Open Access
Add to BibSonomy
Abstract
Chiral metamaterial induced asymmetric transmission (AT) possesses great potential for terahertz (THz) polarization applications, but its design has mainly relied on the conventional trial-and-error forward strategy. Here, based upon massive training samples, we propose a deep-learning-based approach to accelerate the design of chiral metamaterials. The deep learning framework includes two bidirectional networks that allow the model to self-autonomously decipher the nonintuitive relationship between chiral metamaterial structures and their corresponding electromagnetic responses. Our preliminary results show that our model can accurately predict THz responses for any kind of metamaterial structure and inversely retrieve structure parameters from given THz responses. It turns out that the deep-learning-assisted methodology exhibits more efficient design performance than the conventional physical-based metamaterial design approach. This work might provide another artificially intelligent design strategy for chiral metamaterials and shed light on the prosperous development of unprecedented THz applications.
© 2022 Optica Publishing Group
1. INTRODUCTION
The terahertz wave, a special electromagnetic wave ranging from 0.1 THz to 10 THz, has attracted extensive research interest for its many applications in communications [1,2], biomedical [3,4], imaging [5–8], energy harvesting [9,10], and nondestructive testing [11,12]. Most of these applications are highly dependent on the interfacial coupling between the incident THz wave and transmission media, which can be strongly manipulated by the delicate design of the subwavelength unit cell of metamaterials. Metamaterials are currently used to achieve novel optical properties, such as a negative refractive index, perfect absorber, and invisible cloak. Among various metamaterials, chiral metamaterials play an important role because they can simultaneously achieve effective polarization control and chirality for incident THz wave. Arising from chiral metamaterials, the AT phenomenon, which refers to the transmission difference between different incident sides, has been widely applied in functional THz polarization devices such as isolators [13,14], circulators [15], and polarization rotators [16]. Nonetheless, most chiral metamaterials with the AT phenomenon are designed by conventional trial-and-error strategies, which make the metadesign process inefficient and restrains the development of meta-based THz devices.
In recent years, many efforts have been devoted to exploring the delicate design strategy of those subwavelength unit cells of chiral metamaterials to promote the research scope of THz polarization devices. Normally, an efficient design framework for chiral metamaterials has two branches: forward predicting electromagnetic responses by given structural parameters of metamaterials and inversely retrieving structural parameters from desired electromagnetic responses. The difference between the off-diagonal elements of the Jones transmission matrix can guide the ultimate design of chiral metamaterials [17]; however, it remains challenging when quantitatively designing a metamaterial structure by giving the targeted AT phenomenon or rapidly predicting THz responses as the structure changes. The typical approach, such as the physical rule-based trial-and-error strategy, heavily depends on the researcher’s prior experience via screening structural parameters through the finite difference simulation to identify electromagnetic responses. From its working mechanism, it is extremely time-consuming and wastes resources to solve Maxwell’s equations iteration by iteration. To overcome these issues, Ji et al. [18] incorporated the topology optimization approach based on genetic algorithms to design a highly efficient and ultra-broadband asymmetric transmission metasurface. It is well known that a stochastic algorithm must be assisted with simulation software during a practical implementation, and the time cost will exponentially increase once the scale grows. Therefore, improving the design efficiency of the metamaterials has become one of the complex issues for the development of metamaterials with intriguing properties.
