Trapping metallic particles using focused Bloch surface waves

Abstract

Metallic particles are promising for applications in various areas, including optical sensing, imaging and electric field enhancement-induced optical and thermal effects. The ability to trap or transport these particles stably will be important in these applications. However, while traditional optical tweezers can trap metallic Rayleigh particles easily, it is difficult to trap metallic mesoscopic/Mie particles because of the strong scattering forces that come from the far-field trapping laser beam. Here we demonstrate that metallic particles can be trapped stably using focused Bloch surface waves that propagate in the near-field region of a dielectric multilayer structure with a photonic band gap. Focused Bloch surface waves can be excited efficiently using an annular beam with azimuthal polarization and a high-numerical-aperture objective. Numerical simulations were performed to calculate the optical forces loaded on a gold particle by focused Bloch surface waves and the results were consistent with those of the experimental observations.

Introduction

Metallic particles have unique properties, including large-scale electric field enhancement that results in strong absorption of light for use in development of photovoltaic devices, surface-enhanced Raman scattering, fluorescence intensity enhancement for single molecule detection, and nonlinear optical effects such as high-harmonic generation. The localized electric field around a metal particle is sensitive to the surrounding environmental medium and this property can be used to fabricate high-performance sensors.^1-7 These potential applications mean that trapping and transportation of metallic particles has become an important technique and optical tweezers are a natural choice because they provide an efficient tool for noncontact trapping of micro-/nanoscale objects that is easy to operate and low in cost.^8,9 In traditional optical tweezers, trapping occurs when the gradient force exceeds the scattering force. Ordinarily, the gradient force attracts the object toward the beam focus, whereas the scattering force pushes the particle away.¹⁰ Metallic Rayleigh particles (with radius a ≪ λ, where λ denotes the incident optical wavelength) can be trapped using traditional optical tweezers because of their small scattering force.^11,12 However, when the size of the metallic particle increases, trapping becomes more difficult because the scattering force strengthens more rapidly than the gradient force. Several special methods based on the use of scattering forces to trap metallic mesoscopic particles (a ~ λ) and Mie particles (a ≫ λ) have been reported, including use of a scanning laser beam,¹³ a laser beam that is focused near the bottom of the particle¹⁴ and an obstructed laser beam.¹⁵ However, these methods require high-precision control of the relative positioning between the focus of the laser beam and the bottom of the metal particle.

It was previously reported that focused surface plasmon polaritons (SPPs) excited on a thin gold film using a combination of a high-numerical-aperture (NA) objective and a radially polarized laser beam can act as plasmonic tweezers to trap metallic particles with diameters exceeding 500 nm.^16,17 However, the heating effect caused by the SPPs associated with the metallic film reduces the trapping stability, particularly when long-term trapping and observation are needed.¹⁸ In addition, experimental experience has shown that thin gold films are not very robust and cannot be reused many times. In this article, optical tweezers are proposed based on focused Bloch surface waves (BSWs) excited on an all-dielectric multilayer structure that can be used for stable trapping of metallic particles with diameters in the 0.5–3.0 μm range. BSWs are electromagnetic surface waves that are excited at the interface between a truncated periodic dielectric multilayer structure with a photonic band gap (PBG) and its surrounding medium. These waves can be considered to be the dielectric analog of SPPs and have similar merits, including surface sensitivity and electric field enhancement, but with lower propagation losses and more choices in terms of polarization states and incident wavelengths.^19-21

Results

Configurations of the all-dielectric multilayer PBG structure and the experimental set-up

