## 3 Simulations and Discussion  
To verify the performance of the proposed 3D PASL, finite element method (FEM) simulations are performed using Comsol Multiphysics 5.2a software. In our simulation, the power of the incident light is 80 mw, the free space wavelength of the incident light is 1310 nm and the dielectric constant of gold at this wavelength is \(\varepsilon_{m}\)= \(-68.53+7.4542i\)[31]. Consequently, the wavelength of the SPP that is launched at the Au-PBS interface is \(\lambda_{spp}=969\)nm, which is calculated from \(\lambda_{spp}=\lambda_{0}\sqrt{\left(\varepsilon_{m}^{\text{Re}}+\varepsilon_{ d}\right)/\left(\varepsilon_{m}^{\text{Re}}\varepsilon_{d}\right)}\), where \(\varepsilon_{m}^{\text{Re}}\) represents the real part of \(\varepsilon_{m}\) and the dielectric constant of the ambient PBS buffer solution is \(\varepsilon_{d}=1.7408\). Figure 1: Schematics of the geometry of a 3D plasmonic Archimedes spiral lens from (**a**) the _side_ view and (**b**) the _top_ view. Cross-sectional intensity distributions of the 3D PASL under normally incident excitation with (**c**) left-hand and (**d**) right-hand circular polarization, where the insets on the upper right represent the corresponding intensity distributions from the top view. FEM simulation parameters: illumination wavelength \(\lambda=1310\) nm, \(r_{0}=1400\) nm, \(gap=800\) nm, _thickness_ = 300 nm, _width_ = 200 nm, and \(h_{0}=625\) nm. The above equations for the 3D AS indicate that the main factor that affects the focal length of the proposed 3D structure is the rise height \(h_{0}\) of the 3D PASL, which can be expected to influence the generated SP near-field. Therefore, we focus here on discussing the effects of this factor on the focal length under excitation by light with left-hand circular polarization. Through numerical analysis, we found that, compared with parameters such as _gap_ and \(r_{0}\), \(h_{0}\) has the most obvious effect on the focal length. We therefore fixed the parameters _gap_, \(r_{0}\) and _width_ at 800 nm, 1400 nm and 200 nm, respectively, and then gradually changed the rise height \(h_{0}\) between successive turns of the spiral from 575 nm to 725 nm in steps of 50 nm. Figure 2a-d shows the intensity distributions (\(\,\mid\,\)E\(\,\mid\,\)\({}^{2}\)) for different \(h_{0}\) values under illumination by incident light with left-hand circular polarization. It was found that the focus position floats above the funnel shape and moves further from the bottom of the funnel with increasing \(h_{0}\). Because the 3D spiral groove rises simultaneously, this also causes the focus to rise. Unlike its counterpart 2D structure composed of three concentric annular rings, a slight asymmetry in the intensity is expected to be observed when the damping of the SPP in both media and the unequal propagation distances from the spiral slot to the focus over the different azimuthal angles are taken into account. The focusing characteristics of the 3D PASL are depicted in Figure 3. The figure shows that the focal length is elongated by 656 nm (from 2765 nm to 3421 nm) when \(h_{0}\) increases by 150 nm. Figure 3b indicates that there is no obvious change in the value of the full width at half maximum. When the type of incident light is switched to plane waves with right-hand circular polarization, a plasmonic vortex field is generated around the origin of the structure that fits the second Bessel function of the first kind, as shown in Figure 4. In addition, when \(h_{0}\) changes, the position of the plasmonic vortex varies with a similar trend to that of the focus under excitation by light with left-hand circular polarization. This longitudinal parameter \(h_{0}\) thus makes the 3D PASL more flexible and the focal point or plasmonic vortex can therefore be adjusted in the longitudinal direction.