Abstract

Inverse lithography technology (ILT) treats photomask design for microlithography as an inverse mathematical problem. We show how the inverse lithography problem can be addressed as an obstacle reconstruction problem or an extended nonlinear image restoration problem, and then solved by a level set time-dependent model with finite difference schemes. We present explicit detailed formulation of the problem together with the first-order temporal and second-order spatial accurate discretization scheme. Experimental results show the superiority of the proposed level set-based ILT over the mainstream gradient methods.

© 2009 Optical Society of America

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  1. A. K.-K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE Press, Bellingham, WA, 2001).
  2. F. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” Proc. SPIE 5377, 1–20 (2004).
    [CrossRef]
  3. 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]
  4. L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): a natural solution for model-based SRAF at 45nm and 32nm,” Proc. SPIE 6607, 660739 (2007).
    [CrossRef]
  5. 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), 14746–14761(2008).
    [CrossRef]
  6. E. Y. Lam and A. K. Wong, “Computation lithography: virtual reality and virtual virtuality,” Opt. Express 17(15), 12259–12268 (2009).
    [CrossRef]
  7. A. K. Wong and E. Y. Lam, “The nebulous hotspot and algorithm variability,” Proc. SPIE 7275, 727509 (2009).
    [CrossRef]
  8. Y. Liu and A. Zakhor, “Optimal binary image design for optical lithography,” in Proc. SPIE 1264, 401–412 (1990).
  9. Y. Liu and A. Zakhor, “Binary and phase-shifting image design for optical lithography,” Proc. SPIE 1463, 382–399 (1991).
    [CrossRef]
  10. S. Sherif, B. Saleh, and R. De Leone, “Binary image synthesis using mixed linear integer programming,” IEEE Trans. Image Process. 4(9), 1252–1257 (1995).
    [CrossRef]
  11. 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).
    [CrossRef]
  12. Y. Granik, “Solving inverse problems of optical microlithography,” Proc. SPIE 5754, 506–526 (2004).
    [CrossRef]
  13. Y. Granik, K. Sakajiri, and S. Shang, “On objectives and algorithms of inverse methods in microlithography,” Proc. SPIE 6349, 63494R (2006).
    [CrossRef]
  14. A. Poonawala and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE 5674, 114–127 (2005).
    [CrossRef]
  15. A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007).
    [CrossRef]
  16. 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).
  17. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007).
    [CrossRef]
  18. X. Ma and G. R. Arce, “PSM design for inverse lithography with partially coherent illumination,” Opt. Express 16(24), 20126–20141 (2008).
    [CrossRef]
  19. V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
    [CrossRef]
  20. A. Poonawala, Y. Borodovsky, and P. Milanfar, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
    [CrossRef]
  21. N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 752032 (2009).
  22. N. Jia, A. K. Wong, and E. Y. Lam, “Robust photomask design with defocus variation using inverse synthesis,” Proc. SPIE 7140, 71401W (2008).
    [CrossRef]
  23. 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. SPIE 6924, 69240T (2008).
    [CrossRef]
  24. S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. 169(2), 463–502 (2001).
    [CrossRef]
  25. D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography I: algorithms and two-dimensional simulations,” J. Comput. Phys. 120(1), 128–144 (1995).
    [CrossRef]
  26. D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography II: three-dimensional simulations,” J. Comput. Phys. 122(2), 348–366 (1995).
    [CrossRef]
  27. D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography III: complex simulations and multiple effects,” J. Comput. Phys. 138(1), 193–223 (1997).
    [CrossRef]
  28. 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).
    [CrossRef]
  29. S. Osher and N. Paragios, Geometric Level Set Methods in Imaging, Vision, and Graphics (Springer Verlag New York, NJ, USA, 2003).
  30. F. Santosa, “A level-set approach for inverse problems involving obstacles,” ESAIM Contröle Optim. Calc. Var. 1, 17–33 (1996).
    [CrossRef]
  31. 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).
    [CrossRef]
  32. 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).
    [CrossRef]
  33. A. K.-K. Wong, Optical Imaging in Projection Microlithography (SPIE Press, Bellingham, WA, 2005).
  34. T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” in Hand-book of Mathematical Models of Computer Vision, pp. 17–32 (Springer Verlag, 2005).
  35. Y. Shen, N. Wong, and E. Y. Lam, “Interconnect thermal simulation with higher order spatial accuracy,” in Proceedings of IEEE Asia Pacific Conference on Circuits and Systems, pp. 566–569 (2008).
  36. S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2002).
  37. A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, “Uniformly high order accurate essentially non-oscillatory schemes III,” J. Comput. Phys. 71, 231–303 (1987).
    [CrossRef]
  38. C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes,” J. Comput. Phys. 77(2), 439–471 (1988).
    [CrossRef]
  39. C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes II,” J. Comput. Phys. 83(1), 32–78 (1989).
    [CrossRef]
  40. M. Minoux, Mathematical Programming: Theory and Algorithms (Wiley, New York, 1986).

