Abstract

Source and mask optimization (SMO) is an important method to improve lithography imaging fidelity. However, constrained by the computational inefficiency, the current SMO method can be used only in clip level applications. In this paper, to our best knowledge, the fast nonlinear compressive sensing (CS) theory is for the first time applied to solve the nonlinear inverse reconstruction problem in SMO. The proposed method simultaneously downsamples the layout pattern in the SMO procedure, which can effectively reduce the computation complexity. The space basis and two-dimensional (2D) discrete cosine transform (DCT) basis are selected to sparsely represent the source pattern and mask pattern, respectively. Based on the sparsity assumption of source and mask pattern, the SMO can be formulated as a nonlinear CS reconstruction problem. A Newton-iteration hard thresholding (Newton-IHTs) algorithm, by taking into account the second derivative of the cost function to accelerate convergence, is innovated to realize nonlinear CS-SMO with high imaging fidelity. Simulation results show the proposed method can significantly accelerate the SMO procedure over a traditional gradient-based method and IHTs-based method by a factor of 9.31 and 7.39, respectively.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (11)

S. Lan, J. Liu, Y. Wang, K. Zhao, and J. Li, “Deep learning assisted fast mask optimization,” Proc. SPIE 10587, 105870H (2018).

S. Wang, S. Baron, N. Kachwala, C. Kallingal, D. Sun, V. Shu, W. Fong, Z. Li, A. Elsaid, J. Gao, J. Su, J. Ser, Q. Zhang, B. Chen, R. Howell, S. Hsu, L. Luo, Y. Zou, G. Zhang, Y. Lu, and Y. Cao, “Efficient full-chip SRAF placement using machine learning for best accuracy and improved consistency,” Proc. SPIE 10587, 105870N (2018).
[Crossref]

H. Vu, S. Kim, J. Word, and Y. Cai, “A novel processing platform for post tape out flows,” Proc. SPIE 10587, 105870R (2018).
[Crossref]

Y. Du, L. Li, J. Zhang, F. Shao, C. Zuniga, and Y. Deng, “A model-based approach for the scattering-bar printing avoidance,” Proc. SPIE 10587, 105870Q (2018).

S. Kobelkov, V. Roizen, S. Rodin, A. Tritchkov, J. Han, and Y. Granik, “Constraint approaches for some inverse lithography problems with pixel-based mask,” Proc. SPIE 10587, 105870I (2018).
[Crossref]

H. Choi and A. Hamouda, “Inverse lithography OPC correction with multiple patterning and etch awareness,” Proc. SPIE 10587, 105870O (2018).
[Crossref]

H. Lee, S. Kim, J. Hong, S. Lee, and H. Han, “Thread scheduling for GPU-based OPC simulation on multi-thread,” Proc. SPIE 10587, 105870P (2018).
[Crossref]

A. Chen, Y. Foong, J. Maeng, N. Jain, and S. McDermott, “Exploration of resist effect in source mask optimization,” Proc. SPIE 10587, 105870J (2018).
[Crossref]

K. Ahi, S. Shahbazmohamadi, and N. Asadizanjani, “Quality control and authentication of packaged integrated circuits using enhanced-spatial-resolution terahertz time-domain spectroscopy and imaging,” Opt. Lasers Eng. 104, 274–284 (2018).
[Crossref]

X. Ma, Z. Wang, H. Lin, Y. Li, G. R. Arce, and L. Zhang, “Optimization of lithography source illumination arrays using diffraction subspaces,” Opt. Express 26(4), 3738–3755 (2018).
[Crossref] [PubMed]

X. Ma, Z. Wang, Y. Li, G. R. Arce, L. Dong, and J. Garcia-Frias, “Fast optical proximity correction method based on nonlinear compressive sensing,” Opt. Express 26(11), 14479–14498 (2018).
[Crossref] [PubMed]

2017 (3)

X. Ma, D. Shi, Z. Wang, Y. Li, and G. R. Arce, “Lithographic source optimization based on adaptive projection compressive sensing,” Opt. Express 25(6), 7131–7149 (2017).
[Crossref] [PubMed]

K. Ahi, “Mathematical Modeling of THz Point Spread Function and Simulation of THz Imaging Systems,” IEEE Trans. Terahertz Sci. Technol. 7(6), 747–754 (2017).
[Crossref]

T. Li and Y. Li, “Lithographic source and mask optimization with low aberration sensitivity,” IEEE Trans. NanoTechnol. 16(6), 1099–1105 (2017).
[Crossref]

2015 (1)

2014 (5)

