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Abstract
Metamaterials are created by arranging small scatterers in a regular array throughout a space to manipulate electromagnetic waves. However, current design methods view metasurfaces as independent meta-atoms, which limits the range of geometrical structures and materials used, and prevents the generation of arbitrary electric field distributions. To address this issue, we propose an inverse design method based on generative adversarial networks (GANs), which includes both a forward model and an inverse algorithm. The forward model utilizes dyadic Green’s function to interpret the expression of non-local response, realizing the mapping from scattering properties to generated electric fields. The inverse algorithm innovatively transforms the scattering properties and electric fields into images and generates datasets with methods in computer vision (CV), proposing an architecture of GAN with ResBlock to achieve the design for the target electric field pattern. Our algorithm improves upon traditional methods, as it achieves greater time efficiency and generates higher quality electric fields. From a metamaterial perspective, our method can find optimal scattering properties for specific generated electric fields. Training results and extensive experiments demonstrate the algorithm’s validity.
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1. Introduction
Metamaterials have gained significant attention in recent years due to their unique ability to manipulate electromagnetic waves, using synthetic materials that do not exist in nature. These materials include photonic band gap (PBG) structures and frequency selective surfaces (FSS) [1]. Metasurfaces, which are two-dimensional electromagnetic metamaterials, have the advantage of taking up less physical space than fully three-dimensional metamaterial structures, allowing for the creation of low-loss structures [2]. These materials have a wide range of potential applications in electromagnetics, spanning from low microwave to optical frequencies. Some of the applications include shielding, low-reflective materials, antennas, electronic switches, resonators, and Terahertz communication, to name a few [3,4]. To obtain the desired bulk behavior, metamaterials are typically engineered by arranging a set of small scatterers in a regular pattern throughout a region of space. This concept can be extended to create a two-dimensional pattern by placing electrically small scatterers on a surface or interface, which is the basic physical model of metasurfaces [5,6].
The current process for designing metamaterials can be summarized into three main steps. First, a series of basic meta-atoms are designed with different but similar geometrical structures and materials. Next, the electromagnetic responses of a single meta-atom are simulated using computer simulations. Finally, suitable meta-atoms are selected and combined from a series of completed designs. Inverse design paradigms have been proposed to facilitate this design process. For example, the Gercherg-Saxton algorithm can be used to find the phase-offset map for a given plane of incident and output electromagnetic waves, allowing for the identification of suitable meta-atom combinations [7]. Another synthesis algorithm proposed in the literature [8] optimizes the surface current needed for each meta-atom to induce the desired field distribution.
However, the above algorithms [7,8] cannot control the near-field power pattern precisely because of the reduction of degrees of freedom (with 3 or 1 degree of freedom) [9] and can not solve non-local interactions such as perfect anomalous reflection and refraction because of ignoring of interactions between multiple meta-atoms [10]. The main reason for the aforementioned problems is that metamaterials have not been viewed as their primary model during the design process. To address this issue, an alternative approach has been employed, which abstracts the metamaterial as scatterers and transforms the design process into one that focuses on designing scattering properties.
The design of electromagnetic scattering properties on subwavelength structures has always been subject to extensive investigations owing to fundamental challenges and promising applications, especially in metamaterial design. The method proposed in the literature [10] only achieves forward modeling from an array of entirely identical isotropic scatterers to the scattered field, unable to design the scattering properties inversely due to the high complexity. The literature [11] and [12] demonstrates super scatterers bypassing the single channel dipole limit with wire bundle and split ring resonators, respectively. pyGDM [13,14] uses the Green’s function method to accurately simulate a wide range of nanostructure geometries. Metasurface simulation distribution strategy [15] is proposed that significantly reduces simulation time for large-area metasurface. Another approach [16] is presented that allows for the quick calculation of the exact multipole decomposition for any illumination. Moreover, an analytical theory [17] is developed that links the properties of the scatterer to its optical response via the lattice coupling matrix. However, the above algorithms [11–17] maximizing the total scattering cross-section cannot generate arbitrary near-field electric distribution due to the limitations of the specific structures. According to the literature [18], determining the complete scattering components of a single scatterer requires measuring the scattered fields in six directions for 12 different incidences. This highlights the complexity of optimizing all the scattering properties of a scatterer array. To address this issue, machine learning methods have been introduced as a natural solution to complete such high-complexity and large-parametric inverse designs, leveraging their ability to identify potential features.
