Since the introduction of optical trapping in 1986 [1], this technology has found widespread applications in fields such as biological and medical sciences [2,3], material engineering [4], and colloidal interactions [5,6]. Conventional optical tweezers, initially designed to manipulate high-refractive-index microscopic particles, typically draw these particles towards regions of high light intensity, such as the focal point of strongly focused optical tweezers [7–18]. As such, these methods are primarily effective for trapping particles with high refractive indices, commonly referred to as high-refractive-index (HRI) particles, due to their inherent light-seeking behavior. However, challenges arise when attempting to capture and manipulate particles with low refractive indices or light-absorbing properties. These particles tend to be repelled by strong light, thus displaying characteristics akin to dark-seeking behavior. To address this, various methods based on computer-generated holograms (CGHs) have been proposed for manipulating low-refractive-index (LRI) particles [19–25]. For instance, successful trapping of LRI particles using a Gaussian vortex beam with a central dark core was demonstrated [19–21]. Subsequently, the manipulation of dark regions between adjacent bright rings of a high-order Bessel beam was shown [22]. However, as particle diameter increases, trapped particles may escape due to the fixed size of dark regions between adjacent bright rings in these methods. An improved method for trapping and rotating LRI microparticles using a quasi-perfect optical vortex, generated by Fourier transforming a high-order quasi-Bessel beam, was proposed [23]. Yet, high-order Bessel beams only allow for the circulation motion of particles and do not offer flexible regulation and arrangement of the particle motion path [24]. Recently, the development of a difference-of-Gaussians trap that can create a dark central core to rigidly trap LRI particles was shown [25]. However, this method requires a unique series of CGHs on the spatial light modulator (SLM) for each particle’s motion, with each hologram corresponding to a fixed capture point, and continuous motion on the trajectory indirectly realized by constantly refreshing the hologram to vary the intensity gradient force, which increases the complexity and inconvenience of the method. The generation of optical patterns and lattices in interferometric beams is crucial for effectively manipulating particles with different properties [26–28]. However, interference beams can only utilize phase interference to generate beams with specific intensity structures, and simultaneously adjusting intensity and phase is challenging. The fixed intensity structure of the light field usually only forms a stationary fixation of particles through the intensity gradient force, without causing particle motion. Employing controllable light intensity structures to simultaneously capture HRI and LRI particles, combined with customized orbital flow density (OFD) to drive particle motion, can greatly improve the feasibility and portability of manipulation.

In this study, we propose a novel class of dual-curvilinear optical vortex beams (DC-OVBs), formed by superimposing two light loops with prescribed OFD variations along trajectories. The DC-OVB structure allows for controllable sizes and shapes of the dark region between the two bright curved beams and the tunable OFD carried by each beam. We experimentally demonstrate the ability of the DC-OVB to manipulate both HRI and LRI particles simultaneously. The generated beam can induce particle motion around the dark regions nestled between the arbitrarily curved bright beams and can even trigger self-rotation of LRI particles while orbiting, facilitated by the carried OFD of the DC-OVB.

Let us first consider a focusing process under the paraxial condition. We want to generate a focal beam that conforms its intensity and phase distribution to a predefined two-dimensional curve path represented by l(t)=[x0(t),y0(t)] with t∈[0,T]. Here the parameter T represents the largest possible value of the variable t that determines the terminals of the curve. The desired focal beam is generated by computing the complex amplitude at the incident plane of a focusing lens, expressed as [29,30] E(x,y)=∫0Tg(t)exp{−iw02[xy0(t)+yx0(t)]}dt,(1)where w0 is a constant. The term g(t) in Eq. (1) determines the amplitude and phase to be applied along the prescribed curve, and is given by g(t)=a(t)exp{i2πm[S(t)S(T)]α}.(2)

