1. Introduction

A metasurface absorber (MA), based on metamaterials, employs precisely designed two-dimensional microstructures to achieve efficient absorption of electromagnetic waves [1,2]. In comparison to traditional absorbing materials, the MAs possess several advantageous characteristics such as reduced thickness, lightweight, enhanced flexibility, and improved controllability. By precisely manipulating the geometric shape and dimensions of microstructures, the spatial modulation of the amplitude and phase of incident waves can be achieved [3], thereby enabling effective absorption within specific frequency ranges. This design approach enables MAs to exhibit outstanding performance features, including high absorption efficiency [4], wide bandwidth [5], and wide-angle response [6]. Various simple geometric shapes, such as square [7], T-shaped [8], cross-shaped [9], and circular [10], have been successfully demonstrated, showcasing favorable absorption properties while facilitating the design and manufacturing processes. Nonetheless, these simple structures lack sufficient geometrical parameters to tune their absorption characteristics due to limited degrees of freedom (DOF) in their morphology. Consequently, exploring novel metasurface structures with complex topologies has emerged as a promising solution to overcome this limitation [11–14].

Despite the increased degrees of freedom in irregular metasurface morphology, their complex geometric structures pose challenges for theoretical analysis using resonant mode. Moreover, conventional numerical simulations are inefficient for optimizing irregular metasurface structures, as they involve multiple parameter optimizations. Optional solutions for optimizing irregular metasurface include topology optimization [15], particle swarm optimization (PSO) [16], and genetic optimization [17]. However, these methods require repeating the optimization process when the input objectives change, limiting their efficiency in rapidly updating designs.

Another approach to optimizing irregular metasurfaces is leveraging deep learning, which has gained attention in recent years for its ability to quickly predict desired electromagnetic structures based on input objectives. Pioneering works have successfully demonstrated the potential of deep learning in optimizing complex metasurface structures [18]. In our previous research, we designed fully connected neural networks and convolutional neural networks to predict metasurface structures by inputting objective absorption properties, and vice versa [19,20]. This approach allows for rapid updating of objective designs and obtaining corresponding prediction results once the deep neural network (DNN) is established and trained, eliminating the need for recalculations.

However, the deep learning method still has certain limitations. Firstly, it requires a substantial amount of time to establish a comprehensive training set consisting of an adequate number of input data samples. The prediction accuracy of the DNN is often influenced by the size of the training set, which in turn depends on the complexity of the irregular structure being analyzed. For instance, in our previous work on predicting 9 × 9 QR code structures without geometric constraints, the DNN required 43,000 sets of data as the training set [19]. To reduce the amount of input data without compromising prediction accuracy, researchers opt to decrease the DOF in irregular structures, thereby reducing the number of geometric parameters requiring optimization. In another study, we achieved this by fixing the vertex coordinates of irregular polygon structures in the x-axis direction, resulting in the reduction of input data to 14,000 while maintaining good prediction capabilities [20]. However, reducing the geometrical parameters of irregular structures may limit the potential absorption characteristics that could be achieved. Secondly, the traditional approach of collecting datasets involves randomly scanning the geometric parameters of irregular structures, which provides diversity but lacks centralization. In other words, if the target designs fall within a specific range with certain features, conventional data collection becomes inefficient. Although some neural network techniques like reinforcement learning and supervised learning have been developed to address this issue, they increase network complexity and require additional parameter tuning [21,22].

In recent years, researchers have explored the combination of PSO with deep learning and applied it to design metamaterial absorbers. For example, by applying an auxiliary DNN for mapping the meta-atom's parameters onto its reflection coefficients, J. Chen et al. used the PSO algorithm to globally search the parameter space of the DNN model and achieved the optimal configuration of the meta-atoms for target absorptivity covering 2.2–18 GHz [23]. The DNN was firstly used to prepare the parameter space of the meta-atoms, and subsequently, the optimal meta-atom structures according to a target absorptivity ware iteratively optimized by the PSO algorithm. However, when the target absorptivity criteria change, it necessitates a complete reiteration of the PSO process, increasing computational overhead. Conversely, if the PSO algorithm is employed first to prepare the dataset that is concentrated within the desired absorptivity range, the DNN can more efficiently predict the meta-atom structures based on input absorptivity. In this way, even with variations in the target spectrum, the DNN can still make a rapid prediction. This design method effectively incorporates the PSO algorithm into deep learning models, thereby offering the potential for flexible functionality in the design of irregular MAs. Nonetheless, attempts in this direction remain limited, warranting further investigation.

