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Curvilinear Mask Optimization for Inverse Lithography Based on B-splines and Delaunay Triangulation

Xiaoru Yi, Junqing Chen

TL;DR

This work introduces a gradient-based inverse lithography approach for curvilinear mask optimization by representing mask boundaries with periodic B-splines. It couples Delaunay triangulation of the spline-defined regions with explicit gradient relations between control points and sampling points to enable efficient optimization. An unconstrained objective with a sigmoid-transformed threshold is minimized to match target image intensities, and the authors provide a complete workflow from region discretization to gradient computation. Numerical experiments across diverse patterns demonstrate the method's adaptability and potential for producing manufacturable, high-fidelity curvilinear masks. The approach reduces edge placement errors and handles multiple regions effectively, with detailed analysis of accuracy and computational cost.

Abstract

In this paper, we propose a novel gradient-based method to optimize curvilinear masks in optical lithography. The mask pattern is represented by periodic B-spline curves. We apply Delaunay triangulation to discretize the domains circled by the spline curves. Subsequently, we establish an explicit relationship between the integral points and the control points of the boundary spline curve. Based on the relationship, we derive explicit formulas of the gradient of the optimization objective function with respect to the coordinates of the control points. Then we propose an inverse lithography algorithm to optimize the curvilinear mask pattern. Finally, the results of the numerical experiments demonstrate the feasibility and extensive adaptability of our method.

Curvilinear Mask Optimization for Inverse Lithography Based on B-splines and Delaunay Triangulation

TL;DR

This work introduces a gradient-based inverse lithography approach for curvilinear mask optimization by representing mask boundaries with periodic B-splines. It couples Delaunay triangulation of the spline-defined regions with explicit gradient relations between control points and sampling points to enable efficient optimization. An unconstrained objective with a sigmoid-transformed threshold is minimized to match target image intensities, and the authors provide a complete workflow from region discretization to gradient computation. Numerical experiments across diverse patterns demonstrate the method's adaptability and potential for producing manufacturable, high-fidelity curvilinear masks. The approach reduces edge placement errors and handles multiple regions effectively, with detailed analysis of accuracy and computational cost.

Abstract

In this paper, we propose a novel gradient-based method to optimize curvilinear masks in optical lithography. The mask pattern is represented by periodic B-spline curves. We apply Delaunay triangulation to discretize the domains circled by the spline curves. Subsequently, we establish an explicit relationship between the integral points and the control points of the boundary spline curve. Based on the relationship, we derive explicit formulas of the gradient of the optimization objective function with respect to the coordinates of the control points. Then we propose an inverse lithography algorithm to optimize the curvilinear mask pattern. Finally, the results of the numerical experiments demonstrate the feasibility and extensive adaptability of our method.

Paper Structure

This paper contains 16 sections, 2 theorems, 75 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Basic theory of optical lithography
  4. Periodic B-spline curve boundary representation
  5. Forward and inverse lithography based on B-spline function
  6. Region approximation and discretization
  7. Numerical imaging formula
  8. Inverse lithography problem
  9. Algorithm
  10. Numerical Results
  11. Example 1.
  12. Example 2
  13. Example 3
  14. Example 4
  15. Example 5
  16. ...and 1 more sections

Key Result

Lemma 3.1

Let $f(x)$ have an $(n+1)$-th derivative, $f^{(n+1)}(x)$, in an interval $[a,b]$. Let $P_n(x)$ be the interpolation polynomial for f(x) with respect to $n+1$ distinct points $x_k,k=0,1,\cdots,n$ in the interval $[a,b]$(i.e. $P_n(x_k)=f(x_k)$ and $x_k\in[a,b]$). Then for each $x\in [a,b]$ there exist such that

Figures (10)

  • Figure 1: Basic representation of an optical lithography imaging system
  • Figure 2: Procedure of polygonal approximation and triangulation
  • Figure 3: Error of polygon approximation
  • Figure 4: Error estimation in one arch
  • Figure 5: explicit relation building
  • ...and 5 more figures

Theorems & Definitions (9)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 3.1
  • Theorem 3.1
  • proof
  • Remark 3.1