In recent years, machine learning and materials design have attracted the attention of many researchers as a new interdisciplinary discipline, especially in the field of metamaterial design, which includes forward modeling based on machine learning [19] as well as inverse design methods [20–25]. Zhang et al. used ResNet to establish a mapping between phase and metasurface meta-atoms [20]. To reduce the size of datasets, Zhu et al. utilize transfer learning based on the Inception V3 framework to establish a mapping between pattern and phase [21]. However, the algorithms mentioned above only optimize the phase as the objective, which again reduces the degrees of freedom and similarly cannot holistically generate arbitrary electric field distribution. Naseri et al. propose a GAN-based inverse design of dual- and triple-layer metasurfaces [22]. A trained neural network has been shown in [23] to efficiently infer the internal fields of complex, three-dimensional nanostructures. Generative neural networks have been used to train and design high-efficiency, topologically complex devices [24]. Conditional deep convolutional generative adversarial networks have been utilized to design nanophotonic antennas that lack predefined shapes or constraints, marking a significant advancement in this field [25]. However, the algorithms focus on optimizing the transmission coefficients rather than the electric field distribution, which limits its effectiveness and is constrained by geometric structure and material limitations.
As shown in Fig. 1, the electric field depicted represents an arbitrary distribution of electric field as the target field (near-field radiation pattern). The scattering array is the basic model of the metasurface, and the metasurface is the material implementation of the scattering array. We propose an ideal scatterer model-based metasurface design process, as following two steps: designing the scattering properties by the desired function, obtaining the optimal realization with omega meta-atom [18] or hybrid magneto-electric particle (HMEP) [19]. Compared to current methods, the above process has three advantages: it does not require pre-designed meta-atoms, has no restrictions on geometrical structures and materials, and is capable of achieving an arbitrary electric field distribution by simultaneously designing the scattering properties (with 36 degrees of freedom). This approach has several applications, such as scanning optical microscopy and subsurface object imaging for achieving super-resolution, compact antenna test range (CATR) for evaluating scattering from objects, and non-invasive microwave hyperthermia for heating and destroying tumors based on the near-field electric distribution [26–30].
Fig. 1. The relationship of forwarding mapping and generative machine learning-based inverse design. $E\left ({\mathbf {r}} \right )$ is the mode of the electric field at each observation point, ${\mathbf {E}_0}\left ( {\mathbf {r}} \right )$ is the the electric fields generated at the observation point by the dipole emitter when scatterers are not considered, ${{\mathop \alpha ^ {\leftrightarrow }}_E}$ is the electric and magnetic polarizability.
We transform the inverse design into a style migration problem in CV by treating the electric field distribution and scattering properties as images and using GAN [31] to complete the complex optimization of a considerable number of parameters.
The main contributions of this work are summarized as follows:
- • We establish a generic model between scattering properties and scattered field and realize the mapping from the basic model of metasurface to the fundamental property of the electric field.
- • We analyze the interactions between multiple scatterers interpretably and illustrate two forms of non-local response between scatterers.
- • We innovatively transform the inverse design of scattering properties into a style migration problem and propose a method to generate image datasets.
- • We proposed a GAN architecture with ResBlock [32] to holistically achieve the design for arbitrary electric field distribution by optimizing the polarizability components of the scatterers array.
2. System model
2.1 Scatterers-observation points model
The proposed model is shown in Fig. 2, and its brief idea can be expressed as follows. The dipole radiation theory [33] and scatterers model are first used to forward model the non-local scattered field [10]. The literature [34] proposed a hybrid magneto-electric particle possessing asymmetric scattering properties, which indicates that it is not comprehensive to consider only the total scattering cross-section or the forward scattering. Therefore, the observation points were introduced to extend to the near-field two-dimensional or three-dimensional electric field distribution.
