Optical trapping-enhanced probes designed by a deep learning approach

Miao Peng; Guangzong Xiao; Xinlin Chen; Te Du; Tengfang Kuang; Xiang Han; Wei Xiong; Gangyi Zhu; Junbo Yang; Zhongqi Tan; Kaiyong Yang; Hui Luo

doi:10.1364/PRJ.517547

Abstract

Realizing optical trapping enhancement is crucial in biomedicine, fundamental physics, and precision measurement. Taking the metamaterials with artificially engineered permittivity as photonic force probes in optical tweezers will offer unprecedented opportunities for optical trap enhancement. However, it usually involves multi-parameter optimization and requires lengthy calculations; thereby few studies remain despite decades of research on optical tweezers. Here, we introduce a deep learning (DL) model to attack this problem. The DL model can efficiently predict the maximum axial optical stiffness of

Si / {Si}_{3} N_{4}

(SSN) multilayer metamaterial nanoparticles and reduce the design duration by about one order of magnitude. We experimentally demonstrate that the designed SSN nanoparticles show more than twofold and fivefold improvement in the lateral (

k_{x}

and

k_{y}

) and the axial (

k_{z}

) optical trap stiffness on the high refractive index amorphous

{TiO}_{2}

microsphere. Incorporating the DL model in optical manipulation systems will expedite the design and optimization processes, providing a means for developing various photonic force probes with specialized functional behaviors.

1. INTRODUCTION

Optical tweezers, which are exquisite displacement and force transducers, are widely applied in quantum optomechanics [1,2], biology [3,4], and nanotechnology [5,6]. Small dielectric particles are typically used as the photonic force probe and often act as handles for high-resolution measurements. However, the force range of these particles is usually around a few pN (with the laser power of tens to hundreds of milliwatts) [7], hindering the development of techniques and studies based on it. The optical force relies on the refractive index mismatch between the photonic force probes and the surrounding media. High-refractive-index particles, such as Si, rutile or anatase ${TiO}_{2}$ , and ${ZrO}_{2}$ , have sizeable refractive index mismatches [8 –11]. However, as the mismatch increases, the instability scattering force increases more strongly than the gradient force.

Diverse solutions have developed to compensate for the adverse effect of scattering on trapping [12 –14]. The simplest way is to modify the particle surface with an antireflection layer. Titanium dioxide-coated titania particles can significantly enhance optical forces to 1 nN [15]. Nevertheless, the microsphere size suffers tight tolerance since a 10% change in the shell size would destabilize the trap [16]. In our previous work, we proposed the ${ZrO}_{2} @ {TiO}_{2}$ core-shell nanoparticles to enhance the optical trapping in the nanoscale [9]. During the study, we found that numerous chemical synthesis experiments are needed to improve the accuracy of particle size. It usually requires rich experience and dedicated effort from experts in material engineering. This synthesis protocol is usually time-consuming and limited by the laboratory environment. Therefore, a probe design scheme with optical trap enhancement and precise size control is urgently needed to mitigate such a barrier.

Recent advances in nanofabrication technology have spurred many breakthroughs in optical metamaterials that can solve the above challenges [17 –19]. Metamaterials are 3D artificial materials with tailored permittivity that are capable of large-volume production with high uniformity. The latest works show they can act as light-driving objects to enhance optical torque, construct optical metavehicles, and even explore solar sails [20 –22]. However, most of these are typically governed by hand-tuning the optical parameters. Manual tuning often takes hundreds or thousands of simulations before finding a practical design. Since each simulation is computationally expensive, this method becomes prohibitively slow as the probe size and complexity grow.

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In contrast to manual tuning, the rapidly emerging deep learning (DL) algorithms can speed up the optimization cycle and introduce remarkable design flexibility. At present, DL algorithms have emerged as a revolutionary and robust methodology in nanophotonics and have been applicated in diverse fields [23 –27]. The critical algorithms of the DL model are artificial neural networks (NNs) [28,29], which take the input of the structural parameter and predict the electromagnetic response of the probe. These NNs replace the computationally expensive finite element simulations in the optimization loop, significantly reducing design time.

This work proposes a DL algorithm to predict the maximum axial optical stiffness of $Si / {Si}_{3} N_{4}$ multilayer metamaterial (SSN) nanoparticles with various geometry parameters. The SSN nanoparticles are fabricated according to the predicted value and characterized in an optical trap by measuring their power spectral density (PSD). The trapping properties of the SSN nanoparticles are superior to homogeneous high refractive index nanoparticles. This DL network can bypass the obstacles set by manual tuning, suggesting a new way to train NN for the optimal design of complex photonic force probes, which is the fundamental motivation of this study.