Transverse-electric (TE) polarized BSWs can be excited on the top surface of the all-dielectric multilayer PBG structure depicted in Fig. 1(a). The exact thicknesses of each dielectric layer and the total number of layers (14 layers) are shown in the figure. Fig. 1(b) shows a schematic diagram of the experimental setup, in which a lens array and a pair of axicons are used to obtain an expanded and collimated ring-shaped beam that then passes through a linear polarizer and a vertex retarder to produce an azimuthally polarized beam. This beam fully filled the rear aperture of a high-NA objective (100×, NA of 1.49) that was used to focus the beam in the sample at a location slightly above the BSW surface. The gold particles (diameters of 0.8–1.5 μm) to be trapped were diffused in a water solution and a drop of this solution was placed on the dielectric multilayer to form the sample. A light-emitting diode was then used to illuminate the sample that was on the top of the multilayer substrate. The reflected trapping beam and the transmitted illumination light were collected using two lenses. The front focal plane (FFP) and the back focal plane (BFP) of the objective were imaged using the two cameras (1 and 2).^22-24 Filter 1 partially rejects the laser beam and allows the illumination light to reach camera 1, which is used to monitor the trapping and transportation of the particles. Filter 2 allows only the reflected laser beam to reach camera 2, which is used to confirm the excitation of the BSWs at the water/multilayer interface. The sample was placed on a high-resolution reference-class piezo stage system that can control the distance between the substrate and the objective (along the Z-axis) precisely.

Fig. 1.

Schematics of dielectric multilayer and experimental setup. (a) Dielectric multilayer acting as the substrate, on which a drop of a water solution containing gold particles was deposited. (b) Experimental setup. The combination of the lens array and the pair of axicons can generate a collimated laser beam with a ring shape (i.e., a hollow beam) from a Gaussian laser beam; this beam then passes through a polarizer and a vertex retarder, resulting in the output of an annular azimuthally polarized beam. The inset graph shows the case where the geometric focus of the incident annular beam was located inside the water solution, which is necessary for formation of the BSW-based virtual probe, as shown in (a). The distance between the geometric focus and the water/multilayer interface is defined as d. (c) BFP image of the laser beam when reflected from the dielectric multilayer, in which the dark ring (labeled BSWs) verifies that the BSWs were excited in all azimuthal directions, including at points P1 and P2 on the inset graph in (b).

In our experiments, if the top surface of the dielectric multilayer was located on the FFP of the objective (where the particles attached to the multilayer structure can be imaged precisely), the sample position was defined as d = 0. When the stage was moved upward, the d value then became positive (in this case, the geometrical focus of the incident ring beam will be located inside the multilayer). When the stage was moved downward, the d value then became negative. In this case, the geometrical focus of the incident ring beam will be located inside the water solution on top of the multilayer (see the inset graph of Fig. 1(b)) and BSWs can be excited in all azimuthal directions on the water/multilayer interface. Because of the focusing properties of the objective, these BSWs form a virtual probe at the center of the surface, as illustrated in Fig. 1(a). For example, BSWs excited at points P1 and P2 will propagate in opposite directions and will interfere in the overlapping regions, thus enhancing the strength of the evanescent field (Fig. 1(b)). This process is similar to the excitation of a plasmonic virtual probe on a thin gold film using a radially polarized incident beam and a high-NA objective.^16,25 The excitation of the BSWs was confirmed using the BFP image of the reflected laser beam, as shown in Fig. 1(c), in which a full dark ring appears. From the diameter of this dark ring and the known NA of the objective, the excitation angle for the BSWs can be determined to be approximately 67°.²⁶

Trapping and transportation of metallic particles using focused BSWs on an all-dielectric multilayer

The virtual probe formed using the focused BSWs was then used to trap a gold particle (diameter of 0.8–1.5 μm), as shown in Fig. 2, where the d value is set at −5 μm. Fig. 2(a)-(d) show successive images recorded using camera 1. The image sequence runs from Fig. 2(a)-(d) with a time interval between images of 3 s. Because of the optical force enabled by the focused BSWs, particle 1 was attracted and then flowed toward the center of the field of view of the objective (center of the image in Fig. 2(d)). To verify that particle 1 was then trapped stably at the center, particle 2, which was originally outside the image field in Fig. 2(a)-(d) and was not trapped, was then translated toward the trapped particle 1 from left to right (see Fig. 2(e)-(h)). During this movement of the substrate (i.e., the dielectric multilayer), particle 1 is always fixed, thus demonstrating that it is trapped stably, and it can then be transported (in two dimensions) to any place on the dielectric multilayer. The real-time process of trapping and manipulation of the gold particle using the focused BSWs is also shown in Movie 1 in the ESI.^† This method is different to the process of trapping using a fixed plasmonic pattern, such as that of a plasmonic antenna, which limits the possibility of object manipulation in one direction.¹⁸ The method proposed in this work benefits from the structureless excitation of the BSWs within a dynamic configuration, which reduces the need to fabricate complex nanometer-scale structures, and can be used to transport the trapped target particle to any location on the dielectric multilayer.