In this study, we present a demonstration of a multi-function metasurface absorber composed of irregular-octagon meta-atoms. These octagons possess vertices with arbitrary coordinates, providing a wide range of DOF in geometry that can enable diverse absorption properties. To achieve rapid inverse design of the octagonal metasurface based on target absorption properties, we combine the PSO algorithm and deep learning methods. Initially, we employ the PSO algorithm to optimize the metasurface structure based on the given target, yielding structures concentrated within the desired performance range. Three-dimensional (3D) finite-difference time-domain (FDTD) simulations are used for all calculations. Subsequently, we utilize the PSO-optimized data to establish an equivalent model using a DNN that relates the metasurface structures to their corresponding absorption properties. Through this approach, we achieve a significant reduction in mean square error (MSE) to 0.0008 and 0.0031 for the spectrum prediction and structure prediction, respectively. This allows for rapid and precise prediction of desired structures based on input absorption spectra, as well as vice versa. Our results demonstrate that the proposed octagonal metasurface structure can exhibit multiple absorption properties according to the input targets. These properties include perfect absorption, multi-peaks absorption, and high absorption with filtering windows. We believe that the proposed design method can contribute to the advancement of design methodologies for complex electromagnetic structures and find applications in the field of metasurfaces and metamaterials.

2. Design and methods

In this study, we focused on the design and investigation of a multi-function MA composed of octagonal meta-atoms. We utilized 3D-FDTD simulations to perform numerical calculations for the proposed structures. Our main objective was to achieve inverse design of these irregular metasurface structures based on specific absorption properties. To accomplish this, we employed a combination of the PSO algorithm and deep learning techniques to establish an equivalent model correlating the metasurface structures with their corresponding absorption spectra. As a result, we were able to rapidly predict both the metasurface structures and their absorption characteristics.

2.1 Schematic of the proposed irregular-octagon metasurface

Figure 1 illustrates the proposed octagonal metasurface, which has a period of 200 nm. This metasurface is constructed as a metal-dielectric-metal (MIM) structure, consisting of Au, SiO₂, and Ti layers. The respective layer thicknesses are 80 nm, 50 nm, and 300 nm. The Ti layer at the bottom acts as an opaque layer for incident light. The geometry of MIM meta-atom comprises eight adjustable vertices denoted as P₁ to P₈, as depicted in Figs. 1(b) and (c). In this configuration, each pair of vertices is positioned within a designated quadrant, with random coordinates assigned within that quadrant. Consequently, the octagonal meta-atom exhibits an irregular topology, offering a significant number of geometric degrees of freedom. The octagonal MA structure proposed in this study can be fabricated on a fused silica substrate using nanofabrication methods [24, 25]. The bottom Ti layer is typically deposited via electron-beam evaporation, while the SiO₂ insulating layer can be achieved through atomic layer deposition. As for the top Au pattern, it is commonly fabricated using electron beam lithography, followed by metal deposition and lift-off procedures.

 

Fig. 1. Schematic of the octagonal metasurface absorber. (a) 3D view of the metasurface structure. (b) 3D view of the meta-atom. The parameters are set to d₁ = 80 nm, d₂ = 50 nm, and d₃ = 300 nm. (c) A top view of the meta-atom structure with eight adjustable vertices, each two vertices are distributed in a quadrant and have random coordinates.

