particles and \(k\) = \(2\pi/\lambda\). Thus, the total induced electric and magnetic dipoles can be derived as [32]:  
\[\left\{\begin{array}{c}\rightarrow\\ p_{i}\\ \rightarrow\\ m_{i}\end{array}\right.=\alpha_{E}\left[\varepsilon_{0}\rightarrow\sum \limits_{E_{0}}\alpha_{ij}\rightarrow\sum\limits_{j\
eq 1}\alpha_{ij}\rightarrow+b_{ij}\left(\rightarrow\sum\limits_{m_{j}} \rightarrow\sum\limits_{n_{ji}}\right)\rightarrow\sum\limits_{n_{ji}}-d_{ij}c \left(\rightarrow\sum\limits_{n_{ji}}\times\rightarrow\sum\limits_{p_{j}} \right)\right] \tag{4}\]  
where \(c=\sqrt{1/\varepsilon_{0}\mu_{0}}\) represents the speed of light in vacuum. When \(a_{ij}\), \(b_{ij}\) and \(d_{ij}\to 0\) with \(r_{ij}\rightarrow\infty\), the electric and magnetic dipoles can be regarded as equivalent to that of a single nanoparticle. When \(r_{ij}\to 0\), the scattering characteristics changes dramatically. According to Equation (4), the electric-to-electric interaction, the electric-to-magnetic interaction, and the magnetic-to-magnetic interaction can be found. The coupled nanoparticles can achieve versatile functions by these induced interactions.