Fig. 2. (a) The general S-O model. (b) The simplified S-O model and the non-local response between multiple scatterers.
Thus, a model is established between the scattering properties at the source points and the electric field distribution at the field points. In spatial terms, the model is represented as a mapping from the array of scatterers to the observation surface.
As is shown in Fig. 2(a), the yellow double-arrow indicates the dipole emitter, while the blue arrow illustrates the overall propagation direction of the electromagnetic wave. The red arrow specifically illustrates the propagation path of the electromagnetic wave in the multiple scattering effect. The black sphere represents the scattering object, and the blue sphere represents the reference point. In this general S-O (scatterers-observation points) model, the electric field intensity at each reference point can be derived from Maxwell’s equations for a monochromatic wave of frequency $\omega$ in a general linear medium characterized by the permittivity $\epsilon$ and permeability $\mu$.
The dyadic Green’s function, an important tool for solving differential equations in electromagnetic fields, can be used to solve equations of the form $\nabla \times \nabla \times \mathop {\mathbf {G}}^ \leftrightarrow \left ( {{\mathbf {r}},{{\mathbf {r}}^\prime }} \right ) - {k^2}\mathop {\mathbf {G}}^ \leftrightarrow \left ( {{\mathbf {r}},{{\mathbf {r}}^\prime }} \right ) = \delta \left ( {{\mathbf {r}} - {{\mathbf {r}}^\prime }} \right )\mathbb {I}$. Noting that Eqs. (1) and (2) have the same form as above, we solve for the electric and magnetic field strengths at the observation point by introducing the dyadic Green’s function, which is generally expressed as
To illustrate that the derivation is independent of scale and frequency. Specifically, we choose a scale factor such that the unitless wave number $\xi = \gamma k$, and thus the scaling of the coordinate system is achieved, and for the convenience of calculation and representation, taking $\gamma = 1/k$. The wave impedance ${Z_0} = 1$ in free space under this unit system by considering the interchangeability of the electric and magnetic fields, which is another significant advantage of using this unitless system. Based on this unitless system, we can obtain the following matrix equation by letting ${\mathbf {r}} = {\mathbf {r}}_i^{'}$ [10]
The response matrix ${\mathbf {R}} \in {{\text {C}}^{6N \times 6N}}$ is for three spatial dimensions, for both electric and magnetic fields and all $N$ scatterers, and is the most numerically demanding part of the model. Moreover, because the inverse matrix of this response matrix cannot be represented analytically, it is challenging to design the scattering properties from the electric field inversely.
By solving this matrix equation, which takes the form of ${\mathbf {Ax}} = {\mathbf {b}}$, we can obtain the electric field strength at each scatterer. The mode of the electric field at each observation point can be derived, which is expressed as
According to Eq. (6), the forward modeling of the S-O model can be completed by traversing the observation surface.
In the S-O model, the observation points should be in the near field of the scatterers because the spatial resolution and information capacity in the truncated domain is much higher than in the far field. Therefore it is possible to achieve complex manipulation of the field distribution in the near region.