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The proposed structures were investigated numerically using the 3D-FDTD method implemented in the Lumerical FDTD solution, a commercial software. In the simulations, periodic boundaries were established in the horizontal directions to account for the periodicity of the structure. Perfectly matched layers (PMLs) were utilized in the longitudinal direction to eliminate reflected waves. To ensure accurate results and computational efficiency, a uniform mesh size of 5 nm was employed throughout the computational region. The refractive index of SiO₂ was set to 1.45, while the dielectric constants of Ti and Au were determined by fitting experimental data using the Drude-Lorentzian model [26]. By designing meta-atom structures with a large number of geometrical DOF, it becomes possible to achieve multiple absorption properties, including perfect absorption, multi-peaks absorption, and absorption with desired filtering windows. Through the combination of 3D-FDTD simulations and the PSO algorithm, a library of absorption spectra and corresponding meta-atom structures was obtained for training the DNN.

2.2 Particle swarm optimization for establishing datasets

As described in the previous subsection, irregular octagons possess a multitude of structural degrees of freedom, making it challenging to obtain satisfactory predictive structures solely through deep learning neural networks. Therefore, we propose an initial pre-optimization using the PSO algorithm to obtain pre-optimized structures. These pre-optimized structures are configured to align the absorption curve approximately within the desired range. Subsequently, neural networks are employed for structure prediction. The PSO algorithm, which is inspired by the foraging behavior of birds, has found widespread applications in the field of optical design [27,28]. The fundamental concept of PSO involves continuously adjusting the position and velocity of each particle to explore the solution space and find the optimal solution. The velocity and position updates are determined based on the historical best solution and the best solution within each particle’s neighborhood. This approach allows for efficient exploration of the solution space and facilitates finding the optimal solution. In the optimization of the MA, the structural parameters are treated as particles, and the integrated absorption spectrum is defined as the objective function. The PSO algorithm optimizes the variables of the MA and evaluates the performance of the absorber using fitness functions. After several iterations of the PSO algorithm, a dataset of pre-optimized structure parameters is obtained. By including the results from each iteration in the dataset, we ensure that there is a diverse range of data points rather than just the optimal solutions. This approach helps to maintain the generalization ability of the DNN model and reduces the risk of overfitting.

The flowchart depicting the PSO iteration process is presented in Fig. 2. Initially, the coordinates of the eight vertices of the octagonal meta-atoms are assigned as particles. Subsequently, appropriate boundaries for particle velocity and position are set based on the computational area utilized in the FDTD simulations. Random initialization is then performed for the particle parameters. Afterwards, the fitness function of each particle is computed. This function is defined by Eq. (1), where [${a_{1,}}{a_2}$], [${b_{1,}}{b_2}$], and [${c_{1,}}{c_2}$] represent different wavelength ranges, and f(x) represents the absorption curve.

 

Fig. 2. (a) Optimization flowchart of PSO algorithm. (b) The concept diagram of fitness function in. The integrals S_A, S_B, and S_C are defined in Eq. (1).

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Fig. 3. Schematic diagram of the DPN and SPN. The input and output are the absorption spectrum and the geometric structure of the metasurface absorber, respectively, in DPN, and vice versa in SPN.

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2.3 Design of the neural network model

To establish the relationship between the octagonal metasurface structure and its absorption properties, we employed a fully connected neural network. This network consisted of two separate components: the design predicting network (DPN) and the spectra predicting network (SPN), as depicted in Fig. 3. Combining these two independent networks formed a verification DNN that enabled us to assess the accuracy of the model.

In our study, we utilized the rectified linear unit (ReLU) activation function and employed the Adam optimizer with a learning rate of 0.001. Additionally, a dropout function with a rate of 0.2 was applied to prevent overfitting. The performance of the fully connected neural network relied on various factors, including parameter settings and the number of hidden layers. In this case, we employed four hidden layers, each containing 600 nodes. The SPN underwent 600 iterations, while the DPN went through 2500 iterations. During each iteration, the parameters of the neural network were optimized by continuously reducing the MSE loss.

