## 2 Theoretical Background  
### GLMT of a Sphere Illuminated by an Arbitrarily Staped Beam  
According to the Generalized Lorenz-Mie theory, the incident and scattered fields of a spherical particle irradiated by a shaped beam can be expressed by the vector spherical wave functions (\(\textbf{M}^{(1)}_{nm}(\textbf{kr})\), \(\textbf{N}^{(1)}_{nm}(\textbf{kr})\)) multiplied by the beam shape coefficients (\(g^{m}_{n,TE}\), \(g^{m}_{n,TM}\)) [42; 43]:  
\[E_{i}=\sum_{n=1}^{\infty}\sum_{m=-n}^{n}c^{pw}_{n}[g^{m}_{n,TM}\textbf{N}^{(1) }_{nm}(\textbf{kr})+ig^{m}_{n,TE}\textbf{M}^{(1)}_{nm}(\textbf{kr})] \tag{1}\]  
\[H_{i}=-\frac{ik}{\omega}\sum_{n=1}^{\infty}\sum_{m=-n}^{n}c^{pw}_{n}[g^{m}_{n, TM}\textbf{M}^{(1)}_{nm}(\textbf{kr})+ig^{m}_{n,TE}\textbf{N}^{(1)}_{nm}( \textbf{kr})] \tag{2}\]  
\[E_{s}=\sum_{n=1}^{\infty}\sum_{m=-n}^{n}c^{pw}_{n}[A^{m}_{n}\textbf{N}^{(3)}_{ nm}(\textbf{kr})+i^{m}_{n}g^{m}_{n,TE}\textbf{M}^{(3)}_{nm}(\textbf{kr})] \tag{3}\]  
\[H_{s}=-\frac{ik}{\omega}\sum_{n=1}^{\infty}\sum_{m=-n}^{n}c^{pw}_{n}[A^{m}_{n} \textbf{M}^{(3)}_{nm}(\textbf{kr})+i^{m}_{n}\textbf{N}^{(3)}_{nm}(\textbf{kr})] \tag{4}\]  
where \(k=2\pi/\lambda\) is the wavenumber and  
\[c^{pw}_{n}=(-i)^{n+1}\frac{2n+1}{n(n+1)} \tag{5}\]\(A_{n}^{m}\) and \(B_{n}^{m}\) in Equations (3) and (4) are the expansion coefficients of the scattered field defined as:  
\[\left\{ \begin{array}{l}A_{n}^{m}=a_{n}g_{n,TM}^{m}\\ B_{n}^{m}=b_{n}g_{n,TE}^{m}\end{array} \right. \tag{6}\]  
\(a_{n}\) and \(b_{n}\) are Mie scattering coefficients and can be derived based on the boundary conditions. The boundary conditions for a uniform and isotropic dielectric sphere, and a charged sphere were obtained in [44], and [45], respectively. Therefore, the specific expressions of their Mie scattering coefficients were derived. For a homogeneous and isotropic dielectric sphere, the Mie scattering coefficients are  
\[\left\{ \begin{array}{l}a_{n}=\frac{j_{n}(x)[mx_{h}(mx)]^{\prime}-m^{2}j_{h}( mx)[x_{h}^{\dagger}(x)]^{\prime}}{j_{h}^{(1)}(x)[mx_{h}(mx)]^{\prime}-m^{2}j_{h}( mx)[x_{h}^{\dagger}(x)]^{\prime}}\\ b_{n}=\frac{j_{n}(x)[mx_{h}(mx)]^{\prime}-j_{h}(mx)[x_{h}(x)]^{\prime}}{j_{h} ^{(1)}(x)[mx_{h}(mx)]^{\prime}-j_{h}(mx)[x_{h}^{\dagger}(x)]^{\prime}}\end{array} \right. \tag{7}\]  
and for a charged sphere,  
\[\left\{ \begin{array}{l}a_{n}=\frac{[x_{h}(x)[mx_{h}(mx)]^{\prime}-m^{2}x_{h }(mx)[x_{h}(x)]^{\prime}-i\omega_{1}\omega_{2}\omega_{3}^{\prime}(mx)]^{ \prime}[x_{h}(mx)]^{\prime}/k}{[x_{h}^{(1)}(x)[mx_{h}(mx)]^{\prime}-m^{2}x_{ h}(mx)[x_{h}^{\dagger}(x)]^{\prime}-i\omega_{1}\omega_{2}\omega_{3}[mx_{h}(mx)]^{ \prime}[x_{h}^{\dagger}(x)]^{\prime}/k}\\ b_{n}=\frac{[mx_{h}(mx)[x_{h}(x)]^{\prime}-mx_{h}(x)[mx_{h}(mx)]^{\prime}+i \omega_{1}\omega_{2}\omega_{3}^{\prime}(mx)]/k}{[mx_{h}(mx)][x_{h}^{\dagger }(x)]^{\prime}-mx_{h}^{\dagger}(x)[mx_{h}(mx)]^{\prime}+i\omega_{1}\omega_{2} \omega_{3}^{\prime}[mx^{2}j_{h}(mx)h_{h}^{\dagger}(x)]/k}\end{array} \right. \tag{8}\]  
where \(m\) is the complex refractive index of the sphere relative to the surrounding vacuum environment. Also, \(\sigma_{s}\) is the conductivity of a sphere carrying electric charges [45; 46]:  
\[\sigma_{s}\approx\left[\frac{x}{2}\frac{\omega_{s}^{2}}{\omega^{2}+\gamma_{s} ^{2}}\Big{(}-1+i\frac{\gamma_{s}}{\omega}\Big{)}\cdot k\right]/i\omega\mu_{0} \tag{9}\]  
and \(\omega_{s}\) is the plasma frequency derived by \(\omega_{s}^{2}=\frac{2q\varphi}{m_{e}a^{2}}\), \(m_{e}\) and \(q\) are the mass and charge of an electron, respectively, i.e., \(m_{e}\) = 9.109 \(\times\) 10\({}^{-31}\) kg and \(q\) = 1.602 \(\times\) 10\({}^{-19}\) C. \(\varphi=\frac{q}{4\pi\varepsilon_{0}a^{2}}\) is the surface electrostatic potential of a charged sphere, \(a\) is the radius of the sphere, and \(\varepsilon_{0}\) is the permittivity of the vacuum.