Currently, deep learning (DL) has achieved great success in information technology [19–22], including computer vision, natural language processing, and objection detection. Numerous works recently have been devoted to excavating the underlying correlation between DL and optics metastructures [23–34]. For example, Malkiel et al. [23] built a bidirectional DL model based on a multiplayer perceptron (MLP) to design plasmonic nanostructures. With sufficient training samples, the model will be simultaneously turned into a fast predictor and inverse design tool of IR H-shaped metallics. Sajedian et al. [24] combined convolution neural networks (CNNs) with recurrent neural networks (RNNs) to improve the optical responses of IR nanostructures. Zhang et al. [26] proposed an approach that uses a genetic algorithm to design the architecture of the artificial neural networks, which has proven to be efficient for the inverse design and performance optimization for the plasmonic waveguide coupled with cavities structures. Li et al. [27] reported a framework that combines Bayesian optimization and deep CNN algorithms to optimize the metallic nanostructures. The self-learning mechanism enables the optical properties in the dataset from weak to strong and iteratively enhances the CNN model. Wang et al. [28] proposed generative adversarial networks (GANs) with elaborately designed loss functions and training strategies to inversely design the ultrawideband anisotropic metasurfaces with full phase properties. Under the guidance of the pretrained forward predictor, the proposed method exhibits high efficiency in automatically designing metasurfaces. Huang et al. [32] applied an MLP-based DL algorithm to inversely engineer THz metamaterial with electromagnetically induced transparency (EIT). By correlating three feature points of the EIT spectrum to the metamaterial structural parameters, the algorithm can effectively simplify the inverse design process of a specific THz metastructure. Pillai et al. [34] proposed a tandem encoder–decoder architecture to solve the inverse single-input, multi-output mapping scenario between the response space and the design space to realize the broadband, near “perfect” THz metamaterial absorber design. All these works emphasize the superiority of DL in designing metastructures over the manual design process for certain complex tasks. Although DL algorithms have acquired advances in designing various optical nanostructures with a specific requirement, the AT properties caused by chiral metamaterials remains less explored in the THz regime.
In this work, we introduce an MLP-based DL scheme composed of two bidirectional neural networks to model planar chiral THz metamaterials with the AT spectrum. Massive metamaterial samples are generated by numerical simulation to train the DL model and make it robust to predict the full THz responses of metamaterials and vice versa. The close match between the simulation results and prediction results reveals that the DL model can accurately capture the nonintuitive complex relationship between structural parameters and full electromagnetic spectra. Overall, our proposed model may be a useful scheme to design an unconventional THz metastructure.
2. MODEL AND DATABASE
As schematically shown in Fig. 1(a), the chiral THz metamaterial structure with a “G” shape, which is verified as a planar metamaterial with giant asymmetric transmission [35], is adopted as the prototype to generate massive samples for our model. The unit cell of metamaterial is defined as a G-type gold patterned on the silicon dioxide substrate with periodic constants along the X and Y directions (${\rm{Px}} = {\rm{Py}} = {{70}}\;{\rm{\unicode{x00B5}{\rm m}}}$) and along the Z direction as an open boundary condition. The thicknesses of the silicon dioxide and gold layer are set as 0.20 µm and 0.05 µm, respectively. The electric conductivity of the gold is set as $\sigma = {4.09} \times {{1}}{{{0}}^{7}}\;{\rm{S}}/{\rm{m}}$, and the permittivity of the silicon dioxide is set as 3.9.
Fig. 1. (a) Schematic representation of chiral THz metamaterial. (b) Illustration of asymmetric transmission of chiral THz metamaterial. The arrow “$^{\overrightarrow {}}$” or “$^{\overleftarrow {}}$” indicates the propagation direction of THz waves.
The frequency domain solver of the commercial software CST Microwave Studio is used to collect the electromagnetic responses of all chiral THz metastructures from 0.1 THz to 3.0 THz. The sampling step is set as 0.0145 THz. From Fig. 1(b), the right circularly polarized (RCP) or left circularly polarized (LCP) wave impinges along the normal direction of planar G-type metamaterial and engenders full electromagnetic spectra ${R_{- -}}$, ${T_{- +}}$, ${T_{+ -}}$, and ${T_{- -}},$ where $R$ and $T$ represent the reflection and transmission, − and $+$ represent the polarization state (LCP or RCP), and the first subscript and the second subscript represent transmitted and incident THz waves, respectively. Here, AT refers to the transmission difference of LCP (or RCP) from different incident sides. As shown in Fig. 1(b), the direct transmission of the metamaterial has the relationship ${\overrightarrow T _{- -}} = {\overleftarrow T _{- -}}$, where the arrows “$^{\overrightarrow {}}$” and “$^{\overleftarrow {}}$” stand for the front and back propagation of THz waves, respectively. The polarization conversion terms are unequal for opposite propagating directions of THz waves as ${\overrightarrow T _{+ -}} \ne {\overleftarrow T _{+ -}}$. Nevertheless, the conversion efficiencies for RCP and LCP are simply interchanged in the opposite directions of wave propagation [17] with ${\overrightarrow T _{- +}} = {\overleftarrow T _{+ -}}$. Therefore, the AT can be evaluated as ${\rm{AT}} = {\overrightarrow T _{+ -}} - {\overleftarrow T _{+ -}} = {\overrightarrow T _{+ -}} - {\overrightarrow T _{- +}}$. In this work, the arrows are eliminated to make the terms concise and readable, and the AT is denoted as ${\rm{AT}} = {T_{+ -}} - {T_{- +}}.$ All structural parameters, including the middle-bar width $d$, border length $l$, side clearance $w$, and side-bar width $t$, are systematically screened to generate all metastructures (13800 cases). Table 1 shows the appropriate sampling interval from the parametrical hyperspace of the G-shape unit cell. Among all collected samples, 75% of them are assigned as the training dataset while the remaining samples are assigned as the test dataset.
