Optical nanostructure radiation pattern analysis method and system

Abstract

The application discloses an optical nanostructure radiation pattern analysis method, relates to the cross technical field of nanophotonics and deep learning, and comprises the following steps: constructing an optical nanostructure library and generating a data set by using a finite difference time domain simulation, and extracting multipole moments from near-field distribution as low-dimensional physical labels; a physical-data hybrid driven neural network is constructed and trained, mapping of features to multipole moments is completed by a neural network layer, and a far-field radiation pattern is reconstructed according to a multipole superposition principle through a physical layer. The application further discloses an optical nanostructure radiation pattern analysis system, and the application can predict a radiation pattern at any angle resolution, does not need to be retrained, and reduces data storage by 99.8% and training time by 92.6% at a 1° resolution, while maintaining a prediction accuracy equivalent to full-wave simulation. The application provides an efficient, interpretable and resolution-independent proxy model for nanophotonic radiation analysis.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of nanophotonics and deep learning, and in particular to a method and system for analyzing the radiation pattern of optical nanostructures. Background Technology

[0002] Nanophotonics focuses on the interaction between light and matter at the nanoscale and is a core support for cutting-edge fields such as data storage, quantum photonics, biochemical sensing, optical metasurfaces, optical computing, and nanofabrication. The analysis of the scattering characteristics of optical nanostructures is a key prerequisite for the effective control of light fields and the efficient design of photonic devices, among which the analysis and optimization of radiation patterns are particularly important.

[0003] Currently, the analysis of the radiation properties of optical nanostructures mainly relies on full-wave numerical simulation methods based on Maxwell's equations, such as the Finite-Difference Time-Domain (FDTD) method and the Finite Element Method (FEM). These methods can provide high-precision analysis results, but they suffer from a huge technical bottleneck: as the complexity of optical nanostructures increases, the number of meshes required for simulation increases dramatically, and the demands on computing power and time grow exponentially, making it difficult to meet the needs of rapid device design. In recent years, purely data-driven deep learning models have been introduced into the field of electromagnetic response prediction. Researchers hope to improve computational efficiency by fitting the mapping relationship between geometric parameters and radiation properties through neural networks (NNs). However, these models have significant drawbacks: first, their training requires massive labeled datasets, resulting in high data acquisition costs; second, these models lack physical interpretability, and the output results have no clear physical meaning, making it difficult to guide device structural optimization; furthermore, these models can only predict radiation patterns at a fixed spatial resolution. If higher resolution results are required, the model needs to be retrained, leading to a significant increase in storage and training costs.

[0004] To address these issues, physical information neural networks (PINs) have been proposed. These networks incorporate physical laws into their loss function, ensuring that model predictions conform to these laws. This improves the physical interpretability and generalization ability of the models to some extent. However, there is still room for improvement in training stability and computational efficiency (see M. Raissi et al., "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations", Journal of Computational Physics, 378, 686-707, 2019). Furthermore, hybrid modeling frameworks that combine prior physical knowledge with neural networks are attracting widespread attention. These methods have successfully overcome the limitations of single-model paradigms by leveraging the complementarity between physical mechanisms and data-driven learning (see X. Zheng et al., "Hybrid Physics-Data-Driven Neural Network for Accurate Modeling of Scattering Problems", IEEE Transactions on Antennas and Propagation, 73, 6826-6838, 2025). However, existing hybrid models have not yet incorporated multipole-expansion electromagnetic physics priors, and still suffer from limitations such as inflexible resolution, low storage efficiency, and low training efficiency in predicting radiation patterns of optical nanostructures.

[0005] Multipole expansion is an effective method for describing the interaction between light and matter, condensing information about electromagnetic radiation and field coupling in charge density distribution into multipole moment parameters (including electric dipoles). Magnetic dipole electric quadrupole Magnetic quadrupole This theory provides suitable physical priors for nanoscale radiation analysis. Its correctness and effectiveness in the field of nanophotonics have been verified, but it has not yet been fully integrated with deep learning to solve the problem of rapid analysis of radiation patterns of optical nanostructures. Therefore, there is an urgent need for a method that integrates multipole expansion physical priors with data-driven deep learning to achieve rapid and accurate prediction of radiation patterns of optical nanostructures with low resource consumption, high physical interpretability, and resolution independence, breaking through the technical bottlenecks of traditional numerical methods and purely data-driven models.

