Super-surface mutual coupling effect optimization method based on deep convolutional neural network

Abstract

The invention discloses a super-surface mutual coupling effect optimization method based on a deep convolutional neural network. The super-surface mutual coupling effect optimization method comprises the following steps of S1, calculating actual phases and amplitude responses of randomly generated element atoms placed between different random adjacent atoms to assemble a training data set; s2, constructing a prediction neural network based on a convolutional neural network architecture to realize rapid prediction of local responses of element atoms under different boundary conditions; s3, the mean square error of the local response predicted by the prediction neural network and the label is minimized by reversely adjusting parameters in the hidden neural layer; s4, comparing the performance of the optimization design with the performance of the initial design from the optimization of the light beam deflector, and verifying the effectiveness of the optimization design. The method can be used for explaining performance reduction of the metasurface caused by mutual coupling, and is further used for optimizing the efficiency after being combined with an optimization algorithm. And the efficiency of the light beam deflector and the efficiency of a super lens based on a traditional method are greatly improved.

Description

Technical Field

[0001] The present invention belongs to the field of electromagnetism and is combined with a machine learning algorithm. Specifically, it is an optimization method for the mutual coupling effect of supersurfaces based on a deep convolutional neural network. Background Art

[0002] Metasurfaces, optical devices with subwavelength structures that can control the properties of light with unprecedented spatial resolution, have attracted great attention in the past decade. With their ability to manipulate the phase, amplitude, and polarization of light, metasurfaces have been used in a variety of applications, ranging from achromatic beam focusing, vortex beam generation, and holography to optical computing and quantum optical information processing.

[0003] Metasurfaces offer a novel and promising platform for realizing compact, large-scale optical devices. Conventional metasurface design methods assume periodic boundary conditions for each element, which is inaccurate in most cases because the near-field coupling effects between elements change when surrounded by non-identical structures. The most widely adopted metasurface design method involves two steps: 1) calculating the amplitude and phase masks required for the desired functionality, fitting to a square or hexagonal grid, and 2) finding the meta-atom with the performance closest to the target for each grid for the final design. Accurate and efficient meta-atom on-demand design methods remain a major challenge for metasurface design. Summary of the invention

[0004] In response to the above-mentioned problems, the present invention provides a method for optimizing the mutual coupling effect of metasurfaces based on a deep convolutional neural network. The process of the method for optimizing the mutual coupling effect of metasurfaces described in the present invention includes: simulating the near-field electric field distribution, calculating the actual phase and amplitude response of the meta-atom, constructing a predictive neural network (PNN) based on a convolutional neural network (CNN) architecture, defining a minimization objective function, and verifying the effectiveness of the optimization method.

[0005] A method for optimizing the mutual coupling effect of a super surface based on a deep convolutional neural network comprises the following steps:

[0006] Step S1: assembling a training dataset by calculating the actual phase and amplitude responses of randomly generated meta-atoms placed between different random neighboring atoms;

[0007] Step S2: constructing a prediction neural network based on a convolutional neural network architecture to achieve rapid prediction of the local response of meta-atoms under different boundary conditions;

[0008] Step S3: Minimize the mean square error between the local response predicted by the prediction neural network and the label by reversely adjusting the parameters in the hidden neural layer;

[0009] Step S4: Starting from the optimized beam deflector, the performance of the optimized design is compared with the performance of the initial design to verify the effectiveness of the optimized design.

[0010] In some embodiments, the training data set includes actual phase and amplitude response data obtained by simulating a model in which four randomly generated different adjacent meta-atoms are set on each side of the target meta-atom.

[0011] In some embodiments, in step S2, the two-dimensional cross-section of the target meta-atom and its neighboring atoms is subjected to six consecutive layers of convolution and pooling processing and then converted into a one-dimensional vector. During this period, hidden features and the influence of different neighboring atoms are extracted and calculated. The output of the CNN is flattened into a one-dimensional vector and then passed through three fully connected layers to generate a prediction result of the complex transmission coefficient of the target meta-atom. In the entire network, except for the last layer without an activation function, the ReLU activation function is applied to each layer.

