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Abstract
In this work, we propose artificial neural networks (ANNs) to predict the optical forces on particles with a radius of 50 nm and inverse-design the subwavelength-grating (SWG) waveguides structure for trapping. The SWG waveguides are applied to particle trapping due to their superior bulk sensitivity and surface sensitivity, as well as longer working distance than conventional nanophotonic waveguides. To reduce the time consumption of the design, we train ANNs to predict the trapping forces and to inverse-design the geometric structure of SWG waveguides, and the low mean square errors (MSE) of the networks achieve 2.8 × 10−4. Based on the well-trained forward prediction and inverse-design network, an SWG waveguide with significant trapping performance is designed. The trapping forces in the y-direction achieve−40.39 pN when the center of the particle is placed 100 nm away from the side wall of the silicon segment, and the negative sign of the optical forces indicates the direction of the forces. The maximum trapping potential achieved to 838.16 kBT in the y-direction. The trapping performance in the x and z directions is also quite superior, and the neural network model has been further applied to design SWGs with a high trapping performance. The present work is of significance for further research on the application of artificial neural networks in other optical devices designed for particle trapping.
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1. Introduction
Since the 21st century, the revolutionary development of optical technology has propelled the advancement of various industries, including chip manufacturing, healthcare, and modern pharmacology [1,2]. The future and strategic technological requirements will drive the limits of light toward the direction of sub-wavelength silicon photonic circuit integration [3]. To achieve sub-wavelength silicon photonic integration, it is necessary to consider the waveguide characteristics from a more fundamental level and utilize the concepts of metamaterials and metadevices for material design on the sub-wavelength scale [4,5]. Optical trapping has attracted great attention due to its great potential in physics, chemistry, life sciences, medicine, and other fields. The optical systems for the trapping of the particles were first founded in 1986 by Ashkin [6]. Later in 1987, Ashkin et al. experimentally studied the feasibility of using a laser beam system to capture and control single medium particles and cells. Recent research has facilitated the development of efficient optical manipulation such as accurate detection [7], sorting [8,9], and controlled movement [10–14] of nanoparticles. There are two methods to achieve optical manipulation, one is the traditional free-space method and the other is the near-field method. The free-space method uses the high focus of the free-space beam to realize the optical trapping [13,15]. However, it is faced with the challenge of dealing with nanoparticles due to the diffraction limit of the laser beam focus. On the contrary, on-chip nanophotonic structures interact with the nanoparticles through the surrounding evanescent fields [8,10,14,16–18], bringing new opportunities for manipulating particles with low power. In addition, it should be noted that parallel trapping of multiple particles is of great practical importance due to the needs of basic pharmaceutical research and biological heterogeneity. Furthermore, subwavelength grating (SWG) waveguides have a larger working distance to trap more particles, and the optical forces caused by Bloch mode propagation along SWG waveguides is analyzed theoretically for the first time in 2017 by Ma [19].
However, calculating the optical trapping forces of a given geometry structure and inverse design of the geometry structure with trapping forces are both challenging and time-consuming tasks [20]. For the traditional optimization method, it is often necessary to fine-tune the geometry and simulate repeatedly to gradually approach the optimal structure. In contrast, data-driven approaches based on deep learning networks applied to optical devices have considerable flexibility and reduce the complexity of design. Hence, to increase the speed of inverse design, a solution should be put forward to illustrate the relationship between the structures of the SWG waveguides and trapping forces. Deep learning has been the most promising method due to its data-driven ability. Artificial neural networks (ANNs) can match the relationship between the input data and the labels with quite high accuracy [21–23], which makes it one of the most widely used machine-learning tools. In recent years, many teams have invested in the design and application of neural networks in optical devices [24–29]. Some of the achievements are shown in the prediction and inverse design of optical response by using neural networks [20–23,30–35]. For example, Liu et al. presented a generative model based on a deep learning network in 2018 and revealed an opportunity to design metasurfaces for tailored optical response in an inverse-design manner [36]. In the same year, Ma et al. proposed a structure of a deep learning model composed of two bidirectional neural networks to achieve the on-demand design of chiral metamaterials. After that, they also applied neural networks in assisting the design of metamaterials and nanophotonic structures [37–39]. As for SWG waveguides, they can well limit light due to their small cross-section and ultra-high index contrast, and they exhibit highly nonlinear, unpredictable relationships between microscopic geometric changes and their optical responses. There are many studies on SWG waveguides in optical response [19,40–50], but few research on particle trapping and neural network combination, it remains a question whether simple ANNs are effective enough to learn from nonlinear and complex near-field data structures to enable direct inverse-design.
