Abstract

Optical proximity correction (OPC) methods are resolution enhancement techniques (RET) used extensively in the semiconductor industry to improve the resolution and pattern fidelity of optical lithography. In pixel-based OPC (PBOPC), the mask is divided into small pixels, each of which is modified during the optimization process. Two critical issues in PBOPC are the required computational complexity of the optimization process, and the manufacturability of the optimized mask. Most current OPC optimization methods apply the steepest descent (SD) algorithm to improve image fidelity augmented by regularization penalties to reduce the complexity of the mask. Although simple to implement, the SD algorithm converges slowly. The existing regularization penalties, however, fall short in meeting the mask rule check (MRC) requirements often used in semiconductor manufacturing. This paper focuses on developing OPC optimization algorithms based on the conjugate gradient (CG) method which exhibits much faster convergence than the SD algorithm. The imaging formation process is represented by the Fourier series expansion model which approximates the partially coherent system as a sum of coherent systems. In order to obtain more desirable manufacturability properties of the mask pattern, a MRC penalty is proposed to enlarge the linear size of the sub-resolution assistant features (SRAFs), as well as the distances between the SRAFs and the main body of the mask. Finally, a projection method is developed to further reduce the complexity of the optimized mask pattern.

© 2011 Optical Society of America

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References

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2009 (2)

X. Ma and G. R. Arce, “Binary mask optimization for forward lithography based on boundary layer model in coherent systems,” J. Opt. Soc. Am. A 26(7), 1687–1695 (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]

2008 (2)

X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A 25(12), 2960–2970 (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] [PubMed]

2007 (3)

K. Kato, Y. Taniguchi, K. Nishizawa, and M. Endo, “Mask rule check using priority information of mask patterns,” Proc. SPIE 6730, 67304F (2007).
[CrossRef]

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

A. Poonawala, and P. Milanfar, “Fast and low-complexity mask design in optical microlithography–an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007).
[CrossRef] [PubMed]

2006 (2)

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

K. Kato, K. Nishizawa, and T. Inoue, “Advanced mask rule check (MRC) tool,” Proc. SPIE 6283, 62830O (2006).
[CrossRef]

2004 (3)

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

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 (1)

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

2000 (1)

S. Kalluri and G. R. Arce, “Fast algorithms for weighted myriad computation by fixed-point search,” IEEE Trans. Signal Process. 48(1), 159–171 (2000).
[CrossRef]

1996 (1)

B. Salik, J. Rosen, and A. Yariv, “Average coherent approximation for partially cohernet optical systems,” J. Opt. Soc. Am. A 13, 2086–2090) (1996).
[CrossRef]

1995 (1)

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

1994 (1)

K. Barner and G. R. Arce, “Permutation filters - a class of nonlinear filters based on set permutations,” IEEE Trans. Signal Process. 42, 782–798 (1994).
[CrossRef]

1992 (1)

Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Trans. Semicond. Manuf. 5(2), 138–152 (1992).
[CrossRef]

1987 (1)

M. P. McLoughlin and G. R. Arce, “Deterministic properties of the recursive separable median filter,” IEEE Trans. Acoust. Speech Signal Process. 35, 98–106 (1987).
[CrossRef]

1982 (1)

B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21(15), 2770–2777 (1982).
[CrossRef] [PubMed]

1980 (1)

K. Ritter, “On the rate of superlinear convergence of a class of variable metric methods,” NumerischeMathematik 35, 293–313 (1980).
[CrossRef]

1972 (2)

H. P. Crowder and P. Wolfe, “Linear convergence of the conjugate gradient method,” IBM J. Res. Develop. 16, 431–433 (1972).
[CrossRef]

A. Cohen, “Rate of convergence of several conjugate gredient algorithms,” SIAM J. Numer. Anal. 9, 248–259 (1972).
[CrossRef]

1964 (1)

R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

1952 (1)

M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).

Arce, G. R.

