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Abstract
Reflection reduction metasurface (RRM) has been drawing much attention due to its potential application in stealth technology. However, the traditional RRM is designed mainly based on trial-and-error approaches, which is time-consuming and leads to inefficiency. Here, we report the design of a broadband RRM based on deep-learning methodology. On one hand, we construct a forward prediction network that can forecast the polarization conversion ratio (PCR) of the metasurface in a millisecond, demonstrating a higher efficiency than traditional simulation tools. On the other hand, we construct an inverse network to immediately derive the structure parameters once a target PCR spectrum is given. Thus, an intelligent design methodology of broadband polarization converters has been established. When the polarization conversion units are arranged in chessboard layout with 0/1 form, a broadband RRM is achieved. The experimental results show that the relative bandwidth reaches 116% (reflection<-10 dB) and 107.4% (reflection<-15 dB), which demonstrates a great advantage in bandwidth compared with the previous designs.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Polarization is a fundamental property of electromagnetic (EM) waves, which describes the time-dependent orientation of the electric field vector spatially. Traditionally, manipulating the polarization of EM waves highly relies on anisotropic material properties, such as Faraday effect or natural birefringent crystals [1]. Due to natural materials’ weak EM response, it takes a long propagation distance to accumulate the phase change, resulting in the bulky size and narrow bandwidth of the device. Metasurface [2,3], composed of periodic or quasi-periodic arrays in a two-dimensional (2D) plane, can provide abrupt phase change [4] and powerful control of EM waves, thus allowing a lot of interesting functions such as sensing [5], absorbers [6,7], invisibility cloaks [8], wavefronts controlling [9,10], and polarization converters [11–13]. Furthermore, in the stealth technology field, polarization conversion metasurface has been used to reduce radar cross section (RCS) [14]. Based on the geometry phase and chessboard configuration, reflected beams are dispersed into various directions deviated from specular reflection, which leads to a backward RCS reduction [15–17]. However, they were designed based on trial-and-error approaches, which is time-consuming and leads to inefficiency [14–17].
In the past decade, deep-learning [18] has made remarkable achievements in speech recognition [19], image recognition [20], self-driving [21] and robot control [22]. In recent years, deep-learning has also contributed to the design of metamaterials and photonic structures [23,24]. Numerous novel designs have been applied to coding metasurface [25], all-dielectric metasurface [26], perfect absorber [27], chiral metamaterials [28,29], invisibility cloak [30,31], holograms [32,33], inverse scattering [34,35] and imaging [36,37]. In addition, transfer learning has been proposed, which transfers the learned knowledge from previous source tasks to a new target task [38]. Transfer learning provides an efficient way to accelerate the design of functional metasurfaces [39,40]. All of these pioneering efforts establish a link between machine learning technology and EM applications in the microwave, terahertz and optical frequencies [41]. Moreover, it tremendously boosts the confidence of researchers and engineers in applying deep-learning to the design of next generation EM functional devices.
Here, we report the design of a broadband RRM based on deep-learning methodology. To achieve the goal, we have designed two neural networks, i.e. a forward prediction network (FPN) and an inverse design network (IDN). The FPN is constructed and trained with the ability to predict the PCR spectrum in a millisecond once the geometric parameters of arbitrary structure are given, which is more advanced than traditional simulation method. To achieve intelligent design, the IDN is constructed and a data preprocessing method is introduced in the IDN to improve performance. The required structure parameters will be automatically generated by the IDN once a target PCR spectrum is given. To verify the validity, we have designed three linear polarization converters with PCR above 0.9, which respectively range from 1.5 to 6 GHz, 2 to 8 GHz, and 2.5 to 10 GHz, reaching a 120% relative bandwidth. Finally, we experimentally demonstrate a broadband RRM with a low specular reflection below -10 dB from 2.1 GHz to 7.9 GHz, which accords well with the simulation result.
2. Design and theoretical analysis
Figure 1(a) conceptually illustrates the reflection reduction mechanism, which is achieved by arranging the polarization conversion units in chessboard layout with 0/1 form. The normally incident EM waves are redirected into four symmetrically oriented directions, thus suppressing the specular reflection. Figure 1(b) shows the schematic of the broadband reflection reduction structure. The top metasurface and the bottom metallic layer are copper, which are separated by a PMI foam (${\varepsilon _r} = 1.06$) spacer with a thickness of h.
Fig. 1. (a) Conceptually demonstration of a broadband reflection reduction metasurface. (b) Schematic of the proposed broadband reflection reduction metasurface. (c) Schematic illustration of the working principle of “0” element and “1” element.
