Optical tweezers [1–3] are three-dimensional traps formed by highly focused laser beams. Optical force and torque emerge from momentum and angular momentum exchange between light and matter, resulting in coupled translation and rotation. Angular trapping further adds one-dimensional spin based on the three-dimensional translation by modulating the properties of the optical field [4–8] or particles [9–13] and has been widely applied in cell biology [14–17], molecular biology [18–22], micro motor [23–29], and optical tweezers in vacuum [30–34]. Janus particles possess two distinct compositions [35,36] and have recently attracted great interest from researchers due to their unique structures and functions. Zong et al. [37] realized the bistable rotation of a gold-coated Janus colloidal particle. Nedev et al. [38] manipulated the reversible displacement of a silica sphere with a gold half-shell along the axis of the laser beam. Peng et al. [27] developed opto-thermoelectric microswimmers based on dielectric-Au Janus particles to achieve in-plane rotation under a temperature-gradient-induced electrical field. Our previous work also proposed a new method for angular trapping based on chemically synthesized Janus microspheres [39,40]. For Janus microspheres, the adjustable size ratio of the compositions implies “programmable” optical properties, yet their great potential for optical manipulation has been neglected. It remains unknown what kinematic characteristics will be exhibited by Janus microspheres with different composition ratios in the optical field. Hence, it is critical to investigate the inner mechanism of how designable ratios affect steady postures and rotational properties.

In this paper, we report the optical manipulation based on Janus microspheres with various size ratios of polystyrene (PS) and polymethyl methacrylate (PMMA). The refractive index distribution of Janus microspheres is quantified by the difference in the optical path between the two compositions along the rotation axis of the microspheres. We employ an efficient electromagnetic scattering computation method, the T-matrix method, to obtain the trapped postures of Janus microspheres. Their rotation radii maintain a good linear relationship with the optical range difference over a specific range. Moreover, at a particular value, the microspheres flip and exhibit a similar “double-lens device” along the optical axis. We develop a robust microfluidic methodology to prepare Janus microspheres with different composition ratios by adjusting the concentration and flow rate of the solution and capture them in optical tweezers. The results are in high agreement with the simulations. Our work provides a systematic and feasible solution for the theoretical analysis of force and torque, designable preparation, and angular trapping of heterogeneous particles. It will greatly improve optical tweezers’ precise manipulation and measurement capabilities and promises to promote the study of multidimensional mechanical properties in biophysics.

The T-matrix method simulates electromagnetic scattering from single or mixed homogeneous particles of arbitrary shape [41–43]. Expand the incident and scattered field as a finite sum of vector spherical wave functions in the coordinate system OlXlYlZl (see Appendix A for full expressions). The expansion coefficients are linearly transited by the matrix T: (pnmqnm)=T(anmbnm),(1)where anm and bnm are the expansion coefficients of the incident field obtained through the point-matching method [44]; p and q are those of the scattered field. The force and torque applied on a particle depend on the changes in momentum and angular momentum of the beam passing through, which are related to expansion coefficients. The formulas for the z-axis components of trapping and torque efficiency, Fz and τz, are included in Appendix B. Calculate the x- and y-axis components in the same formula by rotating the incident field 90 deg around the y axis and x axis, respectively [45,46].

The most important, complex, and time-consuming part is constructing the matrix T. The matrix T depends on the particles’ size, shape, and refractive index distribution. Given the specific optical field, as long as the matrix T is constructed once, we can repeatedly use it to calculate force and torque. When the relative positions and orientations of the incident field and particle change, multiplying the expansion coefficients or the matrix T by the corresponding translation or rotation matrices can calculate the new scattered field [46–49], greatly reducing the computation time. The T-matrix method simulates the trapping trajectory of the particle in the optical field by multiple iterations until the force and torque approach zero. At this point, we acquire the positions and orientations of the trapped particle. It is worth noting that the T-matrix method yields absolute equilibrium states of Janus microspheres. Therefore, the simulation is not really a dynamical process, and only involves the numerical calculation and iteration of optical force and torque.

The extended boundary condition method (EBCM) effectively constructs the matrix T of a homogeneous particle with rotational symmetry [50–52]. Establish a coordinate system OXp′Yp′Zp′, with the particle’s center as the origin and its rotation axis of symmetry as the z axis. Its matrix T is expressed as [52] T=−Qp33(km,kp)[Qp13(km,kp)]−1,(2)where the corner marks 1 and 3 represent the scattered field and incident field, respectively, km=2πnm/λ and kp=2πnp/λ represent the wave number in the medium and microsphere, respectively, and nm and np represent their refractive indices (refer to Appendix C for a detailed calculation of the matrix Q).