2. METHODS

In the Rayleigh regime [30], the gradient force is proportional to $Δ n$ and $W^{3}$ , and the scattering force is proportional to $Δ n^{2}$ and $W^{6}$ . $W$ is the width or diameter of the trapped particles, and $Δ n = n_{eff} - n_{m}$ , where $n_{m}$ and $n_{eff}$ are the refractive indices of the surrounding medium and particles. We select Si ( $n_{1} = 3.48$ ), the naturally occurring material with the highest refractive index, as the primary material to improve the gradient force. Then, ${Si}_{3} N_{4}$ ( $n_{2} = 1.98$ ) is selected as the matching material to decrease the scattering force according to the design of anti-reflection film [31]. That is, $n_{2} = \sqrt n_{m} n_{1}$ . Meanwhile, we use the cuboid instead of the commonly used sphere. This geometry decreases light scattering for a fixed volume of a particle by reducing the surface area encountered by the input laser beam [32]. Using the effective medium theory (EMT) [33], the high refractive index Si and ${Si}_{3} N_{4}$ are alternately superimposed to form multilayer metamaterial particles with high $n_{eff}$ , thereby increasing $Δ n$ to enhance the gradient force.

Figure 1(a) is the schematic diagram of the SSN nanoparticle composed of ${Si}_{3} N_{4}$ (blue) and Si (magenta) with its long axis aligned to the propagation direction ( $z$ -axis) of the trapping beam (light red). The trapping beam is linearly polarized along the $x$ -axis. The cross-section of the SSN nanoparticle in the $x - y$ plane is a square with a width of $W$ . The aspect ratio $A R$ is the ratio of height $H$ to width $W$ . The $ρ$ is the thickness ratio of the higher index material layer $d_{1}$ and the unit layer-pair ( $d_{1} + d_{2}$ ). The value of $W$ ranges from 50 to 500 nm, $A R$ ranges from 1 to 5, and $ρ$ ranges from 0.1 to 0.9. To ensure the structural symmetry of the particles, the thickness of the ${Si}_{3} N_{4}$ films on both sides is half of the normal thickness. The optics axis ( $n_{⊥}$ ) of the SSN nanoparticle aligns with the $y$ -axis, while the axes associated with the higher refractive index ( $n_{∥}$ ) align with the $x$ - and $z$ -axes. The optical birefringence in the $x - y$ plane enables active rotary control around the $z$ -axis (indicated by the blue curved arrow).

$(a) The schematic of the trapped SSN nanoparticle, which is made of Si (magenta) and Si3N4 (blue), with its long axis aligned to the optical axis (z-axis) of the trapping beam (red). The cross section in the x−y plane is a square of width W, and the aspect ratio AR is the ratio of height H to width W. (b) The effective refractive indices and birefringence of the SSN nanoparticles as a function of ρ. (c) Top-view and (d) side-view of SEM images of SSN nanoparticles with ρ=0.2, W=450 nm, and AR=2.$

Figure 1.(a) The schematic of the trapped SSN nanoparticle, which is made of Si (magenta) and ${Si}_{3} N_{4}$ (blue), with its long axis aligned to the optical axis ( $z$ -axis) of the trapping beam (red). The cross section in the $x - y$ plane is a square of width $W$ , and the aspect ratio $A R$ is the ratio of height $H$ to width $W$ . (b) The effective refractive indices and birefringence of the SSN nanoparticles as a function of $ρ$ . (c) Top-view and (d) side-view of SEM images of SSN nanoparticles with $ρ = 0.2$ , $W = 450 nm$ , and $A R = 2$ .

Figure 1(b) shows the relationship of the refractive index ( $n_{o}$ , $n_{e}$ , $n_{eff}$ ), birefractive index ( $n_{r}$ ), and $ρ$ of SSN nanoparticles. The refractive index along the ordinary axes ( $n_{o} = n_{∥} = \sqrt ε_{∥}$ ) exceeds that along the extraordinary axis ( $n_{e} = n_{⊥} = \sqrt ε_{⊥}$ ), indicating that the SSN nanoparticles are with a negative uniaxial birefringence ( $n_{e} < n_{o}$ ). $ε_{∥}$ and $ε_{⊥}$ are the permittivity components parallel and perpendicular to the interfacial surfaces of the SSN nanoparticles, respectively [33]. The effective index [ $n_{eff} = (n_{e} - n_{o}) / 2$ ] varies from 1.98 (that of ${Si}_{3} N_{4}$ ) to 3.48 (that of Si), while the birefringence ( $n_{r} = | n_{e} - n_{o} |$ ) can be tuned between zero and its maximum value ( $n_{r} = 0.403$ when $ρ = 0.6$ ). The top-view and side-view of scanning electron microscope (SEM) images of the SSN nanoparticles with $ρ = 0.2$ , $W = 450 nm$ , and $A R = 2$ are shown in Figs. 1(c) and 1(d), respectively.