Table 1. Design Parameters of the G-Shape Metastructure
Fig. 2. Framework of the DL model to automatically design chiral THz metamaterials. The blocks represent the data or the network weights while the circles represent the network neurons. The model is composed of two bidirectional networks: SN and EN. (a) The data flow in the SN: The forward path contains a TDS module and a TUS module to convert the structural parameters into response spectra, and the inverse path can effectively solve the many-to-one problem in the metamaterial design with the aid of the well-trained forward path of the SN. (b) The structure of the EN: The forward path can realize the highly accurate prediction of THz metamaterial’s AT spectrum, and the inverse path can directly retrieve structural parameters from the desired AT. (c) The combiner can output the weighted sum of retrieval parameters from the SN and EN. (d) Details of the NT layer in the TDS module of SN. fcg1–4, fully connected layer groups with the same architecture; bl, bottleneck layer; usu1–4, upsampling units with the same architecture; fcn, fully connected layer followed by batch normalization layer in the SN; fc, fully connected layer; fcb, fully connected layer followed by batch normalization layer in the EN; tconv, transposed convolutional layer; itp, interpolation; and conv, convolutional layer.
3. DEEP LEARNING METHOD
Figure 2 shows the overall framework of the DL model that comprises a spectra network (SN) and an extended network (EN) with both bidirectional configurations. It clearly shows that the DL model regulates the data flow bidirectionally. On the one hand, the forward paths of two networks are serially connected to guarantee the prediction accuracy and expand the model function. On the other hand, two retrieval outputs are combined with different weights to make up a combiner as a higher-level window due to parallel inverse paths. As highlighted in Fig. 2(a), SN is used to evaluate the nonintuitive relationship between metamaterial structures and full electromagnetic spectra through the forward and inverse design paths. For the G-shape unit cell, design parameters are fixed at four ($d$, $l$, $t$, and $w$), and the interested full spectra are uniformly sampled into 201 discrete points, respectively. Fundamentally, SN mainly solves the regression problem between THz metamaterials’ structural parameters (size ${{4}} \times {{1}}$) and THz response spectra (size ${{201}} \times {{4}}$). The AT from each metamaterial’s full response spectra is chosen as an important indicator to evaluate the designed metamaterial and is explicitly modeled in EN.
