## Abstract

Metamaterial perfect absorbers (MPAs) typically have regularly-shaped unit structures owing to constraints on conventional analysis methods, limiting their absorption properties. We propose an MPA structure with a general polygon-shaped meta-atom. Its irregular unit structure provides multiple degrees-of-freedom, enabling flexible properties, such as dual-band absorption. We constructed a deep neural network to predict the parameters of the corresponding MPA structure with a given absorptivity as input, and vice versa. The mean-square error was as low as 0.0017 on the validation set. This study provides a basis for the design of complicated artificial electromagnetic structures for application in metamaterials and metasurfaces.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Metamaterials are artificial electromagnetic (EM) structures that have been attracting increasing scientific interest because of their novel EM properties. These properties are superior to those of natural materials, and include perfect absorbance [1], negative refractive index [2], asymmetric light transmission [3], inverse Doppler effect [4], and EM wave cloaking [5]. Among these unprecedented characteristics and relevant applications, the metamaterial perfect absorber (MPA), which was demonstrated experimentally in the microwave region by Landy et al. [6], has been investigated extensively, and its operations have been realized within a broad frequency range (from THz to visible light frequencies) during the last decade [7]. Taking advantage of its compact size, high absorptivity, and tunable wavelength, MPA plays an important role in applications such as heat emitters [8], photodetection [9], and sensors [10].

Compared with the conventional photonic absorber whose performance is usually determined by its intrinsic properties, including the chemical constituents and crystalline structures of the materials, the MPA provides an efficient method to modulate and control the light propagation by its artificially periodic structure [11]. The metal–dielectric–metal (MIM) structure is an excellent design of an MPA that can accommodate the EM energy inside and gradually dissipate it. The unit structures of MIM absorbers, also called meta-atoms, are mainly regular and simple geometries, including square [12], rectangular [13], cross [14], and circular [15]. These simple geometries facilitate the design and fabrication process, and have successfully demonstrated good absorbance properties. However, most of these designs exhibit only one resonant absorption peak within a certain wavelength range. These simple structures have low degrees-of-freedom in their morphology; thus, they lack a sufficient number of parameters to tune their absorptivity and operation wavelength characteristics. Several schemes have been proposed and studied to realize multiband absorption, including composite structures, which consist of different simple meta-atoms [16,17] and multiple-layer MIM structures [18,19]. However, these designs complicate fabrication and limit the device size and efficiency. Thus, novel MPA based on complex meta-atoms with irregular structures have not been investigated yet.

Despite the advantage of more degrees-of-freedom in the morphology, the complex meta-atom MPA imposes design difficulties, as the irregular geometry can hardly be analyzed based on the theoretical resonant mode. It is also inconvenient to use conventional numerical simulations because there are a variety of parameters to be optimized for an irregular MIM structure. Although some conventional optimization approaches, such as topology optimization, particle swarm optimization, evolutionary optimization and so on, can be applied to achieve a singular “best” design of complicated artificial EM structures depending on the input optimization goal, an update of the input goal needs to start again a new inverse-design process from beginning [20–23]. Conversely, the emerging neural network algorithm (NNA) has attracted the interest of researchers in recent years because a computational learning model is built based on a huge amount of the input and output data that are used to establish the equivalent data relationship. In this way, the established model is capable of rapidly predicting the optimization structure depending on the input design goal without recalculation. The NNA has been extensively used to address various engineering tasks that are intricate and arduous by conventional approaches, such as medical imaging [24], disaster risk prediction [25], and metamaterial design [26–30], and thus provides a promising solution for the design of complicated EM nanostructures.

To the best of our knowledge, this is the first study to propose and investigate a novel MPA structure with a polygonal meta-atom. The proposed MPA takes advantage of tunable shape parameters, thereby enabling a higher degree-of-freedom in terms of meta-atom modeling, to realize dual-band perfect absorption with a single-layer structure, which is demonstrated by three-dimensional (3D), finite-difference time-domain (FDTD) simulations. Herein, we use the NNA to establish a model that associates the geometry of the MPA with the absorption spectrum to predict the absorption spectrum for a given MPA structure, and vice versa. Furthermore, we improved the NNA to obtain the ideal target model accuracy. As a result, the model accuracy increased by 38%, and the final mean-square error (MSE) of the training and validation sets were reduced to 0.0008 and 0.0017, respectively. Our proposed MPA structure, modeled and trained by the NNA shows novel properties in terms of its capacity to absorb perfectly with flexible and efficient design, and can find applications in the fields of sensors, heat meters, and photovoltaics.