[0006] Existing pure data-driven deep learning models suffer from drawbacks in predicting far-field radiation patterns of optical nanostructures, such as large data requirements, poor physical interpretability, and fixed angular resolution, as well as the high computational cost of traditional full-wave numerical methods. Summary of the Invention

[0007] The technical problem to be solved by the present invention is to overcome the defects of the prior art. The present invention provides a method and system for analyzing the radiation pattern of optical nanostructures. This method can significantly reduce data storage requirements and training time while maintaining high prediction accuracy, and at the same time ensure the high physical interpretability and resolution independence of the model.

[0008] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0009] A method for analyzing the radiation pattern of optical nanostructures according to the present invention includes:

[0010] Step S1: Construct an optical nanostructure library. The optical nanostructures in the library include one or more geometric types. Each optical nanostructure is characterized by geometric parameters, and a type label is assigned to each geometric type.

[0011] Step S2: Construct the PDHDNN dataset, a hybrid physical-data-driven neural network; details are as follows:

[0012] Full-wave electromagnetic simulations were performed on each optical nanostructure in the optical nanostructure library using the finite-difference time-domain method (FDTD) to obtain the near-field electric field distribution under different geometric parameters and incident light frequencies. Multipole moments, including electric dipole moments, were extracted from the near-field electric field distribution and complex refractive index distribution. Magnetic dipole moment Electric quadrupole and magnetic quadrupole moment ;

[0013] Based on the tensor symmetry of each multipole moment and the magnitude analysis of the contribution of each multipole moment component to far-field scattering, the dominant multipole moment component is selected; geometric parameters, incident light frequency, and type labels are used as input features, and the dominant multipole moment component is used as output labels to construct the PDHDNN dataset.

[0014] Step S3: Construct PDHDNN and train PDHDNN using the PDHDNN dataset. PDHDNN includes a neural network layer and a fixed physical layer. The neural network layer is used to receive input features and predict the selected dominant multipole moment components. The fixed physical layer reconstructs the far-field radiation pattern based on the principle of multipole moment superposition and the predicted dominant multipole moment components.

[0015] Step S4: Use the trained PDHDNN to predict the far-field radiation pattern. Input the geometric parameters of the optical nanostructure to be predicted, the incident light frequency, and the type label into the trained neural network layer to obtain the predicted dominant multipole moment component. Input the dominant multipole moment component into a fixed physical layer and reconstruct the far-field radiation pattern with arbitrary angular resolution by changing the observation direction vector.

[0016] As a further optimization of the optical nanostructure radiation pattern analysis method described in this invention, the optical nanostructure in step S1 includes at least one of a cuboid, an elliptical cylinder, and a cross-shaped prism, and the geometric parameters of the optical nanostructure include width. ,length ,high Type tags are encoded in binary.

[0017] As a further optimization of the radiation pattern analysis method for optical nanostructures described in this invention, the method for extracting the multipole moment from the near-field electric field distribution and complex refractive index distribution in step S2 is as follows: based on the induced current density... The electric dipole moment is calculated using volume integrals. Magnetic dipole moment Electric quadrupole and magnetic quadrupole moment ,in: It is the imaginary unit. , For frequency, It is angular frequency. Let n be the vacuum permittivity, and n be the refractive index distribution. This represents the near-field electric field distribution.

[0018] As a further optimization of the radiation pattern analysis method for optical nanostructures described in this invention, the method for screening the dominant multipole moment component in step S2 is as follows: analyze the symmetry satisfied by the multipole moment tensor, i.e., the electric multipole moment tensor. It is symmetrical and satisfies Magnetic multipole moment tensor It is also symmetrical, satisfying... Remove symmetrically repeated redundant components from the multipole moment tensor; calculate the contribution ratio of each non-redundant component to the total scattering cross section, and retain the dominant components whose contribution ratio is much higher than that of other non-redundant components.

[0019] As a further optimization scheme for the radiation pattern analysis method of optical nanostructures described in this invention, in step S2, the input features are normalized to the [0,1] interval by minimum-maximum before training, and the dominant multipole moment components are standardized.

[0020] As a further optimization of the radiation pattern analysis method for optical nanostructures described in this invention, in step S2, the neural network layer in step S3 includes four independent backpropagation neural networks, which respectively predict the electric dipole moment P, the magnetic dipole moment M, and the electric quadrupole moment. Magnetic quadrupole moment The dominant multipole moment component; each backpropagation neural network shares the input format and hyperparameter settings.