In the forward path of SN, structural parameters (size ${{4}} \times {{1}}$) will be transformed into response spectra with a size of ${{201}} \times {{4}}$, which indicates that the dimension of input data is much lower than that of output data and it will be challenging to make the network converge, especially when the spectra have drastic variations around resonant frequencies. To deal with this issue, a two-stage strategy is employed to build an approximate input–bottleneck–output network structure. Concretely, the dimension of input data is first reduced to form bottleneck layers, then gradually increases until it finishes the regression progress. The incorporation of bottleneck layers effectively increases the depth and training weights of SN networks, which will make the deep networks easily converge. Following this strategy, a tensor downsampling (TDS) module and a tconv upsampling (TUS) module are both employed in the forward path, and the last layers of the TDS module serve as the bottleneck layers (each has ${{26}} \times {{1}}$ neurons). A neural tensor (NT) layer, shown in Fig. 2(d), can raise the dimension of the input structural parameters (size ${{4}} \times {{1}}$) to four parallel data (size ${{50}} \times {{1}}$) for a subsequent dimensional reduction operation, and it is first proposed for relation reasoning of the entity vector [36], and later applied to design optics nanostructures [30,37]. Simultaneously, the NT layer can reflect the second-order relationship of a variable couple rather than a linear combination of fully connected layers. In our study, the NT layers are self-contained to describe the interaction of different structural parameters of the G-type pattern with the incident THz wave. The output of the NT layer can be described as
The TDS module is first trained in a supervised manner, where the training labels are the ${{26}} \times {{4}}$ vectors uniformly downsampled from four spectra. These subspectra, which basically represent the full spectra distribution, are used as the ground truth for pretraining the TDS module that includes an NT layer and four parallel layer groups with three fully connected layers. At the start of the TDS module, the structural parameters are initially transformed into four parallel data (size ${{50}} \times {{1}}$) in the NT layer. Then, the parallel data flow into the four parallel layer groups to reduce the dimension to be ${{26}} \times {{1}}$, respectively. At the end of the TDS module, four parallel data are concatenated into four-channel data (size ${{26}} \times {{4}}$) to evaluate the cost with the target subspectra for convenience. After the pretraining is done, the TDS module is frozen and subsequently used in the training procedure of the TUS module. As shown in Fig. 2(a), four channels of data are flowed into individual upsampling units with the same architecture, respectively. An upsampling unit comprises three similar neural layer groups and each has a transposed convolutional layer, an interpolation layer, and a fully connected layer. Here, the transposed convolutional kernel with shape ${{1}} \times {{3}} \times {{1}}$ (${\rm{output}}\;{\rm{channel}} \times {\rm{kernel}}\;{\rm{size}} \times {\rm{input}}\;{\rm{channel}}$) is used in each transposed convolutional layer to extract the local data distribution feature of the inflow data without changing its shape. The linear interpolation layer increases the dimension and ensures the data is continuous and smooth in general, and the following fully connected layer fine-tunes the passing data and guarantees consistency with the ground truth distribution. After the data flow through the above-mentioned layers, the data dimension is transformed into ${{51}} \times {{1}}$, ${{101}} \times {{1}}$, and ${{201}} \times {{1}}$, respectively. Finally, all data from four parallel upsampling units are merged as the ultimate result of the TUS module to evaluate the cost with the target full spectra. After the TUS module is properly trained, two pretrained modules are trained together as a whole forward path of SN to fine-tune the forward network. The activation function in most layers of the SN is chosen to be “tanh,” as the RELU activation function is easy to make neurons invalid when the network has deep layers; that is, regardless of any kind of data input, the output of these layers will not change.
In the inverse path of SN, two typical convolutional neural layers with a ${{4}} \times {{3}} \times {{4}}$ kernel and a ${{1}} \times {{3}} \times {{4}}$ kernel, respectively, are employed followed by three fully connected layers. Arising from the fundamental property of the inverse design that the same response spectra may be aroused by multi-groups of different structural parameters, it’s hard to enforce the deep neural networks to converge during the inverse design process. The inverse network in a full loop with the forward path of SN is trained under the tandem training strategy [37,38]. In our design, only the tandem network loss may retrieve a set of negative parameters that are close to the opposite number of the true value in the dataset, because the “tanh” activation function is an odd function and the NT layer also contains quadratic terms of structural parameters. To make sure the retrieval parameters are close to the dataset, the inverse design loss that plays as a penalty term to enhance the network’s robustness can be evaluated as
4. MODEL EVALUATION
Our model is trained with the Adam optimization algorithm and processed using a graphic card (NVIDIA GeForce GTX 1660Ti) with CUDA 10.0 on the Win10 system. The learning rate periodically decreases with the increment of training epochs, and the model is built on the open-source deep learning framework PyTorch. Three logical steps are implemented to train the forward path of SN. We initially train the TDS module and feed its output into the TUS module for the second training step. After the TDS module and TUS module are well trained, the entire forward path is retrained again as a whole part. After the SN training is done, the network’s performance is evaluated by the test data that do not exist in the training dataset. Figure 3 plots the loss variations for the TUS module and inverse path of SN. The mean square error (MSE) criterion is converged at ${1.94} \times {{1}}{{{0}}^{- 4}}$ for the training dataset and around ${1.47} \times {{1}}{{{0}}^{- 4}}$ for the test dataset in the forward path of SN.
