Abstract

Inverse lithography techniques (ILT) have been extensively used by the semiconductor industry to compensate for the inherent image distortions in optical lithography. However, the iterative ILT optimization procedure requires rather prohibitive time steps leading to poor efficiency with explicit time discretization. In this paper, a semi-implicit time discretization scheme is applied, enabling stable computation of mask synthesis with large time steps. Additive operator splittering (AOS) is implemented with respect to coordinate axes, reducing mask synthesis to consecutive one-dimensional updates represented by tridiagonal linear equations, which is solved efficiently by the Thomas algorithm. Simulation results merit the superiority of the proposed semi-implicit approach with improved convergence performance.

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

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References

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    [Crossref]
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    [Crossref]
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  5. A. Poonawala and P. Milanfar, “Mask design for optical microlithography – an inverse imaging problem,” IEEE. T. Image. Process 16(3), 774–788 (2007).
    [Crossref]
  6. Xu Ma and Gonzalo R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express 17(7), 5783–5793 (2009).
    [Crossref] [PubMed]
  7. X. Ma and G. R. Arce, Computational Lithography, Wiley Series in Pure and Applied Optics, 1st ed. (John Wiley and Sons, 2010).
    [Crossref]
  8. P. Gao, A. Gu, and A. Zakhor, “Optical Proximity Correction with Principal Component Regression,” Proc. SPIE 6924, 69243N (2008).
    [Crossref]
  9. Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE. T. Image. Process 20(10), 2856–2864 (2011).
    [Crossref]
  10. X. Ma, C. Han, Y. Li, L. Dong, and G. R. Arce, “Pixelated source and mask optimization for immersion lithography,” J. Opt. Soc. Am. A 30(1), 112–123 (2013).
    [Crossref]
  11. N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19(20), 19384–19398 (2011).
    [Crossref]
  12. 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]
  13. J. Li and E. Y. Lam, “Robust source and mask optimization compensating for mask topography effects in computational lithography,” Opt. Express 22(8), 9471 (2014).
    [Crossref]
  14. 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]
  15. Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
    [Crossref]
  16. Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
    [Crossref]
  17. Y. Shen, “Level-set based mask synthesis with a vector imaging model,” Opt. Express 25(18), 21775 (2017).
    [Crossref]
  18. Y. Shen, “Lithographic source and mask optimization with narrow-band level-set method,” Opt. Express 26(8), 10065–10078 (2018).
    [Crossref]
  19. F. Peng and Y. Shen, “Source and mask co-optimization based on depth learning methods,” in 2018 China Semiconductor Technology International Conference (CSTIC), (2018), pp. 1–3.
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    [Crossref]
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    [Crossref]
  22. X. Ma, Q. Zhao, H. Zhang, Z. Wang, and G. R. Arce, “Model-driven convolution neural network for inverse lithography,” Opt. Express 26(25), 32565–32584 (2018).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  28. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Science, 1996).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  32. F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29(1), 182–193 (1992).
    [Crossref]
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    [Crossref]
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2018 (3)

2017 (2)

2014 (1)

2013 (3)

2011 (3)

2010 (1)

L. W. Liebmann, S. M. Mansfield, A. K.-K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM. J. Res. Dev. 45(5), 651–665 (2010).
[Crossref]

2009 (2)

2008 (1)

P. Gao, A. Gu, and A. Zakhor, “Optical Proximity Correction with Principal Component Regression,” Proc. SPIE 6924, 69243N (2008).
[Crossref]

2007 (1)

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

2006 (1)

Y. L. Linyong Pang and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE 6283, 6283X (2006).

2003 (1)

D. Barash, T. Schlick, M. Israeli, and R. Kimmel, “Multiplicative operator splittings in nonlinear diffusion: From spatial splitting to multiple timesteps,” J. Math. Imaging. Vis. 19(1), 33–48 (2003).
[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]

1998 (1)

J. Weickert, B. M. T. H. Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE. T. Image. Process 7(3), 398–410 (1998).
[Crossref]

1992 (2)

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29(1), 182–193 (1992).
[Crossref]

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion II,” SIAM J. Numer. Anal. 29(3), 845–866 (1992).
[Crossref]

1990 (1)

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE. T. Pattern. Anal. 12(7), 629 (1990).
[Crossref]

1955 (1)

D. W. Peaceman and J. H. H. Rachford, “The numerical solution of parabolic and elliptic differential equations,” J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955).
[Crossref]

Abrams, D.

