1. Introduction

Optical tweezers has been established as a critical and non-invasive optical force tool applied in a multitude of fields [1–5], including bio-medicine [6,7], quantum optics [8–10], soft condensed matter physics [11,12], and the burgeoning lab-on-a-chip technology [13–17]. Based on evanescent fields generation by dielectric or metallic structures, the near-field trapping method surpasses the diffraction limit of traditional far-field optics and elicits an optical gradient force that draws particles towards the surface [18–20]. Recent advancements have suggested the potential for nanoscale optical trapping through the implementation of focused radially polarized beams on thin metal films [21] and the utilization of plasmonic antennas and apertures [22–24]. However, these methods often require highly focused laser beams or possess limited functional capabilities.

The chiral plasmonic nanostructure, being absence of mirror symmetry, exhibits a robust chiral-optic response to circularly polarized light and is commonly utilized for generating local fields to own wide applications in areas of optical manipulation, quantum communication or optical data storage etc [25–27]. The Archimedean helix presents an ideal shape for the design of chiral nanostructures, having been widely utilized in near-field studies of on-chip vortex generation with geometry-dependent optical angular momentum (OAM) in micron dimensions [26,28–30]. On the opposite, the perfect on-chip focusing property generated by the Archimedean metal disk is rarely investigated, but exploring this property potentially owns the tremendous advantage in realizing selectively on-chip trapping due to spin-orbit interaction.

Here, we numerically propose a metal disk with Archimedean spiral structure to aim at realizing selective trapping of nanoparticles under the excitation of 532 nm circularly polarized (CP) light. Due to the spin-orbit interaction, the high-quality plasmonic hotspot generated on the metal disk originates from the scattering of incident CP light around the geometric shape of Archimedean disk [25,26]. The near-field plasmonic field on the metal disk can be designed by adjusting the geometric parameter flexibly. The radius and screw pitch for the metal disk are optimized to realize the switch modulation of hotspot field generation. The optical force and potential well exerted on the dielectric and metal nanoparticles by the locally excited plasmonic field are numerically analyzed, which illustrates the switchable optical trapping ability by the proposed metal disk in the time and spatial domain. Specially, the optical trapping effect for particles with different sizes are analyzed to examine the underlying dipole and quadrupole mechanisms, demonstrating the inherent role of light-matter interaction in achieving near-field optical trapping.

2. Main text

2.1 Origin of selective trapping force

The principle of plasmonic trapping by the Archimedean metal disk based on the spin-hall interaction is illustrated in Fig. 1. Figure 1(a) gives the schematic of a left-handed circularly polarized (LCP) light normally incident onto the metal disk along + z direction. Due to the function of spin-hall effect, the excited intensity distribution of hotspot and the corresponding optical potential on the disk are depicted in Fig. 1(a). The metal disk is designed with right rotation on the silica substrate for generating the enhanced hotspot field by the excitation of LCP light. The geometric profile of Archimedean metal disk is designed by the formula $r(\varphi )= {r_0} + \varphi \cdot p/2\pi $ in the polar coordinate system, where ${r_0}$, p are the starting radius and pitch of Archimedean metal disk, and the height of the metal disk is labeled as h shown in Fig. 1(a). Here the top line demonstrates the one-dimensional characteristic trapping potential well U of the hotspot, activated by the excitation of circularly polarized light. It exemplifies the unique optical trapping effect intrinsic to the designed metal disk. To satisfy the plasmon focusing condition [26], the pitch p of designed Archimedean metalens is determined by the permittivity of surrounding medium and wavelength of incident light. Figure 1(b) shows the variation curve between the pitch p and refractive index of the medium under the incident light with wavelength 532 nm, it indicates that the metal disk surrounded in the medium with higher permittivity needs a shorter pitch.

 

Fig. 1. (a) Schematic of selective trapping particles by the excited plasmonic hotspot under the incidence of circular polarization light on the Archimedean metal disk. A left-hand polarized light is incident vertically in z- direction onto the gold metal structure. The radius, screw pitch and thickness of the metal disk are labelled as ${r_0}$, $\textrm{p}$ and $\textrm{h}$ separately. The trapping potential well U of hotspot for the microparticles is shown on the metal disk on the silica substrate. (b) The setting of screw pitch p varies with refractive of surrounding media when the wavelength of incident light is 532 nm. (c, d) The amplitude distribution and (e, f) phase diagram with topological charge $l = 0$ and $l = 2$ of the electric field on the Archimedean metal disk under the excitation of left-handed and right-handed circularly polarized light, respectively.