Z. Song, X. Ma, J. Gao, J. Wang, Y. Li, and G. R. Arce, “Inverse lithography source optimization via compressive sensing,” Opt. Express 22(12), 14180–14198 (2014).
[Crossref] [PubMed]

C. Han, Y. Li, L. Dong, X. Ma, and X. Guo, “Inverse pupil wavefront optimization for immersion lithography,” Appl. Opt. 53(29), 6861–6871 (2014).
[Crossref] [PubMed]

Y. Zhang, Z. Dong, G. Ji, and S. Wang, “An improved reconstruction method for CS-MRI based on exponential wavelet transform and iterative shrinkage/thresholding algorithm,” J. Electromagnet. Wave. 28(18), 2327–2338 (2014).
[Crossref]

S. Patterson, Y. C. Eldar, and I. Keidar, “Distributed compressed sensing for static and time-varying networks,” IEEE T. Signal Processing 62(19), 4931–4946 (2014).

W. Lv, E. Y. Lam, H. Wei, and S. Liu, “Cascadic multigrid algorithm for robust inverse mask synthesis in optical lithography,” J. Micro-Nanolith. Mem. 13(2), 023003 (2014).

2013 (1)

T. Blumensath, “Compressed sensing with nonlinear observations and related nonlinear optimization problems,” IEEE Trans. Inf. Theory 59(6), 3466–3474 (2013).
[Crossref]

2012 (2)

P. Liu, Z. Zhang, S. Lan, Q. Zhao, M. Feng, H. Liu, V. Vellanki, and Y. Lu, “A full-chip 3D computational lithography framework,” Proc. SPIE 8326, 83260A (2012).
[Crossref]

X. Ma, Y. Li, and L. Dong, “Mask optimization approaches in optical lithography based on a vector imaging model,” J. Opt. Soc. Am. A 29(7), 1300–1312 (2012).
[Crossref] [PubMed]

2011 (3)

X. Ma and G. R. Arce, “Pixel-based OPC optimization based on conjugate gradients,” Opt. Express 19(3), 2165–2180 (2011).
[Crossref] [PubMed]

K. Lai, M. Gabrani, D. Demaris, N. Casati, A. Torres, S. Sarkar, P. Strenski, S. Bagheri, D. Scarpazza, A. Rosenbluth, D. Melville, A. Wachter, J. Lee, V. Austel, M. Szeto-Millstone, K. Tian, F. Barahona, T. Inoue, and M. Sakamoto, “Design specific joint optimization of masks and sources on a very large scale,” Proc. SPIE 7973, 797308 (2011).
[Crossref]

K. Tian, M. Fakhry, A. Dave, A. Tritchkov, J. Tirapu-Azpiroz, A. Rosenbluth, D. Melville, M. Sakamoto, T. Inoue, S. Mansfield, A. Wei, Y. Kim, B. Durgan, K. Adam, G. Berger, G. Bhatara, J. Meiring, H. Haffner, and B. Kim, “Applicability of global source mask optimization to 22/20 nm node and beyond,” Proc. SPIE 7973, 79730C (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(4), 45601–45609 (2010).
[Crossref]

Y. V. Miklyaev, W. Imgrunt, V. S. Pavelyev, D. G. Kachalov, T. Bizjak, L. Aschke, and V. N. Lissotschenko, “Novel continuously shaped diffractive optical elements enable high-efficiency beam shaping,” Proc. SPIE 7640, 764024 (2010).
[Crossref]

J. T. Carriere, J. Stack, J. Childers, K. Welch, and M. D. Himel, “Advances in DOE modeling and optical performance for SMO applications,” Proc. SPIE 7640, 764025 (2010).
[Crossref]

2009 (3)

2008 (1)

S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE 7140, 714010 (2008).
[Crossref]

2007 (2)

A. Poonawala and P. Milanfar, “Mask design for optical microlithography--an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007).
[Crossref] [PubMed]

L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): a natural solution for model-based SRAF at 45nm and 32nm,” Proc. SPIE 6607, 660739 (2007).
[Crossref]

2006 (3)

Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Micro-Nanolith. Mem. 5(4), 043002 (2006).

E. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

D. Donoho, “Compressive sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

2005 (3)

Y. Granik, “Solving inverse problems of optical microlithography,” Proc. SPIE 5754, 506–526 (2005).
[Crossref]

C. Progler, W. Conley, B. Socha, and Y. Ham, “Layout and source dependent phase mask transmission tuning,” Proc. SPIE 5454, 315–326 (2005).
[Crossref]

R. Socha, X. Shi, and D. LeHoty, “Simultaneous source mask optimization (SMO),” Proc. SPIE 5853, 180–193 (2005).
[Crossref]

2004 (1)

A. Erdmann, T. Fuehner, T. Schnattinger, and B. Tollkuhn, “Towards automatic mask and source optimization for optical lithography,” Proc. SPIE 5377, 646–657 (2004).
[Crossref]

2002 (1)

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlithogr., Microfabr., Microsyst. 1, 12–30 (2002).