X. Ma and G. R. Arce, “Binary mask optimization for forward lithography based on boundary layer model in coherent systems,” J. Opt. Soc. Am. A 26(7), 1687–1695 (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]

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

X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A 25(12), 2960–2970 (2008).
[CrossRef]

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

S. Kalluri and G. R. Arce, “Fast algorithms for weighted myriad computation by fixed-point search,” IEEE Trans. Signal Process. 48(1), 159–171 (2000).
[CrossRef]

K. Barner and G. R. Arce, “Permutation filters - a class of nonlinear filters based on set permutations,” IEEE Trans. Signal Process. 42, 782–798 (1994).
[CrossRef]

M. P. McLoughlin and G. R. Arce, “Deterministic properties of the recursive separable median filter,” IEEE Trans. Acoust. Speech Signal Process. 35, 98–106 (1987).
[CrossRef]

Barner, K.

K. Barner and G. R. Arce, “Permutation filters - a class of nonlinear filters based on set permutations,” IEEE Trans. Signal Process. 42, 782–798 (1994).
[CrossRef]

Cohen, A.

A. Cohen, “Rate of convergence of several conjugate gredient algorithms,” SIAM J. Numer. Anal. 9, 248–259 (1972).
[CrossRef]

Crowder, H. P.

H. P. Crowder and P. Wolfe, “Linear convergence of the conjugate gradient method,” IBM J. Res. Develop. 16, 431–433 (1972).
[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. Develop. 45, 651–665 (2001).
[CrossRef]

Endo, M.

K. Kato, Y. Taniguchi, K. Nishizawa, and M. Endo, “Mask rule check using priority information of mask patterns,” Proc. SPIE 6730, 67304F (2007).
[CrossRef]

Erdmann, A.

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

Farkas, R.

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

Fletcher, R.

R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

Fühner, T.

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

Granik, Y.

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

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

Hestenes, M. R.

M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).

Inoue, T.

K. Kato, K. Nishizawa, and T. Inoue, “Advanced mask rule check (MRC) tool,” Proc. SPIE 6283, 62830O (2006).
[CrossRef]

Kalluri, S.

S. Kalluri and G. R. Arce, “Fast algorithms for weighted myriad computation by fixed-point search,” IEEE Trans. Signal Process. 48(1), 159–171 (2000).
[CrossRef]

Kato, K.

K. Kato, Y. Taniguchi, K. Nishizawa, and M. Endo, “Mask rule check using priority information of mask patterns,” Proc. SPIE 6730, 67304F (2007).
[CrossRef]

K. Kato, K. Nishizawa, and T. Inoue, “Advanced mask rule check (MRC) tool,” Proc. SPIE 6283, 62830O (2006).
[CrossRef]

Kokai, G.

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

Lavin, M.

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop. 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. Develop. 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(9), 1252–1257 (1995).
[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. Develop. 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(2), 138–152 (1992).
[CrossRef]

Ma, X.

X. Ma and G. R. Arce, “Binary mask optimization for forward lithography based on boundary layer model in coherent systems,” J. Opt. Soc. Am. A 26(7), 1687–1695 (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]

X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A 25(12), 2960–2970 (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] [PubMed]

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

Mansfield, S.

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

McLoughlin, M. P.

M. P. McLoughlin and G. R. Arce, “Deterministic properties of the recursive separable median filter,” IEEE Trans. Acoust. Speech Signal Process. 35, 98–106 (1987).
[CrossRef]

Milanfar, P.

A. Poonawala, and P. Milanfar, “Fast and low-complexity mask design in optical microlithography–an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007).
[CrossRef] [PubMed]

Nishizawa, K.

K. Kato, Y. Taniguchi, K. Nishizawa, and M. Endo, “Mask rule check using priority information of mask patterns,” Proc. SPIE 6730, 67304F (2007).
[CrossRef]

K. Kato, K. Nishizawa, and T. Inoue, “Advanced mask rule check (MRC) tool,” Proc. SPIE 6283, 62830O (2006).
[CrossRef]

Poonawala, A.