[0021] As a further optimization scheme for the radiation pattern analysis method of optical nanostructures described in this invention, each backpropagation network employs three hidden layers, with the number of neurons in each hidden layer being 30, 25, and 20, respectively; the activation function of the hidden layers is ReLU; training uses the Adam optimizer, with an initial learning rate of... Every 10 training cycles, the learning rate is multiplied by a decay factor of 0.95.

[0022] As a further optimization of the optical nanostructure radiation pattern analysis method described in this invention, in step S3, the far-field scattering electric field is reconstructed by fixing the physical layer according to the following formula. :

[0023]

[0024] in, For natural index, It is the imaginary unit. Radial distance from the observation point Angular frequency The corresponding wave number, The vacuum permittivity, At the speed of light, This is the unit vector for the far-field observation direction.

[0025] As a further optimization of the optical nanostructure radiation pattern analysis method described in this invention, screening the dominant multipole moment component means: screening according to the magnitude of the contribution of different multipole moment components to far-field scattering, with the dominant multipole moment component contributing more to the total scattering cross section than other components.

[0026] An optical nanostructure radiation pattern analysis system, comprising:

[0027] The nanostructure library building module is used to build an optical nanostructure library. The optical nanostructures in the optical nanostructure library include one or more geometric types. Each optical nanostructure is characterized by geometric parameters and a type label is assigned to each geometric type.

[0028] The FDTD simulation and multipole moment calculation module is used to perform full-wave electromagnetic simulations of each optical nanostructure in the optical nanostructure library using the finite-difference time-domain (FDTD) method. It obtains the near-field electric field distribution under different geometric parameters and incident light frequencies, and extracts multipole moments, including electric dipole moments, from the near-field electric field distribution and complex refractive index distribution. Magnetic dipole moment Electric quadrupole and magnetic quadrupole moment Based on the tensor symmetry of each multipole moment and the magnitude analysis of the contribution of each multipole moment component to far-field scattering, the dominant multipole moment component is selected; geometric parameters, incident light frequency, and type label are used as input features, and the dominant multipole moment component is used as output label to construct the PDHDNN dataset.

[0029] The PDHDNN building module is used to build PDHDNN and train PDHDNN using the PDHDNN dataset. PDHDNN includes a neural network layer and a fixed physical layer. The neural network layer is used to receive input features and predict the selected dominant multipole moment components. The fixed physical layer reconstructs the far-field radiation pattern based on the principle of multipole moment superposition and the predicted dominant multipole moment components.

[0030] The radiation pattern prediction module is used to predict the far-field radiation pattern using a trained PDHDNN. The geometric parameters of the optical nanostructure to be predicted, the incident light frequency, and the type label are input into the trained neural network layer to obtain the predicted dominant multipole moment component. The dominant multipole moment component is input into a fixed physical layer, and the far-field radiation pattern is reconstructed at arbitrary angular resolution by changing the observation direction vector.

[0031] Compared with the prior art, the present invention, employing the above technical solution, has the following technical effects:

[0032] (1) High physical interpretability: By constraining the output of the neural network to a multipole moment with clear physical meaning, and using the multipole moment superimposed physical layer to reconstruct the radiation pattern, the problem of the "black box" driven by pure data is avoided.

[0033] (2) Significantly reduces data storage and training costs: Traditional pure data-driven models require storing high-dimensional far-field radiation pattern data, and the higher the angular resolution, the larger the data volume. This invention only needs to store low-dimensional multi-pole moment labels. At a 1° angular resolution, the data storage volume is reduced by 99.8% and the training time is reduced by 92.6%.

[0034] (3) Angular resolution independent: Traditional neural networks have fixed output dimensions, which leads to resolution lock-in. However, the present invention outputs low-dimensional multipole moments. Once the neural network is trained, the radiation pattern can be reconstructed at any angular resolution by using fixed physical layer analytical formulas without retraining.