Thus, in this paper, we propose ANNs to predict the optical forces on particles with a radius of 50 nm and inverse-design the SWG waveguide structure for trapping. In our work, we focus on the trapping forces in the y direction of the SWG waveguides. We use three-dimensional finite-difference time-domain (3D-FDTD) to create the dataset of ANNs. The simple fully connected neural network is chosen for training, and a low mean square error (MSE) of 2.8 × 10−4 for trapping forces prediction and 8.68 × 10−4 for inverse design are achieved. The trapping forces of the designed structure achieve - 40.39 pN when the distance between the center of each particle and the sidewall of each silicon segment G = 100 nm, and the negative sign of the optical forces indicates the direction of the forces. The inverse-design network has a very high accuracy, and we obtain an SWG waveguide with parameters L = 221 nm, W = 423 nm, H = 212 nm, Λ = 439 nm through this network. We also analyze the trapping forces and potential energy of the particle in three directions according to the 3D-FDTD simulation results. Our results demonstrate the possibility of ANN in solving the prediction and inverse-design problem of the optical trapping forces, and we believe that the proposed method has significant implications for the design of other optical devices in particle trapping.
2. Theory and structure design
2.1 Theoretical research and data set creation
The schematic configuration of an SWG waveguide on silicon is shown in Fig. 1(a), and its side view with particles is shown in Fig. 1(b), which illustrates that each particle is trapped on the side of each silicon segment (nsi = 3.476). To avoid the formation of the standing waves due to the Bragg-grating effect and the opening of a band gap at the operation wavelength, one should choose Λ < λeff / 2, where Λ is the grating period and λeff is the effective wavelength. In this case, the SWG waveguides support a Bloch mode, which can propagate along the segmented waveguides without loss in theory [51]. The distance between the center of each particle and the sidewall of each silicon segment is defined as G. The 3D-FDTD method is used to simulate the propagation of the supported Bloch-mode along the designed SWG waveguides, ten randomly distributed Bloch mode lights are used as the light source, and the electric field distribution is periodical along the propagation distance x. Taking the center of the silicon segment in the dashed box in Fig. 1(a) as the coordinate origin, When the segment length L = 150 nm, the segment width W = 450 nm, the waveguide height H = 220 nm, and the grating pitch Λ = 300 nm, the electric field distribution can be seen in Figs. 1(c)-(e). In this simulation, TE polarization is considered and the electric field distribution (E(x, y, z = 0)) is periodical along the propagation distance x [51]. Moreover, significant evanescent field enhancement is observed near the sidewalls of the silicon segment. This special light field distribution along an SWG waveguide makes it an attractive tool for on-chip nanoparticle manipulation.
Fig. 1. (a) Schematic of the SWG waveguide; (b) side view of the SWG waveguide and nanoparticles. Electric field distribution in the (c) x-y plane (z = 0), (d) x-z plane (y = 0), and (e) y-z plane (x = 0).
