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

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2012 (3)

2011 (8)

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express19, 5511–5521 (2011).
[CrossRef] [PubMed]

N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express19, 19384–19398 (2011).
[CrossRef] [PubMed]

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]

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]

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]

2010 (3)

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]

E. Y. Lam and A. K. Wong, “Nebulous hotspot and algorithm variability in computation lithography,” J. Micro/Nanolith., MEMS, MOEMS9, 033002 (2010).

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]

2009 (4)

2008 (2)

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.

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]

2007 (2)

A. Poonawala and P. Milanfar, “Mask design for optical microlithography— an inverse imaging problem,” IEEE Trans. Image Process.16, 774–788 (2007).
[CrossRef] [PubMed]

D. Noll, “Local convergence of an augmented Lagrangian method for matrix inequality constrained programming,” Optim. Method Softw.22, 777–802 (2007).
[CrossRef]

1974 (1)

R. T. Rockafellar, “Augmented Lagrange multiplier functions and duality in nonconvex programming,” SIAM J. Control12, 268–285 (1974).
[CrossRef]

1969 (1)

M. R. Hestenes, “Multiplier and gradient methods,” J. Optimiz. Theory App.4, 303–320 (1969).
[CrossRef]

Adam, K.

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]

Adrichem, P. V.

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]

Afonso, M. V.

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]

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]

Arce, G. R.

Babcock, C.

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]

Baik, K.-H.

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.

Baron, S.

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]

Bertsekas, D. K.

D. K. Bertsekas, Constrained Optimization and Lagrange Multiplier Method (Academic, 1982).

Bioucas-Dias, J. M.

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]

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]

Bisschop, P. D.

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]

Boyd, S.

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]

Capodieci, L.

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.

Cecil, T.

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.

Chan, S. H.

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]

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.

Chao, H.-Y.

Choy, S. K.

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]

Chu, E.

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]

Chua, G.

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]

Cobb, N. B.

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).

Dam, T.

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.

Deng, Y.

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.

Domnenko, V.

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]

Dover, R.

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]

Eckstein, J.

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]

Fakhry, M.

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]

Feng, M.

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]

Fessler, J. A.

S. Ramani and J. A. Fessler, “Parallel MR image reconstruction using augmented Lagrangian methods,” IEEE Trans. Image Process.30, 694–706 (2011).
[CrossRef]

Figueiredo, M. A. T.

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]

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]

Fischer, T.

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]

Foong, Y.

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]

Gibson, K. B.

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]

Gill, P. E.

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]

Gleason, B.

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.

Goldstein, T.

T. Goldstein and S. Osher, “The split Bregman algorithm for l1 regularized problems,” SIAM J. Imaging Sciences2, 323–343 (2009).
[CrossRef]

Granik, Y.

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]

Gronlund, K.

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]

Hestenes, M. R.

M. R. Hestenes, “Multiplier and gradient methods,” J. Optimiz. Theory App.4, 303–320 (1969).
[CrossRef]

Hsu, S.

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]

Hu, P.

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.

IL, C. B.

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]

Iwase, K.

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]

Jessy, S.

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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.
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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.
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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).
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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.
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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.
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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).
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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).
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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.
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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.

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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]

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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.

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.
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[CrossRef]

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]

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Figures (7)

Fig. 1
Fig. 1

Simulation results of the test pattern with different choices of μ. Top row is the target image, and middle row presents the optimized masks and the corresponding outputs with pattern error (PE) are in the third row. The units of PE are in pixels.

Fig. 2
Fig. 2

Convergence profile of the proposed algorithm using different values of τ.

Fig. 3
Fig. 3

Two test patterns used in experiments: (a) cross gate design and (b) brick poly array. Red and magenta lines mark the critical locations for measuring the process window of the two patterns, respectively.

Fig. 4
Fig. 4

Simulation results of the first test pattern.

Fig. 5
Fig. 5

Simulation results of the second test pattern.

Fig. 6
Fig. 6

Magnified aerial image threshold contours of (a) near-end regions (c) center regions. (b) is one magnified poly pattern with lines for EPE evaluation.

Fig. 7
Fig. 7

Comparison of average process window of (a) cross gate design and (b) brick poly array.

