Abstract

Recently, a set of gradient-based optical proximity correction (OPC) and phase-shifting mask (PSM) optimization methods has been developed to solve for the inverse lithography problem under scalar imaging models, which are only accurate for numerical apertures (NAs) of less than approximately 0.4. However, as lithography technology enters the 45 nm realm, immersion lithography systems with hyper-NA (NA>1) are now extensively used in the semiconductor industry. For the hyper-NA lithography systems, the vector nature of the electromagnetic field must be taken into account, leading to the vector imaging models. Thus, the OPC and PSM optimization approaches developed under the scalar imaging models are inadequate to enhance the resolution in immersion lithography systems. This paper focuses on developing pixelated gradient-based OPC and PSM optimization algorithms under a vector imaging model. We first formulate the mask optimization framework, in which the imaging process of the optical lithography system is represented by an integrative and analytic vector imaging model. A gradient-based algorithm is then used to optimize the mask iteratively. Subsequently, a generalized wavelet penalty is proposed to keep a balance between the mask complexity and convergence errors. Finally, a set of methods is exploited to speed up the proposed algorithms.

© 2012 Optical Society of America

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    [CrossRef]
  33. Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express 19, 5511–5521(2011).
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  34. N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19, 19384–19398 (2011).
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    [CrossRef]
  37. G. M. Gallatin, “High-numerical-aperture scalar imaging,” Appl. Opt. 40, 4958–4964 (2001).
    [CrossRef]
  38. M. Yeung, “Modeling high numerical aperture optical lithography,” Proc. SPIE 922, 149–167 (1988).
  39. D. G. Flagello, “High numerical aperture imaging in homogeneous thin films,” Ph.D. dissertation (University of Arizona, 1993).
  40. T. V. Pistor, “Electromagnetic simulation and modeling with applications in lithography,” Ph.D. dissertation (University of California, Berkeley, 2001).
  41. K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
    [CrossRef]
  42. 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]
  43. T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
    [CrossRef]
  44. 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]
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    [CrossRef]
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2011

2010

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]

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]

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]

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

2009

2008

2007

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, Y. Borodovsky, and P. Milanfar, “Double exposure inverse lithography,” Microlithogr. World 16, 7–9 (2007).

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

A. Poonawala and P. Milanfar, “A pixel-based regularization approach to inverse lithography,” Microelectron. Eng. 84, 2837–2852 (2007).
[CrossRef]

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

2005

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

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]

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

2003

K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[CrossRef]

2002

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, 13–30 (2002).
[CrossRef]

2001

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, “Tcad development for lithography resolution enhancement,” IBM J. Res. Devel. 45, 651–665 (2001).
[CrossRef]

G. M. Gallatin, “High-numerical-aperture scalar imaging,” Appl. Opt. 40, 4958–4964 (2001).
[CrossRef]

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 integer programming,” IEEE Trans. Image Process. 4, 1252–1257 (1995).
[CrossRef]

1994

1992

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

1988

M. Yeung, “Modeling high numerical aperture optical lithography,” Proc. SPIE 922, 149–167 (1988).

1982

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
[CrossRef]

Adam, K.

K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[CrossRef]

Arce, G. R.

Borodovsky, Y.

A. Poonawala, Y. Borodovsky, and P. Milanfar, “Double exposure inverse lithography,” Microlithogr. World 16, 7–9 (2007).

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, 13–30 (2002).
[CrossRef]

Campbell, S. A.

S. A. Campbell, The Science and Engineering of Microelectronic Fabrication, 2nd ed. (Oxford University, 2003).

Chan, S. H.

Cobb, N.

K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[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]

Dunham, T.

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, “Tcad development for lithography resolution enhancement,” IBM J. Res. Devel. 45, 651–665 (2001).
[CrossRef]

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]

Flagello, D. G.

D. G. Flagello, “High numerical aperture imaging in homogeneous thin films,” Ph.D. dissertation (University of Arizona, 1993).

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, 13–30 (2002).
[CrossRef]

Gallatin, G. M.

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 Science, 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]

K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[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]

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]

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, 13–30 (2002).
[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]

Jia, N.

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

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

Kailath, T.

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]

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]

Lai, 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, 13–30 (2002).
[CrossRef]

Lam, E. Y.

Lavin, M.

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, “Tcad development for lithography resolution enhancement,” IBM J. Res. Devel. 45, 651–665 (2001).
[CrossRef]

Leipold, W.

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, “Tcad development for lithography resolution enhancement,” IBM J. Res. Devel. 45, 651–665 (2001).
[CrossRef]

Leone, R.

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

Levenson, M. D.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
[CrossRef]

Li, Y.

