The fabrication of a single advanced-node logic or memory chip typically involves thousands of steps, including wafer preparation, deposition, lithography, etching, polishing, cleaning, and metrology, many of which are repeated throughout the process. Lithography is the most critical step, using a scanner or stepper to transfer circuit patterns from a mask onto a photoresist-coated silicon wafer through exposure imaging. This process exemplifies the technical sophistication of integrated circuit manufacturing and directly determines the minimum printable feature size. 193nm ArF immersion lithographyand 13.5nm EUV lithography are widely used in the semiconductor industry to manufacture integrated circuitsat 28nm node and beyond. The manufacturing process is similar to creating thin lines with a broad brush, which is physically beyond the resolution limit. When the exposure beam passes through the mask patterns, optical proximity effects such as diffraction become highly pronounced. This phenomenon leads to significant discrepancies between the exposed image formed on the photoresist and the original mask layout. This situation, analogous to drawing fine lines with a broad brush, would result in severe imaging distortions and a notable degradation in pattern fidelity.

Computational lithography makes it possible to realize a manufacturing paradigm akin to "drawing fine lines with a broad brush," while ensuring high-volume and high-fidelity integrated circuit production. It has thus emerged as one of the key drivers sustaining Moore's Law. The principle of computational lithography is based on modeling and simulation methodologies that establish connections between the imaging system, which includes the illumination source, mask, and projection optics, and process steps such as photoresist exposure and etching. Through mathematical approaches, including mask pattern decomposition and correction, illumination pupil optimization, and projection lens parameter adjustment, it effectively reduces the impact of optical proximity effects on exposure imaging. This enhancement in lithographic resolution helps address the technological challenge of manufacturing advanced-node chips, which remains difficult even with advanced lithography equipment. The success of computational lithography relies heavily on two main procedures involved, namely, the fast and accurate forward optical imaging simulation and the fast and robust inverse source mask optimization strategy.

Figure1 Fundamental Principles and Key Components of Computational Lithography

Ⅰ Scientific problems and methods

Based on the fundamental characteristics of computational lithography, we have systematically summarized several universal and critical scientific challenges that require urgent solutions. These challenges primarily include: the accurate and efficient modeling and solving of mask 3D effects; the high-precision and efficient modeling of photoresist exposure response behavior; the inherent contradiction between the region size and efficiency in mask optimization; the manufacturability assurance and robust solving for mask optimization; and the accurate and efficient detection of lithographic mask hotspots. In addressing the aforementioned scientific challenges, this research leverages modern mathematical theories, physical optics, and computational optimization methods, while incorporating recent advancements in artificial intelligence and machine learning. The investigation focuses on several key areas: rapid and accurate numerical methods for modeling mask 3D effects in extreme ultraviolet lithography; compact models for photoresist behavior and their efficient calibration techniques; fast computational methods for lithographic exposure imaging; accelerated modeling and solution techniques for vectorial lithography imaging based on convolutional variable separation; efficient and robust inverse optimization methods for source-mask co-design; high-throughput detection of lithographic hotspots; and minimalist representation schemes with efficient optimization methodologies for curvilinear masks. Based on the research above, this work aims to overcome the limitations of traditional computational lithography technologies in terms of computational accuracy, efficiency, memory consumption, process window extension, compatibility with curvilinear masks, adaptability to full-chip operational conditions, and applicability to emerging lithography scenarios. It seeks to explore new computational lithography principles and methodologies for full-chip extreme-short-wavelength lithography manufacturing. The research will provide innovative theoretical foundations and cutting-edge technical solutions for lithography process development in advanced manufacturing contexts, including advanced-node integrated circuit manufacturing and atomic-scale chip fabrication. Ultimately, this effort will support the achievement of fully independent and controllable development in China's computational lithography EDA software domain.