Here S(t)=∫0tdl, and S(T) represents the curve length and serves as a normalization factor. For all cases discussed in this study, we set T=2π. It is important to note that the intensity distribution along the curve is determined by the factor a(t); one special case is when a(t)=[x0′(t)]2+[y0′(t)]2, resulting in a uniformly distributed intensity along the trajectory. Furthermore, the parameter m defines the phase accumulation along the entire curve, providing overall control of the phase along the curves. In the case of closed curves, the parameter m can be understood as the vortex topological charge (TC), i.e., m=∮ℓ∇ψdℓ/2π [31]. The parameter α governs the local phase gradient along the curve and plays a critical role in defining the local motion of the trapped particle along the curve, as will be demonstrated in the particle trapping experiments. Specifically, when α=1, it denotes uniform phase variation along the curve. However, for any other positive value of α not equal to 1, it signifies a non-uniform phase gradient.

In addition, curves can be elegantly represented using the Superformula expression [32]. This expression, given by ℓ(t)=R0[|1acos(n04t)|n2+|1bsin(n04t)|n3]−1/n1, employs a constant parameter R0 across all closed curves and a set of real numbers q=(a,b,n0,n1,n2,n3) capable of generating a variety of closed polygons having different degrees of symmetry.

Normally, stable optical trapping of microscopic particles is realized in a tightly focused field, which deviates from the paraxial condition. Therefore, the preceding design for the curvilinear focal beams needs adjustment to accommodate this tight focusing scenario. We propose to construct DC-OVBs within the focal volume, employing a technique developed in our previous work for creating structured beams [33,34]. The key methodology involves the computation of a CGH using Eq. (1). This CGH serves to produce an incident field, denoted by Ei=E1+E2, which is a composite of two fields, each possessing a unique complex amplitude. These amplitudes are essential in generating two distinct curvilinear beams within the focal volume.

As shown in Fig. 1, the light field E→i incident upon the lens gives rise to a transmitted field E→t at the exit pupil, expressed in the cylindrical coordinate system (ρ,ϕ,z) as follows [35]: E→t(θ,ϕ)=cosθM(θ,ϕ)Ei(x,y)e^p.(3)

Here e^p denotes the unit vector of the incident light polarization and cosθ is responsible for ensuring energy conservation throughout the transmission process [36]. In this work, the incident light is assumed to be x-polarized, although our approach is applicable to other polarizations. The polarization transformation matrix M(θ,ϕ) in Eq. (3) involves three successive geometric rotation operations, namely, Rz(ϕ), Ry(θ), and Rz(−ϕ), where Rn(γ) represents a rotation transformation matrix about the n-axis by the angle γ, acting on the incident field vector as the refracted ray is bent towards focus of the lens [37], and can be expressed as M(θ,ϕ)=[cos2ϕcosθ+sin2ϕ(cosθ−1)sinϕcosϕ−sinθcosϕ(cosθ−1)sinϕcosϕcos2ϕ+sin2ϕcosθ−sinθsinϕsinθcosϕsinθsinϕcosθ].(4)

The light field within the focal volume of the lens can be computed via the Richards-Wolf vectorial diffraction integral, and can furtherly be simplified in terms of Fourier transform as [38] E→f(x,y,z)=F{P(ρ)E→t(θ,ϕ)exp(ikzz)/cosθ}.(5)

Here a multiplicative constant factor has been dropped for clarity, F{·} denotes the two-dimensional Fourier transform, and P(ρ) accounts for the aperture of the lens, being one inside and zero outside the aperture. In Eq. (5), kz is the z-component of the wave vector, and the phase factor exp(ikzz) represents the phase accumulation during propagation along the z-axis.