[0035] (4) High prediction accuracy: At the resonance peak position, the relative error of the predicted scattering cross section is less than 8.2%, and the normalized correlation coefficient of the radiation pattern reaches 0.88~0.94, which is in good agreement with the FDTD full-wave simulation results. Attached Figure Description

[0036] Figure 1 is a comparison diagram of the architecture of the PDHDNN of the present invention and the traditional pure data-driven neural network;

[0037] Figure 2a shows the workflow of FDTD scattering analysis and multipole moment decomposition;

[0038] Figure 2b A comparison of the scattering cross section of the multipole moment superposition with the baseline results;

[0039] Figure 2c A comparison of the radiation pattern obtained by superimposing multiple pole moments with the baseline results;

[0040] Figure 3a shows the specific network structure of PDHDNN;

[0041] Figure 3b shows a comparison between the scattering cross section and radiation pattern predicted by PDHDNN and the true values ​​of FDTD;

[0042] Figure 4a shows the verification of the radiation pattern prediction capabilities of PDHDNN and traditional methods at different angular resolutions;

[0043] Figure 4b compares the data requirements of PDHDNN and traditional methods at different resolutions;

[0044] Figure 4c shows a comparison of training time between PDHDNN and traditional methods at different resolutions. Detailed Implementation

[0045] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0046] Specifically, this invention relates to a Physics-Data Hybrid Driven Neural Network (PDHDNN) that combines precise multipole expansion physical priors with deep learning, for the rapid and accurate prediction of far-field radiation patterns of all-dielectric nanostructures in the optical near-infrared band.

[0047] This invention provides a PDHDNN for analyzing the radiation pattern of optical nanostructures, specifically targeting the far-field scattering radiation pattern analysis of silicon-based optical nanostructures at optical frequencies. It integrates multi-pole expansion electromagnetic physics priors with data-driven deep learning. Figure 1This diagram illustrates a comparison between the physical data hybrid driven neural network of this invention and the traditional pure data driven method. Traditional data-driven neural networks output radiation results at a fixed resolution, which suffers from problems such as large data requirements, complex network structure, and poor physical interpretability. The method of this invention (PDHDNN) has low data requirements, high training efficiency, and good physical interpretability. The specific implementation steps are as follows:

[0048] Step 1: Constructing an optical nanostructure library

[0049] Three typical silicon-based optical nanostructures (cubic prism, elliptical cylinder, and cross-prism) were selected to construct an optical nanostructure library covering the parameter range of practical applications. The geometric parameter of this optical nanostructure is width. ,length ,high (See Figure 1 ), parameter range set to =330~390 nm =680~740 nm =280~340 nm, parameter step size of 10 nm, a total of 7×7×7=343 different structural configurations were generated. For each type of nanostructure, the FDTD dataset selects the corresponding frequency band based on its structural geometry characteristics, and samples 31 frequency points evenly within the frequency band. Specifically, the sampling frequency band for elliptical cylindrical nanostructures is 242~312 THz, while the sampling frequency bands for cuboid prisms and cross-shaped prisms are 182~258 THz and 210~260 THz, respectively. The above frequency bands are selected to cover the main resonant characteristics of various nanostructures, and this set of frequency bands is used uniformly throughout the training, verification, and testing of the neural network. To distinguish different structural types, corresponding type labels (label 1, label 2) are assigned to cuboids, elliptical cylinders, and cross-shaped prisms, where cuboids are (1,1), elliptical cylinders are (1,0), and cross-shaped prisms are (0,1); structural parameters are removed. =350 nm =700 nm The 300 nm configuration was used for independent performance verification of PDHDNN.

[0050] Step 2: Construct the PDHDNN dataset, a hybrid physical-data neural network.

[0051] Full-wave electromagnetic simulations of 343 configurations in an optical nanostructure library were performed using the finite-difference time-domain (FDTD) method, generating a simulation dataset. The simulation parameters were set strictly according to electromagnetic laws, specifically:

[0052] In the numerical simulation, the incident wave is an x-polarized plane wave with an amplitude of 1 V / m propagating along the -z direction. The solution domain is a three-dimensional cube with a side length of 1000 nm, and the optical nanostructure is placed at its origin. A perfectly matched layer (PML) is wrapped around the solution domain to eliminate the influence of boundary reflections on the simulation results. A structured hexahedral mesh with a uniform step size of Δx=Δy=Δz= 20 nm is used to ensure the stability of the numerical simulation. At the same time, for each structural configuration and each incident light frequency, the three-dimensional near-field electric field distribution is recorded in the form of a 4D matrix. and complex refractive index distribution ,See Figure 2a The complex refractive index takes into account the frequency dispersion characteristics of silicon at optical frequencies, which is consistent with the actual physical scenario.

