1. Introduction

Controlling the scattering is a key subject in theory and application of electromagnetic (EM). If one can arbitrarily manipulate the scattering field or radar cross section (RCS) of the scatterer, it will be possible to disguise the object as nothing or others. In the past, the emphasis of research is focused on RCS reduction for radar stealth or electromagnetic compatibility (EMC) applications. Since the scattering field depends on the geometry, material properties, and impedance loading of the object, the traditional manners to reduce the scattering field include shaping of the target geometry, impedance loading, radar absorbing materials, and active methods [1]. Recently, another scattering manipulating scheme including invisibility cloak [2], radar illusion [3], carpet cloak [4,5], and mantle cloak [6] has also been provided based on transformation optics theory.

On the other hand, scattering augmentation has also been widely used in both military and civil research fields. To name a few, a RCS-augmented scatterer serving as a radar decoy can improve the survivability of true target [7]. With enhanced retroreflection, the echoes of small vehicles, such as lifeboats and dinghies might be increased, which can facilitate rescue or navigation safety. Another urgent demand is how to enhance the backscattering echo of chipless radio-frequency identification (RFID) tag which will greatly improve the rate of identification for IoT applications. Unfortunately, relatively less attention is concentrated on the scheme for scattering enhancement.

Retroreflector is a device or surface that reflects incident EM waves back to its incoming direction by enhancing the backward scattering. So far, devices like corner reflector and Luneburg lens are implemented to realize retroreflection. However, due to their bulky and nonplanar structures, these devices inevitably have limitations in miniaturization and integration with other components. To deal with the above problems, metasurface has been introduced to this field.

In the past decade, metasurface has emerged as one of the most potential candidates for EM wave control devices, it can provide fascinating capabilities for EM control such as perfect absorption, anomalous reflection, wavefront shaping, and dispersion engineering [8–12]. Metasurfaces are typically made of planar dense arrays with subwavelength unit cells. In principle, EM waves can be manipulated flexibly by controlling the magnitude and phase of the unit cells. In 2014, Cui et al. proposed a new concept of coding metasurface by arranging the corresponding coding units as a predesigned coding matrix [13]. The coding units are obtained by encoding the discrete phases of unit cells in binary numbers. The coding metasurface is of more freedom than the traditional metasurface, it can form various scattering patterns due to constructive/destructive interference or phase cancellation for RCS enhancement or reduction in designated directions. For instance, in [14], Su et al. have proposed a checkerboard metasurface for super-wideband RCS reduction based on diffuse reflection theory and phase cancellation principle. In [15], Cheng et al. have proposed a nonperiodic metasurface used for retroreflection of oblique incident wave. In [16], feng et al. have proposed a method based on phase gradient metasurface to realize RCS enhancement by reflecting the incident wave to the desired direction.

Currently, the traditional design method of metasurface still relies on the trial-and-error method and the intuition of the designer [17–20]. With the increasing complexity of metasurface structure, especially the coding metasurface which has an extremely high complexity, this conventional approach gradually shows its limitations. The repeated calculations require a lot of computation resources and effort of the designer. Thus, the optimization algorithm has been applied to the inverse design of metasurface by some researchers.

The optimization algorithms such as topology optimization [21], genetic algorithm (GA) [22], and gradient-based algorithm [23] contain iterative searching steps and rely on estimate methods such as full-wave numerical simulation in each step to produce intermediate results that help optimize the searching strategy. For instance, in [24], Mahdad Mansouree et al. have proposed an adjoint-optimization-based design technique to design large-scale metasurface with lower computational cost. In [25], Lin et al. have demonstrated the topology optimization of freeform large-area optical metasurfaces in two and three dimensions. The optimization algorithm can effectively replace the manpower to do the repetitive work, however, the random-search nature of such stochastic algorithms still causes some problems when the number of optimization parameters increase. The first problem is that the initialization of the algorithm affects the optimized result to a great extent. A good initialization may lead to a fast convergence, while a bad initialization may heavily delay the convergence or even make the optimization not converge. Another problem is that the optimization algorithm may converge to the local optimal solution rather than the global optimal solution. The final problem is that the optimized design can only work in certain circumstances, once the design goal is modified, the design will need to be re-optimized.

