Abstract

This paper proposes to use the a-priori knowledge of the target layout patterns to design data-adaptive compressive sensing (CS) methods for efficient source optimization (SO) in lithography systems. A set of monitoring pixels are selected from the target layout based on blue noise random patterns. The SO is then formulated as an under-determined linear problem to improve image fidelity according to the monitoring pixels. Adaptive projections are then designed, based on the a-priori knowledge of the target layout, in order to further reduce the dimension of the optimization problem, while trying to retain the SO performance. Different from traditional CS methods, adaptive projections are constructed directly from the target layout data via a nonlinear thresholding operation. Adaptive projections are proved to achieve superior SO performance over the random projections. This paper also studies and compares the impact of different sparse representation bases on the SO performance. It is shown that the discrete cosine transform (DCT), spatial and Haar wavelet bases are good choices for source representation.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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2015 (5)

L. Wang, S. Li, X. Wang, G. Yan, and C. Yang, “Source optimization using particle swarm optimization algorithm in optical lithography,” ACTA OPTICA SINICA 35(4), 0422002 (2015).
[Crossref]

H. Jiang and T. Xing, “A method of source optimization to maximize process window,” Laser & Optoelectronics Progress 52, 101101 (2015).
[Crossref]

X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and G. R. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
[Crossref]

C. Han, Y. Li, X. Ma, and L. Liu, “Robust hybrid source and mask optimization to lithography source blur and flare,” Appl. Opt. 54(17), 5291–5302 (2015).
[Crossref] [PubMed]

A. P. Cuadros, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture optimization for compressive X-ray tomosynthesis,” Opt. Express 23(25), 32788–32802 (2015).
[Crossref] [PubMed]

2014 (6)

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]

C. Han, Y. Li, L. Dong, X. Ma, and X. Guo, “Inverse pupil wavefront optimization for immersion lithography,” Appl. Opt. 53(29), 6861–6871 (2014).
[Crossref] [PubMed]

H. Arguello and G. R. Arce, “Colored coded aperture design by concentration of measure in compressive spectral imaging,” IEEE Trans. Image Process. 23(4), 1896–1908 (2014).
[Crossref] [PubMed]

X. Guo, Y. Li, L. Dong, L. Liu, X. Ma, and C. Han, “Parametric source-mask-numerical aperture co-optimization for immersion lithography,” J. Micro/Nanolith. MEMS MOEMS 13(4), 043013 (2014).
[Crossref]

L. Wei and Y. Li, “Hybrid approach for the design of mirror array to produce freeform illumination sources in immersion lithography,” Optik 125, 6166–6171 (2014).
[Crossref]

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: an introduction,” IEEE Signal Processing Magazine 31(1), 105–115 (2014).
[Crossref]

2013 (2)

2012 (2)

2011 (3)

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2× node logic,” Proc. SPIE 8166, 81662A (2011).
[Crossref]

J. Yu and P. Yu, “Gradient-based fast source mask optimization (SMO),” Proc. SPIE 7973, 797320 (2011).
[Crossref]

S. Hansen, “Source mask polarization optimization,” J. Micro/Nanolith. MEMS MOEMS 10(3), 033003 (2011).
[Crossref]

2010 (3)

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Process. 19(1), 264–270 (2010).
[Crossref]

L. Wei, “Multi-class blue noise sampling,” ACM Transactions on Graphics 29(4), 157–166 (2010).
[Crossref]

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

2009 (3)

J. F. Cai, S. Osher, and Z. Shen, “Linearized bregman iterations for compressed sensing,” Mathematics of Computation 78(267), 1515–1536 (2009).
[Crossref]

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of global source optimization in optical lithography,” Proc. SPIE 7274, 72740C (2009).
[Crossref]

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

2006 (2)

E. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52(2), 489–509 (2006).
[Crossref]

D. Donoho, “Compressive sensing,” IEEE Trans. Inform. Theory 52(4), 1289–1306 (2006).
[Crossref]

2005 (1)

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[Crossref]

2004 (2)

Y. Granik, “Source optimization for image fidelity and throughput,” J. Microlith. Microfab. Microsyst. 3(4), 509–522 (2004).