X. Ma, G. R. Arce, and Y. Li, “Optimal 3D phase-shifting masks in partially coherent illumination,” Appl. Opt. 50, 5567–5576 (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]

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

Liebmann, L.

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, “Tcad development for lithography resolution enhancement,” IBM J. Res. Devel. 45, 651–665 (2001).
[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]

Ma, X.

Mansfield, S.

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, “Tcad development for lithography resolution enhancement,” IBM J. Res. Devel. 45, 651–665 (2001).
[CrossRef]

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, “A pixel-based regularization approach to inverse lithography,” Microelectron. Eng. 84, 2837–2852 (2007).
[CrossRef]

A. Poonawala, Y. Borodovsky, and P. Milanfar, “Double exposure inverse lithography,” Microlithogr. World 16, 7–9 (2007).

A. Poonawala and P. Milanfar, “Double exposure mask synthesis using inverse lithography,” J. Micro/Nanolith. 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).

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

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, 13–30 (2002).
[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]

Pati, Y. C.

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]

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]

T. V. Pistor, “Electromagnetic simulation and modeling with applications in lithography,” Ph.D. dissertation (University of California, Berkeley, 2001).

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, Y. Borodovsky, and P. Milanfar, “Double exposure inverse lithography,” Microlithogr. World 16, 7–9 (2007).

A. Poonawala and P. Milanfar, “A pixel-based regularization approach to inverse lithography,” Microelectron. Eng. 84, 2837–2852 (2007).
[CrossRef]

A. Poonawala and P. Milanfar, “Double exposure mask synthesis using inverse lithography,” J. Micro/Nanolith. 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).

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

Rosenbluth, A. E.

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, 13–30 (2002).
[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 integer programming,” IEEE Trans. Image Process. 4, 1252–1257 (1995).
[CrossRef]

Schellenberg, F.

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

F. Schellenberg, Selected Papers on Resolution Enhancement Techniques in Optical Lithography (SPIE Press, 2004).

Shen, Y.

Sherif, S.

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

Simpson, R. A.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
[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, 13–30 (2002).
[CrossRef]

Socha, R. J.

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Figures (7)

Fig. 1.
Fig. 1.

Comparison of intensity distributions calculated by the scalar imaging model, the vector imaging model, and Prolith software. (a) Comparison of intensity distributions based on a binary mask and (b) comparison of intensity distributions based on an alternating PSM.

Fig. 2.
Fig. 2.

Imaging formation based on the vector imaging model.

Fig. 3.
Fig. 3.

Forward imaging process of the optical lithography system.

Fig. 4.
Fig. 4.

Performance comparison between the OPC optimization algorithms based on the scalar and vector imaging models. Top row (from left to right) shows (a) target pattern, (b) optimized OPC based on the scalar imaging model, and (c) optimized OPC based on the vector imaging model. Bottom row shows the printed images corresponding to the input masks in the top row. Gray and white represent 0 and 1, respectively.

Fig. 5.
Fig. 5.

Performance comparison between the PSM optimization algorithms based on the scalar and vector imaging models. Top row (from left to right) shows (a) initial PSM, (b) optimized PSM based on the scalar imaging model, and (c) optimized PSM based on the vector imaging model. Bottom row shows the printed images corresponding to the input masks in the top row. Black, gray, and white represent 1 , 0 and 1, respectively.

Fig. 6.
Fig. 6.

Simulations based on another target pattern. Top row (from left to right) shows (a) target pattern, (b) optimized OPC based on the vector imaging model, (c) initial PSM, and (d) optimized PSM based on the vector imaging model. Bottom row shows the printed images corresponding to the input masks in the top row. Black, gray, and white represent 1 , 0, and 1, respectively.

Fig. 7.
Fig. 7.

Performance comparison between the PSM optimization algorithms with WP and GWP. Top row (from left to right) shows (a) optimized PSM of the first pattern using WP, (b) optimized PSM of the first pattern using GWP, (c) optimized PSM of the second pattern using WP, and (d) optimized PSM of the second pattern using GWP. Bottom row shows the printed images corresponding to the input masks in the top row. Black, gray, and white represent 1 , 0, and 1, respectively.