During the training process of the neural network, it is common to preprocess the input data in order to reduce the disparity between the input values and the output results and ensure the reliability of the predictions. In the case of DNNs, the absorptivity range for different spectral ranges is naturally set from 0 to 1. However, the geometric parameters of various octagonal structures may have different value ranges. Therefore, appropriate data preprocessing methods are necessary to normalize the structural data. In this study, we employed a scaling technique to transform all the original geometric parameters into normalized values ranging from 0 to 1 using the following formulas:

3. Results and discussion

After numerous iterations, the MSE of the SPN in the verification process can be reduced to 0.0008, indicating excellent prediction accuracy. Similarly, the MSE of the DPN is reduced to 0.0031 after thousands of iterations, also demonstrating good accuracy. The utilization of these two networks allows for the rapid and precise determination of absorption properties by inputting specific metasurface structures, as well as the reverse process of designing MAs with complex structures. Consequently, our design method not only offers great flexibility in studying MA structures with high DOF but also significantly reduces the overall design time. This provides researchers with an efficient tool to explore and optimize the structures of MAs, facilitating further advancements in this field.

3.1 Perfect absorption

The octagonal MA proposed in our study exhibits several advantages, including high absorptivity and narrow-band characteristics. These qualities make it suitable for a wide range of applications in fields such as communication, sensing, and radar. To validate the perfect absorption functionality of our proposed structure, we input an absorption spectrum with a 99.1% absorptivity at a wavelength of 858 nm into the DPN, which is depicted as the orange curve in Fig. 4(a). The DPN accurately predicted the corresponding octagonal metasurface structure, as shown in the insert in Fig. 4(a). To verify the prediction accuracy of the DPN, we performed 3D-FDTD simulations to calculate the absorption spectrum of the predicted structure. The simulated spectrum, indicated by the green curve in Fig. 4(a), achieved a 99.1% absorption peak at 833 nm. This result closely matched the input spectrum, demonstrating that our proposed octagonal structure and the designed DNN enabled rapid design of perfect absorbers. Additionally, we employed the SPN to predict the spectrum of the octagonal structure predicted by the DPN, further verifying the effectiveness of the SPN. The predicted spectrum, exhibiting a 93.2% absorption peak at 864 nm, is displayed as the blue curve in Fig. 4(a). This prediction also aligned well with the other two curves, providing further evidence of the accuracy and reliability of our proposed design approach.

 

Fig. 4. Prediction of the octagonal metasurface structure from the input perfect-absorption spectra in (a) and (b). In each case, the input data are the absorption spectra (orange curve), and the output data are the predicted octagonal structure (see inserts) by DPN. The spectra of the octagonal structure by SPN prediction and 3D-FDTD simulation are plotted in blue curve and green curve, respectively.

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To further validate the prediction accuracy of the DNNs, we continued to input different target spectra and examined the performance of our approach in achieving perfect absorption. This was done to ensure that the success of the octagonal metasurface structure in achieving perfect absorption was not an isolated result. For instance, we introduced another spectrum into the DPN from the validation dataset. This spectrum exhibited a 98.5% absorption peak at 1080 nm, as shown by the orange curve in Fig. 4(b). The DPN accurately predicted the corresponding octagonal metasurface structure, as depicted in the inset plot. Additionally, the SPN predicted a spectrum with 95.6% absorption at 1082 nm, represented by the blue curve. To verify the accuracy of these predictions, we conducted 3D-FDTD simulations for the predicted octagonal structure. The resulting absorption spectrum showed a 96.6% absorption peak at 1079 nm, which closely matched both the input spectrum and the SPN-predicted spectrum. These findings further confirm the potential of the octagonal metasurface structure in achieving perfect absorption and demonstrate the accuracy and effectiveness of our designed DPN and SPN.

3.2 Multi-peaks absorption

In addition to single-peak perfect absorption, multi-peaks absorbers have wide-ranging applications. For instance, a tunable all-dielectric absorber consisting of grating structures was designed for multi-peaks absorption in THz band, enabling the potential applications such as chlorpyrifos sensing [29]. For another instance, a metasurface structure with four-fold rotationary symmetry was proposed and designed. It realized high absorption at three specific frequencies in GHz band, exhibiting high sensitivity to changes in the surrounding environment [30]. These absorbers offer high sensitivity and selective absorption, making them valuable in sensor and detector applications.

