Metasurface lens structure optimization method

Abstract

The invention discloses a metasurface lens structure optimization method, and belongs to the technical field of optical imaging. The method comprises the following steps: firstly, acquiring a metasurface lens structure data set and an electromagnetic response data set corresponding to the metasurface lens structure data set, constructing a Fourier neural operator network model, and then training the Fourier neural operator model by using the metasurface lens structure data set and the electromagnetic response data set. The Fourier neural operator network model establishes a mapping relation between a metasurface lens structure data set and an electromagnetic response data set, so that the Fourier neural operator network model replaces a traditional Maxwell solver to calculate electromagnetic response and perform gradient descent iteration; and generating metasurface lens structure data corresponding to the design target. The calculation speed of the electromagnetic response can be remarkably improved through the Fourier neural operator network, rapid convergence can be achieved through gradient descent, and therefore the efficiency of optimizing the metasurface lens structure can be remarkably improved through the scheme.

Description

Technical Field

[0001] The present invention relates to the technical field of optical imaging, and particularly to a method for optimizing the structure of a metasurface lens. Background Art

[0002] A metasurface is an artificially designed two-dimensional material whose structural unit size is smaller than the wavelength of light and is composed of sub-wavelength structural units arranged periodically or aperiodically. These structural units can interact with the incident electromagnetic wave to control the reflection, refraction, polarization, phase, absorption, etc. of the electromagnetic wave. Compared with traditional optical elements formed by traditional optical processing of natural materials such as CNC and grinding and polishing, the metasurface has many advantages, such as an ultra-thin thickness, light weight, easy integration, and a flexible control ability for electromagnetic waves. The optimization of the metasurface lens structure refers to optimizing the height, structural unit size, and other characteristics of the metasurface lens to make it have better optical properties. However, analyzing the propagation of light in sub-wavelength structures requires solving Maxwell's equations, and obtaining an accurate electromagnetic response takes a lot of time. This makes it very difficult to design and optimize metasurface structures.

[0003] The existing methods for optimizing the metasurface lens structure can be roughly divided into two categories:

[0004] (1) First, calculate the correspondence between the structure and electromagnetic response of different structural units to obtain information such as the reflection, refraction, and phase of a single structural unit, and then use methods similar to the iterative Fourier transform algorithm to obtain the desired phase information, and pair it through the previously obtained structure-electromagnetic response relationship to finally obtain a suitable metasurface structure.

[0005] (2) Regard the metasurface as a whole for electromagnetic simulation, and use an evolutionary algorithm to optimize and iterate the metasurface structure. The specific process is as follows:

[0006] a. Generate a set of initial structure data D0, and use electromagnetic simulation algorithms such as RCWA (Rigorous Coupled Wave Analysis) or FDTD (Finite-Difference Time-Domain) to obtain the electromagnetic response E0 of the initial structure parameter design D0;

[0007] b. Use evolutionary algorithms such as particle swarm optimization and genetic algorithms to perform iterative optimization on the structure data D i based on E i to obtain new structure data D i+1 ;

[0008] c. Use electromagnetic simulation algorithms such as RCWA / FDTD to obtain the electromagnetic response E of the new structure data D i+1 ofi+1 ;

[0009] d. Repeat steps b - c until the structural data that meets the design objectives is obtained.

[0010] Both of the above two methods need to solve Maxwell's equations during the calculation of electromagnetic response. Since the first method only calculates the electromagnetic response of a single structural unit, it cannot consider the perturbations of adjacent structural units, and the spliced electromagnetic simulation structure cannot accurately reflect the optical properties of the real world. Therefore, this method has low accuracy and poor optimization effect for the metasurface lens with complex structures. The second method calculates the electromagnetic response of the entire metasurface, so the optimization result is based on an accurate electromagnetic simulation algorithm. However, due to the use of an evolutionary algorithm, multiple structures need to be iteratively optimized. And because solving Maxwell's equations for large-scale metasurfaces is very slow, this method requires a lot of time. Summary of the Invention

[0011] The technical problem to be solved by the present invention is: aiming at the above-mentioned defects of the prior art, to provide a method for optimizing the structure of a metasurface lens, so as to improve the problems of slow calculation and low efficiency of traditional optimization methods.

[0012] To achieve the above object, the present invention provides a method for optimizing the structure of a metasurface lens, the method comprising the following steps:

[0013] Step S1, obtaining a metasurface lens structure data set and the corresponding electromagnetic response data set;

[0014] Step S2, constructing a Fourier neural operator network model, and training the model using the metasurface lens structure data set and the electromagnetic response data set, and fixing the model parameters;

[0015] Step S3, according to the evaluation function corresponding to the design objective, using the Fourier neural operator network model equipped with the fixed model parameters to calculate the electromagnetic response and perform gradient descent iteration to generate the metasurface lens structure data corresponding to the design objective.

[0016] In the method for optimizing the structure of a metasurface lens provided by the present invention, the specific method of step S2 is:

[0017] Step S21: Construct a Fourier neural operator network model, where the Fourier neural operator network model includes a feature extraction network and multiple Fourier layers connected in series to the output end of the feature extraction network in sequence; the feature extraction network is composed of a residual network and a convolutional network, and each Fourier layer includes a Fourier transform module, a convolutional module, an inverse Fourier transform module, a linear transformation module, an activation function, and a Fourier kernel integral operator; the feature extraction network extracts a metasurface lens structure feature matrix from the metasurface lens structure data, and then restores it to an electromagnetic field simulation result after serial calculation through the multiple Fourier layers; the Fourier kernel integral operator is used to perform a convolution operation with the Fourier transform result of the metasurface lens structure feature matrix.