A. Erdmann, T. Fühner, T. Schnattinger, and B. Tollkühn, “Towards automatic mask and source optimization for optical lithography,” Proc. SPIE 5377, 646–657 (2004).
[Crossref]

2003 (2)

D. L. Lau, R. Ulichney, and G. R. Arce, “Blue and green noise halftoning models,” IEEE Signal Processing Magazine 20(4), 28–38 (2003).
[Crossref]

D. L. Lau, R. Ulichney, and G. R. Arce, “Blue and green-noise halftoning models,” IEEE Signal Processing Magazine 20(4), 28–38 (2003).
[Crossref]

2002 (1)

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

2000 (1)

D. L. Lau, G. R. Arce, and N. C. Gallagher, “Digital color halftoning with generalized error diffusion and multichannel green-noise masks,” IEEE Trans. Image Process. 9(5), 923–935 (2000).
[Crossref]

1998 (1)

D. L. Lau, G. R. Arce, and Neal C. Gallagher, “Green-noise digital halftoning,” Proc. IEEE 86(12), 2424–2444 (1998).
[Crossref]

1992 (1)

1988 (1)

R. Ulichney, “Dithering with blue noise,” Proc. IEEE 76(1), 56–79 (1988).
[Crossref]

Adrichem, P. V.

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2× node logic,” Proc. SPIE 8166, 81662A (2011).
[Crossref]

Arce, G. R.

X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and G. R. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
[Crossref]

A. P. Cuadros, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture optimization for compressive X-ray tomosynthesis,” Opt. Express 23(25), 32788–32802 (2015).
[Crossref] [PubMed]

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]

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: an introduction,” IEEE Signal Processing Magazine 31(1), 105–115 (2014).
[Crossref]

H. Arguello and G. R. Arce, “Colored coded aperture design by concentration of measure in compressive spectral imaging,” IEEE Trans. Image Process. 23(4), 1896–1908 (2014).
[Crossref] [PubMed]

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]

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Process. 19(1), 264–270 (2010).
[Crossref]

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

D. L. Lau, R. Ulichney, and G. R. Arce, “Blue and green-noise halftoning models,” IEEE Signal Processing Magazine 20(4), 28–38 (2003).
[Crossref]

D. L. Lau, R. Ulichney, and G. R. Arce, “Blue and green noise halftoning models,” IEEE Signal Processing Magazine 20(4), 28–38 (2003).
[Crossref]

D. L. Lau, G. R. Arce, and N. C. Gallagher, “Digital color halftoning with generalized error diffusion and multichannel green-noise masks,” IEEE Trans. Image Process. 9(5), 923–935 (2000).
[Crossref]

D. L. Lau, G. R. Arce, and Neal C. Gallagher, “Green-noise digital halftoning,” Proc. IEEE 86(12), 2424–2444 (1998).
[Crossref]

D. L. Lau and G. R. Arce, Modern Digital Halftoning, 2st ed. (CRC, 2008).
[Crossref]

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

Arguello, H.

A. P. Cuadros, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture optimization for compressive X-ray tomosynthesis,” Opt. Express 23(25), 32788–32802 (2015).
[Crossref] [PubMed]

H. Arguello and G. R. Arce, “Colored coded aperture design by concentration of measure in compressive spectral imaging,” IEEE Trans. Image Process. 23(4), 1896–1908 (2014).
[Crossref] [PubMed]

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: an introduction,” IEEE Signal Processing Magazine 31(1), 105–115 (2014).
[Crossref]

Bagheri, S.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of global source optimization in optical lithography,” Proc. SPIE 7274, 72740C (2009).
[Crossref]

Bekaert, J.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Bisschop, P. D.

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2× node logic,” Proc. SPIE 8166, 81662A (2011).
[Crossref]

Bouma, A.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Bouman, W.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Brady, D. J.

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: an introduction,” IEEE Signal Processing Magazine 31(1), 105–115 (2014).
[Crossref]

Bukofsky, S.

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Burger, M.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[Crossref]

Cai, J. F.

J. F. Cai, S. Osher, and Z. Shen, “Linearized bregman iterations for compressed sensing,” Mathematics of Computation 78(267), 1515–1536 (2009).
[Crossref]

Candés, E.

E. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52(2), 489–509 (2006).
[Crossref]

Carin, L.

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: an introduction,” IEEE Signal Processing Magazine 31(1), 105–115 (2014).
[Crossref]

Chao, H. Y.

Chen, C. C.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of global source optimization in optical lithography,” Proc. SPIE 7274, 72740C (2009).
[Crossref]

Corliss, D.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Cuadros, A. P.

Dong, L.

X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and G. R. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
[Crossref]

X. Guo, Y. Li, L. Dong, L. Liu, X. Ma, and C. Han, “Parametric source-mask-numerical aperture co-optimization for immersion lithography,” J. Micro/Nanolith. MEMS MOEMS 13(4), 043013 (2014).
[Crossref]

C. Han, Y. Li, L. Dong, X. Ma, and X. Guo, “Inverse pupil wavefront optimization for immersion lithography,” Appl. Opt. 53(29), 6861–6871 (2014).
[Crossref] [PubMed]

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]

Donoho, D.

D. Donoho, “Compressive sensing,” IEEE Trans. Inform. Theory 52(4), 1289–1306 (2006).
[Crossref]

Drieenhuizen, B.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Endendijk, W.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Engelen, A.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Erdmann, A.

A. Erdmann, T. Fühner, T. Schnattinger, and B. Tollkühn, “Towards automatic mask and source optimization for optical lithography,” Proc. SPIE 5377, 646–657 (2004).
[Crossref]

Fonseca, C.

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Fühner, T.

A. Erdmann, T. Fühner, T. Schnattinger, and B. Tollkühn, “Towards automatic mask and source optimization for optical lithography,” Proc. SPIE 5377, 646–657 (2004).
[Crossref]

Gallagher, N. C.

D. L. Lau, G. R. Arce, and N. C. Gallagher, “Digital color halftoning with generalized error diffusion and multichannel green-noise masks,” IEEE Trans. Image Process. 9(5), 923–935 (2000).
[Crossref]

Gallagher, Neal C.

D. L. Lau, G. R. Arce, and Neal C. Gallagher, “Green-noise digital halftoning,” Proc. IEEE 86(12), 2424–2444 (1998).
[Crossref]

Gao, J.

X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and G. R. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
[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]

Gil, D.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of global source optimization in optical lithography,” Proc. SPIE 7274, 72740C (2009).
[Crossref]

Goldfarb, D.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[Crossref]

Granik, Y.

Y. Granik, “Source optimization for image fidelity and throughput,” J. Microlith. Microfab. Microsyst. 3(4), 509–522 (2004).

Gronlund, K.

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2× node logic,” Proc. SPIE 8166, 81662A (2011).
[Crossref]

Guo, X.

X. Guo, Y. Li, L. Dong, L. Liu, X. Ma, and C. Han, “Parametric source-mask-numerical aperture co-optimization for immersion lithography,” J. Micro/Nanolith. MEMS MOEMS 13(4), 043013 (2014).
[Crossref]

C. Han, Y. Li, L. Dong, X. Ma, and X. Guo, “Inverse pupil wavefront optimization for immersion lithography,” Appl. Opt. 53(29), 6861–6871 (2014).
[Crossref] [PubMed]

Han, C.

C. Han, Y. Li, X. Ma, and L. Liu, “Robust hybrid source and mask optimization to lithography source blur and flare,” Appl. Opt. 54(17), 5291–5302 (2015).
[Crossref] [PubMed]

X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and G. R. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
[Crossref]

X. Guo, Y. Li, L. Dong, L. Liu, X. Ma, and C. Han, “Parametric source-mask-numerical aperture co-optimization for immersion lithography,” J. Micro/Nanolith. MEMS MOEMS 13(4), 043013 (2014).
[Crossref]

C. Han, Y. Li, L. Dong, X. Ma, and X. Guo, “Inverse pupil wavefront optimization for immersion lithography,” Appl. Opt. 53(29), 6861–6871 (2014).
[Crossref] [PubMed]

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]

Hansen, S.

S. Hansen, “Source mask polarization optimization,” J. Micro/Nanolith. MEMS MOEMS 10(3), 033003 (2011).
[Crossref]

Hibbs, M.

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Hsu, S.

