Multidimensional optical tweezers synthetized by rigid-body emulated structured light
The manipulation of structured light with more extended degrees of freedom (DoFs) and in higher dimensions has recently attracted increasing attention owing to its advanced applications [1,2], which are significant among optical trapping or tweezer technologies with functional structures and increased precision [3,4]. The original optical tweezers can trap particles only at the intersection of six laser beams; the particles are held in a magnetic field by extrusion produced by opposite beams . Since Ashkin discovered that particles can be trapped by gradient forces of light using only one laser beam , optical tweezers have been applied in biosciences , nanotechnology [8–14], and force measurement  as non-contact and noninvasive tools. Usually, conventional single-beam optical tweezers trap particles by using a Gaussian beam. The movement of the trapped particles was performed by moving the sample stage  or changing the position of the light spot , which endows the optical tweezers with two DoFs (2D manipulation).
In the recent decade, two main techniques have been introduced in optical tweezers to significantly improve the flexibility of manipulation in higher dimensions, that is, digital holography and structured light. Digital holography refers to the acquisition and processing of holograms by a digital sensor array. Holographic optical tweezers were proposed by refreshing phase masks to change the focus position and provide dynamic and digital manipulation modes. These optical tweezers provide flexible manipulation with three DoFs. However, holographic optical tweezers use algorithms to move the optical trap so that its positioning in space depends on spatial parametric equations and the rotation matrix, and it is difficult to freely operate in 3D space because of the limitation of the parametric equation . Holographic optical tweezers rely on the apertures of the mask and elements, which limits the precision of their manipulation [19,20]. Note that the tomographic mold for optical trapping (TOMOTRAP) is a distinguished method that controls the particle’s attitude by analyzing the 3D refractive index of the sample and calculating the corresponding control light field . However, this method is time consuming to determine the sample’s 3D refractive index distribution. Moreover, the trapping capability of this method is constrained by the size of the particles being captured (particle diameters, 2–8 μm).
Owing to the recent emergence of digitally controlled structured light, optical tweezers based on structured light have been endowed with novel capabilities of an optical wrench involving a gradient force, which facilitates novel manipulation modes . Moreover, the structured light based on polarization modulation has been utilized for aligning asymmetric particles [23–26]. In addition to 3D positional trapping, they realized alignment of asymmetric particles. However, it is still a challenge to manipulate particles with three or more DoFs. On the other hand, a vortex beam possesses a spiral phase front that enables the transfer of orbital angular momentum (OAM) to particles, facilitating their rotation and translation. Hence, advanced structured optical tweezers have become a hot topic in optics and photonics [3,4,27]. Considering the rotation of the trapped particle on the structured beam, the existing structured optical tweezers can achieve a fourth DoF of rotation around the optical axis in addition to the three DoF displacements in space [3,28–41]. However, it is almost impossible to further improve the DoFs of manipulation based on the existing schemes. This limitation stems from the traditional view that a trapped particle’s motion appears as a point-like motion in 3D () space. Unfortunately, this perspective does not encompass the complete range of motion modes, which is why six-axis robotic arms and drones have been developed. While we cannot break out of the 3D framework, we can adopt a new approach that views the previous particle model as a rigid-body model. By leveraging novel forms of structured light, we can construct innovative optical tweezers with more DoFs and higher dimensions from this new perspective.
To achieve this goal, we introduce the concept of rigid-body mechanics in optical tweezers and propose 6D structured optical tweezers by tailoring structured light to simulate rigid-body mechanics. This structured light acts as a toroidal ring structure consisting of two semicircular arms with opposing OAMs and an interfering tip to trap particles. When the beam comes into contact with a particle, the particle is trapped at the tip by the ring arm. The combination of the coordinate transformation technique and Fourier transform theorem allows arbitrary rotation and displacement in free space. Six independent DoFs were achieved, and the motions of a six-axis rigid body were conducted, including waves, sway, heave, roll, pitch, and yaw. Therefore, toroidal light can move particles in free space and customize the particle trajectory using the Euler angle in rigid-body mechanics. Finally, we set up an experimental device and designed two experiments to verify the six-DoF modes of motion along arbitrary on-demand spatial trajectories.