The working principle of polarization conversion units of “0” element and “1"element are illustrated in Fig. 1(c). The incident E-field along y-polarization can be considered as a combination of two perpendicular components as ${\vec{E}_i} = ({\hat{u}{E_{iu}} + \hat{v}{E_{iv}}} ){e^{\textrm{j}\phi }}$ . Accordingly, the reflected E-field can be expressed as ${\vec{E}_r} = ({\hat{u}|{{r_u}} |{e^{\textrm{j}{\phi_u}}}{E_{iu}} + \hat{v}|{{r_v}} |{e^{\textrm{j}{\phi_v}}}{E_{iv}}} ){e^{\textrm{j}\phi }}$ . The asymmetric feature of the metasurface will introduce a phase difference ($\Delta \phi = |{{\phi_u} - {\phi_v}} |$) between ${r_u}$ and ${r_v}$. When $|{{r_u}} |\approx |{{r_v}} |$ and $\Delta \phi = {180^ \circ }$ are satisfied, the reflected field ${E_r}$ will be along the x-direction, thus realizing a $90^\circ$ polarization conversion. In general, polarization conversion ratio (PCR) is adopted to evaluate the efficiency of the polarization conversion. PCR is defined as $PCR = {r^2}_{xy}/({{r^2}_{xy} + {r^2}_{yy}} )$.
As shown in Fig. 1(c), element “0” is rotated 90°to get element “1”. Reflected waves from element “0” and element “1” have the same amplitude but opposite phase, which can be arranged in chessboard layout to realize reflection reduction. For a metasurface with $N \times N$ lattices, the scattering far-field can be quantitatively calculated according to Eq. (1) [42].
3. Deep-learning modeling
Figure 2 demonstrates the process of the forward prediction procedure. It consists of two modules: data gathering and neural network training. To train the FPN, a large number of structure parameters of the unit cell and the corresponding PCR spectra are required to build up the dataset. Here, a series of structure parameters are randomly generated and utilized to calculate the corresponding PCR with commercial software CST. Specifically, there are 5 structure parameters (h, p, r, w, g). We set h ≤ 20 mm and p ≤ 40 mm, respectively. To ensure that the physical model can be implemented, r, w, and g should be smaller than p/2 and are randomly generated. The PCR spectrum from 1 to 10 GHz is described with a 1001-dimensional vector. After establishing the dataset, we divide it into training, validation and test dataset, with 4000, 500 and 500 samples, respectively. In the training of FPN, our deep-learning platform is the TensorFlow 2.6 (GPU version) in python 3.7 built on a data science platform Anaconda 3. The integrated development environment (IDE) is PyCharm Community 2021.2.2 and the program is running on a graphics processing unit (GPU) Nvidia RTX3080Ti. The optimizer of neural network is Adam, where the learning rate and the batch size are set as 0.0003 and 64, respectively.
The input of the FPN is a 5-dimensional vector (h, p, r, w, g) and the output is a 1001-dimensional vector (PCR spectrum). The designed neural network consists of five fully connected layers of size {128, 512, 1024, 2048, 1001} and the first four layers are hidden layers activated by rectified linear unit (ReLu) function. Then a bias is added to each neuron, which can be described as a linear transformation
where the $x_i^j$ is the number i neuron in layer j, ${x^{j - 1}}$ is a column vector representing all the neurons in layer j-1, $w_i^j$ is the weight parameter vector connecting ${x^{j - 1}}$ and $x_i^j$, and $b_i^j$ is the bias parameter of neuron $x_i^j$. In order to develop the ability to deal with nonlinear tasks, ReLu function is used as the activation function and expressed asMean square error (MSE) is adopted as the loss function, which can be expressed as
When the MSE of training dataset and validation dataset become close to each other, it means that the network converges well. After training, the final MSE of the training dataset and test dataset are 0.0026 and 0.0049, respectively. Subsequently, we input the structure parameters of test dataset to predict the corresponding PCR spectra for comparison. In Fig. 3, four samples are chosen randomly, which validates the good performance of the trained FPN. We can see that the predicted PCR spectra (the red dash curve) by FPN match well with the simulated PCR spectra (the blue solid curve).
Once the structure parameters are given, the FPN may quickly obtain the PCR spectrum. Inversely, when a target spectrum is given, can we derive the required structure parameters at once? Traditionally, the required structure parameters are obtained based on trial-and-error approaches, which is time-consuming and leads to inefficiency. Here, we create an inverse design network (IDN) that can instantly anticipate required structure parameters once the PCR spectrum is given.
We use five fully connected layers to construct the IDN of size {4096, 1024, 256, 64, 5}. The first four layers are hidden layers activated by ReLu function, and the last layer is 5-dimensional vector (h, p, r, w, g). The input is a 1001-dimensional vector (PCR spectrum). We need to consider a fundamental property of the inverse design model that the same PCR spectra may be created by different structure parameters. This non-unique response-to-design mapping [44] leads to conflicting issues and becomes a significant challenge in training IDN. To deal with the problem, we use a tandem network structure, as shown in Fig. 4(a). The loss function of IDN is loss1 and the loss function of FPN is loss2, respectively. Both loss functions are defined as MSE. The sum of loss1 and loos2 is a final loss function to train the whole network.