II. Computationallithographyalgorithmsoftwaredevelopment andapplication

(1) Rapid and precise numerical methods for modeling mask 3D effects in EUV lithography

As a reflective photomask fundamentally distinct from conventional transmissive counterparts, the extreme ultraviolet (EUV) lithography mask comprises a Mo/Si multilayer stack and a patterned absorber layer made of tantalum-based materials such as TaN/TaBN. Given that the absorber's depth and width are several times larger than the exposure wavelength, the mask exhibits pronounced 3D effects primarily caused by diffraction from the absorber structures and the shadowing effect under oblique illumination. Consequently, accurate and efficient modeling of these effects is indispensable in lithographic simulation to ensure both precision and computational efficiency in predicting the aerial image. Furthermore, directly applying the thin-mask approximation, widely used in deep ultraviolet lithography imaging modeling, or employing perturbation-based corrections derived from it often leads to insufficient simulation accuracy. To address this limitation, we have developed novel near-field modeling and solution methods tailored for different EUV lithography application scenarios. By leveraging the weak optical contrast between the complex refractive index of absorber materials and that of vacuum, we have created efficient and accurate near-field computation algorithms, including a modified Born series-based mask model and a vectorial beam propagation-based mask model. Additionally, we introduced a pseudo-boundary condition method based on angular spectrum propagation to enhance memory utilization and computational efficiency. The resulting solution achieves accuracy comparable to rigorous finite-difference time-domain methods while improving computational speed by two orders of magnitude. This advancement completely resolves the long-standing challenge of balancing accuracy, efficiency, and memory usage in large-scale lithography imaging simulations, thereby providing a core unit technology for full-chip lithography imaging simulation and mask optimization.

Fig.2 Near-field modeling methodologies for EUV lithography masks: (a) modified Born series-based approach; (b) vectorial beam propagation method.

(2) Compactresistmodel anditsfast，accuratecalibrationmethodology

Rapid, accurate, and generalizable photoresist modeling and calibration techniques represent an indispensable component of optical proximity correction (OPC). To address the challenges of nonlinear compact resist modeling and calibration for full-chip OPC applications, this study introduces a cascaded quadratic Wiener-Padé network-based approach capable of modeling both positive- and negative-tone development processes. By incorporating multiple kernel functions capable of characterizing photoresist physicochemical effects into the Padé formulation, along with classical Wiener multiple convolution structures, and replacing traditional high-order approximations with multi-stage low-order cascades, the method achieves low-order fitting of complex nonlinear photoresist response behaviors. Furthermore, we propose a photoresist model calibration methodology that integrates surrogate model-assisted genetic algorithms with constrained quadratic convex optimization. A two-stage calibration strategy based on constrained quadratic convex optimization effectively mitigates overfitting while enabling fast and accurate global calibration of model coefficients. The introduction of surrogate models into the genetic algorithm replaces computationally expensive processes, resulting in rapid, precise, and automated parameter calibration. This approach provides an industry-standard solution, featured by speed, accuracy, and generalizability, for current mainstream-node photoresist simulation and optimization challenges.

Fig.3 A compact photoresist model based on a cascaded quadratic Wiener-Padé network

(3) Fast computation method for lithographic exposure imaging

A kernel decomposition method for imaging systems based on circular sampling functions

The four-dimensional cross transmission coefficient matrix of the imaging system is projected into the space of circular sampling functions to obtain a projection coefficient matrix. Singular value decomposition is applied to this matrix, and the resulting vectors are combined with the circular sampling functions to derive the system kernels. This approach avoids direct singular value decomposition of the four-dimensional cross transmission coefficient matrix, significantly reducing computational time. Moreover, the obtained kernels possess analytical forms, allowing flexible configuration of different resolutions.

A rapid aerial image computation method for single elementary mask patterns

To accelerate aerial image calculation during optical proximity correction, a convolution table is precomputed and stored based on elementary mask patterns and the system kernels. Polygonal masks in integrated circuits are decomposed into superpositions of shifted elementary mask patterns. The aerial image of the full mask is then efficiently synthesized by referencing the precomputed table. This method is particularly suitable for aerial image computation in optical proximity correction workflows.

Fig.4 Rapid computational methods for lithographic exposure imaging: (a) first six kernels obtained through circular sampling function decomposition; (b) schematic illustrating mask pattern representation using elementary mask segments.

