3D mask simulation and lithographic imaging using physics-informed neural networks

Background: The increasing demands on computational lithography and computational imaging in the design and optimization of lithography processes necessitate rigorous modeling of EUV light diffracted from the mask. Traditional electromagnetic field (EMF) solvers are inefficient for large-scale technology problems, while deep neural networks rely on a huge amount of expensive rigorously simulated or measured data. Aim: In order to overcome these constraints, we explore the potential of physics-informed neural networks (PINN) as a promising solution for addressing complex optical problems in the field of EUV lithography. Approach: We extend the existing MaxwellNet to simulate the light diffraction from typical reflective EUV masks. The coupling of the predicted diffraction spectrum with image simulations enables the evaluation of PINN performance in predicting relevant lithographic metrics and typical mask 3D effects. Results: The results of modeling near- and far-field diffraction using PINN showcase a good performance in terms of convergence behavior, stability, accuracy, and a significant speed-up (up to ×10000) compared to the rigorous 3D mask simulation using an established numerical EMF solver. In contrast to other machine learning approaches, PINN is able to accurately simulate the near field, learns the involved physics, and captures the optical and mask-induced 3D effects. PINNs can predict lithographic process windows with sufficient accuracy. Conclusions: Differently from numerical solvers, once trained, generalized PINN can simulate light scattering in several milliseconds without re-training and independently of problem complexity. This opens up the capabilities for partially coherent imaging simulations without the Hopkins approach, source optimization, and fast investigation of mask 3D effects.


INTRODUCTION
The increasing demands on computational lithography and computational imaging in the design and optimization of lithography processes necessitate an appropriate approach for correct modeling of the EUV light diffracted from the mask.The traditional Kirchoff approach assumes an infinitely thin mask and a constant phase of light, so it falls short of capturing the complexities of real-world scenarios.In reality, the mask absorber must be several wavelengths thick to absorb a sufficient amount of light.Modeling of EUV light interaction with such a thick mask comprises various optical and imaging effects that cannot be adequately predicted by a Kirchhoff approach.This includes light amplitude and phase deformation at the mask, mask-induced 3D effects [4], polarization effects [5], wave aberrations [6], and others.To effectively simulate the scattering and diffraction of EUV light from a typical EUV mask and comprehend the corresponding effects, rigorous electromagnetic field (EMF) modeling approaches are essential.They provide a more reliable foundation for computational lithography and imaging processes.Efficient rigorous modeling of mask diffraction in the vicinity of the EUV mask requires solving the scattered field computation problem, which has traditionally been approached through numerical approximations of Maxwell's partial differential equations (PDE).In recent decades, numerical solvers have emerged as the state-of-the-art method for modeling the interaction of light with nanostructures.Rigorous solvers for mask simulation typically employ iterative algorithmic techniques based on time and/or space discretization of Maxwell's equations and domain decomposition.As an example, Finite-Difference Time-Domain (FDTD) method was used for the calculation of reflected field to explore the effects of mask topography and multilayer coating defects on aerial images of reflective masks [7].A 2D Helmholtz PDE was iteratively solved via Finite Element Method (FEM) to simulate latent image formation illuminated by partially coherent light [8].Proposed in the early eighties, rigorous coupled wave analysis (RCWA) [9] and its derivative waveguide method [10] are still actively utilized by researchers and lithography engineers for mask computation.For example, fast fully rigorous defect simulation and defect repair simulation can be performed using an extended RCWA approach [11,12].The extensive use of computational lithography and computational imaging for the design and optimization of lithography processes puts high demands on the rigorous simulation of light diffraction from the mask.This comprises high modeling accuracy, reduced simulation time, and modeling of larger mask areas with design-relevant layouts.Numerical solvers often experience challenges related to increased consumption of computational resources as resolution, domain size, and problem complexity expand.As an alternative approach, data-driven deep learning networks can learn the correlation between input and output even without solving Maxwell's equations.In recent years, deep neural networks have been employed as proficient approximators in the realm of optical problem-solving, particularly in the modeling of lithographic imaging [13].An illustrative instance involves the deep learning approaches for modeling the EUV lithographic imaging taking into account 3D mask effects and EUV process variations [14] or emulating the mask-topographyinduced modifications of the diffraction spectrum [15,16].Most of the works relate to data-driven learning of the input-to-output mapping: 2D-to-3D aerial image prediction [14], mask-to-diffraction matrix calculation [1], etc.In the domain of the given scattered field computation problem, traditional deep learning networks would be forced to learn the relationship between the input mask instance and the output scattered electromagnetic field.Nevertheless, this approach turns out to be inconvenient due to several reasons.First, traditional data-driven training is typically supervised by a huge amount of simulated or measured data [1].Rigorously simulated data is computationally expensive since it relies on the numerical solver with the aforementioned computational limitations.Meanwhile, training data collected experimentally is not always available.Secondly, the physics behind Maxwell's equations, which could contain valuable information, is completely ignored behind the data-driven networks.In summary, numerical EMF solvers are inefficient for large-scale technology problems, while deep neural networks rely on a huge amount of expensive rigorously simulated or measured data.In order to overcome these constraints, we explore the potential of physics-informed neural networks (PINN) [2] as a novel promising solution for addressing complex optical problems in the field of EUV lithography.Consequently, this work is structured as outlined below: Section 2 presents a description of the scattered field computation problem and the physical constraints of the simulation setup.The following Section 3 provides details on the general idea of PINN and its implementation for 3D mask simulation.Section 4 contains results revealing the PINN performance in lithographic imaging.Finally, in Section 5, the findings of the study are summarized, accompanied by an outlook on their potential application to other EUV lithography scenarios.