Fig. 2.

Trapping and transporting gold particles with diameters of 0.8–1.5 μm. (a–d) The gold particle (particle 1, labeled by the red circle) was first attracted and then moved to the center of the virtual probe. (e–h) The stage supporting the multilayer is moved from left to right. The trapped particle (particle 1) can then be transported to other locations on the multilayer. In contrast, the untrapped particle (particle 2, labeled by the yellow circle) moves with the stage and the multilayer. Particle 1 was fixed at the same position in images (e–h) during the movement of the stage, demonstrating that it was trapped by the focused BSWs.

Our experimental results also demonstrate that, in addition to the gold particles with diameters of 0.8–1.5 μm, larger gold particles with diameters of 1.5–3.0 μm and smaller gold particles with diameters of 0.5–0.8 μm can all be trapped (see Fig. S1 in the ESI^†) and transported (see Fig. S2 in the ESI^†) using the focused BSWs. In addition, a dielectric (polystyrene) particle with a diameter of 2 μm can also be trapped and transported using the focused BSWs, as shown in Fig. S3 in the ESI;^† this demonstrates that the proposed technique can be used universally for both dielectric and metal particles.

As noted in the description of the experimental setup, the distance between the dielectric multilayer and the objective can be tuned using the Physik Instrumente (PI) stage along the Z-axis, which results in a different d value (the d value is defined as the distance between the geometric focus and the water/multilayer interface, as illustrated in the inset graph in Fig. 1(b)). In the experiment, if the d value is positive and very large, stable trapping of the gold particle is not observed because the virtual probe cannot be formed on the water/multilayer interface in this case.²⁵ In addition to the negative d value of −5 μm used in Fig. 2, we also used other d values including d = −3 μm and d = −7 μm, for which the geometrical focus of the incident ring laser beam will be located inside the water solution, as illustrated in the inset graph in Fig. 1(b). In these two cases, gold particles with diameters of 0.8–1.5 μm can again be trapped, as shown in Fig. S4 in the ESI.^† Fig. 3(a)-(d) (for d = −3 μm) and Fig. 3(e)-(h) (for d = −7 μm) show that the trapped particles can be translated to other locations on the multilayer structure by moving the PI stage. During this movement, trapped particle 1 always remains stable on the images while the untrapped particle (particle 2) is shifted from left to right. The real-time processes of trapping and transportation of the gold particles using the focused BSWs in the cases where d = −3 μm and d = −7 μm are shown in Movies 2 and 3, respectively, in the ESI.^†

Fig. 3.

Trapping and transporting gold particles with diameters of 0.8–1.5 μm at different defocus planes (i.e., different values of d). (a–d) In these images, d was set at d = −3 μm. (e–h) In these images, d was set at d = −7 μm. The magenta ring represents the BSWs excitation ring. The geometric focus of the incident ring-shaped beam was located inside the water solution, and d is the distance between the geometric focus and the water/multilayer interface (as shown in the inset graph in Fig. 1(b)). Particle 1 is trapped by the focused BSWs. The stage that supports the multilayer moves from left to right. Particle 2 is not trapped and thus moves with the stage and the multilayer.

These results demonstrate that trapping and transportation of a gold particle using focused BSWs does not require precise control of the relative positions of the focus and the particle. This differs from the previously reported method that required the focus of the trapping laser beam to be located precisely at the bottom of the metal particle. The reason for this increased tolerance for the defocus distance (the d value) can be explained using the inset graph in Fig. 1(b). When the distance between the geometric focus and the water/multilayer interface increased, the distance between points P1 and P2 also increased; however, the propagation losses of the BSWs are low, which means that the excited BSWs from points P1 and P2 can still interfere with each other and form the virtual probe. This method provides easy access to the trapped sample because of the large distance tolerance between the objective and the trapped sample. However, when d was set at d = −7 μm, the distance between points P1 and P2 can reach approximately 32 μm (Fig. 1(b)), which is much greater than the propagation lengths of SPPs (on a water/gold interface) at the same wavelength. This means that the trapping region of the focused BSWs will be larger than that of focused SPPs.