[0018] Step S22: Use the metasurface lens structure data set and the corresponding electromagnetic response data set to train the Fourier neural operator network model.

[0019] Step S23: Verify the Fourier neural operator network model.

[0020] In the metasurface lens structure optimization method provided by the present invention, the specific method of step S3 is as follows:

[0021] Step S31: Initialize the metasurface lens structure data D0.

[0022] Step S32: For the given metasurface lens structure data D i , use the Fourier neural operator network model equipped with the fixed model parameters to calculate the electric field response data E i ;

[0023] Step S33: Calculate the loss function according to the design objective evaluation function. If the loss function converges or the number of iterations exceeds the maximum number, then exit. Otherwise, use the neural network backpropagation mechanism to calculate the differential ΔD of the metasurface lens structure data i ;

[0024] Step S34: Update the metasurface lens structure data D i+1 = D i - αΔD i , and jump to step S32, where α is the learning rate.

[0025] In the metasurface lens structure optimization method provided by the present invention, in step S22, the output of the Fourier neural operator network model is the Fourier space value of the predicted electric field, and the loss function is the L2 space distance between the predicted value and the true value; in step S23, the output of the Fourier neural operator network model is the real space value of the predicted electric field, and the loss function is the L1 space distance between the predicted value and the true value.

[0026] In the method for optimizing the metasurface lens structure provided by the present invention, the parameter optimization range of the Fourier kernel integral operator is restricted within an elliptical region.

[0027] In the method for optimizing the metasurface lens structure provided by the present invention, the metasurface is a periodic symmetric structure, and the parameter optimization range of the Fourier operator kernel is restricted within a quadrant of an elliptical region.

[0028] In the method for optimizing the metasurface lens structure provided by the present invention, the specific method of step S1 is as follows:

[0029] Step S11, initialize the metasurface lens structure D10;

[0030] Step S12, for a given metasurface lens structure D1 i , use the rigorous coupled-wave analysis method to calculate the electromagnetic response E1 in the forward direction i ; record the metasurface lens structure D1 i and the electromagnetic response E1 i ;

[0031] Step S13, calculate the loss function according to the design objective evaluation function. If the loss function converges or the number of iterations exceeds the maximum number, output the recorded metasurface lens structure D1 i and the electromagnetic response E1 i as training data and exit. Otherwise, calculate the differential ΔD1 of the metasurface lens structure according to the differential approximation equation i ;

[0032] Step S14, update the metasurface lens structure D1 i+1 = D1 i - α1ΔD1 i , and jump to step S12, where α1 is the learning rate.

[0033] In the method for optimizing the metasurface lens structure provided by the present invention, the differential approximation equation is: the differential approximation equation obtained by decomposing the characteristic matrix in the rigorous coupled-wave analysis method through the Wirtinger integral and the regularization term.

[0034] 10. In the method for optimizing the metasurface lens structure provided by the present invention, the generation process of the differential approximation equation is as follows:

[0035] Given an eigenvalue decomposition problem: ΦΛ = ΜΦ

[0036] M is the input matrix, and Λ and Φ are the eigenvalues and eigenvectors;

[0037] Define the differentiable eigen - decomposition operator as \(T:\mathcal{T}(M)=\Lambda,\Phi\), and the coupled - wave analysis operation based on the eigen - decomposition result \((T)\) as \(f:f(\Lambda,\Phi,\cdots)=E\). That is, through the eigen - decomposition results \(\Lambda,\Phi\) and other differentiable intermediate steps, the target electromagnetic response \(E\) can be finally obtained. The differential equation of the eigen - decomposition operator can be obtained by the chain rule as follows:

[0038]

[0039] where \(\epsilon\) is a regularization term used to handle the case where \(M\) is a defective matrix.

[0040] In the method for optimizing the metasurface lens structure provided by the present invention, the metasurface is the metasurface of a laser beam splitter, and the evaluation function includes the zero - order diffraction intensity and the diffraction spot uniformity.

[0041] The present invention has the following beneficial effects: First, the present invention obtains a metasurface lens structure data set and the corresponding electromagnetic response data set, constructs a Fourier neural operator network model, and then uses the metasurface lens structure data set and the electromagnetic response data set to train the Fourier neural operator model, so that the Fourier neural operator network model establishes a mapping relationship between the metasurface lens structure data set and the electromagnetic response data set. Thus, the Fourier neural operator network model is used to replace the traditional Maxwell solver to calculate the electromagnetic response and perform gradient - descent iteration to generate metasurface lens structure data corresponding to the design target. The calculation speed of the electromagnetic response can be significantly improved through the Fourier neural operator network, and rapid convergence can be achieved through gradient descent. Therefore, this solution can significantly improve the efficiency of optimizing the metasurface lens structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] The drawings described herein are used to provide a further understanding of the present invention, and constitute a part of the present invention. The schematic embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0043] Figure 1 is a schematic structural diagram of a three - dimensional metasurface lens in some embodiments of the present invention.