1995 (2)

S. Sherif, B. Saleh, and R. De Leone, “Binary image synthesis using mixed linear integer programming,” IEEE Trans. Image Process. 4(9), 1252–1257 (1995).
[Crossref] [PubMed]

T. J. Ypma, “Historical development of the Newton-Raphson method,” SIAM Rev. 37(4), 531–551 (1995).
[Crossref]

1992 (1)

Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE T. Semiconduct. M. 5(2), 138–152 (1992).
[Crossref]

1980 (1)

J. Nocedal, “Updating Quasi-Newton matrices with limited storage,” Math. Comput. 35(151), 773–782 (1980).
[Crossref]

Abrams, D.

L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): a natural solution for model-based SRAF at 45nm and 32nm,” Proc. SPIE 6607, 660739 (2007).
[Crossref]

Adam, K.

K. Tian, M. Fakhry, A. Dave, A. Tritchkov, J. Tirapu-Azpiroz, A. Rosenbluth, D. Melville, M. Sakamoto, T. Inoue, S. Mansfield, A. Wei, Y. Kim, B. Durgan, K. Adam, G. Berger, G. Bhatara, J. Meiring, H. Haffner, and B. Kim, “Applicability of global source mask optimization to 22/20 nm node and beyond,” Proc. SPIE 7973, 79730C (2011).
[Crossref]

Ahi, K.

K. Ahi, S. Shahbazmohamadi, and N. Asadizanjani, “Quality control and authentication of packaged integrated circuits using enhanced-spatial-resolution terahertz time-domain spectroscopy and imaging,” Opt. Lasers Eng. 104, 274–284 (2018).
[Crossref]

K. Ahi, “Mathematical Modeling of THz Point Spread Function and Simulation of THz Imaging Systems,” IEEE Trans. Terahertz Sci. Technol. 7(6), 747–754 (2017).
[Crossref]

Arce, G. R.

Asadizanjani, N.

K. Ahi, S. Shahbazmohamadi, and N. Asadizanjani, “Quality control and authentication of packaged integrated circuits using enhanced-spatial-resolution terahertz time-domain spectroscopy and imaging,” Opt. Lasers Eng. 104, 274–284 (2018).
[Crossref]

Aschke, L.

Y. V. Miklyaev, W. Imgrunt, V. S. Pavelyev, D. G. Kachalov, T. Bizjak, L. Aschke, and V. N. Lissotschenko, “Novel continuously shaped diffractive optical elements enable high-efficiency beam shaping,” Proc. SPIE 7640, 764024 (2010).
[Crossref]

Austel, V.

K. Lai, M. Gabrani, D. Demaris, N. Casati, A. Torres, S. Sarkar, P. Strenski, S. Bagheri, D. Scarpazza, A. Rosenbluth, D. Melville, A. Wachter, J. Lee, V. Austel, M. Szeto-Millstone, K. Tian, F. Barahona, T. Inoue, and M. Sakamoto, “Design specific joint optimization of masks and sources on a very large scale,” Proc. SPIE 7973, 797308 (2011).
[Crossref]

Bagheri, S.

K. Lai, M. Gabrani, D. Demaris, N. Casati, A. Torres, S. Sarkar, P. Strenski, S. Bagheri, D. Scarpazza, A. Rosenbluth, D. Melville, A. Wachter, J. Lee, V. Austel, M. Szeto-Millstone, K. Tian, F. Barahona, T. Inoue, and M. Sakamoto, “Design specific joint optimization of masks and sources on a very large scale,” Proc. SPIE 7973, 797308 (2011).
[Crossref]

Barahona, F.

K. Lai, M. Gabrani, D. Demaris, N. Casati, A. Torres, S. Sarkar, P. Strenski, S. Bagheri, D. Scarpazza, A. Rosenbluth, D. Melville, A. Wachter, J. Lee, V. Austel, M. Szeto-Millstone, K. Tian, F. Barahona, T. Inoue, and M. Sakamoto, “Design specific joint optimization of masks and sources on a very large scale,” Proc. SPIE 7973, 797308 (2011).
[Crossref]

Baron, S.