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Based on the spin-orbit coupling effect under the incidence of circular polarized light, the topological charge of plasmonic field on the metal disk contributes from the sum of the geometric phase and spin angular momentum. Among them, the integral geometrical phase along the edge of metal disk leads to is $2m\mathrm{\pi },\; m$ is defined by $m = \frac{p}{{{\lambda _{spp}}}}$, with ${\lambda _{spp}} = {\lambda _0}\sqrt {{\varepsilon _d} + {\varepsilon _m}/{\varepsilon _d}{\varepsilon _m}} $ is the wavelength of surface plasmonic polaritons (SPP) on the surface of disk, where ${\varepsilon _d}$ and ${\varepsilon _m}$ are the permittivity of dielectric medium and metal separately [26]. The resonant plasmonic hotspot condition is ${\phi _{\textrm{total}}}\; = 2\mathrm{\pi }({m + {\sigma_ \pm }} )$, in our case it should be 0, thus m = -${\sigma _ \pm }$. Here, ${\sigma _ + } ={+} 1$ denotes the spin quantum number of incident light with right-handed circular polarization (RCP), and ${\sigma _ - } ={-} 1$ for left-handed circular polarization (LCP). As the same convention, the designed metal disk with right rotation corresponds to positive topological charge (m = 1) [26,30]. For satisfying above condition, it requires $p = {\lambda _{spp}} = 484\; \textrm{nm}$ under the LCP incident light with the wavelength of 532 nm, which ${\varepsilon _d} = 1$, $\; {\varepsilon _m} ={-} 5.6 + 2.2i$, ${r_0} = 300\; \textrm{nm},{\; \; \textrm{and}}\; \; h = 200\; \textrm{nm}$ are adapted in Fig. 1(c-f).

Here, all data are calculated by three-dimensional finite-difference time-domain (FDTD) simulations (Lumerical FDTD Solutions). In the FDTD model, the minimum grid size is set as 2 nm in the region near material interfaces, and perfectly matched layers are placed around the whole simulation area. For demonstrating the selective hotspot excitation by different chiral light, Figs. 1(c-f) show the amplitude and phase distribution of the excited electric field on the designed Archimedean disk. The gold disk is chosen since it is an advantageable plasmonic material at the visible light wavelength, and is chemically stable in the solution environment required in the potential application of optical tweezers.

The topological charge of generated plasmonic field on the metal disk is calculated to be equal to multiple of 2π that the phase varies in one full circle around the central phase singularity as shown in the Figs. 1(e) and 1(f). It can be seen that the spiral structure can focus the incident LCP light into focused hotspot field with topological charge $l = 0$ at the center of the metalens, as the electric field and phase distribution indicating in Figs. 1(c, e). Comparatively, an imperfect plasmonic vortex with $l = 2$ is excited by the RCP incidence light as shown in Figs. 1(d, f). And it can be seen that the maximum intensity of hotspot in Fig. 1(c) is much higher than that of plasmonic vortex in Fig. 1(d). The simulation results verify that the near field generation by the Archimedean disk is controllable between the plasmonic hotspot and imperfect vortex by adjusting the CP of incident light.

To investigate the stating radius on the influence of electric field distribution on the Archimedean metal disk, which is related with its selective trapping ability of nanoparticles, Figs. 2(a-f) show the intensity distribution variation of electric field with different radius ${r_0} = {100\textrm{nm},\; 150\textrm{nm},\; 200\textrm{nm}}$ under the excitation of LCP, RCP incident light, respectively.

 

Fig. 2. The intensity distribution of electric field on the Archimedean metal disk with different radius r₀= 100 nm, r₀= 150 nm, r₀= 200 nm under the incidence of (a-c) LCP light and (d-f) RCP light with wavelength of 532 nm, and the resonance screw pitch is p = 341 nm set in water. The in-line intensity distribution across (g) y direction and (h) x direction for the excitation of LCP and RCP respectively.

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Here the pitch of plasmonic metalens is designed of $p = 341nm$ for considering the application medium of optical tweezers in water, which is smaller than above design in air. Figures 2(a-c) reveal the three designed metal disks generate similar plasmonic hotspot profiles by the LCP light excitation. While an expanded radius slightly enhances the hotspot's peak intensity of electric field, as depicted in the Fig. 2(g) about the intensity distribution along y-direction. This effect results from the higher spin-orbit coupling efficiency along the spiral nanostructure boundary for a larger starting radius. Under the excitation of RCP light, Figs. 2(d-f) depict the plasmonic vortex generation around the metal disk center for above three starting radius cases. For this scenario, a diminishing initial radius enhances the energy concentration at the metal disk corners as shown in the Fig. 2(h) about the intensity distribution of electric field along the x-direction, which will trigger optical trapping under the RCP light excitation but weakens selective trapping ability of the designed metal disk. Consequently, the designed metal disk with a smaller starting radius like ${r_0} = {100\textrm{nm}\; or\; }{r_0} = \textrm{150nm}$ may exhibit lower selective trapping capabilities. Thus, the subsequent analysis of optical trapping by the metalens will be conducted with a starting radius ${r_0} = \textrm{200nm}$ and $p = \textrm{341nm}$, taking into account the solution environment.