[0044] Figure 2 is a schematic diagram of the steps of the method for optimizing the metasurface lens structure provided by the embodiments of the present invention.

[0045] Figure 3 is a structural diagram of the Fourier neural operator network model provided by the embodiments of the present invention.

[0046] Figure 4 is Figure 3 a structural diagram of the feature extraction network in

[0047] Figure 5 isFigure 3 Structural diagram of a single Fourier layer in

[0048] Figure 6 Schematic diagram of the parameter optimization range of the common Fourier kernel integral operator K.

[0049] Figure 7 Schematic diagram of the elliptical Fourier kernel integral operator K provided by an embodiment of the present invention.

[0050] Figure 8 Schematic diagram of the 1 / 4 elliptical Fourier kernel integral operator K provided by an embodiment of the present invention.

[0051] Figures 9 - 10 Comparison chart of the training and validation errors of different Fourier kernel integral operators K.

[0052] Figure 11 Schematic diagram of the differentiable coupled-wave analysis method provided by an embodiment of the present invention.

[0053] Figure 12 Comparison of the time required for one iteration of the differentiable coupled-wave analysis method using an RTX2000 Ada GPU and an i7-13700H CPU.

[0054] Figure 13 Schematic diagram of using the differentiable coupled-wave analysis method in combination with a Fourier neural operator network.

[0055] Figure 14 Comparison of the electric field strength simulation results obtained using the traditional RCWA algorithm and the neural network algorithm of this application is shown.

[0056] Figure 15 Schematic diagram of the effect of parallel optimization of multiple initial designs. Detailed implementation manners

[0057] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0058] The general idea of the present invention is as follows: construct a Fourier neural operator network model including a feature extraction network and multiple cascaded Fourier layers, obtain a dataset of metasurface lens structures and their corresponding electromagnetic responses, use the dataset to train the Fourier neural operator network model to replace the traditional Maxwell solver, and then based on the evaluation function corresponding to the design target, use the trained Fourier neural operator network model to calculate the electromagnetic response and perform gradient descent iteration, finally generating metasurface lens structure data corresponding to the design target, so as to realize the optimization of the metasurface lens structure.

[0059] The embodiments of the present invention will be further described in detail below with reference to the accompanying drawings of the specification. It should be understood that the embodiments described herein are only used to illustrate and explain the present invention, and are not used to limit the present invention.

[0060] As Figure 1 shown is a schematic structural diagram of a three-dimensional metasurface lens according to some embodiments of the present invention. The metasurface lens includes a substrate and a plurality of structural units located on one surface of the substrate. The size of the structural units is smaller than the wavelength of light, and the plurality of structural units are periodically arranged on the surface of the substrate. The specific form, structure, and distribution position of the substrate and the structural units are designed according to specific applications, and are not limited to Figure 1 the structure and form shown. The metasurface lens structure data described below is a digital representation of the metasurface lens structure features.

[0061] The technical solution of the present invention can be widely applied to the design and optimization of metasurface lenses in different application scenarios, including but not limited to optical imaging, optical sensing, beam shaping, optical communication, etc. By simply modifying the evaluation function, the design and optimization of metasurface lenses with different application requirements can be achieved. The design target of the metasurface lens is the optical characteristics that the metasurface lens equipped with the target design parameters should exhibit, and the optical characteristics vary according to the application scenario. In some embodiments of the present invention, the design target includes realizing laser beam splitting. In some other embodiments of the present invention, the design target includes focusing at one or more specific positions. In some other embodiments of the present invention, the design target includes forming a specific light field or electromagnetic field.

[0062] Hereinafter, taking the metasurface lens as a diffractive laser beam splitter as an example, its structure optimization method will be described in detail.

[0063] As Figure 2 shown, the embodiments of the present invention provide a metasurface lens structure optimization method, characterized in that the method includes the following steps:

[0064] Step S1, obtain a metasurface lens structure dataset and its corresponding electromagnetic response dataset.

[0065] In some embodiments of the present invention, the dataset of the metasurface lens structure and the corresponding electromagnetic response dataset are obtained through a traditional Maxwell solver. Specifically, the dataset of the metasurface lens structure and the corresponding electromagnetic response dataset can be generated by simulation software.

[0066] Step S2: Construct a Fourier neural operator network model, and use the dataset of the metasurface lens structure and the electromagnetic response dataset to train the model, and fix the model parameters.

[0067] PINN (Physics-Informed Neural Networks) is an attempt to use neural networks to replace traditional partial differential equation solvers recently. Among them, the Fourier neural operator network (FNO) uses the Fourier transform to replace the convolution operation to learn the operator mapping between function spaces, and is naturally suitable for solving high-resolution problems with periodic boundary conditions. An embodiment of the present invention proposes a method for optimizing the metasurface structure by using a Fourier neural operator network to replace the traditional Maxwell solver, which improves the accuracy of structure optimization.