In the past decade, machine learning has been successfully applied in many areas such as computer vision, natural language processing, speech recognition, and face recognition [26–28]. As one of the most powerful tools of machine learning, deep learning has excellent generalization ability and generative ability. Taking advantage of its capability for solving non-intuitive problems, some researchers have proposed different approaches to design the metasurface [29–32]. In [33], Xu et al. employed a tandem network to obtain a bifunctional metasurface dealing with photons and phonons. In [34], Iman Sajedian et al. used a double deep Q-learning network (DDQN) to find the right material type and the optimal geometrical design for metasurface holograms to reach high efficiency. In [35], Sensong An et al. presented a conditional generative network that can generate metasurface designs based on different performance requirements. In the deep-learning-based method, once the inverse design model is constructed, it can automatically and immediately generate the target design according to the requirements, which will significantly facilitate the design process without the manpower to intervene. However, the deep neural network (DNN) heavily relies on its training data, the generalization ability of the DNN is mostly determined by the training data. For a simple structure, it is easy to obtain a mass of training data that the DNN can be easily trained to fulfill its task. Nevertheless, as the complexity of the problem increases, it will be much more difficult to acquire amounts of valid data to train the DNN.

In this paper, a deep-learning-assisted genetic algorithm (DLAGA) is proposed to find the optimal design of the metasurface retroreflector (MRF). As is shown in Fig. 1, this method is composed of three modules, a generative model named as conditional variational auto-encoder (CVAE), a predicting neural network (PNN) and an optimization module based on GA. The CVAE is trained to generate the satisfactory coding matrix of MRF according to the input condition, and it is employed to guarantee the good initialization of GA. The PNN can predict the monostatic RCS fastly and accurately, therefore, it is used to evaluate the fitness of each offspring in the GA. With the assistance of CVAE and PNN, the GA is more likely to find the optimal solution. We also proposed an efficient data collection method that can obtain valid data rapidly. It can greatly accelerate the establishment of the training data set for neural networks. To showcase the performance of the proposed method, we optimized four MRF with excellent performance.

 

Fig. 1. Schematic of the machine-learning-assisted method.

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2. Results and discussion

As presented in [36,37], the total reflected energy from the metasurface upon the incidence of a plane wave is the sum of the reflections from all unit cells. The scattered field pattern can be represented by the superposition of the initial excitation phase and phase response of each unit cell [15]:

2.1 Data collection method

The direction of the incident wave can be described by angles $\theta$ and $\varphi$ in spherical coordinates. The RCS of the metasurface are both determined by the incident angle and the phase pattern of the unit cells. The MRF is arranged as a $16 \times 16$ array wihch composed of four types of unit cells. To simply demonstrate the robustness and effectiveness of our inverse design algorithm, we restrict the incident wave within the plane $\varphi =0$ with $\theta$ in the range of 0$^\circ$~90$^\circ$, hence, the arrangement of the unit cells is only change along one direction and remain unchanged in the other direction. We also assume that the unit cells of the metasurface have a simple structure which is depicted in Fig. 2. The unit cell is designed as square metallic patch with width $w$ which is printed on an F4B dielectric substrate. This dielectric substrate has a thickness of $h$=2mm, dielectric constant $\varepsilon _{r}$=2.65 and loss tangent $tan\delta$=0.001. The periodicity of the unit cell is $p$=5mm. A metal ground is placed on the bottom of the dielectric substrate to block the incident wave from transmission. From the initial simulation study, we found that when the designed patch widths are 4.9, 4.4, 4 and 0.5 mm, the reflective amplitude of four elements is close to 1 at 11 GHz while their phase difference is approximately 90$^\circ$ as shown in Fig. 2(c) and Fig. 2(d). According to [13], these four types of unit cells can be used to define a 2-bit coding metasurface with coding sequence ’1’, ’2’, ’3’, ’4’ respectively. Although the unit cells are only change along one direction which means the MRF can be described as a $16 \times 1$ coding sequence, there are still $4^{16}$ possible coding matrices which form an extremely large solution space. To obtain a well-trained deep learning model, effective data collection is quite demanded.