In this study, unlike composite and multilayer structures [31], we focused on designing a single-layer metasurface composed of octagonal structures capable of achieving multi-peaks absorptions. To demonstrate its effectiveness, we selected a spectrum from the validation dataset as the design target. This spectrum, illustrated in Fig. 5(a), exhibited absorptivity peaks of 85.1% at 458 nm, 95.8% at 650 nm, and 94.2% at 1133 nm. The DPN successfully predicted the corresponding metasurface structure based on the input spectrum, as depicted in the inset plot in Fig. 5(a). Additionally, the numerically calculated absorption spectrum using 3D-FDTD simulations and the predicted spectrum by the SPN were plotted as the green and blue curves, respectively, in Fig. 5(a). The calculated spectrum demonstrated three absorption peaks: 84.1% at 457 nm, 93.5% at 663 nm, and 95.4% at 1120 nm. Similarly, the predicted spectrum showed a similar absorption curve with peaks of 78.8% at 457 nm, 90.7% at 654 nm, and 94.6% at 1132 nm, respectively. The close alignment between the simulated and predicted spectra confirms not only the accuracy of the DNNs but also the multi-peak absorption capability of the octagonal metasurface structure.

 

Fig. 5. (a) Prediction of the octagonal metasurface structure from the input multi-peak absorption spectrum (orange curve). The octagonal structure predicted by DPN is drawn in the insert. The spectra of the predicted metasurface structure by SPN and 3D-FDTD simulations are plotted in blue curve and green curve, respectively. (b) The magnetic field intensity distributions in the xy plane crossing the center of the precited octagonal structure, corresponding to peak 1, peak 2 and peak 3 in (a), respectively. In each case, the dashed line marks the structure of the octagon meta-atom.

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Additionally, we conducted an investigation into the electromagnetic (EM) modes within the octagonal structure depicted in Fig. 5(a). We analyzed the magnetic field intensity distributions in the xy-plane, which intersected the center of the octagonal meta-atoms, corresponding to the three peaks illustrated by the green curve. These distributions are presented in Fig. 5(b). It can be observed that the field intensity of peak 1 exhibits more energy leakage outside the octagonal structure, indicating lower absorptivity compared to the other two peaks. Interestingly, all three magnetic intensity distributions exhibit irregular mode patterns that are primarily localized at the vertices or sides of the octagon, rather than conforming to the regular mode distributions typically observed in conventional metasurface structures [32,33]. This finding suggests that the proposed octagonal structure possesses the potential to support various absorption characteristics, potentially allowing for the design of absorbers with specific properties through the use of irregular metasurface structures. Consequently, our method offers a promising approach for flexibly designing and tailoring absorption properties using such irregular structures.

3.3 High absorption with filtering window

Due to their unique materials and structures, MAs can selectively absorb electromagnetic waves within specific frequency ranges. Through careful design, these absorbers can achieve high absorption over a broad frequency range while maintaining a filtering window in desired wavelength ranges. This filter-like MA holds great potential in applications such as wireless communication systems [34], stealth technology [35], and so on.

In our study, we employed the PSO algorithm to optimize the parameters of the octagonal structures. By defining a specific objective function, as described in Section 2.2, we created a dataset with various spectral data located near the desired spectrum, including the required filtering window. In this study, the term “filtering window” is employed to describe the specific wavelength range of light that is reflected by the MA instead of being absorbed. This approach ensured that the training dataset for the DPN and SPN had sufficient data to establish an equivalent model between the metasurface structure and its absorption properties. As a result, we only utilized 14,000 sets of spectra as the training dataset in this study, achieving MSE values of 0.0031 for DPN and 0.0008 for SPN.

In comparison to our previous work, where we used a similar network structure to predict polygonal metasurface structures, there were some differences. The polygonal structure also had eight vertices, with fixed x-coordinate values to reduce the DOF in the metasurface geometry. However, in the previous study, we still needed to design a more complex DNN using the K-fold cross-validation method to improve the accuracy of the neural networks. This was necessary because the training data in that study was collected randomly without preprocessing, resulting in lower learning efficiency for the DNN. In contrast, the combination of DNN and the PSO algorithm in this work significantly improved efficiency and enabled flexible predictions of specific absorption characteristics.