[0068] In an embodiment of the present invention, the Fourier neural operator model is trained using the dataset of the metasurface lens structure and the electromagnetic response dataset, so that the Fourier neural operator network model establishes a mapping relationship between the dataset of the metasurface lens structure and the electromagnetic response dataset, thereby using the Fourier neural operator network model to replace the traditional Maxwell solver for electromagnetic simulation to accelerate the metasurface lens structure optimization process. Compared with the strict electromagnetic simulation algorithm that takes 30 seconds to 10 minutes, the electromagnetic simulation using the Fourier neural operator network can be completed within 1 second, and the efficiency is greatly improved. In addition, taking advantage of the characteristic that the Fourier neural operator network can perform parallel operations on a single GPU, multiple candidate structures can be optimized simultaneously to achieve the effect of global optimization and avoid falling into local optimal solutions.

[0069] In an embodiment of the present invention, the specific method of the step S2 is as follows:

[0070] Step S21, construct a Fourier neural operator network model, where the Fourier neural operator network model includes a feature extraction network and a plurality of Fourier layers connected in series in sequence at the output end of the feature extraction network; the feature extraction network is composed of a residual network and a convolutional network, and each Fourier layer includes a Fourier transform module, a convolutional module, an inverse Fourier transform module, a linear transformation module, an activation function, and a Fourier kernel integral operator; the feature extraction network extracts a metasurface lens structure feature matrix from the metasurface lens structure data, and then restores it to an electromagnetic field simulation result after serial calculation through the plurality of Fourier layers; the Fourier kernel integral operator is used to perform a convolution operation with the Fourier transform result of the metasurface lens structure feature matrix.

[0071] As Figure 3 shown is the structural diagram of the Fourier neural operator network model provided by the embodiment of the present invention. The metasurface structure serves as the input of the feature extraction network, and the output of the feature extraction network is connected to the first Fourier layer. After the plurality of Fourier layers are connected in series, electromagnetic field data is obtained. A new metasurface structure is obtained based on the electromagnetic field data and enters the next iteration. The embodiment of the present invention has no special limitation on the specific number of Fourier layers.

[0072] As Figure 4 shown is the network structure of the feature extraction network, including a plurality of convolutional layers, a normalization layer, an activation function, a pooling layer, and a plurality of residual blocks mixed between the layers. After the structure matrix of the metasurface lens passes through the feature extraction network, a feature matrix is obtained. Figure 4 This is a preferred embodiment of the present invention, and the present invention is not limited thereto.

[0073] As Figure 5 shown is the structure of a single Fourier layer. There are two paths in a single Fourier layer. In the first path, the metasurface lens structure feature matrix is subjected to a Fourier transform to obtain a frequency-domain feature matrix. The frequency-domain feature matrix performs a convolution operation with the Fourier kernel integral operator K in the Fourier space, and then an inverse Fourier transform is performed to obtain a first real-space eigenvector; in the second path, the metasurface lens structure feature matrix is linearly transformed in the real space using a linear transformation vector W to obtain a second real-space eigenvector. Finally, the first real-space eigenvector and the second real-space eigenvector are added and then processed through an activation function to obtain the output of a single Fourier layer. Assume that the input and output of a single Fourier layer are v i and v i+1 , and the mathematical principle of a single Fourier layer is as follows:

[0074] v i+1 (x) = σ(W·v i (x) + (Γ(K)v i )(x))

[0075] Among them, x is the specific position in the feature vector space, σ is the activation function, and in the embodiments of the present invention, the sigmoid equation is taken. W is the trainable linear transformation vector, K is the trainable Fourier kernel integral operator, and Γ(K)v i = F -1 (K·F(v i ))), where F and F -1 are the Fourier transform and the inverse Fourier transform respectively.

[0076] The design of the Fourier layer ensures the periodicity of the electromagnetic simulation, that is, when a periodic structure is input, a periodic electromagnetic field is surely output, making it easier for the neural network to converge. In addition, the Fourier kernel integral operator K only retains low-frequency information and filters out high-frequency information, thereby playing a role in filtering noise. As Figure 6 shown in the figure is a schematic diagram of the parameter optimization range of the common Fourier kernel integral operator K. It is a rectangular area, the middle white area is the trainable parameter area, and the black area is the high-frequency filtering area. Since the frequency components in the diagonal direction of the rectangular low-pass filter decay slowly, some unnatural ringing and checkerboard effects will appear in the simulation results. Therefore, in the embodiments of the present invention, as Figure 7 shown in the figure, the parameter optimization range of the Fourier kernel integral operator K is restricted within an elliptical area, improving the accuracy of the electromagnetic simulation results, further ensuring the double rotational symmetry and reflection symmetry of the neural network simulation results, and enabling the optical element to obtain better optical performance.

[0077] The symmetric Fourier kernel integral operator K satisfies K i,j = K -i,j = K i,-j = K -i,-j , and the anti-symmetric Fourier kernel integral operator K satisfies K i,j = -K -i,j = -K i,-j = K -i,-j , where i and j are the abscissa and ordinate in the coordinate system as Figures 6 - 8 shown in the figure respectively. From the above formulas, it can be seen that the symmetric Fourier kernel integral operator and the anti-symmetric Fourier kernel integral operator K are only defined in one quadrant where i≥0 and j≥0, and the values in the remaining quadrants can be deduced. Therefore, only 1 / 4 of the area can be trained to reduce the number of training parameters and improve the training efficiency.