 

Fig. 2. The 2-bit coding metasurface. (a) An example of the coding metasurface. The ’1’, ’2’, ’3’ and ’4’ elements (from left to right) are realized by square metallic patches with different sizes.

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Data collection has always been a sore point of the machine-learning-based method in the design of metasurfaces. In [38], Tao shan et al. used GA to compute the element codes of metasurface based on the form of the scattering wave. However, GA can only get one batch of labeled data through one optimization process, which is inefficient owing to the computation time wasted on the repeated samples.

To address the above-mentioned problems, an efficient data collection method is proposed based on the customized GA which is named as memory genetic algorithm (MGA). MGA is customized to select the qualified samples which have significant enhanced effect on retroreflection. The enhanced effect of retroreflection is assessed through the monostatic RCS of the metasurface, the figure of merit (FoM) of the metasurface is described as:

The flowchart of the MGA is shown in Fig. 3(b). As is shown in the figure, we assigned two datasets named total dataset (TDS) and the qualified dataset (QDS). TDS is used to store all the samples produced during the optimization process, and QDS is used to store the qualified samples. The procedure of MGA is as follows. A population of N chromosomes are initialized by randomly selecting N samples from the QDS. The fitness of each chromosome is computed by the simulation software, and the qualified chromosomes are stored in the QDS. If the number of generations is less than 20, two chromosomes will be selected according to the fitness value. The single-point crossover operator is applied to the selected chromosomes to produce an offspring. Thereafter, uniform mutation operator is applied to the produced offspring. The mutated offspring are placed in the new population. The selection, crossover, and mutation operations will be repeated on the current population until the new population is complete. After each offspring is produced, the MGA will check each sample in this offspring, if the sample is in the TDS, the corresponding fitness will be fetched from TDS directly and pass the simulation step. The MGA will continue producing data until the amount of data is sufficient. Compared to the classical GA, the MGA has the advantage of avoiding repetitive evaluations of the same samples, which can effectively accelerate the speed of optimization process and data generation.

 

Fig. 3. Schematic of the GA and MGA.

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Through MGA, 60000 samples of the coding metasurface have been collected, which all of them will be used to train the PNN. Among all the collected data, only 16000 of them are qualified and assigned as the training data set of the CVAE.

2.2 Forward prediction

Since the above defined FoM is calculated from the RCS of metasurface, iteratively RCS calculation will slow down the whole optimization process. To accelerate the evaluation process during the optimization, a surrogate model named as PNN is constructed. The PNN is a fast data-driven DNN that can predict the RCS of the metasurface according to its coding matrix, the high accuracy and short response time make it a perfect surrogate model for numerical simulation in the inverse design. DNN is an artificial neural network with many hidden layers, where each hidden layer consists of multiple hidden neurons. It has been recognized to be powerful at representing very complex input-output relationships [39,40]. Since the relation between the metasurface and its RCS is complicated, it is a good choice to use DNN to uncover the hidden relation between the metasurface and its RCS.

The predicting process is shown in Fig. 4(a). As shown in the figure, based on the design, the meatsurface is encoded as the coding matrix which is 16-dimensional vector, it is fed into the PNN as the inputs. After the calculation of the PNN, it will output the corresponding predicted RCS under 121 sampling angles(−60$^\circ$−60$^\circ$). The detail of the proposed PNN is shown in Fig. 5(d). Theoretically, the fitting performance of the neural network is proportional to the number of hidden layers and neurons [39,41], but as the network deepens, a series of problems will arise, such as gradient exploding or gradient vanishing, which may cause invalid calculations and other wastes of computing resources, and greatly increase the time for model training. Different numbers of hidden layers and neurons are tried in the experiment to achieve the best performance of the model without increasing the computational burden. In order to accelerate the training process of PNN, the batch normalization is added behind the input layer and each hidden layer of the neural network. The batch normalization can effectively avoid the vanishing gradient problem and increase the generalization ability of the neural network [42]. LeakyReLU activation function is used as the activation function for neurons in the hidden layer. For the PNN training process, the least-squares error function, which is also known as mean squared error (MSE), is used as the loss function. The MSE is expressed as

 

Fig. 4. (a) Schematic of the predicting process. (b) The MSELoss of the training process.

