Abstract

Computational lithography is nowadays playing an indispensible role in improving the imaging performance of optical lithography systems. This paper develops a new and powerful approach to computational lithography by introducing an information theoretical channel modeling in partially coherent lithography systems. A statistical model is built up based on the lithography imaging model to characterize the information transfer between the mask and print images. Then, this paper calculates the optimal information transfer (OIT) in partially coherent lithography systems, and derives the theoretical limit of image fidelity for optical proximity correction (OPC), which is used extensively in computational lithography. Finally, the proposed information theoretical approaches are applied to improve the OPC solutions obtained by the gradient-based algorithm. A set of simulations are provided to verify the proposed information theoretical model and approaches.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. A. K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE, 2001).
    [Crossref]
  2. X. Ma and G. R. Arce, Computational Lithography, 1st ed. (John Wiley and Sons, 2010).
    [Crossref]
  3. Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Trans. Semicond. Manuf. 5(2), 138–152 (1992).
    [Crossref]
  4. Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlith. Microfab. Microsyst. 5(4), 043002 (2006).
  5. 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]
  6. X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A 25(12), 2960–2970 (2008).
    [Crossref]
  7. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007).
    [Crossref] [PubMed]
  8. N. B. Cobb and Y. Granik, “Dense OPC for 65nm and below,” Proc. SPIE 5992, 599259 (2005).
    [Crossref]
  9. P. M. Martin, C. J. Progler, G. Xiao, R. Gray, L. Pang, and Y. Liu, “Manufacturability study of masks created by inverse lithography technology (ILT),” Proc. SPIE 5992, 599235 (2005).
    [Crossref]
  10. A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” Proc. SPIE 6154, 61543H (2006).
    [Crossref]
  11. A. Poonawala, B. Painter, and C. Kerchner, “Model-based assist feature placement for 32nm and 22nm technology nodes using inverse mask technology,” Proc. SPIE 7488, 748814 (2009).
    [Crossref]
  12. Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
    [Crossref]
  13. N. Jia and E. Y. Lam, “Machine learning for inverse lithography: Using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 045601 (2010).
    [Crossref]
  14. J. Yu and P. Yu, “Impacts of cost functions on inverse lithography patterning,” Opt. Express 18(8), 23331–23342 (2010).
    [Crossref] [PubMed]
  15. X. Ma and G. R. Arce, “Pixel-based OPC optimization based on conjugate gradients,” Opt. Express 19(3), 2165–2180 (2011).
    [Crossref] [PubMed]
  16. 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]
  17. X. Ma, Z. Song, Y. Li, and G. R. Arce, “Block-based mask optimization for optical lithography,” Appl. Opt. 52(14), 3351–3363 (2013).
    [Crossref] [PubMed]
  18. X. Ma, G. R. Arce, and Y. Li, “Optimal 3D phase-shifting masks in partially coherent illumination,” Appl. Opt. 50(28), 5567–5576 (2011).
    [Crossref] [PubMed]
  19. 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]
  20. M. L. Rieger, “Communication theory in optical lithography,” J. Micro/Nanolith. MEMS MOEMS 11(1), 013003 (2012).
    [Crossref]
  21. X. Ma, H. Zhang, Z. Wang, Y. Li, G. R. Arce, J. Garcia-Frias, and L. Zhang, “Information theoretical aspects in coherent optical lithography systems,” Opt. Express 25(23), 29043–29057 (2017).
    [Crossref]
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  23. 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]
  24. 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]
  25. W. Lv, Q. Xia, and S. Liu, “Mask-filtering-based inverse lithography,” J. Micro/Nanolith. MEMS MOEMS 12(4), 043003 (2013).
    [Crossref]
  26. M. Born and E. Wolf, Principles of Optics, Cambridge University (1999).
    [Crossref]
  27. R. Wilson, Fourier Series and Optical Transform Techniques in Contemporary Optics (Wiley, 1995).
  28. P. Yu, S. X. Shi, and D. Z. Pan, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Micro/Nanolith. MEMS MOEMS 6(3), 031004 (2007).
    [Crossref]
  29. S. Jiang, X. Ma, and A. Zakhor, “A recursive cost-based approach to fracturing,” Proc. SPIE 7973, 79732P (2011).
    [Crossref]

2017 (2)

2014 (1)

2013 (3)

W. Lv, Q. Xia, and S. Liu, “Mask-filtering-based inverse lithography,” J. Micro/Nanolith. MEMS MOEMS 12(4), 043003 (2013).
[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]