We use the finite element method (FEM) solver COMSOL Multiphysics 5.6 to design the SSN nanoparticle. The optical force is retrieved by integrating the Maxwell stress tensor (MST) on a virtual surface enclosing the nanoparticle [34]. We use the grid search algorithm to determine the optimal $ρ$ to reduce the number of optimized parameters. The $k_{z}$ for different $ρ$ values is plotted in Fig. 2. The black pixels mean that the SSN nanoparticles cannot be trapped in these parameters. Each parameter search range is $ρ = 0.1 - 0.9$ , $W = 50 - 500 nm$ , and $A R = 1 - 5$ , with the pixel size of $Δ ρ = 0.1$ , $Δ W = 10 nm$ , $Δ A R = 0.2$ . The nanoparticles with $ρ = 0.2$ , $0.4 - 0.9$ are trappable over the entire range of $W$ and $A R$ . Although the $k_{z}$ of the $ρ = 0.4 - 0.9$ nanoparticles can be maximized by selecting nanoparticle sizes in the bright yellow region at the bottom of the map, in practice their utilization in 3D trapping is difficult. Due to variations in nanoparticle sizes during manufacturing, some batches of particles will lie outside of the bright yellow region, leading to unstable trapping. Therefore, we selected nanoparticles with $ρ = 0.2$ in the DL optimization process. In particular, the nanoparticles with $ρ = 0.1$ and 0.3 include continuous nontrappable regions displayed as black pixels in Figs. 2(a) and 2(c). For the sake of “intelligence,” we select $ρ = 0.3$ to verify the accuracy and applicability of the DL network. The maximum value can still be found accurately in the discontinuous parameter space.

Figure 2.The relationship between the axial stiffness $k_{z}$ of SSN nanoparticles and the width $W$ , aspect ratio $A R$ under different $ρ$ values. (a) $ρ = 0.1$ , (b) $ρ = 0.2$ , (c) $ρ = 0.3$ , (d) $ρ = 0.4$ , (e) $ρ = 0.5$ , (f) $ρ = 0.6$ , (g) $ρ = 0.7$ , (h) $ρ = 0.8$ , (i) $ρ = 0.9$ . The black pixels in the maps indicate the nanoparticle sizes that cannot be trapped in 3D due to excessive scattering forces.

We perform the DL network based on the NN and particle swarm optimization (PSO) hybrid optimization (NN-PSO) algorithm to find the optimal $W$ and $A R$ . First, $W$ and $A R$ are divided into several grids. As shown in Fig. 3, a random grid (e.g., $W = 250 - 300 nm$ , $A R = 4 - 5$ , $Δ W = 1 nm$ , $Δ A R = 0.1$ , $M = 561$ ) is selected as the training set $D_{1} (W_{i}, A R_{i}, k_{z i}), i = 1, 2, \dots, M$ , serving as the input layer of NN. Subsequently, we establish a mapping between $W$ , $A R$ , and $k_{z}$ to obtain the NN model. Then, we randomly select $W_{i}^{'}$ and $A R_{i}^{'}$ in $D_{1}$ and plug them into the NN model to quickly predict the corresponding $k_{z i}^{'}$ . These $k_{z i}^{'}$ are regarded as the particles of PSO and grouped up as the initial population. Each $k_{z i}^{'}$ has velocity $v_{i}$ and position $p_{i}$ ( $W_{i}^{'}$ , $A R_{i}^{'}$ ). The $v_{i}$ determines how fast the $k_{z i}^{'}$ moves, and the $p_{i}$ determines the direction in which the $k_{z i}^{'}$ moves. We calculate the fitness $Δ k_{z i}$ of each $k_{z i}^{'}$ . The $Δ k_{z i}$ represents the difference between the current $k_{z i}^{''}$ and the previous $k_{z i}^{'}$ [35], $Δ k_{z i} = \sum_{W_{i}^{'}} \sum_{A R_{i}^{'}} | k_{z i}^{''} (p_{i} (W_{i}^{'}, A R_{i}^{'})) - k_{z i}^{'} (p_{i} (W_{i}^{'}, A R_{i}^{'})) | .$ (1)