To assess the effectiveness of the DPN and SPN, we selected high-absorption spectra with filtering windows as inputs to the networks, as illustrated in Fig. 6. The orange curves in Fig. 6 represent the input spectra, while the predicted structures by DPN are depicted in the insets. The simulated spectra of the predicted structures are shown as green curves, and the predicted spectra of the structures by SPN are represented by blue curves. These three subplots collectively confirm the capability of the octagonal metasurface to achieve high absorption with filtering windows. Furthermore, they demonstrate that both DPN and SPN provide accurate predictions, as the simulated and predicted spectra align well.

 

Fig. 6. Prediction of octagonal metasurface structures from the input spectra that has high absorption bands with different filtering windows in (a)-(c), respectively. In each case, the input absorption spectrum is plotted in orange curve, and the output octagonal structure is drawn in the insert. The spectra of the octagonal structure by SPN prediction and 3D-FDTD simulation are plotted in blue curve and green curve, respectively.

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For instance, in Fig. 6(a), the input spectrum exhibits high absorption within the wavelength range of 400 nm to 586 nm and 1418 nm to 2187 nm. However, it allows a filtering window between these two absorption bands. The filtering window has a central wavelength of 940 nm and a full width at half maximum (FWHM) of 670 nm. Within the wavelength range of 700 nm to 1010 nm, the reflectivity exceeds 80%. It is evident that the three curves closely match each other in both the high absorption band and the filtering window band, providing strong evidence for the effectiveness of the DNNs.

Moreover, we examined the flexibility of the absorption function by modifying the FWHM of the filtering windows. Figures 6(b) and (c) depict two spectra with wider filtering windows compared to Fig. 6(a). The DPN successfully predicted the desired structures capable of supporting these input spectra. The simulation results and SPN predictions not only confirm the accuracy of the DPN but also highlight the versatility of the octagonal metasurface in achieving absorption bands with different filtering windows.

For instance, in Fig. 6(b), a filtering window centered at 1104 nm with an FWHM of 962 nm is achieved. In this case, the absorption bands from 400 nm to 600 nm and from 1755nm to 2500 nm exhibit high absorptivity above 80%. Even if we further widen the filtering window, the octagonal metasurface can still be designed to achieve the desired spectrum, as demonstrated in Fig. 6(c). Here, the spectrum has a filtering window centered at 1294 nm with an FWHM of 1160 nm. Within the wavelength range of 874 nm to 1520 nm, it achieves reflectivity above 80%. Notably, within the range of 977 nm to 1270 nm, the reflectivity can even reach 90%. These examples showcase the diverse functions of the proposed irregular metasurface with octagonal meta-atoms.

4. Conclusion

In conclusion, we have introduced a metasurface absorber with an irregular structure composed of octagonal meta-atoms. This structure offers multiple DOFs with its eight tunable vertices, allowing for the realization of various absorption properties through modifications of the unit structures. Through extensive 3D-FDTD simulations, we have demonstrated the capability of these metasurface absorbers to support different absorption functions, including perfect absorption, multi-peaks absorption, and high absorption with a filtering window. To overcome the challenges associated with designing complex octagonal structures with numerous geometrical parameters, we have developed DNNs in conjunction with an auxiliary PSO algorithm. The PSO algorithm was utilized to pre-optimize the metasurface structure, establishing a dataset within the desired absorptivity range. Subsequently, the DNNs were trained to establish an equivalent model between the metasurface structures and their corresponding absorption spectra. The DPN was employed to predict the inverse design of the metasurface structure for a target absorption spectrum, achieving a MSE loss of 0.0031. Conversely, the SPN accurately predicted the absorption spectrum for a known octagonal structure, attaining an MSE loss of 0.0008. By combining the DNNs and the PSO algorithm, our proposed metasurface absorbers can be efficiently designed for various functions. We believe that this methodology provides a foundation for rapidly and accurately designing complex electromagnetic structures, which can find applications in the fields of metasurfaces and metamaterials.