Mask 3D (M3D) effects, arising from light scattering by 3D absorbers atop reflective multilayers, strongly influence the imaging performance of extreme ultraviolet (EUV) lithography. This study shows that M3D effects-traditionally considered detrimental-can be harnessed to improve imaging fidelity in high-NA and hyper-NA EUV, under certain conditions. We performed fully rigorous source-mask optimizations (SMO) to compare the achievable performance of 3D masks with that of Kirchhoff-type flat masks and to identify the best imaging solutions for both. In-depth analyses of near-fields, diffraction, and aerial images explain the advantages of 3D dark-field masks: tighter, higher near-field intensity peaks due to light guiding in low-n absorbers and more efficient coupling into diffraction orders within the projection NA. Realizing these benefits requires split-pupil exposure or other techniques to offset contrast loss. Our simulation results indicate that advances in 3D-mask-aware SMO/ILT and mask technology can further unlock M3D benefits.

Background: The lithographic imaging performance of extreme ultraviolet (EUV) lithography is limited by the efficiency of light diffraction and contrast fading caused by 3D mask effects. The dual monopole concept has been proposed by Joern-Holger Franke to mitigate contrast fading for line-space (L/S) patterns. Aim: We employ various modeling techniques to investigate the extendibility of dual monopole or split pupil exposures (SPs) to dense arrays of contacts on dark field and light field masks using different mask absorber options. Approach: First, a semi-analytic model is introduced to understand the relevant imaging mechanisms of split pupil exposures for L/S patterns. Next, we apply the split pupil exposure to a regular array of contact holes on a dark field mask. A multi-objective optimization approach helps to identify general trends and specific solutions. Analysis of the near fields of the light reflected from the mask for these particular solutions provides further insights into the imaging mechanisms of split pupil exposures and the different behavior of dark field (DF) and light field (LF) masks. Investigations for several mask absorber materials, tonalities, source fillings, and target sizes demonstrate the application of SP to different use-case scenarios. Results: Our simulations indicate that split pupil exposures benefit 1D (L/S) and 2D (arrays of contacts/pillars) features. The achievable gain compared with a single exposure (SE) depends on tonality, source filling, absorber material, and target size. The application of SP significantly impacts source mask optimization (SMO). SP affects optical proximity correction (OPC) and optimum source shape and may even modify the optimum absorber thickness. The combination of low-n absorbers, SP, and multi-objective SMO enables the identification of the best imaging solutions and pushes low k1 high-numerical aperture (NA) imaging to its ultimate limits. Conclusions: Split pupil exposures can provide a promising addition to the toolbox of resolution enhancement techniques for low k(1) high-NA lithography and unleash the full potential of low-n/low-k absorber materials.

Mask3D-induced effects, including orientation-dependent image asymmetries, non-telecentricity, pitch-dependent best focus, and image blur, are increasingly important for EUV imaging. To improve the fundamental understanding of these effects and their impact on the optical resolution limit of high NA and hyper NA EUV lithography, this paper investigates the imaging of lines/spaces (L/S) with a pitch of 9 nm using an ideal fictive diffraction-limited projection system with a NA of 0.85. The results of our simulations suggest that mask3D effects will not limit the achievable imaging performance of high NA and hyper NA EUV systems. Comparisons of rigorous mask simulations with results obtained by a Kirchhoff (flat) mask model indicate that mask3D effects do not necessarily negatively impact EUV imaging. Absorber patterns for the smallest pitches behave like volume gratings. Such volume gratings exhibit a significant dependency of the diffracted light on the illumination direction. In contrast to thin gratings, volume gratings enable a more flexible distribution of light between diffraction orders. Based on the improved understanding of the involved imaging mechanisms, one could take advantage of mask3D effects to enhance the imaging performance. The opportunities for such innovative solutions depend on the limitations of mask fabrication, which are not considered in this discussion.

