## 3 Simulations and Discussion
Next, we change the incident light to a right-hand circularly polarized plane wave and calculate the resulting optical force and potential. When a 1-\(\upmu\)m-diameter microsphere leaves the origin to travel from \(-1.5\)\(\upmu\)m to \(1.5\)\(\upmu\)m along the \(y\)-axis/\(x\)-axis, the optical forces that are exerted on the microsphere are as shown in Figure 6a-c. We found that when the microsphere is located outside the plasmonic \"doughnut\", the optical forces will drag the microsphere into the ring. Note here that because of the asymmetry of the AS, the plasmonic vortex field is not a completely uniform ring. Therefore, the curves of Fx and Fy are not symmetrical with \(x\)=0 or \(y\)=0. In addition, as shown in Figure 6b,d, when the plasmonic vortex is generated by excitation with right-hand circularly polarized light, the depth of optical potential is greater than \(30k_{B}T\), i.e., the plasmonic vortex can trap the nanoparticle within the hollow of the doughnut shape. Normally, spherical microparticles are commonly used as the objects to be captured and manipulated in optical trapping experiments. In addition, _E. coli_ bacteria, cubic crystals, and ring particles can also be manipulated. Another very interesting nonspherical shape for use in these experiments is a cylinder, such as micro-nano metal rods or carbon nanotubes. These nonspherical particles provide more dynamic information than spherical particles, particularly in the case of optical vortex trapping. Therefore, we also studied the optical torque acting when a cylindrical micro-rod is lying at the center of the 3D PASL at a rotation angle \(\theta\). As is shown in Figure 7, the cylindrical micro-rod is in the \(x\)-\(z\) plane and rotated around the \(y\)-axis. Here, the rotation angle \(\theta\) is defined as the angle between the main axis of the cylinder and the \(z\)-axis, i.e., the angle between the dotted line and the \(z\)-axis in Figure 7. The parameters of the 3D-PASL in this case are \(h_{0}=625\) nm and \(gap=800\) nm. The cross-sectional radius of the cylinder is 200 nm, and its height is 1400 nm. The cylinder is assumed lying at the center of the focus (under left-hand circularly polarized excitation) or the plasmonic vortex (under right-hand circularly polarized excitation). The optical torque \(T\) that is exerted on the cylinder has also been calculated by integrating the Maxwell stress tensor \(\hat{T}\) over a surface that encloses the object under study, as follows:  
Figure 3.0  
Figure 3.1 Optical trapping force exerted on a microsphere under left-hand circularly polarized light excitation. Please note that the focus is defined as the origin here. (**a**) Calculated optical trapping force in the \(x\) and \(y\) directions when the microsphere moves on the \(x\)-axis. (**b**) Calculated trapping potential when the microsphere moves on the \(x\)-axis. (**c**) Calculated optical trapping force in the \(x\) and \(y\) directions when the microsphere moves on the \(y\)-axis. (**d**) Calculated trapping potential when the microsphere moves on the \(y\)-axis. (**e**) Calculated optical forces in the \(x\), \(y\) and \(z\) directions when the microsphere moves on the \(z\)-axis. (**f**) Calculated trapping potential in the \(z\)-direction. Figure 8b, we found that the optical torque in the \(y\)-direction reached a maximum when the rotation angle was approximately \(\pm 45^{\circ}\), and the optical force then tends to rotate the rod towards the horizontal direction. However, the rotation angle at the maximum torque is different under the left-hand circularly polarized excitation conditions, this is because of the different beam waists of the focal point and the plasmonic vortex.