A large numerical aperture lithographic objective vector diffraction field calculation method based on PINN

Abstract

The application discloses a large numerical aperture lithography objective vector diffraction field calculation method based on PINN, comprising the following steps: obtaining a three-dimensional vector diffraction light field of a large NA lithography objective through simulation, reconstructing the three-dimensional vector diffraction light field at a sampling point position, and generating corresponding complex electric field distribution matrix data; performing physical purification on the complex electric field distribution matrix data to construct a corresponding data set; constructing a corresponding Helmholtz equation as a constraint equation based on the light field tight focusing characteristics of the large NA lithography objective; constructing an initial model, training the initial model by using the data set in combination with the constraint equation, and obtaining a PINN model for predicting complex electric field data; and inputting the spatial coordinates of a target to be predicted into the PINN model to obtain complex electric field data. The method provided by the application can significantly improve the calculation efficiency of the vector diffraction field while maintaining high simulation accuracy, and is suitable for fast modeling and analysis of three-dimensional vector light fields of large numerical aperture lithography objectives.

Description

Technical Field

[0001] This invention belongs to the field of lithography objective imaging technology, specifically relating to a method for rapid calculation of the diffraction field of the mirror image vector of a large numerical aperture lithography machine. Background Technology

[0002] Photolithography machines are core equipment in integrated circuit manufacturing, with projection lithography machines playing a crucial role. During lithography, the projection lens of the lithography machine needs to image the mask pattern onto the silicon wafer with high fidelity; therefore, the imaging quality of the lithography lens largely determines the quality of the lithography. To achieve image quality control of the lithography lens, a diffraction propagation model capable of accurately describing the propagation process of the image-space light field must be established. Especially for large-nanometer (NA) lithography lenses (e.g., 193nm, 1.35NA) used in advanced node lithography machines, the diffracted light is strongly compressed by the tight focusing effect, and polarization components are continuously coupled, resulting in complex three-dimensional vector evolution characteristics of the image-space light field. Therefore, to accurately describe this vector evolution process, a vector diffraction propagation model covering the entire image-space path needs to be constructed. In applications such as aberration detection, OPC, and SMO, the vector diffraction propagation model must ensure ultra-high (e.g., sub-nanometer) simulation accuracy while supporting rapid simulation speeds for multi-layer, multiple calculations within the image-space. Therefore, large NA lithography objectives require a high-precision and high-speed method for calculating vector diffraction fields.

[0003] Prior technique 1 (Rosenbluth AE, et al. "Fast calculation of images for high numerical aperture lithography". Proc. SPIE, 5377: 615–628 (2004)) based on Sort Coherent System Stacking (SOCS) provides a vectorized fast imaging method for large-area lithography. It combines factors such as polarization and birefringence into a unified imaging kernel, enabling OPC to complete the calculation of the single-layer image plane light field in about minutes while maintaining about 90–95% (compared to rigorous electromagnetic simulation) of computational accuracy. However, since only a two-dimensional planar light field at a specified depth can be obtained at a time, constructing the global (three-dimensional) light field of the image plane requires layer-by-layer calculation, resulting in a significant increase in computational load and a decrease in computational speed.

[0004] Prior technique two (NIU Zhiyuan, LI Sikun, LIU Yang. "Rigorous simulation model of double-Ronchi shearing interferometry on lithographic tools". Applied Optics, 62(18): 4759-4765 (2023)) provides a high-precision vector imaging model based on Rayleigh–Sommerfeld diffraction integral, achieving wavelet aberration reconstruction error of approximately 0.01 nm RMS. To address the significant computational burden of point-by-point vector integration, this method utilizes GPU acceleration, drastically reducing the generation time of a single light field from hours to minutes. However, if this point-by-point integration method is extended to a larger field of view or higher sampling density, the computational burden increases quadratically, making further speed improvements difficult.