S. Wang, S. Baron, N. Kachwala, C. Kallingal, D. Sun, V. Shu, W. Fong, Z. Li, A. Elsaid, J. Gao, J. Su, J. Ser, Q. Zhang, B. Chen, R. Howell, S. Hsu, L. Luo, Y. Zou, G. Zhang, Y. Lu, and Y. Cao, “Efficient full-chip SRAF placement using machine learning for best accuracy and improved consistency,” Proc. SPIE 10587, 105870N (2018).
[Crossref]

Berger, G.

K. Tian, M. Fakhry, A. Dave, A. Tritchkov, J. Tirapu-Azpiroz, A. Rosenbluth, D. Melville, M. Sakamoto, T. Inoue, S. Mansfield, A. Wei, Y. Kim, B. Durgan, K. Adam, G. Berger, G. Bhatara, J. Meiring, H. Haffner, and B. Kim, “Applicability of global source mask optimization to 22/20 nm node and beyond,” Proc. SPIE 7973, 79730C (2011).
[Crossref]

Bhatara, G.

K. Tian, M. Fakhry, A. Dave, A. Tritchkov, J. Tirapu-Azpiroz, A. Rosenbluth, D. Melville, M. Sakamoto, T. Inoue, S. Mansfield, A. Wei, Y. Kim, B. Durgan, K. Adam, G. Berger, G. Bhatara, J. Meiring, H. Haffner, and B. Kim, “Applicability of global source mask optimization to 22/20 nm node and beyond,” Proc. SPIE 7973, 79730C (2011).
[Crossref]

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H. Vu, S. Kim, J. Word, and Y. Cai, “A novel processing platform for post tape out flows,” Proc. SPIE 10587, 105870R (2018).
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S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE 7140, 714010 (2008).
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J. T. Carriere, J. Stack, J. Childers, K. Welch, and M. D. Himel, “Advances in DOE modeling and optical performance for SMO applications,” Proc. SPIE 7640, 764025 (2010).
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S. Wang, S. Baron, N. Kachwala, C. Kallingal, D. Sun, V. Shu, W. Fong, Z. Li, A. Elsaid, J. Gao, J. Su, J. Ser, Q. Zhang, B. Chen, R. Howell, S. Hsu, L. Luo, Y. Zou, G. Zhang, Y. Lu, and Y. Cao, “Efficient full-chip SRAF placement using machine learning for best accuracy and improved consistency,” Proc. SPIE 10587, 105870N (2018).
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Chen, L.

S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE 7140, 714010 (2008).
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C. Progler, W. Conley, B. Socha, and Y. Ham, “Layout and source dependent phase mask transmission tuning,” Proc. SPIE 5454, 315–326 (2005).
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K. Tian, M. Fakhry, A. Dave, A. Tritchkov, J. Tirapu-Azpiroz, A. Rosenbluth, D. Melville, M. Sakamoto, T. Inoue, S. Mansfield, A. Wei, Y. Kim, B. Durgan, K. Adam, G. Berger, G. Bhatara, J. Meiring, H. Haffner, and B. Kim, “Applicability of global source mask optimization to 22/20 nm node and beyond,” Proc. SPIE 7973, 79730C (2011).
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T. Blumensath and M. E. Davies, “Iterative hard thresholding for compressed sensing,” Appl. Comput. Harmon. Anal. 27(3), 265–274 (2009).
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Deng, Y.

Y. Du, L. Li, J. Zhang, F. Shao, C. Zuniga, and Y. Deng, “A model-based approach for the scattering-bar printing avoidance,” Proc. SPIE 10587, 105870Q (2018).

Dong, L.

Dong, Z.

Y. Zhang, Z. Dong, G. Ji, and S. Wang, “An improved reconstruction method for CS-MRI based on exponential wavelet transform and iterative shrinkage/thresholding algorithm,” J. Electromagnet. Wave. 28(18), 2327–2338 (2014).
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Durgan, B.

K. Tian, M. Fakhry, A. Dave, A. Tritchkov, J. Tirapu-Azpiroz, A. Rosenbluth, D. Melville, M. Sakamoto, T. Inoue, S. Mansfield, A. Wei, Y. Kim, B. Durgan, K. Adam, G. Berger, G. Bhatara, J. Meiring, H. Haffner, and B. Kim, “Applicability of global source mask optimization to 22/20 nm node and beyond,” Proc. SPIE 7973, 79730C (2011).
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S. Wang, S. Baron, N. Kachwala, C. Kallingal, D. Sun, V. Shu, W. Fong, Z. Li, A. Elsaid, J. Gao, J. Su, J. Ser, Q. Zhang, B. Chen, R. Howell, S. Hsu, L. Luo, Y. Zou, G. Zhang, Y. Lu, and Y. Cao, “Efficient full-chip SRAF placement using machine learning for best accuracy and improved consistency,” Proc. SPIE 10587, 105870N (2018).
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Erdmann, A.