[0053] Based on the multipole expansion method, the three-dimensional near-field electric field distribution obtained in step 2 is... and complex refractive index distribution To calculate the induced current density The formula is:

[0054]

[0055] in: It is the imaginary unit. , It is angular frequency. It's frequency. Let be the vacuum permittivity, and n be the refractive index. Electric dipole moment P, magnetic dipole moment M, electric quadrupole moment... Magnetic quadrupole moment The calculation model is as follows:

[0056]

[0057]

[0058]

[0059]

[0060] in, , , λ is the wavelength in a vacuum. Represents the Kronecker function. The spherical Bessel function is defined as follows:

[0061]

[0062] in It is a Bessel function of the first kind. Relative error and normalized correlation coefficient (NCC) are used as evaluation metrics, where the NCC is calculated using the following formula:

[0063]

[0064] In the formula, , X and Y represent the PDHDNN predicted value and the FDTD simulated true value, respectively, with X and Y being the sample means of the predicted and true values, respectively. Further, based on the scattering contribution factor... Determine whether to include an octet or higher-order multipole moment in a fixed physical layer:

[0065]

[0066] It is the total scattering cross section calculated by FDTD full-wave. It is the scattering cross section corresponding to the electric dipole moment. It is the scattering cross section corresponding to the magnetic dipole moment. It is the scattering cross section corresponding to the electric quadrupole moment and It is the scattering cross section corresponding to the magnetic quadrupole moment, when If the correlation coefficient between the reconstructed radiation pattern and the full-wave simulation result is close to 1 and the normalized correlation coefficient is higher than a preset threshold (e.g., 0.95), the physical layer of the four multipole moments is confirmed to be sufficient; otherwise, an octet and higher-order multipole terms are introduced into the physical layer.

[0067] Step 3: Construct PDHDNN and train PDHDNN using the PDHDNN dataset.

[0068] PDHDNN's neural network layers are multi-output backpropagation neural networks. Its core function is to learn the mapping relationship between the geometric parameters of the optical nanostructure, the incident light frequency, and the multipole moments. The specific structural design is as follows:

[0069] The overall network structure consists of four independent backpropagation subnetworks, each corresponding to an electric dipole moment. Magnetic dipole moment Electric quadrupole Magnetic quadrupole moment The four sub-networks share the input layer and hyperparameters but are trained independently; the input layer has 6 input nodes, which are the geometric parameters. , , Incident light frequency And two structural type tags, namely tag1 and tag2, such as Figure 3aAs shown, all input parameters are normalized to [0,1] to eliminate the impact of parameter scale differences on training; training uses a 3-layer hidden layer structure with 30, 25, and 20 neurons respectively, and the ReLU activation function is used in the hidden layers to alleviate the gradient vanishing problem in deep neural network training; the output of the output layer consists of four dominant multipole moment components, i.e. , , , The magnitude of this type of component is about six orders of magnitude higher than that of the other components, while the remaining components are ignored due to their negligible contribution. At the same time, dimensionality reduction is achieved by utilizing tensor symmetry, i.e., the electric multipole moment tensor. It is symmetrical and satisfies Magnetic multipole moment tensor It is also symmetrical, satisfying... This further reduces model complexity and standardizes the multipole moment parameters, thereby stabilizing the model training process.

[0070] When partitioning the dataset, the simulation dataset generated in step 2 was randomly divided into training, validation, and test sets in an 8:1:1 ratio to ensure the model's generalization ability. The model was trained using the Adam optimizer with an initial learning rate of [missing information]. The learning rate is multiplied by a decay factor of 0.95 every 10 training epochs, with a maximum of 1000 training epochs. A setting is also set so that the mean squared error (MSE) of the validation set is lower than... When the model training has converged, training is stopped.

[0071] The physical layer is calculated using the formula from the far-field scattering electric field calculation model:

[0072]

[0073] The formula for calculating the total scattering cross section is:

[0074]

[0075] in, For natural index, It is the imaginary unit. Radial distance from the observation point Angular frequency The corresponding wave number, Let the electric field amplitude of the plane wave be denoted as . The vacuum permittivity, At the speed of light, This is the unit vector in the far-field observation direction. Using this model, low-dimensional multipole moment parameters can be converted into a high-dimensional far-field scattered electric field distribution, thus obtaining the radiation pattern, and only the unit vector needs to be adjusted. This allows the generation of radiation patterns with arbitrary spatial resolution.