When a nanoparticle is close to an optical waveguide, it can potentially be trapped due to the optical forces generated by the optical field. The optical force F can be estimated by integrating the time-independent Maxwell stress tensor < TM > on the external surface S enclosing the nanoparticle [52]:
Figures 2(a) and 2(b) show the dependence of the optical forces on the size of the SWG waveguide segment. In Fig. 2(a), the curves with similar shapes and different colors represent the optical forces of SWG waveguides of different sizes, and the sizes of the SWG waveguides segment are chosen as L = 230 nm, H = 220 nm, Λ = 400 nm and the value of W ranges from 520 nm to 620 nm with a step of 20 nm. The blue dotted line in Fig. 2(a) is fitted with the peaks of several optical force curves. It should be noted that the negative sign of the optical forces indicates the direction of forces. In this process, the value of optical forces decreases from - 16.33 pN to - 11.56 pN when the input power is 1 mW. Additionally, as the size of the SWG waveguide increases, the wavelength experiences a redshift of approximately 80 nm. To further investigate this relationship, we vary the segment length L while keeping H = 220 nm, Λ = 400 nm, and W range from 520 nm to 620 nm with a step of 20 nm to get the peak fitting of the optical force curve groups. Four groups of force peaks are shown in Fig. 2(b), with the segment length L ranging from 200 nm to 230 nm with a step of 10 nm. It can be seen that the peak of the optical force curves gradually increases and the wavelength redshifts accordingly with the size of the SWG waveguides increases. However, the relationship between the size of the SWG waveguides and the force curve is nonlinear, which means that adjusting the structural parameters conventionally to reach a target force peak would require a significant amount of time and resources. Therefore, it is of great interest to find a way to quickly calculate the relationship between the force-wavelength and the structural parameters of SWG waveguides and to be able to determine the corresponding structural parameters based on the desired force-wavelength values.
Fig. 2. Optical force FY changes with different sizes of the SWG waveguides. (a) Optical force curves (solid line) and peak trend fitting (blue dotted line) of SWG waveguides with different W (G = 100 nm). (b) Four groups of peak trend fitting lines of SWG waveguides with different L (G = 100 nm). The range of values of W in each set of the curve is the same as in Fig. 2(a). The values of L, H, and Λ are shown in the legend.
In conventional practices, the design begins with intuition and experience followed by iterative optimization, with the hope of optimizing the optical force and getting the desired SWG waveguide. However, this approach presents significant challenges due to the sophisticated and nonlinear relationship. To address this issue, we propose using artificial ANNs to design an SWG waveguide with high-performance trapping forces. For the dataset of ANN, we randomly vary L from 80 nm to 380 nm, W from 400 nm to 700 nm, H from 200 nm to 250 nm, and Λ from 380 nm to 500 nm, which is a total of 2056 groups of sampled data.
2.2 Artificial neural network for forward prediction
In this part, we describe the construction of the ANN for the forward prediction. The architecture of the ANN typically consists of one input layer, several hidden layers, and one output layer. For the input layer, there are four nodes denoting L, W, H, and Λ, while the output layer contains nodes representing the entire optical force curve. The optical force curve is described in two parts. To observe the peak value of the optical force curve, the first part of the data is the force peak value and the corresponding wavelength (Fm, Vm). And to provide a complete description of the information of the optical force curve, the other part is the complete optical force curve, which is discretized into 100 points labeled F1 - Fn, where n is taken to be 100, and the wavelength ranges from 1300 nm to 1700nm. The schematic of the forward prediction ANN is shown in Fig. 3. To generate a model with high accuracy, we train the network multiple times, varying the number of neuron nodes, learning rates, and hidden layers. For the networks in our work, rectified linear unit (ReLU) is chosen as the activation function, and Adam optimizer [21] is chosen as the optimization algorithm. We choose MSE as the loss function and mean absolute percentage error (MAPE) to quantify the accuracy of the trained models. The definitions of the two indicators are described in Eqs. (3) and (4). In the equations, Factual represents the labels, and Fpredict is the output from the network.
To optimize the structure of the ANN for better adaptability to the data characteristics of SWG waveguides, we construct an ANN with higher accuracy by adjusting the learning rate, the number of neurons, and the number of hidden layers. Figures 4(a)–4(b) display the results of our comparison between different ANN structures.
Fig. 4. (a) MSE changes with different numbers of neuron nodes; (b) MAPE changes with different learning rates.
Figure 4(a) shows the dependency of MAPE on the number of neuron nodes of the network with two hidden layers. After testing various numbers of neurons, we found that 100 neurons yielded the best performance. With 100 neurons, both MSE and MAPE reached a relatively low level. Increasing the number of neurons beyond 100 would only increase computational complexity without significantly reducing MSE or MAPE. Therefore, we ultimately set the number of neuron nodes to 100. Figure 4(b) shows the dependency of MAPE on the depth of the network and the learning rate when the number of neurons per layer is 100. MAPE decreases as the number of hidden layers increases. However, the three-hidden-layer network only shows a slight advantage compared to the two-hidden-layer network but costs a relatively longer training time. For the two-hidden-layer network, there are 20,902 parameters and the training time is 149.83s. As for the influence of learning rate on MAPE, we can see that MAPE decreases at first and then increases when the learning rate increases due to poor convergence with too small or too large learning rates. The lowest MAPE of 0.537% is achieved when the learning rate is 0.001 with two hidden layers. The MSE in this circumstance is 2.8 × 10−4 and 8.68 × 10−4 for the training and testing set, which shows the high accuracy of the network.
