Abstract

Recently, a set of generalized gradient-based optical proximity correction (OPC) optimization methods have been developed to solve for the forward and inverse lithography problems under the thin-mask assumption, where the mask is considered a thin 2D object. However, as the critical dimension printed on the wafer shrinks into the subwavelength regime, thick-mask effects become prevalent, and thus these effects must be taken into account in OPC optimization methods. OPC methods derived under the thin-mask assumption have inherent limitations and perform poorly in the subwavelength regime. This paper focuses on developing model-based forward binary mask optimization methods that account for the thick-mask effects of coherent imaging systems. The boundary layer (BL) model is exploited to simplify and characterize the thick-mask effects, leading to a model-based OPC method. The BL model is simpler than other thick-mask models, treating the near field of the mask as the superposition of the interior transmission areas and the boundary layers. The advantages and limitations of the proposed algorithm are discussed, and several illustrative simulations are presented.

© 2009 Optical Society of America

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References

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2009 (2)

X. Ma and G. R. Arce, “Psm design for inverse lithography using illumination with samll partial coherence factor,” Proc. SPIE 7274, 727437 (2009).
[CrossRef]

X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization,” Opt. Express 17, 5783-5793 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (3)

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

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Proc. SPIE 6520, 65200U (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]

2006 (2)

L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ilt): What is the impact to the photomask industry?” Proc. SPIE 6283, 62830X-1 (2006).
[CrossRef]

A. Erdmann, P. Evanschitzky, G. Citarella, T. Fühner, and P. D. Bisschop, “Rigorous mask modeling using waveguide and FDTD methods: An assessment for typical hyper NA imaging problems,” in Proc. SPIE 6283, 628319 (2006).
[CrossRef]

2004 (4)

J. Tirapu-Azpiroz and E. Yablonovitch, “Fast evaluation of photomask near-fields in subwavelength 193 nm lithography,” Proc. SPIE 5377, 1528-1535 (2004).
[CrossRef]

Y. Granik, “Illuminator optimization methods in microlithography,” Proc. SPIE 5524, 217-229 (2004).
[CrossRef]

A. Erdmann, R. Farkas, T. Fuhner, B. Tollkuhn, and G. Kokai, “Towards automatic mask and source optimization for optical lithography,” Proc. SPIE 5377, 646-657 (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 (1)

J. Tirapu-Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611-1619 (2003).
[CrossRef]

2002 (1)

K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlithogr. Microfabr. Microsyst. 1, 253-269 (2002).
[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. Dev. 45, 651-665 (2001).
[CrossRef]

1996 (2)

1995 (1)

S. Sherif, B. Saleh, and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Trans. Image Process. 4, 1252-1257 (1995).
[CrossRef] [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, 2438-2452 (1994).
[CrossRef]

A. Wong and A. R. Neureuther, “Mask topography effects in projection printing of phase shift masks,” IEEE Trans. Electron Devices 41, 895-902 (1994).
[CrossRef]

1993 (1)

C. M. Yuan, “Calculation of one-dimension lithographic aerial images using the vector theory,” IEEE Trans. Electron Devices 40, 1604-1613 (1993).
[CrossRef]

1992 (2)

L. Lam, S. W. Lee, and C. Y. Suen, “Thinning methodologies--a comprehensive survey,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 869-885 (1992).
[CrossRef]

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

1982 (1)

Abrams, D.

L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ilt): What is the impact to the photomask industry?” Proc. SPIE 6283, 62830X-1 (2006).
[CrossRef]

Adam, K.

K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlithogr. Microfabr. Microsyst. 1, 253-269 (2002).
[CrossRef]

K. Adam, “Domain decomposition methods for the electromagnetic simulation of scattering from three-dimensional structures with applications in lithography,” Ph.D. thesis (University of California, Berkeley, 2001).

Arce, G. R.

Bisschop, P. D.

A. Erdmann, P. Evanschitzky, G. Citarella, T. Fühner, and P. D. Bisschop, “Rigorous mask modeling using waveguide and FDTD methods: An assessment for typical hyper NA imaging problems,” in Proc. SPIE 6283, 628319 (2006).
[CrossRef]

Born, M.

M. Born and E. Wolfe, Principles of Optics (Cambridge U. Press, 1999).

Burchard, P.

J. Tirapu-Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611-1619 (2003).
[CrossRef]

Campbell, S. A.

S. A. Campbell, The Science and Engineering of Microelectronic Fabrication2nd ed. (Publishing House of Electronics Industry, Beijing, China, 2003).