Next, $Δ k_{z i}$ is compared to find the optimal solution of the $k_{z i}^{'}$ , which is also called the individual extremum $P_{i}$ . We share the $P_{i}$ with the population and find the global optimal solution $G_{j}$ , where $j$ represents the training set number. If the requirements are satisfied, the iteration is ended. Otherwise, the $v_{i}$ and $p_{i}$ of each $k_{z i}^{'}$ are updated. The algorithm enters the next iteration until the results converge to the optimal point. The $v_{i}$ and $p_{i}$ are as follows: $v_{i} = w \times v_{i} + c_{1} \times rand (0, 1) \times (P_{i} - p_{i} (W_{i}, A R_{i})) + c_{2} \times rand (0, 1) \times (G_{j} - p_{i} (W_{i}, A R_{i})),$ (2) $p_{i + 1} (W_{i + 1}, A R_{i + 1}) = p_{i} (W_{i}, A R_{i}) + v_{i},$ (3)where $w$ is the inertia factor set to 0.5, $c_{1}$ and $c_{2}$ are the learning factors, with values of 0.5, and rand (0, 1) is a random number in the interval (0, 1).

Figure 3.Architecture of the DL network based on the NN-PSO algorithm, where the input is the size parameters and the output is the $k_{z}$ .

3. RESULTS AND DISCUSSION

A. Estimation of the NN-PSO Model

We evaluate the performance of the NN-PSO model in predicting the optimal $k_{z}$ of SSN nanoparticles with $ρ = 0.2$ and 0.3, with the results shown in Figs. 4(a) and 4(b). Figure 4(a) shows the optimization process for the SSN nanoparticles at $ρ = 0.2$ . White arrows indicate that we need five steps to find the globally optimal solution (blue dot). The circled numbers (① to ⑥) indicate that only six grids need to be computed to find the optimal $k_{z}$ . The optimal $k_{z}$ is $1.7 pN {μm}^{- 1} {mW}^{- 1}$ , and the optimal parameters are $W = 450 nm$ , $H = 1125 nm$ , and $A R = 2.5$ .

Figure 4.The $k_{z}$ optimization process for $ρ = 0.2$ (a) and 0.3 (b), respectively. The arrow indicates the route to find the optimal solution. The circle numbers indicate the number of the grid to be calculated in that route, and the blue dot indicates the optimal solution.

Figure 4(b) shows the optimization process for the SSN nanoparticles at $ρ = 0.3$ . Black arrows indicate that we only need two steps to find the optimal $k_{z}$ . We randomly select grid ① as the initial training set. Since the larger $k_{z}$ in grid ② is near the right edge, grid ② is chosen as the next training set. The larger $k_{z}$ in grid ② is concentrated at the bottom of the lower boundary, indicating the maximum value below grid ②. We calculate grid ③ and grid ④, respectively. The results show that the global optimal solution is in grid ③ (blue dot). The predicted $k_{z}$ is $1. 6 pN {μm}^{- 1} {mW}^{- 1}$ , and the optimal parameters are $W = 383 nm$ , $H = 1110.7 nm$ , $A R = 2.9$ .

B. Trapping Properties of the SSN Nanoparticles

Guided by the results above, we fabricated and characterized the SSN nanoparticles with $ρ = 0.2$ , whose sizes are $W = 450 nm$ , $H = 1125 nm$ , and $A R = 2.5$ , and the $n_{eff}$ is 2.24. The fabrication process is shown in Appendix A. The SEM images of these SSN nanoparticles are shown in Figs. 1(c) and 1(d). To measure the trapping properties of the SSN nanoparticles, we then optically trapped them in a standard optical tweezers system [36,37]. The laser is a linearly polarized fundamental mode Gaussian beam along the $x$ direction with power and wavelength as 100 mW and 980 nm, respectively. An oil-immersed objective with a numerical aperture (NA) of 1.4 is used to ensure the stable trap of nanoparticles. We developed a homemade sample chamber to match the working distance of the oil-immersed objective. (The details are supplemented in Appendix A.) We detected the $x$ - and $y$ -positions of the trapped nanoparticle by quadrant detectors (QPDs) and measured the axial ( $z$ -) displacement by a differential detection unit. The optical stiffness is calculated using the power spectrum analysis method [38]. For comparison, we trapped a high refractive index (1.75) amorphous ${TiO}_{2}$ microsphere with a radius of 600 nm. (The trapping efficiency is highest at this radius; see Appendix B.)

Figure 5.The $x$ - (a), $y$ - (b), and $z$ - (c) axis power spectral density curves for two types of particles (SSN and ${TiO}_{2}$ ) measured in water.