2.2 Analysis of optical trapping force by Archimedean metalens

In order to illustrate the selective optical trapping effect, the optical force and trapping potential distribution for nanoparticles, induced by the plasmonic field on the upper surface of the designed Archimedean metalens, are analyzed and plotted in Fig. 3. Figs. 3(a) and 3(b) show the optical force ${F_x}$ and ${F_z}$ along different position in x-axis for dielectric particle with diameter d = 200 nm and refractive index n = 1.8, under the LCP and RCP incidence light, respectively. Here, the optical force in y-direction is neglected since it has the same performance due to the symmetry for the electric field distribution as shown in Fig. 1. The thickness of metal disk is set to h = 100 nm, 200 nm, 300 nm to optimize the parameters of Archimedean metalens for selective plasmonic trapping. The optical forces on the particles are calculated using the Maxwell stress tensor method by [19]

 

Fig. 3. The optical force analysis for 200 nm dielectric particle along the direction of (a) x- and (b) z- axis variation with the thickness of metal disk being 100 nm, 200 nm and 300 nm. The potential well of electric field and corresponding intensity distribution of electric field (inset) on the Archimedean metal disk under the excitation of (c) LCP and (d) RCP light. The intensity distribution of electric field for the array Archimedean disks with the same pitch and radius designed in the water (the metal disks labeled 1 and 4 are with right rotation and the disks 2, 3 with left rotation) excited by the (e) LCP and (f) RCP light separately.

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From Figs. 3(a) and 3(b), it can be seen that the optical force around the hotspot field excited by LCP light is much larger than that of a disrupted vortex field excited by RCP light, and there is about five times magnitude difference for the force ${F_z}$ specially. The direction of optical force ${F_x}$ and ${F_z}$ around the hotspot are pointing to the center and towards the surface of metal disk, where the plasmonic force performs as the trapping effect of focused Gaussian beams in typical optical tweezers system [21,31]. It should be noted that the thickness of metal disk has fundamental impaction on the optical trapping ability for hotspot field excited by LCP light, and the thickness of 200 nm has the best trapping effect. This can be explained that the metal disk with thinner thickness has higher scattering force for the particle due to more scattering light propagating around the boundary of the metal disk. While the over thicker disk will attenuate the efficiency of spin-orbit coupling greatly, which causes the optical force to decrease either.

For further illustrating the optical trapping effect, the optical potential distribution formed by the optical force are depicted in Fig. 3(c) and 3(d), which reveal the stability of optical trap. The optical potential can be calculated by $U({{r_1}} )={-} \mathop \smallint \limits_\infty ^{{r_1}} F(r )dr$, where ${r_1}$ is the distance to the hotspot center on the proposed metalens and the force ${F_\infty }$ is regarded as 0. It can be found that the negative potential in the center of metal disk excited by LCP light forms a deep well capable of trapping particles. The depth of trapping potential well is about $- 12{K_b}T$ (${K_b}$ is the Boltzmann constant, temperature T = 300 K), which could overcome the Brownian motion of particles and satisfy the condition of stable optical trapping [18,27]. While the negative optical potential excited by RCP light along the truncated ring is weak and the maximum potential depth is about −3 ${K_b}T$. Such a potential is insufficient for the stable trapping or rotation of nanoparticles. Thus, the proposed metalens can realize the purely trapping on/off in time-domain by simply adjusting the circular polarization of incident light. The corresponding intensity distribution of electric field in the XY plane, displayed in the lower insets of Figs. 3(c) and (d), reveals that the hotspot field is localized in size of about 100 nm, which has the intrinsic ability to realize the selective trapping of nanoparticle. It’s noticed that the optical potential depth by the designed spiral disk is larger than that of spiral slit [28], it’s considered that the spiral disk has higher efficiency to spin-orbit couple light to generate hotspot field relative to the limited light transmitting through the slit of helix. Furthermore, the Archimedean disks can be designed with different rotation in the array distribution as shown in Figs. 3(e) and 3(f) (the metal disks labeled 1 and 4 are of right rotation with positive topological charge m = 1 and the disks 2, 3 are of left rotation with negative topological charge m = −1), it can be seen that the designed array metal disks can generate the electric field distribution of hotspot and imperfect vortex field simultaneously under the excitation of either LCP or RCP incident light, which can potentially realize the selective trapping excited by the circularly polarized light in the spatial domain. On the other hand, compared to the plasmonic field generated by the spiral slit propagates tens of microns along the gold film and will easily interfere with each other in the potential application of array selective trapping [28], the array plasmonic fields by our proposed array metal disks have the advantage for spatial selective trapping without interfering with each other since the array disks’ structures are disconnected with each other as shown in the Figs. 3(e) and 3(f). Therefore, it can be figured out that the designed Archimedean spiral disk potentially owns the ability of selective trapping or sorting of nanoparticle in both the time and spatial domain, which has very important applications like in bio-medical region that needs high precision.