A. Poonawala, and P. Milanfar, “Fast and low-complexity mask design in optical microlithography–an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007).
[CrossRef] [PubMed]

Rabbani, M.

B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21(15), 2770–2777 (1982).
[CrossRef] [PubMed]

Reeves, C. M.

R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

Ritter, K.

K. Ritter, “On the rate of superlinear convergence of a class of variable metric methods,” NumerischeMathematik 35, 293–313 (1980).
[CrossRef]

Rosen, J.

B. Salik, J. Rosen, and A. Yariv, “Average coherent approximation for partially cohernet optical systems,” J. Opt. Soc. Am. A 13, 2086–2090) (1996).
[CrossRef]

Saleh, B.

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

Saleh, B. E. A.

B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21(15), 2770–2777 (1982).
[CrossRef] [PubMed]

Salik, B.

B. Salik, J. Rosen, and A. Yariv, “Average coherent approximation for partially cohernet optical systems,” J. Opt. Soc. Am. A 13, 2086–2090) (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]

Sherif, S.

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

Stiefel, E.

M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).

Taniguchi, Y.

K. Kato, Y. Taniguchi, K. Nishizawa, and M. Endo, “Mask rule check using priority information of mask patterns,” Proc. SPIE 6730, 67304F (2007).
[CrossRef]

Tollkühn, B.

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

Wolfe, P.

H. P. Crowder and P. Wolfe, “Linear convergence of the conjugate gradient method,” IBM J. Res. Develop. 16, 431–433 (1972).
[CrossRef]

Wong, A.

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

Yariv, A.

B. Salik, J. Rosen, and A. Yariv, “Average coherent approximation for partially cohernet optical systems,” J. Opt. Soc. Am. A 13, 2086–2090) (1996).
[CrossRef]

Zakhor, A.

Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Trans. Semicond. Manuf. 5(2), 138–152 (1992).
[CrossRef]

Appl. Opt. (1)

B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21(15), 2770–2777 (1982).
[CrossRef] [PubMed]

Comput. J. (1)

R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

IBM J. Res. Develop. (2)

H. P. Crowder and P. Wolfe, “Linear convergence of the conjugate gradient method,” IBM J. Res. Develop. 16, 431–433 (1972).
[CrossRef]

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

IEEE Trans. Acoust. Speech Signal Process. (1)

M. P. McLoughlin and G. R. Arce, “Deterministic properties of the recursive separable median filter,” IEEE Trans. Acoust. Speech Signal Process. 35, 98–106 (1987).
[CrossRef]

IEEE Trans. Image Process. (2)

A. Poonawala, and P. Milanfar, “Fast and low-complexity mask design in optical microlithography–an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007).
[CrossRef] [PubMed]

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

IEEE Trans. Semicond. Manuf. (1)

Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Trans. Semicond. Manuf. 5(2), 138–152 (1992).
[CrossRef]

IEEE Trans. Signal Process. (2)

S. Kalluri and G. R. Arce, “Fast algorithms for weighted myriad computation by fixed-point search,” IEEE Trans. Signal Process. 48(1), 159–171 (2000).
[CrossRef]

K. Barner and G. R. Arce, “Permutation filters - a class of nonlinear filters based on set permutations,” IEEE Trans. Signal Process. 42, 782–798 (1994).
[CrossRef]

J. Microlithogr., Microfabr, Microsyst. (1)

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

J. Opt. Soc. Am. A (3)

B. Salik, J. Rosen, and A. Yariv, “Average coherent approximation for partially cohernet optical systems,” J. Opt. Soc. Am. A 13, 2086–2090) (1996).
[CrossRef]

X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A 25(12), 2960–2970 (2008).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Optical lithography system with partially coherent illuminations.

Fig. 2
Fig. 2

Approximated forward process model.

Fig. 3
Fig. 3

Performance comparison between the SD method and the CG method; (a) desired pattern ; (b) optimized OPC pattern from the SD method; (c) optimized OPC pattern from the CG method; (d) the output pattern when (a) is used as input; (e) the output pattern when (b) is used as input; (f) the output pattern when (c) is used as input.