Tables (3)

Tables Icon

Table 1 Pseudo-code of mask optimization procedure

Tables Icon

Table 2 Pseudo-code of source optimization procedure

Tables Icon

Table 3 Comparison of performance and convergence rate

Equations (25)

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z a l = 1 P λ l H ˜ l * m 2 ,
z a = I s s .
z = 1 1 + e α ( z a t r ) ,
𝒟 x ( m ) = 𝒱 ( M x M ) and 𝒟 y ( m ) = 𝒱 ( M y M ) ,
𝒟 = [ 𝒟 x 𝒟 y ] .
μ 2 z z 0 2 2 + 𝒟 ( m z 0 ) 1 .
minimize m f 1 ( m ) = μ 2 z z 0 2 2 + v 1 subject to v = 𝒟 ( m z 0 ) , 0 m 1 .
L ρ ( m , v , d ) = μ 2 z z 0 2 2 + v 1 d T [ v 𝒟 ( m z 0 ) ] + ρ 2 v 𝒟 ( m z 0 ) 2 2 .
m k + 1 = argmin m μ 2 z z 0 2 2 d k T [ v k 𝒟 ( m z 0 ) ] + ρ 2 v k 𝒟 ( m z 0 ) 2 2 subject to 0 m 1 ,
v k + 1 = argmin v v 1 d k T [ v 𝒟 ( m k + 1 z 0 ) ] + ρ 2 v 𝒟 ( m ) k + 1 z 0 2 2 ,
d k + 1 = d k ρ [ v k + 1 𝒟 ( m k + 1 z 0 ) ] ,
L ρ ( m , v , d ) m = μ α Re { l = 1 P λ l ( H ¯ l * [ ( z z 0 ) z ( 1 z ) ( H ˜ l * m ) ] ) } + 𝒟 T ( d k ) ρ 𝒟 T ( v k ) + ρ 𝒟 T 𝒟 ( m ) ρ 𝒟 T 𝒟 ( z 0 ) ,
v k + 1 = max { | d k ρ + 𝒟 ( m k + 1 z 0 ) | 1 ρ , 0 } sgn [ d k ρ + 𝒟 ( m k + 1 z 0 ) ] ,
sgn ( x ) = { 1 x > 0 0 x = 0 1 x < 0 .
minimize s f 2 ( s ) = μ 2 z z 0 2 2 + v 1 subject to v = 𝒟 ( s ) , s 0 .
L ρ ( s , v , d ) = μ 2 z z 0 2 2 + v 1 d T [ v 𝒟 ( s ) ] + ρ 2 v 𝒟 ( s ) 2 2 .
s k + 1 = argmin s μ 2 z z 0 2 2 d k T [ v k 𝒟 ( s ) ] + ρ 2 v k 𝒟 ( s ) 2 2 subject to s 0 ,
v k + 1 = argmin v v 1 d k T [ v 𝒟 ( s k + 1 ) ] + ρ 2 v 𝒟 ( s k + 1 ) 2 2 ,
d k + 1 = d k ρ [ v k + 1 𝒟 ( s k + 1 ) ] .
L ρ ( s , v , d ) s = μ α I s T [ ( z z 0 ) z ( 1 z ) ] + 𝒟 T ( d k ) ρ 𝒟 T ( v k ) + ρ 𝒟 T 𝒟 ( s ) .
v k + 1 = max { | d k ρ + 𝒟 ( s k + 1 ) | 1 ρ , 0 } sgn ( d k ρ + 𝒟 ( s k + 1 ) ) .
minimize s f ( s ) + r g ( u ) subject to u = g ( s ) , s 0 .
ρ = { ρ , if v k + 1 𝒟 ( m k + 1 z 0 ) 2 η τ ρ , otherwise ,
L ρ ( m , v , d ) m = μ 2 z z 0 2 2 m d T [ v 𝒟 ( m z 0 ) ] m + ρ 2 v 𝒟 ( m z 0 ) 2 2 m = μ 2 [ 2 α ( z z 0 ) z ( 1 z ) z a m ] + 𝒟 T d + ρ 2 [ v 𝒟 ( m z 0 ) ] T [ v 𝒟 ( m z 0 ) ] m = μ α Re { l = 1 P λ l ( H ¯ l * [ ( z z 0 ) z ( 1 z ) ( H ˜ l * m ) ] ) } + 𝒟 T ( d ) ρ 𝒟 T ( v ) + ρ 𝒟 T 𝒟 ( m ) ρ 𝒟 T 𝒟 ( z 0 ) .
L ρ ( s , v , d ) s = μ 2 z z 0 2 2 s d T [ v 𝒟 ( s ) ] s + ρ 2 v 𝒟 ( s ) 2 2 s = μ 2 [ 2 α ( z z 0 ) z ( 1 z ) z a s ] + 𝒟 T d + ρ 2 [ 2 v T 𝒟 ( s ) + s T 𝒟 T 𝒟 ( s ) ] s = μ α I s T [ ( z z 0 ) z ( 1 z ) ] + 𝒟 T ( d ) ρ 𝒟 T ( v ) + ρ 𝒟 T 𝒟 ( s ) .

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