Abstract

Level-set based inverse lithography technology (ILT) treats photomask design for microlithography as an inverse mathematical problem, interpreted with a time-dependent model, and then solved as a partial differential equation with finite difference schemes. This paper focuses on developing level-set based ILT for partially coherent systems, and upon that an expectation-orient optimization framework weighting the cost function by random process condition variables. These include defocus and aberration to enhance robustness of layout patterns against process variations. Results demonstrating the benefits of defocus-aberration-aware level-set based ILT are presented.

© 2011 OSA

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  3. O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).
  4. S. Shioiri and H. Tanabe, “Fast optical proximity correction: analytical method,” Proc. SPIE 2440, 261–269 (1995).
  5. 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).
  6. Y. Liu and A. Zakhor, “Optimal binary image design for optical lithography,” Proc. SPIE 1264, 401–412 (1990).
  7. Y. Liu and A. Zakhor, “Binary and phase-shifting image design for optical lithography,” Proc. SPIE 1463, 382–399 (1991).
  8. 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).
    [PubMed]
  9. Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A 11(9), 2438–2452 (1994).
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    [PubMed]
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  16. A. Poonawala, Y. Borodovsky, and P. Milanfar, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
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  24. A. Marquina and S. Osher, “Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal,” SIAM J. Sci. Comp. 22, 387–405 (2000).
  25. L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE6924, 69240T (2008).
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    [PubMed]
  28. E. Y. Lam and A. K. Wong, “Nebulous hotspot and algorithm variability in computation lithography,” J. Micro/Nanolithogr. MEMS MOEMS 9(3), 033002 (2010).
  29. 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).
  30. Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. of SPIE 7748, 77481U (2010).
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  32. J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 2000).
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    [PubMed]
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  39. P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the Extended Nijboer-Zernike theory,” Proc. SPIE5754, 263 (2005).

2010 (3)

E. Y. Lam and A. K. Wong, “Nebulous hotspot and algorithm variability in computation lithography,” J. Micro/Nanolithogr. MEMS MOEMS 9(3), 033002 (2010).

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).

Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. of SPIE 7748, 77481U (2010).

2009 (4)

2008 (4)

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

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).

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).

S. H. Chan, A. K. Wong, and E. Y. Lam, “Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography,” Opt. Express 16(19), 14746–14760 (2008).
[PubMed]

2007 (3)

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

A. Poonawala, Y. Borodovsky, and P. Milanfar, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).

2005 (1)

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

2001 (2)

S. Osher and F. Santosa, “Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum,” J. Comput. Phys. 171(1), 272–288 (2001).

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

2000 (1)

A. Marquina and S. Osher, “Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal,” SIAM J. Sci. Comp. 22, 387–405 (2000).

1996 (1)

F. Santosa, “A level-set approach for inverse problems involving obstacles,” ESAIM Contröle Optim. Calc. Var. 1, 17–33 (1996).

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).
[PubMed]

S. Shioiri and H. Tanabe, “Fast optical proximity correction: analytical method,” Proc. SPIE 2440, 261–269 (1995).

1994 (2)

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A 11(9), 2438–2452 (1994).

1991 (1)

Y. Liu and A. Zakhor, “Binary and phase-shifting image design for optical lithography,” Proc. SPIE 1463, 382–399 (1991).

1990 (1)

Y. Liu and A. Zakhor, “Optimal binary image design for optical lithography,” Proc. SPIE 1264, 401–412 (1990).

1982 (1)

1976 (1)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66(3), 207–211 (1976).

1953 (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. of the Royal Soc. of London 217A(1130), 408–432 (1953).

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).

Adalsteinsson, D.

J. A. Sethian and D. Adalsteinsson, “An overview of level set methods for etching, deposition, and lithography development,” IEEE Trans. Semicond. Manuf.10, 167–184 (1997).

Arce, G. R.

Baik, K.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE6924, 69240T (2008).

Bollepalli, S.

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon PressOxford, 1980).

Borodovsky, Y.

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).

A. Poonawala, Y. Borodovsky, and P. Milanfar, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).

Braat, J.

P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the Extended Nijboer-Zernike theory,” Proc. SPIE5754, 263 (2005).

Cecil, T.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE6924, 69240T (2008).

Chan, S. H.

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).