[0076] like Figure 2b As shown, the total scattering cross section reconstructed based on four multipole moments is in high agreement with the direct calculation results of FDTD. Figure 2c The reconstruction effect of the radiation pattern was further demonstrated. The radiation pattern corresponding to the two peaks of the elliptical cylinder and the reference NCC value reached 0.96 and 0.92, respectively, which verified the accuracy of the physical layer.

[0077] Step 4: Use the trained PDHDNN to predict the far-field radiation pattern.

[0078] The geometric parameters of the silicon-based optical nanostructure to be analyzed ( / / Incident light frequency The PDHDNN, trained by inputting structural type labels, outputs electric dipole moments through four sub-networks of the neural network layers. Magnetic dipole moment Electric quadrupole Magnetic quadrupole moment The dominant component is then input into the far-field scattering electric field calculation model of the physical layer, and the unit vector of the far-field observation direction is adjusted. It can generate far-field scattering electric field distributions with arbitrary spatial resolution, thereby obtaining the radiation pattern of optical nanostructures. The entire prediction process does not require additional training, the calculation speed is much faster than FDTD simulation, and the storage and computing costs are extremely low.

[0079] This invention also provides a PDHDNN system for implementing the above method. This system is modularly designed and includes a nanostructure library construction module, an FDTD simulation and multipole moment calculation module, a PDHDNN construction module, and a radiation pattern prediction module connected sequentially. The functions of each module are as follows:

[0080] The nanostructure library construction module is used to build a library of medium optical nanostructures of different types and geometric parameters, and assign type labels to different structure types. The FDTD simulation and multipole moment calculation module is used to perform FDTD electromagnetic simulation on the structures in the optical nanostructure library to obtain the near-field electric field distribution and complex refractive index distribution. Based on the multipole expansion method, the electric dipole moment, magnetic dipole moment, electric quadrupole moment, and magnetic quadrupole moment are calculated from the near-field data, and the dominant multipole moment component is selected as the output label according to the magnitude of the contribution of each component to far-field scattering. The physics layer construction module establishes a far-field scattering electric field calculation model based on the multipole moment superposition formula, and reconstructs the dominant multipole moment component predicted by the neural network into a far-field radiation pattern. The PDHDNN model training module is used to build the neural network layer, divide the simulation dataset into training set / validation set / test set, train and validate the PDHDNN, and obtain the trained model. The radiation pattern prediction module is used to receive the geometric parameters, incident light frequency, and type label of the optical nanostructure to be analyzed, input the trained PDHDNN, and output a far-field radiation pattern with arbitrary spatial resolution.

[0081] Example Effect Verification

[0082] The performance of the PDHDNN of this invention was compared with that of a traditional pure data-driven neural network, and the verification results are as follows:

[0083] Regarding resolution independence, PDHDNN can predict radiation patterns at different angular resolutions such as 20°, 10°, and 1°, while traditional pure data-driven models can only predict results at a fixed resolution set during training and cannot adapt to changes in resolution. Figure 4a As shown; in terms of data requirement comparison, PDHDNN's dataset requirement is fixed at 0.34GB and is independent of resolution; the traditional pure data-driven model requires 273.28GB of data at 1° resolution, meaning PDHDNN can save 99.8% of storage. Figure 4b As shown; in terms of training time comparison, PDHDNN's training time is fixed at 37.28 minutes and is independent of resolution. Traditional pure data-driven models have significantly longer training times at 1° resolution than PDHDNN. PDHDNN can save 92.6% of training time. Figure 4c As shown; in terms of prediction accuracy, the location of the scattering cross-section resonance peak predicted by PDHDNN is completely consistent with the FDTD results, as shown in the figure. Figure 3bThe relative error of the main peak shown is less than 8.2%, and the NCC of the radiation pattern reaches as high as 0.94, with prediction accuracy significantly better than the traditional pure data-driven model. Table 1 lists the prediction accuracy comparison: at 1° resolution, the relative error of the peak of the present invention is 5.8% (Case 1) and 8.2% (Case 2), while the pure data-driven model is 5.1% and 8.4%, with comparable accuracy; at 10° resolution, the error of the pure data-driven model rises to 13.9% and 17.6%, while the present invention maintains its original accuracy, and the NCC value is stable above 0.91~0.93.