2.2 Two forms of interactions between multiple scatterers
To illustrate the effects of the dipole emitter and scatterers in Eq. (6) on the generated electric field, as well as the interactions between the scatterers, we consider a special case consisting of a dipole emitter and two scatterers. By letting $n=2$, we can obtain the generated electric field. In this case, ${\mathbf {E}}\left ( {{\mathbf {r}}_i^{'}} \right )$ and ${\mathbf {H}}\left ( {{\mathbf {r}}_i^{'}} \right )$ are, respectively the generated electric field can be expressed as
As shown in Fig. 2, the analysis of Eqs. (8)–(9) allows us to understand how the interaction between the two scatterers arises. In the 2-scatterers-system, the interaction can be divided into two kinds: one is comprised of scattering from ${\mathbf {r}}_1^{'}$ to ${\mathbf {r}}_2^{'}$ then back again, and this is what is expressed as the second power of the dyadic Green’s function shaped as ${\mathop {\mathbf {G}}^ {\leftrightarrow } }\left ( {{\mathbf {r}}_1^{'},{\mathbf {r}}_2^{'}} \right ){\mathop {\mathbf {G}}^ {\leftrightarrow } }\left ( {{\mathbf {r}}_2^{'},{\mathbf {r}}_1^{'}} \right )$, which is multiplied by its own ${{\mathbf {E}}_0}\left ( {{\mathbf {r}}_i^{'}} \right )$ or ${{\mathbf {H}}_0}\left ( {{\mathbf {r}}_i^{'}} \right )$; and the second is comprised of scattering from ${\mathbf {r}}_1^{'}$ to ${\mathbf {r}}_2^{'}$ then back again and again, and this is what is expressed as the third power of the dyadic Green’s function shaped as ${\mathop {\mathbf {G}}^ {\leftrightarrow } }^{2}\left ( {{\mathbf {r}}_1^{'},{\mathbf {r}}_2^{'}} \right ){\mathop {\mathbf {G}}^ {\leftrightarrow } }\left ( {{\mathbf {r}}_2^{'},{\mathbf {r}}_1^{'}} \right )$, which is multiplied by the ${{\mathbf {E}}_0}\left ( {{\mathbf {r}}_i^{'}} \right )$ or ${{\mathbf {H}}_0}\left ( {{\mathbf {r}}_i^{'}} \right )$ of another scatterer.
This understanding can be extended to the N-scatterers-system, where the electric field component generated by any scatterer will include the electric field from itself and then scattered from itself and the electric field from itself and then scattered by others. And the semi-analytical form of ${\mathbf {E}}\left ( {{\mathbf {r}}_i^{'}} \right )$ and ${\mathbf {H}}\left ( {{\mathbf {r}}_i^{'}} \right )$ in N-scatterers-system can be expressed as
3. Generative machine learning-based inverse design
3.1 Designing of scattering properties – electric field distribution dataset and generating adversarial network
As the complexity analysis in section 2, to design scattering properties of an N-scatterers-system, we need to calculate the inverse of a response matrix with $6N \times 6N$ elements. And the resulting expression will contain $6(N - 1) \times 6(N - 1) \times 2N$ components. To solve this inverse design problem with high computational complexity and many parameters, we innovatively transform the scattering properties and electric field distribution into pictures, thus transforming the problem into a style migration problem in CV. Further, a generative adversarial network is used as a powerful tool to generate scattering properties images from the target electric field images.
The general S-O model does not limit a specific arrangement of scatterers or observation points, which can be in abstract two-dimensional or three-dimensional form. However, for practical application, the scatterer array and observation point surface can be considered two-dimensional rectangular planes to derive a simplified S-O model.
The dipole emitter is similarly located at the origin of the Cartesian coordinate system, i.e., $r_0^{'} = (0,0,0)$. The model contains a total of $N$ scatterers and satisfies $N = {n^{2}}$, located at ${\mathbf {r}}_{ij}^{'} = (\frac {n}{2}d_x^{'} - id_x^{'},\frac {n}{2}d_y^{'} - id_y^{'},0),i,j = 1,2, \ldots,n$, where $d_x^{'}$ and $d_y^{'}$ are the spacing of scatterers in the x-axis and y-axis directions, respectively; similarly, the model contains a total of $M$ observation points and satisfies $M = {m^{2}}$, located at ${{\mathbf {r}}_{ij}} = (\frac {n}{2}{d_x} - i{d_x},\frac {n}{2}{d_y} - i{d_y},{d_z}),i,j = 1,2, \ldots,m$. And the distance between two parallel planes is taken as $5\lambda$ to meet the near-field requirements, and $\lambda = 2\pi$ in the unitless system.