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Fig. 5. (a) Structure of AE. (b) Structure of VAE. (c) Structure of the proposed CVAE and its parametric form of input and output.

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During the training process, among all the collected data, 55000 samples are assigned as the training data set, while the remaining 5000 samples are assigned as validation data set. The validation data set does not directly participate in the training process of the PNN, but it is used to monitor the quality of the PNN model during training and to determine the stop criteria for the training process. It can also help adjust the hyper-parameters of the training. The learning rate is 0.00001, the batch size is 128, and the epoch number is 50. After the training, to verify the validity and accuracy of PNN, 300 sets of coding matrices are sent to the PNN and full-wave simulation software at the same time. Note that those samples are generated randomly and selected to make sure that they are not included in the training data set for training the PNN, the MSE between the simulation results and the predicted results is 3.24. And the corresponding loss-epoch number graphs is shown in Fig. 4(b). The corresponding average error rate of the predicting result is only 3.9%, which is computed by

2.3 Conditional variational auto-encoder

To generate the target MRF according to the requirement, a CVAE is constructed and trained as the generative model. The CVAE is a variant of the variational auto-encoder (VAE) which is originated from auto-encoder (AE). The general idea of AE is to train an encoder and a decoder to learn the best encoding-decoding scheme using an iterative optimization process. The general structure of AE is depicted in Fig. 5(a). A VAE can be treated as an AE whose training is regularized to avoid overfitting and ensure good properties of the latent space, which enable the generative ability. Instead of encoding the input as a single point in the latent space, VAE encodes the input as a distribution over the latent space. Compared to the AE, the VAE has the ability to generate contents. However, the content generated by the VAE is random, which means that we cannot control the contents generated by the VAE. To generate controllable content, we propose a CVAE based on the VAE. In the proposed CVAE, conditions are added in the input of the encoder and decoder. With the assistance of the added conditions, the CVAE can produce specified samples based on different requirements.

The details of the DNNs for the CVAE’s encoder and decoder are shown in Fig. 5(d), the structure of the CVAE is determined after multiple experiments to make sure its good performance and low computational burden. To design the MRF working in arbitrary direction, the half-power angle range is treated as the enhancement direction of the MRF. The half-power angle range represents the angle range between the upper angle where the RCS value is 3 dB down from maximum and the lower angle where RCS value is 3 dB down. The parametric form of the input and output of the CVAE is shown in Fig. 5(c). The half-power angle range is assigned as the condition term that is combined with the coding matrix fed into the encoder. The output of the encoder is the mean and covariance tensor that describes the standard normal distribution. Based on the mean and covariance tensor, a latent encoded vector can be obtained through the reparameterize method [43], then the latent encoded vector and the condition are input to the decoder together. The decoder tries to restore the management sequence of unit cells in the metasurface. The loss function used to train the CVAE is defined as

 

Fig. 6. The predicting and simulation results of the generated design, the blue curve represents the simulation result, the red curve represents the predicting result of the PNN, the target angle range and the actual angle range of the design are shown in the upper left of each inset, the actual angle range is covered by the yellow area.

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2.4 Deep learning assisted global optimization

The MRFs generated by CVAE have shown obvious enhancement of the retroreflection, but the performance of the MRFs generated by CVAE is limited by the training data. In the application of retroreflector, a better performance is always necessary, if the design of MRF with the optimal performance is required, the CVAE can not guarantee the MRF has the best performance. Therefore, a method that can break through the limitation of training data is highly desired.

