Abstract

Gradient-based phase-shifting mask (PSM) optimization methods have emerged as an important tool in computational lithography to solve for the inverse lithography problem under the thin-mask assumption, where the mask is considered a thin two-dimensional object. 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 PSM optimization methods. Thick-mask effects are particularly aggravated and pronounced in etching profiles with abrupt discontinuities and trench depths. PSM methods derived under the thin-mask assumption have inherent limitations and perform poorly in the subwavelength scenario. This paper focuses on developing three-dimensional PSM optimization methods that can overcome the thick-mask effects in lithography systems with partially coherent illumination. The boundary layer model is exploited to simplify and characterize the thick-mask effects, leading to a gradient-based PSM optimization method. Several illustrative simulations are presented.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2011 (2)

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

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

2009 (2)

2008 (2)

2007 (1)

2006 (2)

A. Poonawala and P. Milanfar, “Opc and psm design using inverse lithography: A non-linear optimization approach,” Proc. SPIE 6154, 1159–1172 (2006).

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,” Proc. SPIE 6283, 628319 (2006).
[CrossRef]

2004 (3)

J. Tirapu-Azpiroz and E. Yablonovitch, “Fast evaluation of photomask near-fields in sub-wavelength 193 nm lithography,” Proc. SPIE 5377, 1528–1535 (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]

Y. Granik, “Solving inverse problem of optical microlithography,” Proc. SPIE 5754, 506–526 (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. Microfabrication 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 (1)

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 (1)

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

1982 (2)

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, 2770–2777 (1982).
[CrossRef] [PubMed]

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

Adam, K.

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

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,” Proc. SPIE 6283, 628319 (2006).
[CrossRef]

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 Fabrication, 2nd ed. (Publishing House of Electronics Industry, 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,” 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,” Proc. SPIE 6283, 628319 (2006).
[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,” Proc. SPIE 6283, 628319 (2006).
[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,” Proc. SPIE 6283, 628319 (2006).
[CrossRef]

Granik, Y.

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

Kailath, T.

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]

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]

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 29, 1828–1836 (1982).
[CrossRef]

Li, Y.

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

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, “Opc and psm design using inverse lithography: A non-linear optimization approach,” Proc. SPIE 6154, 1159–1172 (2006).

Neureuther, A. R.

K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlithogr. Microfabrication 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]

Pati, Y. C.

Pierrat, C.

C. Pierrat, A. Wong, and S. Vaidya, “Phase-shifting mask topography effects on lithographic image quality,” in International Electron Devices Meeting (IEDM ’92), Technical Digest (1992), pp. 53–56.

Poonawala, A.

A. Poonawala and P. Milanfar, “Opc and psm design using inverse lithography: A non-linear optimization approach,” Proc. SPIE 6154, 1159–1172 (2006).

Rabbani, M.

Saleh, B. E. A.

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, 2004).

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 29, 1828–1836 (1982).
[CrossRef]

Strojwas, A. J.

Tanabe, H.

Tirapu-Azpiroz, J.

J. Tirapu-Azpiroz and E. Yablonovitch, “Fast evaluation of photomask near-fields in sub-wavelength 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).

Vaidya, S.

C. Pierrat, A. Wong, and S. Vaidya, “Phase-shifting mask topography effects on lithographic image quality,” in International Electron Devices Meeting (IEDM ’92), Technical Digest (1992), pp. 53–56.

Viswanathan, N. S.

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

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

C. Pierrat, A. Wong, and S. Vaidya, “Phase-shifting mask topography effects on lithographic image quality,” in International Electron Devices Meeting (IEDM ’92), Technical Digest (1992), pp. 53–56.

Wong, A. K.

A. K. Wong, Resolution Enhancement Techniques, Vol. 1 (SPIE, 2001).
[CrossRef]

Yablonovitch, E.

J. Tirapu-Azpiroz and E. Yablonovitch, “Fast evaluation of photomask near-fields in sub-wavelength 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]

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 (3)

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices 29, 1828–1836 (1982).
[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. M. Yuan, “Calculation of one-dimension lithographic aerial images using the vector theory,” IEEE Trans. Electron Devices 40, 1604–1613 (1993).
[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. Micro/Nanolith. MEMS MOEMS (1)

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

J. Microlithogr. Microfabrication Microsyst. (1)

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

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

Opt. Express (4)

Proc. SPIE (6)

Y. Granik, “Solving inverse problem of optical microlithography,” Proc. SPIE 5754, 506–526 (2004).
[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. 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,” Proc. SPIE 6283, 628319 (2006).
[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 and E. Yablonovitch, “Fast evaluation of photomask near-fields in sub-wavelength 193 nm lithography,” Proc. SPIE 5377, 1528–1535 (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]

Other (7)

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

A. K. Wong, Resolution Enhancement Techniques, Vol. 1 (SPIE, 2001).
[CrossRef]

X. Ma and G. R. Arce, Computational Lithography, 1st ed., Wiley Series in Pure and Applied Optics (Wiley, 2010).
[CrossRef]

S. A. Campbell, The Science and Engineering of Microelectronic Fabrication, 2nd ed. (Publishing House of Electronics Industry, 2003).