## 2. Schematic of the proposed MPA

Figure 1 shows a schematic of the proposed, polygonal MIM structure which consists of Al_{2}O_{3} and Ag. The thickness of the Ag layer at the bottom of the unit, which is opaque to the incident light, was 80 nm. The geometry of the MIM unit is designed as a general polygon with eight tunable vertices (P_{1}–P_{8}), as shown in Fig. 1. The position of each vertex is aligned evenly along the x-axis, but can be modified along the y-axis, thus achieving a flexible polygonal topology with multiple degrees-of-freedom. For each polygonal unit, the thicknesses of the Ag and Al_{2}O_{3} layers were 30 nm and 12 nm, respectively. The period of the unit was 250 nm in both the x and y directions. This proposed MIM structure can be fabricated on a fused silica substrate by standard nanofabrication techniques [31,32]. The bottom Ag layer is assumed to be deposited using electron-beam evaporation, while the Al_{2}O_{3} dielectric layer can be deposited by atomic layer deposition. The top Ag pattern can be fabricated using electron beam lithography, followed by metal deposition and lift-off procedures.

The 3D-FDTD method was used to investigate numerically the proposed structure with the Lumerical FDTD solution-based, commercial software package. The experimentally obtained dielectric constant values of Ag were fitted with the Drude–Lorentzian model, and the refractive index of Al_{2}O_{3} was set to 1.75 [33]. In the simulation, the period boundaries were set in the X and Y directions, while the perfectly matched layers (PML) were used on the boundaries in the Z direction to absorb the reflective waves. The mesh size of the computational area was uniformly set to 2 nm, which guarantees good convergence and rapid calculation time.

The absorbance of the incident transverse magnetic (TM) polarized light was studied numerically and calculated from the simple relation *A* = 1 − *T* − *R*, where *A* is the absorbance, *T* is the transmittance and *R* is the reflectance. By designing the unit structure with multiple degrees-of-freedom (Fig. 1), both single peak and double-peak absorptions can be achieved within a certain wavelength range. Using FDTD simulations, absorbance spectral data and the corresponding structural parameters (for example, P_{1}–P_{8}) are used to train the neural network. The trained neural network can successfully predict the absorption spectrum of a general polygonal MIM structure, and design a demanded absorber structure according to a given spectrum, as discussed in detail in the following sections.

## 3. Design of the neural network model

To realize the flexible topology of the proposed meta-atom in Fig. 1, we set eight parameters to design its geometrical structure. Subsequently, 3D-FDTD simulations were conducted to obtain the corresponding absorption spectrum; for simplicity, but without loss of generality, each spectrum was recorded as 300 frequency points. We construct and train our dataset on the AMD EPYC 7302 16 - Core CPU using the python 3. 7 as the programming language, and TensorFlow 2. 3. 0 as the development framework. To investigate and optimize the connection between the complex MPA structure in Fig. 1 and its absorption spectrum, we designed fully connected neural networks to establish the model, including the design predicting network (DPN) and spectra predicting network (SPN). These neural networks were trained separately to ensure accuracy. By combining the two networks, it can be used as a verification neural network to verify the accuracy of the model, as shown in Fig. 2.

First, we used the DPN and SPN in Fig. 2 for the first-step verification. In this study, we used dense layers with a rectified linear unit activation function and the Adam optimizer. The epochs were set to 300. By reducing continually, the MSE losses, the parameters of the neural network were optimized once every iteration.

We then focused on the improvement of the NNA of the DPN to obtain accurate structural parameters. We used the K-fold (leave-one-out) cross-validation method to fully train the models. These models can reduce the overfitting and obtain sufficient effective information from the limited data [34]. The value of K was selected to be equal to five so that it matched the model to training set efficiently and quickly. Meanwhile, the effect of a fully connected neural network depends on the parameter setting and number of hidden layers. By optimizing a variety of settings, we combined the effects of several models with four, five, and six hidden layers, and built a new neural network to guarantee the balance between the linear effect and convergence of the model, as shown in Fig. 3.

