Optical manipulation of ratio-designable Janus microspheres

Yulu Chen; Cong Zhai; Xiaoqing Gao; Han Wang; Zuzeng Lin; Xiaowei Zhou; Chunguang Hu

doi:10.1364/PRJ.517601

1. INTRODUCTION

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.

2. THEORY AND SIMULATION

A. Basic Theory of the $T$ -Matrix Method

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 $O_{l} X_{l} Y_{l} Z_{l}$ (see Appendix A for full expressions). The expansion coefficients are linearly transited by the matrix $T$ :

(\begin{matrix} p_{nm} \\ q_{nm} \end{matrix}) = T (\begin{matrix} a_{nm} \\ b_{nm} \end{matrix}),

where

a_{nm}

and

b_{nm}

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,

F_{z}

and

τ_{z}

, 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.

B. Extended Boundary Condition Method

The extended boundary condition method (EBCM) effectively constructs the matrix $T$ of a homogeneous particle with rotational symmetry [50–52]. Establish a coordinate system $O X_{p}^{'} Y_{p}^{'} Z_{p}^{'}$ , 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 = - Q_{p}^{33} (k_{m}, k_{p}) {[Q_{p}^{13} (k_{m}, k_{p})]}^{- 1},

where the corner marks 1 and 3 represent the scattered field and incident field, respectively,

k_{m} = 2 π n_{m} / λ

and

k_{p} = 2 π n_{p} / λ

represent the wave number in the medium and microsphere, respectively, and

n_{m}

and

n_{p}

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 $n_{1}$ and the core is $n_{2}$ , the matrix $T$ is expressed as [53,54]

T = [T_{1} - Q_{1}^{31} T_{2} {(Q_{1}^{13})}^{- 1}] {[I + Q_{1}^{11} T_{2} {(Q_{1}^{13})}^{- 1}]}^{- 1},

where

I

is the identity matrix,

T_{1} = - Q_{1}^{33} (k_{m}, k_{1}) \cdot {[Q_{1}^{13} (k_{m}, k_{1})]}^{- 1}

is the matrix of the shell, and

T_{2} = - Q_{2}^{33} (k_{1}, k_{2}) {[Q_{2}^{13} (k_{1}, k_{2})]}^{- 1}

is the core, with

k_{1}

and

k_{2}

denoting the wave number in the shell and core, respectively. For a Janus microsphere, which is chemically synthesized by polystyrene (PS,

n_{1} = 1.57

) and polymethyl methacrylate (PMMA,

n_{2} = 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

T_{2}

needs to be translated along the

z

axis:

T_{2} = T (- n_{PS} d) T_{2} T (n_{PS} d),

where

d

is the distance between the centers of the PMMA part and the microsphere.

T (n_{PS} d)

and

T (- n_{PS} d)

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.

Fig. 1. Schematic illustration of (a) concentric bilayer microsphere, (b) eccentric bilayer microsphere, and (c) trapped Janus microsphere. (d) Optical path difference of Janus microspheres of the radius 4 μm with different ratios.

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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 $X O Y$ 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 = n_{PS} \cdot h_{PS} - n_{PMMA} \cdot h_{PMMA},

where

h_{PS}

and

h_{PMMA}

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.

C. Simulation Results

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, $n_{m}$ , 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 $(x_{0}, y_{0}, z_{0}) = (0, 0, 0) μm$ ; the initial orientation $(θ_{0}, φ_{0}) = (45, 45) \deg$ .

Table 1. Initial Setup of the Incident Field and Janus Microspheres

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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., $θ = 97.9 \deg$ ) and strictly parallel to the light polarization direction (i.e., $φ = 90 \deg$ ).

Fig. 2. Changes in (a) positions and (b) orientations of the Janus microsphere with ratio PS:PMMA = 5:1 during trapping. Changes in trapped microsphere (c) positions and (d) orientations with light polarization directions.