Comparison between trapping using focused BSWs and trapping using focused SPPs

Our previous work demonstrated that focused plasmonic waves enable stable trapping of metallic particles in the mesoscopic/Mie particle size range.¹⁶ The experiments in this work have demonstrated that the focused BSWs can also trap metallic particles stably. These results have led us to compare the stabilities of the two trapping techniques. The experiments for trapping of gold particles using the focused BSWs on a dielectric multilayer and the focused SPPs on a thin gold film were performed individually here. For excitation of the focused SPPs, the incident laser beam was radially polarized, while the laser beam for excitation of the focused BSWs was azimuthally polarized. The power of the incident laser beam was kept constant at approximately 10 mW. The diameter of the gold particles was again in the 0.8–1.5 μm range. In the experiments, we attempted to trap particles of nearly the same size using the two techniques. When a gold particle was trapped by either the focused BSWs or the focused SPPs, a high-speed complementary metal–oxide-semiconductor (CMOS) camera (MV-CA-003-21UM, Hikvision, China) was used to measure the motion of the trapped particle positions; this method is widely used to measure the positional displacement of trapped particles and analyze the trapping stiffness.^27,28

Fig. 4(a) and (b) present the retrieved position distributions for single trapped gold particles on a dielectric multilayer and on a gold film, respectively; these distributions were derived from a series of images captured using the high-speed CMOS camera (98 images per second). Each point on Fig. 4(a) and (b) was derived from a single captured image. The distributions of these points indicate the stability of the trapping technique. A more concentrated distribution of the points indicates greater trapping stability. To describe the trapping stabilities of the two techniques quantitatively, particle count vs. position curves are shown in Fig. 4(c) and (d) that correspond to the results in Fig. 4(a) and (b), respectively. Here, the particle count (the Y-axis in Fig. 4(c) and (d)) means the number of times that the trapped gold particle appears at the same displaced position relative to the center (where the center is defined as X = 0 and Y =0 μm in Fig. 5(a) and (b)). The two curves can be fitted using a Gaussian function,²⁹ from which the full width at half-maximum (FWHM) values of the two curves in Fig. 4(c) and (d) can be derived as 0.565 μm (for the gold particle trapped using focused BSWs) and 0.746 μm (for the gold particle trapped using focused SPPs), respectively. The trapping stiffness of each gold particle can also be derived from these two curves, with values of 0.0768 PN μm⁻¹ (when using focused BSWs) and 0.0496 PN μm⁻¹ (when using focused SPPs).^27-29 It is known that a smaller FWHM (or greater stiffness) corresponds to better trap stability, so the gold particle can be trapped more stably when using the focused BSWs rather than the focused SPPs.

Fig. 4.

Position tracking and stability analysis of gold particles trapped using focused BSWs and using focused SPPs. The diameter of the gold particle is approximately 1 μm. (a) Position distribution of the gold particle trapped using the focused BSWs. (b) Position distribution of the gold particle trapped using the focused SPPs. (c) Curve of counts vs. position (hollow dots) fitted using a Gaussian curve (red line), which represents the data shown in (a). (d) Curve of counts vs. position fitted using a Gaussian curve, which represents the data shown in (b). The particle count here (Y-axis on (c) and (d)) represents the number of times that this trapped gold particle appears in the same displaced position relative to the center (where the center is defined as the mid-point between points P1 and P2 in Fig. 1(b)).

Fig. 5.

