Optical conveyor belts can transport small particles via optical forces along a predesigned path, with applications spanning a wide range of fields including biomedicine [1–5], enantiomer separation [6–10], and nanomaterials science [11–13]. It was initially created based on standing waves, in which the particles are transported by time-varying intensity-gradient forces [14], but this first-generation belt only allows for a straight-line delivery. By contrast, tailoring the phase distribution of fields opens the opportunity to realize micro-manipulation by the optical phase-gradient force or orbital (energy) flow [15–17]. A typical mechanical effect of this force is the particle orbiting around the axis of a phase vortex beam [18–20]. With the method of the generalized perfect optical vortex (GPOV) [21,22], the transportation path can be customized so that swinging the particles for obstacle avoidance is possible during the delivery process [23,24]. However, such customization is usually achieved at the sacrifice of planar orbital flow, with a high risk of breaking off the transportation, especially at the large-curvature regions. To address this issue, recent efforts have been paid to optimize the distribution of field phase gradient ∇Φ [25–28], but retrieving the uniform orbital flow pO requires simultaneous consideration of both the phase gradient and the intensity I, as pO∝I∇Φ.

In this contribution, we propose a method for creating shape-customizable optical conveyor belts, with an orbital flow evenly distributed along the particle delivery path. By integrating random phases and the equidistant sampling method into the conventional GPOV generation algorithm, we can sculpture the optical vortex beam to obtain a uniform distribution of the phase gradient and intensity. We demonstrate particle delivery of steady velocity along a variety of curvature-varying paths. This method can be extended to a dual-belt configuration for the transportation of large particles. Our method employs a pure phase modulation technique, which accelerates the hologram computation speed and improves the energy utilization rate compared to the traditional complex amplitude light field shaping strategy [29].

In Fig. 1, the reflective phase-only liquid crystal spatial light modulator (SLM) is illuminated by an x-polarized monochromatic plane wave of wavelength λ. The phase-modulated beam is focused at the back focal plane of the lens with a focal length f. This configuration requires a computer-generated hologram (CGH) to be addressed on the SLM at the front focal plane, whose complex amplitude is expressed as follows [25]: H(xj,yj)=∬U(xm,ym)exp[i2πλf(xjxm+yjym)]dxmdym,(1)where U(xm,ym) represents the output target field; xj and yj denote the coordinates of the jth pixel on the SLM, while xm and ym represent the spatial position coordinates of the mth focus point in the output field. Mathematically, H(xj,yj) can also be expressed as the sum of individual plane waves with weighted factors U(xm,ym). Therefore, the integral operation can be replaced by summation calculation [30]: H(xj,yj)=∑m=1MU(xm,ym)exp[i2πλf(xjxm+yjym)+iψm].(2)

However, it should be noted that the SLM modulation capability is limited to phase manipulation only, whereas the amplitude information of H(xj,yj) projected onto the SLM is lost. This results in a significant disparity in intensity uniformity between the generated light field U0(xm,ym) and the desired light field U(xm,ym). M represents the total number of focus points. To mitigate this disparity, a random phase ψm=d·Rand(0,1) is introduced at each focus point, where 0≤d≤2π represents the modulation depth and Rand (0, 1) is a uniform random number in the range [0, 1]. Notably, with the increase of modulation depth, the uniformity of light intensity is remarkably improved. However, the gain in uniformity via this method comes with trade-offs in the energy utilization rate, so it is necessary to find a compromise between them for practical applications. In the Cartesian coordinate, the transmission path is depicted by the parametric equations [31]: {x(θ)=r2{acosθ−(−1)bncos[(−1)baθ]},y(θ)=r2{asinθ−(−1)bnsin[(−1)baθ]},(3)where a=[r1+(−1)br2]/r2, and r1 and r2 are the radii of the fixed and rolling circles, respectively, which determine the size and shape of the curve. The exponent parameter b is set to either 0 or 1. When b=0, the equation represents an epicycloid; otherwise, it corresponds to a hypocycloid. The n serves to adjust the curvature of the curve, and it assumes a value within the range [0, 1]. The parameter θ represents the angle value corresponding to the spatial position coordinates of the output field.