2.1 A. Principles and Methods
In conventional optical tweezers, the trapped particle is treated as a mass point, the light and particle are treated separately, and only the particle motion state is studied with a maximum of three dimensions, as shown in Fig. 1(a). In our proposed scheme, trapping is treated as a rigid body in which the beam and particle are analyzed as a whole using rigid-body mechanics, as shown in Fig. 1(b). When particles are trapped by gradient forces, they can be moved by moving the beam as holographic optical tweezers  or by moving the sample chamber . It seems certain that the particle moving from one point to another can be decomposed into components of motion along the , , and axes. The motion starting at (, , ) can be expressed as
Based on these concepts, as shown in Fig. 1(b), rigid-body optical tweezers are composed of two semicircular vortex beams, represented by two different colors. The two parts carry opposite topological charges (TCs) and produce an interference point at one of the junctions to form a tweezers’ tip to trap particles. The structured light and trapped particle are then considered as a combined rigid body. Its generation and control methods are discussed in detail in Appendix A. Consequently, no matter where the particle is within the range of the rigid-body optical tweezers, it will eventually be trapped at the tip of the tweezers. We treat the combination of the structured beam and particle as a rigid body during manipulation. Therefore, the entire optical tweezers system is endowed with the properties of a rigid body, and the motion of the particle can be regarded as the motion of a fixed point on the rigid body. The surge , sway , and heave of the system can be precisely controlled by the Fourier phase-shift theorem and additional spherical waves as presented in Appendix A.
Fig. 1. Concepts of (a) conventional optical tweezers for manipulating mass-point particles and (b) rigid-body emulated structured light tweezers for manipulating irregular objects with full-degree-of-freedom six-axis motion.
To describe and regulate these poses better, we exploit the universal rotation theory of rigid-body Euler angles. Figure 2 shows a schematic of the surge, sway, heave, roll, pitch, and yaw. The six DoFs are indicated by arrows in the subgraphs in different colors. Each subgraph represents the motion of toroidal light for a specific DoF. The Euler angles consist of a nutation angle , precession angle , and spin angle , which correspond to the roll, yaw, and pitch in Fig. 2, respectively. Up to now, optical tweezers have been successfully endowed with these three additional rotation dimensions. According to Schaller’s theorem , the displacement of a rigid body can be produced by translation along its screw axis (Mozzi axis) followed by a rotation around an axis parallel to that screw axis. In our case, all rigid-body motions are conducted in two steps: by determining the displacements (, , ) and then executing the rotations (, , ). Finally, the rotation is evenly interpolated into the displacement. It should be noted that the three rotational DoFs do not conform to the law of exchange, which indicates that the rotation order affects the final result. Therefore, we can realize the rigid-body motion of toroidal light involving trapped particles in space, i.e., 6D optical tweezers. Subsequently, we select several paths for verification in the experiments.
Fig. 2. Schematic of six independent DoFs of the rigid-body mechanics with the structured light trapped particle, including surge, sway, heave, roll, pitch, and yaw.
2.2 B. Experimental Setup
To verify the 6D optical tweezers, we set up an experimental device as shown in Fig. 3. The laser beam was amplified by lenses L1 () and L2 () and adjusted into an approximate parallel beam. The 532 nm beam was then intercepted into a flat-top beam by a small aperture A1. Through polarizer P1, linearly polarized light illuminated the spatial light modulator (SLM, HOLOEYE, pixel size, ; resolution, pixels) for modulation. The modulated beam was filtered using a system comprising L3 () and L4 (). Finally, toroidal light (st order diffraction) was obtained, which was focused through MO1 (1.2 NA, ) to the sample chamber for particle manipulation. The illumination was red LED light focused through the microscopic objective MO2 (0.4 NA, ). Finally, the charge-coupled device (CCD, Basler acA1600-60gc; pixel size, ) recorded the motion of the particles. Polarizer P2 was used to eliminate the green light in the CCD. Yeast cells were selected as samples to highlight their application in biosciences. Toroidal light is generated by holography, and the mask is loaded or switched in the SLM in real time to manipulate the cells.
Fig. 3. Schematic of the experimental setup. L1, concave lens; L2–L5, convex lenses; P1, P2, polarizers; A1, A2, apertures; SLM, spatial light modulator; M1–M4, mirrors; CCD, charge-coupled device; MO1, MO2, micro-objectives. (a) Phase mask diagram, (b) intensity image of captured particle at
plane, and (c) 3D state model diagram of the captured particle.
