Abstract

Level-set based inverse lithography technology (ILT) treats photomask design for microlithography as an inverse mathematical problem, interpreted with a time-dependent model, and then solved as a partial differential equation with finite difference schemes. This paper focuses on developing level-set based ILT for partially coherent systems, and upon that an expectation-orient optimization framework weighting the cost function by random process condition variables. These include defocus and aberration to enhance robustness of layout patterns against process variations. Results demonstrating the benefits of defocus-aberration-aware level-set based ILT are presented.

© 2011 Optical Society of America

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  4. S. Shioiri, and H. Tanabe, “Fast optical proximity correction: analytical method,” Proc. SPIE 2440, 261–269 (1995).
  5. L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): a natural solution for model-based SRAF at 45nm and 32nm,” Proc. SPIE 6607, 660739 (2007).
  6. Y. Liu, and A. Zakhor, “Optimal binary image design for optical lithography,” Proc. SPIE 1264, 401–412 (1990).
  7. Y. Liu, and A. Zakhor, “Binary and phase-shifting image design for optical lithography,” Proc. SPIE 1463, 382–399 (1991).
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  16. A. Poonawala, Y. Borodovsky, and P. Milanfar, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
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  25. L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE 6924, 69240T (2008).
  26. Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
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2010 (3)

E. Y. Lam, and A. K. Wong, ““Nebulous hotspot and algorithm variability in computation lithography,” J. Micro/ Nanolithogr,” MEMS MOEMS 9(3), 033002 (2010).

N. Jia, and E. Y. Lam, “Machine learning for inverse lithography: Using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 045601 (2010).

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

2009 (4)

2008 (4)

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

S. H. Chan, A. K. Wong, and E. Y. Lam, “Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography,” Opt. Express 16(19), 14746–14760 (2008).
[PubMed]

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

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE 6924, 69240T (2008).

2007 (3)

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

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

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

2005 (2)

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

P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the Extended Nijboer-Zernike theory,” Proc. SPIE 5754, 263 (2005).

2004 (1)

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

2001 (2)

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

S. Osher, and F. Santosa, “Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum,” J. Comput. Phys. 171(1), 272–288 (2001).

2000 (1)

A. Marquina, and S. Osher, “Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal,” SIAM J. Sci. Comput. 22, 387–405 (2000).

1997 (1)

J. A. Sethian, and D. Adalsteinsson, “An overview of level set methods for etching, deposition, and lithography development,” IEEE Trans. Semicond. Manuf. 10, 167–184 (1997).

1996 (1)

F. Santosa, ““A level-set approach for inverse problems involving obstacles,” ESAIM Contr¨ole Optim,” Calc. Var. 1, 17–33 (1996).

1995 (2)

S. Shioiri, and H. Tanabe, “Fast optical proximity correction: analytical method,” Proc. SPIE 2440, 261–269 (1995).

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

1994 (2)

Y. C. Pati, and T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A 11(9), 2438–2452 (1994).

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

1991 (1)

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

1990 (1)

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

1982 (1)

1976 (1)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66(3), 207–211 (1976).

1953 (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. 217A(1130), 408–432 (1953).

Abrams, D.

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

Adalsteinsson, D.

J. A. Sethian, and D. Adalsteinsson, “An overview of level set methods for etching, deposition, and lithography development,” IEEE Trans. Semicond. Manuf. 10, 167–184 (1997).

Arce, G. R.

Baik, K.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE 6924, 69240T (2008).

Bollepalli, S.

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

Borodovsky, Y.

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

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

Braat, J.

P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the Extended Nijboer-Zernike theory,” Proc. SPIE 5754, 263 (2005).

Cecil, T.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE 6924, 69240T (2008).

Chan, S. H.

Chen, D.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE 6924, 69240T (2008).

Cui, Y.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE 6924, 69240T (2008).

Dai, G.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE 6924, 69240T (2008).

Dam, T.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE 6924, 69240T (2008).

De Leone, R.

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

Dirksen, P.

P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the Extended Nijboer-Zernike theory,” Proc. SPIE 5754, 263 (2005).

Fedkiw, R. P.

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

Garofalo, J. G.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Henderson, R. C.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. 217A(1130), 408–432 (1953).

Hu, B.

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

Hu, P.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE 6924, 69240T (2008).

Janssen, A.

P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the Extended Nijboer-Zernike theory,” Proc. SPIE 5754, 263 (2005).

Jia, N.

N. Jia, and E. Y. Lam, “Machine learning for inverse lithography: Using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 045601 (2010).

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

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

Kailath, T.

Kostelak, R. L.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Lam, E. Y.

N. Jia, and E. Y. Lam, “Machine learning for inverse lithography: Using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 045601 (2010).

E. Y. Lam, and A. K. Wong, ““Nebulous hotspot and algorithm variability in computation lithography,” J. Micro/ Nanolithogr,” MEMS MOEMS 9(3), 033002 (2010).

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

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

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

Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).

S. H. Chan, A. K. Wong, and E. Y. Lam, “Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography,” Opt. Express 16(19), 14746–14760 (2008).
[PubMed]

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

Leeuwestein, A.

P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the Extended Nijboer-Zernike theory,” Proc. SPIE 5754, 263 (2005).

Liu, Y.

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

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

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

Low, K. K.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Ma, X.

Marquina, A.

A. Marquina, and S. Osher, “Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal,” SIAM J. Sci. Comput. 22, 387–405 (2000).

Milanfar, P.

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

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

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

Noll, R. J.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66(3), 207–211 (1976).

Osher, S.

S. Osher, and F. Santosa, “Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum,” J. Comput. Phys. 171(1), 272–288 (2001).

