Abstract

With continuous shrinking of critical dimension (CD) and the application of immersion lithography system to technology nodes 22nm and beyond, the vector nature of electromagnetic fields propagating from mask to wafer plane cannot be ignored, rendering mask synthesis under scalar imaging model inadequate. In this paper, we develop a level-set based optimization framework for mask synthesis with a vector imaging model. The forward model of vector image formation is established, and then the photomask synthesis is addressed as an inverse imaging problem whose variational level-set reformulation is represented by a stable time-dependent model, which is solved by employing conjugate gradient methods of the cost function and readily available finite-difference schemes. Experimental results demonstrate pronounced performance in terms of pattern fidelity and edge placement error, together with notable computation acceleration and better convergence performance.

© 2017 Optical Society of America

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References

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  22. N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19(20), 19384–19398 (2011).
    [Crossref] [PubMed]
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    [Crossref]
  24. J. Li and E. Y. Lam, “Robust source and mask optimization compensating for mask topography effects in computational lithography,” Opt. Express 22(8), 9471–9485 (2014).
    [Crossref] [PubMed]
  25. J. Li and E. Y. Lam, “Joint optimization of source, mask, and pupil in optical lithography,” Proc. SPIE 9052, 90520S (2014).
    [Crossref]
  26. X. Wu, S. Liu, J. Li, and E. Y. Lam, “Efficient source mask optimization with Zernike polynomial functions for source representation,” Opt. Express 22(4), 3924–3937 (2014).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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  30. Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 1–8 (2010).
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    [Crossref]
  34. D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
    [Crossref]
  35. 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]
  36. Y. Shen, “Level-set based ILT with a vector imaging model,” in Proceedings of 2017 China Semiconductor Technology International Conference (CSTIC), pp. 1–3 (2017).
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    [Crossref]
  38. S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. 169(2), 463–502 (2001).
    [Crossref]
  39. W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pac. J. Optim 2(1), 35–58 (2006).

2015 (1)

2014 (4)

W. Lv, S. Liu, X. Wu, and E. Y. Lam, “Illumination source optimization in optical lithography via derivative-free optimization,” J. Opt. Soc. Am. A 31(12), 19–26 (2014).
[Crossref]

J. Li and E. Y. Lam, “Robust source and mask optimization compensating for mask topography effects in computational lithography,” Opt. Express 22(8), 9471–9485 (2014).
[Crossref] [PubMed]

J. Li and E. Y. Lam, “Joint optimization of source, mask, and pupil in optical lithography,” Proc. SPIE 9052, 90520S (2014).
[Crossref]

X. Wu, S. Liu, J. Li, and E. Y. Lam, “Efficient source mask optimization with Zernike polynomial functions for source representation,” Opt. Express 22(4), 3924–3937 (2014).
[Crossref] [PubMed]

2013 (3)

2012 (1)

2011 (2)

2010 (2)

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 1–8 (2010).

2009 (4)

N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
[Crossref]

2008 (3)

V. Singh, K. Toh, and B. Yan, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

X. Ma and G. R. Arce, “PSM design for inverse lithography with partially coherent illumination,” Opt. Express 16(24), 20126–20141 (2008).
[Crossref] [PubMed]

N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” Proc. SPIE 7140, 71401W (2008).
[Crossref]

2007 (3)

A. Poonawala, P. Milanfar, and B. Yan, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
[Crossref]

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007).
[Crossref] [PubMed]

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]

2006 (1)

W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pac. J. Optim 2(1), 35–58 (2006).

2005 (1)

A. Poonawala and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE 5674, 114–127 (2005).
[Crossref]

2003 (1)

K. Adam, Y. Granik, A. Torres, and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[Crossref]

2001 (2)

S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. 169(2), 463–502 (2001).
[Crossref]

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

2000 (1)

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[Crossref]

1994 (1)

Adam, K.

K. Adam, Y. Granik, A. Torres, and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[Crossref]

Arce, G. R.

Baik, K. H.

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Cecil, T.

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

Chan, S. H.

A. K. Wong, E. Y. Lam, and S. H. Chan, “Inverse Synthesis of Phase-Shifting Mask for Optical Lithography,” in Proceedings of OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007).