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2× node logic,” Proc. SPIE 8166, 81662A (2011).
[Crossref]

Iwase, K.

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2× node logic,” Proc. SPIE 8166, 81662A (2011).
[Crossref]

Jiang, H.

H. Jiang and T. Xing, “A method of source optimization to maximize process window,” Laser & Optoelectronics Progress 52, 101101 (2015).
[Crossref]

Jürgens, D.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Kazinczi, R.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Kittle, D. S.

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: an introduction,” IEEE Signal Processing Magazine 31(1), 105–115 (2014).
[Crossref]

Krasnoperova, A.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of global source optimization in optical lithography,” Proc. SPIE 7274, 72740C (2009).
[Crossref]

Laenens, B.

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2× node logic,” Proc. SPIE 8166, 81662A (2011).
[Crossref]

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Lai, K.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of global source optimization in optical lithography,” Proc. SPIE 7274, 72740C (2009).
[Crossref]

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Lam, E.

Lau, D. L.

D. L. Lau, R. Ulichney, and G. R. Arce, “Blue and green-noise halftoning models,” IEEE Signal Processing Magazine 20(4), 28–38 (2003).
[Crossref]

D. L. Lau, R. Ulichney, and G. R. Arce, “Blue and green noise halftoning models,” IEEE Signal Processing Magazine 20(4), 28–38 (2003).
[Crossref]

D. L. Lau, G. R. Arce, and N. C. Gallagher, “Digital color halftoning with generalized error diffusion and multichannel green-noise masks,” IEEE Trans. Image Process. 9(5), 923–935 (2000).
[Crossref]

D. L. Lau, G. R. Arce, and Neal C. Gallagher, “Green-noise digital halftoning,” Proc. IEEE 86(12), 2424–2444 (1998).
[Crossref]

D. L. Lau and G. R. Arce, Modern Digital Halftoning, 2st ed. (CRC, 2008).
[Crossref]

Li, J.

Li, S.

L. Wang, S. Li, X. Wang, G. Yan, and C. Yang, “Source optimization using particle swarm optimization algorithm in optical lithography,” ACTA OPTICA SINICA 35(4), 0422002 (2015).
[Crossref]

Li, Y.

X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and G. R. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
[Crossref]

C. Han, Y. Li, X. Ma, and L. Liu, “Robust hybrid source and mask optimization to lithography source blur and flare,” Appl. Opt. 54(17), 5291–5302 (2015).
[Crossref] [PubMed]

C. Han, Y. Li, L. Dong, X. Ma, and X. Guo, “Inverse pupil wavefront optimization for immersion lithography,” Appl. Opt. 53(29), 6861–6871 (2014).
[Crossref] [PubMed]

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]

L. Wei and Y. Li, “Hybrid approach for the design of mirror array to produce freeform illumination sources in immersion lithography,” Optik 125, 6166–6171 (2014).
[Crossref]

X. Guo, Y. Li, L. Dong, L. Liu, X. Ma, and C. Han, “Parametric source-mask-numerical aperture co-optimization for immersion lithography,” J. Micro/Nanolith. MEMS MOEMS 13(4), 043013 (2014).
[Crossref]

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]

Li, Z.

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2× node logic,” Proc. SPIE 8166, 81662A (2011).
[Crossref]

Liu, L.

C. Han, Y. Li, X. Ma, and L. Liu, “Robust hybrid source and mask optimization to lithography source blur and flare,” Appl. Opt. 54(17), 5291–5302 (2015).
[Crossref] [PubMed]

X. Guo, Y. Li, L. Dong, L. Liu, X. Ma, and C. Han, “Parametric source-mask-numerical aperture co-optimization for immersion lithography,” J. Micro/Nanolith. MEMS MOEMS 13(4), 043013 (2014).
[Crossref]

Liu, S.

Ma, X.

McIntyre, Greg

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Melville, D.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of global source optimization in optical lithography,” Proc. SPIE 7274, 72740C (2009).
[Crossref]

Mitsa, T.

Molless, A.

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Morgenfeld, B.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of global source optimization in optical lithography,” Proc. SPIE 7274, 72740C (2009).
[Crossref]

Mulder, M.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Noordman, O.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Nuenen, C.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Osher, S.