Based on the simplified S-O model, this paper proposes a method to generate scattering properties-electric field distribution dataset (S-E dataset). The input data of the model is the polarization rate of each scatterer, which can be expressed as
Uniformly distributed random functions generate the real and imaginary parts to ensure randomness, and the generated polarization rate is verified to satisfy $\operatorname {Im} \left [ {\mathop \alpha ^ {\leftrightarrow } {_{E,H}^{ - 1}} } \right ] \geqslant - \mathbb {I}{k^3}/(6\pi \varepsilon )$. Therefore, Eq. (6) can be traversed to generate the modes of the electric field distribution at all observation points, thus generating one-to-one correspondence between the scattering properties and the electric field distribution data.
The scattering properties do not satisfy the arrangement of images naturally. With the development of GAN, in which CNN gradually implements the generator and discriminator, this implies the introduction of convolutional kernels. Therefore, we must ensure spatial correlation when generating scattering properties images to satisfy the convolution kernel’s characteristic that focuses only on local features. The mapping relationship in the dataset is shown in Fig. 3, the shape of the convolution kernel arranges the polarizability components of each scatterer, and the position corresponds to the scatterer’s location.
The GAN consists of the generator and the discriminator, where the generator is trained to produce outputs that cannot be distinguished as "real" images by the adversarially trained discriminator. And the latter is trained to detect the " fake" images of the generator as well as possible. The mapping relationship between the random noise vector $z$ to the output image $y$ can be expressed as $G:z \to y$, and its loss function of it can be expressed as
Based on the standard GAN model, the main idea of cGAN [35] is to find the mapping relationship between both the input data $x$ and the random noise vector $z$ to the output image $y$ through training, which can be denoted as $G:\{ x,z\} \to y$ accordingly, and its loss function can be denoted as
Similarly, the optimization objective of cGAN can be expressed as
Pix2pix [36] is a variant of cGAN proposed for the implementation of image-to-image translation, and this method makes further changes to the loss function compared to cGAN. It can be expressed as
whose introduction allows $D$ to be near the ground truth output in an L1 sense. Further, this addition allows the method to implement image-to-image translation and learn a loss function to train this mapping.3.2 Training configuration and setup
In the actual generation of the dataset, we take $n = 8$ and $m = 32$ so that the electric field distribution generated by the $8 \times 8$ scatterers is inscribed by the $32 \times 32$ observation points. During the process, let $d_x^{'} = d_y^{'} = \frac {\lambda }{2}$ and ${d_x} = {d_y} = \frac {\lambda }{4}$. Based on the method of the S-E dataset, we generated datasets of various sizes and divided them accordingly. The size of the generated dataset in this paper contains 5000 (dataset I) and 10,000 (dataset II) pairs of images, of which 75% are classified as training set, 20% are classified as validation set and 5% are classified as test set.
Based on the concept and basic framework of pix2pix, we proposed the matching generator and discriminator network structures for the practical problem of scattering properties design, as shown in Fig. 4. We made specific modifications to the network architecture of pix2pix to address the practical problem of designing scatterers parameters. Specifically, we introduced the idea of using resblocks from the ResNet [26] achitecture. This allowed us to enhance the network’s ability to csolve the optimization difficulty problem and gradient disappearance phenomenon in image generation, improving the network’s ability to handle complex image-to-image translation tasks involving scatterers, where $x$ represents the scattering parameters as input data, $z$ represents the generated electric field distribution, and $y$ represents random noise to prevent overfitting and improve the model’s ability to generalize to new data.
Fig. 4. The architecture of GAN. (a) The architecture of the generator network. (b) The architecture of the discriminator network.
The generator network consists of 5 parts, from input to output: data input block, 2 downsampling blocks, 9 ResBlock blocks, 2 upsampling blocks, and data output block, respectively.
Compared with the generator network, the discriminator network has a relatively simple design and is a 5-block convolutional neural network. The first four blocks include the convolutional layer, BatchNorm layer, and activation function Relu layer; the fifth block includes the convolutional layer and activation function sigmoid. The size of the convolutional kernel used in the discriminator network is 4.