2009 (3)

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

A. K. Wong and E. Y. Lam, “The nebulous hotspot and algorithm variability,” Proc. SPIE 7275, 727509 (2009).
[CrossRef]

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

2008 (5)

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

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. SPIE 6924, 69240T (2008).
[CrossRef]

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), 14746–14761(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]

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

2007 (4)

A. Poonawala, Y. Borodovsky, and P. Milanfar, “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]

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

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

2006 (1)

Y. Granik, K. Sakajiri, and S. Shang, “On objectives and algorithms of inverse methods in microlithography,” Proc. SPIE 6349, 63494R (2006).
[CrossRef]

2005 (1)

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

2004 (2)

Y. Granik, “Solving inverse problems of optical microlithography,” Proc. SPIE 5754, 506–526 (2004).
[CrossRef]

F. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” Proc. SPIE 5377, 1–20 (2004).
[CrossRef]

2001 (3)

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]

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

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).
[CrossRef]

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).
[CrossRef]

1997 (2)

D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography III: complex simulations and multiple effects,” J. Comput. Phys. 138(1), 193–223 (1997).
[CrossRef]

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).
[CrossRef]

1996 (1)

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

1995 (3)

D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography I: algorithms and two-dimensional simulations,” J. Comput. Phys. 120(1), 128–144 (1995).
[CrossRef]

D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography II: three-dimensional simulations,” J. Comput. Phys. 122(2), 348–366 (1995).
[CrossRef]

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

1994 (1)

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).
[CrossRef]

1991 (1)

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

1990 (1)

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

1989 (1)

C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes II,” J. Comput. Phys. 83(1), 32–78 (1989).
[CrossRef]

1988 (1)

C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes,” J. Comput. Phys. 77(2), 439–471 (1988).
[CrossRef]

1987 (1)

A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, “Uniformly high order accurate essentially non-oscillatory schemes III,” J. Comput. Phys. 71, 231–303 (1987).
[CrossRef]

Abrams, D.

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

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).
[CrossRef]

D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography III: complex simulations and multiple effects,” J. Comput. Phys. 138(1), 193–223 (1997).
[CrossRef]

D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography I: algorithms and two-dimensional simulations,” J. Comput. Phys. 120(1), 128–144 (1995).
[CrossRef]

D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography II: three-dimensional simulations,” J. Comput. Phys. 122(2), 348–366 (1995).
[CrossRef]

Arce, G. R.

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

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. SPIE 6924, 69240T (2008).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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

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. SPIE 6924, 69240T (2008).
[CrossRef]

Chakravarthy, S.

A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, “Uniformly high order accurate essentially non-oscillatory schemes III,” J. Comput. Phys. 71, 231–303 (1987).
[CrossRef]

Chan, S. H.

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), 14746–14761(2008).
[CrossRef]

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).

Chan, T.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” in Hand-book of Mathematical Models of Computer Vision, pp. 17–32 (Springer Verlag, 2005).