To generate a GPOV with a uniform phase gradient, it is imperative to achieve equidistant sampling along the conveyor belt curve, which involves several steps. First, the number of target points on the curve is determined. Then, the circumference of the curve is calculated using the arc length formula. After evenly distributing the curve’s circumference according to the number of target points, the length from the starting sampling point to each sampling point is computed. Next, the difference between the actual sampled trajectory length and the pre-calculated target length is calculated. When this difference equals zero, the corresponding angular value is derived by performing derivative operations on the sampled length. Finally, to obtain the target coordinates, the angular values are substituted into parametric equations that describe the transport path, and the vortex phase is appended to the transport path by Eq. (2) to achieve a uniform phase gradient for the customized conveyor belt.

The tightly focused field at the objective lens focal plane, as described by Eq. (2), can be characterized using the Richards–Wolf integral. Since our method involves only pure phase modulation, the amplitude information of the output light field is disregarded during phase recovery. Consequently, diffraction sidelobes are observed in the tight-focused optical field, resulting in a dark region between the sidelobe and the main lobe on both sides. Metallic particles can be trapped within this dark region where the scattering and gradient forces acting on the gold particle achieve equilibrium. Subsequently, they exhibit orbital motion influenced by the phase vortex flow. The anticipated generalized vortex beam could serve as a versatile tool for manipulating metal particles. To validate its trapping performance, we can calculate the time-averaged optical force exerted on the particle by integrating the Maxwell stress tensor over the surface S surrounding the particle [32–34]: ⟨F⟩=⟨∮sn·Tds⟩,(4)where n is the outwardly directed normal unit vector to the surface, S is a closed surface containing the particle, and T denotes the time-averaged Maxwell stress tensor: T=ε1EE+μ1HH−12(ε1E2+μ1H2)I.(5)Here, ε1 and μ1 are the permittivity and permeability of the ambient medium, and I denotes the unit dyadic. E=Einc+Esca and H=Hinc+Hsca denote the total fields outside the particle, including the incident fields (Einc and Hinc) and the scattered fields (Esca and Hsca). The scattering problem can be solved using the Mie theory [35–38].

The proposed GPOV conveyor belts were generated based on the holographic optical tweezers system (Fig. 2). The linearly polarized beam with wavelength λ=1064nm was output from a continuous wave fiber laser (VFLS-1064-B-SF-HP, Connet Laser Technology Co., Ltd., China) with power adjustable from 0 to 5 W. The linearly polarized beam was expanded through a telescope system consisting of convex lenses L1 and L2 to form a parallel beam. After passing through a half-wave plate (HWP) and a polarizing beam splitter (PBS), the input beam becomes horizontally polarized. A 96° isosceles right-angle reflecting prism was employed to couple the incident beam onto the SLM (1920×1080pixels, 8 μm pixel pitch; PLUTO-2-NIR-049, Holoeye Photonics AG, Germany), loaded with a predesigned CGH [Fig. 2, inset (a)]. A 4-f system consisting of lenses L3 and L4 relayed the SLM plane to the back focal plane of the objective lens (O1, 100×, NA 1.4, oil-immersion, CFI Plan Apo, Nikon Inc., Japan). Then, a tightly focused GPOV beam was produced at the focal plane of the objective lens. The O1 was also employed for the object imaging. A CCD camera (2048×2048pixels, pixel pitch 5.5 μm, frame rate 90 frames per second; Point Grey GS3-U3-41C6M-C, FLIR System Inc., USA) was used to monitor and record the manipulation process, with a near-infrared filter (F) positioned in front of it to eliminate residual laser light passing through the dichroic mirror (DM).

The system’s aberration was fitted with Zernike polynomials and corrected using the SLM. The shape of a Laguerre-Gaussian (LG) mode with a topological charge of ℓ=1 was taken as an optimization goal, and the weights of the Zernike polynomials were finely tuned to fit the system’s aberration [39,40]. After obtaining the aberration correction hologram, the SLM was addressed with it so that the aberration of the system could be corrected. To prepare samples, gold particles (gold micron-particles diameter of 0.5–3 μm, Thermo Fisher Scientific Inc., USA) and aluminum particles (micron-particles diameter of 8–13 μm, Thermo Fisher Scientific Inc., USA) were individually dispersed in distilled water. A sample chamber with a depth of ∼125μm was fabricated using a glass slide, a cover glass with a thickness of 0.17 mm, and double-sided tape. Subsequently, the particle suspension was introduced into this chamber and sealed with an adhesive.