3. RESULTS AND DISCUSSION
Owing to the advantages of the proposed 6D optical tweezers, we could freely customize the motion trajectories of the trapped particles. In this case, three rotational DoFs are expressed by the angle and three displacement DoFs are expressed in micrometers. A set of experiments was designed to verify the capacity of the proposed 6D optical tweezers. In the first experiment, yeast cells with asymmetrical sizes were chosen to confirm the beam’s capability in controlling the direction within the plane (see
Fig. 4. Steering ability of the beam was verified by manipulating the yeast cells. (a1)–(a5) Images recorded during the experiment, and the parameter of the beam is the spin angle
. (b1)–(b5) Schematics of the 3D model corresponding to the first row. (c1)–(c5) Images recorded when the yeast cells were manipulated by the spin angle and displacement along the axis . (d1)–(d5) Schematics of the 3D model corresponding to the third row. (e1)–(e5) Images recorded during the experiment of performing a 180º flip of the yeast cells, and the parameter of the beam is the nutation angle . (f1)–(f5) Schematics of the 3D model corresponding to the fifth row.
Fig. 5. (a) 3D model schematic of the TOMOTRAP method. (b) 3D model schematic of the rigid-body emulated optical tweezers. (c) Schematic of the operation of a 3D model of rigid-body optical tweezers. (d) Yeast cells are manipulated by the spin angle
, nutation angle , and displacement along the axis .
Execution of rigid-body motion of multiple particles is the outstanding advantage of the proposed 6D optical tweezers. For example, of two particles’ rigid-body motion, as depicted in Fig. 5(a), particles 1 and 2 are manipulated from to , and to in 3D space, respectively. During motion, particles 1 and 2 maintain the positions relationship of a rigid body. This is very difficult for previous manipulation techniques, especially the TOMOTRAP method  due to single particle manipulation. As shown in Fig. 5(b), it is easy to execute the same manipulation using the proposed rigid-body emulated optical tweezers. To verify the other DoFs and illustrate the directional change of the rigid body, we capture two particles at the same time and design a more complex 3D cycloid path, which is shown in Fig. 5 (for details, see
The capacity of the proposed 6D optical tweezers was greater than that of the conventional tweezers. The trapping range can be controlled by adjusting the radius of toroidal light, whereas the velocity of the particles can be controlled by adjusting the change frequency of the mask or the interval between changes in the DoF parameters (see Appendix B). The experiment of capturing yeast in different planes by changing the precession angle is described in Appendix B. Furthermore, the optical trap stiffness was measured as a standard to evaluate the performance of the proposed optical tweezers. Considering generality, polystyrene spheres were selected for the drag test. The power of the beam before entering the microscope objective prevails, and the test results indicate that the trap stiffness of the particle is proportional to the laser power for a specific particle. When the laser power is 30 mW, the optical trap stiffness is approximately 1.34 pN/μm, which is sufficient to support it to drive most biological cells without causing thermal damage. The details are presented in Appendix B.
For discussion, it is beneficial for readers to compare and review the proposed method with previous manipulation techniques, such as polarization modulation [23–26] and TOMOTRAP.
4. CONCLUSIONS AND OUTLOOK
Our experiments demonstrated the feasibility of using 6D optical tweezers. We also measure the performance parameters of 6D optical tweezers in Appendix B, which lays the foundation for their applications in other fields. In our case, based on the demonstration of trapped particles, only two trajectories were illustrated, but the six-DoF settings of the optical tweezers (in addition to making toroidal light perfectly parallel to the optical axis) were arbitrary. Thus, the velocity and trajectory of the particles can be customized according to researchers’ desires and requirements. In our experiment, the orientation of the multi-particle system can be controlled significantly, and we can achieve a large angle between the toroidal structured beam and the plane (enough to flip the particle 360°; see Appendix B and
The rigid-body-emulated tweezers in toroidal structured light provide extra DoFs for optical tweezers and achieve 6D optical tweezers. The proposed 6D optical tweezers can simultaneously generate multiple interference points with controllable spacing and are not limited to circular beams. Our method can be extended to novel advanced optical tweezers employing other kinds of complex structured light, such as torus knots [43–45], super-toroids , and ray–wave vector beams [47–49], which will lead to higher-dimensional control. This scheme is a fundamental platform, based on which even higher-dimensional (7D) optical tweezers can be developed by involving time as an additional DoF. The structured beam on the spatiotemporal scale can cause particles to execute arbitrary circular motions under certain circumstances [50–52], and more dimensions subjoin the optical tweezers by introducing space–time optics. The self-accelerating wave envelopment causes arbitrary temporal control acceleration for the optical field, which creates longitudinally invariant DoFs for optical manipulation [53–55]. Moreover, the introduction of symmetry and nonlinearity of increasingly complex structured light will provide more DoFs and lead to novel applications for complex tweezers and multi-particle manipulation [56–59]. In summary, we devised a basic method to generate higher-dimensional optical tweezers of arbitrary rigid-body objects, paving the way for studying more extreme light–matter interactions in the future.
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