SCATTERING PROBLEM
In this study, we are addressing a typical problem of scattered field computation.Instead of solving the full set of Maxwell's equations, we aim to compute the electric field distribution in the space domain.Therefore, we solve a 3D scalar Helmholtz equation within a computational domain, as presented in Figure 1.The material difference between EUV materials and the surrounding air is small, corresponding to weak refraction and scattering effects.For 1D mask structures (e.g., line-space pattern), there will be also no polarization effects, so the considered scalar formulation of the Helmholtz equation remains valid.However, scattering from 2D mask patterns (e.g., pillar) assumes a generation of additional polarization components.In the EUV lithography, these polarization effects are weak and, therefore, neglected in the domain of this work.Simulations are done using a monochromatic light with a wavelength λ of 13.5 nm, used in EUV lithography systems.The mask is illuminated by plane waves from different directions.Our study focuses on the use case involving a real 3D EUV mask, where the geometry is represented by a real EUV absorber on the top of the multilayer.The mask feature consists of two materials: antireflective coating presented by TaBO layer deposited on the top of TaBN absorber.Below the TaBN layer, there is a thin Ru capping layer followed by a MoSi multilayer.In the domain of this work, the use cases include 3D masks with varying absorber geometries and fixed multilayer settings.For these investigations, we use the MoSi multilayer with a constant period of 7.0 nm, comprising forty alternating layers of 3.0 nm Mo and 4.0 nm Si.The ultimate goal is to develop a comprehensive PINN-based EMF solver with good generalization capabilities toward relevant physical lithographic parameters.In this work, lithographic simulations are shown in detail for a horizontal line (Figure 1).The results of a contact hole simulation are provided in the Appendix.Boundary conditions are essential constraints that describe the processes within a simulation domain to correctly find the unique solution for a given physics problem.To reduce the problem to a finite computation domain, we apply the technique of perfectly-matched layer (PML) [17] on the top and bottom domain boundaries.PML works as an artificial absorbing layer to prevent the back reflection of outgoing light.These PMLs perfectly absorb outgoing waves regardless of the polarization state and incidence angle on the boundaries.Since a lithographic mask has a periodic structure with features repeated over a certain pitch p, we implement periodic boundary conditions in the horizontal direction.Classic periodic boundary conditions simply copy the electric fields at one edge of the simulation region and re-inject them at the other edge [18].However, when the propagation is at an angle, the fields from one period to the next are not exactly periodic: they will be out of phase by a distinct value.Floquet-Bloch boundary conditions [19] are very similar to periodic ones, but when copying the fields from one edge to the other, they also apply a phase correction to the fields: where e ipk bloch represents the phase correction term depending on the incident angle.For example, to satisfy the Floquet-Bloch boundary conditions in x-direction, we multiply the electric field values on the right and left domain boundaries by e −ipk bloch and e ipk bloch , respectively.