To validate the superiority of the GPOV conveyor belt in comparison to the conventional GPOV transmission, we simulated four optical patterns by modifying the parameter equations of the transmission path, when the topological charge is ℓ=12. We conducted a quantitative comparative verification of their intensity and phase gradient uniformity. These shapes are depicted in Fig. 3, and for simplicity, we have designated them as diamond (I), pentagon (II), hypocycloid quadrilateral (III), and hypocycloid hexagon (IV). In the context of tight focusing, Fig. 3(a) illustrates the superior uniformities in intensity and phase profiles generated by our method for the four distinct types of POV patterns (types of I–IV). Conversely, Fig. 3(b) reveals that the conventional integral method, without the optimization of intensity and phase, yields POV patterns characterized by intensely deteriorated intensity noise and abrupt phase gradient transitions at inflection points. The detrimental optical potential trap, resulting from their combined effects, is almost inevitable during transmission. In the scenario where M=300 and the hologram size is 1080×1080pixels, the time required to generate a hypocycloid hexagon phase hologram is approximately 1.29 s. We conducted ten calculations by using MATLAB software in a computer equipped with an 11th Gen Intel Core i7-11700F at 2.50 GHz CPU and subsequently averaged the results. In comparison to the traditional integration method, which requires 11.67 s, our approach achieves a computational speed that is nearly nine times faster. Figures 3(c) and 3(d) elucidate the light intensity distribution along the normalized transmission path length for the optimized and non-optimized methods, respectively. Figures 3(e) and 3(f) present the phase distribution of different types of POV generated by the two methods as they vary with transmission length, offering corresponding trends in phase gradient changes. Comparative analysis indicates that our method can yield a relatively uniform light field intensity and a linearly changing phase distribution. Finally, Fig. 3(g) conducts a mean square error (MSE) analysis on the overall intensity and phase distribution of the four types of POV light fields produced by the two methods. The results indicate that the mean square error of intensity fluctuations for the optimized method is less than 0.075, and the phase gradient MSE fluctuations are less than 0.2. This low-fluctuation light field provides a crucial guarantee for the smooth operation of metal particles on an optical conveyor belt.

In the simulation, we assume the input laser power P=1W, the wavelength λ=1064nm, the objective lens NA=1.4, the medium refractive index n1=1.33, the gold particle radius R=0.5μm, and the refractive index n2=0.26+6.97i. To generate a hypocycloid hexagon POV conveyor belt, the fixed circular radius and rolling circular radius mentioned in Eq. (3) are set as r1=1.5μm and r2=9μm, respectively, with a topological charge of ℓ=20 and a curvature n=0.5. Figure 4(a) illustrates the distribution of transverse optical forces exerted on a gold sphere at the focal plane of the POV belt. The arrows denote the direction and magnitude of the optical force, while the background represents the intensity distribution of the focused field. The inset shows the force distribution of a red box magnified three times. From the force distribution, it can be observed that opposite optical forces are present on either side of the main lobe of the light field, enabling particle trapping.