[0084] Table 1. Comparison of prediction accuracy between the present invention and pure data-driven neural networks.

[0085]

[0086] (Note: Case 1 corresponds to a cross-shaped prism structure, and Case 2 corresponds to a cuboid structure; NCC is the normalized correlation coefficient.)

[0087] Variations and extensions of the present invention

[0088] The above embodiments of the present invention are only applicable to silicon-based optical nanostructures and four dominant multipole moments. In practical applications, the present invention can be modified and extended in the following ways, all of which fall within the scope of protection of the present invention:

[0089] 1. Material System Expansion: This invention is not limited to silicon-based materials. When extending the framework to other dielectric materials (such as titanium dioxide, gallium nitride, etc.) or metal-dielectric hybrid systems, it is only necessary to incorporate the dispersion parameters of the corresponding materials in the dataset generation stage (step S2), recalculate the induced current density, and extract the multipole moment. The neural network architecture and physical layers do not need to be fundamentally changed.

[0090] 2. Wider Geometric Expansion: The type labels of this invention use binary encoding. Currently, two binary bits are used to distinguish three geometric types, with the remaining encoding bits reserved for future expansion. When more geometric types need to be included, the coverage of the nanostructure library can be expanded by increasing the number of bits in the label or by adopting more efficient encoding methods (such as one-hot encoding or learnable geometric embedding).

[0091] 3. The incident conditions are extended to include any polarization state and incident angle of the incident light, thus improving the applicability of the model.

[0092] 4. Introduction of higher-order multipole moments: When the electrical size of nanostructures increases or the structure becomes more complex, the quadruple multipole cutoff may be insufficient. In this case, the scattering contribution factor can be calculated. Assess the adequacy of the cutoff, if If the deviation from 1 is significant, an octuplet (magnetic octuplet moment, electric octuplet moment) and higher-order multipole terms are introduced into the physical layer, and additional subnetworks are trained accordingly for prediction. The framework of this invention is not architecturally limited to four multipoles; the truncation in the current embodiment is a trade-off between accuracy and efficiency for the studied electrical size range.

[0093] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for analyzing the radiation pattern of optical nanostructures, characterized in that, include: Step S1: Construct an optical nanostructure library. The optical nanostructures in the library include one or more geometric types. Each optical nanostructure is characterized by geometric parameters, and a type label is assigned to each geometric type. Step S2: Construct the PDHDNN dataset, a hybrid physical-data-driven neural network; details are as follows: Full-wave electromagnetic simulations were performed on each optical nanostructure in the optical nanostructure library using the finite-difference time-domain method (FDTD) to obtain the near-field electric field distribution under different geometric parameters and incident light frequencies. Multipole moments, including electric dipole moments, were extracted from the near-field electric field distribution and complex refractive index distribution. Magnetic dipole moment Electric quadrupole and magnetic quadrupole moment ; Based on the tensor symmetry of each multipole moment and the magnitude analysis of the contribution of each multipole moment component to far-field scattering, the dominant multipole moment component is screened. The PDHDNN dataset is constructed by using geometric parameters, incident light frequency, and type labels as input features and dominant multipole moment components as output labels. Step S3: Construct PDHDNN and train PDHDNN using the PDHDNN dataset. PDHDNN includes a neural network layer and a fixed physical layer. The neural network layer is used to receive input features and predict the selected dominant multipole moment components. The fixed physical layer is based on the principle of multipole moment superposition and uses the predicted dominant multipole moment components to reconstruct the far-field radiation pattern. Step S4: Use the trained PDHDNN to predict the far-field radiation pattern. Input the geometric parameters of the optical nanostructure to be predicted, the incident light frequency, and the type label into the trained neural network layer to obtain the predicted dominant multipole moment components. By inputting the dominant multipole moment components into a fixed physical layer, the far-field radiation pattern can be reconstructed at arbitrary angular resolution by changing the observation direction vector.

2. The method for analyzing the radiation pattern of an optical nanostructure according to claim 1, characterized in that, In step S1, the optical nanostructure includes at least one of a cuboid, an elliptical cylinder, and a cross-shaped prism. The geometric parameters of the optical nanostructure include width. ,length ,high Type tags are encoded in binary.