Calculated lateral and longitudinal forces vs. particle positions. Lateral (F_x) and longitudinal (F_z) forces exerted on a metallic particle in focused BSW-based tweezers at different lateral positions (a) and at different longitudinal positions (b). In (a), the gap between the gold particle and the multilayer is maintained at 20 nm. In (b), the particle is located at X = 0 μm and the gap (along the Z-axis) increases. Distributions of the electric field intensity of the focused BSWs (c, e, f) and SPPs (d) with a gold particle on the multilayer and the gold film. Here, X = 0 μm, X = 0.2 μm and X = 1 μm are the lateral positions of the gold particle (at the center of the sphere). The red arrows in (c–f) represent the direction of the force exerted on the gold particle in each case. The white dashed lines represent the water/multilayer and water/gold interfaces. The gold particle is 1 μm in diameter and its position is represented by a white dashed-line circle. X = 0 and Z = 0 are defined as coordinates of the center of the focused BSWs or SPPs.

Numerical simulations of the BSWs and the optical force on a gold particle produced by focused BSWs

To derive the trapping mechanism produced by the focused BSWs, the electric field properties of the focused BSWs were analyzed using numerical simulations. Fig. S5(a)^† shows the angle-dependent reflection curve obtained from the dielectric multilayer structure when there were no gold particles on the multilayer. The sharp dip that appears at 66.68° on this reflectance curve verifies that BSWs can be excited on the dielectric multilayer using TE-polarized light, which is consistent with the experimental results derived from the BFP image (67°; see Fig. 1(c)). This dip is much narrower than the surface plasmon resonance dip in the corresponding reflection curve from the thin gold film, meaning that the propagation losses of the BSWs are much lower than those of the SPPs.¹ The electric field intensity distribution in this dielectric multilayer when the angle of incidence was fixed at 66.68° (the excitation angle for the BSWs) was calculated as shown in Fig. S5(b);^† the results show that the BSWs are evanescent at the interface between the water and the dielectric multilayer and have a strong field in the top dielectric layer.

In our experiments, the incident laser beam was azimuthally polarized and was focused onto the dielectric multilayer using an objective. Because of the axial symmetry of the objective, the incident light would then be TE-polarized relative to the plane of the dielectric multilayer at all azimuthal directions. BSWs can then be excited in all these directions and would propagate toward the center of the focal plane. To show the resulting electric field distributions of the focused BSWs, full electromagnetic simulations were performed using the finite-difference time-domain (FDTD) method.^30,31 The azimuthally polarized beam was constructed using the Richards and Wolf vector integral.³² To eliminate the effects of directly transmitted light, an apodization function was used to prevent any light with an angle of incidence that was smaller than a critical angle from passing through the structure. The water/multilayer interface was set to be 1 μm below the geometric focus (d = −1 μm). Fig. S5(c) and (d)^† show the electric field distributions of the focused BSWs on the X–Y plane and on the plane oriented perpendicular to the multilayer structure (X–Z plane), respectively. It is clear that a virtual probe was formed at the center that decayed rapidly along the Z-axis. Unlike the virtual probe formed by the focused SPPs, which had a solid center, virtual probe formed by the BSWs is donutshaped with a black hole, where the diameter of this hole is approximately 278 nm. This difference is due to the different polarizations of the SPPs and BSWs. For the BSWs, the main electric field components are E_x and E_y, whereas the main electric field component for the SPPs is E_z.¹⁶

When a gold particle was placed on the dielectric multilayer, it was then subject to the optical force induced by the focused BSWs. A three-dimensional FDTD simulation was used along with Maxwell’s stress tensor (MST) to calculate this force.^30,31 The diameter of the gold particle was set at 1 μm, while the wavelength and polarization of the incident laser beam and the structural parameters of the dielectric multilayer and the gold film were all the same as those used in the experiments. As shown in Fig. 5(a), when the gap between the gold particle and the dielectric multilayer was set at 20 nm, the lateral force (F_x, acting along the X-axis as defined in Fig. 5(c)) increases with increasing distance between the gold particle and the center of the focused BSWs (where this center was defined as shown in Fig. 1(b) as the mid-point between points P1 and P2; the center is also defined as shown in Fig. 5(c) as point (0,0)). F_x reaches a maximum at X = 0.2 μm. Then, F_x decreases and reaches a minimum at X =1 μm. It is clearly shown that this lateral force F_x turns from a positive force into a negative force. The positive force causes the gold particle to be pushed away from the center of the focused BSWs, while the negative force causes the particle to be attracted toward the center. Therefore, we can determine that the trapping site for the gold particle is located between the positions of X = 0.2 μm and X =1 μm, where the lateral force F_x is zero. As shown in Fig. 5(b), when the gold particle was located at the center of the focused BSWs, the lateral force remains constant at zero, although the gap between the gold particle and the multilayer structure (along the Z-axis, as shown in Fig. 5(c)) increases; this can be attributed to the axial symmetry of the focused BSWs. Although the lateral force F_x is also zero at the center of the focused BSWs (X =0, Z = 0), this position may not be the preferred trapping site for the gold particle because, when the particle is shifted slightly from the center, the lateral force F_x becomes positive and the particle will then be pushed away.

