## 3 Theoretical Model
### Mie Theory  
Compared with the Rayleigh approximation, Mie theory, proposed by [PERSON] in 1908, was deduced based on the solution of the Maxwell equations. It assumes particles are ideal spheres and adopts the spherical Bessel function to obtain related parameters. The main objective of Mie theory is to address the issue of absorption and scattering of light by spherical particles. Despite its limitations, Mie theory can accurately describe the effects of small particles that are not immediately apparent. According to Mie theory, particle attenuation is primarily characterized by the extinction cross-section, scattering cross-section, and absorption cross-section, which can be separately expressed as [28]:  
\[\left\{ \begin{array}{c}C_{e}(r,\lambda,m)=\frac{\lambda^{2}}{2\pi}\sum\limits _{n=1}^{\infty}(2n+1)\rm Re(\emph{a}_{n}+\emph{b}_{n}),\\ \\ C_{s}(r,\lambda,m)=\frac{\lambda^{2}}{2\pi}\sum\limits_{n=1}^{\infty}(2n+1) \Big{(}|\emph{a}_{n}|^{2}+|\emph{b}_{n}|^{2}\Big{)},\\ \\ C_{a}(r,\lambda,m)=C_{e}(r,\lambda,m)-C_{s}(r,\lambda,m).\end{array} \right. \tag{12}\]  
The efficiency factors for the extinction, scattering, and absorption are defined as:  
\[\left\{ \begin{array}{c}Q_{e}(r,\lambda,m)=\frac{C_{e}(r,\lambda,m)}{\pi r^{ 2}},\\ \\ Q_{s}(r,\lambda,m)=\frac{C_{s}(r,\lambda,m)}{\pi r^{2}},\\ \\ Q_{a}(r,\lambda,m)=\frac{C_{s}(r,\lambda,m)}{\pi r^{2}}.\end{array} \right. \tag{13}\]Therefore, it is evident that:  
\[\left\{ \begin{array}{c}Q_{e}(x,m)=\frac{2}{x^{2}}\sum\limits_{n=1}^{\infty}{(2n+ 1)\text{Re}(a_{n}+b_{n})},\\ Q_{s}(x,m)=\frac{2}{x^{2}}\sum\limits_{n=1}^{\infty}{(2n+1)\Big{(}|a_{n}|^{2}+| b_{n}|^{2}\Big{)}},\\ Q_{a}(x,m)=Q_{e}(x,m)-Q_{s}(x,m),\end{array} \right. \tag{14}\]  
where \(m\) is the refractive index of the particle relative to the surrounding medium, \(m=m_{1}+m_{2}\). As for the size parameter of the particles, \(x=kr\), and \(k=2\pi/\lambda\). \(r\) is the radius of the particle, and \(\lambda\) is the length of the incident waves. If the scattering coefficients \(a_{n}\) and \(b_{n}\) are known, all measurable quantities related to scattering and absorption, such as cross-sections and scattering matrix elements, can be determined. By applying boundary conditions for magnetic and electric fields, expanding the incident, internal, and scattered magnetic fields into series, and using expressions for vector harmonics, a set of linear equations is established, from which the scattering coefficients \(a_{n}\) and \(b_{n}\) can be calculated [28]:  
\[a_{n}=\frac{\psi^{\prime}_{n}(mx)\psi_{n}(x)-m\psi_{n}(mx)\psi^{\prime}_{n}(x) }{\psi^{\prime}_{n}(mx)\xi_{n}(x)-m\psi_{n}(mx)\xi^{\prime}_{n}(x)}, \tag{15}\]  
\[b_{n}=\frac{m\psi^{\prime}_{n}(mx)\psi_{n}(x)-\psi_{n}(mx)\psi^{\prime}_{n}(x) }{m\psi^{\prime}_{n}(mx)\xi_{n}(x)-\psi_{n}(mx)\xi^{\prime}_{n}(x)}, \tag{16}\]  
where \(\psi_{n}(x)\) and \(\xi_{n}(x)\) are the \(n\)-order spherical Bessel functions of the first kind and the first type of the \(n\)-order spherical Bessel functions of the third kind, respectively. \(\psi^{\prime}_{n}(x)\) and \(\xi^{\prime}_{n}(x)\) are the derivatives of \(\psi_{n}(x)\) and \(\xi_{n}(x)\), respectively. The \(n\)-order spherical Bessel functions of the first kind and the third kind are calculated as:  
\[\psi_{n}(x)=xj_{n}(x)=\sqrt{\frac{\pi x}{2}}J_{n+0.5}(x) \tag{17}\]  
\[\xi_{n}(x)=xh^{(1)}_{n}(x)=\sqrt{\frac{\pi x}{2}}H^{(1)}_{n+0.5}(x) \tag{18}\]  
where \(x\) is the size parameter of the particle, determined by the particle radius and the wavelength of the incident wave; \(J_{n+0.5}(x)\) is a Bessel function of the first kind of half-integer order; and \(H^{(1)}_{n+0.5}(x)\) is a Bessel function of the first one of the third kind of the half-integer order. Functions \(j_{n}(z)\) and \(h^{(1)}_{n}(z)=j_{n}(z)+iy_{n}(z)\) are the spherical Bessel functions (\(z=x\) or \(mx\)) of order \(n\) (\(n=1,2,\ldots\)). The derivative of the spherical Bessel function is represented as [34]:  
\[\left\{