Fig. 3. TUS module and the SN inverse design loss variation during the training process. The light blue lines represent the original loss while the navy blue lines represent the smoothed loss variation.
Figure 4 demonstrates the effectiveness of the well-trained model with two randomly selected samples from the test dataset. It clearly shows that the predicted spectra from SN, as shown in Figs. 4(a) and 4(b), agree well with the CST simulated response spectra, which implies the powerful design ability of SN in the prediction of electromagnetic spectra concerning their typical structures. Figures 4(b) and 4(d) display the AT profile through the formula ${\rm{AT}} = {T_{+ -}} - {T_{- +}}$, where ${T_{+ -}}$ and ${T_{- +}}$ are predicted from SN. The calculated AT is quite close to the target AT spectra with trivial fluctuance. To further improve the AT prediction accuracy, EN is added to bridge structural parameters with AT spectra with the aid of SN. The forward path of EN can transfer the calculated AT from SN to target AT (size ${{51}} \times {{1}}$), while the inverse path explicitly converts the given AT to structural parameters. The data dimension evolves from a large size to a small one in the EN. Figure 2(b) shows five traditional fully connected layers in both the forward and inverse path of EN. We input the calculated AT of two random examples into the forward path of the trained EN and obtain the predicted AT (size ${{51}} \times {{1}})$ displayed in blue dots in Figs. 4(b) and 4(d). The inset tables show the difference between the desired simulated AT and predicted AT from EN. The prediction error of the AT of the two samples is no more than 15%. With the aid of the EN model, the MSE of the AT prediction in the test dataset drops from ${1.52} \times {{1}}{{{0}}^{- 5}}$ to ${{9}} \times {{1}}{{{0}}^{- 6}}$, indicating the significant prediction ability of the forward path of the EN model.
Fig. 4. Forward prediction of two samples in test dataset by DL model. (a),(b) Prediction results from the SN, where the dotted lines represent the prediction results from the SN and the solid lines represent the target spectra from the test dataset. (c),(d) Additional prediction results from the EN, where the red lines represent the desirable AT spectra and the green lines are the results directly calculated from the SN predict spectra, and the blue lines are the output by the EN. The insets represent metamaterial structures randomly selected from the test dataset.
Fig. 6. CST simulation from design parameters of the SN and the combiner. The inset metamaterial structures are retrieved from the SN.
To evaluate the inverse design performance of the DL model, we also randomly select two groups of spectra in the test dataset as desired responses. Since SN and EN are both bidirectional networks, they can simultaneously retrieve a set of structural parameters. For the inverse path, a combiner is used to output the weighted sum of retrieval parameters from the SN and EN, where two weights are obtained from two neurons trained by the given data. The retrieved design parameters shown in Fig. 5 indicate that the combiner might have a similar performance as the SN. To intuitively illustrate the accuracy of the designed spectra (full spectra and AT spectra) profiles, the retrieval metamaterial structures (from SN and combiner) are simulated again in CST software in Fig. 6. The MSEs of the two samples are listed in Table 2. The close results of full spectra and AT profiles indicate that the combiner may be a better choice than SN for structural parameters when designing metamaterials.
Table 2. Comparison of SN and Combiner Retrieval Results
Fig. 7. Inverse design by the DL model. (a) and (c) Desired and simulated AT spectra. The insets list the retrieval structure parameters. The light blue lines represent the AT curves referenced to our desired AT. The green dots are the uniform discrete points used to input into the inverse design path of the EN. (b) and (d) Predicted full response spectra along with the full-wave simulations with the retrieval parameters.