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. SPIE 6924, 69240T (2008).
[CrossRef]

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. SPIE 6924, 69240T (2008).
[CrossRef]

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. SPIE 6924, 69240T (2008).
[CrossRef]

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. SPIE 6924, 69240T (2008).
[CrossRef]

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).
[CrossRef]

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]

Engquist, B.

A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, “Uniformly high order accurate essentially non-oscillatory schemes III,” J. Comput. Phys. 71, 231–303 (1987).
[CrossRef]

Esedoglu, S.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” in Hand-book of Mathematical Models of Computer Vision, pp. 17–32 (Springer Verlag, 2005).

Fedkiw, R.

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2002).

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]

Granik, Y.

Y. Granik, K. Sakajiri, and S. Shang, “On objectives and algorithms of inverse methods in microlithography,” Proc. SPIE 6349, 63494R (2006).
[CrossRef]

Y. Granik, “Solving inverse problems of optical microlithography,” Proc. SPIE 5754, 506–526 (2004).
[CrossRef]

Harten, A.

A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, “Uniformly high order accurate essentially non-oscillatory schemes III,” J. Comput. Phys. 71, 231–303 (1987).
[CrossRef]

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).
[CrossRef]

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. SPIE 6924, 69240T (2008).
[CrossRef]

Jia, N.

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).
[CrossRef]

Kailath, T.

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).
[CrossRef]

Lam, E. Y.

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

A. K. Wong and E. Y. Lam, “The nebulous hotspot and algorithm variability,” Proc. SPIE 7275, 727509 (2009).
[CrossRef]

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).
[CrossRef]

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), 14746–14761(2008).
[CrossRef]

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).

Y. Shen, N. Wong, and E. Y. Lam, “Interconnect thermal simulation with higher order spatial accuracy,” in Proceedings of IEEE Asia Pacific Conference on Circuits and Systems, pp. 566–569 (2008).

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]

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, 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).
[CrossRef]

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

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

Ma, X.

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

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]

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).
[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]

A. Poonawala, Y. Borodovsky, and P. Milanfar, “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]

Minoux, M.

M. Minoux, Mathematical Programming: Theory and Algorithms (Wiley, New York, 1986).

Osher, S.

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).
[CrossRef]

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

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).
[CrossRef]

C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes II,” J. Comput. Phys. 83(1), 32–78 (1989).
[CrossRef]

C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes,” J. Comput. Phys. 77(2), 439–471 (1988).
[CrossRef]

A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, “Uniformly high order accurate essentially non-oscillatory schemes III,” J. Comput. Phys. 71, 231–303 (1987).
[CrossRef]

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2002).

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

Pang, L.

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. SPIE 6924, 69240T (2008).
[CrossRef]

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

Paragios, N.

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

Park, F.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” in Hand-book of Mathematical Models of Computer Vision, pp. 17–32 (Springer Verlag, 2005).

Pati, Y. C.

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).
[CrossRef]

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. SPIE 6924, 69240T (2008).
[CrossRef]

Poonawala, A.

A. Poonawala, Y. Borodovsky, and P. Milanfar, “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]

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

Sakajiri, K.

Y. Granik, K. Sakajiri, and S. Shang, “On objectives and algorithms of inverse methods in microlithography,” Proc. SPIE 6349, 63494R (2006).
[CrossRef]

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).
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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).
[CrossRef]

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

Schellenberg, F.

F. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” Proc. SPIE 5377, 1–20 (2004).
[CrossRef]

Sethian, J. A.

D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography III: complex simulations and multiple effects,” J. Comput. Phys. 138(1), 193–223 (1997).
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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).
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D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography II: three-dimensional simulations,” J. Comput. Phys. 122(2), 348–366 (1995).
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D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography I: algorithms and two-dimensional simulations,” J. Comput. Phys. 120(1), 128–144 (1995).
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Shang, S.

Y. Granik, K. Sakajiri, and S. Shang, “On objectives and algorithms of inverse methods in microlithography,” Proc. SPIE 6349, 63494R (2006).
[CrossRef]

Shen, Y.