We propose an alternative data-free deep learning method using a physics-informed neural network (PINN) to enable more efficient computation of light diffraction from 3D optical metasurfaces, modeling of corresponding polarization effects, and wavefront manipulation. Our model learns only from the governing physics represented by vector Maxwell's equations, Floquet-Bloch boundary conditions, and perfectly matched layers (PML). PINN accurately simulates near-field and far-field responses, and the impact of polarization, meta-atom geometry, and illumination settings on the transmitted light. Once trained, the PINN-based electromagnetic field (EMF) solver simulates light scattering response for multiple inputs within a single inference pass of several milliseconds. This approach offers a significant speed-up compared to traditional numerical solvers, along with improved accuracy and data independence over data-driven networks. (c) 2025 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

Two-photon lithography (TPL) has received significant attention for its ability to induce localized polymerization with spatial resolution below 100 nm. Beyond conventional 2D and 2.5D lithography, TPL utilizes a focused laser to initiate two-photon polymerization (TPP), enabling the fabrication of full 3D structures. Despite its potential, TPL's application is currently hindered by limited throughput, highlighting the need for advanced simulation models to explore technical options for improvement and optimization. One of the primary challenges in accurately modeling the TPL process lies in accounting for the complex interplay of photochemical reactions and oxygen diffusion. Detailed simulation approaches often require extensive computational resources, limiting their practicality. In this work, we propose a generalized compact model that balances chemical and physical fidelity with computational efficiency. This model integrates essential chemical and physical parameters and incorporates diffusion-reaction dynamics through a simplified yet accurate framework. To ensure practical applicability, we propose a straightforward way to calibrate a derived formula that minimizes the computational cost while enabling simulation across varying exposure times. Through calibration of parameters using experimental data, the model demonstrates enhanced predictive accuracy.

Metasurfaces have become a key focus in research and are applied in numerous fields because of their exceptional capability to control electromagnetic waves across microwave to optical frequencies. These artificial sheet materials have the advantages of lightweight and ability to control wave propagation both on the surface and in the surrounding free space. The complexity of fabricating metasurfaces via two-photon lithography (TPL) is addressed through sophisticated modeling. Critical to the success of TPL is the ability to predict the effects of the fabrication process on the final product. This paper introduces three distinct modeling approaches that vary in complexity and predictive capabilities. We evaluate the performance and limitations of a simple threshold model, a compact model and a full model of polymerization. Through application examples, we demonstrate how these models can guide the fabrication of metasurfaces.

Designing high-performance optical metasurfaces traditionally relies on computationally expensive iterative simulations and is often limited by the scalability of gradient-based neural networks that struggle with high-dimensional inverse design tasks. To address these challenges, we present a data-free machine learning framework that combines a physics-informed neural network (PINN) with reinforcement learning (RL) for the inverse design of transmission-type all-dielectric metasurfaces. The PINN learns to model light diffraction directly from Maxwell's equations and boundary conditions, serving as a fast and accurate surrogate model that replaces computationally expensive full-wave 3D numerical electromagnetic simulations within the RL loop. The RL agent optimizes metasurface geometries by trial and error, receiving rewards based on the mean absolute error between target optical spectra and spectra predicted by the PINN for each proposed design. In our experiments, the RL agent outperforms conventional gradient-free optimizers in both accuracy and final design quality. This approach paves the way for data-efficient, physically reliable inverse design of next-generation optical devices.

Rigorous modeling of extreme-ultraviolet (EUV) masks-using techniques such as rigorous coupled-wave analysis (RCWA)-is critical for advancing semiconductor manufacturing, as it accurately captures the complex electromagnetic interactions within multilayer and absorber structures. While these numerical methods are indispensable for quantitative predictions, they often provide limited physical insight into the underlying optical phenomena. To address this, Mesilhy et al.1 introduced an analytical waveguide-based model that treats the vacuum gap between absorbers as an isolated waveguide, offering a more intuitive understanding of modal behavior. In this work, we build on that model by automating the detection of all supported modes and quantifying their individual contributions to the total field using overlap integrals. We further extend the model to periodic line-space masks using grating theory, capturing lateral mode coupling and Bloch-periodic field behavior. This extended formulation captures the modulation of diffraction efficiencies with absorber thickness and reveals how refractive index contrast governs energy redistribution among diffraction orders. Comparisons with RCWA show excellent agreement, particularly for thin absorbers, demonstrating the improved accuracy of the extended model and establishing the analytical approach as a complementary tool to rigorous methods for fast mask evaluation and design exploration.