Overview
The recently introduced PINN concept [20] combines the fast inference and generalization capabilities of deep learning algorithms with explicit supervision by established physical laws, particularly PDEs, boundary, and initial conditions.PINNs can seamlessly integrate the information from the PDE by embedding its residual f (x, y, z) into the loss function of the neural network.For example, physics-driven loss can be minimized as mean squared error (MSE) where N x , N y and N z are the number of grid points in x-, y-and z-direction, respectively.After the loss or the PDE residual is computed, the weights are updated via backpropagation similar to traditional deep learning.At the backward pass, the gradients are computed with respect to the network weights.The network is trained by minimizing the loss via a gradient-based optimizer and finally converging to a unique solution to the given physical problem.
In the field of nano-optics and photonics, PINNs have already been applied to solve both various forward and inverse problems.The application of PINN in optics leverages the power of deep learning and physics-based modeling to advance various aspects of optical science and engineering.For example, Chen et al. [21] retrieved multiple optical parameters from nanostructures by solving inverse scattering problems using PINNs.Both [21] and [22] addressed a cloaking problem associated with the design of electromagnetic metamaterials using PINN.
An overview of deep learning approaches, including PINN, as applied for the acceleration of nanophotonics simulations, is detailed in the work of Wiecha et al. [23].Lim and Psaltis [3] developed a convolution-based PINN, called MaxwellNet, for the prediction of the electromagnetic field distributions and inverse design of microaspheric lenses.Gigli et al. [24] extended and adapted MaxwellNet to solve a nonlinear scattering problem and predict the optical Kerr effect for aspheric micro-lenses.Zheng and Zhao [25] proposed a photolithographic simulator blended with physics-informed modeling and data-driven training to address the 'design-to-manufacturing' gap in computational optics.Our first attempt [26] to employ PINN driven exclusively by physics in the domain of EUV lithography involved a simulation of the near-and far-field diffraction of EUV light from typical EUV absorber patterns.The goal of this work is to implement a full 3D PINN-based mask solver generalized toward relevant lithographic parameters.To enable better generalization capabilities we employed a convolution-based PINN approach.In contrast to fully connected PINN (FC-PINN), convolution-based deep learning can directly capture complex spatial correlations between geometric structures and provide a multi-resolution representation of the underlying solution field [27].Parameter sharing via filter-based convolution operations positions convolution-based PINN favorably for addressing large-scale and high-dimensional problems.Given these considerations, convolutionbased PINN emerges as a natural choice for achieving a generalized solution for variations of mask geometries.More detailed performance comparisons of FC-PINN and convolution-based PINN are described in [28,29].
Our approach involves the adaptation of the existing MaxwellNet prototype [3] leveraging the U-Net [30] architecture.Ultimately, we extend the MaxwellNet to accommodate the specific requirements of photomask applications.

PINN for 3D mask simulation
For a given 3D scattering problem, we employ the residual of the full Helmholtz equation (Eq.( 1)) as the physics-driven loss (Eq.( 3)) and calculate it along with the material distribution n(x, y, z).
We decompose the total electric field and represent three field components in Helmholtz PDE separately: The scattered field denoted as E s , carries the information about the EUV light reflected from the mask and scattered forward from the absorber into the multilayer.This light will be used for further lithographic imaging, as elucidated in Subsection 4.2.On the other hand, the incident field E i encapsulates details about the illumination employed, e.g., as plane wave characterized by an amplitude E 0 , wavenumber k 0 , incident angle ϕ, azimuthal angle θ, etc.The total electric field E encompasses near-field information, including the phase profile and intensity distribution.The training pipeline of PINN is schematically described in Figure 2. By discretizing the electric and magnetic field vectors on the Yee grid [31], the derivatives of Maxwell's PDE can be computed using finite differences approximation (FDA).In particular, the field derivatives can be discretized as 3D convolutions with two non-trainable filters with shifted elements, corresponding to the two staggered grids in the Yee scheme.A more detailed description of the FDA scheme on Yee grid follows the reference [3].Before the computation of derivatives, the PINN output is padded to maintain the original domain size after FDA convolutions.For example, we use zero padding in the z-direction to encode the PML physics.Left and right domain boundaries are explicitly padded with phase-corrected field values according to the Floquet-Bloch theorem described in Section 2. FDA convolutions cut off the padded regions and ultimately, the PDE derivatives are informed about the physics on the boundaries and outside the domain.Finally, the residual is backpropagated to update the network weights and iteratively obtain the pixel-based solutions of the scattered electric field.