X. Ma, Z. Song, Y. Li, and G. R. Arce, “Block-based mask optimization for optical lithography,” Appl. Opt. 52(14), 3351–3363 (2013).
[Crossref] [PubMed]

2012 (2)

2011 (3)

2010 (2)

N. Jia and E. Y. Lam, “Machine learning for inverse lithography: Using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 045601 (2010).
[Crossref]

J. Yu and P. Yu, “Impacts of cost functions on inverse lithography patterning,” Opt. Express 18(8), 23331–23342 (2010).
[Crossref] [PubMed]

2009 (2)

A. Poonawala, B. Painter, and C. Kerchner, “Model-based assist feature placement for 32nm and 22nm technology nodes using inverse mask technology,” Proc. SPIE 7488, 748814 (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 (1)

2007 (3)

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]

P. Yu, S. X. Shi, and D. Z. Pan, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Micro/Nanolith. MEMS MOEMS 6(3), 031004 (2007).
[Crossref]

2006 (2)

A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” Proc. SPIE 6154, 61543H (2006).
[Crossref]

Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlith. Microfab. Microsyst. 5(4), 043002 (2006).

2005 (2)

N. B. Cobb and Y. Granik, “Dense OPC for 65nm and below,” Proc. SPIE 5992, 599259 (2005).
[Crossref]

P. M. Martin, C. J. Progler, G. Xiao, R. Gray, L. Pang, and Y. Liu, “Manufacturability study of masks created by inverse lithography technology (ILT),” Proc. SPIE 5992, 599235 (2005).
[Crossref]

1992 (1)

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

1982 (1)

Arce, G. R.

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]

X. Ma, H. Zhang, Z. Wang, Y. Li, G. R. Arce, J. Garcia-Frias, and L. Zhang, “Information theoretical aspects in coherent optical lithography systems,” Opt. Express 25(23), 29043–29057 (2017).
[Crossref]

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]

X. Ma, Z. Song, Y. Li, and G. R. Arce, “Block-based mask optimization for optical lithography,” Appl. Opt. 52(14), 3351–3363 (2013).
[Crossref] [PubMed]

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

X. Ma, G. R. Arce, and Y. Li, “Optimal 3D phase-shifting masks in partially coherent illumination,” Appl. Opt. 50(28), 5567–5576 (2011).
[Crossref] [PubMed]

X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A 25(12), 2960–2970 (2008).
[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]

X. Ma and G. R. Arce, Computational Lithography, 1st ed. (John Wiley and Sons, 2010).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, Cambridge University (1999).
[Crossref]

Cobb, N. B.

N. B. Cobb and Y. Granik, “Dense OPC for 65nm and below,” Proc. SPIE 5992, 599259 (2005).
[Crossref]

Dong, L.

Gao, J.

Garcia-Frias, J.

Granik, Y.

Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlith. Microfab. Microsyst. 5(4), 043002 (2006).

N. B. Cobb and Y. Granik, “Dense OPC for 65nm and below,” Proc. SPIE 5992, 599259 (2005).
[Crossref]

Gray, R.

P. M. Martin, C. J. Progler, G. Xiao, R. Gray, L. Pang, and Y. Liu, “Manufacturability study of masks created by inverse lithography technology (ILT),” Proc. SPIE 5992, 599235 (2005).
[Crossref]

Jia, N.

N. Jia and E. Y. Lam, “Machine learning for inverse lithography: Using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 045601 (2010).
[Crossref]

Jiang, S.

S. Jiang, X. Ma, and A. Zakhor, “A recursive cost-based approach to fracturing,” Proc. SPIE 7973, 79732P (2011).
[Crossref]

Kerchner, C.

A. Poonawala, B. Painter, and C. Kerchner, “Model-based assist feature placement for 32nm and 22nm technology nodes using inverse mask technology,” Proc. SPIE 7488, 748814 (2009).
[Crossref]

Lam, E. 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]

N. Jia and E. Y. Lam, “Machine learning for inverse lithography: Using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 045601 (2010).
[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]

Li, Y.

Liu, S.

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]

W. Lv, Q. Xia, and S. Liu, “Mask-filtering-based inverse lithography,” J. Micro/Nanolith. MEMS MOEMS 12(4), 043003 (2013).
[Crossref]

Liu, Y.

P. M. Martin, C. J. Progler, G. Xiao, R. Gray, L. Pang, and Y. Liu, “Manufacturability study of masks created by inverse lithography technology (ILT),” Proc. SPIE 5992, 599235 (2005).
[Crossref]

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

Lv, W.

