Abstract

Pixel-based optical proximity correction (PBOPC) methods have been developed as a leading-edge resolution enhancement technique (RET) for integrated circuit fabrication. PBOPC independently modulates each pixel on the reticle, which tremendously increases the mask’s complexity and, at the same time, deteriorates its manufacturability. Most current PBOPC algorithms recur to regularization methods or a mask manufacturing rule check (MRC) to improve the mask manufacturability. Typically, these approaches either fail to satisfy manufacturing constraints on the practical product line, or lead to suboptimal mask patterns that may degrade the lithographic performance. This paper develops a block-based optical proximity correction (BBOPC) algorithm to pursue the optimal masks with manufacturability compliance, where the mask is shaped by a set of overlapped basis blocks rather than pixels. BBOPC optimization is formulated based on a vector imaging model, which is adequate for both dry lithography with lower numerical aperture (NA), and immersion lithography with hyper-NA. The BBOPC algorithm successively optimizes the main features (MF) and subresolution assist features (SRAF) based on a modified conjugate gradient method. It is effective at smoothing any unmanufacturable jogs along edges. A weight matrix is introduced in the cost function to preserve the edge fidelity of the printed images. Simulations show that the BBOPC algorithm can improve lithographic imaging performance while maintaining mask manufacturing constraints.

© 2013 Optical Society of America

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References

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2013

2012

X. Ma, Y. Li, and L. Dong, “Mask optimization approaches in optical lithography based on a vector imaging model,” J. Opt. Soc. Am. A 29, 1300–1312 (2012).
[CrossRef]

K. Kato, Y. Taniguchi, T. Inoue, and K. Kadota, “Novel MRC algorithms using GPGPU,” Proc. SPIE 8441, 84410R (2012).
[CrossRef]

2011

Y. Ping, X. Li, S. Jang, D. Kwa, Y. Zhang, and R. Lugg, “Tolerance-based OPC and solution to MRC-constrained OPC,” Proc. SPIE 7973, 79732M (2011).
[CrossRef]

T. Cecil, C. Ashton, D. Irby, L. Luan, D. H. Son, G. Xiao, X. Zhou, D. Kim, and B. Gleason, “Enhancing fullchip ILT mask synthesis capability for IC manufacturability,” Proc. SPIE 7973, 79731C (2011).
[CrossRef]

X. Ma, S. Jiang, and A. Zakhor, “A cost-driven fracture heuristics to minimize sliver length,” Proc. SPIE 7973, 79732O (2011).
[CrossRef]

J. Yu and P. Yu, “Choosing objective functions for inverse lithography patterning,” Proc. SPIE 7973, 79731N (2011).
[CrossRef]

X. Ma and Y. Li, “Resolution enhancement optimization methods in optical lithography with improved manufacturability,” J. Micro/Nanolith. MEMS MOEMS 10, 023009 (2011).
[CrossRef]

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

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

2010

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

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

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

D. Peng, P. Hu, V. Tolani, and T. Dam, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[CrossRef]

2009

N. Jia, A. K. Wang, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).
[CrossRef]

B. Kim, S. S. Suh, S. G. Woo, H. Cho, G. Xiao, D. H. Son, D. Irby, D. Kim, and K. Baik, “Inverse lithography technology (ILT) mask manufacturability for full-chip device,” Proc. SPIE 7488, 748812 (2009).
[CrossRef]

2008

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

Y. Zhou and Y. Li, “Optimization of double bottom antireflective coating for hyper numerical aperture lithography,” Acta Opt. Sin. 28, 472–477 (2008).
[CrossRef]

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

2007

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

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

A. Poonawala and P. Milanfar, “Double-exposure mask synthesis using inverse lithography,” J. Micro/Nanolithogr. MEMS MOEMS 6, 043001 (2007).
[CrossRef]

2006

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

A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: a non-linear optimization approach,” Proc. SPIE 6154, 1159–1172 (2006).
[CrossRef]

2005

M. Totzeck, P. Graüpner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

2004

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

2001

2000

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[CrossRef]

1995

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

1992

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

Arce, G. R.

Ashton, C.

T. Cecil, C. Ashton, D. Irby, L. Luan, D. H. Son, G. Xiao, X. Zhou, D. Kim, and B. Gleason, “Enhancing fullchip ILT mask synthesis capability for IC manufacturability,” Proc. SPIE 7973, 79731C (2011).
[CrossRef]

Baik, K.

B. Kim, S. S. Suh, S. G. Woo, H. Cho, G. Xiao, D. H. Son, D. Irby, D. Kim, and K. Baik, “Inverse lithography technology (ILT) mask manufacturability for full-chip device,” Proc. SPIE 7488, 748812 (2009).
[CrossRef]

Cecil, T.