We numerically investigated a variety of the proposed MPA with different meta-atom structures using 3D-FDTD simulations, and obtained a large amount of data. All datasets included the absorption spectra and relevant geometrical parameters. Among them, most data groups were fed in the neural network as a training set, and the remaining data were used as the validation set to evaluate the final effect of the model. For example, we fed the absorption spectrum, which is a matrix of N rows and 300 columns, in each model to obtain the average predicted parameters of the structure, which is a matrix of N rows and 8 columns. Herein, N is the number of training data points. Subsequently, we stacked the results predicted by the three models together, fed the stack data (in the form of a matrix of N rows and 24 columns) in the second network with one hidden layer to obtain the final (improved) result, which is a matrix of N rows and 8 columns. The second network was designed to allocate the weight of each model by choosing the better part of each model. Note that the validation set will not appear in the training set. Thus, it can be used to verify the prediction effect of unknown data. Once the DPN and SPN are established and trained, they can rapidly predict the corresponding MPA structure with an input absorptivity, and vice versa, showing superiority to the conventional optimization approaches.

## 4. Results and discussion

#### 4.1 Effects of the spectra predicting network

We used a network with four hidden layers, and then randomly selected a set of geometrical parameters to describe a general polygonal-based MPA structure, as defined in Fig. 1. The selected parameters P_{1}–P_{8} are 30 nm, 40 nm, 90 nm, 60 nm, 70 nm, 80 nm, 100 nm, and 50 nm, respectively, so that the initial MPA structure was obtained as the input data, as shown in the inset of Fig. 4(a). Subsequently, the SPN predicts the absorption spectrum (orange line in Fig. 4(a)), as the output data of the input structure. For comparison, the 3D-FDTD simulation result of the given structure was also calculated (blue curve in Fig. 4(a)). We can see that the predicted result of the network fits well the simulation result, and demonstrates the effectiveness of the neural network. Note that the input MPA structure in Fig. 4(a) is chosen randomly without any optimization; thus, its spectrum does not show a perfect absorption peak.

In addition, we investigated the MSE loss of the SPN, as shown in Fig. 4(b). This shows that after several hundreds of iterations, the MSE loss can be reduced to values as low as 0.0008 in our validation set, thus indicating good prediction accuracy. Using our designed neural network, the properties of arbitrary MPA structures proposed in this study can be obtained quickly and correctly within a few seconds; thus, the design process of the complex MPA structure with a general polygonal meta-atom becomes efficient and convenient. As a result, the novel properties of the proposed MPA structure with multiple degrees-of-freedom can be studied exhaustively.

#### 4.2 Effects of the improved design predicting network

Compared with the SPN, the MSE loss of the DPN is higher. Thus, we can improve its NNA using the network designed in Fig. 3. The prediction ability of the model usually depends on the number of training data. Therefore, we investigated the MSE loss of different training data, as listed in Table 1. With an increasing number of input training data, the MSE losses of models A, B, and C are reduced, thus indicating that a sufficient number of training data is important to achieve feature mining and higher accuracy.

Furthermore, the MSE of the improved DPN is significantly reduced for all the cases in Table 1, thus demonstrating the effectiveness of the designed NNA of the DPN. For example, in the case in which 14000 groups of data are in the training data, and the remaining 1000 groups are in the validation set, the initial MSEs of the DPN with any simple model A, B, and C, are 0.0035, 0.0033, and 0.0042, respectively. With the improved NNA, the MSE of the predicted data in the validation set was 0.0026, which yielded a reduction in the range of 21–38%. Note that in this study, we used 14000 groups of data as training data.