W. Lv, Q. Xia, and S. Liu, “Mask-filtering-based inverse lithography,” J. Micro/Nanolith. MEMS MOEMS 12(4), 043003 (2013).
[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.

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]

X. Ma, H. Zhang, Z. Wang, Y. Li, G. R. Arce, J. Garcia-Frias, and L. Zhang, “Information theoretical aspects in coherent optical lithography systems,” Opt. Express 25(23), 29043–29057 (2017).
[Crossref]

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]

X. Ma, Z. Song, Y. Li, and G. R. Arce, “Block-based mask optimization for optical lithography,” Appl. Opt. 52(14), 3351–3363 (2013).
[Crossref] [PubMed]

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]

X. Ma, G. R. Arce, and Y. Li, “Optimal 3D phase-shifting masks in partially coherent illumination,” Appl. Opt. 50(28), 5567–5576 (2011).
[Crossref] [PubMed]

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

S. Jiang, X. Ma, and A. Zakhor, “A recursive cost-based approach to fracturing,” Proc. SPIE 7973, 79732P (2011).
[Crossref]

X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A 25(12), 2960–2970 (2008).
[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]

X. Ma and G. R. Arce, Computational Lithography, 1st ed. (John Wiley and Sons, 2010).
[Crossref]

Martin, P. M.

P. M. Martin, C. J. Progler, G. Xiao, R. Gray, L. Pang, and Y. Liu, “Manufacturability study of masks created by inverse lithography technology (ILT),” Proc. SPIE 5992, 599235 (2005).
[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 and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” Proc. SPIE 6154, 61543H (2006).
[Crossref]

Painter, B.

A. Poonawala, B. Painter, and C. Kerchner, “Model-based assist feature placement for 32nm and 22nm technology nodes using inverse mask technology,” Proc. SPIE 7488, 748814 (2009).
[Crossref]

Pan, D. Z.

P. Yu, S. X. Shi, and D. Z. Pan, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Micro/Nanolith. MEMS MOEMS 6(3), 031004 (2007).
[Crossref]

Pang, L.

P. M. Martin, C. J. Progler, G. Xiao, R. Gray, L. Pang, and Y. Liu, “Manufacturability study of masks created by inverse lithography technology (ILT),” Proc. SPIE 5992, 599235 (2005).
[Crossref]

Poonawala, A.

A. Poonawala, B. Painter, and C. Kerchner, “Model-based assist feature placement for 32nm and 22nm technology nodes using inverse mask technology,” Proc. SPIE 7488, 748814 (2009).
[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, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” Proc. SPIE 6154, 61543H (2006).
[Crossref]

Progler, C. J.

P. M. Martin, C. J. Progler, G. Xiao, R. Gray, L. Pang, and Y. Liu, “Manufacturability study of masks created by inverse lithography technology (ILT),” Proc. SPIE 5992, 599235 (2005).
[Crossref]

Rabbani, M.

Rieger, M. L.

M. L. Rieger, “Communication theory in optical lithography,” J. Micro/Nanolith. MEMS MOEMS 11(1), 013003 (2012).
[Crossref]

Saleh, B. E. A.

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

Shi, D.

Shi, S. X.

P. Yu, S. X. Shi, and D. Z. Pan, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Micro/Nanolith. MEMS MOEMS 6(3), 031004 (2007).
[Crossref]

Song, Z.

Wang, J.

Wang, Z.

Wilson, R.

R. Wilson, Fourier Series and Optical Transform Techniques in Contemporary Optics (Wiley, 1995).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, Cambridge University (1999).
[Crossref]

Wong, A. K.

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

Wong, N.

Wu, X.

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]

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]

W. Lv, Q. Xia, and S. Liu, “Mask-filtering-based inverse lithography,” J. Micro/Nanolith. MEMS MOEMS 12(4), 043003 (2013).
[Crossref]

Xiao, G.

P. M. Martin, C. J. Progler, G. Xiao, R. Gray, L. Pang, and Y. Liu, “Manufacturability study of masks created by inverse lithography technology (ILT),” Proc. SPIE 5992, 599235 (2005).
[Crossref]

Yu, J.

Yu, P.

J. Yu and P. Yu, “Impacts of cost functions on inverse lithography patterning,” Opt. Express 18(8), 23331–23342 (2010).
[Crossref] [PubMed]

P. Yu, S. X. Shi, and D. Z. Pan, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Micro/Nanolith. MEMS MOEMS 6(3), 031004 (2007).
[Crossref]

Zakhor, A.