[0005] Prior technique 3 (Guo Y, et al. "Mask3D-compatible full-vectorial Hopkinsimaging for optical lithography". Optica, 12(7): 924–934 (2025)) provides a fast vector lithography imaging modeling method based on Hopkins by constructing a high-dimensional vectorized TCC and combining it with Lanczos decomposition. It can reduce the computation time of a single-layer light field to about 1–8 seconds while maintaining an aberration accuracy of about 1–2% (relative to the vector Abbe reference). However, this method requires multi-region decomposition and FFT convolution operations on the high-dimensional cross-transfer coefficients. The computational load will increase with the increase of resolution or the number of light source regions, and the computational speed is still limited. Summary of the Invention

[0006] The purpose of this invention is to provide a method for calculating the vector diffraction field of a large numerical aperture lithography objective lens based on PINN. This method leverages the computational acceleration advantage of neural networks to improve the simulation speed of the vector diffraction field, while ensuring the high fidelity and interpretability of the prediction results.

[0007] To achieve the objectives of this invention, the following technical solution is provided: a method for calculating the vector diffraction field of a large numerical aperture lithography objective lens based on PINN, comprising the following steps:

[0008] The three-dimensional vector diffraction light field of the large NA lithography objective is obtained by simulation. The three-dimensional vector diffraction light field is reconstructed by sampling point positions to generate the corresponding complex electric field distribution matrix data.

[0009] Physical purification is performed on the complex electric field distribution matrix data to construct a corresponding dataset, which includes objective lens incident light parameters, spatial coordinates of all sampling points, and complex electric field distribution matrix data of complex electric field data under spatial coordinates.

[0010] Based on the light field tight focusing characteristics of the large NA lithography objective, the corresponding Helmholtz equation is constructed as a constraint equation.

[0011] An initial model is constructed, and the initial model is trained using the dataset and constraint equations to obtain a PINN model for predicting complex electric field data.

[0012] The spatial coordinates of the target to be predicted are input into the PINN model to obtain the complex electric field data under the corresponding spatial coordinates.

[0013] This invention achieves integrated modeling of "physical constraints + data-driven" by embedding physical equations into neural networks. By combining data purification and dynamic weight optimization of physical constraints, it solves the problems of low computational efficiency and low computational accuracy of existing methods.

[0014] Specifically, the complex electric field distribution matrix is ​​constructed using each sampling point as a unit, and each sampling point contains the complex electric field values ​​of the three polarization components of the vector light field at that location.

[0015] Specifically, the physical purification process involves filtering complex electric field distribution matrix data by judging the magnitude of the field strength, spatial importance, and gradient changes of the matrix data.

[0016] Specifically, the expression for the constraint equation is as follows:

[0017] ;

[0018] Among them, among them, The vector complex electric field distribution in image space, Vacuum wavenumber, It is the wavelength of light in a vacuum. is the relative permittivity.

[0019] Specifically, during the training process, the objective lens incident light parameters and the spatial coordinates of the sampling points are used as inputs, and the corresponding complex electric field distribution matrix data are used as the prediction results for evaluation.

[0020] Specifically, during training, the parameters of the initial model are updated using a composite loss function, which includes data supervision loss and constraint equations.

[0021] Specifically, during the training process, a dynamic balancing physical weight strategy is used to adjust the weights of the composite loss function. The dynamic balancing physical weight strategy is divided into two stages. In the first stage, training or fine-tuning is performed only based on data supervision loss. In the second stage, the weights of the constraint equations in the composite loss function are gradually increased.

[0022] Specifically, the incident light parameters of the objective lens include the incident light wavelength, amplitude, phase, and polarization state.

[0023] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0024] This invention introduces the physical laws of vector diffraction of the projection lens of a large NA lithography machine into neural network training. Compared with traditional calculation methods based on physical laws (see prior art 1 to 3), the calculation time of the model of this invention can be shortened from minutes / hours to milliseconds while maintaining similar calculation accuracy. Attached Figure Description

[0025] Figure 1 This is a flowchart of the method for calculating the vector diffraction field of a large numerical aperture lithography objective lens based on PINN provided in this embodiment;

[0026] Figure 2 The diagram shows the model network structure and training process provided in this embodiment. Detailed Implementation

[0027] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0028] like Figure 1 As shown, this embodiment provides a method for calculating the vector diffraction field of a large numerical aperture lithography objective lens based on PINN, which aims to improve the efficiency of simulation prediction.