Tables (1)

Tables Icon

Table 1. Performance Comparison of Different Mask Optimization Approaches

Equations (62)

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[ E x E y E z ] = T [ E E ] ,
T = [ β ρ α γ ρ α ρ β γ ρ 0 ρ ] ,
E⃗ i = T E⃗ i .
E = E i B M ,
B ( m , n ) = exp ( j 2 π β s m × p i x e l λ ) exp ( j 2 π α s n × p i x e l λ ) , m , n = 1 , 2 , , N ,
E wafer ( α s , β s ) = 2 π n w R F 1 { V } ( B M ) ,
V ( m , n ) = [ β 2 + α 2 γ 1 γ 2 α β 1 + γ α α β 1 + γ β 2 + α 2 γ 1 γ 2 β α β γ ] , m , n = 1 , 2 , , N ,
U = { 1 f 2 + g 2 NA m λ 0 elsewhere .
E p wafer ( α s , β s ) = H p ( B M ) , p = x , y , z ,
H p = 2 π n w R F 1 { V p } , p = x , y , z ,
I ( α s , β s ) = p = x , y , z H p ( B M ) 2 2 ,
I = 1 N s α s β s p = x , y , z H p α s β s ( B α s β s M ) 2 2 ,
F = d ( Z , Z ˜ ) = d ( T { M } , Z ˜ )
M ^ = arg min M N × N d ( T { M } , Z ˜ ) .
sig ( x ) = 1 1 + exp [ a ( x t r ) ] ,
Z = sig { 1 N s α s β s p = x , y , z H p α s β s ( B α s β s M ) 2 2 } .
z⃗ m = 1 1 + exp [ a N s α s β s p = x , y , z ( n = 1 N 2 H p , m n α s β s b⃗ n α s β s m⃗ n ) 2 + a t r ] , m = 1 , , N 2 ,
m ^ = arg min m ^ { F ( m ) } ,
F ( m⃗ ) = z ˜ z⃗ 2 2 = m = 1 N 2 ( z ˜ m z⃗ m ) 2 ,
m ̲ n = f ( ω⃗ n ) , n = 1 , , N 2 ,
ω ^ = arg min ω⃗ { F ( ω⃗ ) } = arg min ω⃗ { m = 1 N 2 ( z⃗ m 1 1 + exp [ a N s α s β s p = x , y , z ( n = 1 N 2 H p , m n α s β s b⃗ n α s β s f ( ω⃗ n ) ) 2 + a t r ] ) 2 } .
m⃗ n = f ( ω⃗ n ) = 1 + cos ω⃗ n 2 , n = 1 , , N 2 .
m⃗ n = f ( ω⃗ n ) = cos ω⃗ n , n = 1 , , N 2 .
F ( Ω ) = 4 a N s f ( Ω ) α s β s p = x , y , z Real [ ( B α s β s ) * ( ( H p α s β s ) * { [ H p α s β s ( B α s β s M ) ] ( Z ˜ Z ) Z ( 1 Z ) } ) ] ,
f ( Ω ) = 1 2 sin Ω .
f ( Ω ) = sin Ω .
Ω m + 1 = Ω m s Ω F ( Ω ) ,
m ^ d , n = Γ ( m ^ n t m ) , n = 1 , 2 , , N 2 ,
m ^ d , n = Γ ( m ^ n t m ) Γ ( m ^ n t m ) , n = 1 , 2 , , N 2 ,
Er = Z ˜ Z b 2 2 = Z ˜ Γ ( 1 N s α s β s p = x , y , z H p α s β s ( B α s β s M b ) 2 2 t r ) 2 2 .
F ( Ω ) = 4 a N s f ( Ω ) α s β s p = x , y , z Real [ ( B α s β s ) * ( ( H p α s β s ) * { E p wafer ( α s , β s ) ( Z ˜ Z ) Z ( 1 Z ) } ) ] .
F ( Ω ) = 4 a N s f ( Ω ) α s β s p = x , y , z Real [ ( B α s β s ) * F 1 { F [ ( H p α s β s ) * ] F [ E p wafer ( α s , β s ) ( Z ˜ Z ) Z ( 1 Z ) ] } ] .
F ( Ω ) = 4 a N s f ( Ω ) α s β s p = x , y , z Real [ ( B α s β s ) * F 1 { 2 π n w R V p α s β s * C F [ E p wafer ( α s , β s ) ( Z ˜ Z ) Z ( 1 Z ) ] } ] ,
I = 1 N s α s β s p = x , y , z F 1 { 2 π n w R V p α s β s F ( B α s β s M ) } 2 2 .
m ^ = arg min m ^ { F ( m⃗ ) + γ W R W ( m⃗ ) } ,
M 0 = Γ ( M t m ) sig ( M t m ) = 1 1 + exp ( a M + a t m ) ,
M 180 = Γ ( M t m ) sig ( M t m ) = 1 1 + exp ( a M + a t m ) .
R GW = 1 2 ( R GW 0 + R GW 180 ) = 1 2 ( i = 1 N / 2 j = 1 N / 2 h 0 i j 2 + v 0 i j 2 + d 0 i j 2 + i = 1 N / 2 j = 1 N / 2 h 180 i j 2 + v 180 i j 2 + d 180 i j 2 ) ,
h x i j = M x ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) M x ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) + M x ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) M x ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
v x i j = M x ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) + M x ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) M x ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) M x ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