J. F. Cai, S. Osher, and Z. Shen, “Linearized bregman iterations for compressed sensing,” Mathematics of Computation 78(267), 1515–1536 (2009).
[Crossref]

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[Crossref]

Parker, K. J.

Peitsch, C.

Romberg, J.

E. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52(2), 489–509 (2006).
[Crossref]

Rosenbluth, A. E.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of global source optimization in optical lithography,” Proc. SPIE 7274, 72740C (2009).
[Crossref]

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Schnattinger, T.

A. Erdmann, T. Fühner, T. Schnattinger, and B. Tollkühn, “Towards automatic mask and source optimization for optical lithography,” Proc. SPIE 5377, 646–657 (2004).
[Crossref]

Shen, Y.

Shen, Z.

J. F. Cai, S. Osher, and Z. Shen, “Linearized bregman iterations for compressed sensing,” Mathematics of Computation 78(267), 1515–1536 (2009).
[Crossref]

Singh, R. N.

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Socha, R.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Song, Z.

Streutker, G.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Tao, T.

E. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52(2), 489–509 (2006).
[Crossref]

Tian, K.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of global source optimization in optical lithography,” Proc. SPIE 7274, 72740C (2009).
[Crossref]

Tirapu-Azpiroz, J.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of global source optimization in optical lithography,” Proc. SPIE 7274, 72740C (2009).
[Crossref]

Tollkühn, B.

A. Erdmann, T. Fühner, T. Schnattinger, and B. Tollkühn, “Towards automatic mask and source optimization for optical lithography,” Proc. SPIE 5377, 646–657 (2004).
[Crossref]

Trauter, B.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Ulichney, R.

D. L. Lau, R. Ulichney, and G. R. Arce, “Blue and green noise halftoning models,” IEEE Signal Processing Magazine 20(4), 28–38 (2003).
[Crossref]

D. L. Lau, R. Ulichney, and G. R. Arce, “Blue and green-noise halftoning models,” IEEE Signal Processing Magazine 20(4), 28–38 (2003).
[Crossref]

R. Ulichney, “Dithering with blue noise,” Proc. IEEE 76(1), 56–79 (1988).
[Crossref]

Verbeeck, J.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

Wang, J.

Wang, L.

L. Wang, S. Li, X. Wang, G. Yan, and C. Yang, “Source optimization using particle swarm optimization algorithm in optical lithography,” ACTA OPTICA SINICA 35(4), 0422002 (2015).
[Crossref]

Wang, X.

L. Wang, S. Li, X. Wang, G. Yan, and C. Yang, “Source optimization using particle swarm optimization algorithm in optical lithography,” ACTA OPTICA SINICA 35(4), 0422002 (2015).
[Crossref]

Wang, Z.

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Process. 19(1), 264–270 (2010).
[Crossref]

Wei, L.

L. Wei and Y. Li, “Hybrid approach for the design of mirror array to produce freeform illumination sources in immersion lithography,” Optik 125, 6166–6171 (2014).
[Crossref]

L. Wei, “Multi-class blue noise sampling,” ACM Transactions on Graphics 29(4), 157–166 (2010).
[Crossref]

Wong, A. K.

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

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

Xing, T.

H. Jiang and T. Xing, “A method of source optimization to maximize process window,” Laser & Optoelectronics Progress 52, 101101 (2015).
[Crossref]

Xu, J.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[Crossref]

Yan, G.

L. Wang, S. Li, X. Wang, G. Yan, and C. Yang, “Source optimization using particle swarm optimization algorithm in optical lithography,” ACTA OPTICA SINICA 35(4), 0422002 (2015).
[Crossref]

Yang, C.

L. Wang, S. Li, X. Wang, G. Yan, and C. Yang, “Source optimization using particle swarm optimization algorithm in optical lithography,” ACTA OPTICA SINICA 35(4), 0422002 (2015).
[Crossref]

Yin, W.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
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Yu, J.

J. Yu and P. Yu, “Gradient-based fast source mask optimization (SMO),” Proc. SPIE 7973, 797320 (2011).
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Yu, J. C.

Yu, P.

Zimmermann, J.