Y. Shen, N. Wong, and E. Y. Lam, “Interconnect thermal simulation with higher order spatial accuracy,” in Proceedings of IEEE Asia Pacific Conference on Circuits and Systems, pp. 566–569 (2008).

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).
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C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes II,” J. Comput. Phys. 83(1), 32–78 (1989).
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C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes,” J. Comput. Phys. 77(2), 439–471 (1988).
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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).
[CrossRef]

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).
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Wagner, S.

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

Wong, A. K.

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

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

A. K. Wong and E. Y. Lam, “The nebulous hotspot and algorithm variability,” Proc. SPIE 7275, 727509 (2009).
[CrossRef]

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), 14746–14761(2008).
[CrossRef]

N. Jia, A. K. Wong, and E. Y. Lam, “Robust photomask 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).
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Wong, A. K.-K.

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

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

Wong, N.

Y. Shen, N. Wong, and E. Y. Lam, “Interconnect thermal simulation with higher order spatial accuracy,” in Proceedings of IEEE Asia Pacific Conference on Circuits and Systems, pp. 566–569 (2008).

Yip, A.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” in Hand-book of Mathematical Models of Computer Vision, pp. 17–32 (Springer Verlag, 2005).

Zakhor, A.

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

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

ESAIM Contröle Optim. Calc. Var. (1)

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

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. (2)

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

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

IEEE Trans. Semicond. Manuf. (1)

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).
[CrossRef]

in Proc. SPIE (1)

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

J. Comput. Phys. (8)

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

D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography I: algorithms and two-dimensional simulations,” J. Comput. Phys. 120(1), 128–144 (1995).
[CrossRef]

D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography II: three-dimensional simulations,” J. Comput. Phys. 122(2), 348–366 (1995).
[CrossRef]

D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography III: complex simulations and multiple effects,” J. Comput. Phys. 138(1), 193–223 (1997).
[CrossRef]

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).
[CrossRef]

A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, “Uniformly high order accurate essentially non-oscillatory schemes III,” J. Comput. Phys. 71, 231–303 (1987).
[CrossRef]

C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes,” J. Comput. Phys. 77(2), 439–471 (1988).
[CrossRef]

C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes II,” J. Comput. Phys. 83(1), 32–78 (1989).
[CrossRef]

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

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).
[CrossRef]

Opt. Express (4)

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

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

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), 14746–14761(2008).
[CrossRef]

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

Proc. SPIE (12)

A. K. Wong and E. Y. Lam, “The nebulous hotspot and algorithm variability,” Proc. SPIE 7275, 727509 (2009).
[CrossRef]

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

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

F. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” Proc. SPIE 5377, 1–20 (2004).
[CrossRef]

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

A. Poonawala, Y. Borodovsky, and P. Milanfar, “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, “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).
[CrossRef]

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. SPIE 6924, 69240T (2008).
[CrossRef]

Y. Granik, “Solving inverse problems of optical microlithography,” Proc. SPIE 5754, 506–526 (2004).
[CrossRef]

Y. Granik, K. Sakajiri, and S. Shang, “On objectives and algorithms of inverse methods in microlithography,” Proc. SPIE 6349, 63494R (2006).
[CrossRef]

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

SIAM J. Sci. Comp. (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).
[CrossRef]

Other (8)

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

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” in Hand-book of Mathematical Models of Computer Vision, pp. 17–32 (Springer Verlag, 2005).

Y. Shen, N. Wong, and E. Y. Lam, “Interconnect thermal simulation with higher order spatial accuracy,” in Proceedings of IEEE Asia Pacific Conference on Circuits and Systems, pp. 566–569 (2008).

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2002).

M. Minoux, Mathematical Programming: Theory and Algorithms (Wiley, New York, 1986).

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

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 Press, Bellingham, WA, 2001).

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Figures (3)

Fig. 1.
Fig. 1.