Two-photon polymerization (TPP) enables high-resolution additive manufacturing at the microscale, but accurate control of voxel shape remains a challenge due to nonlinear exposure dynamics and complex resist behavior. We present a physics-based differentiable model of the TPP process that includes radical generation, oxygen quenching, polymerization and development. The model is implemented in PyTorch and allows optimization of laser power using gradient-based backpropagation. Experimental validation is performed using a setup with a diffractive optical element (DOE) that enables parallel voxel exposure. This parallelization improves throughput but introduces proximity effects when several voxels are written close to each other. We show that the optimization technique compensates for these effects, resulting in better agreement between simulated and printed structures.

Two-photon lithography (TPL) has emerged as a pivotal technique for fabricating highly intricate micro- and nanoscale structures with diverse applications, ranging from photonics and microfluidics to biomedical engineering. Despite its significant advantages, the manufacturing process is often compromised by defects such as structural collapse and deformation. These issues predominantly arise due to capillary forces exerted by solvents during the development stage, significantly affecting the structural integrity, quality, and throughput of the fabricated components. To address these challenges, this study systematically investigates the mechanical phenomena underpinning these defects, leveraging multiphysics simulations to identify their origins and propose effective modeling strategies. A dual-framework approach was employed to achieve these objectives. First, a structural analysis was performed to determine the critical aspect ratio that would help predict a collapse. This model was validated via finite element simulations. Second, a rheological flow model was formulated to simulate the polymer melt behavior. The models exhibit good agreement with the experimental data and provide useful pointers to the fabricators.

The Multi-Trigger Resist (MTR), a novel negative tone photoresist, shows significant potential for extreme ultraviolet (EUV) lithography, offering lower dose to size (DtS) requirements compared to chemically amplified resists (CARs) and metal oxide resists (MORs). This research develops and validates a stochastic model of the Multi-Trigger Resist process. The model accounts for statistical variations in photon distribution, secondary electron generation, and molecular distribution within the photoresist. First simulation results show a good reproduction of experimental data. Current studies refine the existing MTR model with updated calibration data and expands the analysis to better understand roughness effects. Simulation results closely reproduce experimental trends, capturing critical dimension (CD) and line-edge roughness (LER) behaviors. An exploratory study of PAG loading reveals that increasing PAG concentration improves sensitivity and reduces roughness up to an intermediate optimum, while under-loading raises dose requirements and produces a LER upturn, most pronounced at 28-nm pitch. These results define a practical PAG loading window that balances DtS and LER performance, providing guidance for future MTR formulation and process optimization.

Conventional lithography simulation often treats mask manufacturing as an ideal process that introduces no distortions, which leads to inaccurate patterning and yield predictions since the mask making process introduces distortions. Approaches to model the distortions from the mask process and their corrections often take a blackbox form,1, 2 which limits the capabilities of end-to-end modeling, disallowing integrated lithography simulations and mask-aware process corrections. The aim of this work is to develop efficient and differentiable "whitebox" models for the mask process that help yield prediction and improve design for manufacturability through enabling end-to-end optimization. Our approach involves pre-processing SEM images of the manufactured mask and converting them into a raster format to allow differentiable pixel loss formulations. We then fit a mask process from mask input design to manufactured SEM using accurate machine learning models. We illustrate that such mask process model can be integrated in end-to-end lithography simulations to improve the accuracy of wafer pattern predictions with minimal overhead and can be used for efficient sensitivity analysis of the mask. Moreover, the utilization of differentiable modeling for all process steps including the mask process is instrumental in enabling effective end-to-end lithography modeling and optimization.