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Fakhry, M.

K. Tian, M. Fakhry, A. Dave, A. Tritchkov, J. Tirapu-Azpiroz, A. Rosenbluth, D. Melville, M. Sakamoto, T. Inoue, S. Mansfield, A. Wei, Y. Kim, B. Durgan, K. Adam, G. Berger, G. Bhatara, J. Meiring, H. Haffner, and B. Kim, “Applicability of global source mask optimization to 22/20 nm node and beyond,” Proc. SPIE 7973, 79730C (2011).
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Feng, M.

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Fuehner, T.

A. Erdmann, T. Fuehner, T. Schnattinger, and B. Tollkuhn, “Towards automatic mask and source optimization for optical lithography,” Proc. SPIE 5377, 646–657 (2004).
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Granik, Y.

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Haffner, H.

K. Tian, M. Fakhry, A. Dave, A. Tritchkov, J. Tirapu-Azpiroz, A. Rosenbluth, D. Melville, M. Sakamoto, T. Inoue, S. Mansfield, A. Wei, Y. Kim, B. Durgan, K. Adam, G. Berger, G. Bhatara, J. Meiring, H. Haffner, and B. Kim, “Applicability of global source mask optimization to 22/20 nm node and beyond,” Proc. SPIE 7973, 79730C (2011).
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Ham, Y.

C. Progler, W. Conley, B. Socha, and Y. Ham, “Layout and source dependent phase mask transmission tuning,” Proc. SPIE 5454, 315–326 (2005).
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Hamouda, A.

H. Choi and A. Hamouda, “Inverse lithography OPC correction with multiple patterning and etch awareness,” Proc. SPIE 10587, 105870O (2018).
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Han, H.

H. Lee, S. Kim, J. Hong, S. Lee, and H. Han, “Thread scheduling for GPU-based OPC simulation on multi-thread,” Proc. SPIE 10587, 105870P (2018).
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Han, J.

S. Kobelkov, V. Roizen, S. Rodin, A. Tritchkov, J. Han, and Y. Granik, “Constraint approaches for some inverse lithography problems with pixel-based mask,” Proc. SPIE 10587, 105870I (2018).
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S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE 7140, 714010 (2008).
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A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlithogr., Microfabr., Microsyst. 1, 12–30 (2002).

Himel, M. D.

J. T. Carriere, J. Stack, J. Childers, K. Welch, and M. D. Himel, “Advances in DOE modeling and optical performance for SMO applications,” Proc. SPIE 7640, 764025 (2010).
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Hong, J.

H. Lee, S. Kim, J. Hong, S. Lee, and H. Han, “Thread scheduling for GPU-based OPC simulation on multi-thread,” Proc. SPIE 10587, 105870P (2018).
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S. Wang, S. Baron, N. Kachwala, C. Kallingal, D. Sun, V. Shu, W. Fong, Z. Li, A. Elsaid, J. Gao, J. Su, J. Ser, Q. Zhang, B. Chen, R. Howell, S. Hsu, L. Luo, Y. Zou, G. Zhang, Y. Lu, and Y. Cao, “Efficient full-chip SRAF placement using machine learning for best accuracy and improved consistency,” Proc. SPIE 10587, 105870N (2018).
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S. Wang, S. Baron, N. Kachwala, C. Kallingal, D. Sun, V. Shu, W. Fong, Z. Li, A. Elsaid, J. Gao, J. Su, J. Ser, Q. Zhang, B. Chen, R. Howell, S. Hsu, L. Luo, Y. Zou, G. Zhang, Y. Lu, and Y. Cao, “Efficient full-chip SRAF placement using machine learning for best accuracy and improved consistency,” Proc. SPIE 10587, 105870N (2018).
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S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE 7140, 714010 (2008).
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Imgrunt, W.

Y. V. Miklyaev, W. Imgrunt, V. S. Pavelyev, D. G. Kachalov, T. Bizjak, L. Aschke, and V. N. Lissotschenko, “Novel continuously shaped diffractive optical elements enable high-efficiency beam shaping,” Proc. SPIE 7640, 764024 (2010).
[Crossref]

Inoue, T.