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Rotate the polarization direction around the $z$ axis and set it to [ $\cos α$ $\sin α$ ], 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

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For the Janus microsphere with the ratio 1:5, as shown in Figs. 3(a) and 3(b), its final position $(x, y, z) = (0, 0, 1.651) μm$ , and the pitch angle is 0 deg. Although the azimuth angle converges, the value of the azimuth angle at any point on the $z$ axis has no real physical meaning and is therefore not discussed. The interface is perpendicular to the light propagation direction. When rotating the light polarization direction, the position and pitch angle of the microsphere remain unchanged, as shown in Figs. 3(c) and 3(d). Although the Janus microsphere can be stably captured, it cannot achieve the controlled rotation based on the light polarization direction. Summarized from Tables 2 and 3, for microspheres with the radius of 4 μm, when the volume ratios are from 5:1 to 1:2, the Janus microsphere can theoretically perform controllable rotational manipulation, while these with ratios from 1:3 and 1:5 present a “two-lens device” along the light propagation direction.

Fig. 3. Changes in (a) positions and (b) orientations of the Janus microsphere with ratio PS:PMMA = 1:5 during trapping. Changes in trapped microsphere (c) positions and (d) orientations with light polarization directions.

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Table 3. Steady Postures of Janus Microspheres with PS:PMMA Ratios of 1:2 to 1:5

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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 $6 λ$ 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.

Fig. 4. (a) Positions and (b) orientations of Janus microspheres vary with the ratios (or the optical path difference). (c), (d) Alternative steady postures when initial orientation changes.

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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

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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.

3. METHOD

A. Preparation of Janus Microspheres

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.

Fig. 5. (a) Preparation of the Janus microspheres. (b) Histogram of the Janus microspheres’ diameters and fitted normal distribution curve in orange. The illustrations of the Janus microsphere with the ratio 5:1 under (c) optical microscope and (d) scanning electron microscope. The scale is 2 μm.

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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.

B. Experimental Setups

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 $63 \times / 1.20 W$ 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 ( $24 μm \times 50 μm \times 170 μm$ ) and a cover glass ( $18 μm \times 18 μm \times 170 μm$ ). 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.

Fig. 6. (a) Schematic diagram of the optical tweezers system. (b) Change in the light polarization direction after passing through the half-wave plate.

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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 $2 β$ 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.

4. RESULTS

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 $1.05 g \cdot cm$ ⁻³ and that of PMMA is $1.19 g \cdot cm$ ⁻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 $\pm 3 \deg$ 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.

Fig. 7. (a) Free and (b) trapped Janus microspheres with different ratios. The scale is 2 μm. (c) Positions and (d) orientations of Janus microspheres vary with the ratios (or the optical path difference).

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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 $25 \deg \cdot s^{- 1}$ (i.e., the rotation speed of the light polarization direction is $50 \deg \cdot s^{- 1}$ ) 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.

Fig. 8. Changes in (a) positions and (b) orientations of Janus microspheres with ratio 5:1 when light polarization direction rotates. (c) Changes in positions of Janus microspheres with ratios 5:1, 4:1, 2:1, and 1:1. The black curves are fitted circles, and the legend indicates rotation radii. (d) Statistical results for rotation radii. Changes in (e) positions and (f) orientations of Janus microspheres with ratio 1:5 when light polarization direction rotates.

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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.

Fig. 9. Coordinate systems of the optical field and the microsphere.

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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.

5. CONCLUSION

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.

APPENDIX A: EXPANSION OF OPTICAL FIELD

Establish a coordinate system $O_{l} X_{l} Y_{l} Z_{l}$ with the center of the optical trap as the origin and the light propagation direction as the $+ z$ axis to expand the incoming and outgoing fields [45,52]:

E_{in} = \sum_{n = 1}^{N_{\max}} \sum_{m = - n}^{n} a_{nm}^{in / out} M_{nm}^{(2)} (k r) + b_{nm}^{in / out} N_{nm}^{(2)} (k r),

E_{out} = \sum_{n = 1}^{N_{\max}} \sum_{m = - n}^{n} p_{nm}^{in / out} M_{nm}^{(1)} (k r) + q_{nm}^{in / out} N_{nm}^{(1)} (k r),

where

k

is the wave number in the medium,

r

is the radius vector of the minimum enclosing sphere,

N_{\max} = k r + 3 \sqrt[3]{k r}

is the truncation factor,

a_{nm}^{in / out}

,

b_{nm}^{in / out}

and

p_{nm}^{in / out}

,

q_{nm}^{in / out}

are the expansion coefficients of the incoming and outgoing fields, respectively [44],