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).

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

C. Pierrat, A. Wong, and S. Vaidya, “Phase-shifting mask topography effects on lithographic image quality,” in International Electron Devices Meeting (IEDM ’92), Technical Digest (1992), pp. 53–56.

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Figures (5)

Fig. 1
Fig. 1

Optical lithography system with partially coherent illuminations.

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, and a and b are the width and height of the entire opening area respectively.

Fig. 3
Fig. 3

Approximated forward imaging process based on the BL model under partially coherent illumination.

Fig. 4
Fig. 4

PSM optimization for type I optical lithography system. NA = 0.68 and λ = 248 nm . (a) the initial mask; (b) the optimized PSM based on thin-mask approximation; (c) the optimized PSM based on BL model. (d), (e) and (f) show the aerial images corresponding to (a), (b) and (c), respectively. In the mask patterns, black, gray and white represent 1 , 0 and 1, respectively.

Fig. 5
Fig. 5

PSM optimization for type II optical lithography system. NA = 0.85 and λ = 193 nm . (a) the initial mask; (b) the optimized PSM based on thin-mask approximation; (c) the optimized PSM based on BL model. (d), (e) and (f) show the aerial images corresponding to (a), (b) and (c), respectively. In the mask patterns, black, gray and white represent 1 , 0 and 1, respectively.

Tables (2)

Tables Icon

Table 1 Boundary Widths, Transmission Coefficients, and Corresponding Minimum Opening Sizes of the BL Model