2.3 Artificial neural network for inverse-designing
In this part, we aim to design an SWG waveguide with a superior optical force to trap the nanoparticles using the inverse-design network. The network model in Fig. 3 allows us to match the geometric information of SWG waveguides with the optical forces. This model can then be efficiently employed to predict the optical force curve of the SWG waveguides within a short time. To design the corresponding structure parameters (L, W, H, Λ) quickly according to the target optical force curve F1 - Fn (n = 100), this section will introduce how to build a neural network for inverse design.
The network is constructed by cascading the pre-training forward prediction network behind the inverse-design network to overcome the issue of data inconsistency. Initially, our design target was only the force peak value and the corresponding wavelength (Fm, Vm). To further improve the accuracy of the design, we added the complete optical force curve F1 - Fn predicted by the forward network to the input layer of the reverse network, as shown in Fig. 5. The performance of the network can be evaluated by comparing the difference between the input and output layers of the cascaded network. The entire process of the inverse-design portion can be summarized as follows: Firstly, the target force peak value data is directly input into the input layer of the inverse network. The output layer of the inverse network will then generate structural parameters (L, W, H, Λ), which are subsequently fed into the input layer of the forward network. After computation by the forward prediction neural network, a complete optical force curve F1 - Fn will be ultimately output. This entire optical force curve is then added to the input layer of the inverse network, and iterative optimization is conducted on this target optical force curve until an output SWG waveguide structure that meets the design objectives of this study is achieved. Finally, the intermediate layer of the cascaded network outputs the structure parameters (L, W, H, Λ).
Fig. 5. Structure of the cascaded ANN. The pretrained forward prediction network is combined after the inverse-design network to explore the network’s accuracy more efficiently.
After comparing the network performance with different numbers of hidden layer neurons, we set the number of neurons in each hidden layer to 300, ReLU is chosen as the activation function, Adam optimizer is chosen as the optimization algorithm, and the training time for the forward network and the inverse-design network is 149.83s and 201.36s, respectively, as shown in Table 1.
Table 1. Hyperparameters for the cascaded artificial neural network
After completing the construction of the ANN, we randomly select two sets of SWG structures to evaluate the prediction performance of the forward prediction ANN. We input the structural parameters (L, W, H, Λ) = (260, 690, 220, 400) nm and the optical force peak (Fm, Vm) = (− 7.41 pN, 1557.91 nm) of the test set (data not used for ANN model training) into the forward prediction neural network. The output peak value is (− 7.58 pN, 1558.07 nm), with an error of (− 0.17 pN, 0.16 nm). As Fig. 6(c) shows, the optical forces and wavelengths from the ANN prediction (Desired) and 3D FDTD simulation (Simulated) match well. The correlation coefficient between the predicted value and actual simulation value in the test set is calculated and shown in Figs. 6(a) and 6(b). The correlation coefficient ρ is 0.991720 and 0.981425, respectively, which proves the very high accuracy of the inverse-design network. In this case, the learning curves for training and prediction are shown in Fig. 6(d). As the number of training epochs increases, the MSE gradually decreases and the learning curve becomes smoother, indicating that the forward ANN can accurately fit the relationship between the dataset after training without overfitting. The MSE for the training and test sets are 2.8 × 10−4 and 8.68 × 10−4, respectively, demonstrating the high precision of the network. The learning curves in Fig. 6(d) have already converged, further proving that the forward ANN we constructed can predict the optical force curves generated when capturing 50 nm PS nanoparticles with a very high accuracy rate based on the structural parameters of SWG waveguides.