[0029] First, the three-dimensional vector diffraction light field of the large NA lithography objective was calculated using electromagnetic field numerical simulation tools to obtain the complex electric field distribution. EThe data consists of an n×m dimensional matrix (x,y,z); where n is the number of sampling points and m is the number of complex electric field forms in the three polarization directions of the vector field, divided into real and imaginary parts, totaling m=6 (3*2) channels. The numerical simulation tools include, but are not limited to, any rigorous electromagnetic solver. The simulation tools can output the complex electric field (real / imaginary) or amplitude / phase at each sampling point in the image space as labels.

[0030] Physical purification is performed on the complex electric field distribution matrix data to construct a corresponding dataset. The dataset includes incident light parameters, spatial coordinates of all sampling points, and complex electric field distribution matrix data of the complex electric field data under the spatial coordinates.

[0031] Physical purification involves using a strategy that considers the magnitude of the field strength, spatial importance, and gradient changes in the matrix data to ensure both full coverage and preservation of key features in the training set. First, regular grid sampling is performed on the image space to guarantee full coverage. Then, layered sampling is performed based on field strength or synthetic quality to ensure that various field strength levels are represented in a balanced manner. Finally, dense sampling (e.g., at a ratio of 40% / 40% / 20%) is performed on high-field, transition band, and high-gradient regions to enhance the learning ability of hotspots, boundaries, and high-frequency details.

[0032] The total sample size is 10. 3 -10 6 For large NA objectives, the preferred image space sampling interval is ≤λ / (2·NA).

[0033] Based on the tight focusing characteristics of large NA objectives, the vector Helmholtz equation is used as a physical constraint:

[0034]

[0035] Among them, among them, The vector complex electric field distribution in image space, Vacuum wavenumber, It is the wavelength of light in a vacuum. Let L be the relative permittivity. In PINN, the residuals of the above equations are used as the physical loss term L_Phys, and the first / second derivatives are calculated using automatic differentiation. Define the residual vector. ,make Furthermore, the model is trained to approach zero as much as possible, so that the model results follow physical rules.

[0036] Design the initial structure of the neural network, construct a loss function that is simultaneously subject to data and physical constraints, and build the PINN model.

[0037] like Figure 2As shown, the backbone of this model employs a multilayer feedforward neural network (MLP) to map the normalized object space coordinates with the incident light parameters and objective lens parameters into complete vector complex electric field components. Specifically, the network input includes three-dimensional spatial coordinates (x, y, z), wavelength λ, amplitude A, phase φ, and polarization state (which can be represented using Jones vectors or equivalent parameterization). The input is first normalized (coordinates are mapped to [-1, 1], and wavelength / amplitude / phase / polarization are linearly normalized according to the reference scale) to improve numerical conditions. The network consists of several fully connected layers (example structure: input -> 256 -> 512 -> 512 -> 256 -> 128 -> output). Each hidden layer uses the Tanh or SIREN activation function and is initialized with Xavier. The output layer is a linear unit, and the output is the real and imaginary parts of the three components of the vector complex electric field, Ex, Ey, and Ez.

[0038] The loss function L_Total is defined as: L_Total = ω 1*L_Data + ω 2*L_Phys, where: L_Data is the data loss term (MSE of the real / imaginary components of the complex electric field at the training points); L_Phys is the physics loss term; and ω 1. ω 2 represents an adjustable weight. The first / second derivatives and Laplace derivatives required in the residual terms are calculated using automatic differentiation (autograd) to ensure accurate differentiation.

[0039] During training, the dataset is fed into the initial model, and the training uses the Adam optimizer (e.g., with an initial learning rate of 1e). -3 ~1e -4 The training epochs are 100-500, and the batch size is preferably 16-1024. During the process, the physical weights are dynamically balanced to optimize the synergistic effect of data and physical constraints. After iteration, a PINN model that can be used for efficient prediction of three-dimensional vector complex electric fields is obtained.

[0040] The dynamic balancing of physical weights includes: employing a phased training strategy (Phase 1: data-driven fine-tuning or weakly physical-constrained pre-training; Phase 2: gradually increasing the physical weights). ω 2. Joint training can be conducted, and the data and physical terms can be balanced using a weight scheduling function or an adaptive weight update algorithm (such as dynamic weighting based on the loss ratio or the sliding window mean method). During training, the physical residuals can be calculated only on a subset of samples to reduce computational cost.