M_{nm}^{(2)}

and

N_{nm}^{(2)}

are vector spherical wave functions of the extended inward-propagating TE and TM fields, and

M_{nm}^{(1)}

and

N_{nm}^{(1)}

are those of the outward-propagating fields:

M_{nm}^{(1, 2)} (k r) = N_{n} h_{n}^{(1, 2)} (k r) C_{nm} (θ, φ),

N_{nm}^{(1, 2)} (k r) = \frac{h_{n}^{(1, 2)} (kr)}{k r N_{n}} P_{nm} (θ, φ) + N_{n} [h_{n - 1}^{(1, 2)} (k r) - \frac{n h_{n}^{(1, 2)} (kr)}{kr}] B_{nm} (θ, φ),

where

h_{n}^{(1, 2)} (k r)

are spherical Hankel functions of the first and second kind,

N_{n} = 1 / \sqrt{n (n + 1)}

are normalization constants, and C_nm

(θ, φ)

, B_nm

(θ, φ)

, and P_nm

(θ, φ)

are the vector spherical harmonics:

C_{nm} (θ, φ) = \nabla \times [r Y_{n}^{m} (θ, φ)] = \hat{θ} \frac{im}{\sin θ} Y_{n}^{m} (θ, φ) - \hat{φ} \frac{\partial}{\partial θ} Y_{n}^{m} (θ, φ),

B_{nm} (θ, φ) = r \nabla Y_{n}^{m} (θ, φ) = \hat{θ} \frac{\partial}{\partial θ} Y_{n}^{m} (θ, φ) + \hat{φ} \frac{im}{\sin θ} Y_{n}^{m} (θ, φ),

P_{nm} (θ, φ) = {r Y}_{n}^{m} (θ, φ),

where

Y_{n}^{m} (θ, φ)

is the normalized scalar spherical harmonic, and

\hat{θ}

and

\hat{φ}

are unit vectors of pitch and azimuth, whose components are the angular partial derivatives of

Y_{n}^{m} (θ, φ)

.

Since the wave function has the singularity at the origin, define the regular vector spherical wave functions without singularities:

R g M_{nm}^{(3)} (k r) = \frac{1}{2} [M_{nm}^{(1)} (k r) + M_{nm}^{(2)} (k r)],

R g N_{nm}^{(3)} (k r) = \frac{1}{2} [N_{nm}^{(1)} (k r) + N_{nm}^{(2)} (k r)] .

Replacing the spherical Hankel functions in Eqs. (A3) and (A4) with the spherical Bessel functions $j_{n} (k r) = [h_{n}^{(1)} (k r) + h_{n}^{(2)} (k r)] / 2$ can calculate $R g M$ and $R g N$ . The incident field is the whole optical field in the absence of a scatterer, while the scattered field is the changes in the optical field due to the presence of a scatterer:

E_{inc} = \sum_{n = 1}^{N_{\max}} \sum_{m = - n}^{n} a_{nm}^{inc / scat} R g M_{nm}^{(3)} (k r) + b_{nm}^{inc / scat} R g N_{nm}^{(3)} (k r),

E_{scat} = \sum_{n = 1}^{N_{\max}} \sum_{m = - n}^{n} p_{nm}^{inc / scat} M_{nm}^{(1)} (k r) + q_{nm}^{inc / scat} N_{nm}^{(1)} (k r),

where

a_{nm}^{inc / scat}

,

b_{nm}^{inc / scat}

and

p_{nm}^{inc / scat}

,

q_{nm}^{inc / scat}

are the expansion coefficients of the incident and scattered fields, respectively.

T^{inc / scat}

is the transition matrix:

(\begin{matrix} p_{nm}^{inc / scat} \\ q_{nm}^{inc / scat} \end{matrix}) = T^{inc / scat} (\begin{matrix} a_{nm}^{inc / scat} \\ b_{nm}^{inc / scat} \end{matrix}) .