For a concentric bilayer microsphere, as shown in Fig. 1(a), assuming the refractive index of the shell is n1 and the core is n2, the matrix T is expressed as [53,54] T=[T1−Q131T2(Q113)−1][I+Q111T2(Q113)−1]−1,(3)where I is the identity matrix, T1=−Q133(km,k1)⋅[Q113(km,k1)]−1 is the matrix of the shell, and T2=−Q233(k1,k2)[Q213(k1,k2)]−1 is the core, with k1 and k2 denoting the wave number in the shell and core, respectively. For a Janus microsphere, which is chemically synthesized by polystyrene (PS, n1=1.57) and polymethyl methacrylate (PMMA, n2=1.49), as shown in Fig. 1(b), the dark blue part is PS as the shell, the dashed line is the virtual shell, and the light blue part is PMMA as the core. Thus, the Janus microsphere is equivalent to an eccentric bilayer microsphere with rotational symmetry, the matrix T of which can also be calculated using Eq. (3). Two differences worth noting, first, the enclosed surface surrounding the PMMA core is no longer a complete sphere, but rather a spherical crown surface with an interface and, second, due to the noncoincidence of the centers of the shell and core, their matrices T are not in a unified coordinate system, and T2 needs to be translated along the z axis: T2=T(−nPSd)T2T(nPSd),(4)where d is the distance between the centers of the PMMA part and the microsphere. T(nPSd) and T(−nPSd) are translation matrices in the positive and negative directions along the z axis, respectively. The detailed derivation process can be referred to Appendix D. Since the matrix T of PMMA is translated in the PS, the optical path needs to be multiplied by the refractive index of PS.

As shown in Fig. 1(c), the yellow arrow indicates the normal vector perpendicular to the interface of the Janus microsphere. The pitch angle θ is the angle of the normal vector with the +z axis (i.e., light propagation direction), while the azimuth angle φ is the angle of its projection on the XOY plane with the +x axis (i.e., light polarization direction), which describe the orientation of the Janus microsphere. The volume ratios between the PS and PMMA parts are adjustable during preparation. To quantify the refractive index distribution, we define the optical path difference along the rotation axis between PS and PMMA parts as shown in Fig. 1(d): Δ=nPS·hPS−nPMMA·hPMMA,(5)where hPS and hPMMA are heights of PS and PMMA crowns, respectively.

The Janus microsphere discussed in this paper is an ideal model: a sphere with a planar interface. The corresponding closed-surface equations need to be established for other nonideal cases, such as curved interfaces or Janus ellipsoids. Theoretically, the T-matrix method can simulate the electromagnetic scattering of particles with more complex structures.

As listed in Table 1, set the incident field as the Gaussian beam propagating along the +z axis and linearly polarized along the x axis, with the power P of 1 W and the wavelength λ of 1064 nm; the numerical aperture (NA) of the objective is 1.2. The refractive index of the medium, nm, is 1.33; the radius of the Janus microspheres is 4 μm; the volume ratios of PS to PMMA spherical crowns range from 5:1 to 1:5. The initial position of the Janus microspheres (x0,y0,z0)=(0,0,0)μm; the initial orientation (θ0,φ0)=(45,45)deg.Table 1.

Initial Setup of the Incident Field and Janus Microspheres

Here, we discuss two opposite ratios, PS:PMMA = 5:1 and 1:5, as examples. For the Janus microsphere with the ratio 5:1, as shown in Figs. 2(a) and 2(b), the final position (x,y,z)=(0,0.238,0.769)μm. The interface is approximately parallel to the light propagation direction (i.e., θ=97.9deg) and strictly parallel to the light polarization direction (i.e., φ=90deg).

Rotate the polarization direction around the z axis and set it to [cosαsinα], where α is the angle between the polarization direction and the +x axis. Figures 2(c) and 2(d) display the changes in microsphere’s postures at different polarization directions. The Janus microsphere rotates synchronously around the z axis. Its position on the z axis, rotation radius (i.e., the distance between the center of the microsphere and the trap), and pitch angle remain constant. The azimuth angle is strictly equal to 90 deg − α; i.e., the interface is always parallel to the light polarization direction. Ratios 4:1–1:1 all perform similarly, especially maintaining good synchronization with the light polarization direction, as shown in Table 2.Table 2.