T. Cecil, C. Ashton, D. Irby, L. Luan, D. H. Son, G. Xiao, X. Zhou, D. Kim, and B. Gleason, “Enhancing fullchip ILT mask synthesis capability for IC manufacturability,” Proc. SPIE 7973, 79731C (2011).
[CrossRef]

Cho, H.

B. Kim, S. S. Suh, S. G. Woo, H. Cho, G. Xiao, D. H. Son, D. Irby, D. Kim, and K. Baik, “Inverse lithography technology (ILT) mask manufacturability for full-chip device,” Proc. SPIE 7488, 748812 (2009).
[CrossRef]

Dam, T.

D. Peng, P. Hu, V. Tolani, and T. Dam, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[CrossRef]

Dittmann, O.

M. Totzeck, P. Graüpner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Dong, L.

Flagello, D.

M. Totzeck, P. Graüpner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Gallatin, G. M.

Gleason, B.

T. Cecil, C. Ashton, D. Irby, L. Luan, D. H. Son, G. Xiao, X. Zhou, D. Kim, and B. Gleason, “Enhancing fullchip ILT mask synthesis capability for IC manufacturability,” Proc. SPIE 7973, 79731C (2011).
[CrossRef]

Göhnermeier, A.

M. Totzeck, P. Graüpner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 2nd ed.(McGraw-Hill, 1996).

Granik, Y.

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

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

Graüpner, P.

M. Totzeck, P. Graüpner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Han, C.

Heil, T.

M. Totzeck, P. Graüpner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Hu, P.

D. Peng, P. Hu, V. Tolani, and T. Dam, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[CrossRef]

Inoue, T.

K. Kato, Y. Taniguchi, T. Inoue, and K. Kadota, “Novel MRC algorithms using GPGPU,” Proc. SPIE 8441, 84410R (2012).
[CrossRef]

Irby, D.

T. Cecil, C. Ashton, D. Irby, L. Luan, D. H. Son, G. Xiao, X. Zhou, D. Kim, and B. Gleason, “Enhancing fullchip ILT mask synthesis capability for IC manufacturability,” Proc. SPIE 7973, 79731C (2011).
[CrossRef]

B. Kim, S. S. Suh, S. G. Woo, H. Cho, G. Xiao, D. H. Son, D. Irby, D. Kim, and K. Baik, “Inverse lithography technology (ILT) mask manufacturability for full-chip device,” Proc. SPIE 7488, 748812 (2009).
[CrossRef]

Jang, S.

Y. Ping, X. Li, S. Jang, D. Kwa, Y. Zhang, and R. Lugg, “Tolerance-based OPC and solution to MRC-constrained OPC,” Proc. SPIE 7973, 79732M (2011).
[CrossRef]

Jia, N.

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

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

N. Jia, A. K. Wang, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).
[CrossRef]

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

Jiang, S.

X. Ma, S. Jiang, and A. Zakhor, “A cost-driven fracture heuristics to minimize sliver length,” Proc. SPIE 7973, 79732O (2011).
[CrossRef]

Kadota, K.

K. Kato, Y. Taniguchi, T. Inoue, and K. Kadota, “Novel MRC algorithms using GPGPU,” Proc. SPIE 8441, 84410R (2012).
[CrossRef]

Kamenov, V.

M. Totzeck, P. Graüpner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Kato, K.

K. Kato, Y. Taniguchi, T. Inoue, and K. Kadota, “Novel MRC algorithms using GPGPU,” Proc. SPIE 8441, 84410R (2012).
[CrossRef]

Kim, B.

B. Kim, S. S. Suh, S. G. Woo, H. Cho, G. Xiao, D. H. Son, D. Irby, D. Kim, and K. Baik, “Inverse lithography technology (ILT) mask manufacturability for full-chip device,” Proc. SPIE 7488, 748812 (2009).
[CrossRef]

Kim, D.

T. Cecil, C. Ashton, D. Irby, L. Luan, D. H. Son, G. Xiao, X. Zhou, D. Kim, and B. Gleason, “Enhancing fullchip ILT mask synthesis capability for IC manufacturability,” Proc. SPIE 7973, 79731C (2011).
[CrossRef]

B. Kim, S. S. Suh, S. G. Woo, H. Cho, G. Xiao, D. H. Son, D. Irby, D. Kim, and K. Baik, “Inverse lithography technology (ILT) mask manufacturability for full-chip device,” Proc. SPIE 7488, 748812 (2009).
[CrossRef]

Krähmer, D.