Fig. 4. (a) Schematic diagram of inverse design procedure. (b) Data preprocessing in inverse design procedure.
From Figs. 3(c) and 3(d) we can see some peaks at the high frequency of the spectrum, which does not contribute to the polarization conversion bandwidth. Moreover, it will greatly affect the accuracy and convergence of the network. As a result, we devised a data preprocessing approach to eliminate these high-frequency peaks from entire spectra, as shown in Fig. 4(b). We can see that the two peaks between 8 GHz and 10 GHz have been removed, which does not affect the effective bandwidth at all.
Due to the data preprocessing, the datasets have changed and the FPN of Fig. 4(a) has to be trained again before starting inverse design procedure. Interestingly, the FPN performs better after data preprocessing, showing an MSE of 0.001 in the training dataset and 0.002 in the test dataset. After convergence, the MSE of IDN is 0.25.
As shown in Fig. 5(a), the input is a target PCR spectrum, and the output is the required design parameters. To further assess the accuracy of the design method, with these design parameters we simulated the PCR assisted by CST. Through the comparison between the target PCR and the simulated PCR, the effectiveness of the design method can be verified. We build three polarization converters with a 120% relative bandwidth, as shown in Fig. 5(b). Their effective bandwidths (PCR > 90%) are 1.5-6 GHz, 2-8 GHz, and 2.5-10 GHz, respectively. Based on IDN, the structure parameters of the three design samples are obtained as shown in Tab. 1. We can see that the thickness of the dielectric layer h increases as the working frequency moves to low frequency. Then we simulated the three design samples with CST, as shown in Fig. 5(c). We can see that Fig. 5(c) accords well with Fig. 5(b), which validates the effectiveness of the proposed method.
Fig. 5. (a) Schematic diagram of inverse design process. (b) Three targeted PCR spectrum. (c) The PCR spectrum of the three designs produced by CST.
Table 1. Structure parameters of the three design samples
To understand the mechanism of the proposed broadband polarization converter, we take Design 2 in Table 1 as an example to analyze the surface current distributions. Figure 6 shows the surface current distributions of the top metasurface and bottom metal plate at resonant frequencies of 2.3 GHz, 4.6 GHz, 7.3 GHz and 8.0 GHz, respectively. We can see that the surface currents of the top layer are antiparallel to the background sheet at 2.3 GHz and 4.6 GHz, which results in a magnetic resonance. In contrast, the surface currents of the top layer are parallel to the background sheet at 7.3 GHz and 8.0 GHz, which results in an electric resonance. The four resonances are significant to achieve high efficiency and broadband polarization conversion. Actually, the broadband property is attributed to the superposition of multiple PCR peaks.
Fig. 6. Surface current distributions on the metasurface and bottom metal plate at different resonance frequencies. (a) 2.3 GHz; (b) 4.6 GHz; (c) 7.3 GHz; (d) 8.0 GHz.
4. Fabrication and experiment
To experimentally validate the design, we have fabricated the sample of “Design 2” in Table 1. A 300 mm ×300 mm sample has been fabricated using screen printing technology, which is shown in the inset of Fig. 7(a). Figure 7(a) shows the experimental setup, where a pair of horn antennas working from 1 GHz to 18 GHz is connected to a Vector Network Analyzer (Agilent 8720ET). Figure 7(b) shows the comparison between measured and simulated reflection results, implying a good agreement. In particular, the measured result shows a reflection lower than -10 dB in the frequency of 2.1 GHz to 7.9 GHz, and a reflection lower than -15 dB in the frequency of 2.2 GHz to 7.3 GHz, demonstrating an excellent performance.
To further demonstrate the better performance of our design, a comparison with other designs reported in the literature is shown in Table 2. We can see that the relative bandwidth of our design achieves 116% with the reflection below -10 dB. In particular, the relative bandwidth of our design still achieves 107.4% with the reflection below -15 dB, which shows a great advantage compared with the previous designs [45–52].
Table 2. Comparison with other designs
5. Summary
In summary, we have proposed an ultra-broadband reflection reduction metasurface based on deep-learning methodology. Deep-learning methodology builds a bridge between the structure parameters of the metasurface and corresponding EM response. On one hand, when the structure parameters are given, its PCR spectrum can be obtained immediately. On the other hand, for a given target PCR spectrum, the corresponding structure parameters can be obtained instantly. Thus, an intelligent design method for RRM has been established, which may be extended to the design of high-temperature RRM [52]. To experimentally validate our methodology, we have fabricated and measured a broadband RRM, which shows a great advantage of bandwidth compared with the previous designs [45–52].
Funding
Strategic research and consulting project of Chinese Academy of Engineering (2022-XY-127); National Natural Science Foundation of China (52021001).
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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