Y. L. Linyong Pang and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE 6283, 6283X (2006).

Alvarez, L.

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion II,” SIAM J. Numer. Anal. 29(3), 845–866 (1992).
[Crossref]

Arce, G. R.

Arce, Gonzalo R.

Barash, D.

D. Barash, T. Schlick, M. Israeli, and R. Kimmel, “Multiplicative operator splittings in nonlinear diffusion: From spatial splitting to multiple timesteps,” J. Math. Imaging. Vis. 19(1), 33–48 (2003).
[Crossref]

Born, M.

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

Catté, F.

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29(1), 182–193 (1992).
[Crossref]

Coll, T.

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29(1), 182–193 (1992).
[Crossref]

Conte, S. D.

S. D. Conte and C. deBoor, Elementary Numerical Analysis (McGraw-Hill Science, 1972).

deBoor, C.

S. D. Conte and C. deBoor, Elementary Numerical Analysis (McGraw-Hill Science, 1972).

Dong, L.

Dunham, T. G.

L. W. Liebmann, S. M. Mansfield, A. K.-K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM. J. Res. Dev. 45(5), 651–665 (2010).
[Crossref]

Gao, P.

P. Gao, A. Gu, and A. Zakhor, “Optical Proximity Correction with Principal Component Regression,” Proc. SPIE 6924, 69243N (2008).
[Crossref]

Garcia-Frias, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Science, 1996).

Gu, A.

P. Gao, A. Gu, and A. Zakhor, “Optical Proximity Correction with Principal Component Regression,” Proc. SPIE 6924, 69243N (2008).
[Crossref]

Han, C.

Israeli, M.

D. Barash, T. Schlick, M. Israeli, and R. Kimmel, “Multiplicative operator splittings in nonlinear diffusion: From spatial splitting to multiple timesteps,” J. Math. Imaging. Vis. 19(1), 33–48 (2003).
[Crossref]

Jia, N.

Kimmel, R.

D. Barash, T. Schlick, M. Israeli, and R. Kimmel, “Multiplicative operator splittings in nonlinear diffusion: From spatial splitting to multiple timesteps,” J. Math. Imaging. Vis. 19(1), 33–48 (2003).
[Crossref]

Lam, E. Y.

Lavin, M. A.

L. W. Liebmann, S. M. Mansfield, A. K.-K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM. J. Res. Dev. 45(5), 651–665 (2010).
[Crossref]

Leipold, W. C.

L. W. Liebmann, S. M. Mansfield, A. K.-K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM. J. Res. Dev. 45(5), 651–665 (2010).
[Crossref]

Li, J.

Li, Y.

Liebmann, L. W.

L. W. Liebmann, S. M. Mansfield, A. K.-K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM. J. Res. Dev. 45(5), 651–665 (2010).
[Crossref]

Linyong Pang, Y. L.

Y. L. Linyong Pang and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE 6283, 6283X (2006).

Lions, P.-L.

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion II,” SIAM J. Numer. Anal. 29(3), 845–866 (1992).
[Crossref]

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29(1), 182–193 (1992).
[Crossref]

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]

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]

Lv, W.

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.

Ma, Xu

Malik, J.

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE. T. Pattern. Anal. 12(7), 629 (1990).
[Crossref]

Mansfield, S. M.

L. W. Liebmann, S. M. Mansfield, A. K.-K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM. J. Res. Dev. 45(5), 651–665 (2010).
[Crossref]

Milanfar, P.

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

Morel, J.-M.

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29(1), 182–193 (1992).
[Crossref]

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion II,” SIAM J. Numer. Anal. 29(3), 845–866 (1992).
[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 N. Paragios, Geometric Level Set Methods in Imaging, Vision, and Graphics (Springer, 2003).

Paragios, N.

S. Osher and N. Paragios, Geometric Level Set Methods in Imaging, Vision, and Graphics (Springer, 2003).

Peaceman, D. W.

D. W. Peaceman and J. H. H. Rachford, “The numerical solution of parabolic and elliptic differential equations,” J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955).
[Crossref]

Peng, F.