[0012] In some embodiments, in step S3, the collected training data is randomly divided into a training set and a test set, which contain 70% and 30% of the total training data, respectively. During the training process, the local response predicted by the PNN is compared with the label, and the mean square error is calculated, which is minimized by reversely adjusting the parameters in the hidden neural layer.

[0013] In some embodiments, in step S4, a beam deflector composed of 1×4 atomic elements is optimized, the phase targets are set to 60, 150, 240 and 330 degrees, the amplitude target is set to 1, the optimization is started from the initial design, and the objective function is minimized, which is defined as:

[0014]

[0015] Where N is the number of atoms in the device, and A i is the local phase and amplitude response and are the target phase and amplitude responses, respectively.

[0016] The invention has the following beneficial effects: it can be used to explain the degradation of the performance of the metasurface caused by mutual coupling, and can be further used to optimize its efficiency after being combined with an optimization algorithm. It has also made great progress in the efficiency of beam deflectors and metalenses based on traditional methods. In addition, it has demonstrated the correlation between the performance of the metasurface and its design errors caused by mutual coupling, and this is not constrained by specific specifications. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1A flow chart of an optimization method for the mutual coupling effect of metasurfaces based on a deep convolutional neural network is shown in some examples of the present disclosure.

[0018] Figure 2a Schematic diagram of meta-atoms composed of silicon nanoblocks in some examples of the present disclosure is shown.

[0019] Figure 2b Shows Figure 2a Shown are the transmittance phase and amplitude responses of meta-atoms with different lengths dx and widths dy at a wavelength of 1.55 μm.

[0020] Figure 2c 1000 randomly generated target meta-atoms are shown, with each side of the target meta-atoms surrounded by 10 random neighboring meta-atoms.

[0021] Figure 2d It is shown that different numbers of neighboring meta-atoms (N) are considered in the simulation to derive the accurate EM response of each target meta-atom.

[0022] Figure 2e The average amplitude and phase errors as a function of N are shown.

[0023] Figure 3 A diagram of the prediction neural network architecture is shown.

[0024] Figure 4a The average MSE of the training and test sets during training is shown.

[0025] Figure 4b Six sets of results are randomly selected from the test set and demonstrated.

[0026] Figure 5a A workflow diagram of an optimization method for a beam deflector composed of 1×4 meta-atomic elements is shown.

[0027] Figure 5b The cyclic and local responses of the initial and final optimized designs are shown.

[0028] Figure 5c The phase error and amplitude of the optimized design compared to the initial design are shown.

[0029] Figure 5d The value of the objective function during the optimization process is shown.

[0030] Figure 6a The simulated electric field (real part) of the initial design is shown.

[0031] Figure 6b The target phase, local phase, and phase error are shown for each meta-atom.

[0032] Figure 6c The optimized design of the simulated electric field is shown.

[0033] Figure 6d The target phase, local phase and phase error are shown.

[0034] Figure 6e The value of the objective function during the optimization process is shown.

[0035] Figure 6f Shown are the simulated electric fields along the optical (Z) axis for the initial and optimized designs using x-polarization incidence.

[0036] Figure 7a The target phase, local phase, and phase error of each meta-atom in the initial design are shown.

[0037] Figure 7b An optimized design is shown.

[0038] Figure 7c Shown are the simulated electric fields along the optical (Z) axis for the initial and optimized designs using x-polarized incidence; the inset contains a magnified view of the e-field near the focus.

[0039] Figure 7d The simulated electric field (real part) of the optimized design and the enlarged view of the electric field near the focus are shown.

[0040] Figure 8a Schematic diagram showing free-form meta-atoms composed of a high refractive index material on a low refractive index substrate.

[0041] Figure 8b Amplitude and phase MAEs for a specified number of distinct adjacent meta-atoms per side are shown.

[0042] Figure 9a The average MSE of the training set (red curve) and the test set (blue curve) during the training process is shown. Figure 9b Six randomly selected sets of results from the demonstration test set are shown.

[0043] Fig.10a Shown are the periodic phase and amplitude responses of a 400 meta-atom design generated by the fully trained GAN model, as well as the MSE and efficiency of a 1000-beam deflector consisting of 1000 meta-atoms.