Citarella, G.

A. Erdmann, P. Evanschitzky, G. Citarella, T. Fühner, and P. D. Bisschop, “Rigorous mask modeling using waveguide and FDTD methods: An assessment for typical hyper NA imaging problems,” in Proc. SPIE 6283, 628319 (2006).
[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. Dev. 45, 651-665 (2001).
[CrossRef]

Erdmann, A.

A. Erdmann, P. Evanschitzky, G. Citarella, T. Fühner, and P. D. Bisschop, “Rigorous mask modeling using waveguide and FDTD methods: An assessment for typical hyper NA imaging problems,” in Proc. SPIE 6283, 628319 (2006).
[CrossRef]

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

Evanschitzky, P.

A. Erdmann, P. Evanschitzky, G. Citarella, T. Fühner, and P. D. Bisschop, “Rigorous mask modeling using waveguide and FDTD methods: An assessment for typical hyper NA imaging problems,” in Proc. SPIE 6283, 628319 (2006).
[CrossRef]

Farkas, R.

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

Fuhner, T.

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

Fühner, T.

A. Erdmann, P. Evanschitzky, G. Citarella, T. Fühner, and P. D. Bisschop, “Rigorous mask modeling using waveguide and FDTD methods: An assessment for typical hyper NA imaging problems,” in Proc. SPIE 6283, 628319 (2006).
[CrossRef]

Granik, Y.

Y. Granik, “Illuminator optimization methods in microlithography,” Proc. SPIE 5524, 217-229 (2004).
[CrossRef]

Kailath, T.

Kokai, G.

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

Lam, L.

L. Lam, S. W. Lee, and C. Y. Suen, “Thinning methodologies--a comprehensive survey,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 869-885 (1992).
[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. Dev. 45, 651-665 (2001).
[CrossRef]

Lee, S. W.

L. Lam, S. W. Lee, and C. Y. Suen, “Thinning methodologies--a comprehensive survey,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 869-885 (1992).
[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. Dev. 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] [PubMed]

Liebmann, L.

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

Liu, Y.

L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ilt): What is the impact to the photomask industry?” Proc. SPIE 6283, 62830X-1 (2006).
[CrossRef]

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

Lucas, K.

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. Dev. 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] [PubMed]

Neureuther, A. R.

K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlithogr. Microfabr. Microsyst. 1, 253-269 (2002).
[CrossRef]

A. Wong and A. R. Neureuther, “Mask topography effects in projection printing of phase shift masks,” IEEE Trans. Electron Devices 41, 895-902 (1994).
[CrossRef]

Pan, D. Z.

P. Yu and D. Z. Pan, “Tip-opc: a new topological invariant paradigm for pixel based optical proximity correction,” in Proceedings of the ACM/IEEE International Conference on Computer-Aided Design (ICCAD) (IEEE, 2007), pp. 847-853.

Pang, L.

L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ilt): What is the impact to the photomask industry?” Proc. SPIE 6283, 62830X-1 (2006).
[CrossRef]

Pati, Y. C.

Pierrat, C.

C. Pierrat, A. Wong, and S. Vaidya, “Phase-shifting mask topography effects on lithographic image quality,” in IEEE International Electron Devices Meeting, Technical Digest (IEEE, 1992), pp. 53-56.

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] [PubMed]

A. Poonawala, “Mask design for single and double exposure optical microlithography: an inverse imaging approach,” Ph.D. thesis (University of California, Santa Cruz, 2007).

Rabbani, M.

Rosen, J.

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] [PubMed]

Saleh, B. E. A.

Salik, B.

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, Resolution Enhancement Techniques in Optical Lithography (SPIE Press, 2004).

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] [PubMed]

Strojwas, A. J.

Suen, C. Y.

L. Lam, S. W. Lee, and C. Y. Suen, “Thinning methodologies--a comprehensive survey,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 869-885 (1992).
[CrossRef]

Tanabe, H.

Tirapu-Azpiroz, J.

J. Tirapu-Azpiroz and E. Yablonovitch, “Fast evaluation of photomask near-fields in subwavelength 193 nm lithography,” Proc. SPIE 5377, 1528-1535 (2004).
[CrossRef]

J. Tirapu-Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611-1619 (2003).
[CrossRef]

J. Tirapu-Azpiroz, “Analysis and modeling of photomask near-fields in sub-wavelength deep ultraviolet lithography with optical proximity corrections,” Ph.D. thesis (University of California, Los Angeles, 2004).

Tollkuhn, B.

