## 3 Simulations and Discussion
As a result,  
Figure 2: Normalized E-field at the \(Y\)-\(Z\) plane under excitation by left-hand circularly polarized light with \(gap=800\) nm for (**a**) \(h_{0}=575\) nm, (**b**) \(h_{0}=625\) nm, (**c**) \(h_{0}=675\) nm, and (**d**) \(h_{0}=725\) nm. The insets on the upper right represent the corresponding intensity distributions from the top view. the proposed 3D PASL is capable of creating a multi-functional application that will represent an aggressive expansion from the capabilities of the 2D PASL. We also investigate the feasibility of trapping and rotation of microparticles using the 3D PASL, as compared with the feasibility of using the previous 2D PASL. It is assumed that the microparticle to be used is a polystyrene sphere with a diameter of 1 \(\upmu\)m that is lying on the focal plane. Because of the three-dimensional structure of the PASL, we also analyzed the optical force acting in the \(z\)-direction, which helps to overcome the problem of particle manipulation on the surface when using the 2D PASL. The Maxwell stress tensor \(\hat{T}_{ij}\) is used to calculate the optical force (F) and the optical trapping potential based on Equations (2)-(4) below. An integral stress tensor is applied over the entire spherical surface to obtain the optical force acting on the microsphere. The optical trapping potential is given by integrating the optical force acting on the microsphere along a specific direction. Figure 4: Normalized E-field at the Y-Z plane under right-hand circularly polarized light excitation with \(gap=800\) nm when (**a**) \(h_{0}=575\) nm, (**b**) \(h_{0}=625\) nm, (**c**) \(h_{0}=675\) nm, and (**d**) \(h_{0}=725\) nm. Figure 3: Focusing properties of 3D PASL with left-hand circularly polarized light excitation. The intensity profiles are shown (**a**) along the positive \(z\)-axis and (**b**) on the focal plane of the structure with gradually increasing values of \(h_{0}\). \[\hat{T}_{ij}=\left[\epsilon E_{i}E_{j}+\mu_{0}H_{i}H_{j}-\frac{1}{2}\Big{(}\epsilon E ^{2}+\mu_{0}H^{2}\Big{)}\delta_{ij}\right] \tag{2}\]  
\[F=\oint_{S}n\hat{T}_{ij}dS \tag{3}\]  
\[U=\int Fdl \tag{4}\]  
Here, \(n\) is the unit vector in the normal direction of the closed surface that surrounds the target space. When the sphere moves on the \(x\)-axis in the focal plane, the optical forces that are exerted on the microsphere in the \(x\)-direction (\(Fx\)) and the \(y\)-direction (\(Fy\)) are as shown in Figure 5a. Obviously, \(Fx\) is much stronger than \(Fy\), and \(Fx\) plays a major role in pulling the microsphere back towards the center of the focus. When the sphere moves on the \(y\)-axis in the focal plane, Figure 5c indicates that the optical force exerted on the microsphere in the \(y\)-direction (\(Fy\)) plays a major role and also pulls the microsphere back towards the center of the focus. In the vertical direction, as shown in Figure 5e, when the microsphere is below the focal point, the optical force that is exerted on the microsphere is positive along the \(z\)-axis, while the optical force that is exerted on the microsphere is negative along the \(z\)-axis when the microsphere is located above the focal point; this also means that the stable position of such a 1-\(\upmu\)m-diameter microsphere is the focus. These results are consistent with the principle of optical tweezers, in either the lateral focal plane or the vertical focal plane. The calculated trapping potentials in the \(x\)-direction and the \(y\)-direction are shown in Figure 5b,d, respectively. Normally, we consider \(10k_{B}T\) as the threshold for stable optical trapping [15], where \(k_{B}\) is the Boltzmann constant and \(T\) is the thermodynamic temperature. We can show here that the potential well is about \(250k_{B}T\) in depth, which means that the formed plasmonic focus can capture a polystyrene sphere with a diameter of 1 \(\upmu\)m stably when the power of the incident light is 80 mw. So conventional lasers can generate sufficient optical force for stable optical trapping.