(4) Vector imaging simulations using the convolution-variation separation method

Efficient optical imaging simulation with process variations

We propose a novel method, termed Convolution-Variation Separation (CVS), to enable efficient yet accurate optical image simulation across a wide range of process variations. Derived from first principles via a series expansion, the CVS method decomposes the optical imaging model into a set of predetermined basis functions, weighted by corresponding expansion coefficients. These basis functions remain independent of process variations and can be precomputed and stored, whereas the coefficients depend solely on the variation conditions. Applications in defocus and aberration simulations, particularly in robust inverse lithography technology and lens aberration metrology, demonstrate the efficacy and key advantages of the CVS approach.

Vector imaging simulation based on efficient representation of mask transmittance function

We introduce a generalized method for efficient representation of the incident-angle-dependent mask transmittance function (MTF) of thick masks. The method decomposes the MTF into a series expansion comprising predetermined basis functions weighted by corresponding expansion coefficients. Since the basis functions are independent of incident angles, they can be pre-computed and stored offline. In contrast, the expansion coefficients depend solely on incident angles and can be rapidly evaluated online. Simulations of near-field and optical imaging demonstrate that the proposed method achieves excellent accuracy and significant computational speedup. This approach offers a promising pathway to accelerate thick mask simulation substantially without compromising accuracy, showing great potential for application in OPC and inverse lithography technology (ILT).

Fig.5 Modeling approach for lithographic imaging based on convolution-variation separation: (a) simulation results of decomposed and efficient lithographic imaging under the presence of wavefront aberrations; (b) vectorial aerial image simulation based on mask transmittance functions at multiple incidence angles.

(5) Fast and robust inverse source-mask optimization algorithms

Cascadic multigrid algorithm for fast inverse mask synthesis

We propose a cascaded multigrid (CMG) algorithm for fast inverse mask synthesis, which starts from a relatively coarse mask grid and refines it iteratively in stages, so as to achieve a significant increase in speed without compromising numerical accuracy. As a result, our algorithm achieves more than a fourfold increase in speed over conventional methods that synthesize a mask on a fixed fine grid.

Mask manufacturability enhancement methods.

We propose a level-set method to solve the inverse mask synthesis problem, where the boundary of the mask pattern is iteratively evolved instead of the mask itself. It results in a smooth mask contour. In addition, we propose a new regularization framework for inverse lithography that directly regularizes masks by applying a mask filtering technique to improve computational efficiency and enhance mask manufacturability. This technique differs from the conventional regularization method, which regularizes a mask by incorporating various penalty functions into the cost function. A specifically designed mask filter is applied directly to the mask during optimization, effectively removing undesirable features such as islands, broken lines, and grayscale areas. This ensures the manufacturability of the mask throughout each iteration.

Statistical strategy for robust inverse mask synthesis

As critical dimensions continue to shrink, integrated circuit pattern density increases significantly, while lithographic process variations grow more severe. To synthesize masks that remain robust under such variations, we optimize the average wafer performance across expected process fluctuations. This methodology explicitly accounts for process variability in the optimization framework. Furthermore, we analyze how arbitrary statistical distributions of process variations influence the characteristics of the synthesized mask patterns.

Derivative-free optimization for source optimization under a rigorous simulation model.

We present a source optimization (SO) methodology for optical lithography using a rigorous simulation model that accounts for critical effects, including the vector nature of light and mask topography. Our approach introduces a novel source pattern representation characterized by a compact yet complete parameter set. A derivative-free optimization (DFO) technique is then employed to optimize these parameters. Unlike gradient-based methods, the proposed DFO approach requires no closed-form model formulation and is agnostic to the specific form of the cost function.

Fig.6 Fast and robust inverse source-mask optimization: (a) optimization results using multi-grid-based mask optimization; (b) intermediate mask generated with mask filtering during the optimization process; (c) optimized mask and corresponding process window under different process conditions; (d) source optimization based on a derivative-free optimization method.