Figure 2 presents three variants of elliptical DC-OVBs generated using the specified parameters, q=(1,1,2,10,6,6). The radius parameter R0 of the inner ellipse is set to 0.1, while for the outer ellipse, R0 is set to 0.5. The symbols m1 and m2 represent the phase TC of the inner and outer curved beams, respectively. The sequential columns, from left to right in Fig. 2, display the x-, y-, and z-components of the electrical field in the focal plane. Amplitude and phase distributions are visualized using a colormap, allowing the identification of phase TCs by the number of colored petals on the inner and outer loops. To emphasize the impact of α on the phase gradient along the curves, distinct α values are assigned to the three elliptical DC-OVBs: α=1 for the first and third rows, and α=8 for the second row. The uniform phase gradient for α=1 and the non-uniform phase gradient for α=8 are discernible by the first column of Fig. 2. Given that the incident light is x-polarized, the x-component dominates the focal field, accompanied by a negligible y-component and a very weak z-component [39]. The pronounced strength contrast among these three components becomes evident in the height distribution profile (depicted by the green line in Fig. 2) of their absolute values along the x-axis.

The momentum flow density in the focal field governs the dynamical behavior of microparticles situated within the focal field, thereby influencing the efficacy of capturing and manipulating these particles using optical tweezers. Given this significance, further calculations and analysis of the momentum flow density in the focal field, as generated by the described scheme, are imperative. The time-averaged momentum density is calculated by utilizing the Poynting energy flow vector [40–42] and can be expressed as follows: p→=Im[Ef*→×(∇×E→f)],(6)which can be further decomposed into orbital and spin parts p→=p→o+p→s: p→o=Im[Ef*→·(∇)Ef*→],p→s=∇×Im[Ef*→×Ef*→]/2.(7)

The z-component of the light’s angular momentum density is J→z=r→×p→⊥, which, according to Eq. (7), can also be decomposed into the orbital and spin components, denoted respectively as J→zo=r→×p→⊥o,J→zs=r→×p→⊥s,(8)where the symbol ⊥ denotes the transverse component. Here r→=xx^+yy^+zz^ is the position vector represented in the Cartesian coordinate system, and x^, y^, and z^ denote the unit vectors along the x-, y-, and z-axes, respectively. In the optical potential well under consideration, an LRI particle is subjected to a scattering force, which can be expressed as the force vector [43] F→∝−12∇|Ef|2+β∑j=x,y,z|Efj|∇φj,(9)where the first term on the right-hand side of Eq. (9) denotes the intensity gradient force, indicating that LRI particles experience a repelling force due to strong light, and the second term is a generalization of the phase gradient force for arbitrary polarizations. Here φj denotes the phase of the component field Efj, and β is a scaling factor. In subsequent calculations, β is approximated as one for simplicity.

Figure 3 displays the flow density distributions corresponding to the dual-ellipse focal field presented in Fig. 2: (first column) intensity, (second column) tangential ϕ-component of momentum density, (third column) radial ρ-component of momentum density, and (fourth column) optical force distribution of LRI particles. Due to the sharp, steeply distributed intensity [29,34], a robust potential barrier is formed within the dark area between the two ellipses. This barrier will effectively trap the LRI particles, preventing their escape due to the sharp and confining nature of the curves.

In the first row of Fig. 3, with the parameters set to α=1, m1=0, and m2=30, the momentum density distribution in the transverse plane is uniform along the outer curve trajectory, but its direction is tangent to the curve. This will impart a counterclockwise driving force near the outer curve due to the momentum density acting on captured particles. Conversely, along the inner curve trajectory, the momentum density magnitude is zero, rendering the nearby particles unaffected by the momentum. From the transverse force distribution (shown in the right-most column of Fig. 3), it can also be observed that LRI particles will experience tangential and radial forces on the outer curve, while on the inner curve, only the radial force provided by the intensity gradient is present. The second row of Fig. 3 displays a different scenario wherein the uniform distribution of momentum density is disrupted, with the inner curve also bearing momentum. In this case, the settings of α=8, m1=5, and m2=30 lead to a cyclic variation in the angular momentum density magnitude for both the inner and outer ellipses, ranging from a maximum in the horizontal direction to a minimum in the vertical direction. The arbitrary distribution of momentum density on these curves significantly increases the degrees of freedom in particle manipulation. In contrast, the third row of Fig. 3 demonstrates inner and outer loops with different signs of m1 and m2, where m1=−5, m2=40, and α=1. The inner ellipse provides a driving force for LRI particles along the curve’s tangent in the clockwise direction, while the outer ellipse imparts a force in the counterclockwise direction. This distribution is clearly visible in the enlarged view within the white dashed box in the force vector diagram shown in the fourth column of Fig. 3. Under the combined action of the positive (counterclockwise) momentum flow on the outer ring and the negative (clockwise) momentum flow on the inner ring, the LRI particles straddling the inner and outer orbits experience a counterclockwise torque, resulting in a counterclockwise rotation. In other words, the orbital flow originating from the phase gradient along the inner and outer paths, rather than spin flow, can cause the LRI particles to self-rotate, which will be demonstrated in the following experiment.