S. H. Chan and E. Y. Lam, “Inverse image problem of designing phase shifting masks in optical lithography,” in Proceedings of IEEE International Conference on Image Processing, pp. 1832–1835 (2008).

Chen, D.

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Cobb, N. B.

K. Adam, Y. Granik, A. Torres, and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[Crossref]

Dam, T.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Dong, L.

Dunham, T. G.

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

Erdmann, A.

X. Wu, S. Liu, A. Erdmann, and E. Y. Lam, “Incorporating photomask shape uncertainty in computational lithography,” in Proceedings of SPIE Advanced Lithography, p. 97800Q (2016).

Fedkiw, R. P.

S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. 169(2), 463–502 (2001).
[Crossref]

Flagello, D. G.

D. G. Flagello, “High Numerical Aperture Imaging in Homogeneous Thin Films,” Ph.D. thesis, The University of Arizona (1993).

Gleason, B.

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

Granik, Y.

K. Adam, Y. Granik, A. Torres, and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[Crossref]

Hager, W. W.

W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pac. J. Optim 2(1), 35–58 (2006).

Han, C.

He, L.

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Hu, P.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Jia, N.

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
[Crossref] [PubMed]

N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19(20), 19384–19398 (2011).
[Crossref] [PubMed]

N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).

N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” Proc. SPIE 7140, 71401W (2008).
[Crossref]

Kailath, T.

Lam, E. Y.

X. Wu, S. Liu, W. Lv, and E. Y. Lam, “Sparse nonlinear inverse imaging for shot count reduction in inverse lithography,” Opt. Express 23(21), 26919 (2015).
[Crossref] [PubMed]

X. Wu, S. Liu, J. Li, and E. Y. Lam, “Efficient source mask optimization with Zernike polynomial functions for source representation,” Opt. Express 22(4), 3924–3937 (2014).
[Crossref] [PubMed]

W. Lv, S. Liu, X. Wu, and E. Y. Lam, “Illumination source optimization in optical lithography via derivative-free optimization,” J. Opt. Soc. Am. A 31(12), 19–26 (2014).
[Crossref]

J. Li and E. Y. Lam, “Robust source and mask optimization compensating for mask topography effects in computational lithography,” Opt. Express 22(8), 9471–9485 (2014).
[Crossref] [PubMed]

J. Li and E. Y. Lam, “Joint optimization of source, mask, and pupil in optical lithography,” Proc. SPIE 9052, 90520S (2014).
[Crossref]

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

J. Li, S. Liu, and E. Y. Lam, “Efficient source and mask optimization with augmented Lagrangian methods in optical lithography,” Opt. Express 21(7), 8076–8090 (2013).
[Crossref] [PubMed]

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
[Crossref] [PubMed]

N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19(20), 19384–19398 (2011).
[Crossref] [PubMed]

Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 1–8 (2010).

N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).

Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
[Crossref]

N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” Proc. SPIE 7140, 71401W (2008).
[Crossref]

X. Wu, S. Liu, A. Erdmann, and E. Y. Lam, “Incorporating photomask shape uncertainty in computational lithography,” in Proceedings of SPIE Advanced Lithography, p. 97800Q (2016).

S. H. Chan and E. Y. Lam, “Inverse image problem of designing phase shifting masks in optical lithography,” in Proceedings of IEEE International Conference on Image Processing, pp. 1832–1835 (2008).

A. K. Wong, E. Y. Lam, and S. H. Chan, “Inverse Synthesis of Phase-Shifting Mask for Optical Lithography,” in Proceedings of OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007).

Lavin, M. A.

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

Leipold, W. C.

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

Li, J.

Li, Y.

Liebmann, L. W.

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

Liu, S.