[0044] Fig.10b The local responses corresponding to all meta-atoms in the beam deflector are shown.

[0045] Fig.10c Shows from Fig.10a Simulated electric fields for the two deflectors chosen in , and the local response of each meta-atom for the two designs. DETAILED DESCRIPTION

[0046] The present invention will be further described in detail below in conjunction with the accompanying drawings.

[0047] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions of the present invention will be clearly and completely described below in combination with the specific embodiments of the present application and the corresponding drawings; it is obvious that the described embodiments are only part of the embodiments of the present invention, not all of the embodiments.

[0048] Figure 1 A flow chart of an optimization method for the mutual coupling effect of metasurfaces based on a deep convolutional neural network is shown in some examples of the present disclosure.

[0049] The present invention provides a method for optimizing the mutual coupling effect of a super surface based on a deep convolutional neural network, including the following contents:

[0050] Step S1: The training dataset is assembled by computing the actual phase and amplitude responses of randomly generated meta-atoms placed between different random neighboring atoms.

[0051] Figure 2a Schematic diagram of meta-atom composed of silicon nanoblocks in some examples of the present disclosure is shown. The design is constructed of rectangular high-refractive index (n=3.67) polysilicon nanoblocks on a low-refractive index (n=1.45) fused silica substrate. The wavelength is set to 1.55μm, and the lattice constant and the height of the nanoblock are fixed to 800nm ​​and 270nm, respectively.

[0052] Figure 2b The transmittance phase and amplitude responses of meta-atoms with different lengths dx and widths dy at a wavelength of 1.55 μm are shown. By selecting the length and width of each nanoblock, this meta-atom design can cover the entire 2π phase while maintaining high transmittance, which is critical for most optical applications.

[0053] In some examples of the present disclosure, the overall structure is first modeled, wherein the model includes the target meta-atom and the adjacent atoms arranged on both sides of the target atom, and then the equivalent source (J) of the target atom is obtained. S , M S ) and exported to a new simulation file to measure its local phase and amplitude response. For example, the commercial software CST can be used for modeling, and the equivalent source (J S , M S ).

[0054] For simplicity, the study can be restricted to one dimension, which means that there are only coupling effects along the x-axis and periodic boundary conditions are set along the y-axis. Considering that the mutual coupling effect decreases with increasing distance, it is necessary to determine the number of neighboring meta-atoms that need to be considered in each simulation. Figure 2c 1000 randomly generated target meta-atoms are shown, with each side of the target meta-atom surrounded by 10 random neighboring meta-atoms. Figure 2c , we randomly generated a set of meta-atoms consisting of a target meta-atom surrounded by 10 different meta-atoms on each side, and calculated the phase error and amplitude error. We then repeated this experiment on 1000 different sets of meta-atoms and calculated the average phase error and amplitude error. Figure 2d It is shown that different numbers of neighboring meta-atoms (N) are considered in the simulation to derive the exact EM response of each target meta-atom. The structure without considering different neighboring meta-atoms (periodic boundary conditions), i.e. the structure when N=0, is first simulated, and the number of neighboring atoms considered in the simulation is gradually increased to study how the local response of the target meta-atom changes. This experiment can be repeated on 1000 different groups of meta-atoms and the average phase error and amplitude error are calculated. Figure 2e The average amplitude and phase errors as a function of N are shown. The local response of all 10 neighboring meta-atoms on each side is considered as a reference point of zero error, because the improvement in accuracy is negligible when adding additional neighboring meta-atoms beyond this point. The simulation results for the target meta-atom with periodic boundaries (referred to as "periodic response") have significant amplitude and phase mean absolute errors (MAE) of 0.33 degrees and 59.14 degrees compared to when the number of neighboring meta-atoms (N) is 10. The error gradually decreases as the number of neighboring meta-atoms increases. Figure 2e The large error values ​​in are caused by the possible phase gradient abrupt changes in the randomly arranged meta-atoms. According to the results, even considering only the nearest single meta-atoms on both sides of the target greatly improves the accuracy of its EM response.

