## Appendix A  
Considering a spherical particle with radius \(a\) and relative complex refractive index \(m\) illuminated by a vector Bessel-Gaussian vortex beam. The coordinate systems of the particle and beam are o-xyz and O-XYZ, respectively (Figure A1). Here, \(\alpha\) is the angle between the wave vector \(\mathbf{k}\) of the plane wave component and the transmission axis \(Z\), and \(\beta\) is the azimuthal angle. The coordinates of the beam center O in the particle coordinate system are \((x_{0},y_{0},z_{0})\) indicating the relative position of the beam and particle. The expression of the electric field of a vector Bessel-Gaussian beam with order \(l\) at the source plane is  
\[\mathbf{E}(\rho,\varphi,0)=\mathbf{e_{u}}E_{0}l(\gamma\rho)\exp(-\frac{\rho^{2 }}{w\varphi^{2}})e^{il\varphi} \tag{1}\]where, \(\rho=\sqrt{x^{2}+y^{2}}\), \(\varphi=\arctan(y/x)\), \(\theta_{\rm b}\) is the conic angle and \(\gamma=k\sin\theta_{b}\). \(w_{0}\) is the waist radius of the incident beam, \(J_{l}(x)\) is the Bessel function of order \(l\). \(\mathbf{e_{u}}\) represents the vector state of the beam and the subscript \(u\) indicates the polarization type. In the angular spectrum theory of light, the light field on any plane can be considered as a linear superposition of many plane waves with different amplitudes and propagation directions. Therefore, from the definition, Equation (A1) can be expressed as  
\[P(k_{x},k_{y},k_{z=0})=\int\limits_{-\infty}^{+\infty}\int\limits_{-\infty}^{+ \infty}E(x,y,0)\exp[-i(k_{x}x+k_{y}y)]dxdy\] (A2)  
where, \(k_{z}=k\cos\alpha\), \(k_{\rho}=k\sin\alpha\), \(k_{x}=k_{\rho}\cos\beta\) and \(k_{y}=k_{\rho}\sin\beta\). Using straightforward mathematical transformations, the derivation result of Equation (A2) is  
\[P(k_{x},k_{y},k_{z=0})=P(k_{\rho},\beta)=E_{0}(-1)^{l+1}\pi{w_{0}}^{2}e^{il( \beta+\frac{\pi}{2})}\exp\left(-\frac{({k_{\rho}}^{2}+\gamma^{2}){w_{0}}^{2}} {4}\right)I_{l}(\frac{k_{\rho}\gamma{w_{0}}^{2}}{2})\] (A3)  
\(I_{l}(x)\) is the modified Bessel function of order \(l\). For the details of the derivation process, see [40]. For a beam on an arbitrary plane, the expression of the electric field changes to  
\[\mathbf{E}(x,y,z)=\mathbf{E}(\alpha,\beta)=\frac{1}{4\pi^{2}}k^{2}\int\limits_ {0}^{2\pi}\int\limits_{0}^{\pi/2}\mathbf{e_{u}}P(k_{\rho},\beta)e^{i\mathbf{k }\cdot\mathbf{r}}\sin\alpha\cos ad\beta\] (A4)  
Taking the beam position into account, i.e., the beam's location is (\(x_{0}\), \(y_{0}\), \(z_{0}\)) instead of (0, 0, 0). Correspondingly, the above equation becomes  
\[\mathbf{E}(\alpha,\beta)=\frac{1}{4\pi^{2}}k^{2}\int\limits_{0}^{2\pi}\int \limits_{0}^{\pi/2}\mathbf{e_{u}}e^{i\mathbf{k}\cdot\mathbf{r}}P(k_{\rho}, \beta)e^{i\mathbf{k}\cdot\mathbf{r}_{0}}\sin\alpha\cos ad\beta\] (A5)  
In Equation (A5), \(r_{0}=\sqrt{z_{0}^{2}+\rho_{0}^{2}}\), \(\rho_{0}=\sqrt{{x_{0}}^{2}+\[g_{n,TE}^{m,u}=\left\{ \begin{array}{ll}\frac{-in(n+1)}{\pi^{+}(2n+1)}p_{mn}^{u} \quad\quad m\geq 0\\ \frac{1}{(-1)^{m|}}\frac{(n+m)!}{(n-m)!}\frac{-in(n+1)}{!^{n+1}(2n+1)}p_{mn}^{u} \quad m<0\end{array} \right. \tag{11}\]  
In the simulation part, y, right circular, and radial polarization are discussed. Therefore, the derivation results of the beam shape coefficients for these three polarization states are given. 1. Y polarization \[\left\{ \begin{array}{ll}p^{p}{}_{mn}\\ q^{q}{}_{mn}\end{array} \right\}=I\left\{\int_{0}^{\pi/2}\exp(-\frac{(k_{p}^{2}+ \gamma^{2})w^{2}}{4})I_{l}(\frac{k_{p}\gamma w^{2}}{2})\times\left[\cos a\left\{ \begin{array}{ll}\pi_{nm}(\cos a)\\ \pi_{mn}(\cos a)\end{array} \right\}I_{-}-i\left\{ \begin{array}{ll}\pi_{mn}( \cos a)\\ \pi_{mn}(\cos a)\end{array}