3. The method for analyzing the radiation pattern of an optical nanostructure according to claim 1, characterized in that, The method for extracting the multipole moment from the near-field electric field distribution and complex refractive index distribution in step S2 is as follows: based on the induced current density... The electric dipole moment is calculated using volume integrals. Magnetic dipole moment Electric quadrupole and magnetic quadrupole moment ,in: It is the imaginary unit. , For frequency, It is angular frequency. Let n be the vacuum permittivity, and n be the refractive index distribution. This represents the near-field electric field distribution.

4. The method for analyzing the radiation pattern of an optical nanostructure according to claim 1, characterized in that, The method for selecting the dominant multipole moment component in step S2 is as follows: analyze the symmetry satisfied by the multipole moment tensor, i.e., the electric multipole moment tensor. It is symmetrical and satisfies Magnetic multipole moment tensor It is also symmetrical, satisfying... Remove symmetrically repeated redundant components from the multipole moment tensor; calculate the contribution ratio of each non-redundant component to the total scattering cross section, and retain the dominant components whose contribution ratio is much higher than that of other non-redundant components.

5. The method for analyzing the radiation pattern of an optical nanostructure according to claim 1, characterized in that, In step S2, before training, the input features are min-max normalized to the [0,1] interval, and the dominant multipole moment components are standardized.

6. The method for analyzing the radiation pattern of an optical nanostructure according to claim 1, characterized in that, In step S3, the neural network layer includes four independent backpropagation neural networks, which respectively predict the electric dipole moment P, the magnetic dipole moment M, and the electric quadrupole moment. Magnetic quadrupole moment The dominant multipole moment component; each backpropagation neural network shares the input format and hyperparameter settings.

7. The method for analyzing the radiation pattern of an optical nanostructure according to claim 1, characterized in that, Each backpropagation network uses three hidden layers, with 30, 25, and 20 neurons in each layer; the activation function for the hidden layers is ReLU; training uses the Adam optimizer, with an initial learning rate of [missing value]. Every 10 training cycles, the learning rate is multiplied by a decay factor of 0.

95.

8. The method for analyzing the radiation pattern of an optical nanostructure according to claim 1, characterized in that, In step S3, the fixed physical layer reconstructs the far-field scattered electric field according to the following formula. : ； in, For natural index, It is the imaginary unit. Radial distance from the observation point Angular frequency The corresponding wave number, The vacuum permittivity, At the speed of light, This is the unit vector for the far-field observation direction.

9. The method for analyzing the radiation pattern of an optical nanostructure according to claim 1, characterized in that, Screening for dominant multipole moment components refers to selecting components based on their contribution to far-field scattering, with the dominant multipole moment component contributing more to the total scattering cross section than other components.

10. A radiation pattern analysis system for optical nanostructures, characterized in that, include: The nanostructure library building module is used to build an optical nanostructure library. The optical nanostructures in the optical nanostructure library include one or more geometric types. Each optical nanostructure is characterized by geometric parameters and a type label is assigned to each geometric type. The FDTD simulation and multipole moment calculation module is used to perform full-wave electromagnetic simulations of each optical nanostructure in the optical nanostructure library using the finite-difference time-domain (FDTD) method. It obtains the near-field electric field distribution under different geometric parameters and incident light frequencies, and extracts multipole moments, including electric dipole moments, from the near-field electric field distribution and complex refractive index distribution. Magnetic dipole moment Electric quadrupole and magnetic quadrupole moment ; Based on the tensor symmetry of each multipole moment and the magnitude analysis of the contribution of each multipole moment component to far-field scattering, the dominant multipole moment component is selected; geometric parameters, incident light frequency, and type labels are used as input features, and the dominant multipole moment component is used as output labels to construct the PDHDNN dataset. The PDHDNN building block is used to build PDHDNN and train PDHDNN using the PDHDNN dataset. PDHDNN includes neural network layers and fixed physical layers. The neural network layers are used to receive input features and predict the filtered dominant multipole moment components. The far-field radiation pattern is reconstructed by fixing the physical layer based on the principle of superposition of multiple moments and combining the predicted dominant multiple moment components. The radiation pattern prediction module is used to predict the far-field radiation pattern using a trained PDHDNN. The geometric parameters of the optical nanostructure to be predicted, the incident light frequency, and the type label are input into the trained neural network layer to obtain the predicted dominant multipole moment components. By inputting the dominant multipole moment components into a fixed physical layer, the far-field radiation pattern can be reconstructed at arbitrary angular resolution by changing the observation direction vector.

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