## 2 Theory  
The scattering physical model and spatial relation are shown in Figure 1a, and the measurement environment is depicted in Figure 1b. The beam coordinate system and target coordinate system are represented by \(o_{i}(x_{i},y_{i},z_{i})\) and \(o(x,y,z)\), respectively. To illustrate the characteristics of the Bessel vortex wave, the amplitude and phase profiles of an \(x\)-polarized electric field distributed along a latitude line on a sphere with a radius of 150 mm are shown in Figure 2. The amplitude of the \(x\)-polarized component was much larger than that of the cross-polarized \(y\)-component. Furthermore, a small longitudinal \(z\) component field appeared with an increase in the propagation distance. What is more noteworthy is that the amplitude, phase, and polarization of the incident Bessel vortex electric field were different at each selected facet element, which led to its inapplicability for the traditional PO method in scattering. Therefore, the ASE combined with PO is an effective approach to address this problem. Figure 1: Physical model of the Bessel wave and sphere (**a**). Measurement scene of the scattering near field in the anechoic chamber (**b**). For any triangular facet element _j-th_, the normal unit vector can be expressed as  
\[\hat{n}_{j}=\frac{p_{j3}^{\rightarrow}p_{j1}\times p_{j2}^{\rightarrow}p_{j1}}{ \left|p_{j3}^{\rightarrow}p_{j1}\times p_{j2}^{\rightarrow}p_{j1}\right|} \tag{1}\]  
where \(p_{j1},p_{j2},p_{j3}\) are the three vertex coordinates of the _j-th_ triangular facet element. The angle between the incident direction of any decomposed plane wave \(\overset{\rightarrow}{k_{1}^{\prime}}\) and the normal direction vector \(\hat{n}_{j}\) is defined as:  
\[\theta_{j}=\arccos(\frac{\overset{\rightarrow}{k_{1}^{\prime}}\cdot\hat{n}_{ j}}{\left|\overset{\rightarrow}{k_{1}^{\prime}}\cdot\hat{n}_{j}\right|}) \tag{2}\]  
The expressions \(\hat{e}_{1j}=\hat{e}_{2j}\times\overset{\rightarrow}{k_{1}^{\prime}}\Big{/} \Big{|}\hat{e}_{2j}\times\overset{\rightarrow}{k_{1}^{\prime}}\Big{|},\hat{e} _{2j}=\overset{\rightarrow}{k_{1}^{\prime}}\times\hat{n}_{j}\Big{/}\Big{|} \overset{\rightarrow}{k_{1}^{\prime}}\times\hat{n}_{j}\Big{|}\) are the parallel and perpendicular polarized unit vector at each principal plane. Taking the ideal conductor as an example, according to the boundary conditions of \(\hat{n}\times\overset{\rightarrow}{E}^{\rightarrow}=0\), the expression of the scattered electric field of the decomposed plane wave on the _j-th_ facet can be obtained as [20]:  
\[ \begin{split}\overset{\rightarrow}{E}_{s}^{pw_{j}}& =\frac{ikq_{0}}{4\pi}\frac{\exp(-ikr)}{r}\hat{r}\times\int\limits_{ s1}2(\hat{n}\times H_{\text{oam}}^{i})\times\hat{r}\exp(ik\hat{r}\cdot r^{ \prime})ds\end{split}  \tag{3}\] \[=-\frac{ik}{2\pi}\frac{\exp(ikr)}{r}A_{PW}\hat{r}\times\Big{\{} \hat{r}\times[\hat{f}_{oam}\cdot\hat{e}_{2j}\cos\theta_{j}\hat{e}_{2j}+\hat{f} _{oam}\cdot\hat{e}_{1j}(\hat{n}\times\hat{e}_{2j})]\Big{\}}\\ &\times\int\limits_{s1}\exp[ik(\hat{k}_{1}^{\prime}-\hat{r}) \cdot\overset{\rightarrow}{r}^{\prime}]ds^{\prime}\end{split}\]  
where \(A_{PW}=E_{0}\exp[il\text{arctan}(k_{yi}/k_{xi})]/(i^{n}k\sin\alpha_{0})\) and \(\hat{f}_{oam}=[1,0,0]\) are the complex amplitude and _x_-polarization of the incident Bessel wave, \(\alpha_{0}\) is the half cone angle, and \(k_{yi}=k\cdot\sin\alpha_{0}\sin\phi,k_{xi}=k\cdot\sin\alpha_{0}\cos\phi\). Therefore, the scattering solution of the Bessel wave upon the target can be characterized as the integral summation of all scattered fields from the decomposed plane waves:  
\[\overset{\rightarrow}{E}_{s}^{\rightarrow\alpha am}=\iint\limits_{k_{\rho}} \sum\limits_{j=1}^{N_{x}}\overset{\rightarrow}{E}_{s}^{pw_{j}}Q(k_{x},k_{y}) dk_{x}dk_{y} \tag{4}\]  
Figure 2: Amplitude (**a**) and phase (**b**) of the incident field at different facet elements. where \(Q(k_{x},k_{y})=\cos\theta_{j}+k_{x}/k_{y}\sin\theta_{j}\) is generated by the conversion of \(dk_{x}\) and \(dk_{y}\) in coordinate systems.