F. Peng and Y. Shen, “Source and mask co-optimization based on depth learning methods,” in 2018 China Semiconductor Technology International Conference (CSTIC), (2018), pp. 1–3.

Peng, Y.

Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE. T. Image. Process 20(10), 2856–2864 (2011).
[Crossref]

Perona, P.

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE. T. Pattern. Anal. 12(7), 629 (1990).
[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]

Poonawala, A.

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

Rachford, J. H. H.

D. W. Peaceman and J. H. H. Rachford, “The numerical solution of parabolic and elliptic differential equations,” J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955).
[Crossref]

Romeny, B. M. T. H.

J. Weickert, B. M. T. H. Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE. T. Image. Process 7(3), 398–410 (1998).
[Crossref]

Schlick, T.

D. Barash, T. Schlick, M. Israeli, and R. Kimmel, “Multiplicative operator splittings in nonlinear diffusion: From spatial splitting to multiple timesteps,” J. Math. Imaging. Vis. 19(1), 33–48 (2003).
[Crossref]

Shen, Y.

Y. Shen, “Lithographic source and mask optimization with narrow-band level-set method,” Opt. Express 26(8), 10065–10078 (2018).
[Crossref]

Y. Shen, “Level-set based mask synthesis with a vector imaging model,” Opt. Express 25(18), 21775 (2017).
[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]

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

F. Peng and Y. Shen, “Source and mask co-optimization based on depth learning methods,” in 2018 China Semiconductor Technology International Conference (CSTIC), (2018), pp. 1–3.

Shi, D.

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]

Viergever, M. A.

J. Weickert, B. M. T. H. Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE. T. Image. Process 7(3), 398–410 (1998).
[Crossref]

Wang, Y.

Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE. T. Image. Process 20(10), 2856–2864 (2011).
[Crossref]

Wang, Z.

Weickert, J.

J. Weickert, B. M. T. H. Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE. T. Image. Process 7(3), 398–410 (1998).
[Crossref]

J. Weickert, Scale-Space Theory in Computer Vision (Springer, 1997).

Wolf, E.

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

Wong, A. K.-K.

L. W. Liebmann, S. M. Mansfield, A. K.-K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM. J. Res. Dev. 45(5), 651–665 (2010).
[Crossref]

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

A. K.-K. Wong, Optical Imaging in Projection Microlithography (SPIE Press, 2005).
[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]

Yu, Z.

Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE. T. Image. Process 20(10), 2856–2864 (2011).
[Crossref]

Zakhor, A.

P. Gao, A. Gu, and A. Zakhor, “Optical Proximity Correction with Principal Component Regression,” Proc. SPIE 6924, 69243N (2008).
[Crossref]

Zhang, H.

Zhang, J.

Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE. T. Image. Process 20(10), 2856–2864 (2011).
[Crossref]

Zhao, Q.

IBM. J. Res. Dev. (1)

L. W. Liebmann, S. M. Mansfield, A. K.-K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM. J. Res. Dev. 45(5), 651–665 (2010).
[Crossref]

IEEE. T. Image. Process (3)

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

Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE. T. Image. Process 20(10), 2856–2864 (2011).
[Crossref]

J. Weickert, B. M. T. H. Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE. T. Image. Process 7(3), 398–410 (1998).
[Crossref]

IEEE. T. Pattern. Anal. (1)

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE. T. Pattern. Anal. 12(7), 629 (1990).
[Crossref]

J. Math. Imaging. Vis. (1)

D. Barash, T. Schlick, M. Israeli, and R. Kimmel, “Multiplicative operator splittings in nonlinear diffusion: From spatial splitting to multiple timesteps,” J. Math. Imaging. Vis. 19(1), 33–48 (2003).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Soc. Ind. Appl. Math. (1)

D. W. Peaceman and J. H. H. Rachford, “The numerical solution of parabolic and elliptic differential equations,” J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955).
[Crossref]

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 (11)

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

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]

Y. Shen, “Level-set based mask synthesis with a vector imaging model,” Opt. Express 25(18), 21775 (2017).
[Crossref]

Y. Shen, “Lithographic source and mask optimization with narrow-band level-set method,” Opt. Express 26(8), 10065–10078 (2018).
[Crossref]