In addition to the lateral force, the gold particle is also subject to the longitudinal force (F_z) that is oriented perpendicular to the multilayer structure (along the Z-axis). As shown in Fig. 5(a) and (b), the longitudinal force is positive at all times. However, as shown in our previous work,¹⁶ the longitudinal force that was applied on the gold particle by the focused SPPs was negative, which could then attract the particle to the gold film and thus realize longitudinal trapping of the particle. To determine the reasons for this difference, the electric field distributions of the focused BSWs and the focused SPPs with gold particles on their substrates were simulated as shown in Fig. 5(c)-(f). The gold particle with the diameter of 1 μm is located 20 nm above the multilayer (Fig. 5(c), (e), and (f)), or 20 nm above the gold film (Fig. 5(d), where the gold film thickness was set at 45 nm). The forces that acted on the different points of the particles were also simulated, as indicated by the red arrows near the particle surface. The red arrows that start from the center of the sphere in Fig. 5(c)-(f) denote the resultant force acting on the particle and show that the Z-component of the force in the case of the focused BSWs is positive, while that in the case of the focused SPPs is negative. These electric field distributions show that there is a hot spot or a strong electric field in the gap between the gold particle and the gold film (Fig. 5(d)) that will induce enhancement of the optical gradient force and thus attract the particle to the film.¹⁶ As a result, the force F_z must be negative. In contrast, for the focused BSWs, the strongest electric field was located inside the top layer and the resulting gradient force was thus weak (Fig. 5(c), (e) and (f)). Because of the high scattering properties of the gold particle, the scattering force produced by the focused BSWs must be high and would then induce a positive F_z. Therefore, the longitudinal trapping of the gold particle by the focused BSWs can be attributed to the balance between the positive F_z and the gravitational force of the gold particle (which is negative).

The electric field distributions shown in Fig. 5(c)-(f) can also be used to explain why the focused BSWs can trap the gold particle more stably than the focused SPPs, as indicated by the experimental results (Fig. 4). While the strong electric field inside the gap between the gold particle and the gold film will generate a sufficiently large gradient force to attract the particle, it will also generate considerable heat because of the absorption properties of the metallic materials; this heat will certainly affect the stability of the trapped particle, particularly in cases of long-term trapping.

Conclusions

BSWs have specific properties that differentiate them from SPPs. They are not subject to the losses caused by metal absorption, which thus allows for BSW resonances with high quality factors and long propagation lengths. The dielectric multilayer structures required for BSWs are more stable than thin metal films and can be used stably multiple times. In addition, dielectric substrates are used almost universally for applications in cell biology or biomedical optics, rather than metal substrates. Therefore, in the past few years, BSWs have been widely used in sensing applications and have formed a basis for two-dimensional integrated optics and for enhancement of the Goos–Hänchen effect and fluorescence or Raman scattering signals, among other applications.^33-39 BSWs have also been used as novel tools to push micro-sized dielectric beads.⁴⁰ However, to the best of our knowledge, there have been no previous reports on how to trap either metallic particles or dielectric particles using BSWs. Our work here demonstrates that focused BSWs can be used as optical tweezers to trap and transport both metallic and dielectric particles of various sizes. Because of the much lower propagation losses of BSWs, the method proposed here also provides easy access to the trapped sample because of the large distance tolerance between the trapping objective and the sample; for example, the dielectric multilayer can move upward or downward on a micrometer-level scale, such as from d = −3 μm to d = −7 μm. When compared with the tweezers formed using focused SPPs, the tweezers formed by the focused BSWs are more stable because less heat is generated on the dielectric multilayer when compared with that on the metal film. In addition, the longer propagation length of BSWs when compared with that of SPPs means that the trapping region of the focused BSWs is also larger.