Fig. 4
Fig. 4

Convergence comparison of the SD method (black solid line), the CG method (blue dashed line), the CG method with MRC penalty (green dotted line), and the projection method (red dot-dashed line).

Fig. 5
Fig. 5

Influence of the wavelet penalty weight on the optimized OPCs and their output patterns; (a) optimized OPC pattern with γW = 0.025; (b) optimized OPC pattern with γW = 0.035; (c) optimized OPC pattern with γW = 0.045; (d) the output pattern when (a) is used as input; (e) the output pattern when (b) is used as input; (f) the output pattern when (c) is used as input.

Fig. 6
Fig. 6

Performance comparison between the the CG method with MRC penalty and the projection method; (a) optimized OPC pattern from the CG method with MRC penalty; (b) the output pattern when (a) is used as input; (c) optimized OPC pattern from the projection method; (d) the output pattern when (c) is used as input.

Fig. 7
Fig. 7

Influence of the MRC filter dimension on the optimized OPCs and their output patterns; (a) optimized OPC pattern with g = 13×3; (b) optimized OPC pattern with g = 15×5; (c) optimized OPC pattern with g = 17×7; (d) the output pattern when (a) is used as input; (e) the output pattern when (b) is used as input; (f) the output pattern when (c) is used as input.

Fig. 8
Fig. 8

Scheme of the projection method.

Tables (1)

Tables Icon

Table 1 Comparison of performance and convergence

Equations (41)