It is important to note that the first through third rows of Fig. 3 feature different m values, yet the total intensity distribution of the focusing field, including the inner and outer loops, remains constant. This momentum density design paradigm enables simultaneous manipulation of HRI and LRI particles.

Figure 4 presents a detailed decomposition of the angular momentum corresponding to the light fields depicted in Figs. 2 and 3. The first column displays the z-component of the total angular momentum density, while the second column shows the distribution of the z-components of the orbital and spin angular momentum densities. It can be observed that the orbital angular momentum density plays a major role in the motion of particles, while the spin angular momentum density is small. Notably, the z-component (magnitude or direction) of the orbital angular momentum density corresponds to the inner and outer elliptical trajectories, determined by the topological charges on their respective paths.

In our experimental configuration, as illustrated in Fig. 5, we integrate an inverted microscope (Nikon, TE2000-U) with a 4−f complex light field generator featuring a reflective spatial light modulator (SLM). This SLM is programmed with a computer-generated hologram (CGH), calculated using Eq. (1), and implemented via a blazing-grating method [44]. A horizontally polarized and collimated green laser beam with a wavelength of 532 nm (Coherent, Verdi-v5) is incident on the SLM (Holoeye Leto, pixel pitch of 6.4 μm, pixel number of 1920×1080). The beam, once reflected off the SLM, enters a 4−f system comprising lenses L1 and L2. At the focal plane of L1, the +1st-order diffracted beam is selected by a spatial filter for passage. The output beam from the 4−f system is then redirected into an objective lens (Nikon Plan, 1.1 NA, 100× magnification). This objective lens produces the desired DC-OVB with the prescribed intensity and phase distribution at its focal region, creating a trap in a sample cell that contains both LRI particles (silver-coated hollow glass spheres with a mean diameter of 10 μm) and HRI particles (polystyrene spheres with a diameter of 1.3 μm) dispersed in deionized water. The scene of trapped particles under white light illumination (Nikon, Halogen) is imaged by the objective lens on a charge-coupled device (CCD) camera (Nikon, Digital Sight DS-Ri1, pixel pitch of 6.45 μm, pixel number of 1280×1024). Hollow glass beads are typically considered LRI particles because, when the shell thickness is sufficiently thin, the average refractive index of these beads is lower than that of the surrounding solution [19–25]. To further ensure the role of hollow glass beads as LRI particles, we specifically select silver-coated hollow glass spheres (Dantecdynamics Co., Item No. 80A7001). These semi-transparent spheres partially absorb light, aligning more closely with the desired LRI characteristics [45,46]. The schematic diagram of silver-coated hollow glass spheres (S-HGSs) used in our experiment is depicted in Fig. 6.