To further understand the dynamics of gold particles in the GPOV conveyor belts, we perform quantitative calculations in Fig. 4(b), where Itot represents the intensity of the optical field (red curve), and Fx (green curve) and Fy (blue curve) represent the horizontal force and vertical force (transport force) exerted on the gold particle, respectively. Upon examining the trend of curves Itot and Fx in the x direction, it becomes evident that the gold particle has equilibrium positions on both the inner and outer sides of the deviated light field, with trapping positions identified at x1=7.51μm and x2=8.57μm, respectively. Notice that Fy is non-zero at these two locations, and its magnitude and direction are closely related to the topological charge in the optical field. As a result, when trapped at either of these two trapping points, the gold particle tends to rotate about the beam axis. However, along the axial direction, the gold particle is subjected to an axial scattering force Fz (cyan curve) in the positive z-direction, resulting in no axial equilibrium position. In the following experiments, we typically utilized the surface pressure exerted by the cover glass as well as the self-gravity of particles to counterbalance this axial scattering force, resulting in stable movement of the gold particle within the focal plane. Careful consideration of the horizontal force Fx reveals [see Fig. 4(c)] that for 0.3μm<R<0.8μm, the particle can be trapped on the inner and outer sides of the hexagonal-like field intensity, as indicated by the black dashed line delineating the trap location. For example, the red dot indicates the equilibrium position of particles with a radius of R=0.5μm, which are effectively trapped inside and outside the main lobe of the optical field. Remarkably, our method exhibits universality in trapping other metallic particles with similar dimensions. Figures 4(d) and 4(e) demonstrate the influence of laser power on the transport force at a topological charge of ℓ=20, as well as the effect of topological charge on the transport force at an incident power of P=1W. The results reveal that the increase in laser power significantly enhances the transport force, thereby accelerating the particle transportation speed. Concurrently, the increase in phase gradient also promotes a higher speed of particle transport.

To validate the flexible manipulation of micron-sized metallic particles using generated GPOVs, we employed the rhombic, pentagonal, and Archimedean spiral perfect vortex beam to perform precise rotation and transportation operations on gold particles with diameters ∼1.4μm. In addition, by ingeniously superimposing curved beams of varying sizes, we created a triple-track hyperbolic hexagonal conveyor belt that enables efficient rotation and transport of multiple metallic particles on different tracks. Furthermore, a customized dual-wavy channel and a dual-track pentagonal conveyor belt enable the stable transport of metal particles with a diameter of more than 2 μm.

Because accurately measuring the laser power in the focal plane of the objective lens remains a challenge in the experiment, consequently, we measure the incident laser power P at the back pupil plane of the objective lens. Unless otherwise specified, all laser powers refer to the powers at the back pupil plane of the objective lens.

A single quasi-spherical gold particle with a diameter of 1.4 μm is trapped in the rhombic conveyor belt (with r1=2μm, r2=8μm, b=−1, P=1W, and ℓ=20). Figure 5(a) presents screenshots from the video sequence and time-lapse images of manipulating the gold particle. The rhombic conveyor belt confines the gold particle to the light field periphery where it rotates uniformly and stably counterclockwise at an average rotation speed of 24.87 μm/s (see Visualization 1 for details). By adjusting r1 to 1.8 μm and r2 to 9 μm while keeping the topological charge ℓ unchanged, we transform the conveyor belt structure from rhombic to pentagonal configuration. Figure 5(b) illustrates the rotation of gold particles by the pentagonal transporting region. The metallic particles exhibit a uniform and stable counterclockwise rotation within the inner area of the main lobe of the pentagonal optical belt, at an average rotation speed of 37.62 μm/s (see Visualization 2). To visualize the interaction between conveyor belts and particles over time, we select frames at equal intervals representing static positions of particles, which are then synthesized into a time-lapse image using colormap to identify different particle positions during rotation along their trajectory within the optical trap field. These visualization procedures were performed using the Fiji ImageJ software’s temporal color code module. Figure 5(c) presents the video frames of particles transported by a pentagonal belt calculated by the conventional integral method, and the distribution of transverse optical forces acting on gold particles in the x-y plane. The uneven force field at the turning point of the pentagonal region (inset of the central area) changes the optical angular momentum flow, which leads to an increase in the residence time of particles at the turning point and affects the transmission efficiency (see Visualization 3).

Modulating the parametric equation can convert a closed curve into an open Archimedean spiral. This optical pattern, characterized by chiral intensity distribution, is extensively employed in fabricating chiral micro-nano structures and particle micro-manipulation. By setting a topological charge of ℓ=50 and laser power of P=1W, we successfully achieved a stable long-distance anti-clockwise transport of gold particles (0.58 μm in diameter) along the Archimedean spiral, with a path circumference of 157.2 μm and an average speed of 56.79 μm/s. Figure 6 presents a screenshot of this process and a time-lapse image. Here, the direction of particle transport can be flexibly controlled by changing the topological charge sign, with further details provided in Visualization 4 and Visualization 5.

