## Efficient source and mask optimization with augmented Lagrangian methods in optical lithography |

Optics Express, Vol. 21, Issue 7, pp. 8076-8090 (2013)

http://dx.doi.org/10.1364/OE.21.008076

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### Abstract

Source mask optimization (SMO) is a powerful and effective technique to obtain sufficient process stability in optical lithography, particularly in view of the challenges associated with 22nm process technology and beyond. However, SMO algorithms generally involve computation-intensive nonlinear optimization. In this work, a fast algorithm based on augmented Lagrangian methods (ALMs) is developed for solving SMO. We first convert the optimization to an equivalent problem with constraints using variable splitting, and then apply an alternating minimization method which gives a straightforward implementation of the algorithm. We also use the quasi-Newton method to tackle the sub-problem so as to accelerate convergence, and a tentative penalty parameter schedule for adjustment and control. Simulation results demonstrate that the proposed method leads to faster convergence and better pattern fidelity.

© 2013 OSA

**OCIS Codes**

(110.3960) Imaging systems : Microlithography

(110.5220) Imaging systems : Photolithography

(110.1758) Imaging systems : Computational imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: January 31, 2013

Manuscript Accepted: March 8, 2013

Published: March 27, 2013

**Citation**

Jia Li, Shiyuan Liu, and Edmund Y. Lam, "Efficient source and mask optimization with augmented Lagrangian methods in optical lithography," Opt. Express **21**, 8076-8090 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8076