[0078] For special applications such as diffractive laser beam splitters, users usually hope to obtain a symmetric structure. For the incident light perpendicular to the plane, the symmetric structure generates symmetric or anti-symmetric electromagnetic responses. Therefore, as Figure 8 shown in the figure, in some embodiments of the present invention, the metasurface is a periodic symmetric structure, and the parameter optimization range of the Fourier kernel integral operator K is restricted within one quadrant of an elliptical area.

[0079] Step S22: Train the Fourier neural operator network model using the metasurface lens structure dataset and the corresponding electromagnetic response dataset.

[0080] In an embodiment of the present invention, in the Fourier layer, the Fourier kernel integral operator K and the linear transformation vector W are trainable parameters; the Fourier kernel integral operators K and the linear transformation vectors W in the multiple Fourier layers are independent of each other. During the model training process, the Fourier kernel integral operator K and the linear transformation vector W are adaptively adjusted.

[0081] In an embodiment of the present invention, in step S22, the output of the Fourier neural operator network model is the Fourier space value of the predicted electric field, and the loss function is the L2 space distance between the predicted value and the true value. The L2 space distance is |x1 - x2|^2, which is differentiable everywhere and has a smooth curve, making it easy to train. Therefore, it is used as the calculation method of the loss function during training. However, the present invention does not limit the specific implementation of the loss function during training.

[0082] Step S23: Verify the Fourier neural operator network model.

[0083] In an embodiment of the present invention, in step S23, the output of the Fourier neural operator network model is the real space value of the predicted electric field, and the loss function is the L1 space distance between the predicted value and the true value. The L1 space distance is |x1 - x2|, which is more intuitive. Therefore, it is used as the calculation method of the loss function during the verification phase. However, the present invention does not limit the specific implementation of the loss function during verification.

[0084] As Figures 9 - 10 shown is the comparison of the training and verification processes of the 1 / 4 elliptical Fourier kernel integral operator K as Figure 8 shown and the rectangular Fourier kernel integral operator K as Figure 6 shown. The "symmetric / elliptical calculation kernel" in the legend corresponds to the 1 / 4 elliptical Fourier kernel integral operator K in Figure 8 , and the "random / square calculation kernel" corresponds to the rectangular Fourier kernel integral operator K in Figure 6 . The training set contains 1900 pairs of structure - electric field data, and the test set contains 100 pairs of structure - electric field data. In the training phase, the neural network output is the Fourier space value of the predicted electric field, and the loss function is the L2 space distance between the predicted value and the true value. In the verification phase, we perform an inverse Fourier transform on the Fourier electric field output by the network to obtain the electric field representation in the real space. The loss function in the verification phase is the L1 space distance between the electric field predicted value and the true value. From the data, it can be seen that using the 1 / 4 elliptical Fourier kernel integral operator K can enable the neural network to obtain better convergence performance and perform better on the test set.

[0085] Step S3: According to the evaluation function corresponding to the design goal, use the Fourier neural operator network model with the fixed model parameters to calculate the electromagnetic response and perform gradient descent iteration to generate the metasurface lens structure data corresponding to the design goal.

[0086] The evaluation function is a quantification of the optical properties of the metasurface. Different evaluation functions can be defined for different metasurface design goals. In the embodiments of the present invention, for the application of a diffractive laser beam splitter, the zero-order diffraction intensity and the diffraction spot uniformity are selected as the evaluation functions, and the optimization goal is to improve the diffraction spot uniformity as much as possible and reduce the zero-order diffraction intensity. The zero-order diffraction intensity refers to the light intensity at the center position in the diffraction pattern. It is the result of the concentrated superposition of all the non-diffracted light rays and is usually the brightest point in the diffraction pattern. When light waves or other waves encounter obstacles, such as slits, gratings, or crystals, diffraction occurs, and the diffracted light waves form a specific pattern called the diffraction pattern. This pattern consists of a series of bright and dark fringes or spots, and their positions and intensities depend on the structure of the obstacle and the wavelength of the incident wave. The zero-order diffraction corresponds to the non-diffracted light rays, which directly pass through the obstacle or are reflected without changing their propagation direction. In the diffraction pattern, these light rays converge at the center position to form the brightest spot or fringe, which is the zero-order diffraction spot. The formula for the zero-order diffraction intensity is as follows, where Eff is the intensity of each diffraction spot:

[0087]

[0088] The zero-order outgoing light is the light with an outgoing direction parallel to the incident direction, and the zero-order diffraction intensity is the ratio of the energy of the zero-order outgoing light to the average energy of the outgoing light at all angles.

[0089] The formula for the diffraction spot uniformity is as follows:

[0090]

[0091] In the embodiments of the present invention, step S3 includes:

[0092] Step S31: Initialize the metasurface lens structure data D0;

[0093] Step S32: For the given metasurface lens structure data D i , use the Fourier neural operator network model with the fixed model parameters to forward calculate the electric field response data E i ;

[0094] Step S33: Calculate the loss function according to the design objective evaluation function. If the loss function converges or the number of iterations exceeds the maximum number, then exit. Otherwise, use the neural network backpropagation mechanism to calculate the differential ΔD of the metasurface lens structure data i ;

[0095] Step S34: Update the metasurface lens structure data D i+1 = D i - αΔD i , and jump to step S32, where α is the learning rate.

[0096] In the embodiments of the present invention, the neural network and differential calculation are implemented with the aid of the modern neural network framework Pytorch.