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

A. Marquina, and S. Osher, “Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal,” SIAM J. Sci. Comput. 22, 387–405 (2000).

Otto, O. W.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Pang, L.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE 6924, 69240T (2008).

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

Pati, Y. C.

Peng, D.

L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE 6924, 69240T (2008).

Pierrat, C.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

Poonawala, A.

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

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

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

Rabbani, M.

Saleh, B.

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

Saleh, B. E. A.

Santosa, F.

S. Osher, and F. Santosa, “Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum,” J. Comput. Phys. 171(1), 272–288 (2001).

F. Santosa, ““A level-set approach for inverse problems involving obstacles,” ESAIM Contr¨ole Optim,” Calc. Var. 1, 17–33 (1996).

Schellenberg, F.

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

Sethian, J. A.

J. A. Sethian, and D. Adalsteinsson, “An overview of level set methods for etching, deposition, and lithography development,” IEEE Trans. Semicond. Manuf. 10, 167–184 (1997).

Shen, Y.

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

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E. Y. Lam, and A. K. Wong, ““Nebulous hotspot and algorithm variability in computation lithography,” J. Micro/ Nanolithogr,” MEMS MOEMS 9(3), 033002 (2010).

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Yuan, C.-M.

O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).

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L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE 6924, 69240T (2008).

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

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Figures (4)

Fig. 1
Fig. 1

Common illumination sources: (a) conventional, (b) annular, and (c) dipole.

Fig. 2
Fig. 2

(a) Target pattern of size 101 × 101. (b) Output pattern under circular source, resulting in a pattern error of 45 pixels. (c) Output pattern under annular source, resulting in a pattern error of 116 pixels. (d) Output pattern under dipole source, resulting in a pattern error of 144 pixels.

Fig. 3
Fig. 3

Simulation of lithographic imaging with different mask patterns computed using level-set based ILT. The first column denotes the input U(x), the second column Iaerial(x), and the third column I(x). Rows (a), (b) and (c) use the derived pattern under circular illumination, annular illumination, and dipole illumination as input, resulting in pattern errors of 5, 24, and 45 pixels respectively.

Fig. 4
Fig. 4

Performances of the proposed level-set based statistical method with aberration variations. (a) focus-aware input mask pattern computed using the statistical method. (b) coma-aware input mask pattern computed using the statistical method. (c) Comparison of pixel errors under different focus errors. (d) Comparison of pixel errors under different coma.

Equations (21)

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I ( x ) = 𝒯 { U ( x ) } ,
U ^ ( x ) = argmin U ( x ) d ( I 0 ( x ) , 𝒯 { U ( x ) } ) ,
sig ( U ( x ) ) = 1 1 + e a ( U ( x ) t r ) ,
I aerial ( x ) = U * ( x 1 ) U ( x 2 ) γ ( x 1 x 2 ) H * ( x x 1 ) H ( x x 2 ) d x 1 d x 2 ,
γ ( x ) = m Γ m e j ω 0 m x ,
Γ m = 1 G 2 A γ γ ( x ) e j ω 0 m x d x ,
I aerial ( x ) = m Γ m | U ( x ) * H m ( x ) | 2 ,
H m ( x ) = H ( x ) e j ω 0 m x .
I ( x ) = sig ( m Γ m | U ( x ) * H m ( x ) | 2 ) .
U ( x ) = { U int for { x : ϕ ( x ) < 0 } U ext for { x : ϕ ( x ) > 0 } ,
F ( U ) = 1 2 | 𝒯 ( U ) I 0 | 2 .
ϕ t = | ϕ | α ( x , t ) ,
α ( x , t ) = J ( U ) ( 𝒯 ( U ) I 0 ) = 1 2 U ( I I 0 ) 2 = 1 2 U ( sig ( m Γ m | U * H m | 2 ) I 0 ) 2 = a { m Γ m H m * [ ( I 0 I ) I ( 1 I ) ( H m * U ) ] } ,
Φ ( ρ , θ ) = n , m c n m R n m ( ρ ) cos m θ ,
( H ) = ( H 0 ) × e j Φ ,
U optimal = min U { I I 0 2 2 } ,
ϕ t = | ϕ | α ( x , t ) ,
α ( x , t ) = 1 2 U { ( I I 0 ) 2 } = 1 2 U { ( sig ( m Γ m | U * H m | 2 ) I 0 ) 2 } = a × { m Γ m H m * [ ( I 0 I ) I ( 1 I ) ( H m * U ) ] } .
c 2 0 = 2 π λ z ( 1 ( 1 N A 2 ) ) z π N A 2 λ ,
j c 2 0 ρ 2 j π z N A 2 λ ρ 2 = j π z N A 2 λ [ ( u λ N A ) 2 + ( v λ N A ) 2 ] = j π λ z ( u 2 + v 2 ) = j π λ z [ ( m 1 N Δ x ) 2 + ( n 1 N Δ x ) 2 ] = j π λ z m 2 + n 2 ( N Δ x ) 2 ,
j c 3 1 ( 3 ρ 3 2 ρ ) cos θ = j c 3 1 [ 3 ( ( u λ N A ) 2 + ( v λ N A ) 2 ) 3 2 ( u λ N A ) 2 + ( v λ N A ) 2 ] cos θ = j c 3 1 [ 3 λ 3 N A 3 ( u 2 + v 2 ) 3 2 2 λ N A ( u 2 + v 2 ) 1 2 ] cos θ = j c 3 1 [ 3 λ 3 ( N Δ x N A ) 3 ( m 2 + n 2 ) 3 2 2 λ N Δ x N A ( m 2 + n 2 ) 1 2 ] cos θ .

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