To achieve the global optimal design of the metasurface, we proposed the DLAGA method. The schematic of this method is demonstrated in Fig. 7(a). In this method, the initial population of the GA is produced by the well-trained CVAE according to the requirement. This brings two advantages, it is assured that the enhancement direction of all initial population produced by CVAE can match the requirement. In this way, the offspring of the initial generation will easily inherit this feature. Secondly, the initial population produced by the CVAE will have better fitness compared to completely random initialization. As the initial population has high fitness, their offspring are more likely to have higher fitness. To understand this from the aspect of the solution space, it can be considered that the CVAE has compressed the searching space to enable the GA to achieve the global optimization. To accelerate the evaluation of each offspring, at the evaluation step, the PNN is employed to quickly predict the RCS of the offspring and pass it to the evaluation agent of GA. After the evaluation of each generation, the algorithm will decide whether the optimal solution has been found. The sign of the optimal solution has been found is that the sample with the highest fitness keeps invariant for a certain generation. If the GA finds a satisfactory solution, the optimal solution will be cross-validated by the full-wave numerical simulation, otherwise, a new offspring will be produced and sent to the PNN. This operation is repeated until the optimal solution is found or this optimization reaches the max iteration number.

 

Fig. 7. The design schematic and the optimal results. (a) The schematic of the DLAGA method. (b) The distribution of the solution space for GA and DLAGA. (c) The optimal results and their corresponding coding matrix. (d)The retroreflection enhanced effect of the above optimal results, the blue curve represents the RCS of the metal with equivalent size, the black curve represents the simulation results, the red curve represents the predicted results of the PNN, the target direction and the actual angle range of the design which covered by the yellow area is shown in the upper left of each inset.

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To show the validity of DLAGA, it has been employed to design several MRFs that work in different directions. The population size is set to 20, and the optimal solution will be the sample with the highest fitness keeps invariant for 20 generations. In Fig. 7(c,d), four optimized metasurfaces working in different directions and the enhanced effect of their retroreflection have been demonstrated. Further more, We have optimized 480 MRFs through both DLAGA and simple GA under the same condition, and the purpose is to maximize the FoM of MRFs. So we compared the $RCS_{r}$ and the $RCS_{m}$ of these MRFs. The distribution of the optimized results for simple GA and DLAGA is demonstrate in the $RCS_{m}$-$RCS_{r}$ scatter diagram which is shown in Fig. 7(b). The x-axis of the scatter diagram is $RCS_{r}$ and the y-axis is $RCS_{m}$, the red points represent the MRFs optimized by the DLAGA and the black points represent the MRFs optimized by the GA. The results optimized by DLAGA mainly distributed in the upper left corner of the scatter diagram and the results optimized by simple GA is distributed dispersively in the scatter diagram. It means the results optimized by DLAGA generally have better performance compared to the results optimized by simple GA. And among all the optimized MRFs, the best samples are optimized by the DLAGA which means DLAGA is more likely to find the optimal solution of the MRFs.

3. Conclusion

In this paper, we have proposed an arbitrary MRF design method. A machine-learning-assisted global optimization approach has been proposed to search the optimal design of the metasurface by utilizing the evolution optimization algorithm on the searching space compressed by the CVAE. A prediction model and a generative model are constructed and trained to uncover the relationship between the metasurface and their RCS. We also modified the GA and proposed an effective data collection method, which can significantly accelerate the construction of the data set.

Compared with the traditional method mentioned previously, our inverse design method avoids the vast full-wave numerical simulation and realizes the automatically on-demand design of MRF in few seconds. We have also shown that by applying the optimization algorithm directly on the basis of the compressed design space, it is possible to achieve efficient optimization of the metasurfaces with high performance.

The work we have done in this paper has proved the potential of this machine-learning-assisted optimization method in future work, and there remain some areas for future research to augment the utility of this method. Firstly, add more constraints to the samples generated by the MGA, this may enhance the generative ability of the CVAE. Secondly, applying the design scheme to higher dimensions that will increase the degrees of freedom to optimize the objectives at different azimuths which enable the coding metasurface to control the enhanced direction in a larger region.

We also expect that this machine-learning-assisted optimization design method can be applied to not only the coding metasurface but other EM devices, such as antennas, integrated optical circuit devices. This method can also extend to optical frequencies for plasmonic or dielectric materials. This can be a possible future direction of research.