Simulation of lithographic imaging with different mask patterns computed using different spatial schemes. The first column denotes the input U(x), the second column I aerial(x), and the third column I(x). Row (a) uses the target circuit pattern I 0 as input. Row (b) uses the pattern derived by our level set algorithm, with first-order temporal accuracy and ENO1 spatial accuracy. Row (c), (d) and (e) are similar to (b), but with ENO2, ENO3 and WENO spatial accuracy, respectively.

Fig. 2.
Fig. 2.

Simulation of lithographic imaging with different mask patterns computed using different temporal schemes. The first column denotes the input U(x), the second column I aerial(x), and the third column I(x). Row (a) uses the target circuit pattern I 0 as input. Row (b) uses the pattern derived by our level set algorithm, with ENO2 spatial and first-order temporal accuracy. Row (c) and (d) are similar to (b), but with second-order and third-order temporal accuracy, respectively.

Fig. 3.
Fig. 3.

Simulation of lithographic imaging with different mask patterns computed using gradient method and level set approach. The first column denotes the input U(x), the second column I aerial(x), and the third column I(x). Row (a) uses the target circuit pattern I0 as input. Row (b) uses the pattern derived by gradient method. Row (c) uses the target pattern derived by our approach with ENO2 spatial and first-order temporal accuracy. Row (d) uses the target pattern derived by gradient method with target circuit pattern in Fig 1.

Tables (3)

Tables Icon

Table 1. Normalized Computation Time (against that in Fig. 1(c) using ENO2) and Pattern Error (pixel difference) in Fig. 1

Tables Icon

Table 2. Normalized Computation Time (against that in Fig. 2(b) using first-order temporal accuracy) and Pattern Error (pixel difference) in Fig. 2

Tables Icon

Table 3. Table 3. Normalized Computation Time (computation time in Fig. 3(b) and Fig. 3(d) using gradient method normalized against that in Fig. 3(c) and Fig. 1(c) using the proposed method, respectively) and Pattern Error (pixel difference) in Fig. 3

Equations (23)

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

I ( x ) = { U ( x ) } ,
Û ( x ) = arg min U ( x ) d ( I 0 ( x ) , { U ( x ) } ) ,
I aerial ( x ) = H ( x ) * U ( x ) 2 ,
sig ( U ( x ) ) = 1 1 + e a ( U ( x ) t r )
I ( x ) = { U ( x ) } = sig ( I aerial ( x ) ) = sig ( H ( x ) * U ( x ) 2 ) .
𝓡 1 ( U ) = Δ U L 2 = Ω ( U x ) 2 + ( U Y ) 2 d x ,
󑓡 2 ( U ) = U L 2 = Ω ( 2 U x 2 + 2 U Y 2 ) 2 d x ,
𝓡 3 ( U ) = Ω U d x = Ω ( U x ) 2 + ( U y ) 2 d x .
minimize Ω U d x
subject to Ω ( sig ( H * U 2 ) I 0 ) 2 d x = ε ,
λ · ( U U ) + α ( x ) = 0 ,
1 2 ( Ω ( sig ( H * U 2 ) I 0 ) 2 d x Ω ε ) = 0 ,
α ( x ) = 1 2 U ( sig ( H * U 2 ) I 0 ) 2
= α { H * [ ( I 0 sig ( H * U ) ) sig ( H * U ) ( 1 ( H * U ) ) ( H * U ) ] } ,
min U Ω ( λ U + 1 2 ( sig ( H * U 2 ) I 0 ) 2 ) d x ,
U t = α ( x , t ) + λ · ( U U ) ,
U t = U α ( x , t ) + λ U · ( U U ) .
U ( x ) = { U int for { x : ϕ ( x ) < 0 } U int for { x : ϕ ( x ) > 0 }
F ( U ) = 1 2 𝒯 ( U ) I 0 2 .
δ ϕ + ∇ϕ · δ x = 0 ,
δ x = α ( x , t ) ϕ ϕ .
α ( x , t ) = J ( U ) T ( 𝓣 ( U ) I 0 ) ,
ϕ t = ϕ α ( x , t ) .

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