The SSN nanoparticles are negative uniaxial birefringent crystals, and the birefringence-originated torque constrains only one rotational degree of freedom (RDOF). The drawback is that unconstrained RDOF may introduce unexpected angular fluctuations into the displacement signal. The geometric anisotropy provided by a rod shape can compensate for this drawback and sufficiently constrain the three RDOFs (Fig. 10 of Appendix C). Further, the influence of spin on $k_{z}$ is analyzed (Fig. 11 of Appendix C). Figure 5(c) shows an optical beating at $2 f_{γ}$ when the nanoparticle rotates at a spin frequency of $f_{γ} (= 1276 Hz)$ . We remark that higher harmonics of the fundamental rotation frequency of $f_{γ}$ are also observed in the PSD signal. The Cr fragment or impurity remaining on the SSN nanoparticle surface leads to the uneven distribution of the scattered and transmitted light, resulting in variations in the photodiode signal.

4. CONCLUSION

In this study, we have introduced the deep learning algorithm in the design of optical trapping-enhanced probes. The deep learning algorithm provides a time-saving way to replace the computationally expensive FEM simulations in the optimization loop, reducing the design duration by about one order of magnitude. Notably, we have achieved at least a twofold and fivefold improvement in the lateral ( $k_{x}$ and $k_{y}$ ) and the axial ( $k_{z}$ ) optical trap stiffness, respectively, compared to the amorphous ${TiO}_{2}$ microsphere in water.

We envision several compelling extensions of this work. For example, one can precisely control the particle size using advanced thin-film deposition and lithography techniques. This strategy can bypass the limitation set by the size tolerance originating from the chemical synthesis scheme [16], suggesting a new way to fabricate probes on the nanoscale. On the other hand, the biochemical inertness, insolubility of $Si / {Si}_{3} N_{4}$ nanoparticles, together with their optical trap enhancement characteristics, will facilitate their application in in vivo manipulation of cells [39] or in optofluidics [40,41]. The rotation effect of multilayer metamaterial makes it possible to use them as micro-motors to construct hydrodynamic optical tweezers for biological targeted therapy [42]. Significantly, benefiting from the deep learning algorithm, compared with the manual tuning, this work opens the door to a highly efficient design of photonic force probes, spanning from the nanoscale to the microscale. Together, optical trap enhancement with deep learning algorithms makes multilayer metamaterial nanoparticles up-and-coming candidates for optomechanical systems in science and engineering.

Acknowledgment

Acknowledgment. The authors gratefully acknowledge the technical support on metamaterial processing from Peiguang Yan and Zhenyuan Shang at Shenzhen University.

APPENDIX A: FABRICATION PROCESS OF SSN NANOPARTICLES

1 cm \times 1 cm

Figure 6.The fabrication process of SSN nanoparticles. (The legend is the color coding for different materials.)

Figure 7.Characterization of SSN nanoparticles. (a) and (b) SEM images of the sample shown In step (8). (d) Height of the unit cell as measured by AFM.

Figure 8.To determine the exact dissolution time of the Cr sacrificial layer. (a) 47 s, (b) 80 s, (c) 120 s, (d) 180 s, (e) 7 min. (f) A small drop of DI water is dropped on the surface of the samples.

We developed the homemade sample chamber to trap the SSN nanoparticles. First, a standard slide is affixed with a “doughnut” shaped 3M tape with a thickness of 100 μm. The particle dispersion is then drawn with a syringe and dripped inside the “doughnut” ring. Finally, a custom cover glass with a thickness of 500 μm is covered, and the channel round is sealed with vacuum grease to prevent the sample from drying out. The chamber is filled with the SSN nanoparticle solution without any air bubbles.

APPENDIX B: TRAPPING EFFICIENCY OF THE AMORPHOUS TiO2 MICROSPHERE

{TiO}_{2}

Figure 9.The trapping efficiency as a function of the radius of the amorphous ${TiO}_{2}$ microsphere. The blue and red curves are the axial ( $Q_{z}$ ) and lateral ( $Q_{x}$ ) trapping efficiency, respectively.

APPENDIX C: ANGULAR TRAPPING BEHAVIOR OF SSN NANOPARTICLES

τ_{x}

Figure 10.The relationship between torques and the angular displacement for two shapes of SSN nanoparticles. $ρ = 0.2$ , $τ_{x}$ (a), $τ_{y}$ (b), $τ_{z}$ (c); $ρ = 0.3$ , $τ_{x}$ (d), $τ_{y}$ (e), and $τ_{z}$ (f). The blue and red curves represent cylindrical and rectangular SSN nanoparticles, respectively.

Figure 11.Relationship between rotation angle $γ$ and $k_{z}$ when $ρ$ is 0.2 and 0.3, respectively.

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