Figure 7(a) displays a CCD-recorded image depicting the capture of LRI particles by a DC-OVB featuring dual-focused bright rings. These rings, outlined by dashed yellow lines, are characterized by the parameter set q=(1,1,4,−0.5,2,2), with inner and outer diameters of 2.7 μm and 13.6 μm, respectively. Note that the difference in diameter between the inner and outer bright rings determines the width of the dark channel between the inner and outer rings, which is dictated by the size of the particles that need to be captured. Therefore, for particles with smaller diameters, the difference between the diameters of the inner and outer bright rings can be reduced. The forces influencing particle behavior are denoted as Fϕ, representing the tangent force originating from the OFD responsible for particle rotation, and Fρ, symbolizing the repulsive force exerted by the bright ring on the LRI particle, directed towards the dark area. In this specific configuration, particles are confined in the annular dark region between the two bright rings due to the repulsive forces, with the tangent forces propelling the particles along the annular trajectory. It is crucial to note that the two tangential forces, Fρ1 and Fρ2, provided by the inner and outer rings, can exhibit clockwise, counterclockwise, or varying directions. Figure 7(b) demonstrates the effect of transverse OFD carried by each of the inner and outer rings on the motion of the trapped particles in the experiment. Both the inner and outer rings effectively regulate particle motion velocity. Four inner-loop transverse OFD structures are designed with inner loops m1=5, 1, 0, and −1. Notably, in situations featuring high values of TCs associated with small inner curves, particularly when |m1|≥5 in our experimental configuration, the pixelated structure of the SLM will lead to a deterioration of the quality of the generated light field. This degradation proves detrimental to the optical trapping of particles.

These dual-ring structures are used to investigate the change of motion of captured LRI particles as the outer-loop transverse OFD is gradually increased from m2=0 to 40. To maintain experimental consistency and measure particle motion efficiently, multiple measurements are repeated in each set of experiments. The effective linear velocity was calculated by using focal field power as a normalization factor, measuring the variation of focal field power due to slight changes in the hologram while keeping outgoing laser power constant. Under identical outer-loop OFD, the linear velocity of particles increases with the inner-loop OFD (or m1). For each set of inner-loop OFD structures, linear velocity exhibits a linear increase as the outer-loop OFD (or m2) increases. The slower velocity when the inner and outer ring OFD directions are opposite is due to the cancellation of the positive and negative OFDs.

The primary feature of DC-OVBs is their unique ability to concurrently trap both HRI and LRI particles. Below, we will substantiate this capability through optical manipulation experiments involving both particle types. We designed three optical potential wells with varying curve shapes to manipulate both HRI and LRI particles. Table 1 presents q parameter values for circular, elliptical, and quadrilateral curve types, along with corresponding time periods and settings for m1, m2, and α values. These correspond to the ring curve featured in Visualization 1, the ellipse curve demonstrated in Visualization 2, and the quadrilateral curve showcased in Visualization 3 and Visualization 4. The direction of the OFD in the curve is determined by the sign of m, where positive values indicate a counterclockwise direction and negative values indicate a clockwise direction. The parameter α determines the uniformity of the OFD distribution along the curve, with α=1 resulting in linear accumulation (uniform distribution), while other non-zero values of α lead to a non-uniform distribution.Table 1.

Parameters of Curves to be Constructed

The first experiment involves a circular DC-OVB designed to propel both LRI and HRI particles along their respective circular trajectories. This DC-OVB comprises dual bright rings, delineated by yellow dashed lines in Fig. 8. The black dashed line represents the predetermined trajectory of the LRI particle. As expected, an LRI particle is confined within and moves along a circular trajectory between the inner and outer rings, with black arrows indicating the direction of motion. Simultaneously, HRI particles with a diameter of 1.3 μm are confined within and move along bright rings, as denoted by yellow arrows indicating the direction of motion. In the first row of Fig. 8, we set m1=−5 for the inner ring’s curve and m2=−40 for the outer ring’s curve. This configuration results in clockwise OFD directions along dual rings. Consequently, both HRI and LRI particles exhibit clockwise motion. In contrast to the first row, the second row of Fig. 8 features a smaller absolute value of m2 for the outer ring (changing from −40 to −30), resulting in slower orbital motion for LRI particles orbiting along the annulus trajectory as well as HRI particles orbiting along the outer ring. The time required for an LRI particle to complete one cycle increases from 0.36 to 1.08 s due to the diminished OFD supplied by the outer ring. In the third row of Fig. 8, the OFD carried by the inner and outer rings are oriented in opposite directions. Consequently, the HRI particles on the inner and outer rings move clockwise and counterclockwise, respectively. The period of the LRI particle further extends to 1.28 s. It is worth noting that careful observation reveals that, due to the opposite directions of the OFD of the inner and outer bright rings, the LRI particles situated between the inner and outer rings experience the involvement of OFDs in different directions. This results in the self-rotation of the LRI particles while they orbit, as illustrated by the red dashed ring arrows, contrasting with the purely orbital movement of HRI particles. It should also be noted that the simultaneous manipulation of HRI and LRI particles is not entirely independent, even though both types of particles can be simultaneously captured and exhibit different motion behaviors. This is evidenced by the fact that LRI particles can self-rotate while orbiting, whereas HRI particles can only orbit.

