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Abstract
Optical chirality is highly demanded for biochemical sensing, spectral detection, and advanced imaging, however, conventional design schemes for chiral metamaterials require highly computational cost due to the trial-and-error strategy, and it is crucial to accelerate the design process particularly in comparably simple planar chiral metamaterials. Herein, we construct a bidirectional deep learning (BDL) network consists of spectra predicting network (SPN) and design predicting network (DPN) to accelerate the prediction of spectra and inverse design of chiroptical response of planar chiral metamaterials. It is shown that the proposed BDL network can accelerate the design process and exhibit high prediction accuracy. The average process of prediction only takes ∼15 ms, which is 1 in 40000 compared to finite-difference time-domain (FDTD). The mean-square error (MSE) loss of forward and inverse prediction reaches 0.0085 after 100 epochs. Over 95.2% of training samples have MSE ≤ 0.0042 and MSE ≤ 0.0044 for SPN and DPN, respectively; indicating that the BDL network is robust in the inverse deign without underfitting or overfitting for both SPN and DPN. Our founding shows great potentials in accelerating the on-demand design of planar chiral metamaterials.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Chirality refers to the structural property of being non-superimposable onto its own mirror image by any symmetry axis [1]. Chiral objects are ubiquitous in nature, ranging from molecules at the nanoscale to gastropod shells at the macroscale [2–5]. However, limited by the minimal difference in the chiroptical response of natural molecules, chirality of natural materials is usually weak [6]. To resolve this issue, chiral metamaterials are proposed to enhance chiroptical responses due to their unique electromagnetic properties, interesting design schemes include optical activities [7,8], asymmetric transmission (AT) effects [9–11] and circular dichroism (CD) [12–14]. In particular, CD, as a manifestation of chirality, can be described by the differential reflection of right-handed circularly polarized light (RCP) and left-handed circularly polarized light (LCP) in spectral response of the chiral structure [15], which is promising for various applications such as circularly polarized light detection [16], charity sensing [17,18], spin-selective light absorption [19,20] and circularly polarized light emission [21–23].
Unfortunately, conventional design approaches for the chiroptical responses of chiral metamaterials, such as finite-element method (FEM), finite-difference time-domain (FDTD) method, and finite integration technique (FIT), are computationally expensive and severely time-consuming due to their iterative simulation processes. To overcome the time-consuming problem, deep learning (DL) has emerged as a powerful computational tool because it can train a model from the collected dataset and predict the result within milliseconds of high accuracy [24,25]. It is shown that DL exhibits huge advantages for the designs of diverse optical devices, such as fibers [26], microscopes [27], absorbers [28,29] and functional elements [30–32]. Particularly, the DL framework can be used to accelerate the on-demand design of chiral metamaterials due to its efficient data-driven model [33]. By constructing the DL-assisted network, versatile chiroptical responses such as AT [34], CD [35,36], chiral sensing [37], vortex focusing [38], and chiral wavefront control [39] can be realized. However, previous DL-based design schemes are mainly focused on the chiroptical responses of the complicated three-dimensional (3D) chiral metamaterials, and flexible design of the chiroptical responses within the comparably simple planar metasurfaces is still rare.
In this work, we propose a bidirectional deep learning (BDL) network to achieve both the forward and inverse predictions of chiroptical responses of the planar chiral metasurfaces. The forward network is aiming to predict the chiroptical response according to the established geometric parameters with high accuracy comparable to numerical simulations, while the inverse network works on retrieving the structural geometry parameters of the chiral metasurfaces based on the reflection (both RCP and LCP) and CD spectra. The planar chiral metasurfaces consist of dagger-shaped silver (Ag) arrays and an Ag mirror separated by a dielectric spacer, and highly efficient CD enhancement can be realized through the efficient data-driven model of the BDL network. During the training process, the loss for the BDL network can reach 0.0085 after 100 epochs. The average process of prediction only takes ∼15 ms, which is 1 in 40000 compared to FDTD simulations. Our work provides important insights for flexible design of planar chiral metamaterials, which may be helpful in the study of complex optical chirality through DL networks.
2. Design principle
Figure 1 shows the schematic diagram of the proposed planar metasurface, which consists of dagger-shaped Ag arrays and Ag substrate separated by a spacer of silicon dioxide (SiO2). Depending on the structure, the metasurface can function as a chiral meta-mirror, enabling selective reflection of designated circularly polarized light without reversing its handedness yet high absorption of the other polarization state at a certain wavelength [40,41]. The thickness h1 of Ag arrays is 64 nm, whereas the period of the unit cell is fixed as 320 nm and 420 along the x- and y-axis, respectively. The thickness h2 of SiO2 spacer is set as 110 nm. Transmission are blocked due to the optically thick Ag substrate with thickness of 100 nm. Therefore, the structure of chiral metasurface is determined by four design parameters, which contains the widths of vertical bar (w1) and horizontal bar (w2), the distance (d) of the vertical bar translating to the left, and the twisted angle (θ) of the horizontal bar. As transmission is blocked by the Ag substrate, the CD response can be defined as the relative difference of reflection for the RCP and LCP lights, that is, CD = (RLCP-RRCP)/(RLCP + RRCP), where RRCP and RLCP represent the reflection of the RCP and LCP lights, respectively. In simulation, the simulated reflection responses of the planar chiral metamaterials are calculated by using the FDTD software of Ansys Lumerical. Periodic boundary conditions are used in the x and y direction, perfectly matched layers are used in the z direction, and the grid size is chosen as 5 nm to ensure the accuracy of the calculation.