M. Totzeck, P. Graüpner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Kwa, D.

Y. Ping, X. Li, S. Jang, D. Kwa, Y. Zhang, and R. Lugg, “Tolerance-based OPC and solution to MRC-constrained OPC,” Proc. SPIE 7973, 79732M (2011).
[CrossRef]

Lam, E. Y.

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

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

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

N. Jia, A. K. Wang, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).
[CrossRef]

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

Leone, R.

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

Li, X.

Y. Ping, X. Li, S. Jang, D. Kwa, Y. Zhang, and R. Lugg, “Tolerance-based OPC and solution to MRC-constrained OPC,” Proc. SPIE 7973, 79732M (2011).
[CrossRef]

Li, Y.

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, 112–123 (2013).
[CrossRef]

X. Ma, Y. Li, and L. Dong, “Mask optimization approaches in optical lithography based on a vector imaging model,” J. Opt. Soc. Am. A 29, 1300–1312 (2012).
[CrossRef]

X. Ma and Y. Li, “Resolution enhancement optimization methods in optical lithography with improved manufacturability,” J. Micro/Nanolith. MEMS MOEMS 10, 023009 (2011).
[CrossRef]

Y. Zhou and Y. Li, “Optimization of double bottom antireflective coating for hyper numerical aperture lithography,” Acta Opt. Sin. 28, 472–477 (2008).
[CrossRef]

Liu, Y.

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

Luan, L.

T. Cecil, C. Ashton, D. Irby, L. Luan, D. H. Son, G. Xiao, X. Zhou, D. Kim, and B. Gleason, “Enhancing fullchip ILT mask synthesis capability for IC manufacturability,” Proc. SPIE 7973, 79731C (2011).
[CrossRef]

Lugg, R.

Y. Ping, X. Li, S. Jang, D. Kwa, Y. Zhang, and R. Lugg, “Tolerance-based OPC and solution to MRC-constrained OPC,” Proc. SPIE 7973, 79732M (2011).
[CrossRef]

Ma, X.

Milanfar, P.

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

A. Poonawala and P. Milanfar, “Double-exposure mask synthesis using inverse lithography,” J. Micro/Nanolithogr. MEMS MOEMS 6, 043001 (2007).
[CrossRef]

A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: a non-linear optimization approach,” Proc. SPIE 6154, 1159–1172 (2006).
[CrossRef]

Neureuther, A. R.

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[CrossRef]

Peng, D.

D. Peng, P. Hu, V. Tolani, and T. Dam, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[CrossRef]

Ping, Y.

Y. Ping, X. Li, S. Jang, D. Kwa, Y. Zhang, and R. Lugg, “Tolerance-based OPC and solution to MRC-constrained OPC,” Proc. SPIE 7973, 79732M (2011).
[CrossRef]

Pistor, T. V.

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[CrossRef]

Poonawala, A.

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

A. Poonawala and P. Milanfar, “Double-exposure mask synthesis using inverse lithography,” J. Micro/Nanolithogr. MEMS MOEMS 6, 043001 (2007).
[CrossRef]

A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: a non-linear optimization approach,” Proc. SPIE 6154, 1159–1172 (2006).
[CrossRef]

Ruoff, J.

M. Totzeck, P. Graüpner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Saleh, B.

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

Shen, Y.

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

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

Sherif, S.

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

Socha, R. J.

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[CrossRef]

Son, D. H.

T. Cecil, C. Ashton, D. Irby, L. Luan, D. H. Son, G. Xiao, X. Zhou, D. Kim, and B. Gleason, “Enhancing fullchip ILT mask synthesis capability for IC manufacturability,” Proc. SPIE 7973, 79731C (2011).
[CrossRef]

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Figures (13)

Fig. 1.
Fig. 1.

Illustration of different types of OPC approaches: (a) The target pattern to be projected on the wafer, and the corresponding (b) rule-based OPC, (c) edge-based OPC, and (d) pixel-based OPC masks.

Fig. 2.
Fig. 2.

Critical parameters concerned for mask manufacturability: (a) the minimum lateral sizes of MFs and SRAFs, and the minimum space between MFs and SRAFs, (b) the height and arm lengths of the jog.

Fig. 3.
Fig. 3.

Flowchart of the BBOPC algorithm.

Fig. 4.
Fig. 4.

Mask formation process. From left to right: (a) basis block W, (b) coefficient matrix Θ, and the (c) mask pattern M, where the black and white represent opaque areas and openings, respectively.

Fig. 5.
Fig. 5.

Flowcharts for the (a) MFO procedure, (b) “Method 1,” and (c) “Method 2.”

Fig. 6.
Fig. 6.

