Local Level Set Based Mask Optimization with Semi-Implicit Discretization

Yijiang Shen; Xiaopeng Wang; Yanzhou Zhou; Zhenrong Zhang

doi:10.3788/AOS202141.0911004

Journals >Acta Optica Sinica >Volume 41 >Issue 9 >Page 0911004 > Article

Acta Optica Sinica
Vol. 41, Issue 9, 0911004 (2021)

Local Level Set Based Mask Optimization with Semi-Implicit Discretization

Yijiang Shen^1、*, Xiaopeng Wang¹, Yanzhou Zhou¹, and Zhenrong Zhang²

Author Affiliations

¹School of Automation, Guangdong University of Technology, Guangzhou, Guangdong 510006, China

²School of Computer, Electronics and Information, Guangxi University, Nanning, Guangxi 530004, China

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DOI: 10.3788/AOS202141.0911004 Cite this Article Set citation alerts

Yijiang Shen, Xiaopeng Wang, Yanzhou Zhou, Zhenrong Zhang. Local Level Set Based Mask Optimization with Semi-Implicit Discretization[J]. Acta Optica Sinica, 2021, 41(9): 0911004 Copy Citation Text

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Fig. 1. Schematic of vector imaging formation

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Numerical simulation results. (a) Annular source J; (b) desired pattern I0

Fig. 2. Numerical simulation results. (a) Annular source J; (b) desired pattern I₀

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Fig. 3. Lithographic imaging performance with multi-density mask pattern I₀. (a) Simulation results without OPC and I₀ as illuminated mask; (b) conventional level-set MO; (c) semi-implicit MO; (d) localized semi-implicit MO

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Fig. 4. Pattern error versus simulation time for desired pattern I₀

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Fig. 5. Simulation experiment running time

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Fig. 6. Optimization of monitoring pixels. (a) Begin; (b) end

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Fig. 7. Monitoring pixel counting in x and y coordinates with respect to iteration

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Fig. 8. Time to solve linear system of equations in semi-implicit and localized semi-implicit MO approaches with respect to iteration

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Mask synthesis with semi-implicit discretization
Input is target pattern I₀
Output is synthesized mask M
1. Initialize M as I₀, ω=arccos(2\|M\|-1), according to Eq. (3)
2. Set K as maximum iteration number
3. Start iteration with k=0
4. While k<K
5. Construct partial differential equation according to Eq. (5)
6. Label pixels in x coordinate with $\|ω_{x}\|$ >e_thr as monitoring pixels
7. Compute linear system of equations [E-2τA_x(ω^k)]u^k⁺¹=ω^k for u^k⁺¹
8. Label pixels in x coordinate with $\|ω_{y}\|$ >e_thr as monitoring pixels
9. Compute linear system of equations [E-2τA_x(ω^k)]v^k⁺¹=ω^k for v^k⁺¹
10. Average ω^k⁺¹=0.5(u^k⁺¹+v^k⁺¹)
11. Update \|M^k⁺¹\|= $\frac{1 + \cos (ω^{k + 1})}{2}$ according to Eq. (3)
12. k=k+1
13. end while
14. Return M=M^k⁺¹

Table 1. Main work flow of proposed approach

Yijiang Shen, Xiaopeng Wang, Yanzhou Zhou, Zhenrong Zhang. Local Level Set Based Mask Optimization with Semi-Implicit Discretization[J]. Acta Optica Sinica, 2021, 41(9): 0911004

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