BSWs are also highly sensitive to environmental changes on surfaces and this property has been used to develop a high-performance sensor based on monitoring of the angular shift of the reflected light from the dielectric multilayer.⁴¹ In our work, camera 2 is used to record the BFP image of the objective, which can be used to monitor the angular shift in the BSW dip in the reflectance from the multilayer;⁴² our setup can thus also act as a sensing instrument and monitor the changes in the trapped particle in real-time. This can be achieved in both the visible and near-infrared wavelength regions.

Methods

Experimental setup

A schematic of the experimental setup is shown in Fig. 1. The laser beam is produced using a solid-state laser at a wavelength of 671 nm. The angle of the cone of the axicon is 10.0° (AX2510-A, Thorlabs Inc., USA). The axicon pair can convert the expanded laser beam into a linearly collimated ring-shaped beam. The vertex retarder, which was zero-order vortex half-wave retarder; m = 1 (Thorlab, USA), converts the linearly polarized ring beam into an azimuthally polarized ring beam. The samples were placed on a high-resolution reference-class piezo stage system (PInano Cap XYZ Piezo System, Physik Instrumente (PI), Germany) that can control the distance between the substrate and the objective precisely (Fig. 5); this system can also be used to adjust the relative distance between the untrapped beads and the excitation region (Fig. 3). The high-NA objective was obtained from Olympus, Japan. Two cameras were used to record the FFP and BFP images; the first was the Retiga 6000 (QImaging, Canada) and the second was the Neo sCMOS (Andor, UK).

Sample preparation

The dielectric multilayers were fabricated by plasma-enhanced chemical vapor deposition (PlasmaPro System 100, Oxford Instruments, UK) of SiO₂ and Si₃N₄ layers on a standard microscope cover glass (0.17 mm thickness) at a vacuum pressure of <0.1 mTorr and a temperature of 300 °C. In this case, SiO₂ was the low-refractive-index (L) dielectric and Si₃N₄ was the high-refractive-index (H) dielectric. These dielectric layers had thicknesses of 100 and 70 nm, respectively. In total, 14 dielectric layers were deposited on top of each other. The top SiO₂ layer thickness was approximately 490 nm. A gold film with a thickness of approximately 45 nm was deposited using an electron beam evaporator. The gold particles were purchased from Alfa Aesar Inc. The modes were: 44636-gold powder, spherical, APS 0.5–0.8 micron, 99.96+% (metals basis); 39817-gold powder, spherical, APS 0.8–1.5 micron, 99.96+% (metals basis); and 39818-gold powder, spherical, APS 1.5–3.0 micron, 99.96+% (metals basis) (https://www.alfa.com/zh-cn/search/?q=7440-57-5&page=3). The gold particles were diluted in water and a drop of the mixed solution was then placed on the dielectric multilayer.

Numerical simulations

A three-dimensional finite-difference time-domain (FDTD) model (Lumerical Inc.) was used to simulate the electric field distribution of the focused BSWs and Maxwell’s stress tensor (MST) was then used to calculate the forces acting on the gold particles. In these simulations, the refractive indices of gold, Si₃N₄ and SiO₂ at the wavelength of 671 nm were set at n_Au = 0.14 + i × 3.81, n_Si3N4 = 2.65 + i × 10⁻³ and n_SiO2 = 1.49 + i × 10⁻⁵, respectively. The refractive indices of the glass and water were n_glass = 1.515 and n_water = 1.33, respectively. The thicknesses of the periodic layers of Si₃N₄ and SiO₂ and the top defect layer were 70 nm, 100 nm and 490 nm, respectively.