## Plasmonic Archimedes Spiral Lens  
Normally, the geometry of an Archimedes spiral (AS) is expressed using polar coordinates as \(r(\phi)=r_{0}+(\phi*gap)/2\pi\), where \(r\) is the distance from the AS to the origin O, \(r_{0}\) is the initial spiral radius, \(\phi\) is the azimuthal angle in radians, and _gap_ represents the horizontal distance between successive turns of the spiral. Here, we extrude the two-dimensional Archimedean spiral into three dimensions along the \(z\)-axis using the relation \(z=h_{0}*\phi/2\pi\), where \(h_{0}\) is the rise height of the 3D AS when \(\phi\) grows by \(2\pi\). Obviously, if \(h_{0}=0\), then the 3D structure reverts to its 2D counterpart. In this manuscript, we design a 3D PASL, as shown in Figure 1. The 3D PASL is fabricated on a silicon dioxide substrate that has a funnel-shaped depression, which is plated with a 300-nm-thick gold film. A 3D Archimedes spiral slot is then engraved by swirling on the inner sidewall of the funnel-shaped gold film. Please note that both the bottom of the silicon dioxide substrate and gold film are closed to avoid the directly transmitted light interfering with the plasmonic field. The ambient medium in the funnel and above the gold film is buffer solution (such as phosphate buffer saline, abbreviated PBS, a water-based pH-adjusted blend of ultrapure-grade phosphate buffers and saline solutions. The contents of PBS buffer are determined by the requirements of concentration of salt in the buffer and pH in practical application. The pH of most common composition of PBS is ~7.4. ), which is feasible to represent the real cell trapping environment. The slot is approximately 200 nm wide and starts at \(r_{0}=1.4~{}\upmu\)m, with its azimuthal angle \(\phi\) ranging from 0 to \(6\pi\), i.e., the completed helix has three turns. When the circularly polarized light propagating along the \(z\)-axis illuminates the 3D PASL from the bottom of the SiO\({}_{2}\) substrate, the incident light wave is coupled to the SPP modes at the entrances to the AS slot. SPPs, as evanescent waves, propagate along the radial direction, either inwards or outwards, to the center of the PASL. Because of the decoupling effect of the 3D spiral slots, some of the SPPs are scattered into free space and are then converted into propagating waves when they encounter other out-of-plane spiral grooves. This is the main point by which our 3D polycyclic AS structure is quite different from its 2D counterpart. These propagating waves interfere and produce high-intensity focal spots or plasmonic vortices at the center of the structure, as shown in Figure 1c,d, which will be discussed in the next section. In this paper, the optical excitation of this PASL involves the use of left-hand or right-hand circularly polarized plane waves, which carry spin angular momentum \(s=1\) or \(s=-1\), respectively. Because of the complexity of the three-dimensional structure, we changed the traditional two-dimensional Archimedes spiral electric field distribution equation [30] and we only study the electric field distribution on the central axis of the funnel. When illuminated by light with left-hand circular polarization, the SP field generated along the \(z\)-direction on the central axis of the structure can be expressed in cylindrical coordinates as:  
\[E_{Z}(h)=E_{0}\int e^{i(\phi+\frac{\phi+h_{0}}{\lambda_{d}})}e^{-ik_{ SPP}\cdot\cos\theta\cdot r_{d}}\sqrt{r^{2}(\phi)+\frac{gap^{2}+h_{0}^{2}}{4\pi^{2}}}d\phi \tag{1}\]Here, \(h\) represents the height of a specified point A on the central axis of the structure, \(k_{spp}\) is the wavenumber of the generated SPP, \(r_{d}\) is the distance from a point on the spiral slot to the point A on the central axis of the structure, and \(\lambda_{d}\) is the wavelength of the incident light in the SiO\({}_{2}\). \(k_{spp}\cos\theta\) is a component of the SPP wave vector and \(\phi h_{0}/\lambda_{d}\) is the additional phase caused by the increase in the spiral's height.