5. INVERSE DESIGN OF CHIRAL THz METAMATERIALS
In a practical procedure of metamaterial design for sensing and imaging, only a few requirements such as resonant frequency and asymmetric ratio (the difference between ${T_{+ -}}$ and ${T_{- +}}$), are initially necessary rather than the full electromagnetic spectra. Particularly, the inverse design scheme also should work when only AT spectra are given, which may tremendously curtail the design time. Considering that the AT in our dataset has no specific pattern and cannot be analytically expressed by a simple formula, two random test samples are expanded to 1.2 times the original one to mimic the desired AT spectra. As long as the prescribed AT spectra are achievable by G-type metamaterial structure, the DL model can retrieve possible structural parameters that match the prescribed requirement of the spectra. The retrieval structure parameters from the EN pass through the entire DL model to predict the full spectra, which agrees well with the full-wave simulation by CST software, as shown in Fig. 7. The good match between the desired spectra and predicted results manifests that the DL model can accelerate the metamaterial design with only a few requirements. Interestingly, the strong asymmetric ratio in Fig. 7(c) is about 12.12% at 2.62 THz, which is around eight times of the original structure [35] where the asymmetric ratio is about 1.51% at 2.52 THz. It implies that the DL model can not only assist in the structure design but also can generate an unexpected structure that is beyond the searching scope of the trial-and-error approach.
6. CONCLUSION
In this work, we have established a deep-learning model to accelerate the design process of THz metamaterials with AT properties. The DL model consists of an SN with a two-module architecture and an EN. Both networks can implement forward prediction and inverse design bidirectionally. On the one hand, this DL model can predict the electromagnetic responses (AT performance) of chiral THz metamaterials with high accuracy. This data-driven approach succeeds over the traditional physical rule-based approach. On the other hand, the model simultaneously solves the inverse problem that can retrieve metamaterial design parameters from specific electromagnetic responses, which makes it facile and flexible to design THz devices with expected chiral functions. It should be also mentioned that, except for targeting the AT property of chiral metamaterial, our proposed model can also be easily extended to intelligently design other types of THz devices with minor modifications. Our work may provide a robust strategy to accelerate the design of THz metamaterials and shed light on its wide applications in optoelectrical devices.
Funding
National Natural Science Foundation of China (52175115, 51805414); Natural Science Foundation of Zhejiang Province (LZ19A020002).
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.
REFERENCES
1. T. Nagatsuma, S. Horiguchi, Y. Minamikata, Y. Yoshimizu, S. Hisatake, S. Kuwano, N. Yoshimoto, J. Terada, and H. Takahashi, “Terahertz wireless communications based on photonics technologies,” Opt. Express 21, 23736–23747 (2013). [CrossRef]
2. J. M. Jornet and I. F. Akyildiz, “Graphene-based plasmonic nano-antenna for terahertz band communication in nanonetworks,” IEEE J. Sel. Areas Commun. 31, 685–694 (2013). [CrossRef]
3. Q. S. Sun, Y. Z. He, K. Liu, S. T. Fan, E. P. J. Parrott, and E. Pickwell-MacPherson, “Recent advances in terahertz technology for biomedical applications,” Quantum Imaging Med. Surg. 7, 345–355 (2017). [CrossRef]
4. X. Yang, X. Zhao, K. Yang, Y. P. Liu, Y. Liu, W. L. Fu, and Y. Luo, “Biomedical applications of terahertz spectroscopy and imaging,” Trends Biotechnol. 34, 810–824 (2016). [CrossRef]