Fig. 6. (a) Correlation map between the desired forces and predicted forces. (b) Correlation map between the desired wavelengths and predicted wavelengths. (c) Comparison of forces and wavelengths between our desired and simulation results. (d) Learning curves for the forward network.
We also randomly select two sets of optical force curves to evaluate the performance of the inverse-design ANN. The two sets of optical force curves are taken from the test set and input into the inverse-design network. The error between the output and original structural parameters of this data set was then calculated to evaluate the performance of the inverse-design network. As shown in Fig. 7(a), the desired and simulated structural parameters of the SWG structural parameters are (346.11, 595.55, 219.95, 401.04) nm and (340, 600, 220, 400) nm, respectively, with an error of (6.11, −4.45, −0.05, 1.04) nm. The MAPE for each parameter is (1.79%, 0.74%, 0.02%, 0.26%). We then calculated all the data in the test set and conducted statistics, as shown in Fig. 7(b). The correlation coefficients between the desired and simulated structural parameters (L, W, H, Λ) are 0.899539, 0.9122, 0.907053, and 0.904686, respectively. This demonstrates the high accuracy of the inverse-design ANN. The learning curve of the inverse network also achieves good convergence within the first 1000 training epochs, as shown in Fig. 7(c) The MSE of the test set reached 1.03 × 10−2, indicating that the inverse network can accurately fit the characteristics of the dataset and successfully design SWG structural parameters based on optical force curves.
Fig. 7. (a) Comparison of structural parameters between our desired and simulation results. (b) Correlation map between the structural parameter and predicted structural parameter. (c) Learning curves for the inverse-design network.
The trained inverse-design network matches the relationship between the geometric information and the trapping forces, making it a useful tool for designing SWG waveguides. In the process of inverse-designing the SWG waveguides, we first input the design objective Vm and Fm into the inverse network. The objective primarily considers the maximum absolute optical forces. Combined with the previous analysis that the absolute optical forces decrease with wavelength redshift, the target wavelength is selected to be around 1350 nm in the target band (1350 nm - 1750nm). The design objective is to optimize the maximum optical forces of 35 pN in the dataset, corresponding to Vm = 1350 nm and Fm = −35 pN. Then we add F1 - Fn to the input layer and repeat the previous process to get the final structure. The structure of the designed SWG waveguide is shown in Fig. 8. The optical force peak and wavelength finally obtained by forward network optimization are respectively Vm = 1328.35 nm and Fm = − 40.39 pN, the parameters of the final designed SWG waveguide are L = 221 nm, W = 423 nm, H = 212 nm, Λ = 439 nm, as shown in Fig. 8. The entire iterative optimization process of using ANN for the inverse-design of the SWG waveguide takes only 300 seconds, whereas the traditional method of adjusting structural parameters to find the optimal solution would require at least a month. Moreover, the optical force achieved through ANN optimization is significantly better than the trapping performance of the SWG waveguide set by the conventional method (the maximum value is − 27 pN in the same direction [19]). The trapping performance comparison of the different structure types is shown in Table 2. Here, the performance comparison of these different structure types is not strict because the size of particles in these systems is different. When the external conditions remain unchanged, the smaller the particle size is, the smaller the optical trapping force acting on the particle is. Based on the above analysis, our ANN network can accurately fit the relationship between structure and trapping performance and can find the combination of multi-dimensional data structure parameters with better performance in high accuracy. This greatly reduces the complexity of designing the structure.
Fig. 8. Schematic of the SWG waveguide. (a) Schematic of the x-z plane. (b) Schematic of the y-z plane.
Table 2. Trapping performance comparison of the different designs
The application of ANN models in the design process shows superior performance in optimizing optical forces and has the added benefit of avoiding times of trial and error, saving masses of time for simulation. All the calculations are carried out on the computer with a 2.71-GHz Intel Core i5-7200U CPU. The time consumed is recorded using the datetime module supported by Python. It takes around 3500s using 3D FDTD simulation to calculate the whole optical force curve while it takes 201s to train the inverse-design network and it can predict 300 sets of geometric parameters in 0.01s. This means that in the future, we can save hours of simulation time in the design of SWG waveguides, making it crucial to improve design time and speed. The speed of predicting optical forces can be quantified as 108 times faster in the entire design process.
