TokMan:Tokenize Manhattan Mask Optimization for Inverse Lithography

We sincerely thank the reviewer for the detailed and thoughtful feedback. Below, we provide clarification and responses to the raised questions and concerns.

Q1&2: What is a clearer explanation of “lithography-aware points segmentation” and the specific criteria for "spatial alignment casting" and corner detection?

Thank you for highlighting the need for further clarification. The lithography-aware points segmentation module is the foundation of our sequence formulation. The key motivation stems from the fact that, under a fixed forward lithography model, the printability of a given layout pattern is influenced not only by its own shape but also by the spatial arrangement of nearby patterns.

Our segmentation algorithm converts polygonal edges into a dense sequence of axis-aligned points by incorporating both intrinsic and contextual lithographic considerations:

Corner-based refinement: Corners are known to suffer from corner rounding due to diffraction. To mitigate this, we apply fine-grained segmentation near detected corners, increasing sampling resolution in these regions to allow for more precise correction.

Spatial alignment casting: For a given edge, we detect other parallel edges in its lithographic interaction zone (a configurable spatial window). The corner vertices of these nearby features are then projected onto the edge being segmented. These projections act as segmentation anchors, embedding cross-feature contextual awareness into the point sequence.

Long-segment partitioning: For extended edges that exceed a lithography-aware printable length threshold, we insert uniformly spaced anchor points, ensuring compliance with minimum printable feature rules.

This process yields a point sequence that not only represents the geometric structure of a polygon, but also encodes interactions and proximity effects from surrounding patterns, making it lithography-aware by construction.

Q3: How sensitive is overall performance to segmentation parameters?

The segmentation parameters—such as corner refinement granularity, alignment radius, and minimum printable length—are selected to reflect realistic manufacturing constraints based on industrial guidelines.

While the model is relatively robust to minor variations in these settings, overly coarse segmentation (e.g., skipping fine corner refinement or disabling spatial alignment casting) can degrade model performance, particularly in terms of Edge Placement Error (EPE), due to insufficient resolution in geometrically or optically sensitive regions.

Conversely, excessive segmentation granularity may lead to longer sequences and increased computational burden, without proportional gain in accuracy. In our experiments, we tuned these parameters empirically to balance manufacturability, simulation accuracy, and training stability. We will add a supplementary ablation of segmentation granularity in the revised version.

Q4: Lack sufficient detail for reproduction for segmentation algorithm; Lack of theoretical analysis

Our segmentation pipeline is described in detail in Section 4.1, where we explicitly introduce three key components: (1) corner rounding mitigation through dense segment splits at polygon corners, (2) spatial alignment casting that ensures connectivity across neighboring edges by projecting parallel corner anchors, and (3) minimum printable segment enforcement that guarantees DFM compliance. These rules are visually illustrated in Figure 3a and are grounded in real lithographic considerations. We also provide extensive descriptions in the Introduction (lines 13–16, 68–72) to motivate this segmentation from both geometric and manufacturing perspectives.

Our work is primarily empirical, but we do motivate the effectiveness of tokenization in Sections 1 and 4.2. The key insight is that Manhattan layouts exhibit strong geometric structure and low entropy, making them highly compatible with sequence modeling. By operating on axis-aligned point tokens rather than pixel grids, our model inherits inductive biases that align with manufacturable mask design. This results in better generalization, stability, and manufacturability, as shown empirically.

Q5: Evaluation is limited to ICCAD2013; synthetic training data; potential mismatch in referenced baseline results

These are important concerns, and we appreciate the chance to clarify.

ICCAD2013 Benchmarks: We chose ICCAD2013 because it remains the most widely used open-source benchmark in OPC literature, allowing direct comparison to prior works such as Neural-ILT and Multi-ILT. We fully agree that more diverse and modern datasets would strengthen the evaluation, and we are currently working with industry partners to validate TokMan on more advanced technique nodes.

Synthetic Dataset Generation: While our training dataset is generated by rearranging real polygons from open benchmarks, we preserve authentic design characteristics such as density, edge alignment, and spacing constraints. Empirically, our model generalizes well to unseen real layouts within the ICCAD2013 test set. These properties are governed by standard design rules, ensuring that the synthesized layouts remain representative and manufacturable.