[0097] For the application of a diffractive laser beam splitter, the loss function can be a weighted sum of the zero-order diffraction intensity and the diffraction spot uniformity. However, the present invention is not limited thereto. For example, the loss function can also be changed to increase the zero-order diffraction intensity to achieve the design of a metalens for focusing function.

[0098] As Figure 11 shown, in some other embodiments of the present invention, in step S1, the differentiable coupled-wave analysis method is used to obtain the metasurface lens structure data set and the corresponding electromagnetic response data set. The specific method is as follows:

[0099] Step S11: Initialize the metasurface lens structure D10;

[0100] Step S12: For a given metasurface lens structure D1 i , use the rigorous coupled-wave analysis method to forward calculate the electromagnetic response E1 i ; Record the metasurface lens structure D1 i and the electromagnetic response E1 i . In step S12, when using the rigorous coupled-wave analysis method to calculate the electromagnetic response, the characteristic matrix decomposition adopts a rigorous numerical solution to ensure the accuracy of the calculation results. Record the input and output of the rigorous coupled-wave analysis method to be used as the training data of the Fourier neural operator network, thereby improving the accuracy of the output of the Fourier neural operator network.

[0101] The Rigorous Coupled Wave Analysis (RCWA), also known as the Fourier Modal Method (FMM), is a very effective tool for dealing with electromagnetic field problems of periodic structures such as diffraction gratings. This method expands the electromagnetic field and the dielectric constant of the material into Fourier series, and then solves the Maxwell equations by solving the eigenvalue and eigenvector problems of the matrix. Compared with other numerical calculation methods, RCWA has a fast calculation speed and high accuracy. Given the material, structure, and incident electric field information of the metasurface, RCWA can calculate the reflected and transmitted electric fields, thereby evaluating the optical properties of the given metasurface. This method is commonly used in the forward optimization process of metasurfaces, such as genetic algorithms and particle swarm optimization.

[0102] Step S13, calculate the loss function according to the design objective evaluation function. If the loss function converges or the number of iterations exceeds the maximum number, output the recorded metasurface lens structure D1 i and the electromagnetic response E1 i as training data and exit. Otherwise, calculate the differential ΔD1 of the metasurface lens structure according to the differential approximation equation i ;

[0103] The differential approximation equation is: the differential approximation equation obtained by decomposing and differentiating the characteristic matrix in the Rigorous Coupled Wave Analysis through the Wirtinger integral and the regularization term.

[0104] The reason why the Rigorous Coupled Wave Analysis requires differentiating the eigenvalue decomposition is as follows:

[0105] The core of the Rigorous Coupled Wave Analysis is to expand the electromagnetic field and the periodic structure of the medium into a series of Fourier components through Fourier expansion in order to analyze the Maxwell equations in the frequency domain. The specific steps are as follows:

[0106] 1. Fourier expansion: Represent the dielectric constant and electromagnetic field of the periodic structure as Fourier series.

[0107] 2. Solve the Maxwell equations: In each layer, convert the Maxwell equations into matrix form.

[0108] 3. Intra-layer mode analysis: Obtain the propagation modes inside the dielectric layer through eigenvalue decomposition.

[0109] 4. Boundary condition matching: Match the continuity conditions of the electromagnetic field between layers.

[0110] In the above steps, the eigenvalue has no stable differential in the complex domain. Therefore, in the embodiments of the present invention, the characteristic matrix in the rigorous coupled-wave analysis method is decomposed and differentiated through the Wirtinger integral and the regularization term to obtain a differential approximation equation. The specific calculation principle is as follows:

[0111] Given an eigenvalue decomposition problem: ΦΛ = ΜΦ

[0112] M is the input matrix, Λ and Φ are the eigenvalues and eigenvectors;

[0113] Define the differentiable eigenvalue decomposition operator as T: Τ(Μ) = Λ; Φ, and the coupled-wave analysis operation based on the eigenvalue decomposition result (T) as f: f(Λ, Φ,...) = E, that is, through the eigenvalue decomposition results Λ, Φ and other differentiable intermediate steps, the target electromagnetic response E can be finally obtained; the differential equation of the eigenvalue decomposition operator can be obtained through the chain rule as:

[0114]

[0115] where ∈ is the regularization term used to handle the case where M is a defective matrix. ∈ can take a very small number. In the embodiments of the present invention, ∈ = 1e -8 .

[0116] Compared with forward optimization, gradient descent uses the differential information of the objective function to directly iterate towards the optimal direction, so its convergence speed is much faster than that of the forward optimization algorithm. In the embodiments of the present invention, the solution process of the Maxwell equation is differentiated to implement the optimization process of gradient descent. In the embodiments of the present invention, the technique of Wirtinger calculus is used to extend the eigenvalue decomposition formula for the non-defective real domain to the non-defective complex domain, and a regularization term is introduced to make it adapt to the defective matrix and strengthen its numerical stability, so that the solution process of the Maxwell equation can be differentiated, and thus the gradient descent method can be used to iteratively optimize the metasurface lens structure.

[0117] For the application of a diffractive laser beam splitter, the loss function can also be the weighted sum of the zero-order diffraction intensity and the diffraction spot uniformity, but the present invention is not limited thereto. For example, the loss function can also be changed to increase the zero-order diffraction intensity to realize the design of a superlens with a focusing function.