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

Vaidya, S.

C. Pierrat, A. Wong, and S. Vaidya, “Phase-shifting mask topography effects on lithographic image quality,” in IEEE International Electron Devices Meeting, Technical Digest (IEEE, 1992), pp. 53-56.

Wilson, R.

R. Wilson, Fourier Series and Optical Transform Techniques in Contemporary Optics (Wiley, 1995).

Wolfe, E.

M. Born and E. Wolfe, Principles of Optics (Cambridge U. Press, 1999).

Wong, A.

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

A. Wong and A. R. Neureuther, “Mask topography effects in projection printing of phase shift masks,” IEEE Trans. Electron Devices 41, 895-902 (1994).
[CrossRef]

C. Pierrat, A. Wong, and S. Vaidya, “Phase-shifting mask topography effects on lithographic image quality,” in IEEE International Electron Devices Meeting, Technical Digest (IEEE, 1992), pp. 53-56.

A. Wong, “Rigorous three-dimensional time-domain finite difference electromagnetic simulation,” Ph.D. thesis (University of California, Berkeley, 1994).

Wong, A. K.

A. K. Wong, Resolution Enhancement Techniques (SPIE Press, 2001).
[CrossRef]

Yablonovitch, E.

J. Tirapu-Azpiroz and E. Yablonovitch, “Fast evaluation of photomask near-fields in subwavelength 193 nm lithography,” Proc. SPIE 5377, 1528-1535 (2004).
[CrossRef]

J. Tirapu-Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611-1619 (2003).
[CrossRef]

Yariv, A.

Yu, P.

P. Yu and D. Z. Pan, “Tip-opc: a new topological invariant paradigm for pixel based optical proximity correction,” in Proceedings of the ACM/IEEE International Conference on Computer-Aided Design (ICCAD) (IEEE, 2007), pp. 847-853.

Yuan, C. M.

C. M. Yuan, “Calculation of one-dimension lithographic aerial images using the vector theory,” IEEE Trans. Electron Devices 40, 1604-1613 (1993).
[CrossRef]

Zakhor, A.

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

Appl. Opt. (1)

IBM J. Res. Dev. (1)

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

IEEE Trans. Electron Devices (2)

C. M. Yuan, “Calculation of one-dimension lithographic aerial images using the vector theory,” IEEE Trans. Electron Devices 40, 1604-1613 (1993).
[CrossRef]

A. Wong and A. R. Neureuther, “Mask topography effects in projection printing of phase shift masks,” IEEE Trans. Electron Devices 41, 895-902 (1994).
[CrossRef]

IEEE Trans. Image Process. (2)

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

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

IEEE Trans. Pattern Anal. Mach. Intell. (1)

L. Lam, S. W. Lee, and C. Y. Suen, “Thinning methodologies--a comprehensive survey,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 869-885 (1992).
[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, 138-152 (1992).
[CrossRef]

J. Microlithogr. Microfabr. Microsyst. (1)

K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlithogr. Microfabr. Microsyst. 1, 253-269 (2002).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Express (3)

Proc. SPIE (10)

J. Tirapu-Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611-1619 (2003).
[CrossRef]

J. Tirapu-Azpiroz and E. Yablonovitch, “Fast evaluation of photomask near-fields in subwavelength 193 nm lithography,” Proc. SPIE 5377, 1528-1535 (2004).
[CrossRef]

X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” Proc. SPIE 7140, 71401A (2008).
[CrossRef]

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Proc. SPIE 6520, 65200U (2007).
[CrossRef]

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

L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ilt): What is the impact to the photomask industry?” Proc. SPIE 6283, 62830X-1 (2006).
[CrossRef]

Y. Granik, “Illuminator optimization methods in microlithography,” Proc. SPIE 5524, 217-229 (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]

A. Erdmann, P. Evanschitzky, G. Citarella, T. Fühner, and P. D. Bisschop, “Rigorous mask modeling using waveguide and FDTD methods: An assessment for typical hyper NA imaging problems,” in Proc. SPIE 6283, 628319 (2006).
[CrossRef]

X. Ma and G. R. Arce, “Psm design for inverse lithography using illumination with samll partial coherence factor,” Proc. SPIE 7274, 727437 (2009).
[CrossRef]

Other (11)

C. Pierrat, A. Wong, and S. Vaidya, “Phase-shifting mask topography effects on lithographic image quality,” in IEEE International Electron Devices Meeting, Technical Digest (IEEE, 1992), pp. 53-56.