[0041] The trained PINN model is input with the spatial coordinates of the test set data to predict the complex electric field data (including the real and imaginary parts of the electric field) in the coordinates. The model accuracy is then evaluated by comparing the complex electric field values ​​with those of the test set. In a set of embodiments based on the above-mentioned optimized model parameters, the prediction accuracy of the model within the image-side field of view at twice the focal length, compared to the test set results (electromagnetic field numerical simulation), can reach approximately 90%, within a 4×10⁻⁶ range. 6 The prediction time at a scale of one prediction point is approximately 0.1 seconds.

[0042] When training data is obtained through rigorous electromagnetic field simulation calculations, meaning the data patterns are more consistent with the Helmholtz physical constraint equations, and when a deeper network structure and more training steps are used (1e 5 ~ 2e 5 When using GPUs with higher computing power (higher CUDA core count, memory bandwidth, etc.), the prediction speed of this invention can be further improved.

[0043] Furthermore, the terms "upper," "lower," "inner," "outer," "front," and "rear" are used for descriptive purposes only and should not be construed as indicating or implying relative importance. Unless otherwise specifically stated, the relative steps, numerical expressions, and values ​​of the components and steps set forth in these embodiments do not limit the scope of the invention.

[0044] Of course, the above description is only a specific embodiment of the present invention and is not intended to limit the scope of the present invention. All equivalent changes or modifications made to the structure, features and principles described in the claims of the present invention should be included in the scope of the claims of the present invention.

[0045] Finally, it should be noted that the above-described embodiments are merely specific implementations of the present invention, used to illustrate the technical solutions of the present invention, and not to limit it. The scope of protection of the present invention is not limited thereto. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that any person skilled in the art can still modify or easily conceive of changes to the technical solutions described in the foregoing embodiments within the technical scope disclosed in the present invention, or make equivalent substitutions for some of the technical features; and these modifications, changes, or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention, and should all be covered within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for calculating the vector diffraction field of a large numerical aperture lithography objective lens based on PINN, characterized in that, Includes the following steps: The three-dimensional vector diffraction light field of the large NA lithography objective is obtained by simulation. The three-dimensional vector diffraction light field is reconstructed by sampling point positions to generate the corresponding complex electric field distribution matrix data. Physical purification is performed on the complex electric field distribution matrix data to construct a corresponding dataset. The dataset includes objective lens incident light parameters, spatial coordinates of all sampling points, and complex electric field distribution matrix data of complex electric field data under spatial coordinates. The physical purification filters the complex electric field distribution matrix data by judging the magnitude of the matrix data field strength, spatial importance, and gradient change. Based on the tightly focused optical field characteristics of the large NA lithography objective, a corresponding vector Helmholtz equation is constructed as a constraint equation, and the expression of the constraint equation is as follows: ;in, The vector complex electric field distribution in image space. Vacuum wavenumber, It is the wavelength of light in a vacuum. The relative permittivity is used. An initial model is constructed, and the initial model is trained using a dataset and constraint equations to obtain a PINN model for predicting complex electric field data. During training, the parameters of the initial model are updated using a composite loss function, which includes data supervision loss and constraint equations. A dynamic balancing physical weight strategy is used to adjust the weights of the composite loss function. The dynamic balancing physical weight strategy is divided into two stages. In the first stage, training or fine-tuning is performed only based on data supervision loss. In the second stage, the weights of the physical constraint equations in the composite loss function are gradually increased. The spatial coordinates of the target to be predicted are input into the PINN model to obtain the complex electric field data under the corresponding spatial coordinates.

2. The method for calculating the vector diffraction field of a large numerical aperture lithography objective lens based on PINN according to claim 1, characterized in that, The complex electric field distribution matrix is ​​constructed with each sampling point as a unit, and each sampling point contains the complex electric field values ​​of the three polarization components of the vector light field at that location.

3. The method for calculating the vector diffraction field of a large numerical aperture lithography objective lens based on PINN according to claim 1, characterized in that, During training, the objective incident light parameters and the spatial coordinates of the sampling points are used as inputs, and the corresponding complex electric field distribution matrix data are used as the prediction results for evaluation.

4. The method for calculating the vector diffraction field of a large numerical aperture lithography objective lens based on PINN according to claim 1, characterized in that, The objective lens incident light parameters include incident light wavelength, amplitude, phase, and polarization state.

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