S. H. Chan, A. K. Wong, and E. Y. Lam, “Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography,” Opt. Express 16(19), 14746–14760 (2008).
[PubMed]

S. H. Chan, A. K. Wong, and E. Y. Lam, “Inverse synthesis of phase-shifting mask for optical lithography,” in OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007).

Chen, D.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE6924, 69240T (2008).

Cui, Y.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE6924, 69240T (2008).

Dai, G.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE6924, 69240T (2008).

Dam, T.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE6924, 69240T (2008).

De Leone, R.

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).
[PubMed]

Dirksen, P.

P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the Extended Nijboer-Zernike theory,” Proc. SPIE5754, 263 (2005).

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).

Garofalo, J. G.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 2000).

Henderson, R. C.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. of the Royal Soc. of London 217A(1130), 408–432 (1953).

Hu, B.

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).

Hu, P.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE6924, 69240T (2008).

Janssen, A.

P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the Extended Nijboer-Zernike theory,” Proc. SPIE5754, 263 (2005).

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).

N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 752032 (2009).

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

Kailath, T.

Kostelak, R. L.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Lam, E. Y.

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).

Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. of SPIE 7748, 77481U (2010).

E. Y. Lam and A. K. Wong, “Nebulous hotspot and algorithm variability in computation lithography,” J. Micro/Nanolithogr. MEMS MOEMS 9(3), 033002 (2010).

E. Y. Lam and A. K. Wong, “Computation lithography: virtual reality and virtual virtuality,” Opt. Express 17(15), 12259–12268 (2009).
[PubMed]

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

N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 752032 (2009).

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

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).

S. H. Chan, A. K. Wong, and E. Y. Lam, “Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography,” Opt. Express 16(19), 14746–14760 (2008).
[PubMed]

S. H. Chan, A. K. Wong, and E. Y. Lam, “Inverse synthesis of phase-shifting mask for optical lithography,” in OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007).

Leeuwestein, A.

P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the Extended Nijboer-Zernike theory,” Proc. SPIE5754, 263 (2005).

Liu, Y.

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).

Y. Liu and A. Zakhor, “Binary and phase-shifting image design for optical lithography,” Proc. SPIE 1463, 382–399 (1991).

Y. Liu and A. Zakhor, “Optimal binary image design for optical lithography,” Proc. SPIE 1264, 401–412 (1990).

Low, K. K.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Ma, X.

Marquina, A.

A. Marquina and S. Osher, “Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal,” SIAM J. Sci. Comp. 22, 387–405 (2000).

Milanfar, P.

A. Poonawala, Y. Borodovsky, and P. Milanfar, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).

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

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

Nijboer, B.

B. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis, Groningen University (1942).

Noll, R. J.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66(3), 207–211 (1976).

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).

S. Osher and F. Santosa, “Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum,” J. Comput. Phys. 171(1), 272–288 (2001).

A. Marquina and S. Osher, “Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal,” SIAM J. Sci. Comp. 22, 387–405 (2000).

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

Otto, O. W.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Pang, L.

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).

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE6924, 69240T (2008).

Paragios, N.

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

Pati, Y. C.

Peng, D.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE6924, 69240T (2008).

Pierrat, C.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Poonawala, A.

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

A. Poonawala, Y. Borodovsky, and P. Milanfar, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).

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

Rabbani, M.

Saleh, B.

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).
[PubMed]

Saleh, B. E. A.

Santosa, F.

S. Osher and F. Santosa, “Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum,” J. Comput. Phys. 171(1), 272–288 (2001).

F. Santosa, “A level-set approach for inverse problems involving obstacles,” ESAIM Contröle Optim. Calc. Var. 1, 17–33 (1996).

Schellenberg, F.

F. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” Proc. SPIE5377, 1–20 (2004).

Sethian, J. A.

J. A. Sethian and D. Adalsteinsson, “An overview of level set methods for etching, deposition, and lithography development,” IEEE Trans. Semicond. Manuf.10, 167–184 (1997).

Shen, Y.

Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. of SPIE 7748, 77481U (2010).

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

Sherif, S.

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).
[PubMed]

Shioiri, S.

S. Shioiri and H. Tanabe, “Fast optical proximity correction: analytical method,” Proc. SPIE 2440, 261–269 (1995).

Singh, V.

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).

Tanabe, H.

S. Shioiri and H. Tanabe, “Fast optical proximity correction: analytical method,” Proc. SPIE 2440, 261–269 (1995).

Toh, K.

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).

Vaidya, S.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Vasudev, P. K.