The two sets of expansion coefficients are related, where the incident field includes the outgoing field [60]:

(\begin{matrix} \begin{matrix} a^{in / out} \\ b^{in / out} \end{matrix} \\ \begin{matrix} p^{in / out} \\ q^{in / out} \end{matrix} \end{matrix}) = (\begin{matrix} \begin{matrix} 0.5 a^{inc / scat} \\ 0.5 b^{inc / scat} \end{matrix} \\ \begin{matrix} 0.5 a^{inc / scat} + p^{inc / scat} \\ 0.5 b^{inc / scat} + q^{inc / scat} \end{matrix} \end{matrix}) .

Appendix B: Calculation of Force and Torque

The incoming/outgoing coefficients are employed to calculate the changes in momentum and angular momentum (i.e., optical force and torque) transmitted by the optical field. The normalized force acting on the scatterer along the $z$ axis is [52]

F_{z} = \frac{2}{P} \sum_{n = 1}^{N_{\max}} \sum_{m = - n}^{n} {\frac{m}{n (n + 1)} Re (a_{nm}^{*} b_{nm} - p_{nm}^{*} q_{nm}) - \frac{1}{n + 1} {[\frac{n (n + 2) (n - m + 1) (n + m + 1)}{(2 n + 1) (2 n + 3)}]}^{\frac{1}{2}} \times Re (a_{nm} a_{n + 1, m}^{*} + b_{nm} b_{n + 1, m}^{*} - p_{nm} p_{n + 1, m}^{*} - q_{nm} q_{n + 1, m}^{*}),

while the corresponding normalized torque is

τ_{z} = \frac{1}{P} \sum_{n = 1}^{N_{\max}} \sum_{m = - n}^{n} m ({| a_{nm} |}^{2} + {| b_{nm} |}^{2} - {| p_{nm} |}^{2} - {| q_{nm} |}^{2}),

where

P = \sum_{n = 1}^{N_{\max}} \sum_{m = - n}^{n} {| a_{nm} |}^{2} + {| b_{nm} |}^{2}

is the total power of the optical field.

Appendix C: Calculation of Matrix $Q$

Denote the matrix $Q_{p}^{xy} (k_{m}, k_{p})$ by four partitioned matrices [50]:

Q_{p}^{x y} (k_{m}, k_{p}) = (\begin{matrix} {(Q_{p}^{x y})}_{v u}^{11} & {(Q_{p}^{x y})}_{v u}^{12} \\ {(Q_{p}^{x y})}_{v u}^{21} & {(Q_{p}^{x y})}_{v u}^{22} \end{matrix}),

where the values of

x

and

y

take only 1 and 3, and represent the scattered field and incident field, respectively.

k_{m}

denotes the wave number of the environment, and

k_{p}

denotes the wave number of the particle. The environment and particle mentioned here are relative. For example, when calculating the matrix

T

of PS, the environment is water and the particle is PS sphere, while calculating the matrix

T

of PMMA, the environment is PS and the particle is a PMMA spherical crown. Each submatrix can be eventually converted into the integral of the spherical wave function over the surface enclosing the scatterer, expressed as

(\begin{matrix} \begin{matrix} {(Q_{p}^{xy})}_{vu}^{11} \\ {(Q_{p}^{xy})}_{vu}^{12} \end{matrix} \\ \begin{matrix} {(Q_{p}^{xy})}_{vu}^{21} \\ {(Q_{p}^{xy})}_{vu}^{22} \end{matrix} \end{matrix}) = - i k_{m} k_{p} (\begin{matrix} \begin{matrix} {(J_{p}^{xy})}_{vu}^{21} \\ {(J_{p}^{xy})}_{vu}^{11} \end{matrix} \\ \begin{matrix} {(J_{p}^{xy})}_{vu}^{22} \\ {(J_{p}^{xy})}_{vu}^{12} \end{matrix} \end{matrix}) - i k_{m}^{2} (\begin{matrix} \begin{matrix} {(J_{p}^{xy})}_{vu}^{12} \\ {(J_{p}^{xy})}_{vu}^{22} \end{matrix} \\ \begin{matrix} {(J_{p}^{xy})}_{vu}^{11} \\ {(J_{p}^{xy})}_{vu}^{21} \end{matrix} \end{matrix}) .