M. Mulder, A. Engelen, O. Noordman, G. Streutker, B. Drieenhuizen, C. Nuenen, W. Endendijk, J. Verbeeck, W. Bouman, A. Bouma, R. Kazinczi, R. Socha, D. Jürgens, J. Zimmermann, B. Trauter, J. Bekaert, B. Laenens, D. Corliss, and Greg McIntyre, “Performance of FlexRay, a fully programmable illumination system for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7640, 76401P (2010).
[Crossref]

ACM Transactions on Graphics (1)

L. Wei, “Multi-class blue noise sampling,” ACM Transactions on Graphics 29(4), 157–166 (2010).
[Crossref]

ACTA OPTICA SINICA (1)

L. Wang, S. Li, X. Wang, G. Yan, and C. Yang, “Source optimization using particle swarm optimization algorithm in optical lithography,” ACTA OPTICA SINICA 35(4), 0422002 (2015).
[Crossref]

Appl. Opt. (2)

IEEE Signal Processing Magazine (3)

D. L. Lau, R. Ulichney, and G. R. Arce, “Blue and green noise halftoning models,” IEEE Signal Processing Magazine 20(4), 28–38 (2003).
[Crossref]

D. L. Lau, R. Ulichney, and G. R. Arce, “Blue and green-noise halftoning models,” IEEE Signal Processing Magazine 20(4), 28–38 (2003).
[Crossref]

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: an introduction,” IEEE Signal Processing Magazine 31(1), 105–115 (2014).
[Crossref]

IEEE Trans. Image Process. (3)

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Process. 19(1), 264–270 (2010).
[Crossref]

H. Arguello and G. R. Arce, “Colored coded aperture design by concentration of measure in compressive spectral imaging,” IEEE Trans. Image Process. 23(4), 1896–1908 (2014).
[Crossref] [PubMed]

D. L. Lau, G. R. Arce, and N. C. Gallagher, “Digital color halftoning with generalized error diffusion and multichannel green-noise masks,” IEEE Trans. Image Process. 9(5), 923–935 (2000).
[Crossref]

IEEE Trans. Inform. Theory (2)

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Figures (14)

Fig. 1
Fig. 1

The sketch of optical lithography system.

Fig. 2
Fig. 2

The critical regions for the (a) vertical line-space layout pattern and (b) horizontal block layout pattern.

Fig. 3
Fig. 3

The generation of monitoring pixels and compressive measurements.

Fig. 4
Fig. 4

The simulations of different SO methods based on the vertical line-space layout pattern.

Fig. 5
Fig. 5

Random and adaptive projection matrices (M = 300, L = 25) used for two different layout patterns.

Fig. 6
Fig. 6

The convergence of the PE and contrast for different SO methods, where (a) and (b) are the convergence curves for the vertical line-space layout pattern, while (c) and (d) are the convergence curves for the horizontal block layout pattern.

Fig. 7
Fig. 7

The overlapped PWs obtained by different SO methods based on the (a) vertical line-space layout pattern and (b) horizontal block layout pattern.

Fig. 8
Fig. 8

The locations to measure the PWs for the (a) vertical line-space layout pattern and (b) horizontal block layout pattern.

Fig. 9
Fig. 9

The simulations of different SO methods based on the horizontal block layout pattern.

Fig. 10
Fig. 10

The simulations of CG method based on the vertical line-space layout pattern and horizontal block layout pattern.

Fig. 11
Fig. 11

The simulations of SMO methods.

Fig. 12
Fig. 12

The comparison of overlapped PWs obtained by SO and SMO methods based on the (a) vertical line-space layout pattern and (b) horizontal block layout pattern.

Fig. 13
Fig. 13

The simulations of SO and SMO methods using line-space layout pattern at 14nm technology node.

Fig. 14
Fig. 14

The coefficients of source patterns on different sparse bases.

Tables (4)

Tables Icon

Table 1 The comparison of average mutual-coherence metrics between the random and adaptive projection methods.

Tables Icon

Table 2 The average PEs, EPEs, NILSs, contrasts and runtimes of different SO methods.

Tables Icon

Table 3 The average PEs, EPEs, NILSs, contrasts and runtimes of CG methods.

Tables Icon

Table 4 The average PEs, EPEs, NILSs and contrasts of the adaptive projection method using different sparse bases.