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

Xu Ma and Gonzalo R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express 17(7), 5783–5793 (2009).
[Crossref] [PubMed]

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]

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]

X. Ma, Q. Zhao, H. Zhang, Z. Wang, and G. R. Arce, “Model-driven convolution neural network for inverse lithography,” Opt. Express 26(25), 32565–32584 (2018).
[Crossref]

Proc. SPIE (3)

P. Gao, A. Gu, and A. Zakhor, “Optical Proximity Correction with Principal Component Regression,” Proc. SPIE 6924, 69243N (2008).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Schematic of forward lithography.
Fig. 2
Fig. 2 From left to right are annular illumination source J with σin = 0.6 and σout = 0.9, desired target pattern I01 and desired target pattern I02.
Fig. 3
Fig. 3 Simulation results with I01 as target pattern illuminated by J in Fig. 2. Columns from left to right: the illuminated input mask pattern M, the aerial image Ia and the wafer image I. Rows (a), (b) and (c) present the simulation results with the desired pattern I01, and synsthesized mask with the SGD method with optimal step-size τ1 = 0.1 and the synthesized mask with the proposed approach with time-step τ1 = 0.85 as the inputs, respectively.
Fig. 4
Fig. 4 Convergence performance using the SGD method with τ1 = 0.1 and the proposed approach with τ1 = 0.1, 0.25, 0.4, 0.55, 0.7, 0.85.
Fig. 5
Fig. 5 Time efficiency of the proposed approach with τ1 = 0.85 and the SGD method with τ1 = 0.1 in Fig. 3.
Fig. 6
Fig. 6 Simulation results with I02 as target pattern illuminated by J in Fig. 2. Columns from left to right: the illuminated input mask pattern M, the aerial image Ia and the wafer image I. Rows (a), (b) and (c) present the simulation results with the desired pattern I02, and synsthesized mask with the SGD method with optimal step-size τ1 = 0.1 and the synthesized mask with the proposed approach with time-step τ1 = 0.85 as the inputs, respectively.
Fig. 7
Fig. 7 Time efficiency of the proposed approach with τ1 = 0.85 and the SGD method with τ1 = 0.1 in Fig. 6.

Equations (15)

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I a = 1 J sum ( α s , β s ) J ( α s , β s ) p = x , y , z | H p α s β s ( B α s β s M ) | 2 ,
I = 𝒯 { M } = sig ( I a ) .
{ t ω = ( g ( | ω | 2 ) | ω | ) Ω ( 𝒯 { M } I 0 ) 2 d x = ε ,
g ( s ) = { 1 ( s 0 ) 1 exp [ 3.315 ( s / λ 1 ) 4 ] ( s > 0 ) ,
λ ( g ( | ω | 2 ) | ω | ) + α ( x ) = 0 ,
α ( x ) = 1 2 ω ( I I 0 ) 2 = a sin ω 2 J sum ( α s , β s ) J ( α s , β s ) p = x , y , z Real [ ( B α s β s ) * ( ( H p α s β s ) * { E p α s β s ( I I 0 ) I ( 1 I ) } ) ]
λ = α ( x ) 2 2 [ Ω α ( x ) ( g ( | ω | 2 ) | ω | ) d x ] 1 ,
t ω = α ( x , t ) + λ ( g ( | ω | 2 ) | ω | ) ,
ω i k + 1 = ω i k + τ λ j 𝒩 ( i ) g j k + g i k 2 h 2 ( ω j k + 1 ω i k ) + τ α ( i , k ) ,
ω k + 1 = ω k + τ α k + τ l { x , y } A l ( ω k ) ω k + 1 ,
a i j l ( ω k ) = { g i k + g j k 2 h 2 [ j 𝒩 l ( i ) ] n 𝒩 l ( i ) g i k + g n k 2 h 2 ( j = i ) 0 ( else ) ,
ω k + 1 = ( I τ l { x , y } A l ( ω k ) ) 1 ( ω k + τ α k ) ,
ω k + 1 = ( I + τ l { x , y } A l ( ω k ) ) ω k + τ α k
ω k + 1 = 1 2 L = 1 2 ( I L τ A l ( ω k ) ) 1 ( ω k + τ α ( x , k ) ) ,
ω k + 1 = ω k + τ 1 α k ,

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