Steady Postures of Janus Microspheres with PS:PMMA Ratios of 5:1 to 1:1

In the x-axis linearly polarized optical trap, the steady postures of Janus microsphere vary with ratios (PS:PMMA = 5:1–1:5), as shown in Figs. 4(a) and 4(b). According to the energy minimum principle, the PS part with higher refractive index overlaps with the higher intensity region of the optical field as much as possible. The electric field intensity extends most in the light propagation direction (z axis), then in the light polarization direction (x axis), and finally in the y axis. Hence, at the ratios of 5:1 to 1:2, the center of the optical trap is within the PS part and the interface of the microsphere is as parallel as possible to the polarization direction; i.e., the azimuth angle is strictly equal to 90 deg. The field intensity and the microsphere structure are symmetric about the x axis, so the position x remains zero. As the PS part decreases, its center moves away from the center of the microsphere, so the position y increases. In addition, the pitch angle decreases, allowing the PS part to coincide with the area of high field intensity as much as possible. More interestingly, after replacing the ratio with optical path difference, the position y varies linearly between 6λ and 0 (roughly the ratio 5:1 to 1:1). This means that within this linear interval, we can reverse design and prepare microspheres based on the manipulation properties of interest. At the ratios of 1:3 to 1:5, the overlap of the whole sphere with the optical field is prioritized due to the small PS part. As a result, the orientation of microspheres flips, showing a “double lens” along the z axis.

Changing the initial orientation of the microspheres, we can get another steady state as shown in Table 4 and Figs. 4(c) and 4(d). It is not difficult to find that for ratios of 5:1 to 1:2, the positions y take opposite values, and the other parameters remain unchanged. For the ratios of 1:3 to 1:5, the orientations of the microspheres flip 180 deg along the z axis.Table 4.

Alternative Steady Postures of Janus Microspheres with PS:PMMA Ratios of 5:1 to 1:5

In summary, the T-matrix method, after finite calculations in optical force and torque (no more than 50 min for each ratio), yields the final positions and orientations of Janus microspheres with different component ratios in the linearly polarized optical trap, as well as the rotational manipulation properties based on the light polarization direction. The trapped postures of microspheres result from the effect of field intensity and refractive index distributions. The T-matrix method provides a fast and convenient way to simulate the electromagnetic scattering of such heterogeneous particles in the optical field, which shows great potential for analyzing the optical force and torque exerted on particles with complex structures.

Microfluidic and phase separation prepare the Janus microspheres [55–59]. In the self-made microfluidic chip, the aqueous sodium dodecyl sulfate (SDS, AR, Tianjin Yuanli Chemical Co., Ltd.) solution is the continuous phase; the polystyrene (PS, general type I, Aladdin) and polymethyl methacrylate (PMMA, heat resistant injection grade, Aladdin) are dissolved in the organic solvent trichloromethane (TM, AR, Tianjin Yuanli Chemical Co., Ltd.) to form the dispersed phase. As shown in Fig. 5(a) and Visualization 1, at the capillary tip, the dispersed phase is “sheared” into mixed droplets by the continuous phase. The volatilization of TM promotes the phase separation of PS and PMMA to structure solid Janus microspheres. The size and the ratio can be adjusted by regulating the flow rate and concentration of the solution.

The shape of Janus particles in Figs. 5(c) and 5(d) is close to sphere. The interface of the microsphere appears to be “curved” because the orientation of the microsphere is random. Due to the precipitation of SDS during drying, it adheres to the microsphere and silicon chip. The microsphere’s surface appears less smooth in Fig. 5(d). The average diameter of 300 Janus microspheres is 8.67 μm, with a coefficient of variation (CV) of 14.19%, roughly conforming to the normal distribution.

The steady postures and rotational characteristics of Janus microspheres with different PS:PMMA ratios are verified on the self-built optical tweezers system, as shown in Fig. 6(a). After beam collimation and expansion (NT59-129, Edmund Optics), the Gaussian laser beam (J20I-BL-106C, Spectra Physics) is focused by the objective lens (C-Apochromat 63×/1.20W Corr UV-VIS-IR, ZEISS) to form a linearly polarized optical trap. The polarizer regulates the magnitude of beam power and fixes the output polarization direction. A three-dimensional piezoelectric precision displacement stage (P-500, Physik Instrumente) controls the movement of the sample cell near the focal plane. The illumination path is coupled to the laser path via two dichroic mirrors (DMs, Precision Photonics). A CCD camera observes and acquires real-time images of trapped microspheres, facilitating the subsequent image processing and extraction of positions and orientations. The sample cell consists of a slide (24μm×50μm×170μm) and a cover glass (18μm×18μm×170μm). Use a pipette to inject an appropriate amount of Janus microsphere solution, seal the slides with epoxy resin, and fix the cell on the stage.