S. Jiang, X. Ma, and A. Zakhor, “A recursive cost-based approach to fracturing,” Proc. SPIE 7973, 79732P (2011).
[Crossref]

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

Zhang, H.

Zhang, L.

Appl. Opt. (3)

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]

IEEE Trans. Semicond. Manuf. (1)

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

J. Micro/Nanolith. MEMS MOEMS (3)

M. L. Rieger, “Communication theory in optical lithography,” J. Micro/Nanolith. MEMS MOEMS 11(1), 013003 (2012).
[Crossref]

W. Lv, Q. Xia, and S. Liu, “Mask-filtering-based inverse lithography,” J. Micro/Nanolith. MEMS MOEMS 12(4), 043003 (2013).
[Crossref]

P. Yu, S. X. Shi, and D. Z. Pan, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Micro/Nanolith. MEMS MOEMS 6(3), 031004 (2007).
[Crossref]

J. Microlith. Microfab. Microsyst. (1)

Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlith. Microfab. Microsyst. 5(4), 043002 (2006).

J. Opt. (1)

N. Jia and E. Y. Lam, “Machine learning for inverse lithography: Using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 045601 (2010).
[Crossref]

J. Opt. Soc. Am. A (2)

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 (7)

Proc. SPIE (5)

S. Jiang, X. Ma, and A. Zakhor, “A recursive cost-based approach to fracturing,” Proc. SPIE 7973, 79732P (2011).
[Crossref]

N. B. Cobb and Y. Granik, “Dense OPC for 65nm and below,” Proc. SPIE 5992, 599259 (2005).
[Crossref]

P. M. Martin, C. J. Progler, G. Xiao, R. Gray, L. Pang, and Y. Liu, “Manufacturability study of masks created by inverse lithography technology (ILT),” Proc. SPIE 5992, 599235 (2005).
[Crossref]

A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” Proc. SPIE 6154, 61543H (2006).
[Crossref]

A. Poonawala, B. Painter, and C. Kerchner, “Model-based assist feature placement for 32nm and 22nm technology nodes using inverse mask technology,” Proc. SPIE 7488, 748814 (2009).
[Crossref]

Other (4)

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

X. Ma and G. R. Arce, Computational Lithography, 1st ed. (John Wiley and Sons, 2010).
[Crossref]

M. Born and E. Wolf, Principles of Optics, Cambridge University (1999).
[Crossref]

R. Wilson, Fourier Series and Optical Transform Techniques in Contemporary Optics (Wiley, 1995).

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Figures (8)

Fig. 1
Fig. 1 (a) The sketch of an optical lithography system and (b) the pixelated OPC method.
Fig. 2
Fig. 2 The imaging model and information channel model of partially coherent lithography systems. The lithography system is regarded as an information channel, and the mask and print image are its input and output signals, respectively.
Fig. 3
Fig. 3 The masks and print images used to calculate the probability transfer matrix T.
Fig. 4
Fig. 4 The relationship between mutual information and image fidelity. In the top row, (a), (b) and (c) show the single pixel, macro pixel and two adjacent macro pixels, respectively. In the bottom row, the target layout can be (d) perfectly covered, (e) incompletely covered or (f) overcompletely covered by the macro pixels.
Fig. 5
Fig. 5 Simulations of pixelated OPC algorithm for two layout patterns. In the top row, (a) and (e) show the target patterns for “Layout 3” and “Layout 4”, respectively. (b) and (f) show the OPC masks for “Layout 3” and “Layout 4”, respectively. The bottom row shows the print images of the masks in the top row.
Fig. 6
Fig. 6 Convergence curves for the cost function and the mutual information for the two layout patterns.
Fig. 7
Fig. 7 Probability distributions of the p ^ (left column), OPC masks (middle column) and refined masks (right column). The top and bottom rows illustrate the probability distributions for “Layout 3” and “Layout 4”, respectively.
Fig. 8
Fig. 8 Refined masks and print images obtained by modifying the masks produced by OPC using the information theoretical approach.

Tables (2)

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Table 1 Theoretical limits of image fidelity, minimum PEs obtained by pixelated OPC, and the improved PEs obtained according to the optimal distributions.

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Table 2 Proposed method to refine the OPC solutions.