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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), 045601 (2010).

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Figures (4)

Fig. 1
Fig. 1

Common illumination sources: (a) conventional, (b) annular, and (c) dipole.

Fig. 2
Fig. 2

(a) Target pattern of size 101 × 101. (b) Output pattern under circular source, resulting in a pattern error of 45 pixels. (c) Output pattern under annular source, resulting in a pattern error of 116 pixels. (d) Output pattern under dipole source, resulting in a pattern error of 144 pixels.

Fig. 3
Fig. 3

Simulation of lithographic imaging with different mask patterns computed using level-set based ILT. The first column denotes the input U(x), the second column Iaerial(x), and the third column I(x). Rows (a), (b) and (c) use the derived pattern under circular illumination, annular illumination, and dipole illumination as input, resulting in pattern errors of 5, 24, and 45 pixels respectively.

Fig. 4
Fig. 4

Performances of the proposed level-set based statistical method with aberration variations. (a) focus-aware input mask pattern computed using the statistical method. (b) coma-aware input mask pattern computed using the statistical method. (c) Comparison of pixel errors under different focus errors. (d) Comparison of pixel errors under different coma.

Equations (21)

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I ( x ) = 𝒯 { U ( x ) } ,
U ^ ( x ) = argmin U ( x ) d ( I 0 ( x ) , 𝒯 { U ( x ) } ) ,
sig ( U ( x ) ) = 1 1 + e a ( U ( x ) t r ) ,
I aerial ( x ) = U * ( x 1 ) U ( x 2 ) γ ( x 1 x 2 ) H * ( x x 1 ) H ( x x 2 ) d x 1 d x 2 ,
γ ( x ) = m Γ m e j ω 0 m x ,
Γ m = 1 G 2 A γ γ ( x ) e j ω 0 m x d x ,
I aerial ( x ) = m Γ m | U ( x ) * H m ( x ) | 2 ,
H m ( x ) = H ( x ) e j ω 0 m x .
I ( x ) = sig ( m Γ m | U ( x ) * H m ( x ) | 2 ) .
U ( x ) = { U int for { x : ϕ ( x ) < 0 } U ext for { x : ϕ ( x ) > 0 } ,
F ( U ) = 1 2 | 𝒯 ( U ) I 0 | 2 .
ϕ t = | ϕ | α ( x , t ) ,
α ( x , t ) = J ( U ) ( 𝒯 ( U ) I 0 ) = 1 2 U ( I I 0 ) 2 = 1 2 U ( sig ( m Γ m | U * H m | 2 ) I 0 ) 2 = a { m Γ m H m * [ ( I 0 I ) I ( 1 I ) ( H m * U ) ] } ,
Φ ( ρ , θ ) = n , m c n m R n m ( ρ ) cos m θ ,
( H ) = ( H 0 ) × e j Φ ,
U optimal = min U { I I 0 2 2 } ,
ϕ t = | ϕ | α ( x , t ) ,
α ( x , t ) = 1 2 U { ( I I 0 ) 2 } = 1 2 U { ( sig ( m Γ m | U * H m | 2 ) I 0 ) 2 } = a × { m Γ m H m * [ ( I 0 I ) I ( 1 I ) ( H m * U ) ] } .
c 2 0 = 2 π λ z ( 1 ( 1 N A 2 ) ) z π N A 2 λ ,
j c 2 0 ρ 2 j π z N A 2 λ ρ 2 = j π z N A 2 λ [ ( u λ N A ) 2 + ( v λ N A ) 2 ] = j π λ z ( u 2 + v 2 ) = j π λ z [ ( m 1 N Δ x ) 2 + ( n 1 N Δ x ) 2 ] = j π λ z m 2 + n 2 ( N Δ x ) 2 ,
j c 3 1 ( 3 ρ 3 2 ρ ) cos θ = j c 3 1 [ 3 ( ( u λ N A ) 2 + ( v λ N A ) 2 ) 3 2 ( u λ N A ) 2 + ( v λ N A ) 2 ] cos θ = j c 3 1 [ 3 λ 3 N A 3 ( u 2 + v 2 ) 3 2 2 λ N A ( u 2 + v 2 ) 1 2 ] cos θ = j c 3 1 [ 3 λ 3 ( N Δ x N A ) 3 ( m 2 + n 2 ) 3 2 2 λ N Δ x N A ( m 2 + n 2 ) 1 2 ] cos θ .

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