The verification function was realized by splicing the networks of the improved DPN and the SPN. After full training, the final MSE of the training set and the validation set were reduced to 0.0008 and 0.0017, respectively. This yielded an excellent prediction performance. For example, we selected a single-peak absorption spectrum with 99.3% absorptivity at 653 nm as the input for the improved DPN, as shown by the blue curve in Fig. 5(a). The improved DPN predicted the corresponding MPA structure as the output, as shown in the inset of Fig. 5(a). The geometrical parameters P_{1}–P_{8} were 41, 45, 87, 61, 69, 77, 22, and 73 nm, respectively. We also used the SPN to obtain the spectrum of the output MPA structure, as shown by the orange curve in Fig. 5(a). This achieved an absorptivity of 96.1% at 651 nm and fitted well to the input spectrum. For comparison, we also conducted a 3D-FDTD simulation to calculate the absorption spectrum of the output structure predicted by the improved DPN, as shown in Fig. 5(a) (black curve). As shown, the simulated spectrum exhibits an absorptivity of 99.5% at 649 nm and fits well the input spectrum, thus demonstrating the appropriateness of the improved DPN.

Further, we investigated the multiband absorption of the proposed MPA structure. Instead of composite and multilayer MIM structures, the MPA structure in this study took advantage of the double-peak absorption by only using a single layer. Its meta-atom was designed as a general polygon with multiple degrees-of-freedom so that the multiple absorption can be obtained and tuned. For example, we demonstrated double-peak absorption of the proposed MPA structure by the improved DPN, which is illustrated in detail in Fig. 5(b). The input spectrum achieved an absorptivity of 96.3% at 651 nm and an absorptivity of 97.3% at 694 nm. The predicted structural parameters P_{1}–P_{8} were 42, 40, 10, 63, 41, 97, 20, and 57 nm, respectively. The predicted spectrum of the output MPA structure had absorptivities equal to 93.8% and 92.6% at 651 nm and 692 nm, respectively, while the simulated spectrum had absorptivities equal to 93.8% and 99.1% at 644 nm and 684 nm, respectively. The prediction process was the same as that of the single-peak absorption in Fig. 5(a), thus indicating both the effectiveness of the proposed MPA structure and the improved DPN.

#### 4.3 Absorption property of the MPA structure in wide-angle incidence

Finally, we investigated the absorption property of the proposed MPA structure in wide-angle incidence cases. For the same MPA structure corresponding to Fig. 5(b), we set a group of incidences at different incident angles from −50° to 50°, and numerically calculated the absorptivity based on 3D-FDTD simulations. Figure 6 shows the absorption properties of the double peaks as functions of the incident angle. Note that the case with normal incidence is the case in Fig. 5(b). At an increasing angle of incidence, the absorptivity of Peak 2 did not change significantly and remained above 96%, while the absorptivity of Peak 1 decreased by a greater extent, as shown in Fig. 6(a). The wavelength of Peak 2 remained almost unchanged at different incident angles, as plotted in Fig. 6(b), thus indicating a good tolerance for the incidence. As a comparison, the wavelength of Peak 1 shows a variation of a few nanometers, but it still exhibits good absorption characteristics over a wide range of incident angles.

## 5. Conclusion

In this study, we proposed a novel MPA structure with general polygonal meta-atoms. Its irregular geometry has multiple degrees-of-freedom and enables unique absorption properties by tuning the meta-atom structures. We conducted 3D-FDTD simulations to investigate the MPA structure, and demonstrated that it can realize dual-band perfect absorption with a single-layer structure. Further, to minimize the difficulty associated with the study of the complex MPA structure with various geometrical parameters, we designed fully connected neural networks, including SPN and DPN, to establish the connection between the MPA structure and its absorption spectrum. For a given structure, its absorption spectrum can be predicted quickly by the SPN with an MSE loss of 0.0008. Conversely, for a known absorption spectrum, the corresponding MPA structure can be predicted by the improved DPN efficiently, with a low MSE loss of 0.0026. Using our designed neural networks, the proposed MPA structures can be designed efficiently to have novel properties. These structures are promising for a wide range of applications, such as sensors, heat meters, and photovoltaics. We believe that our method provides a promising route to design complicated artificial EM structures rapidly and accurately and can facilitate the design methodology and applications of functional metasurfaces.

## Funding

National Natural Science Foundation of China (91750202, 11922406); National Key Research and Development Program of China (2018YFA0306200, 2017YFA0303700).

## Acknowledgments

We thank Mr. Yong Chen and Mr. Hanning Wang for the fruitful discussion and help during this work.

## 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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