X. Wu, S. Liu, W. Lv, and E. Y. Lam, “Sparse nonlinear inverse imaging for shot count reduction in inverse lithography,” Opt. Express 23(21), 26919 (2015).
[Crossref] [PubMed]

W. Lv, S. Liu, X. Wu, and E. Y. Lam, “Illumination source optimization in optical lithography via derivative-free optimization,” J. Opt. Soc. Am. A 31(12), 19–26 (2014).
[Crossref]

X. Wu, S. Liu, J. Li, and E. Y. Lam, “Efficient source mask optimization with Zernike polynomial functions for source representation,” Opt. Express 22(4), 3924–3937 (2014).
[Crossref] [PubMed]

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

J. Li, S. Liu, and E. Y. Lam, “Efficient source and mask optimization with augmented Lagrangian methods in optical lithography,” Opt. Express 21(7), 8076–8090 (2013).
[Crossref] [PubMed]

X. Wu, S. Liu, A. Erdmann, and E. Y. Lam, “Incorporating photomask shape uncertainty in computational lithography,” in Proceedings of SPIE Advanced Lithography, p. 97800Q (2016).

Lv, W.

X. Wu, S. Liu, W. Lv, and E. Y. Lam, “Sparse nonlinear inverse imaging for shot count reduction in inverse lithography,” Opt. Express 23(21), 26919 (2015).
[Crossref] [PubMed]

W. Lv, S. Liu, X. Wu, and E. Y. Lam, “Illumination source optimization in optical lithography via derivative-free optimization,” J. Opt. Soc. Am. A 31(12), 19–26 (2014).
[Crossref]

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Ma, X.

Mansfield, S. M.

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

Milanfar, P.

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]

A. Poonawala, P. Milanfar, and B. Yan, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
[Crossref]

A. Poonawala and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE 5674, 114–127 (2005).
[Crossref]

Neureuther, A. R.

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[Crossref]

Osher, S.

S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. 169(2), 463–502 (2001).
[Crossref]

Pang, L.

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

Pati, Y. C.

Peng, D.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

Pistor, T. V.

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[Crossref]

T. V. Pistor, “Electromagnetic simulation and modeling with applications in lithography,” Ph.D. thesis, University of California, Berkeley (2001).

Poonawala, A.

A. Poonawala, P. Milanfar, and B. Yan, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
[Crossref]

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]

A. Poonawala and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE 5674, 114–127 (2005).
[Crossref]

Shen, Y.

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
[Crossref] [PubMed]

Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 1–8 (2010).

Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
[Crossref]

Y. Shen, “Level-set based ILT with a vector imaging model,” in Proceedings of 2017 China Semiconductor Technology International Conference (CSTIC), pp. 1–3 (2017).

Singh, V.

V. Singh, K. Toh, and B. Yan, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

Sinn, R.

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

Slonaker, S.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Socha, R. J.

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[Crossref]

Toh, K.

V. Singh, K. Toh, and B. Yan, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

Tolani, V.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Torres, A.

K. Adam, Y. Granik, A. Torres, and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[Crossref]

Tyminski, J.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Wong, A. K.

N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).

N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” Proc. SPIE 7140, 71401W (2008).
[Crossref]

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

A. K. Wong, E. Y. Lam, and S. H. Chan, “Inverse Synthesis of Phase-Shifting Mask for Optical Lithography,” in Proceedings of OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007).

Wong, A. K.-K.

A. K.-K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE, Bellingham, WA, 2001).
[Crossref]

A. K.-K. Wong, Optical Imaging in Projection Lithography (SPIE, Bellingham, WA, 2005).
[Crossref]

Wong, N.

Wu, X.

X. Wu, S. Liu, W. Lv, and E. Y. Lam, “Sparse nonlinear inverse imaging for shot count reduction in inverse lithography,” Opt. Express 23(21), 26919 (2015).
[Crossref] [PubMed]

W. Lv, S. Liu, X. Wu, and E. Y. Lam, “Illumination source optimization in optical lithography via derivative-free optimization,” J. Opt. Soc. Am. A 31(12), 19–26 (2014).
[Crossref]

X. Wu, S. Liu, J. Li, and E. Y. Lam, “Efficient source mask optimization with Zernike polynomial functions for source representation,” Opt. Express 22(4), 3924–3937 (2014).
[Crossref] [PubMed]

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

X. Wu, S. Liu, A. Erdmann, and E. Y. Lam, “Incorporating photomask shape uncertainty in computational lithography,” in Proceedings of SPIE Advanced Lithography, p. 97800Q (2016).

Xia, Q.

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Xiao, G.

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Yan, B.

V. Singh, K. Toh, and B. Yan, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

A. Poonawala, P. Milanfar, and B. Yan, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
[Crossref]

Yeung, M. S.