Simulations of MFO and WMFO for the dense line-space target pattern. Top row (from left to right) shows: (a) target pattern, optimized MF patterns using MFO algorithm (b) without and (c) with the usage of weight matrix Π, and (d) MF pattern after smoothing the risk jogs. Black and white represent the opaque areas and mask openings, respectively. Bottom row shows the printed images corresponding to the input masks in the top row. Black and white represent 0 and 1, respectively.

Fig. 7.
Fig. 7.

Element values weight matrix Π for the (a) line-space and the (b) complex target patterns. The gray and white represent the element values of 1 and 1.6, respectively.

Fig. 8.
Fig. 8.

Simulations of MFO and WMFO for a more complex target pattern. Top row (from left to right) shows: (a) target pattern, optimized MF patterns using MFO algorithm (b) without and (c) with the usage of weight matrix Π, and (d) MF pattern after smoothing the risk jogs. Black and white represent the opaque areas and mask openings, respectively. Bottom row shows the printed images corresponding to the input masks in the top row. Black and white represent 0 and 1, respectively.

Fig. 9.
Fig. 9.

Flowchart of risk jog smooth method.

Fig. 10.
Fig. 10.

SRAF seeds for the (a) line-space pattern and (b) complex target pattern. Gray, white and black represent the optimized MFs, SRAF seeds, and the opaque regions.

Fig. 11.
Fig. 11.

Simulations of SRAFO for both line-space pattern and complex patterns. Top row (from left to right) shows: the SRAFO results (a) before and (b) after applying the risk jog smooth method for the line-space pattern, and the SRAFO results (c) before and (d) after applying the risk jog smooth method for the complex pattern. Black and white represent opaque areas and mask openings, respectively. Bottom row shows the printed images corresponding to the input masks in the top row. Black and white represent 0 and 1, respectively.

Fig. 12.
Fig. 12.

Comparison of PBOPC and BBOPC algorithms. Top row (from left to right) shows: (a) PBOPC and (b) BBOPC masks for the line-space target, as well as (c) PBOPC and (d) BBOPC masks for the complex target. Black and white represent the opaque areas and mask openings, respectively. Bottom row shows the printed images corresponding to the input masks in the top row. Black and white represent 0 and 1, respectively.

Fig. 13.
Fig. 13.

Simulations of the PBOPC algorithm combined with the risk jog smooth method for both line-space and complex patterns. From left to right: (a) the mask obtained by the PBOPC algorithm and risk jog smooth method for the line-space target and (b) its corresponding print image, and (c) the mask obtained by the PBOPC algorithm and risk jog smooth method for the complex target and (d) its corresponding print image.

Tables (1)

Tables Icon

Table 1. Performance Comparison Among Target Pattern, BBOPC and PBOPC, Where RJSM Represents Risk Jog Smooth Method

Equations (14)

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M=Γ{WΘ1}.
Z=T{M}=Γ{1Nsxsysp=x,y,zHpxsys(BxsysΓ{WΘ1})22tr},
Θ^=argminΘRN×ND=argminΘRN×N(F+γRR),
F=m=1Nn=1NΠ(m,n)×[Z(m,n)Z˜(m,n)]2=Π(ZZ˜)22
F(Θ)=4a2NsW[F˜(Θ)sig{WΘ,1}(1sig{WΘ,1})],
F˜(Θ)=xsysp=x,y,zReal[(Bxsys)*((Hpxsys)*°{[Hpxsys(Bxsyssig{WΘ,1})]Π(Z˜Z)Z(1Z)})],
F˜(Θ)=xsysp=x,y,zReal[(Bxsys)*F1{2πnwRVpxsys*CF[Epwafer(xs,ys)Π(Z˜Z)Z(1Z)]}],
Θ0(m,n)={0.9/mnW(m,n)if[WZ˜](m,n)<mnW(m,n)1if[WZ˜](m,n)=mnW(m,n),m,n=1,2,,N,
Θbk+1=Γ{Θk+10.5};Mbk+1=Γ{WΘbk+11}.
Θk+1(m,n)={1ifΘk+1(m,n)>1Θk+1(m,n)if0<Θk+1(m,n)<10ifΘk+1(m,n)<0,m,n=1,2,,N.
βk=F(Θk+1)22F(Θk)22;Pk+1=F(Θk+1)+βkPk.
C˜Θ(m,n)={1ifM˜b(m,n)is a boundary pixel0otherwise,m,n=1,2,,N,
ΘS0(m,n)={1ifthe distance between ΘS0(m,n) and MF is equal toϵseed0otherwise,
F˜(Θ)=xsysp=x,y,zReal{(Bxsys)*[(Hpxsys)*°({Hpxsys[Bxsys(Mbm+sig{WSΘS,1})]}(Z˜Z)Z(1Z))]},

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