K. Lai, M. Gabrani, D. Demaris, N. Casati, A. Torres, S. Sarkar, P. Strenski, S. Bagheri, D. Scarpazza, A. Rosenbluth, D. Melville, A. Wachter, J. Lee, V. Austel, M. Szeto-Millstone, K. Tian, F. Barahona, T. Inoue, and M. Sakamoto, “Design specific joint optimization of masks and sources on a very large scale,” Proc. SPIE 7973, 797308 (2011).
[Crossref]

K. Tian, M. Fakhry, A. Dave, A. Tritchkov, J. Tirapu-Azpiroz, A. Rosenbluth, D. Melville, M. Sakamoto, T. Inoue, S. Mansfield, A. Wei, Y. Kim, B. Durgan, K. Adam, G. Berger, G. Bhatara, J. Meiring, H. Haffner, and B. Kim, “Applicability of global source mask optimization to 22/20 nm node and beyond,” Proc. SPIE 7973, 79730C (2011).
[Crossref]

Jain, N.

A. Chen, Y. Foong, J. Maeng, N. Jain, and S. McDermott, “Exploration of resist effect in source mask optimization,” Proc. SPIE 10587, 105870J (2018).
[Crossref]

Ji, G.

Y. Zhang, Z. Dong, G. Ji, and S. Wang, “An improved reconstruction method for CS-MRI based on exponential wavelet transform and iterative shrinkage/thresholding algorithm,” J. Electromagnet. Wave. 28(18), 2327–2338 (2014).
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N. Jia and E. Y. Lam, “Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 45601–45609 (2010).
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Kachalov, D. G.

Y. V. Miklyaev, W. Imgrunt, V. S. Pavelyev, D. G. Kachalov, T. Bizjak, L. Aschke, and V. N. Lissotschenko, “Novel continuously shaped diffractive optical elements enable high-efficiency beam shaping,” Proc. SPIE 7640, 764024 (2010).
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Kachwala, N.

S. Wang, S. Baron, N. Kachwala, C. Kallingal, D. Sun, V. Shu, W. Fong, Z. Li, A. Elsaid, J. Gao, J. Su, J. Ser, Q. Zhang, B. Chen, R. Howell, S. Hsu, L. Luo, Y. Zou, G. Zhang, Y. Lu, and Y. Cao, “Efficient full-chip SRAF placement using machine learning for best accuracy and improved consistency,” Proc. SPIE 10587, 105870N (2018).
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Kallingal, C.

S. Wang, S. Baron, N. Kachwala, C. Kallingal, D. Sun, V. Shu, W. Fong, Z. Li, A. Elsaid, J. Gao, J. Su, J. Ser, Q. Zhang, B. Chen, R. Howell, S. Hsu, L. Luo, Y. Zou, G. Zhang, Y. Lu, and Y. Cao, “Efficient full-chip SRAF placement using machine learning for best accuracy and improved consistency,” Proc. SPIE 10587, 105870N (2018).
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Keidar, I.

S. Patterson, Y. C. Eldar, and I. Keidar, “Distributed compressed sensing for static and time-varying networks,” IEEE T. Signal Processing 62(19), 4931–4946 (2014).

Kim, B.

K. Tian, M. Fakhry, A. Dave, A. Tritchkov, J. Tirapu-Azpiroz, A. Rosenbluth, D. Melville, M. Sakamoto, T. Inoue, S. Mansfield, A. Wei, Y. Kim, B. Durgan, K. Adam, G. Berger, G. Bhatara, J. Meiring, H. Haffner, and B. Kim, “Applicability of global source mask optimization to 22/20 nm node and beyond,” Proc. SPIE 7973, 79730C (2011).
[Crossref]

Kim, S.

H. Vu, S. Kim, J. Word, and Y. Cai, “A novel processing platform for post tape out flows,” Proc. SPIE 10587, 105870R (2018).
[Crossref]

H. Lee, S. Kim, J. Hong, S. Lee, and H. Han, “Thread scheduling for GPU-based OPC simulation on multi-thread,” Proc. SPIE 10587, 105870P (2018).
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Kim, Y.

K. Tian, M. Fakhry, A. Dave, A. Tritchkov, J. Tirapu-Azpiroz, A. Rosenbluth, D. Melville, M. Sakamoto, T. Inoue, S. Mansfield, A. Wei, Y. Kim, B. Durgan, K. Adam, G. Berger, G. Bhatara, J. Meiring, H. Haffner, and B. Kim, “Applicability of global source mask optimization to 22/20 nm node and beyond,” Proc. SPIE 7973, 79730C (2011).
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Kobelkov, S.

S. Kobelkov, V. Roizen, S. Rodin, A. Tritchkov, J. Han, and Y. Granik, “Constraint approaches for some inverse lithography problems with pixel-based mask,” Proc. SPIE 10587, 105870I (2018).
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Lai, K.