M. S. Yeung, “Modeling High Numerical Aperture Optical Lithography,” in Proceedings of Microlithography Conferences, pp. 149–167 (1988).

Zhang, H.

W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pac. J. Optim 2(1), 35–58 (2006).

IBM J. Res. Develop (1)

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

IEEE Trans. Image Process. (1)

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]

J. Comput. Phys. (1)

S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. 169(2), 463–502 (2001).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Vac. Sci. Technol. B (1)

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Opt. Express (9)

J. Li, S. Liu, and E. Y. Lam, “Efficient source and mask optimization with augmented Lagrangian methods in optical lithography,” Opt. Express 21(7), 8076–8090 (2013).
[Crossref] [PubMed]

X. Wu, S. Liu, J. Li, and E. Y. Lam, “Efficient source mask optimization with Zernike polynomial functions for source representation,” Opt. Express 22(4), 3924–3937 (2014).
[Crossref] [PubMed]

J. Li and E. Y. Lam, “Robust source and mask optimization compensating for mask topography effects in computational lithography,” Opt. Express 22(8), 9471–9485 (2014).
[Crossref] [PubMed]

X. Wu, S. Liu, W. Lv, and E. Y. Lam, “Sparse nonlinear inverse imaging for shot count reduction in inverse lithography,” Opt. Express 23(21), 26919 (2015).
[Crossref] [PubMed]

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007).
[Crossref] [PubMed]

X. Ma and G. R. Arce, “PSM design for inverse lithography with partially coherent illumination,” Opt. Express 16(24), 20126–20141 (2008).
[Crossref] [PubMed]

Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
[Crossref]

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
[Crossref] [PubMed]

N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19(20), 19384–19398 (2011).
[Crossref] [PubMed]

Pac. J. Optim (1)

W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pac. J. Optim 2(1), 35–58 (2006).

Proc. SPIE (12)

J. Li and E. Y. Lam, “Joint optimization of source, mask, and pupil in optical lithography,” Proc. SPIE 9052, 90520S (2014).
[Crossref]

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[Crossref]

A. Poonawala, P. Milanfar, and B. Yan, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
[Crossref]

N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” Proc. SPIE 7140, 71401W (2008).
[Crossref]

N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).

Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 1–8 (2010).

K. Adam, Y. Granik, A. Torres, and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[Crossref]

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

V. Singh, K. Toh, and B. Yan, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

A. Poonawala and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE 5674, 114–127 (2005).
[Crossref]

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

Other (9)

A. K.-K. Wong, Optical Imaging in Projection Lithography (SPIE, Bellingham, WA, 2005).
[Crossref]

M. S. Yeung, “Modeling High Numerical Aperture Optical Lithography,” in Proceedings of Microlithography Conferences, pp. 149–167 (1988).

A. K. Wong, E. Y. Lam, and S. H. Chan, “Inverse Synthesis of Phase-Shifting Mask for Optical Lithography,” in Proceedings of OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007).

S. H. Chan and E. Y. Lam, “Inverse image problem of designing phase shifting masks in optical lithography,” in Proceedings of IEEE International Conference on Image Processing, pp. 1832–1835 (2008).

A. K.-K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE, Bellingham, WA, 2001).
[Crossref]

X. Wu, S. Liu, A. Erdmann, and E. Y. Lam, “Incorporating photomask shape uncertainty in computational lithography,” in Proceedings of SPIE Advanced Lithography, p. 97800Q (2016).

D. G. Flagello, “High Numerical Aperture Imaging in Homogeneous Thin Films,” Ph.D. thesis, The University of Arizona (1993).

T. V. Pistor, “Electromagnetic simulation and modeling with applications in lithography,” Ph.D. thesis, University of California, Berkeley (2001).

Y. Shen, “Level-set based ILT with a vector imaging model,” in Proceedings of 2017 China Semiconductor Technology International Conference (CSTIC), pp. 1–3 (2017).