Equations (34)

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I = m Γ m | h m ( r ) M ( r ) | 2 ,
Γ m = 1 D 2 A γ γ ( r ) exp ( j ω 0 m r ) d r ,
h m ( r ) = h ( r ) exp ( j ω 0 m r ) .
h ( r ) = J 1 ( 2 π r NA / λ ) 2 π r NA / λ ,
Z = Λ { I t r } ,
N x = i = 1 K x i , N y = i = 1 K y i .
p m = P r { N x = m } , q n = P r { N y = n } ,
q = T p .
P r { N y = n | N x = m } = # { N x = m ; N y = n } n = 0 K # { N x = m ; N y = n } .
E n ( y ) = n = 0 K { [ P r { N y = n } ( n K ) ] log 2 [ P r { N y = n } ( n K ) ] ( n K ) } = n = 0 K m = 0 K P r { N y = n , N x = m } [ log 2 P r { N y = n } log 2 ( n K ) ] = n = 0 K m = 0 K P r { N y = n , N x = m } log 2 P r { N y = n } + n = 0 K m = 0 K P r { N y = n , N x = m } log 2 ( n K ) .
E n ( y | x ) = m = 0 K P r { N x = m } [ n = 0 K ( P r { N y = n | N x = m } ( n K ) log 2 P r { N y = n | N x = m } ( n K ) ( n K ) ) ] = n = 0 K m = 0 K P r { N y = n , N x = m } log 2 P r { N y = n | N x = m } ( n K ) = n = 0 K m = 0 K P r { N y = n , N x = m } log 2 P r { N y = n | N x = m } + n = 0 K m = 0 K P r { N y = n , N x = m } log 2 ( n K ) .
I ( x ; y ) = E n ( y ) E n ( y | x ) = n = 0 K m = 0 K P r { N y = n , N x = m } ( log 2 P r { N y = n } log 2 P r { N y = n | N x = m } ) = n = 0 K m = 0 K P r { N y = n | N x = m } P r { N y = m } ( log 2 q n log 2 P r { N y = n | N x = m } ) = n = 0 K m = 0 K T n m p m [ log 2 ( u = 0 K T n u p u ) log 2 T n m ] ,
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 ) ,
a = a K I ( x ; y ) .
K I ( x ; y ) = Π ,
PE min = { min { ( a CD ) L t / ( 2 a 2 ) , A t / a 2 } if a > CD 0 if a CD ,
PE min = { min { ( a CD ) L t / ( 2 a 2 ) , A t / a 2 } if a > CD [ a ( a mod a ) ] L t / ( 2 a 2 ) if a CD and ( a mod a ) a / 2 ( a mod a ) L t / ( 2 a 2 ) if a CD and ( a mod a ) < a / 2 .
I ( x ; y ) K Π 2 ,
F ( p ) = { n = 0 K m = 0 K T n m p m [ log 2 ( u = 0 K T n u p u ) log 2 T n m ] + K Π 2 } 2 ,
q ˜ T p ,
p ^ = arg min p { n = 0 K m = 0 K T n m p m [ log 2 ( u = 0 K T n u p u ) log 2 T n m ] + K Π 2 } 2 , s . t . 0 p m 1 for m , m = 1 K p m = 1 , and q ˜ T p .
p ^ = arg min p F ( p ) , s . t . 0 p m 1 for m ,
F ( p ) = { n = 0 K m = 0 K T n m p m [ log 2 ( u = 0 K T n u p u ) log 2 T n m ] + K Π 2 } 2 + ω 1 ( m = 0 K p m 1 ) 2 + ω 2 q ˜ T p 2 2 ,
p m = 1 + cos θ m 2 .
θ ^ = arg min θ F ( θ ) ,
θ i + 1 = θ i step F ( θ ) ,
I ^ = n = 0 K m = 0 K T n m p ^ m [ log 2 ( u = 0 K T n u p ^ u ) log 2 T n m ] .
Z sigmoid { I , t r } = 1 1 + exp [ a r ( I t r ) ] ,
F 1 = { n = 0 K m = 0 K T n m p m [ log 2 ( u = 0 K T n u p u ) log 2 T n m ] + K Π 2 } 2 ,
F 2 = ( m = 0 K p m 1 ) 2 ,
F 3 = q ˜ T p 2 2 .
F 1 p m = 2 { n = 0 K m = 0 K T n m p m [ log 2 ( u = 0 K T n u p u ) log 2 T n m ] + K Π 2 } n = 0 K T n m { [ log 2 ( u = 0 K T n u p u ) log 2 T n m ] + p m T n m ( u = 0 K T n u p u ) ln 2 } = 2 { n = 0 K m = 0 K T n m p m ( log 2 q n log 2 T n m ) + K Π 2 } n = 0 K T n m { log 2 q n log 2 T n m + p m T n m q n ln 2 } .
F 2 p m = 2 ( m = 0 K p m 1 ) .
F 3 | p = 2 T T T p 2 T T q ,

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