K. Lai, M. Gabrani, D. Demaris, N. Casati, A. Torres, S. Sarkar, P. Strenski, S. Bagheri, D. Scarpazza, A. Rosenbluth, D. Melville, A. Wachter, J. Lee, V. Austel, M. Szeto-Millstone, K. Tian, F. Barahona, T. Inoue, and M. Sakamoto, “Design specific joint optimization of masks and sources on a very large scale,” Proc. SPIE 7973, 797308 (2011).
[Crossref]

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlithogr., Microfabr., Microsyst. 1, 12–30 (2002).

Lam, E. Y.

W. Lv, E. Y. Lam, H. Wei, and S. Liu, “Cascadic multigrid algorithm for robust inverse mask synthesis in optical lithography,” J. Micro-Nanolith. Mem. 13(2), 023003 (2014).

N. Jia and E. Y. Lam, “Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 45601–45609 (2010).
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Lan, S.

S. Lan, J. Liu, Y. Wang, K. Zhao, and J. Li, “Deep learning assisted fast mask optimization,” Proc. SPIE 10587, 105870H (2018).

P. Liu, Z. Zhang, S. Lan, Q. Zhao, M. Feng, H. Liu, V. Vellanki, and Y. Lu, “A full-chip 3D computational lithography framework,” Proc. SPIE 8326, 83260A (2012).
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Lee, H.

H. Lee, S. Kim, J. Hong, S. Lee, and H. Han, “Thread scheduling for GPU-based OPC simulation on multi-thread,” Proc. SPIE 10587, 105870P (2018).
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Lee, J.

K. Lai, M. Gabrani, D. Demaris, N. Casati, A. Torres, S. Sarkar, P. Strenski, S. Bagheri, D. Scarpazza, A. Rosenbluth, D. Melville, A. Wachter, J. Lee, V. Austel, M. Szeto-Millstone, K. Tian, F. Barahona, T. Inoue, and M. Sakamoto, “Design specific joint optimization of masks and sources on a very large scale,” Proc. SPIE 7973, 797308 (2011).
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Lee, S.

H. Lee, S. Kim, J. Hong, S. Lee, and H. Han, “Thread scheduling for GPU-based OPC simulation on multi-thread,” Proc. SPIE 10587, 105870P (2018).
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Li, J.

S. Lan, J. Liu, Y. Wang, K. Zhao, and J. Li, “Deep learning assisted fast mask optimization,” Proc. SPIE 10587, 105870H (2018).

Li, L.

Y. Du, L. Li, J. Zhang, F. Shao, C. Zuniga, and Y. Deng, “A model-based approach for the scattering-bar printing avoidance,” Proc. SPIE 10587, 105870Q (2018).

Li, T.

T. Li and Y. Li, “Lithographic source and mask optimization with low aberration sensitivity,” IEEE Trans. NanoTechnol. 16(6), 1099–1105 (2017).
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X. Ma, Z. Wang, H. Lin, Y. Li, G. R. Arce, and L. Zhang, “Optimization of lithography source illumination arrays using diffraction subspaces,” Opt. Express 26(4), 3738–3755 (2018).
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Figures (14)

Fig. 1
Fig. 1 Imaging formation based on the vector imaging model [45].
Fig. 2
Fig. 2 Flowchart for proposed SMO method.
Fig. 3
Fig. 3 Target layout at 45nm technology node.
Fig. 4
Fig. 4 Simulations of different SMO methods using horizontal block layout pattern at 45nm technology node.
Fig. 5
Fig. 5 The convergence curves of different SMO method using horizontal block pattern at 45nm technology node.
Fig. 6
Fig. 6 The runtime of different SMO methods using horizontal block pattern at 45nm technology node.
Fig. 7
Fig. 7 Sparse coefficients of the mask pattern on different sparse bases.
Fig. 8
Fig. 8 Simulations of different sparse basis using Newton-IHTs algorithm and horizontal block layout pattern at 45nm technology node.
Fig. 9
Fig. 9 Simulations of different SMO methods using vertical line-space layout pattern at 45nm technology node.
Fig. 10
Fig. 10 The convergence curves of different SMO method using vertical line-space pattern at 45nm technology node.
Fig. 11
Fig. 11 The runtime of different SMO methods using vertical line-space pattern at 45nm technology node.
Fig. 12
Fig. 12 Simulations of different SMO methods using a complex layout pattern at 45nm technology node.
Fig. 13
Fig. 13 The convergence curves of different SMO method using a complex pattern at 45nm technology node.
Fig. 14
Fig. 14 The runtime of different SMO methods using a complex pattern at 45nm technology node.