The Superformula expression allows for the creation of various complex curve types. Figures 9 and 10 depict the concurrent capture experiments of HRI and LRI particles along elliptical and quadrilateral trajectories. In Fig. 9, the values of (m1, m2) in the first, second, and third rows are set to (5, 40), (−5, −30), and (−5, 40), respectively. Similar to Fig. 8, the presence of opposite topological charges on the inner and outer curves triggers a self-rotation of the LRI particles while they orbit, as illustrated by the red dashed ring arrows in the third row of Figs. 9 and 10. Particularly in the case of the quadrilateral curve trajectory in the third row of Fig. 10, LRI particles rotate for more than 5.68 s before commencing orbital motion, due to the presence of corners in the quadrilateral curve. It can be observed that there is a jitter error in the motion trajectory of the LRI particles relative to the designed trajectory, mainly because the width of the dark-zone channel formed by the inner and outer curved trajectories we set is slightly larger than the diameter of the LRI particles, resulting in a radial oscillation of the LRI particles within the wider dark-zone channel, although they generally move along the predetermined trajectory.

Following the experiments for demonstrating the simultaneous manipulation of HRI and LRI particles along various trajectories, subsequent experiments focus on investigating the impact of the phase gradient on the motion of particles. As expressed by Eq. (2), the phase change control parameter α dictates the phase gradient along the curve trajectory, thereby affecting the motion rate of particles. When α=1, particles move uniformly along the curve, maintaining a constant linear velocity. For α>1 (<1), particles undergo acceleration (deceleration) along the curve. Figure 11 demonstrates experimental results of the variable-speed motion of particles along the quadrilateral DC-OVB, with m1=5 and m2=40 configured for the inner and outer loops. The first, second, and third rows showcase three quadrilateral DC-OVBs with corresponding α values of 1, 0.7, and 1.2, respectively. Time periods for LRI particles to complete one revolution around the curve are represented by τ1, τ2, and τ3 in the three rows, respectively. Subfigures in each row present instantaneous particle images at five different moments, each occurring after every quarter of one period. The white dotted arrow in the figure signifies the movement of the LRI particle, whereas the black dashed circle marks the particle’s position in the previous moment.

In this study, we propose an effective solution for manipulating both HRI and LRI particles using OFD. The constructed, arbitrarily-shaped DC-OVB possesses the capability to simultaneously capture HRI and LRI particles. Our theoretical analysis and experimental results validate the efficacy of DC-OVB in controlling particle motion. Furthermore, this research demonstrates how altering particle motion behaviors can be achieved by manipulating OFD, emphasizing the regulation of instantaneous particle velocities through phase gradient adjustments.

Our method has been designed for the two-dimensional manipulation of particles, establishing a foundational research platform for concurrently manipulating HRI and LRI particles. We believe that the principles underlying our approach can be expanded to enable three-dimensional manipulation. Furthermore, the proposed method can serve as a functional module, enhancing both efficiency and flexibility when integrated with microfluidics [47], particle assembly [48], drug delivery [49], and other techniques [50]. The outcomes of this research hold promise for advancing the application and development of controllable optofluidic tools and biomedical technologies.