Background: To enable the manufacturing of advanced semiconductor devices, EUV lithography has been continuously shrinking the lateral dimensions of the mask and features. Resulting complex diffraction, polarization, and oblique illumination effects require rigorous modeling of EUV light diffracted from the mask. Traditional electromagnetic field (EMF) solvers are inefficient for large field-of-view simulations, while deep neural networks rely on a huge amount of expensive rigorously simulated or measured data. [1] Aim: Building upon our prior research [2], which revealed the PINN's potential and enabled sufficient aerial image simulation of small features, this study aims to broaden the scope of PINN's applications. Specifically, we extend this work towards the investigation of polarization effects and variable illumination settings. Approach: The established PINN model [2] was employed for EUV mask simulations under various illumination directions to investigate the orientation dependence of lithographic patterning. Expressing the residual of 3D Maxwell's equations, we effectively decoupled the transverse electric (TE) and transverse magnetic (TM) modes of EUV light. Employing a vectorial formulation of the wave equation, we investigated the ability of the PINN approach to predict the generation of additional electric field components resulting from the scattering at the absorber edges. Results: PINN accurately predicts the polarization effects that are relatively small, but still notable close to the absorber edges and inside the multilayer. The generalized PINN-based solver, adapted for arbitrary illumination settings, demonstrates the significant impact of the illumination direction and exposure wavelength on the reflected EUV light. PINN captures asymmetries in the near field caused by off-axis illumination and the significant drop in the intensity of the reflected light for larger incidence angles. Conclusions: By pushing the application limits of the existing PINN approach, we demonstrated its capability to accurately model oblique illumination effects, polychromatic effects, and even the weak polarization effects in the EUV spectral range. Different from numerical solvers, a universal PINN-based mask solver can simulate light scattering response in several milliseconds for arbitrary mask geometries and illumination settings without re-training.

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[1]. Aim: In order to overcome these constraints, we explore the potential of physics-informed neural networks (PINN)[2] as a promising solution for addressing complex optical problems in the field of EUV lithography. Approach: We extend the existing MaxwellNet[3] 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 x 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.

The lithographic imaging performance of contact holes is limited by the efficient use of light and contrast fading caused by 3D mask effects. "Split pupil" exposures have been proposed to mitigate contrast fading for linespace-patterns.1 We present a simulation study investigating the extendibility of split pupil exposures to dense arrays of contacts on dark field and light field masks using different mask absorber options. Our simulations indicate that the combination of split pupil exposures and low-n/low-k absorbers can offer comfortable imaging performance for arrays of 10 nm square contacts with a pitch of 20 nm on a dark field mask. These results indicate the potential of combining low-n absorbers and split pupil exposure strategies to enable high-NA EUV lithography to reach its ultimate optical resolution limits.

Multi-photon polymerization (MPP) is widely recognised as promising approach for the fabrication of fully 3D micro- or nano-metric structures, occurring for example in 3D optical metasurfaces. However, increases in fabrication speed by parallelizing the write process are required to facilitate industrial scale application. We present our work on the adaptation of a photolithography simulator, Dr LITHO, to model the DOE and SLM parallelised MPP process in a novel photoplotter currently developed in the EU Fabulous project, including the calibration of key simulator parameters using feedback from the parallel MPP fabrication of different test structures.

This paper addresses the development and modeling of a novel negative-tone photoresist, the Multi-Trigger Resist (MTR), aimed at enhancing extreme ultraviolet (EUV) lithography. The primary objectives are to create a stochastic model of the photoresist process flow and to analyze the behavior of MTR under various conditions. The model incorporates statistical fluctuations in photon distribution, secondary electron generation, and molecular distribution within the photoresist. Calibration with experimental data demonstrates the model's capability in predicting critical dimension (CD), line width roughness (LWR), and dose-to-size (DtS) metrics. Results indicate that higher MTR loading and quencher addition improve performance consistency. Further investigation into different exposure types, electron blur lengths, and photoresist absorbance coefficients reveals the intricate balance required for optimizing lithographic outcomes. The findings underscore the model's utility in guiding the design of next-generation photoresists and highlight areas for future research, including the expansion of calibration datasets and the application to various lithographic features.