5. B. B. Hu and M. C. Nuss, “Imaging with terahertz waves,” Opt. Lett. 20, 1716–1718 (1995). [CrossRef]
6. B. Li, R. Wang, J. J. Ma, and W. H. Xu, “Research on crop water status monitoring and diagnosis by terahertz imaging,” Front. Phys. 8, 419 (2020). [CrossRef]
7. Y. F. Xu, H. B. Hao, D. S. Citrin, X. Y. Wang, L. Y. Zhang, and X. F. Chen, “Three-dimensional nondestructive characterization of delamination in GFRP by terahertz time-of-flight tomography with sparse Bayesian learning-based spectrum-graph integration strategy,” Composites, Part B 225, 109285 (2021). [CrossRef]
8. J. T. Li, J. Li, C. L. Zheng, S. L. Wang, M. Y. Li, H. L. Zhao, J. H. Li, Y. T. Zhang, and J. Q. Yao, “Dynamic control of reflective chiral terahertz metasurface with a new application developing in full grayscale near field imaging,” Carbon 172, 189–199 (2021). [CrossRef]
9. M. Dragoman and M. Aldrigo, “Graphene rectenna for efficient energy harvesting at terahertz frequencies,” Appl. Phys. Lett. 109, 113105 (2016). [CrossRef]
10. M. Pierobon, J. M. Jornet, N. Akkari, S. Almasri, and I. F. Akyildiz, “A routing framework for energy harvesting wireless nanosensor networks in the terahertz band,” Wireless Netw. 20, 1169–1183 (2014). [CrossRef]
11. Y. F. Xu, X. D. Fang, S. T. Fan, L. Y. Zhang, R. Q. Yan, and X. F. Chen, “Double Gaussian mixture model-based terahertz wave dispersion compensation method using convex optimization technique,” Mech. Syst. Signal Process. 164, 108223 (2022). [CrossRef]
12. Y. Xu, X. Wang, L. Zhang, R. Yan, and X. Chen, “Terahertz nondestructive quantitative characterization for layer thickness based on sparse representation method,” NDT&E Int. 124, 102536 (2021). [CrossRef]
13. Y. Y. Ji, F. Fan, Z. Y. Tan, and S. J. Chang, “Terahertz nonreciprocal isolator based on magneto-plasmon and destructive interference at room temperature,” Front. Phys. 8, 334 (2020). [CrossRef]
14. Z. Y. Tan, F. Fan, D. Zhao, Y. Y. Ji, J. R. Cheng, and S. J. Chang, “High-efficiency terahertz nonreciprocal one-way transmission and active asymmetric chiral manipulation based on magnetoplasmon/dielectric metasurface,” Adv. Opt. Mater. 9, 2002216 (2021). [CrossRef]
15. M. Liu, Q. Xu, X. Y. Chen, E. Plum, H. Li, X. Q. Zhang, C. H. Zhang, C. W. Zou, J. G. Han, and W. L. Zhang, “Temperature-controlled asymmetric transmission of electromagnetic waves,” Sci. Rep. 9, 4097 (2019). [CrossRef]
16. T. T. Lv, X. Y. Chen, G. H. Dong, M. Liu, D. M. Liu, C. M. Ouyang, Z. Zhu, Y. X. Li, C. Y. Guan, J. G. Han, W. L. Zhang, S. Zhang, and J. H. Shi, “Dual-band dichroic asymmetric transmission of linearly polarized waves in terahertz chira metamaterial,” Nanophotonics 9, 3235–3242 (2020). [CrossRef]
17. R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80, 153104 (2009). [CrossRef]
18. W. Y. Ji, T. Cai, G. M. Wang, H. P. Li, C. Y. Wang, H. S. Hou, and C. B. Zhang, “High-efficiency and ultra-broadband asymmetric transmission metasurface based on topologically coding optimization method,” Opt. Express 27, 2844–2854 (2019). [CrossRef]
19. Z. Y. Zhong, L. Sun, and Q. Huo, “An anchor-free region proposal network for Faster R-CNN-based text detection approaches,” Int. J. Doc. Anal. Recogn. 22, 315–327 (2019). [CrossRef]
20. I. Sutskever, O. Vinyals, and Q. V. Le, “Sequence to sequence learning with neural networks,” in NIPS'14: Proceedings of the 27th International Conference on Neural Information Processing Systems (MIT Press, 2014), pp. 3104–3112.
21. F. Gao, C. G. Yang, Y. S. Ge, S. F. Lu, and Q. K. Shao, “Dense Receptive Field Network: a backbone network for object detection,” in Artificial Neural Networks and Machine Learning – ICANN 2019: Image Processing (Springer, 2019), pp. 105–118.