Training process
The input of the PINN has the following format (B, C, N x , N y , N z ), where B and C correspond to the number of samples in a batch and image channels, respectively.The EUV materials are described by complex numbers, where k is an imaginary part containing the attenuation and n is a real part that accounts for refraction.Therefore, we use input images with two channels corresponding to two material maps: refractive index n distribution and extinction coefficient k distribution.In this way, a 3D mask is represented in the form of a voxel volume comprising complex refractive indices of the materials.Since the predicted electric field also has a complex notation, real and imaginary parts of different field components are separated into distinct output channels, as shown in Figure 3.At each training step, the mask geometries are generated randomly within user-defined parameter ranges.Subsequently, resulting distributions of the refractive index are used as input for the training process.Such a dynamic, on-the-fly sample generation introduces notable advantages to the training process.Firstly, the on-the-fly approach eliminates the need to store extensive datasets, particularly in the context of 3D samples.Secondly, by generating samples on-the-fly, training is initiated promptly without the need for loading time, contributing to increased efficiency in the training workflow.Thirdly, the on-the-fly approach explores a wider parameter range with longer training, providing enhanced generalization capabilities of the PINN model.The Hopkins approach, or shift-invariance of the diffracted light versus the illumination direction, cannot be used for rigorous modeling and correct EUV imaging simulation.An image simulation without the Hopkins approach [13] approach means that mask diffraction has to be computed for several representative illumination directions.In this case, the fully rigorous modeling for various incidence angles imposes additional computational demands for mask and image modeling [13].To address these requirements, a developed PINN model has to account for fast simulation using multiple illumination directions.Convolution layers of U-Net-based PINN effectively extract features from the random geometries/materials encoded in the input images.However, an illumination angle is inherently a scalar floating-point value, and its representation in an image-based format is not immediately straightforward.Subsequently, we explored the strategies to tune the network with incident and azimuthal angles.Following the successful experience of [32], we extended U-Net by the implementation of the tuning network (Figure 3) with trainable weights.Our tuning network brings the tunable scalar parameter to the main U-Net model controlling the second convolutional layer so that the physical loss may simultaneously be minimized for different illumination directions.The convolutions are transposed in the decoder pathway.We avoid unintended biases in the transposed convolutional layers that may hinder the training or generalization of the model.The weights for convolution layers are initialized using a uniform distribution.The gradients whose norm is bigger than 0.001 are clipped.The summary of network hyperparameters found by tuning is shown in Table  The learning rate depends on the problem's complexity, while resolution defines appropriate network depth.The training and inference time for 2D and 3D models are presented in Table 2.  Once trained, PINN provides an inferred solution within ms, showing significant speedup (up to x10000) with respect to the waveguide method [33], which we employed as a numerical reference solution for accuracy evaluation.

Near-and far-field simulation
From the real and imaginary parts of the numerically computed or predicted complex electromagnetic field, one can extract amplitude/intensity and phase information.To numerically assess the difference between the reference solution computed with the waveguide method and the PINN prediction, we compute mean absolute percentage error (MAPE) and root-mean-square error (RMSE) from the image difference.Figure 5 shows the near-field simulation of EUV light propagation in the vicinity of the EUV mask with reflective multilayer.The comparison of amplitude distributions simulated by the waveguide method and predicted PINN exhibits a good accuracy of the PINN, with a deviation of 0.91% from the reference.Electromagnetic fields obtained with PINN capture the given physics and optical effects: mask shadowing effects, partial penetration of EUV light into the reflective multilayer, and phase deformation by the EUV absorber.For example, mask shadowing occurs due to the height of the absorber structures and the non-telecentric illumination at the mask level.Mask shadowing belongs to typical mask 3D effects that are simulated and discussed in Subsection 4.3.The most significant standing waves on the left and right of the TaBN absorber are caused by interference of incident light and the light reflected from the top of the multilayer.The amplitude decreases from top to bottom due to the notable absorption by the mask absorber and multilayer.The similarity of the optical properties of all materials in the EUV wavelength spectrum and surrounding vacuum results in weak scattering and refraction effects.A duty ratio of 1:1, corresponding to the dense arrangement of mask features, exhibits a strong interaction of EUV light scattered and diffracted in the neighboring domains.PINN accurately predicts the resulting interference patterns within a multilayer, proving a correct implementation of periodic boundary conditions. .Only the reflected near field obtained above the absorber is relevant for further far-field computation and lithographic imaging.This corresponds to EUV light that is reflected from the mask, subsequently observed at the entrance pupil of the EUV projection system and ultimately analyzed in the Fourier domain.In this case, predicted near field at XY -cross-section at position z = 0 nm has MAPE XY = 0.18% deviation.In contrast to other machine learning approaches [14,1], PINN is able to accurately predict the near field.Notably, PINN operates without supervision by reference data and learns only from the governing physics.This enhances its interpretability and potentially makes PINNs more sample-efficient compared to traditional deeplearning methods that heavily rely on target data for training.Nevertheless, it is difficult to derive quantitative conclusions for further lithographic imaging relying only on the near field.A far field can be obtained by near-to-far-transformation using the Fourier transform of the complexvalued near field.Far-field diffraction analysis typically includes the computation of diffraction efficiencies and phase values of discrete diffraction orders contributing to image formation.The phase deformation and asymmetry in the near field are transferred to the far field resulting in the asymmetric diffraction behavior.Based on phase spectra (Figure 6 (b)), relative errors in phase differences δ(∆ phase ): 0.07% between −1 st and 0 th orders and 0.01% between 0 th and +1 st orders.Subsection 4.3 will demonstrate the correct prediction of important mask 3D effects by PINN.