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### References

- B. Küchler, A. Shamsuarov, T. Mülders, U. Klostermann, S.-H. Yang, S. Moon, V. Domnenko, and S.-W. Park, “Computational process optimization of array edges,” in Optical Microlithography XXV, W. Conley, ed. (2012), vol. 8326 of Proc. SPIE, p. 83260H. [CrossRef]
- X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express17, 5783–5793 (2009). [CrossRef] [PubMed]
- M. Fakhry, Y. Granik, K. Adam, and K. Lai, “Total source mask optimization: high-capacity, resist modeling, and production-ready mask solution,” in Photomask Technology 2011, W. Maurer and F. E. Abboud, eds. (2011), vol. 8166 of Proc. SPIE, p. 81663M. [CrossRef]
- N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express19, 19384–19398 (2011). [CrossRef] [PubMed]
- S. K. Choy, N. Jia, C. S. Tong, M. L. Tang, and E. Y. Lam, “A robust computational algorithm for inverse photomask synthesis in optical projection lithography,” SIAM J. Imaging Sciences5, 625–651 (2012). [CrossRef]
- Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE Trans. Image Process.20, 2856–2864 (2011). [CrossRef] [PubMed]
- Y. Deng, Y. Zou, K. Yoshimoto, Y. Ma, C. E. Tabery, J. Kye, L. Capodieci, and H. J. Levinson, “Considerations in source-mask optimization for logic applications,” in Optical Microlithography XXIII, M. V. Dusa and W. Conley, eds. (2010), vol. 7640 of Proc. SPIE, p. 7640J.
- D. Zhang, G. Chua, Y. Foong, Y. Zou, S. Hsu, S. Baron, M. Feng, H.-Y. Liu, Z. Li, S. Jessy, T. Yun, C. Babcock, C. B. IL, R. Stefan, A. Navarra, T. Fischer, A. Leschok, X. Liu, W. Shi, J. Qiu, and R. Dover, “Source mask optimization methodology (SMO) and application to real full chip optical proximity correction,” in Optical Microlithography XXV, W. Conley, ed. (2012), vol. 8326 of Proc. SPIE, p. 83261V. [CrossRef]
- K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2X node logic,” in Photomask Technology 2011, W. Maurer and F. E. Abboud, eds. (2011), vol. 8166 of Proc. SPIE, p. 81662A. [CrossRef]
- J.-C. Yu, P. Yu, and H.-Y. Chao, “Fast source optimization involving quadratic line-contour objectives for the resist image,” Opt. Express20, 8161–8174 (2012). [CrossRef] [PubMed]
- J. Li, Y. Shen, and E. Y. Lam, “Hotspot-aware fast source and mask optimization,” Opt. Express20, 21792–21804 (2012). [CrossRef] [PubMed]
- Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express17, 23690–23701 (2009). [CrossRef]
- L. Pang, G. Xiao, V. Tolani, P. Hu, T. Cecil, T. Dam, K.-H. Baik, and B. Gleason, “Considering MEEF in inverse lithography technology (ILT) and source mask optimization (SMO),” in Photomask Technology, H. Kawahira and L. S. Zurbrick, eds. (2008), vol. 7122 of Proc. SPIE, p. 71221W.
- Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express19, 5511–5521 (2011). [CrossRef] [PubMed]
- J.-C. Yu and P. Yu, “Gradient-based fast source mask optimization (SMO),” in Optical Microlithography XXIV, M. V. Dusa, ed. (2011), vol. 7973 of Proc. SPIE, p. 797320. [CrossRef]
- N. Jia and E. Y. Lam, “Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis,” J. Opt.12, 045601 (2010). [CrossRef]
- S. H. Chan, A. K. Wong, and E. Y. Lam, “Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography,” Opt. Express16, 14746–14760 (2008). [CrossRef] [PubMed]
- S. H. Chan and E. Y. Lam, “Inverse image problem of designing phase shifting masks in optical lithography,” in IEEE International Conference on Image Processing, (2008), p. 1832–1835.
- E. Y. Lam and A. K. Wong, “Computation lithography: Virtual reality and virtual virtuality,” Opt. Express17, 12259–12268 (2009). [CrossRef] [PubMed]
- E. Y. Lam and A. K. Wong, “Nebulous hotspot and algorithm variability in computation lithography,” J. Micro/Nanolith., MEMS, MOEMS9, 033002 (2010).
- J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).
- M. R. Hestenes, “Multiplier and gradient methods,” J. Optimiz. Theory App.4, 303–320 (1969). [CrossRef]
- M. Powell, “A method for nonlinear constraints in minimization problems,” in Optimization, R. Fletcher, ed. (1969), Academic, p. 283–298.
- M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process.20, 681–695 (2011). [CrossRef]
- S. Ramani and J. A. Fessler, “Parallel MR image reconstruction using augmented Lagrangian methods,” IEEE Trans. Image Process.30, 694–706 (2011). [CrossRef]
- S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn.3, 1–124 (2011). [CrossRef]
- R. T. Rockafellar, “Augmented Lagrange multiplier functions and duality in nonconvex programming,” SIAM J. Control12, 268–285 (1974). [CrossRef]
- N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. thesis, Univ. of California at Berkeley, Berkeley, California (1998).
- A. K. Wong, Optical Imaging in Projection Microlithography (SPIE, 2005). [CrossRef]
- A. Poonawala and P. Milanfar, “Mask design for optical microlithography— an inverse imaging problem,” IEEE Trans. Image Process.16, 774–788 (2007). [CrossRef] [PubMed]
- T. Goldstein and S. Osher, “The split Bregman algorithm for l1 regularized problems,” SIAM J. Imaging Sciences2, 323–343 (2009). [CrossRef]
- M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process.19, 2345–2356 (2010). [CrossRef] [PubMed]
- J. L. Morales and J. Nocedal, “Remark on ‘algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization’,” ACM Trans. Math Software23, 550–560 (2011).
- S. H. Chan, R. Khoshabeh, K. B. Gibson, P. E. Gill, and T. Q. Nguyen, “An augmented Lagrangian method for total variation video restoration,” IEEE Trans. Image Process.20, 14746–14760 (2011). [CrossRef]
- D. Noll, “Local convergence of an augmented Lagrangian method for matrix inequality constrained programming,” Optim. Method Softw.22, 777–802 (2007). [CrossRef]
- D. K. Bertsekas, Constrained Optimization and Lagrange Multiplier Method (Academic, 1982).

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