[0118] Step S14, update the metasurface lens structure D1 i+1 = D1 i -α1ΔD1 i , and jump to step S12, where α1 is the learning rate.

[0119] Specifically, in the embodiments of the present invention, the differentiable coupled-wave analysis method is implemented by means of the modern neural network framework Pytorch, realizing a fast and accurate differentiable Maxwell solver, and while generating training data, it also realizes the optimization of the initial structure data. In addition to being used as a framework, Pytorch also realizes the functions of GPU call and matrix operation differentiation.

[0120] Furthermore, in the embodiments of the present invention, an Nvidia GPU is used to accelerate the calculation process of the differentiable coupled-wave analysis method. The Nvidia GPU has obvious speed advantages in matrix operations. For example, Figure 12 as shown in the comparison of the time required for one iteration of the differentiable coupled-wave analysis method using an RTX2000 Ada (Laptop) GPU and an i7-13700H CPU. After using the Nvidia GPU, a speed increase of more than 5 times is achieved.

[0121] The differentiable coupled-wave analysis method of the embodiments of the present invention can also be independently used for the optimization of the metasurface lens structure. It uses the rigorous coupled-wave analysis method to accurately calculate the electromagnetic response during the forward propagation process, and obtains an approximate solution of the differential result of the characteristic matrix through a differential approximation equation during the reverse propagation process, and uses it as the gradient value in the gradient descent iteration. Compared with the iterative optimization through the evolutionary algorithm mentioned in the background technology, its calculation efficiency is also greatly improved.

[0122] For the application of a laser beam splitter, the traditional rigorous coupled-wave analysis method takes 10 minutes for one calculation, 2 minutes after GPU acceleration, and only 1 second using the Fourier neural operator network. The speed of metasurface structure optimization is increased by at least a hundred times.

[0123] For example, Figure 13 as shown in the schematic diagram of using the differentiable coupled-wave analysis method in combination with the Fourier neural operator network. In the first stage, the differentiable coupled-wave analysis method is used to optimize the metasurface lens structure using the gradient descent method while calculating the solution of the rigorous Maxwell equation. The results of the coupled-wave analysis will be stored as training data to train the Fourier neural operator network. In the second stage, the Fourier neural operator network is used instead of the traditional Maxwell solver to implement the gradient descent optimization algorithm. The neural network has the characteristics of fast speed and parallelism, and can optimize different structures simultaneously, thus realizing global optimization. In addition, the advantage of neural network parallelization can simultaneously optimize the target design and the redundant design with added random noise, and can realize the optimization and improvement of the tolerance of the metasurface processing error in the design.

[0124] Figure 14The figure shows a comparison of the electric field strength simulation results obtained using the traditional RCWA algorithm and the neural network algorithm of the present application. Among them, the left part is the electric field strength obtained by the traditional RCWA algorithm, the middle part is the electric field strength obtained by the Fourier neural operator network model of the embodiment of the present invention, and the right part is the error value between the left and right parts. Figure 14 It is the verification visualization result of a pair of structure-electric field data in the test data. The test data contains a total of 100 pairs of structure-electric field data, and the average error is <5%. The trained neural network can generate electric field simulation results with an accuracy similar to that of the RCWA algorithm within 1 second. Thus, it can be seen that using the Fourier operator to replace the RCWA can effectively accelerate the optimization of the metasurface structure.

[0125] In addition, the parallelization feature of the neural network makes it possible to optimize the designs of multiple different initial structures simultaneously, thus achieving the effect of global optimization. As Figure 15 shown, in the embodiment of the present invention, multiple initial designs (p0, p1, p2, p3, p4) are optimized simultaneously, and the one with the best result is selected from them to avoid the optimization process falling into a local optimal solution and achieve the purpose of global optimization. In the figure, the ordinate represents the uniformity of the diffraction spot, and the abscissa represents the number of iterations.

[0126] The present invention has the following beneficial effects: The present invention first obtains a metasurface lens structure dataset and the corresponding electromagnetic response dataset, constructs a Fourier neural operator network model, and then uses the metasurface lens structure dataset and the electromagnetic response dataset to train the Fourier neural operator model, so that the Fourier neural operator network model establishes a mapping relationship between the metasurface lens structure dataset and the electromagnetic response dataset, thereby using the Fourier neural operator network model to replace the traditional Maxwell solver to calculate the electromagnetic response and perform gradient descent iteration to generate metasurface lens structure data corresponding to the design target. The calculation speed of the electromagnetic response can be significantly improved through the Fourier neural operator network, and rapid convergence can be achieved through gradient descent. Therefore, this solution can significantly improve the efficiency of optimizing the metasurface lens structure.

[0127] In addition, the parameter optimization range of the Fourier kernel integral operator in the present invention is restricted to an elliptical region, improving the accuracy of the electromagnetic simulation results, further ensuring the double rotational symmetry and reflection symmetry of the neural network simulation results, and enabling the optical element to obtain better optical performance. For a periodic symmetric structure, the parameter optimization range of the Fourier operator kernel is restricted to a quadrant of an elliptical region, reducing the number of model training parameters and improving the model training efficiency.

[0128] In addition, the differential coupled wave analysis method is used to obtain the training dataset of the Fourier neural operator network model, improving the accuracy and generation efficiency of the dataset.