Lithographic imaging
The use of the predicted diffraction spectrum as input for image simulations enables the evaluation of PINN performance in terms of relevant lithographic metrics.Cross-section plots (Figure 7 (b)) and process windows (Figure 7 (c)) are used to numerically assess the PINN performance in lithographic imaging.The illumination settings depend on the feature type, e.g., a quasar with four poles is most appropriate for illumination of contact holes and pillars and dipole illumination for line-space patterns.We used a dipole with the following illumination parameters (Table 3) to create an image of the horizontal line, which was shown in Figure 5.A quasar with the same pole parameters was used for illumination to obtain the image of an 18 nm contact hole, as demonstrated in the Appendix.
The use of PINNs in lithographic imaging simulations is demonstrated for the EUV projection system with an NA of 0.33.The source is composed of discrete source points to employ mask illumination from different directions.We use five noHopkins points per pole for the rigorous mask simulation (see Figure 7(a)).The image is obtained by incoherent superposition of images obtained by all discrete source points.Our PINN model was adapted for arbitrary illumination settings accounting for the angular support of the multilayer.Such a model can address the computational challenges associated with partially coherent imaging simulations without Hopkins assumption.In contrast to numerical rigorous mask solvers, a trained PINN exhibits a notable advantage: the imaging simulation time is almost independent from the number of used noHopkins points.The image cross-section predicted by PINN shows the deviation of RMSE = 4.5E-5 nm.The pronounced shadowing effect for horizontal lines can be also observed in the extracted cross-sections.The asymmetric illumination of the horizontal lines results in a small image shift to the right.This can be attributed to the differences in the near field at poles and corresponds to one of the mask-induced 3D effects elaborated on in Subsection 4.3.The overlapping green area almost completely covers the ellipses of both process windows (Figure 7 (c)).This indicates that PINNs can predict lithographic process windows with sufficient accuracy.The relevant lithographic metrics are extracted from the reference threshold-to-size level.Intensity-related metrics (CD, NILS) and phase/focus-related metrics (best focus NILS, depth of focus NILS) predicted by PINN for 20 nm horizontal line (wafer scale) are inserted in Table 4.The small absolute and relative errors both for near-, and far-field diffraction metrics as well as lithographic metrics indicate the potential of PINN for use in accurate image simulation for lithography applications.

Mask 3D effects
Mask 3D effects have to be analyzed and properly mitigated to obtain high contrast and correctly placed images in lithographic imaging.To reduce the impact of the 3D mask effect different mitigation strategies are applied comprising a proper mask focus, optical proximity correction (OPC), optimization of asymmetric source shapes and assist features, alternative absorber materials, etched/shifted multilayer stacks, anamorphic large NA systems, etc. [4] Mask shadowing and phase shift were demonstrated in Figure 5 and Figure 6, respectively.In this part of the work, we demonstrate the PINN performance to predict such typical mask 3D effects as non-telecentricity, contrast fading, and best focus shifts.To achieve this, we employ a PINN-based 3D mask solver parametrized towards several physical parameters simultaneously.Specifically, we use our trained model for fast exploration of relevant lithographic metrics as a response to variations of absorber thickness, feature size, pitch, and illumination directions.