Besides, the above network structure has two key features: the generator with skips and the Markovian discriminator (PatchGan). Generator with skips means adding skip connections between each layer $i$ and layer $n - i$, where $n$ is the total number of layers. Its advantage is to facilitate sharing of some underlying information between the input and output of the image-to-image translation model. Markovian discriminator means dividing the image into patches, determining the truth or falsity of each patch separately, and finally taking the average. Its advantage is better judgment for high-frequency parts of the image, such as edges.
The GAN is built by Pytorch 1.10.0 in Python 3.7. The integrated development environment is Anaconda 3 in Ubuntu 18.04 Operation System. The GPU configuration is NVIDIA TITAN V with 12GB video memory. And the training configurations are shown in Table 1.
Table 1. Training configurations
4. Results and discuss
4.1 Model validation and simulation
Based on the configuration in section 3, the training process’s loss-epoch graphs on S-E dataset I and II are shown in Fig. 5, respectively. It can be found that the model has suitable convergence property on both datasets. It is worth noting that the proposed model shows better convergence on smaller datasets with lower loss, which is due to the occurrence of overfitting, leading to a decrease in the model’s ability to generalize to new data and resulting in poor performance on the testing set. Moreover, the training results are shown in Table 2.
Table 2. Training results
PSNR is the most common and widely used objective measurement method for evaluating picture quality, and a higher index means a better quality of the reconstructed image [37,38], which is expressed as
where ${{MAX}_I}$ is the maximum possible pixel value of the image, generally taken as 255, and $MSE$ is the mean square error per pixel between the original and generated images, generally taken as the average of the three RGB color channels when calculating the average PSNR. After 100 epochs of training, the average PSNR is 51.3414dB and 103.1464dB, respectively, which illustrates the degree of similarity between the generated and the electric field image.Meanwhile, the time efficiency of the proposed method is also tested. The time to design 500, 1000, 5000, 10000 $8 \times 8$ scatterer arrays is 98s, 194s, 973s, 1947s, respectively. And the average time to generate 2304 polarization rate components is less than 0.195s.
After training on the S-E datasets, the proposed GAN achieved excellent performance on both the validation and test sets. To evaluate the performance, we selected relatively complex electric field distributions from the test set of S-E dataset I and II as target electric fields, as shown in Fig. 6(a). The absolute error between the target and generated fields is shown in Fig. 6(c).
Fig. 6. (a) Comparison of target (left) and generated (right) electric fields in the test set, respectively. (b) The target (left) and generated (right) fields with single focusing point and quadruple focusing points, respectively. (c) The significant difference between the target and generated fields of (a). (d) The grayscale image of (b).
The comparison between our proposed method and the methods mentioned in other literature is presented in Table 3. The comparison of the data demonstrates that our method, based on optimized network design, achieves convergence on relatively smaller datasets. More importantly, these methods are all subject to certain geometric constraints. Although some of these methods have increased degrees of freedom by dividing the space into two-dimensional or three-dimensional grids, they still cannot fully characterize the parameters and properties of scatterers. In contrast, the work presented in this paper focuses on optimizing the scatterer parameters, which are fundamental physical quantities. Our proposed method enables the generation of arbitrary electric field distributions without being restricted by any geometric structure. Therefore, in theory, it is applicable to fully three-dimensional structures and arbitrary materials. It is worth noting that due to the differences in network models, various methods may use different metrics to express accuracy or error. In this paper, we utilize PSNR, which is a widely used metric for assessing the quality of generated images. Generally, a PSNR value greater than 50dB is considered to indicate relatively high image quality.
Table 3. Comprision with other works
4.2 Algorithm validation setup and results
In this section, we demonstrate the application of our proposed algorithm to a specific scenario where a desired field strength is formed in the observation region by adjusting the properties of dipole scatterers. As mentioned in Section 3, while the positions of scatterers and observation points can be arbitrary, we choose to restrict them to planar surfaces to facilitate the corresponding experiment. The scatterer array used in this scenario is a rectangle comprising $8 \times 8$ discrete scatterers, with a spacing of ${\lambda }/2$ between adjacent scatterers, resulting in a scattering property dimension of 2304. The observation points surface is a rectangle located $5\lambda$ in front of the scatterer array and consists of $32 \times 32$ points with ${\lambda }/4$ intervals between them, resulting in an observation points dimension of 1024.