d x i j = M x ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) M x ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) M x ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) + M x ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
J ( m⃗ ) = F ( m⃗ ) + γ GW R GW ( m⃗ ) .
R W x ω⃗ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) = f ( ω⃗ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) ) × [ 3 M x ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) M x ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) M x ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) M x ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) ] × a G ( M x ) ,
G ( M x ) = { { 1 1 + exp [ a M 0 ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) + a t m ] } { 1 1 exp [ a M 0 ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) + a t m ] } for x = 0 { 1 1 + exp [ a M 180 ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) + a t m ] } { 1 1 exp [ a M 180 ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) + a t m ] } for x = 180 .
# Rectangle = Int { 1 12 v⃗ [ G M ( G M 2 ) ( G M 4 ) ( 2 G M 3 ) ] v⃗ } ,
E l ent ( α , β ) = γ j λ e j k r r F { E } ,
E b ent ( α , β ) = E l ent ( α , β ) = γ j λ e j k r r F { E } .
E l ext ( α , β ) = c U E b ent ( α , β ) ,
c = r r γ γ n w R .
U = { 1 f 2 + g 2 NA m λ 0 elsewhere .
E l ext ( α , β ) = 1 λ r γ γ n w R U F { E } .
E b ext ( α , β ) = V E l ext ( α , β ) ,
V ( m , n ) = [ β 2 + α 2 γ 1 γ 2 α β 1 + γ α α β 1 + γ β 2 + α 2 γ 1 γ 2 β α β γ ] , m , n = 1 , 2 , , N .
E wafer = 2 π λ r j n w 2 e j k r F 1 { 1 γ E b ext } ,
E wafer ( α s , β s ) = 2 π n w R F 1 { γ γ V U F { E i B M } } .
E wafer ( α s , β s ) = 2 π n w R F 1 { V } ( B M ) ,
F ω⃗ q = 2 f ( ω⃗ q ) × m = 1 N 2 ( z ˜ m 1 1 + exp [ a N s α s β s p = x , y , z ( n = 1 N 2 H p , m n α s β s b⃗ n α s β s m⃗ n ) 2 + a t r ] ) × 1 1 + exp [ a N s α s β s p = x , y , z ( n = 1 N 2 H p , m n α s β s b⃗ n α s β s m⃗ n ) 2 + a t r ] × exp [ a N s α s β s p = x , y , z ( n = 1 N 2 H p , m n α s β s b⃗ n α s β s m⃗ n ) 2 + a t r ] 1 + exp [ a N s α s β s p = x , y , z ( n = 1 N 2 H p , m n α s β s b⃗ n α s β s m⃗ n ) 2 + a t r ] × ( a N s ) × α s β s p = x , y , z [ ( m = 1 N 2 H p , m n α s β s b⃗ n α s β s m⃗ n ) × H p , m q α s β s b⃗ q α s β s + ( m = 1 N 2 H p , m n α s β s b⃗ n α s β s m⃗ n ) * × H p , m q α s β s b⃗ q α s β s ] .
F ( Ω ) = 4 a N s f ( Ω ) α s β s p = x , y , z Real [ ( B α s β s ) * ( ( H p α s β s ) * ° { [ H p α s β s ( B α s β s M ) ] ( Z ˜ Z ) Z ( 1 Z ) } ) ] .
F [ ( H p α s β s ) * ° ] ( m , n ) = q = 0 N 1 l = 0 N 1 H p α s β s * ( N q 1 , N l 1 ) exp ( j 2 π m q N ) exp ( j 2 π n l N ) .
F [ ( H p α s β s ) * ° ] ( m , n ) = x = 0 N 1 y = 0 N 1 H p α s β s * ( x , y ) exp ( j 2 π m ( x 1 ) N ) exp ( j 2 π n ( y 1 ) N ) = exp ( j 2 π m N ) exp ( j 2 π n N ) x = 0 N 1 y = 0 N 1 H p α s β s ( x , y ) exp ( j 2 π m x ) N ) exp ( j 2 π n y N ) = exp [ j 2 π ( m N + n N ) ] F { [ ( H p α s β s ) ] * ( m , n ) } .
F [ ( H p α s β s ) * ° ] = 2 π n w R V p α s β s * C ,
F ( Ω ) = 4 a N s f ( Ω ) α s β s p = x , y , z Real [ ( B α s β s ) * F 1 ( 2 π n w R V p α s β s * C F { E p wafer ( α s , β s ) ( Z ˜ Z ) Z ( 1 Z ) } ) ] .

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