Here,

J

can be expressed as

(\begin{matrix} {(J_{p}^{xy})}_{vu}^{11} \\ {(J_{p}^{xy})}_{vu}^{12} \\ {(J_{p}^{xy})}_{vu}^{21} \\ {(J_{p}^{xy})}_{vu}^{22} \end{matrix}) = {(- 1)}^{m} \int d S_{p} \cdot \hat{n} \cdot (\begin{matrix} M_{v}^{y} (k_{p} r, θ, φ) \times M_{\bar{v}}^{x} (k_{m} r, θ, φ) \\ M_{v}^{y} (k_{p} r, θ, φ) \times N_{\bar{v}}^{x} (k_{m} r, θ, φ) \\ N_{v}^{y} (k_{p} r, θ, φ) \times M_{\bar{v}}^{x} (k_{m} r, θ, φ) \\ N_{v}^{y} (k_{p} r, θ, φ) \times N_{\bar{v}}^{x} (k_{m} r, θ, φ) \end{matrix}),

where

u = (n, m), v = (n^{'}, m^{'}), \bar{v} = (n, - m)

are combined indices,

x, y = 1, 3

, and 1 and 3 denote the scattered and incident fields, respectively.

S

is the enclosing surface of the scatterer, and

\hat{n}

is its unit normal vector pointing outward. Substituting Eqs. (A3) and (A4) into Eq. (C3) yields the matrix

Q

. The four matrices

Q

are uncorrelated, and parallel computation improves the simulation speed.

For a homogeneous scatterer with rotational symmetry, when $n + n^{'}$ is odd [50],

{[Q_{p}^{xy} (k_{m}, k_{p})]}_{vu}^{11} = 0, {[Q_{p}^{xy} (k_{m}, k_{p})]}_{vu}^{22} = 0,

and when

n + n^{'}

is even,

{[Q_{p}^{xy} (k_{m}, k_{p})]}_{vu}^{12} = 0, {[Q_{p}^{xy} (k_{m}, k_{p})]}_{vu}^{21} = 0 .

Consequently, the computation of the matrix $Q$ is halved, which can further increase the simulation speed. EBCM is simple and efficient for scatterers with rotational symmetry.

Appendix D: Translation of Matrix $T$

The translation equation of the matrix $T$ is derived from that of the optical field expansion coefficients, which can be found in Refs. [46,47] and the toolbox in Ref. [61].

Figure 9 displays two independent coordinate systems, the optical field coordinate system $O_{l} X Y Z_{l}$ and the homogeneous microsphere coordinate system $O_{p} X Y Z_{p}$ , parallel along the $z$ axis. They define the expansion coefficients of the optical field and the matrix $T$ of the microsphere, respectively. Thus, it is necessary to unify them before proceeding. To calculate the translation matrix of the optical field expansion coefficients, refer to Refs. [46,47] and the MATLAB Toolbox provided in Ref. [61]:

(\begin{matrix} a_{p} \\ b_{p} \end{matrix}) = T (d) (\begin{matrix} a_{l} \\ b_{l} \end{matrix}),

(\begin{matrix} p_{p} \\ q_{p} \end{matrix}) = T (\begin{matrix} a_{p} \\ b_{p} \end{matrix}) = T T (d) (\begin{matrix} a_{l} \\ b_{l} \end{matrix}),

where

a_{l}

and

b_{l}

are the incident field expansion coefficients in

O_{l} X Y Z_{l}

,

a_{p}

and

b_{p}

are those in

O_{p} X Y Z_{p}

,

p_{p}

and

q_{p}

are the scattered field expansion coefficients in

O_{p} X Y Z_{p}

and

T (d)

is the translation matrix of the field expansion coefficients.

T

is the matrix of the microsphere in

O_{p} X Y Z_{p}

. Reverse translation of

p_{p}

and

q_{p}

yields the matrix

T^{'}

in

O_{l} X Y Z_{l}

:

(\begin{matrix} p_{l} \\ q_{l} \end{matrix}) = T (- d) (\begin{matrix} p_{p} \\ q_{p} \end{matrix}) = T (- d) T (\begin{matrix} a_{p} \\ b_{p} \end{matrix}) = T (- d) T T (d) (\begin{matrix} a_{l} \\ b_{l} \end{matrix}) .