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Figures (6)

Fig. 1
Fig. 1 Projection optics in a vector imaging model.
Fig. 2
Fig. 2 (a) The annular illumination source with σin = 0.6 and σout = 0.9. (b) Target pattern I0 of size N = 257. (c) The resist image I based on scalar imaging model with pattern error (PE) 2210 and edge placement error (EPE) 1251. (d) The resist image I based on vector imaging model with PE 2517 and EPE 1277.
Fig. 3
Fig. 3 (a) The synthesized mask using the scalar imaging model. (b) The aerial image Iaerial of (a) using the scalar model. (c) The resist image I of (a) using the scalar model with PE 392. (d) The resist image I of (a) using the vector imaging model with PE 1291.
Fig. 4
Fig. 4 (a) Synthesized masks using the steepest descent level-set methods. (b) The aerial image of the mask in (a). (c) The resist image of the mask in (a) with PE 401 and EPE 152. (d) Synthesized mask using the proposed level-set methods. (e) The aerial image of the mask in (d). (f) The resist image of the mask in (d) with PE 401 and EPE 153.
Fig. 5
Fig. 5 Simulation time by the SD method and the proposed method, for the desired pattern in Fig. 2(b).
Fig. 6
Fig. 6 Convergence of pattern and edge placement error by the SD method and the proposed method, for the desired pattern in Fig. 2(b).

Tables (2)

Tables Icon

Algorithm 1 Mask synthesis with level-set based PRP-CG

Tables Icon

Table 1 Simulation time (hours) in Fig. 5 using the SD method and the proposed method.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

e = z × k | z × k | = [ β ρ α ρ 0 ] , e = k × e = [ α γ ρ β γ ρ ρ ] ,
E = E e + E e = [ e , e ] [ E E ] ,
[ E x E y E z ] = [ β ρ α γ ρ α ρ β γ ρ 0 ρ ] [ E E ] = T [ E E ] .
E ( r ; α s , β s ) = E 0 B M ( r ) ,
B ( k , l ) = e j × 2 π × β s × k N × e j × 2 π × α s × l N , k , l = 0 , 1 , , N 1 .
I aerial ( r ) = 1 N s α s β s p = x , y , z H p α s β s ( B α s β s M ( r ) ) 2 ,
H p = 1 { n m n γ γ h ( α , β ) V p ( α , β , γ ) } , p = x , y , z ,
h ( α , β ) = { 1 α 2 + β 2 N A 0 elsewhere ,
V ( α , β , γ ) = [ β 2 + α 2 γ 1 γ 2 α β 1 + γ α α β 1 + γ α 2 + β 2 γ 1 γ 2 β α β γ ] .
s i g ( x ) = 1 1 + e a ( x t r ) ,
I ( r ) = T { M ( r ) } = s i g ( I aerial ( r ) ) = s i g ( 1 N s α s β s p = x , y , z H p α s β s ( B ) α s β s M ( r ) 2 ) .
M ^ = arg min M N × N d { I 0 , T { M } } ,
M = { m int for { r : ϕ ( r ) < 0 } m ext for { r : ϕ ( r ) > 0 } ,
F ( M ) = 1 2 T { M } I 0 2 ,
ϕ t = | ϕ | v ( r , t ) ,
v ( r , t ) = J { M } T ( T { M } I 0 ) = 1 2 M ( I I 0 ) 2 = 2 a N s α s β s p = x , y , z Real [ ( B α s β s ) * ( ( H p α s β s ) * { [ H p α s β s ( B α s β s M ) ] ( I 0 I ) I ( 1 I ) } ) ] ,
P ( ϕ ) = 1 2 | ϕ | 1 2 ,
ϕ t = | ϕ | v ( r , t ) μ | ϕ | [ Δ ϕ ( ϕ | ϕ | ) ] = | ϕ | g ( r , t ) ,
v ( r , t ) = 1 2 M ( I I 0 ) 2 = 2 a N s α s β s p = x , y , z Real [ ( B α s β s ) * 1 { [ ( ( H p α s β s ) * ] [ ( E p α s β s ) ( I 0 I ) I ( 1 I ) ] ) } ] .
δ t max { | V x | δ x + | V y | δ y } = ϵ ,
V ( r , t k ) = { g ( r , t k ) + η k P R P V ( r , t k 1 ) if k 1 g ( r , t k ) if k = 0 ,
η k P R P = g ( r , t k ) 2 g ( r , t k ) g ( r , t k 1 ) g ( r , t k 1 ) 2 .

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