Equations (21)

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I s = I c c s J .
Y = Φ X = Φ Ψ Θ ,
μ = max { | ϕ i , ψ j | 2 } , i = 1 , 2 , , L and j = 1 , 2 , , N .
μ ϒ = max ψ j ϒ { | ϕ i , ψ j | 2 } = 1 , μ ϒ ¯ = max ψ j ϒ ¯ { | ϕ i , ψ j | 2 } = 0 ,
ϕ i j = sgn ( S j Λ i j ) N ,
μ ¯ ϒ max ψ j ϒ E { | ϕ i , ψ j | 2 } > X 2 N K 2 θ max 2 , μ ¯ ϒ max ψ j ϒ ¯ E { | ϕ i , ϕ j | 2 } 0 .
Θ ^ = min Θ Θ 1 subject to Φ Z s = Φ I s = Φ I c c s J = Φ I c c s Ψ Θ ,
ϕ i j = sgn ( Z s , j Λ i j ) M ,
NILS = CD I con × d I d x | I con ,
NILS = 1 L c { c CD I con × d I d x | I con d c } ,
μ ¯ ϒ = max ψ j ϒ E { | ϕ i , ψ j | 2 } > 1 θ max 2 max ψ j ϒ E { | ϕ i , ψ j θ j | 2 } > 1 K 2 θ max 2 max ψ j ϒ E { | ϕ i , j = 1 K ψ j θ j | 2 } > 1 M K 2 θ max 2 E { i = 1 M | ϕ i , X | 2 } = 1 M K 2 θ max 2 E { Φ X 2 2 } ,
Φ X 2 2 = 1 N i = 1 M [ j = 1 N sgn ( S j Λ i , j ) X j ] 2 .
E ( Δ i ) = X 2 2 + r = 1 N j = 1 , j r N [ T 1 T 2 ] ,
T 1 = X r X j [ P r ( Λ i , r < X r ) P r ( Λ i , j < X j ) + P r ( Λ i , r > X r ) P r ( Λ i , j > X j ) ] ,
T 2 = X r X j [ P r ( Λ i , r > X r ) P r ( Λ i , j < X j ) + P r ( Λ i , r < X r ) P r ( Λ i , j > X j ) ] ,
E ( Δ i ) = X 2 2 + r = 1 N j = 1 , j r N X r X j [ 1 2 Q ( X r / σ Λ 2 + σ X 2 ) 2 Q ( X j / σ Λ 2 + σ X 2 ) + 4 Q ( X r / σ Λ 2 + σ X 2 ) Q ( X j / σ Λ 2 + σ X 2 ) ] ,
1 2 τ r 2 τ j + 4 τ r τ j { 0 : X r X j < 0 0 : X r X j > 0 .
μ ¯ ϒ = max ψ j ϒ E { | ϕ i , ψ j | 2 } > X 2 N K 2 θ max 2 .
μ ¯ ϒ ¯ = max ψ j ϒ ¯ E { | ϕ i , ψ j | 2 } = E { | ϕ ^ , ψ ^ | 2 } = 1 N E { | sgn ( X Λ ) , ψ ^ | 2 } = 1 N E { p = 1 N sgn ( X p Λ ^ p ) ψ ^ p } 2 = 1 N m = 1 N n = 1 N ψ ^ m ψ ^ n E { sgn ( X m Λ ^ m ) sgn ( X n Λ ^ n ) } = 1 N m = 1 N n = 1 N ψ ^ m ψ ^ n [ 1 2 Q ( X m / σ Λ 2 + σ X 2 ) 2 Q ( X n / σ Λ 2 + σ X 2 ) + 4 Q ( X m / σ Λ 2 + σ X 2 ) Q ( X n / σ Λ 2 + σ X 2 ) ] .
Q ( x ) 1 2 1 2 π x .
μ ¯ ϒ ¯ = max ψ j ϒ ¯ E { | ϕ i , ψ j | 2 } 1 N m = 1 N n = 1 N ψ ^ m ψ ^ n 2 X m X n π ( σ Λ 2 + σ X 2 ) = 2 X , ψ ^ 2 N π ( σ Λ 2 + σ X 2 ) = 0 ,

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