We designed the electric field distributions of single and quadruple focusing points as the target fields of the applied model to verify the proposed method’s ability to generate arbitrary electric fields holistically. In the target field with single focusing point, the center coordinate of the focus point is $(0,0)$, and the side length of the rectangular focus region is $3{\lambda }/4$; In the target field with quadruple focusing point, the center coordinates of the focal points are $( \pm 2\lambda, \pm 2\lambda )$, and the side lengths of the rectangular focusing area are ${\lambda }/2$. As is shown in Fig. 6(b). Figure 6(d) shows the grayscale images of the target and generated fields with single and quadruple focusing points, which are the actual images processed by GANs during inverse design. It is worth noting that these images are not represented as heatmaps in three channels, but rather as grayscale images in a single channel.
To quantitatively assess the quality of the generated electric field, we define the signal-to-noise ratio (SNR) as the ratio of the electric field intensity at an observation point to the average electric field intensity over the entire observation surface. And the results demonstrate the respective trends of the target and generated electric fields and the similarity between the two. As shown in Fig. 7, The x-axis in the figure represents the cut field signal-to-noise ratio (SNR) image along the diagonal, with the unit of $\lambda$. The y-axis represents the normalized magnitude of the electric field strength.
Fig. 7. Cuts of normalized SNR graphs. (a) The cut of single focusing electric field SNR image along diagonal. (b) The cut of quadruple focusing electric field SNR image along diagonal.
We have utilized the approach introduced in literature [39,40] to extract the polarizability of an arbitrary dipolar particle by analyzing its computed or measured scattering cross-sections, as shown in Fig. 8(a). This information plays a critical role in optimizing the dimensions and shapes of the omega meta-atoms. By leveraging the obtained polarizabilities as the optimization objective in our proposed GAN ML solution, we are able to fabricate particles with desired scattering performance, as shown in Fig. 8(b).
Fig. 8. (a) The structure of omega particle. (b) The design of omega particle arrays for single-point focusing and four-point focusing. (c) The generated electric fields in HFSS.
To further validate our method, we performed HFSS full-wave electromagnetic simulations to generate the target fields in Fig. 8(c). These simulations were conducted to verify the generated electric fields. The validation through electromagnetic simulations provides additional evidence supporting the reliability and applicability of our proposed method. It not only demonstrates the potential for practical implementation but also affirms our approach’s ability to manipulate electromagnetic waves for various applications.
It is worth seeing that the scattered field distribution generated by the inverse design method proposed in this paper has almost the same distribution as the target field. The primary distortion appears at the center of the electric field since the electric distribution at the center is influenced by all scatterers, requiring consideration of complex superposition effects. However, as the number of scatterers increases, more accurate control of the scattered field can be achieved, significantly reducing the error between the generated and the target electric field.
5. Conclusion
This paper proposes a non-local inverse design method for scattering properties based on GAN. We realize the forward mapping and inverse design between scattering properties and scattered electric field and interpretably illustrate the interaction between multiple scatterers and two forms of non-local response between scatterers. In this process, the scattering properties design is transformed into a style migration problem in CV, holistically realizing the arbitrary electric field distribution by simultaneously optimizing the complex polarizability components of scatterers.
The proposed algorithm is applied to an exemplary scenario, demonstrating the superiority of time efficiency and the quality of generated electric fields. 2304 components were generated in 0.195s, and the PSNR exceeded 100dB.
Funding
Beijing Municipal Natural Science Foundation (L223030, L212028); National Natural Science Foundation of China (61971064, 62271082).
Disclosures
The authors declare no conflict 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.
Supplemental document
See Supplement 1 for supporting content.
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