[0129] The above are only specific embodiments of the present invention and should not be used to limit the scope of the present invention. Equivalent changes made by those of ordinary skill in the art based on this creation, as well as changes well-known to those skilled in the art, should still fall within the scope covered by the present invention.

Claims

1. A method for optimizing a metasurface lens structure, characterized in that: The method comprises the following steps: Step S1, obtaining a metasurface lens structure data set and a corresponding electromagnetic response data set; Step S2, constructing a Fourier neural operator network model, using the metasurface lens structure data set and the electromagnetic response data set to train the model, and fixing the model parameters; Step S3, according to the evaluation function corresponding to the design target, the Fourier neural operator network model equipped with the fixed model parameters is used to calculate the electromagnetic response and perform gradient descent iteration to generate the metasurface lens structure data corresponding to the design target.

2. The method for optimizing the structure of a supersurface lens according to claim 1, wherein: The specific method of step S2 is: Step S21, constructing a Fourier neural operator network model, wherein the Fourier neural operator network model includes a feature extraction network, and a plurality of Fourier layers sequentially connected in series with the output end of the feature extraction network; the feature extraction network is composed of a residual network and a convolutional network, and each of the Fourier layers includes a Fourier transform module, a convolution module, an inverse Fourier transform module, a linear transformation module, an activation function, and a Fourier kernel integral operator; the feature extraction network extracts a metasurface lens structure feature matrix from the metasurface lens structure data, and then restores it to an electromagnetic field simulation result after serial calculation of the plurality of Fourier layers; The Fourier kernel integral operator is used to perform a convolution operation with the Fourier transform result of the metasurface lens structure characteristic matrix; Step S22, training the Fourier neural operator network model using the metasurface lens structure data set and the corresponding electromagnetic response data set; Step S23, verifying the Fourier neural operator network model.

3. The method for optimizing the structure of a supersurface lens according to claim 1, wherein: The specific method of step S3 is: Step S31, initializing the metasurface lens structure data D0; Step S32, for a given super surface lens structure data D i , the Fourier neural operator network model equipped with the fixed model parameters is used to forward calculate the electric field response data E i ; Step S33, calculating the loss function according to the design target evaluation function, if the loss function converges or the number of iterations exceeds the maximum number, exit, otherwise, use the neural network back propagation mechanism to calculate the differential ΔD of the metasurface lens structure data i ; Step S34, updating the super surface lens structure data D i+1 =D i -αΔD i , jump to step S32, where α is the learning rate.

4. The method for optimizing the structure of a supersurface lens according to claim 2, wherein: In step S22, the output of the Fourier neural operator network model is the Fourier space value of the predicted electric field, and the loss function is the L2 spatial distance between the predicted value and the true value; in step S23, the output of the Fourier neural operator network model is the real space value of the predicted electric field, and the loss function is the L1 spatial distance between the predicted value and the true value.

5. The method for optimizing the structure of a supersurface lens according to claim 2, wherein: The parameter optimization range of the Fourier kernel integral operator is limited to an elliptical region.

6. The method for optimizing the structure of a supersurface lens according to claim 5, characterized in that: The hypersurface is a periodically symmetric structure, and the parameter optimization range of the Fourier operator kernel is limited to a quadrant of an elliptical area.

7. The method for optimizing the structure of a supersurface lens according to claim 1, wherein: The specific method of step S1 is: Step S11, initializing the metasurface lens structure D10; Step S12, for a given metasurface lens structure D1 i , using rigorous coupled wave analysis to forward calculate the electromagnetic response E1 i ; Recording of metasurface lens structure D1 i With electromagnetic response E1 i ; Step S13, calculating the loss function according to the design target evaluation function, if the loss function converges or the number of iterations exceeds the maximum number, the recorded metasurface lens structure D1 is i With electromagnetic response E1 i Output as training data and exit, otherwise, calculate the differential ΔD1 of the metasurface lens structure according to the differential approximation equation i ; Step S14, updating the metasurface lens structure D1 i+1 =D1 i -α1ΔD1 i , jump to step S12, where α1 is the learning rate.

8. The method for optimizing the structure of a supersurface lens according to claim 7, wherein: The differential approximate equation is: a differential approximate equation obtained by decomposing the characteristic matrix in the strict coupled wave analysis method through the Wittinger integral and the regularization term.

9. The method for optimizing the structure of a supersurface lens according to claim 8, characterized in that: The differential approximate equation generation process is as follows: Given an eigendecomposition problem: ΦΛ=ΜΦ M is the input matrix, Λ and Φ are the eigenvalues ​​and eigenvectors; Define the differentiable eigendecomposition operator as T: Τ(Μ) = Λ; Φ, the coupled wave analysis operation based on the characteristic decomposition result (T) is f:f(Λ,Φ,...)=E, that is, through the characteristic decomposition results Λ,Φ and other differentiable intermediate steps, the target electromagnetic response E can be finally obtained; the differential equation of the characteristic decomposition operator can be obtained by the chain rule: Among them, ∈ is a regular term, which is used to deal with the case where M is a loss matrix.

10. The method for optimizing the structure of a supersurface lens according to claim 1, characterized in that: The metasurface is a metasurface of a laser beam splitter, and the evaluation function includes zero-order diffraction intensity and diffraction spot uniformity.

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