Tables Icon

Table 2 Runtime Reduction Because of Different Algorithm Acceleration Approaches

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

f ̲ p TE = { 0.52 j : ( γ ̲ p 2 N = 1 and γ ̲ p = 0 ) or ( γ ̲ p + 2 N = 1 and γ ̲ p = 0 ) γ ̲ p : otherwise .
f ̲ p TE = 0.52 j 2 ( 1 γ ̲ p ) ( 1 + γ ̲ p ) γ ̲ p 2 N ( 1 γ ̲ p 2 N ) + 0.52 j 2 ( 1 γ ̲ p ) ( 1 + γ ̲ p ) γ ̲ p + 2 N ( 1 γ ̲ p + 2 N ) + γ ̲ p , p = 1 , 2 , , N 2 ,
f ̲ p TM = 0.52 j 2 ( 1 γ ̲ p ) ( 1 + γ ̲ p ) γ ̲ p 2 ( 1 γ ̲ p 2 ) + 0.52 j 2 ( 1 γ ̲ p ) ( 1 + γ ̲ p ) γ ̲ p + 2 ( 1 γ ̲ p + 2 ) + γ ̲ p , p = 1 , 2 , , N 2 .
f ̲ p TE = { 0.8 j : ( γ ̲ p N = 1 and γ ̲ p = 0 ) or ( γ ̲ p + N = 1 and γ ̲ p = 0 ) 0.30 j : ( γ ̲ p 4 N = 1 and γ ̲ p = 0 ) or ( γ ̲ p + 4 N = 1 and γ ̲ p = 0 ) γ ̲ p : otherwise .
f ̲ p TE = 0.8 j 2 ( 1 γ ̲ p ) ( 1 + γ ̲ p ) γ ̲ p N ( 1 + γ ̲ p N ) + 0.8 j 2 ( 1 γ ̲ p ) ( 1 + γ ̲ p ) γ ̲ p + N ( 1 + γ ̲ p + N ) + 0.3 j 2 ( 1 γ ̲ p ) ( 1 + γ ̲ p ) γ ̲ p 4 N ( 1 γ ̲ p 4 N ) + 0.3 j 2 ( 1 γ ̲ p ) ( 1 + γ ̲ p ) γ ̲ p + 4 N ( 1 γ ̲ p + 4 N ) + γ ̲ p , p = 1 , 2 , , N 2 ,
f ̲ p TM = 0.8 j 2 ( 1 γ ̲ p ) ( 1 + γ ̲ p ) γ ̲ p 1 ( 1 + γ ̲ p 1 ) + 0.8 j 2 ( 1 γ ̲ p ) ( 1 + γ ̲ p ) γ ̲ p + 1 ( 1 + γ ̲ p + 1 ) + 0.3 j 2 ( 1 γ ̲ p ) ( 1 + γ ̲ p ) γ ̲ p 4 ( 1 γ ̲ p 4 ) + 0.3 j 2 ( 1 γ ̲ p ) ( 1 + γ ̲ p ) γ ̲ p + 4 ( 1 γ ̲ p + 4 ) + γ ̲ p , p = 1 , 2 , , N 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 ,
h m ( r ) = h ( r ) exp ( j ω 0 m · r ) .
I = 1 2 m Φ m h m F TE 2 + 1 2 m Φ m h m F TM 2 ,
M ^ = argmin M d ( i , i ˜ ̲ ) .
d = F ( γ ̲ ) = i ̲ i ˜ ̲ 2 2 = p = 1 N 2 ( i ̲ p i ˜ ̲ p ) 2 ,
i ̲ p = 1 2 m Φ m | q = 1 N 2 h p q m f ̲ q TE | 2 + 1 2 m Φ m | q = 1 N 2 h p q m f ̲ q TM | 2 , p = 1 , N 2 ,
E TE m ( x , y ) = F TE ( x , y ) h m ( x , y ) .
E TE m ( x , y ) = E TE m ( x , y ) + [ F TE ( x 0 , y 0 ) F TE ( x 0 , y 0 ) ] × h m ( x x 0 , y y 0 ) .
I ( x , y ) = 1 2 m Φ m | E TE m ( x , y ) | 2 + 1 2 m Φ m | E TM m ( x , y ) | 2 .
h ( r ) = J 1 ( 2 π r NA / λ ) 2 π r NA / λ ,
F = 2 Re { m Φ m { h m * [ ( I ˜ I ) ( h m F TE ) ] [ 0.52 j Γ Γ 2 ( 1 Γ 2 ) + 0.52 j Γ Γ 2 ( 1 Γ 2 ) + 1 ] + { h m * [ ( I ˜ I ) ( h m F TE ) ] } 2 [ 0.26 j ( 1 Γ 2 ) ( 1 + Γ 2 ) ( 1 2 Γ ) ] + { h m * [ ( I ˜ I ) ( h m F TE ) ] } 2 [ 0.26 j ( 1 Γ 2 ) ( 1 + Γ 2 ) ( 1 2 Γ ) ] } + m Φ m { h m * [ ( I ˜ I ) ( h m F TM ) ] [ 0.52 j Γ Γ⃖ 2 ( 1 Γ⃖ 2 ) + 0.52 j Γ Γ 2 ( 1 Γ 2 ) + 1 ] + { h m * [ ( I ˜ I ) ( h m F TM ) ] } 2 [ 0.26 j ( 1 Γ⃖ 2 ) ( 1 + Γ⃖ 2 ) ( 1 2 Γ ) ] + { h m * [ ( I ˜ I ) ( h m F TM ) ] } 2 [ 0.26 j ( 1 Γ 2 ) ( 1 + Γ 2 ) ( 1 2 Γ ) ] } } ,
F = 2 Re m Φ m { h m * [ ( I ˜ I ) ( h m F TE ) ] [ 0.3 j Γ Γ 4 ( 1 Γ 4 ) + 0.3 j Γ Γ 4 ( 1 Γ 4 ) + 0.8 j Γ Γ ( 1 Γ ) + 0.8 j Γ Γ ( 1 Γ ) + 1 ] + { h m * [ ( I ˜ I ) ( h m F TE ) ] } [ 0.4 j ( 1 Γ ) ( 1 + Γ ) ( 1 + 2 Γ ) ] + { h m * [ ( I ˜ I ) ( h m F TE ) ] } [ 0.4 j ( 1 Γ ) ( 1 + Γ ) ( 1 + 2 Γ ) ] + { h m * [ ( I ˜ I ) ( h m F TE ) ] } 4 [ 0.15 j ( 1 Γ 4 ) ( 1 + Γ 4 ) ( 1 2 Γ ) ] + { h m * [ ( I ˜ I ) ( h m F TE ) ] } 4 [ 0.15 j ( 1 Γ 4 ) ( 1 + Γ 4 ) ( 1 2 Γ ) ] } + m Φ m { h m * [ ( I ˜ I ) ( h m F TM ) ] [ 0.3 j Γ Γ⃖ 4 ( 1 Γ⃖ 4 ) + 0.3 j Γ Γ 4 ( 1 Γ 4 ) + 0.8 j Γ Γ⃖ ( 1 Γ⃖ ) + 0.8 j Γ Γ ( 1 Γ ) + 1 ] + { h m * [ ( I ˜ I ) ( h m F TM ) ] } [ 0.4 j ( 1 Γ⃖ ) ( 1 + Γ⃖ ) ( 1 + 2 Γ ) ] + { h m * [ ( I ˜ I ) ( h m F TM ) ] } [ 0.4 j ( 1 Γ ) ( 1 + Γ ) ( 1 + 2 Γ ) ] + { h m * [ ( I ˜ I ) ( h m F TM ) ] } 4 [ 0.15 j ( 1 Γ⃖ 4 ) ( 1 + Γ⃖ 4 ) ( 1 2 Γ ) ] + { h m * [ ( I ˜ I ) ( h m F TM ) ] } 4 [ 0.15 j ( 1 Γ 4 ) ( 1 + Γ 4 ) ( 1 2 Γ ) ] } } .

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