(6) High-efficiency detection of lithographic hotspots

Although the application of resolution enhancement techniques, including OPC, has significantly improved the imaging fidelity of integrated circuit layouts, they cannot entirely eliminate distortions caused by optical proximity effects. Consequently, even after correction, a limited number of defects containing pinching, bridging, opens, and shorts may persist in the exposed patterns. Regions of the mask layout containing such defects are referred to as lithographic hotspots. The presence of these hotspots substantially impacts the final chip functionality and manufacturing yield. To address this issue, this study investigates methods for detecting lithographic hotspots in post-OPC mask patterns prior toactual tape-out. A grid-indexing-based approach for rapid hotspot detection is proposed, which enhances the detection efficiency for soft pinching and soft bridging defects compared to traditional geometry-based algorithms. Furthermore, an improved YOLOv5 deep learning network is introduced for hotspot detection. Building upon the YOLOv5 architecture, a coordinate attention mechanism is incorporated into the backbone network to enhance the model's focus on critical layout regions. The sigmoid-weighted linear unit is employed as the activation function to strengthen the nonlinear representation capability of the model. Moreover, a Scylla intersection- over-union loss function is utilized to quantitatively evaluate bounding box regression loss, thereby improving hotspot detection performance as well as the convergence speed and accuracy of the detection algorithm.

Fig.7 Lithographichotspot detection method using an enhanced YOLOv5 Network.

Representative Papers：

1. S. Guo, H. Chen, C. Mu, S. Zhang, H. Jiang, D. Wei, Y. Sun, and S. Liu, "Mask3D- compatible full-vectorial Hopkins imaging for lithographic modeling," Optica 12, 924-934 (2025).

2. C. Mu, Z. Song, L. Cheng, S. Guo, K. Li, S. Zhang, H. Jiang, D. Wei, Y. Sun, J. Zhu, and S. Liu, "Efficient resist modeling and calibration using a Wiener-Padé formulation and convex optimizations," Opt. Laser Technol. 189, 113022 (2025).

3. C. Mu, L. Cheng, S. Zhang, H. Jiang, D. Wei, Y. Sun, J. Zhu, and S. Liu, "Efficient nonlinear resist modeling by combining and cascading quadratic Wiener systems," Opt. Laser Technol. 183, 112315 (2025).

4. C. Mu, L. Cheng, Z. Song, S. Guo, K. Li, S. Zhang, H. Jiang, D. Wei, J. Zhu, S. Liu. "Surrogate-assisted genetic algorithm for efficient resist calibration," Front. Mech. Eng. 20, 34 (2025).

5. P. He, J. Liu, H. Gu, H. Jiang, and S. Liu, "Modified Born series with virtual absorbing boundary enabling large-scale electromagnetic simulation," Commun. Phys. 7, 383 (2024).

6. P. He, J. Liu, H. Gu, H. Jiang, and S. Liu, "Linearized EUV mask optimization based on adjoint method," Opt. Express 32, 8415-8424 (2024).

7. P. He, J. Liu, H. Gu, J. Zhu, H. Jiang, and S. Liu, "EUV Mask model based on modified Born series," Opt. Express 31, 27797-27809 (2023).

8. W.Lv, S. Liu, X. Wu, and E. Y. Lam, "Illumination source optimization in optical lithography via derivative-free optimization," J. Opt. Soc. Am. A, 31(12), B19-B27 (2014).

9. X. Zhou, C. Zhang, H. Jiang, H. Wei, and S. Y. Liu, "Efficient representation of mask transmittance functions for vectorial lithography simulations," J. Opt. Soc. Am. A, 31(12), B10-B18 (2014).

10. W. Lv, E. Y. Lam, H. Q. Wei, and S. Liu, "Cascadic multigrid algorithm for robust inverse mask synthesis in optical lithography," J. Micro/Nanolith. MEMS MOEMS 13(2), 023003 (2014).

11. S. Liu, X. Zhou, W. Lv, S. Xu, and H. Wei, "Convolution-variation separation method for efficient modeling of optical lithography," Opt. Lett. 38(13), 2168-2170 (2013).

12. W. Lv, Q. Xia, and S. Liu, "Mask-filtering-based inverse lithography," J. Micro/Nanolith. MEMS MOEMS 12(4), 043003 (2013).

13. W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, "Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step," J. Vac. Sci. Technol. B 31(4), 041605 (2013).