non-Telecentricity
Pattern placement error or telecentricity error corresponds to a shift of the feature position through the focus.First, the off-axis illumination of the mask causes a non-telecentricity of the scanner optics, which manifests as a pattern shift on the wafer.Second, taking into account a 3D mask design, both multilayer and absorber are not ideal and contribute to non-telecentricity.Figure 8 (c) shows the results of pattern placement simulation using PINN and waveguide method for dipole illumination scenario.PINN accurately predicts the shifts of image position with an RMSE of 2.0E-2 nm, 3.0E-2 nm and 1.0E-4 nm, corresponding to small-angle (ϕ = 3.134°) pole, large-angle (ϕ = 8.866°) pole, and complete dipole, respectively.The differences between the near fields of the small-angle pole and the large-angle pole result in very different non-telecentric behavior.Asymmetry induced by oblique illumination is transferred from the near field to the imaging.The image of the large-angle pole exhibits a significantly larger shadowing, telecentricity error, and more pronounced shifts of the image position through focus, than the small-angle pole.The resulting telecentricity error of the complete dipole is 2.77 mrad, predicted by PINN with a deviation of 0.72%.

Contrast fading
The complete image of the dipole is incoherently superposed from the images of the individual poles.Shadowing causes a shift between images from the small-angle and large-angle poles.For the dipole illumination setup, the shadowing effect (especially strong for the large-angle pole) contributes to the imbalance of the imagingrelevant diffraction orders.As a result, a superposition of shifted images ultimately leads to the drop in NILS attributed to image blur.The intensity minima of the small-angle and large-angle poles at different positions and different non-telecentricity values cause a further contrast fading, demonstrated in Figure 9. Image blur depends on the balance in diffraction orders and image shifts.By adjusting the linewidth on the mask and thickness of the absorber, it becomes possible to achieve a balance in the diffraction orders, and improved contrast for the image from the single pole.Far-field and imaging results demonstrated PINN capabilities to predict both diffraction order balancing and shifts of image position.Consequently, PINN can be employed to rapidly find an optimal source shape and mask topography to achieve well-balanced diffraction orders.From Figure 9, both the waveguide method and PINN indicate the best contrast at the largest and thickest absorber.This corresponds to a mask with the largest absorber volume and, hence, the highest absorption and sharpest intensity profile in the image plane.However, it is essential to note that increasing absorber height reduces the resulting depth of focus, image intensity, and threshold-to-size, which impacts the throughput.Other absorbers, especially low-n [34] candidates exhibit a much smaller optimum absorber thickness.Although finding an optimal trade-off is necessary, this aspect is beyond the scope of the current work.

Best focus shift
The best focus corresponds to a focus position, where the highest NILS is obtained.This position depends on many physical parameters, such as pitch, absorber geometry, and material.The key driver of best focus shifts is a phase deformation related to mask 3D design and refractive index difference.Phase shifts are caused by the diffraction at the absorber structures: the wavefront proceeds faster inside the absorber compared to the surrounding air.The deformation of the phase in the near field is transferred to the phase deformation in the far field, resulting in asymmetric diffraction behavior.According to Figure 6, PINN can correctly predict a phase shift between orders, therefore we expect a correct simulation of best focus shift.To demonstrate this, we train a PINN model for fast exploration of the best focus shift versus physical parameters (pitch, absorber thickness, feature size).PINN accurately predicts the different focus positions of the lowest blur for every pitch, as shown in Figure 10.