22. G. Hinton, L. Deng, D. Yu, G. E. Dahl, A. R. Mohamed, N. Jaitly, A. Senior, V. Vanhoucke, P. Nguyen, T. N. Sainath, and B. Kingsbury, “Deep neural networks for acoustic modeling in speech recognition,” IEEE Signal Process. Mag. 29, 82–97 (2012). [CrossRef]
23. I. Malkiel, M. Mrejen, A. Nagler, U. Arieli, L. Wolf, and H. Suchowski, “Plasmonic nanostructure design and characterization via deep learning,” Light Sci. Appl. 7, 60 (2018). [CrossRef]
24. I. Sajedian, J. Kim, and J. Rho, “Finding the optical properties of plasmonic structures by image processing using a combination of convolutional neural networks and recurrent neural networks,” Microsyst. Nanoeng. 5, 27 (2019). [CrossRef]
25. W. Ma, F. Cheng, Y. H. Xu, Q. L. Wen, and Y. M. Liu, “Probabilistic representation and inverse design of metamaterials based on a deep generative model with semi-supervised learning strategy,” Adv. Mater. 31, 1901111 (2019). [CrossRef]
26. T. Zhang, J. Wang, Q. Liu, J. Z. Zhou, J. Dai, X. Han, Y. Zhou, and K. Xu, “Efficient spectrum prediction and inverse design for plasmonic waveguide systems based on artificial neural networks,” Photon. Res. 7, 368–380 (2019). [CrossRef]
27. Y. Li, Y. J. Xu, M. L. Jiang, B. W. Li, T. Y. Han, C. Chi, F. Lin, B. Shen, X. Zhu, L. H. Lai, and Z. Y. Fang, “Self-learning perfect optical chirality via a deep neural network,” Phys. Rev. Lett. 123, 213902 (2019). [CrossRef]
28. H. P. Wang, Y. B. Li, H. Li, S. Y. Dong, C. Liu, S. Jin, and T. J. Cui, “Deep learning designs of anisotropic metasurfaces in ultrawideband based on generative adversarial networks,” Adv. Intell. Syst. 2, 2000068 (2020). [CrossRef]
29. C. C. Nadell, B. H. Huang, J. M. Malof, and W. J. Padilla, “Deep learning for accelerated all-dielectric metasurface design,” Opt. Express 27, 27523–27535 (2019). [CrossRef]
30. S. S. An, B. W. Zheng, M. Y. Shalaginov, H. Tang, H. Li, L. Zhou, J. Ding, A. M. Agarwal, C. Rivero-Baleine, M. Kang, K. A. Richardson, T. Gu, J. J. Hu, C. Fowler, and H. L. Zhang, “Deep learning modeling approach for metasurfaces with high degrees of freedom,” Opt. Express 28, 31932–31942 (2020). [CrossRef]
31. E. Ashalley, K. Acheampong, L. V. Besteiro, P. Yu, A. Neogi, A. O. Govorov, and Z. M. Wang, “Multitask deep-learning-based design of chiral plasmonic metamaterials,” Photon. Res. 8, 1213–1225 (2020). [CrossRef]
32. W. Huang, Z. M. Wei, B. Y. Tan, S. Yin, and W. T. Zhang, “Inverse engineering of electromagnetically induced transparency in terahertz metamaterial via deep learning,” J. Phys. D 54, 135102 (2021). [CrossRef]
33. Z. Zhang, D. Han, L. Y. Zhang, X. Q. Wang, and X. F. Chen, “Adaptively reverse design of terahertz metamaterial for electromagnetically induced transparency with generative adversarial network,” J. Appl. Phys. 130, 033101 (2021). [CrossRef]
34. P. Pillai, P. Pal, R. Chacko, D. Jain, and B. Rai, “Leveraging long short-term memory (LSTM)-based neural networks for modeling structure-property relationships of metamaterials from electromagnetic responses,” Sci. Rep. 11, 18629 (2021). [CrossRef]
35. C. J. Liu, Y. Y. Huang, F. R. Hu, E. Yiwen, X. X. Dong, Y. Yang, Y. P. Jin, and X. L. Xu, “Giant asymmetric transmission and circular dichroism with angular tunability in chiral terahertz metamaterials,” Ann. Phys. (Berlin) 532, 2070017 (2020). [CrossRef]
36. R. Socher, D. Chen, C. D. Manning, and A. Y. Ng, “Reasoning with neural tensor networks for knowledge base completion,” in Proceedings of the 26th International Conference on Neural Information Processing Systems (Curran Associates Inc., 2013), Vol. 1, pp. 926–934.
37. W. Ma, F. Cheng, and Y. M. Liu, “Deep-learning-enabled on-demand design of chiral metamaterials,” ACS Nano 12, 6326–6334 (2018). [CrossRef]
38. D. J. Liu, Y. X. Tan, E. Khoram, and Z. F. Yu, “Training deep neural networks for the inverse design of nanophotonic structures,” ACS Photon. 5, 1365–1369 (2018). [CrossRef]