As an extension of GPOVs, multi-orbit transporting and larger metal particle trapping can be achieved by superimposing optical conveyor belts of different orbits and topological charges in phase holograms. This optical conveyor offers extensive tunability, allowing for intricate and adaptable transporting manipulation through adjustments to the size, intensity distribution, interval distance, complex shape, and topological charges carried by different orbits within the optical pattern.

Figure 7(a) shows the single-shot images of three-orbit hypocycloid hexagonal conveyor rotating gold particles with varying orbit sizes determined by r1 and r2. The topological charges, arranged from the innermost to the outermost orbits, are set as 15, 25, and 35. The incident laser power is P=2W. To ensure optimal trapping strength for each ring, the intensity weight factors β are assigned values of 1.00, 1.35, and 1.75, respectively. The diameter of the gold particles ranges from 0.6 to 1.1 μm. Given that the orbital spacing significantly exceeds the dimensions of the trapped particles, crosstalk between orbits is negligible. Consequently, we observe that gold particles on all three orbits rotate counterclockwise. The mean linear velocity of particles in orbits with ℓ=15 is 24.36 μm/s. At ℓ=25, it is 21.61 μm/s, and at ℓ=35, it reaches 28.74 μm/s. The theoretical model predicts the transport forces of 13.16 pN, 12.58 pN, and 13.74 pN for these three orbits, respectively. These values correspond well with the experimentally observed velocity trends, thereby validating the accuracy of our theoretical model. Reversing their topological charge signs, the direction of gold particle rotation can be altered clockwise in Fig. 7(b). To enhance the visualization of particle movement, the dynamic positions of specific particles are circled with different colored rings in Fig. 7. However, due to the unequal optical forces applied to particles within different orbits and sizes, gold particles leap between orbits during rotation due to centrifugal forces. This could be utilized for particle size sorting, as shown in Visualization 6 and Visualization 7.

As discussed in Section 3.B, the robust scattering force of gold particles limits a single optical conveyor belt to capture stably and transport particles approximately less than 2 μm in size, which is also mentioned in our previous report [41,42]. To broaden the particle size range, we demonstrate a dual-conveyor belt that flexibly controls the width of the tunable dark region between the two belts and the phase gradient, facilitating the stable transmission of larger metal particles. In the experiment, we designed a long-distance double-wavy conveyor (ℓ1=ℓ2=40) that successfully achieved unidirectional stable transportation of gold particles with an approximate diameter of 3.5 μm at P=1W. Figure 8(a) presents a long exposure image capturing the transportation process, with the right side depicting a single-frame screenshot (see Visualization 8). In contrast to the noble metal gold particles, the trapping and manipulation of ordinary metal particles are also widely used in practical applications [43–45]. For this reason, we demonstrate the trapping and transporting functions of larger-sized aluminum particles through the tunable dark region between double orbits. With a dual-track pentagon conveyor (ℓ1=20,ℓ2=40,P=1.5W), an ellipsoidal aluminum particle (n2=1.03+9.25i) with a long diameter of 8 μm and a short diameter of 6 μm is captured and transported within the dark region between the two orbits, exhibiting counterclockwise rotation at a speed of 10.45 μm/s. During the orbital motion of particles, a minor spinning behavior occurs due to the uneven distribution of the incident momentum [46], resulting in a slight deformation of its pentagonal trajectory, as illustrated in the time-lapse image presented in Fig. 8(b). Further details are available in Visualization 9.

In summary, we have proposed an efficient approach to generate GPOVs by uniformly selecting target points on a predetermined curve and simultaneously accumulating phase. This method allows the generation of optical patterns with arbitrary shapes, phases, and orbit numbers while optimizing intensity and phase gradient uniformity through simple and fast modification of the phase mask parameters. In our experiments, flexible manipulation of Mie metallic particles has been demonstrated using these novel beams, verifying the feasibility and effectiveness of the proposed optical conveyance system. This complements the previously reported methods for trapping nanoscale Rayleigh metal particles [47–50], providing a straightforward yet powerful tool for trapping and flexibly manipulating full-sized metal particles in the micro-nano regime.