Thus, the translation equation of the matrix $T$ is obtained. Additionally, the translation of PMMA in PS needs to consider the change in optical path length, which is multiplied by $n_{PS}$ :

T^{'} = T (- n_{PS} d) T T (n_{PS} d) .

Funding

National Natural Science Foundation of China (52075383, 61927808); National Key Research and Development Program of China (2022YFF0605501).

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Name	Description
Visualization 1	Janus droplets are prepared by microfluidic and solvent evaporation induces phase separation, ultimately forming Janus solid particles.
Visualization 2	Controlled rotation of Janus microspheres with the ratios 5:1, 4:1, 2:1, and 1:1 following the light polarization direction. In contrast, the Janus microsphere with the ratio 1:5 undergoes restricted Brownian motion.

Incident Field	Power		Polarization	Wavelength	NA
Value	1 W		[1 0]	1064 nm	1.2
Janus Microspheres	Radius	PS:PMMA	Refractive Index	Position	Orientation
Value	1 μm	5:1–1:5	$n_{PS} = 1.49$ , $n_{PMMA} = 1.57$	(0, 0, 0) μm	(45, 45) deg

PS:PMMA	Positions	Orientations
5:1	(0, 0.238, 0.769) μm	(97.9, 90) deg
4:1	(0, 0.311, 0.731) μm	(97.6, 90) deg
3:1	(0, 0.433, 0.691) μm	(99.4, 90) deg
2:1	(0, 0.650, 0.543) μm	(97.2, 90) deg
1:1	(0, 0.949, 0.623) μm	(94.0, 90) deg

PS:PMMA	Positions	Orientations
1:2	(0, 0.981, 1.586) μm	(27.3, 90) deg
1:3	(0, 0, 1.809) μm	(0, /) deg
1:4	(0, 0, 1.790) μm	(0, /) deg
1:5	(0, 0, 1.651) μm	(0, /) deg

PS:PMMA	Positions	Orientations
5:1	(0, −0.238, 0.769) μm	(97.9, 270) deg
4:1	(0, −0.311, 0.731) μm	(97.6, 270) deg
3:1	(0, −0.433, 0.691) μm	(99.4, 270) deg
2:1	(0, −0.650, 0.543) μm	(97.2, 270) deg
1:1	(0, −0.949, 0.623) μm	(94.0, 270) deg
1:2	(0, −0.981, 1.586) μm	(27.3, 270) deg
1:3	(0, 0, −0.744) μm	(180, /) deg
1:4	(0, 0, −0.404) μm	(180, /) deg
1:5	(0, 0, −0.207) μm	(180, /) deg

Optical manipulation of ratio-designable Janus microspheres

Abstract

1. INTRODUCTION

2. THEORY AND SIMULATION

A. Basic Theory of the $T$ -Matrix Method

B. Extended Boundary Condition Method

C. Simulation Results

3. METHOD

A. Preparation of Janus Microspheres

B. Experimental Setups

4. RESULTS

5. CONCLUSION

APPENDIX A: EXPANSION OF OPTICAL FIELD

Appendix B: Calculation of Force and Torque

Appendix C: Calculation of Matrix $Q$

Appendix D: Translation of Matrix $T$

Funding

Disclosures

Data Availability

REFERENCES

Supplementary Material (2)

Data Availability

Cited By

Figures (9)

Tables (4)

Equations (29)

Photonics Research

Abstract

1. INTRODUCTION

2. THEORY AND SIMULATION

A. Basic Theory of the T-Matrix Method

B. Extended Boundary Condition Method

C. Simulation Results

3. METHOD

A. Preparation of Janus Microspheres

B. Experimental Setups

4. RESULTS

5. CONCLUSION

APPENDIX A: EXPANSION OF OPTICAL FIELD

Appendix B: Calculation of Force and Torque

Appendix C: Calculation of Matrix Q

Appendix D: Translation of Matrix T

Funding

Disclosures

Data Availability

REFERENCES

Supplementary Material (2)

Data Availability

Cited By

Figures (9)

Tables (4)

Equations (29)

Photonics Research

A. Basic Theory of the $T$ -Matrix Method

Appendix C: Calculation of Matrix $Q$

Appendix D: Translation of Matrix $T$