3. Analysis of trapping performance of the designed SWG waveguide
In this section, the specific performance of the previously designed SWG waveguide structure on trapping nanoparticles is analyzed. Firstly, the electric field distribution of the final design SWG waveguide is analyzed, as shown in Fig. 9. As previously analyzed, there is a significant electric field enhancement near the sidewalls of the silicon segment. To better observe the electric field localization near the particle capture position, the observation plane of Fig. 9(b) was shifted from y = 0 to y = 300 nm. Figure 9(a) displays the electric field in the x-y plane at z = 0, where the electric field is mainly localized on the side surfaces of the silicon segment. Figure 9(b) displays the electric field in the x-z plane at y = 300 nm, and Fig. 9(c) displays the electric field in the y-z plane at x = 0. Due to the structural discontinuity of the SWG waveguide, the entire structure is supported by a SiO2 substrate. This results in a junction between SiO2 and Si at z = 0, creating a layered electric field.
Fig. 9. Electric field distribution at λ = 1328.35 nm. (a) Electric field distribution of the final design SWG waveguide in the x-y plane (z = 0). (b) Electric field distribution of the final design SWG waveguide in the x-z plane (y = 300 nm). (c) Electric field distribution of the final design SWG waveguide in the y-z plane (x = 0). Here, to observe the electric field near the nanoparticle more clearly, the observation position of the electric field in (b) is set as y = 300 nm.
Since the optical trapping forces acting on particles at different positions are different, we explore the optical trapping forces by changing the position of particles. The above analysis is only for the optical forces in the y direction, in this section, the 50 nm particle is moved along three coordinate axes to study the trapping characteristics of the SWG waveguide designed by the ANN. The wavelength for all simulations and calculations in this section is 1328.35 nm. Figures 10(a)-(c) show the trapping forces in three directions on the 50 nm particle. Figure 10(a) shows the optical forces in the y-direction acting on the 50 nm PS particle when the particles’ position changes from y = 270 nm to y = 640 nm. It can be seen in Fig. 10(a) that since the electric field of the SWG waveguide is localized on the sidewall in the y-direction, the closer the particle is to the silicon section of the grating, the greater the force will be. The trapping force FY is always negative, which is determined by the positional relationship between the PS particle and the electric field. Since the PS particle is located on the positive half-axis of the y-axis, and the hot spot of the electric field is positioned infinitely close to the sidewall of the silicon segment, due to the limiting position of the particle, it will always be attracted towards the hot spot of the electric field. The direction of this attraction has always been towards the negative half-axis of the y-axis. When the particle is infinitely close to the waveguide, FY can reach a maximum of - 60.36 pN at y = 270 nm, x = 0, and z = 106 nm.
Fig. 10. Numerical analysis of the optical trapping force and the trapping potential for the SWG waveguide at λ = 1328.35 nm. All calculations are done for a PS nanoparticle with a radius of 50 nm. (a) FY, (b) FX and (c) FZ are the trapping force along the y-axis (ranging from y = 270 nm to y = 640 nm, x = 0, z = 0), the x-axis (ranging from x = - 225 nm to x = 225 nm, y = 270 nm, z = 0) and z-axis (ranging from z = 60 nm to z = 320 nm, y = 270 nm, x = 0) of the device, respectively. (d), (e) and (f) show the trapping potential obtained by integrating the optical trapping force along the y-axis, x-axis, and z-axis.
Similarly, in Fig. 10(b), the optical force in the x-direction acts on the 50 nm PS particle when the particle’s position changes from x = - 225 nm to x = 225 nm. As can be seen from Fig. 10(b), due to the structural symmetry in the x direction, the zero value of the trapping force FX, occurs at x = 0, and the maximum of FX appears at x = 110 nm. FX varies from a positive maximum 42 pN to a negative maximum - 42 pN when the position of the particle moves from x = - 150 nm, y = 270 nm, z = 106 nm to x = 150 nm, keep y = 270 nm, z = 106 nm. When x > 0, the trapping force FX is negative, indicating that the direction of FX is towards the negative half-axis of the x direction. Therefore, when the particle is away from the center of the x-axis, the trapping force FX will pull the particle back toward the center of the x-axis. Considering that the barrier of the structure restricts the movement of the particles in the z-direction, we only move the particles from z = 60 nm to z = 320 nm (y = 270 nm, x = 0) and explore the trapping performance. When the PS particle moves along the positive direction of the z-axis, combined with the characteristics of the electric field in Fig. 9(b), it can be observed that the direction of FZ changes from positive to negative. As shown in Fig. 10(c), the trapping force varies from a positive maximum of 11.87 pN (z = 80 nm) to a negative maximum of - 31.97 pN (z = 230 nm).