Fig. 1. (a) Schematic diagram of the dagger-shaped planar metamaterials under RCP and LCP illuminations. (b) 3D view of the unit cell of the structure, where w1 and w2 are the widths of the vertical bar and horizontal bar, respectively; d indicates the shift of vertical bar from the center of the unit cell, and θ is the twisted angle of horizontal bar.
Figure 2 shows the BDL network used to achieve the forward and inverse prediction, which consists of an input layer, an output layer and three fully connected hidden layers containing 200, 600, and 800 neurons, respectively. The spectra predicting network (SPN) and design predicting network (DPN) are shown in Fig. 2, two independent networks are trained to predict spectra and retrieve corresponding parameters according to ideal spectra, respectively. The input layer of SPN has 4 neurons corresponding to the four-dimensional parameters (w1,w2,d,θ) of our chiral metamaterials, and the output layer has 600 neurons corresponding to the spectra of RCP, LCP and CD, which can be seen as red and blue dots in Fig. 2, respectively. In the forward path of prediction, structural parameters will be transformed into response spectra after 3 fully connected layers. In the inverse path of prediction, similarly, the output layer contains 600 neurons, with the first 200 neurons representing predicted LCP results, another 200 neurons representing predicted RCP results and others standing for predicted CD responses. We generated almost 12000 sets of simulated data with corresponding parameters via FDTD methods as the dataset. The whole dataset is divided into three parts, including training dataset, validation dataset and testing dataset, where the ratio is roughly 6:3:1. As the problem of prediction is actually a regression problem, the loss function is chosen to be the mean-square error (MSE) loss [42]. The MSE loss can be illustrated as follows:
where N is the number of a batch data, yi is the target label of the training data, and $\bar{y}$ is the spectra predicted by the neural network. In order to implement our regression network, the Pytorch framework is employed [43], which is an open-source DL framework. The training and evaluation processes were performed by using a computer operated on the Win10 system with 2 TB storage memory capacity and 32 GB running memory capacity. The CPU we used is i5-9400F with the dominant CPU frequency of 2.9 GHz, and the graphic card is NVIDIA GeForce GTX 1660Ti with CUDA 11.6. We initially train the SPN module by feeding the structural parameters as the input data. After the SPN is well trained, we feed with previous spectra data to train DPN module. At the ending of the training process, the two networks’ performance needs to be evaluated by the training loss to ensure whether they are well trained.Fig. 2. Schematic diagram of the BDL network consisting of the SPN and DPN. The network has an input layer and an output layer, and 3 fully connected hidden layers with 200, 600, 800 neurons, respectively. Cells in red represent the inputs, whereas gray and blue cells represent the hidden neurons and outputs, respectively. Reflection spectra, CD spectra, and design parameters can be treated as either input or output at specific ports.
3. Results and discussion
To evaluate the effectiveness of the BDL network in the predictions of the spectra and inverse design of planar chiral metamaterials, the operation performance including computation time, training and validation loss are studied, as shown in Fig. 3. In Fig. 3(a), it can be seen that the training and validation loss rapidly decrease for SPN model until they stabilize at 0.0085 after 100 epochs. For validation loss, it is slightly higher than training loss due to the validation dataset has never been used in DL network. Similarly, as shown in Fig. 3(c), for DPN model, the training and validation loss rapidly decrease until epochs reach 100, where the training and validation loss stabilize at 0.0089 and 0.0148, respectively. Note the corresponding losses for both the training and validation in Fig. 3(a) are slightly smaller than those of in Fig. 3(c), this is because that the number of neurons in the input layer of the SPN is significant smaller than those of the DPN. To further demonstrate the efficiency of BDL network, we compare the computation time of the FDTD simulations and BDL network, which is shown in Figs. 3(b) and (d). It can be seen that average computational time for FDTD simulations and BDL network are almost 10 minutes and 15 ms, respectively, which is 1 in 40000 comparable to conventional approaches. According to the comparison, the prediction does not require much computational power, facilitating the accelerating working process in application.
Fig. 3. Operation performance of the BDL network for the predictions of the spectra and inverse design. (a) and (c) are training loss and validation loss for SPN and DPN, respectively. (b) and (d) are the comparison of computation time between FDTD and BDL on 12 different structures for SPN and DPN, respectively.