[0055] In order to strike a balance between simulation accuracy and optimization difficulty (and data collection cost), some embodiments of the present disclosure select N=4 during data collection. The final data set consists of the physical dimensions and local responses of more than 100,000 randomly generated meta-atoms, with 4 randomly generated meta-atoms on each side of each meta-atom. All length (dx) and width (dy) nanoblocks are in the range of [0.25μm, 0.75μm], with a minimum step size of 6.25nm.

[0056] Step S2: Construct a prediction neural network (PNN) based on a convolutional neural network (CNN) architecture to achieve rapid prediction of the local response of meta-atoms under different boundary conditions.

[0057] PNN takes the dimension of the target meta-atom and the dimensions of its neighbors as input and quickly generates predictions of the local response of the target meta-atom. Figure 3 The diagram of the prediction neural network architecture is shown. A 2D cross-section of the target element is extracted and pixelated into a 1152×128 image. The image is binarized, where “1” represents a dielectric region and “0” represents a void. This figure is then processed through 6 consecutive convolutional layers, during which hidden features such as the relationship between the size of the nanoblock and its local response, as well as the influence of its non-identical neighbors, are extracted and calculated. After flattening the output of the CNN into a 1D vector and passing through 3 fully connected layers, the prediction results of the complex transmission coefficient of the target element are generated and ready for evaluation. Throughout the network, a ReLU activation function is applied to each layer except the last layer, which has no activation function.

[0058] Step S3: Minimize the mean square error between the local response predicted by the PNN and the label by inversely adjusting the parameters in the hidden neural layer.

[0059] In some examples of the present disclosure, more than 100,000 sets of training data collected were randomly divided into training sets and test sets, containing 70% and 30% of the total training data, respectively. The test set is used to evaluate the performance of the training network on data that was not used during the training process. During the training process, the local response predicted by the PNN is compared with the label (accurate local response), and the mean square error (MSE) is calculated, and the error is minimized by reversely adjusting the parameters in the hidden neural layer.

[0060] Figure 4a The average MSE of the training set (red curve) and the test set (blue curve) during the training process is shown. Figure 4b Six sets of results are randomly selected from the test set and demonstrated. The target meta-atom is marked in red, while the adjacent meta-atoms are marked in yellow, the periodic response is marked with a green star, and the local response predicted by PNN and calculated by CST are marked with a blue star and a red star, respectively. When the training is completed, the average MSE for the training set is 9e -6 and for the test set 7e -5 ,like Figure 4a As shown in Figure 2, the average prediction standard deviation of each target meta-atom at the target wavelength is 0.005, and the average prediction standard deviation of the phase is 3.15 degrees. Several samples randomly selected from the test set are used to demonstrate the accuracy of the well-trained PNN. Figure 4b As shown, 6 target meta-atoms (marked in red), with 4 different meta-atoms on each side (marked in yellow), are set as the input of the fully trained PNN. The local responses predicted by the PNN are marked with blue stars, while the accurate results calculated by CST are marked in red. The predicted results are in good agreement with the accurate results.

[0061] Step S4: Starting from the optimized beam deflector, the performance of the optimized design is compared with the performance of the initial design to verify the effectiveness of the optimized design.

[0062] Figure 5a-5d This is the optimization data of the beam deflector composed of 1×4 meta-atomic elements described in an embodiment of the present invention. Figure 5a A workflow diagram of the data optimization method for a beam deflector composed of 1×4 meta-atom elements is shown. Figure 5b The periodic and local responses of the initial and final optimized designs are shown, with top views of the two designs shown at the bottom. The dielectric nanoblocks are yellow, while the dark colors represent voids. Figure 5c The phase error and amplitude of the optimized design compared to the initial design are shown. Figure 5d The values ​​of the objective function during the optimization process are shown. For the initial design shown on the left, the periodic response of each meta-atom is marked with a square in the polar coordinate plot, while their corresponding local responses are marked with an empty circle. The phase targets are set to 60, 150, 240, and 330 degrees, respectively, and the amplitude target is set to 1. Although the periodic response is almost perfect, with transmission and exact 90-degree phase shift between each pair of adjacent meta-atoms, their local responses deviate from the design target due to mutual coupling effects, resulting in relatively low efficiency (41.3%). The optimization is then started with this initial design as the starting point, and the objective function is minimized, defined as:

[0063]

[0064] Where N is the number of atoms in the device, and A i is the local phase and amplitude response and The target phase and amplitude responses are shown in Figure 2. After 1000 optimization iterations, the average phase error is reduced from 30.3° (initial design) to 10.4° (optimized design), while the average amplitude remains basically unchanged, indicating the effectiveness of the proposed method.