The corresponding trapping potential acting on the particles could be obtained by integrating the component of the trapping force along the specific path. Figures 10(d)-(f) show the corresponding trapping potential in three directions. The corresponding trapping potential acting on the particle is calculated by integrating the component of the trapping force F along the specific path [52]:
where U is the trapping potential in one direction, F represents the component of total trapping force in the corresponding direction, S is the stability number, kB represents the Boltzmann constant, and T is set to room temperature (300 K). To achieve stable trapping, it is usually required that the depth of the potential energy is greater than 10 kBT [8].According to the simulation results, we can separately calculate the trapping threshold power of three directions, as shown in Figs. 10(d)-(f). In the y-direction, the maximum trapping potential is 838.16 kBT. The maximum trapping potential is 1209.04 kBT in the x-direction and 900.01 kBT in the z-direction. We introduce the stability number to better describe the trapping state of particles. Due to the Brownian motion of particles in the solution, to achieve stable trapping, the depth of potential energy (ΔU) must be greater than 10 kBT. That is, when the stability number is greater than 10, it is considered that particles can be trapped stably. Based on Eq. (6), it can be calculated that the stability numbers of the SWG waveguide are 838.16 in the y-direction, 1209.04 in the x-direction, and 900.01 in the z-direction, respectively, so the stable trapping of particles can be realized with an input intensity of 1 mW/µm2. The structure we designed shows good performance in trapping nanoparticles and can stably trap multiple particles on the side of the silicon in parallel because of the periodic electric field of the SWG waveguide.
4. Conclusion
In this work, well-trained ANNs are structured to predict the optical trapping forces on particles with a radius of 50 nm and inverse-design the geometric structure of the SWG waveguides with trapping forces and corresponding wavelength. The SWG waveguides are applied to particle trapping due to their superior bulk and surface sensitivity, as well as their longer working distance compared to conventional nanophotonic waveguides. The use of ANNs allows us to efficiently design an SWG waveguide with optimal trapping forces and reduce time consumption. The trapping forces of the designed structure achieve-40.39 pN, the negative sign of the optical forces indicates the direction of the forces. In our design, the improvement of speed for predicting the whole trapping forces is remarkable 108 times. For the prediction ANN, the correlation coefficient of trapping forces and the corresponding wavelength between the predicted value and actual simulation value in the testing test are 0.99172 and 0.981425. For the inverse-design network, the correlation coefficients between the desired and simulated structural parameters (L, W, H, Λ) are 0.899539, 0.9122, 0.907053, and 0.904686, respectively. The inverse-design network has a very high accuracy, and we obtain an SWG waveguide with parameters L = 221 nm, W = 423 nm, H = 212 nm, Λ = 439 nm through this network. According to the 3D-FDTD simulation results, the maximum optical trapping forces of the 50 nm particles in three directions are FY = - 60.36 pN, FX = - 42 pN, and FZ = - 31.97 pN. It is calculated that the maximum trapping potential of the 50 nm particles in three directions are 838.16 kBT in the y-direction, 1209.04 kBT in the x-direction, and 900.01 kBT in the z-direction, respectively, which shows superior trapping performance. In conclusion, our results demonstrate the possibility of ANN in learning nonlinear relationships and solving the prediction and inverse-design problem of the optical trapping forces. The networks can help to save masses of time in SWG waveguide design. Furthermore, such implementation proves the way for further study on the inverse-design in other optical devices.
Funding
National Natural Science Foundation of China (62275026, 61431003).
Acknowledgments
The authors would like to thank Zixing Gou, Jinzhi Wang, and Chao Wang for their help.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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