To test the effectiveness of the BDL network, chiroptical responses with different CD peaks are compared with the FDTD simulations, as shown in Fig. 4. In Figs. 4(a) and (b), the solid lines represent the reflection and CD spectra calculated by the FDTD; the dot, triangular and pentagram represent the reflection and CD spectra predicted by SPN, and it only takes about 12 ms to predict spectra. As can be seen in Figs. 4(a) and (b), for the given structural parameters of (w1,w2,d,θ) = (70 nm,80 nm,40 nm,17°), the predicted reflection and CD spectra are in good agreement with those of the FDTD. The MSE loss of the reflection for the LCP RCP, and CD responses are 0.0018, 0.0015, 0.0021, respectively; validating the highly efficient design capability of the SPN in predicting the spectra of planar chiral structures. In Figs. 4(c) and (d), the solid lines represent the input reflection and CD spectra calculated by the FDTD, and the retrieving structural parameters can be predicted as (w1,w2,d,θ) = (75 nm,60 nm,28 nm,17°) through the DPN; also, the results of the DPN are agree well with those of the FDTD. Note the maximal CD of 89.5% from the predictions are larger than those of 3D chiral metamaterials such as twisted [44,45] and stretchable structures [46]; also, it is larger than those of the planar metamaterials consisting of Bragg reflector [47] and dimer-on-mirror metasurface [48], which is advantageous for the flexible control of chiroptical response due to its comparatively simple architecture.
Fig. 4. Evaluation of the BDL network for chiroptical responses with different CD peaks. (a), (b) Comparison of the simulated and predicted spectra (reflection and CD) from SPN model for the given parameters of (w1,w2,d,θ) = (70 nm,80 nm,40 nm,17°). (c), (d) Comparison of the desired and simulated spectra (reflection and CD) from DPN model for the retrieving parameters of (w1,w2,d,θ) = (75 nm,60 nm,28 nm,17°).
In order to investigate the robustness of our SPN and DPN models, we put 300 groups of structural parameters and spectra, which are taken from the testing data and have never been used during the training process, into trained model to test the network accuracy. Relative error [49] is employed to evaluate the similarity of predicted spectra and FDTD simulated spectra, which is given by:
Figure 5 shows the stabilization properties of the forward and inverse prediction. In Figs. 5(a) and (c), the relative error distribution of the SPN and DPN modules are demonstrated for 300 random test samples. As shown in Figs. 5(a) and (c), the average relative errors in the 300 groups of test data are very small, and they are 1.44% and 1.56% for SPN and DPN modules, respectively. In addition, the relative errors of the major test data are smaller than 3.00% for both the SPN and DPN modules, few relative errors are larger than 6.00% due to resonant mutation from data exception [30]. To further test the capability of our SPN and DPN models, we collect the average MSE values between the labeling spectra and the predicted spectra of every training result during the training process. Figures 5(b) and (d) show the histograms of the MSE loss for SPN and DPN modules for evaluation set predictions, where the dashed lines mark the 95.0% confidence borders. As can be seen in Figs. 5(b) and (d), the mean MSE losses of training sets are 0.0014 and 0.0022 for SPN and DPN module. Over 95.2% of these sets have MSE ≤ 0.0042 and MSE ≤ 0.0044 for SPN and DPN, respectively. Some MSE losses are relatively large due to outliers, which are beyond the range of the parameter values in the training data. The relative error distribution and low and consistent MSE values indicate that high prediction accuracy can be achieved by SPN and DPN without underfitting or overfitting.
Fig. 5. Stabilization properties of the SPN and DPN modules. (a) and (c) are the error distribution of 300 random test samples for the SPN and DPN modules, respectively. (b) and (d) are MSE loss histograms for all 7000 training datasets for the SPN and DPN modules, respectively; where mean MSE ≤ 0.0044 for over 95.2% training datasets.
4. Conclusion
In conclusion, we proposed a BDL network to accelerate the prediction and design process for flexible control of chiroptical response of planar chiral metamaterials. Both the traditional FDTD simulations and BDL network are employed to characterize the chiroptical response of the dagger-shaped metamaterials. The forward and backward paths of the proposed BDL network are constructed by the SPN and DPN modules, respectively. The SPN module can predict the reflection and CD spectra according to the input structural parameters, while the DPN module can predict the estimated structural parameters based on the input spectra. Both the SPN and DPN modules predict the reflection and CD responses of the planar chiral metamaterials with high accuracy (MSE ≤ 0.0044) and ultrafast computational speed (∼15 ms), which is 1 in 40000 compared with FDTD simulations. The mean-square error (MSE) loss of forward and inverse prediction reaches 0.0085 after 100 epochs, and the BDL network is robust in prediction without underfitting or overfitting for both the SPN and DPN. In general, the proposed framework can be adapted to the design of other complex chiral metamaterials, and it provides an effective approach for the bidirectional design of optoelectronic devices.
Funding
National Natural Science Foundation of China (62375113).
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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