Equations (30)

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I = 1 J sum x S y S [ J ( x S , y S ) × p = x , y , z | H p x S , y S ( B x S , y S M ) | 2 ] ,
s i g ( x ) = 1 1 + exp [ a ( x t r ) ] ,
Z = s i g ( I ) = 1 1 + exp [ a ( I t r ) ] = Φ ( J , M ) ,
f ( J , M ) = | | Π ( Z ˜ Z ) | | 2 2 = m = 1 N n = 1 N { Π ( m , n ) × [ Z ˜ ( m , n ) Z ( m , n ) ] } 2 ,
f K ( J , M ) = | | Π K ( Z ˜ K Z K ) | | 2 2 = m = 1 N / K n = 1 N / K { Π ( K m , K n ) × [ Z ˜ ( K m , K n ) Z ( K m , K n ) ] } 2 ,
d ( J , M ) = f K ( J , M ) + γ q R q ( M ) + γ w R w ( M ) ,
Ω J = Ψ J J Ψ J T ,
M = 1 + cos Θ 2 ,
Ω M = Ψ M Θ Ψ M T ,
Ω ^ J , Ω ^ M = arg min Ω J , Ω M d ( Ω J , Ω M ) = arg min Ω J , Ω M | | Π K ( Z ˜ K Φ K ( Ω J , Ω M ) ) | | 2 2 + γ q R q ( Ω M ) + γ w R w ( Ω M ) subject to | | Ω J | | 0 S J , | | Ω M | | 0 S M ,
arg min x A f ( x ) ,
x n + 1 = P A S { x n s t e p × f ( x n ) } ,
x n + 1 = P A S { x n s t e p × H n f ( x n ) } ,
H n + 1 = ( V n T V n m T ) H 0 ( V n m V n ) + j = 0 m ρ n m + 1 ( l = 0 m j 1 V n l T ) s n m + j s n m + j T ( l = 0 m j 1 V n l T ) ,
s n = x n x n 1 , t n = f ( x n ) f ( x n 1 ) ,
ρ n = 1 s n T t n , V n = ( E ρ n t n s n T ) ,
d ( Ω J ) = f K ( Ω J ) ,
d ( Ω M ) = f K ( Ω M ) + γ q R q ( Ω M ) + γ w R w ( Ω M ) ,
Ω J n + 1 = P A S J { Ω J n s t e p × H J n d ( Ω J n ) } ,
Ω M n + 1 = P A S M { Ω M n s t e p × H M n d ( Ω M n ) } ,
Δ x 1 2 f c ,
Δ x c = 1 5 f c = 1 5 × λ N A = 1 5 × 193 n m 1.35 = 28.59 n m .
I ( K m , K n ) = 1 J sum x S y S { J ( x S , y S ) × p = x , y , z [ r = 1 N s = 1 N H p x S , y S ( K m r , K n s ) × ( B x S , y S ( r , s ) × M ( r , s ) ) ] 2 } .
I ( K m , K n ) = 1 J sum x S y S ( J ( x S , y S ) × p = x , y , z u = 1 K v = 1 K { a = 0 N / K 1 b = 0 N / K 1 H p x S , y S [ K ( m a ) u , K ( n b ) v ] × [ B x S , y S ( K a + u , K b + v ) × M ( K a + u , K b + v ) ] } 2 ) .
I ( K m , K n ) = 1 J sum x S y S ( J ( x S , y S ) × p = x , y , z u = 1 K v = 1 K { a = 0 N / K 1 b = 0 N / K 1 H p , u v x S , y S ( m a , n b ) × [ B u v x S , y S ( a , b ) × M u v ( a , b ) ] } 2 ) .
I K = 1 J sum x S y S [ J ( x S , y S ) × p = x , y , z u = 1 K v = 1 K | H p , u v x S , y S ( B u v x S , y S M u v ) | 2 2 ] = 1 J sum x S y S [ J ( x S , y S ) × p = x , y , z I p w a f e r ] ,
I p w a f e r = u = 1 K v = 1 K | H p , u v x S , y S ( B u v x S , y S M u v ) | 2 2 .
Z K = s i g ( I K ) = 1 1 + exp [ a ( I K t r ) ] .
f K J ( x S , y S ) = 2 a J sum m = 1 N / K n = 1 N / K { Π ( K m , K n ) × [ Z ˜ ( K m , K n ) Z ( K m , K n ) ] × Z ( K m , K n ) × [ 1 Z ( K m , K n ) ] } × p = x , y , z I p w a f e r .
f K ( J ) = 2 a J sum × 1 N / K × 1 T × [ Π K ( Z ˜ K Z K ) Z K ( 1 Z K ) p = x , y , z I p w a f e r ] × 1 N / K × 1 ,

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