M. Born and E. Wolfe, Principles of Optics (Cambridge U. Press, 1999).

R. Wilson, Fourier Series and Optical Transform Techniques in Contemporary Optics (Wiley, 1995).

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[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Optical lithography system with coherent illumination.

Fig. 2
Fig. 2

BL model under coherent illumination, where the polarization of the electric field is assigned to be in the horizontal direction. w is the width of the boundary areas; a and b are the width and height, respectively, of the entire opening area.

Fig. 3
Fig. 3

Each source point of the partially coherent illumination generates a plane wave impending on the mask plane with incident azimuth angle ϕ and elevation angle θ.

Fig. 4
Fig. 4

(a) Source polarization modes of TE and TM. (b) Approximated source polarization modes of E X and E Y (redrawn based on Fig. 4.11 in [33]).

Fig. 5
Fig. 5

Approximated forward imaging process based on the BL model under coherent illumination, where the polarization of the electric field is assigned to be in the horizontal direction.

Fig. 6
Fig. 6

Desired pattern of the aerial image searched on the wafer.

Fig. 7
Fig. 7

Binary mask optimization based on the BL model for the first type of coherent optical lithography system. λ = 248 nm . Top row (from left to right), the initial mask pattern and the corresponding output aerial image. Middle row (from left to right), the optimized binary mask based on the thin-mask approximation and the corresponding output aerial image. Bottom row, (from left to right) the optimized binary mask based on the BL model and the corresponding output aerial image. Black and white represent 0 and 1, respectively.

Fig. 8
Fig. 8

Binary mask optimization based on the BL model for the second type of coherent optical lithography system. λ = 193 nm . Top row (from left to right), the initial mask pattern and the corresponding output aerial image. Middle row (from left to right), the optimized binary mask based on the thin-mask approximation and the corresponding output aerial image. Bottom row (from left to right), the optimized binary mask based on the BL model and the corresponding output aerial image. Black and white represent 0 and 1, respectively.

Equations (20)

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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 ,
h ( r ) = J 1 ( 2 π r NA λ ) 2 π r NA λ .
I ( r ) = M ( r ) h ( r ) 2 ,
I ( r ) = M ( r ) 2 h ( r ) 2 .
Re { Δ E E } = 4 w d = ( 2 a + 2 b ) w a b = Boundary Layer Area ( real part ) Total Area ,
Im { Δ E E } = η T 2 w b = η T 2 a w a b = η T Boundary Layer Area ( imaginary part ) Total Area ,
e ̂ TE = sin ϕ p ̂ X + cos ϕ p ̂ Y ,
e ̂ TM = sin θ cos ϕ p ̂ X + sin θ cos ϕ p ̂ Y cos θ p ̂ Z ,
D = d ( Z ( x , y ) , Z ̃ ( x , y ) ) = d ( T { M ( x , y ) } , Z ̃ ( x , y ) )
M ̂ ( x , y ) = argmin M ( x , y ) R N × N d ( T { M ( x , y ) } , Z ̃ ( x , y ) ) .
f ̱ p = γ ̱ p , p = 1 , 2 , , N 2 .
f ̱ p = { 0.8 j : ( γ ̱ p N = 1 and γ ̱ p = 0 ) or ( γ ̱ p + N = 1 and γ ̱ p = 0 ) γ ̱ p : otherwise } .
f ̱ p = 0.8 j ( 1 γ ̱ p ) γ ̱ p N + 0.8 j ( 1 γ ̱ p ) γ ̱ p + N + γ ̱ p ,
p = 1 , 2 , , N 2 ,
Z = H { F } 2 .
M ̂ = argmin M d ( H { F } 2 , Z ̃ ) .
z ̱ p = q = 1 N 2 h p q f ̱ q 2 , p = 1 , N 2 ,
D = z ̱ ̃ z ̱ 2 2 = p = 1 N 2 ( z ̱ ̃ p z ̱ p ) 2 ,
D γ ̱ = H T [ ( z ̱ ̃ z ̱ ) H ( γ ̱ ) ] ,
D γ ̱ = 4 Re { H T [ ( z ̱ ̃ z ̱ ) H ( f ̱ ) ] ( 0.8 i γ ̱ + 0.8 i γ ̱ + 1 ) + H T [ ( z ̱ ̃ z ̱ ) H ( f ̱ ) ] 0.8 i ( 1 γ ̱ ) + H T [ ( z ̱ ̃ z ̱ ) H ( f ̱ ) ] 0.8 i ( 1 γ ̱ ) } ,

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