## 3 Numerical Results  
The intensity patterns of different components of both the evanescent wave and the propagating wave are shown in Figures 2 and 3. The intensity patterns of \(x\), \(y\), and \(z\) components of the evanescent wave and propagation wave for different propagation distances are shown in Figure 2. Figure 3 shows the corresponding intensity patterns for different SoP arrangements with variant radial periods (i.e., different \(r_{0}\) in this work). The evanescent wave part dominates the optical field in near field, and is closely related with the initial SoP alignment, as shown in Figures 2 and 3. The evanescent wave will spread out quickly and fade away with increasing propagation distance, as expected. The patterns of the longitudinal components are closely related to the initial SoP arrangements of the RVVB. In particular, the longitudinal components are not negligible in the near field that results in the variation of the total intensity patterns as shown in Figures 2 and 3. The patterns of the longitudinal component rotate with different radial periods of SoP arrangement (i.e., different \(r_{0}\) in this work). Here, we would like to discuss the underlying physics of these phenomena. The radially variant polarization can be regarded as the superposition of the \(x\)- and \(y\)-polarization components with different initial phases, as recognized from Equation (1). The intensity patterns of the x and y components with different SoP are similar but different in phase, as recognized from Equations (7) and (8) and as shown in Figures 2 and 3. On the other hand, the different \(x\)- and \(y\)-component phases will contribute to the variation of the longitudinal component, as recognized from Equation (9). Therefore, the different SoP arrangements result in the different profiles of the longitudinal component, as shown in Figure 3. The manipulation of the longitudinal component shapes [25; 26] has become an essential approach  
Figure 2: (Color online) Intensity patterns of \(x\)-, \(y\)- and \(z\)- components of the evanescent wave and propagating wave in the \(x\)-\(y\) plane for different propagation distances with \(w=0.1\lambda\), \(r_{0}=0.5\lambda\): (a) the evanescent wave; (b) the propagating wave. Upper plots in (**a,b**) transverse (\(x\) or \(y\)) component; lower plots in (**a,b**) \(z\) component. All the images have the same dimension of \(2\times 2\;\lambda^{2}\). because of the fundamental physics and the potential applications for super-resolution, plasmonic beaming, nanofabrication, and 3D imaging, etc. Obviously, these results indicate the longitudinal component patterns can be controlled by the initial SoP arrangement using the above analysis. It is interesting to examine the evolution of the optical SAM-FD in the near field with different SoP arrangements. In particular, the longitudinal component of a vector beam cannot be neglected in the near field, which results in the transverse components of the optical SAM-FD (i.e., \(S_{\mathrm{x}}\), \(S_{\mathrm{y}}\)). The 3D SAM-FD paves a new way for potential applications in the corresponding fields such as manipulation of micro-particles and atoms, nanophotonics, and quantum processing. Hereafter, the optical SAM-FD has been normalized to its initial peak value. It is clearly seen from Figure 4 that the SAM-FD depends on the SoP arrangements. In addition, the transverse (\(x\)- or \(y\)-) components of the SAM-FD distribution (both evanescent wave and propagating wave) break up into two peaks with opposite values. On the other hand, the \(z\)-component of the SAM-FD shape remains the Gaussian profile, as expected.