Potential Mismatch in Baseline Results: All baseline results are obtained using either publicly available open-source implementations or reproduction code that has been acknowledged by the respective authors. Some numerical differences from the originally reported results arise due to differences in evaluation methodology. For instance, in certain official implementations, the mask shot count is computed on a downsampled version of the mask (e.g., from 2048×2048 to 512×512), which we believe does not reflect actual manufacturability and leads to underestimation. We recomputed mask shots at native resolution for fairness. Similarly, our EPE calculation differs slightly from some prior works; we follow a more precise, industry-standard method based on merit points and contour distances, as described in detail in Section 5.1. We believe these choices ensure a more accurate and consistent comparison across all methods.

Q6: Minor issue in Figure 1 ("w/ OPT")

Thank you for spotting this. It should indeed read "w/ OPC." We will correct this in the final version.

We thank the reviewer for their thoughtful feedback and helpful questions. We address the concerns as follows.

Q1: Could you please elaborate on the details of post-processing?

We apologize for not providing sufficient details in the original manuscript. As briefly mentioned in Section 4.3, we apply a lightweight ILT refinement as a post-processing step. Specifically, as illustrated on the right side of Figure 2, we treat the predicted displacements from the DiT model as optimization variables. These displacements are first rasterized into a mask image via a differentiable renderer and then passed through our lithography simulator. The simulated wafer image is compared against the target layout using a differentiable loss, and the resulting gradient is backpropagated to refine the displacements.

Importantly, Manhattan mask optimization is an NP-hard, highly underdetermined problem with a vast solution space. Our approach alleviates this by using a learning-based model to generate high-quality initializations that significantly reduce the computational burden during refinement, while remaining fully compatible with traditional optimization-based industrial flows for seamless downstream integration. The entire pipeline is independently designed to align closely with existing industry practices, making it easy to adopt in real-world settings. While the learning-based model remains the core of our method—responsible for producing manufacturable and high-fidelity initial solutions—the hybrid formulation accelerates convergence and has been particularly well-received in early-stage industrial validation. This fast-generation-plus-refinement strategy has been preliminarily validated on industrial datasets and is actively being tested in production environments.

Q2: Why does increasing the number of decoder layers or attention heads (as in Figure 6) result in degraded performance?

This is a great question. In our ablation study, we observed that while small values of feature dimension, number of attention heads, or decoder layers can lead to underfitting (i.e., insufficient model capacity), simply increasing these parameters does not always yield better results.

When the number of attention heads or decoder layers becomes too large, the model becomes over-parameterized relative to the dataset size and task complexity. This can lead to optimization instability and difficulty in generalization, especially given the statistical nature of the training data and the limited number of correction tokens per sample. In particular, for OPC corrections, the model must resolve fine-grained, spatially localized variations—which excessive global attention capacity can dilute.

This effect is most noticeable with large feature dimensions, where the model may struggle to converge within the given training schedule and may overfit to high-frequency noise. Therefore, careful architectural balancing is critical for achieving both accuracy and generalization. We will clarify this analysis in the ablation section.

Q3: Regarding the comparison with "Fracturing-aware Curvilinear ILT via Circular E-beam Mask Writer"

We appreciate the reviewer’s suggestion to consider broader comparisons. However, we believe that a direct comparison with “Fracturing-aware Curvilinear ILT via Circular E-beam Mask Writer” may not be entirely appropriate due to fundamental differences in target mask formats and downstream manufacturing flows.

Specifically, the referenced work focuses on curvilinear masks designed for circular e-beam writers, whereas TokMan is tailored for Manhattan masks, which remain the dominant format in optical lithography pipelines due to their compatibility with existing EDA tools and manufacturing standards. Comparing across such fundamentally different mask paradigms may not yield meaningful insights, as they operate under different assumptions and optimization goals.

Moreover, from a process integration perspective, the manufacturing flow differs: curvilinear masks typically undergo fracturing before EPC (Electron Proximity Correction), while Manhattan masks require EPC before fracturing. Our work is designed to align with the latter flow, aiming to significantly improve both fidelity and efficiency within the existing Manhattan mask infrastructure widely adopted in industry.

We acknowledge the value of curvilinear ILT approaches in their respective subdomain, but we believe it is more appropriate to view them as addressing a different formulation of the ILT problem, rather than as the SOTA baseline for our target setting. We will consider including a brief discussion of curvilinear ILT approaches in the final version to clarify this distinction and position our work more clearly.

We sincerely thank the reviewer for the constructive comments and suggestions. Below, we address the raised concerns.