The module that controls the light polarization direction consists of a half-wave plate (WPH05M-1064, Thorlabs), a rotating displacement stage (PRM1Z8, Thorlabs), and a servo motor (KDC101, Thorlabs) manipulated by the software Kinesis. The half-wave plate alters the light polarization direction but does not modify other properties of the beam. Assuming that the angle between the polarization direction and the fast axis of the half-wave plate is β, the angle after passing through is still β, but it is rotated by 2β concerning the incident beam, as shown in Fig. 6(b). As the half-wave plate rotates, the polarization direction of the output light rotates at twice the speed. The controlled rotation based on the light polarization direction driven by the half-wave plate is an essential basis for the diverse manipulation of Janus microspheres.

We prepare Janus microspheres with ratios of 5:1, 4:1, 2:1, 1:1, 1:2, 1:3, 1:4, and 1:5. The density of PS is 1.05g·cm⁻³ and that of PMMA is 1.19g·cm⁻³. In Janus microspheres with different ratios (from 5:1 to 1:5), the mass of PMMA part is always greater than that of PS part. As the microspheres freely fall in the vertical sample cell, as shown in Fig. 7(a), the PMMA part is consistently oriented downward. Figure 7(b) presents the trapped postures, with the light propagation direction perpendicular to the paper. The red arrow indicates the light polarization direction, while the white crosses are centers of the optical trap and the red are centers of the Janus microspheres. The orientation of microspheres flips at ratio 1:2, which differs from the theoretical ratio 1:3. The pitch angle of the microsphere with ratio 1:2 is around 27 deg, which is very close to the flipped postures. We mentioned in Section 2.B that the simulation is based on an ideal heterogeneous sphere. However, in reality, the prepared Janus particles may have an ellipsoidal shape and their interfaces may be curved. This can potentially affect the postures of microspheres. Figures 7(c) and 7(d) reveal that from ratios of 5:1 to 1:1 (no 3:1), the distances between the centers of microspheres and the trap increase accordingly. The azimuthal angle is slightly disturbed within ±3deg of the polarization direction. After the microspheres flip, the two centers almost coincide, in perfect agreement with the simulation. Due to the inability to measure the position z and the pitch angle, we will not consider these two parameters.

So far, we can determine that the rotational manipulation of microspheres with ratios 5:1 to 1:1 is feasible. For the ratio 5:1, we set the rotation speed of the half-wave plate to 25deg·s−1 (i.e., the rotation speed of the light polarization direction is 50deg·s−1) to obtain the variation of positions x, y and orientations with the polarization directions, as shown in Figs. 8(a) and 8(b). Visualization 2 displays the rotation at different ratios (5:1, 4:1, 2:1, 1:1), and their circular fitting curves (black) are shown in Fig. 8(c). The rotation of Janus microspheres based on the light polarization direction is circular around the center of the optical trap. The orientations are well synchronized with the polarization directions, consistent with the simulation. Take five sets of data for each ratio to count the rotation radii, as shown in Fig. 8(d), which increase with the decrease in the optical path difference.

For the ratio 1:5, set the same rotation speed. The rotation is shown in Figs. 8(e) and 8(f) and Visualization 2, which is the restricted Brownian motion of the microsphere in the optical trap. The microspheres have flipped, so we do not discuss the azimuth angle.

In summary, the steady postures and rotational properties of Janus microspheres with different PS:PMMA ratios in the linearly polarized optical trap remain consistent with the simulation. Although there is a discrepancy as to whether the Janus microspheres with ratio 1:2 flip or not, the trends of the microspheres’ trapped positions and azimuth angles with different ratios are similar, and the values are basically close to each other. This has proved the reliability of the T-matrix method in calculating the electromagnetic scattering of heterogeneous microspheres in the optical field. In the succeeding work, we will develop the simulation method applicable to microspheres with multiphase and further enrich the theory of manipulated microspheres with more complex structures.

In this paper, we study the steady postures and rotational properties of Janus microspheres with different PS:PMMA ratios in the linearly polarized optical trap. The T-matrix method calculates the optical force and torque applied on the trapped microspheres to obtain their positions and orientations; the self-made Janus microspheres are captured in the optical tweezers system. The variations of microspheres’ postures with ratios (or optical path differences) satisfy the energy minimum principle and remain consistent in simulation and experiment. We have succeeded in providing a systematic and robust strategy for analyzing the manipulated characteristics of Janus microspheres with various ratios. It serves as a new idea for calculating the electromagnetic scattering of multiphase particles with more complex structures and a promising solution for designing and implementing diverse manipulation of particles.