2.3 Effect of particle size and plasmonic resonance mode of gold particle

The optical force exerted on a particle is contingent on the degree of coupling between the particle and the spin-orbit coupled plasmonic hotspot field. The size of the particle plays a crucial role in this process [13,14]. In order to study this effect, we compared the total optical force magnitude $F$ on the center of metal disks composed of either gold or dielectric particles (n = 1.8) with varying particle size, as shown in Fig. 4(a). Firstly, we focus on the result of aforementioned dielectric particle as the blue line plotted in Fig. 4(a), it can be seen that the total optical force magnitude F of dielectric particle increases with particle diameter, reaching a maximum at about 210 nm, and then decreases as the particle size further increased. This decrease is due to the declining coupling efficiency between the particle and the plasmonic field.

 

Fig. 4. The comparison of (a) total optical force F and (b) optical force ratio analysis for dielectric and metal particles with different size. The charge distribution and electric field profile in the YZ plane of Au particle with the diameter of (c) 100 nm and (d) 230 nm, where the diploe and quadrupole effects are excited respectively. Q above the color bar of (c) and (d) represents the quantity of electric charge.

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As a comparison, we also calculated the optical force variation with the particle size for gold particles, the result with red line in Fig. 4(a) shows that the curve of optical force magnitude F have two peaks at particle diameters of 100 nm and 230 nm, which are 3.5 and 2.6 times higher than the peak optical force of dielectric particles being the size of 210 nm, respectively. This is responsible by the magnetic resonance in the particles and has been illustrated in Ref. [32]. However, this peak optical force discrepancy is inherently caused by weaker magnetic component in the plasmonic field on the surface of metal disk as opposed to the stronger electric component.

To demonstrate the underlying resonating mechanisms responsible for the two peaks of gold particle in Fig. 4(a), we investigate the distributions of electric fields and electric charges in the gold particles at each peak. Figure 4(c) and 4(d) demonstrate the results at the first peak and second peak in the YZ plane, respectively. It shows that the electric charge was distributed along the surface of the metal particle, accompanied by electric dipole (quadrupole) resonances, which correspond to resonating electric field distributions around the particle. This indicates that the electric dipole and quadrupole resonances of the gold particle enhance the coupling between the particle and the plasmonic hotspot field, thus leading to the two peaks in the trapping force.

Finally, Fig. 4(b) illustrates the selective trapping capability of different particles by the metalens. It shows the total optical force ratio varies with the particle size excited by LCP light and RCP light for both dielectric and metal particles. The results indicate that the selective trapping ability of both particle types by chiral light modulation is better for smaller particles, and decreases as the particle size increases. Notably, the optical force ratio by LCP and RCP light excitation for gold particles at the first resonance size of 100 nm can reach 30, indicating the strong selective trapping ability by the metalens for nanoparticles, which potentially own important applications in fields such as biomedicine and bio-drugs.

3. Conclusion

In conclusion, a plasmonic optical tweezer based on the Archimedean metal disk has been proposed to realize the selective trapping of nanoparticles through spin-orbit interaction, under the excitation of 532 nm circularly polarized light. The importance of the geometric profile of the Archimedean metal disk are highlighted, including the starting radius, pitch, and height of the disk, in creating the resonant hotspot field. The optical force and potential well exerted on nanoparticles are analyzed, and the results demonstrate the novel selective trapping or sorting ability of nanoparticles by the proposed metal disk in both the time and spatial domain. Specially, we examine the dipole and quadrupole mechanisms underlying the novel optical trapping effect for gold particles of different sizes, which highlights the critical role of light-matter interaction in achieving near-field optical trapping. Overall, this study provides insights into the origin of selective trapping force by the spiral disk, which is a newly proposed member of nanostructure in on-chip optical manipulation of nanoparticles and potentially owns various applications prospect including biosensing, optoelectronics and nano-photonics.