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I ( r ) = M ( r 1 ) M ( r 2 ) γ ( r 1 r 2 ) h * ( r r 1 ) h ( r r 2 ) d r 1 d r 2 ,
γ ( r ) = m Γ m exp ( j ω 0 m r ) ,
Γ m = 1 D 2 A γ γ ( r ) exp ( j ω 0 m r ) d r ,
I ( r ) = m Γ m | M ( r ) h m ( r ) | 2 ,
h m ( r ) = h ( r ) exp ( j ω 0 m r ) .
γ ( r ) = J 1 ( 2 π r / 2 D c u ) 2 π r / 2 D c u D c u 2 D c l 2 J 1 ( 2 π r / 2 D c l ) 2 π r / 2 D c l ,
Γ m = { 4 D c u 2 D c l 2 π D 2 ( D c l 2 D c u 2 ) for D / 2 D c l | m | D / 2 D c u 0 elsewhere ,
h ( r ) = J 1 ( 2 π r N A / λ ) 2 π r N A / λ .
F = d ( Z ( x , y ) , Z ˜ ( x , y ) ) = d ( T { M ( x , y ) } , Z ˜ ( x , y ) )
M ^ ( x , y ) = arg min M ( x , y ) N × N d ( T { M ( x , y ) } , Z ˜ ( x , y ) ) .
sig ( x ) = 1 1 + exp [ a ( x t r ) ] ,
Z = sig { m Γ m | H m { m _ } | 2 } .
M ^ = arg min d M ( sig { m Γ m | H m { m _ } | 2 } , Z ˜ ) .
z _ i = 1 1 + exp [ a m Γ m | j = 1 N 2 h i j m m _ j | 2 + a t r ] , i = 1 , N 2 ,
m ^ _ = arg min m ^ _ { F ( m _ ) } ,
F ( m _ ) = z ˜ _ z _ 2 2 = i = 1 N 2 ( z ˜ _ i z _ i ) 2 ,
m _ j = 1 + cos ( θ _ j ) 2 , j = 1 , N 2 ,
θ ^ _ = arg min θ _ { F ( θ _ ) } = arg min θ _ { i = 1 N 2 ( z ˜ i 1 1 + exp [ a m Γ m | j = 1 N 2 h i j m 1 + cos θ _ j 2 | 2 + a t r ] ) 2 } .
F ( Θ ) = a × sin Θ { m Γ m ( h m ) [ ( Z ˜ Z ) Z ( 1 N × N Z ) ( ( h m ) * M ) ] } + a × sin Θ { m Γ m ( h m ) * [ ( Z ˜ Z ) Z ( 1 N × N Z ) ( h m M ) ] } ,
Θ 0 ( i , j ) = { 4 π 5 for Z ˜ ( i , j ) = 0 π 5 for Z ˜ ( i , j ) = 1 , i , j = 1 , 2 , , N ,
P 0 = F ( Θ 0 ) ,
s k = arg min s F ( Θ k + s P k ) .
Θ k + 1 = Θ k + s k P k .
β k = F ( Θ k + 1 ) 2 2 F ( Θ k ) 2 2 .
p k + 1 = F ( Θ k + 1 ) + β k p k .
m ^ _ b i = U ( m ^ _ i t m ) , i = 1 , , N 2 ,
E = i = 1 N 2 z ^ _ i z _ b i 2 2 = i = 1 N 2 z ^ _ i U ( m Γ m | H m m _ b i | 2 t r ) 2 2 .
m ^ _ = arg min m ^ _ { F ( m _ ) + γ R ( m _ ) } ,
J ( m _ ) = F ( m _ ) + γ Q R Q ( m _ ) + γ W R W ( m _ ) .
R M ( M ) = 1 N × 1 T [ ( M t m N × N ) ( g M ) ] 1 N × 1 ,
R M ( m _ ) = k = 1 N 2 ( t m m _ k ) ( j = 1 N 2 g k , j m _ j ) .
R M ( Θ ) = ( 4.5 2 g M ) ( 0.5 sin Θ ) ,
J ( m _ ) = F ( m _ ) + γ Q R Q ( m _ ) + γ W R W ( m _ ) + γ M R M ( m _ ) ,
R W ( m _ ) = h 11 2 + h 12 2 + h ( N 2 ) ( N 2 ) 2 + v 11 2 + v 12 2 + v ( N 2 ) ( N 2 ) 2 + d 11 2 + d 12 2 + d ( N 2 ) ( N 2 ) 2 ,
h i j = m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) + m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
v i j = m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) + m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
d i j = m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) + m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
R W θ _ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) = 1 2 sin θ _ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) × ( 3 m _ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) m _ ( 2 ( i 1 ) + p 1 ) ( 2 ( j 1 ) + q ) m _ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q 1 ) m _ ( 2 ( i 1 ) + p 1 ) ( 2 ( j 1 ) + q 1 ) ) ,
R M ( m _ ) m _ k = p = 1 N 2 [ ( t m m _ p ) ( j = 1 N 2 g p j m _ j ) ] / m _ k = [ ( t m m _ k ) ( j = 1 N 2 g k j m _ j ) ] / m _ k + p = 1 , p k N 2 [ ( t m m _ p ) ( j = 1 N 2 g p j m _ j ) ] / m _ k = [ ( t m m _ k ) ( j = 1 , j k N 2 g k j m _ j ) ] / m _ k + [ ( t m m _ k ) g k k m _ k ] / m _ k + p = 1 , p k N 2 [ ( t m m _ p ) g p k m _ k ] / m _ k .
R M ( m _ ) m _ k = ( j = 1 , j k N 2 g k j m _ j ) + ( t m m _ k m _ k 2 ) / m _ k + p = 1 , p k N 2 g p k ( t m m _ p ) = ( j = 1 , j k N 2 g k j m _ j ) + t m 2 m _ k ( p = 1 , p k N 2 g k p m _ p ) + t m p = 1 , p k N 2 h p k = 2 ( j = 1 N 2 g k j m _ j ) + 2 g k k m _ k + t m 2 m _ k + t m p = 1 N 2 h p k t m h k k = 2 ( j = 1 N 2 g k j m _ j ) + t m p = 1 N 2 h p k .
R M ( Θ ) = R M ( M ) M ( Θ ) = ( 4.5 2 g M ) ( 0.5 sin Θ ) .

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