[0065] Figure 6a-6f Optimization data of high numerical aperture metals composed of 40 meta-atoms for some examples of the present disclosure are shown. Figure 6a The simulated electric field (real part) of the initial design is shown. Figure 6b The target phase (blue), local phase (red), and phase error (green) are shown for each meta-atom. Figure 6c The optimized design of the simulated electric field is shown. Figure 6d The target phase, local phase and phase error are shown. Figure 6e The value of the objective function during the optimization process is shown. Figure 6fThe simulated electric field along the optical (Z) axis is shown for the initial design (red) and the optimized design (blue) using x-polarized incidence. The advantages of this combined optimization approach become more apparent when applied to larger metasurfaces, such as high numerical aperture (NA) metals. Here, a lens consisting of 40 meta-atoms with a focal length of 10 wavelengths (15.5 μm) was designed using conventional methods. The real part of the transmitted electric field of this lens simulated by CST is shown in Figure 1. Figure 6a As shown in Figure 1, large discontinuities in the wavefront can be observed, as well as unfocused transmitted energy pointing in different directions (marked by red arrows). The local phase response of the meta-atom in the initial design is shown in Figure 1. Figure 6b , where the phase error of each meta-atom is also calculated and plotted. At some points, the phase error between the target and the local response is close to 100 degrees, which leads to phase discontinuities. This initial design is then used as the starting point for optimization using the proposed network and the same objective function as in Eq. After 500 optimization iterations, the local phase response of the optimized meta-atom is shown in Figure 6e As shown, it is closer to the target with an average phase error.

[0066] Figures 7a-7d This is the optimized data of the large metal composed of 200 meta-atoms described in the embodiment of the present invention. Figure 7a The target phase (blue), local phase (red), and phase error (green) of each meta-atom in the initial design are shown. Figure 7b An optimized design is shown. Figure 7c Shown are the simulated electric fields along the optical (Z) axis for the initial design (red) and the optimized design (blue) with x-polarized incidence. The inset includes a zoomed-in view of the e-field near the focus. Figure 7d The simulated electric field (real part) of the optimized design and the enlarged view of the electric field near the focus are shown.

[0067] Figure 8a Schematic diagram showing free-form meta-atoms composed of a high refractive index material on a low refractive index substrate. Figure 8b Amplitude and phase MAEs are shown for a specified number of different adjacent meta-atoms per side, N. N = 0 represents the results obtained with periodic boundaries. N = 2 was used during data collection. Figure 8a This mutual coupling analysis is extended to another metasurface platform, which is fabricated by forming a 1 μm thick film of a high-refractive-index dielectric material (n=5) on a low-refractive-index dielectric material (n=1.4). The period is set to 2.8 × 2.8 μm2, while the spectrum of interest is 5.45 μm. In this case, to introduce more degrees of freedom, the 2D cross section of each meta-atom is defined by a complex image with a resolution of 64 by 64 pixels. Figure 8bAs shown, high-index meta-atoms can confine electromagnetic fields better than low-index meta-atoms and are therefore less prone to mutual coupling.

[0068] Figure 9a-9b The deep learning curves and results described in the embodiments of the present invention are shown. Figure 9a The average MSE of the training set (red curve) and the test set (blue curve) during the training process is shown. Figure 9b Six randomly selected results from the demo test set are shown. During data collection, only the influence of 2 neighbors on each side was considered, which provided the same level of accuracy as the previous setting in the figure. Again, 10,000 sets of data were collected to assemble the training dataset, and the network architecture was slightly modified to accommodate the new input dimension (320 by 64). When training is complete, as shown in Figure 9a As shown, the average MSE of the training set is 2e -5 , the average MSE of the test set is 6e -5 .like Figure 9b As shown, 6 results are randomly selected from the test dataset to show the accuracy of the network.