Q1: Would it be possible to include visualization results for the ablation study of each module to better understand their individual contributions? Could the authors clarify where the improvements occur in visual comparisons, for example using zoomed-in regions?

Thank you for the valuable suggestion. We initially chose not to include extensive visualizations for ablation studies because, in Manhattan mask layouts, even significant numerical improvements (e.g., in EPE) may correspond to visually subtle changes. Unlike natural images, lithographic masks are discrete and highly structured, making qualitative differences harder to perceive. This also highlights an important insight: CV-based models, when properly adapted, can be effectively transferred to non-visual, high-precision domains such as computational lithography.

That said, we agree that focused zoomed-in visualizations can improve interpretability. In the revised version, we will include annotated printed wafer images highlighting localized improvements—for instance, better reconstruction of M-shaped patterns or reduced line-edge deformation in dense regions, which can be seen in Supplementary Figures 2 and 3. These changes, although subtle, are critical for manufacturability and print fidelity.

Thank you for the responses to the raised concerns. I appreciate the discussion on the limitations of visualization in Manhattan mask layouts. That said, after carefully reviewing the rebuttal, I would like to explain why I still lean toward a borderline reject decision:

While I understand that numerical improvements (e.g., in EPE) might be difficult to observe visually, the lack of compelling qualitative evidence still makes it challenging to fully assess the contribution of individual modules. Even subtle differences, if critical to manufacturability, should ideally be more clearly communicated, perhaps with domain-specific visualization strategies. A more rigorous ablation could help build confidence.

Overall, I believe the paper is promising and could reach acceptance with further revision and stronger visual or experimental support. Therefore, I maintain the score at this time.

We sincerely thank the reviewer for the valuable comments. Below, we address each of the raised questions and concerns.

Q1: How does the work customize for patterns other than piecewise constant?

Thank you for raising this important question. In Section 4.1 of our paper, we detail our lithography-aware segmentation algorithm, which currently focuses on Manhattan geometries—i.e., piecewise constant patterns composed of axis-aligned polygon edges. The segmentation process involves three key stages: (1) dense sampling at corners to mitigate rounding artifacts, (2) spatial alignment casting to enhance consistency across neighboring features, and (3) minimum-feature-length–based partitioning to enforce DFM constraints.

While this paper targets Manhattan mask optimization, we believe the segmentation algorithm is highly extensible to non-axis-aligned and even curvilinear masks. For example, diagonal edges can still be segmented uniformly along their linear span. In the case of curved features—such as Bézier or circular arcs—the segmentation can be performed based on curvature, control points, or arc length to ensure optical fidelity.

Exploring such generalizations to non-piecewise-constant layouts represents an exciting direction for future work, particularly in adapting tokenized representations to richer geometric primitives while preserving compatibility with differentiable lithography pipelines.

Q2: Is this work derived from some ideas on inverse problems in imaging? Cite relevant works, if any.

Our work is conceptually grounded in the framework of inverse problems in imaging. Inverse lithography itself is a classical inverse imaging task, where the goal is to infer a mask pattern that, under a forward optical projection model, reproduces a desired resist pattern on the wafer. Beyond the classical formulation, our main novelty lies in recasting inverse lithography as a sequence modeling problem, inspired by the "tokenize everything" paradigm recently adopted in vision and geometry tasks. Works such as MeshGPT [Shue et al., 2023] and EgoEgo [Chi et al., 2023] demonstrate that structured geometric data can be effectively represented and optimized through discrete token sequences. We extend this idea to lithographic mask optimization under Manhattan constraints, enabling our model to operate directly in a manufacturable token space rather than pixel grids. To our knowledge, this is the first work to apply tokenized diffusion transformers to inverse lithography.

Q3: Limitations — The paper could be written better to motivate readers.

We appreciate this suggestion and acknowledge that the manuscript could benefit from clearer motivation and exposition. In future revisions, we will improve the introduction to better highlight the industrial need for scalable, manufacturable OPC solutions and clarify why tokenization is well-suited for lithography. Structured layouts such as Manhattan masks naturally align with discrete token sequences, making our formulation both geometrically meaningful and fabrication-aware. We will make this connection more explicit and warmly welcome any specific suggestions the reviewer may have. Our broader goal is to bring cutting-edge AI techniques—particularly tokenized generative modeling—into semiconductor lithography, and we hope this work helps bridge these two technically rich and impactful domains.