SUMMARY AND OUTLOOK
By integrating physics into a deep learning model, we developed a PINN approach to enable more efficient computation of light diffraction from EUV masks.In this study, we explore for the first time the potential of PINN for 3D mask simulation and lithographic imaging.We implemented a comprehensive PINN-based EMF solver with good generalization capabilities toward relevant physical lithographic parameters, including variation of mask geometries and illumination settings.The results of modeling near-and far-field diffraction using PINN showcase an outstanding performance in terms of convergence behavior, stability, and accuracy.Lithographic process windows predicted with PINN almost completely overlap with the reference results.The good accuracy of the image metrics, which was observed in our simulations, demonstrates the potential of PINN for use in accurate image simulation for computational lithography applications.
Compared to other machine learning approaches, PINN is able to accurately simulate the near field without supervision by large amounts of experimental or rigorously simulated data.PINN learns given physics and captures the optical and mask-induced 3D effects.Differently from numerical solvers, once trained, generalized PINN can simulate light scattering in several milliseconds without re-training.Our model demonstrated significant speed-up (up to ×10000) for the simulation of the same setting.The aforementioned advantages of PINN open up numerous capabilities for other EUV lithography and non-EUV applications.For example, the PINN model adapted for arbitrary illumination can address the computational challenges associated with partially coherent imaging simulations without Hopkins assumption.In contrast to the numerical rigorous mask solvers, the number of noHopkins points has almost no impact on the overall simulation time.Future work will involve the application of PINN to solve inverse lithographic problems.To make PINNs usable for OPC, source mask optimization (SMO), and inverse lithography technology (ILT), we aim to investigate the upscaling of the PINN-based solution for larger areas and more complex geometries, including curvilinear masks.
Appendix A: Simulation of 18 nm contact hole

Figure 1 .
Figure 1.Example of a simulation setup for a scattered light computation problem: refractive index distribution.Use case: horizontal line on the top of MoSi multilayer.Feature size: 80 nm on the mask scale, corresponding to 20 nm on the wafer scale.

Figure 2 .
Figure 2. Schematic description of the PINN training.

Figure 3 .
Figure 3. 3D PINN model with main U-Net and tuning network architectures adapted from[3,32] PINN training job was distributed to two GPU-accelerated machines (NVIDIA A100 80 Gb).The resulting trained PINN model is simultaneously generalized towards absorber geometries and illumination settings.The evolution of the physics-driven loss function over 400K training steps is demonstrated in Figure4.The balance between the training and validation loss curves indicate a robust and stable training process for the PINN model.This close match showcases the model's ability to learn and generalize from the randomly generated dataset without overfitting.As a result of the exponential decay in the learning rate, halving it every 50K epochs, loss spikes gradually decrease and the curve reaches a plateau.

Figure 6
Figure6shows computed and predicted diffraction spectra comprising low diffraction orders of the reflected EUV light.Only several diffraction orders contribute to the final lithographic imaging due to limited NA.Absolute prediction error in diffraction efficiency ranges from 0.4E-3 to 3.1E-3 thereby illustrating PINN accuracy in predicting the ratio between intensities of diffracted light at different orders and the intensity of the incident light.The phase deformation and asymmetry in the near field are transferred to the far field resulting in the asymmetric diffraction behavior.Based on phase spectra (Figure6 (b)), relative errors in phase differences δ(∆ phase ): 0.07% between −1 st and 0 th orders and 0.01% between 0 th and +1 st orders.Subsection 4.3 will demonstrate the correct prediction of important mask 3D effects by PINN.

Figure 7 .
Figure 7. Evaluation of PINN performance in lithographic imaging of 20 nm horizontal line (wafer scale): (a) dipole source; (b) x-cross-section of simulated aerial images; (c) predicted lithographic process windows.

Figure 8 .
Figure 8. Demonstration of non-telecentricity of the EUV system for 20 nm horizontal line with a pitch of 64 nm (wafer scale).(a), (b) simulated near fields for illumination from the center of the large-and small-angle poles, respectively; (c) shifts of the image position for the small-angle pole (left), complete dipole (center), and large-angle pole (right).

Figure 9 .
Figure 9. Image blur: NILS values computed across the absorber geometry ranges: (a) simulation with waveguide approach; (b) PINN prediction; (c) difference with RMSE = 5.4E-2.The feature size on the wafer scale is shown on the y-axis (4× reduction).

Figure 11
Figure 11  demonstrates PINN performance in the simulation of best focus shifts versus variations of absorber geometry.PINN captures the physical effects, such as the swinging behavior of lithography metrics versus thickness variations with the period of ≈ λ/2.These swings are attributed to the phenomenon of double images, arising from the interference of images reflected from multilayer and absorber.

Figure 11 .
Figure 11.Best focus NILS computed across the absorber geometry ranges: (a) simulation with waveguide approach; (b) PINN prediction; (c) difference with RMSE = 1.2 nm.The feature size on the wafer scale is shown on the y-axis (4× reduction).

Figure 13 .
Figure 13.Evaluation of PINN performance in lithographic imaging of an 18 nm contact hole (wafer scale): (a) quasar source; (b) x-cross-section of simulated aerial images; (c) predicted lithographic process windows.