[0069] Figure 10a-10c Shown are the results of free-form meta-atom simulations constructed with high-refractive-index materials. Fig.10a The periodic phase and amplitude responses of 400 meta-atom designs generated by the fully trained GAN model are shown, along with the MSE and efficiency of a 1000 beam deflector consisting of 1000 meta-atoms. The phase targets for each set of 100 meta-atoms are 45° (green), 135° (blue), 225° (cyan), and 315° (red). Several selected examples from each set are shown as insets. Fig.10a To the right of is the MSE and the efficiency of a 1000-beam deflector consisting of 1000 meta-atoms, which are Fig.10a Top view of several deflection plates, one randomly selected from each group of atoms for demonstration purposes, on the left. Fig.10b The local responses corresponding to all meta-atoms in the beam deflector are shown. Fig.10c Shows from Fig.10a The simulated electric fields of the two deflectors selected in , and the local response of each meta-atom in these two designs. It is shown that compared with the simple rectangular meta-atom design, the free-form meta-atom assembly, such as Fig.10c As shown, it is possible to achieve a higher efficiency (86%). At the same time, more complex global optimization methods, such as evolutionary strategies, will be needed to handle the huge design freedom brought by the free-form shaped meta-atoms.

[0070] The above are only some embodiments of the present invention. For those skilled in the art, several modifications and improvements can be made without departing from the creative concept of the present invention, which all belong to the protection scope of the present invention.

Claims

1. A method for optimizing the mutual coupling effect of supersurfaces based on deep convolutional neural networks, characterized in that: The following steps are involved: Step S1: assembling a training dataset by calculating the actual phase and amplitude responses of randomly generated meta-atoms placed between different random neighboring atoms; Step S2: constructing a prediction neural network based on a convolutional neural network architecture to achieve rapid prediction of the local response of meta-atoms under different boundary conditions; Step S3: Minimize the mean square error between the local response predicted by the prediction neural network and the label by inversely adjusting the parameters in the hidden neural layer; Step S4: Starting from the optimized beam deflector, the performance of the optimized design is compared with the performance of the initial design to verify the effectiveness of the optimized design.

2. The method for optimizing the mutual coupling effect of a supersurface based on a deep convolutional neural network according to claim 1, characterized in that: The training data set includes actual phase and amplitude response data obtained by simulating a model in which four randomly generated different adjacent meta-atoms are arranged on both sides of a target meta-atom.

3. The method for optimizing the mutual coupling effect of a supersurface based on a deep convolutional neural network according to claim 1, characterized in that: In step S2, the two-dimensional cross-section of the target meta-atom and its neighboring atoms is subjected to six consecutive layers of convolution and pooling, and then converted into a one-dimensional vector. During this period, the hidden features and the influence of different neighboring atoms are extracted and calculated. The output of the CNN is flattened into a one-dimensional vector and then passed through three fully connected layers to generate the prediction results of the complex transmission coefficient of the target meta-atom. In the entire network, except for the last layer without an activation function, the ReLU activation function is applied to each layer.

4. The method for optimizing the mutual coupling effect of a supersurface based on a deep convolutional neural network according to claim 1, characterized in that: In step S3, the collected training data is randomly divided into a training set and a test set, which contain 70% and 30% of the total training data respectively. During training, the local responses predicted by the PNN are compared with the labels and the mean squared error is calculated, which is minimized by inversely adjusting the parameters in the hidden neural layers.

5. The method for optimizing the mutual coupling effect of a supersurface based on a deep convolutional neural network according to claim 1, characterized in that: In step S4, the beam deflector composed of 1×4 meta-atomic elements is optimized, and the phase targets are set to 60, 150, 240, and 330 degrees, respectively, and the amplitude target is set to 1. The optimization starts with the initial design and minimizes the objective function, which is defined as: Where N is the number of atoms in the device, and A i is the local phase and amplitude response and are the target phase and amplitude responses, respectively.

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