Abstract

Recently, a set of generalized gradient-based optical proximity correction optimization methods have been developed to solve for the inverse lithography problem under coherent illumination. Most practical lithography systems, however, operate under partially coherent illumination. This paper focuses on developing gradient-based binary mask optimization methods that account for the inherent nonlinearities of partially coherent systems. Two nonlinear models are used in the optimization. The first relies on a Fourier representation that approximates the partially coherent system as a sum of coherent systems. The second model is based on an average coherent approximation that is computationally faster. To influence the solution patterns toward more desirable manufacturability properties, wavelet regularization is added to the optimization framework.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. K. Wong, Resolution Enhancement Techniques, Vol. 1 (SPIE Press, 2001).
    [CrossRef]
  2. S. A. Campbell, The Science and Engineering of Microelectronic Fabrication, 2nd ed. (Publishing House of Electronics Industry, 2003).
  3. F. Schellenberg, “Resolution enhancement technology: The past, the present, and extensions for the future,” Proc. SPIE 5377, 1-20 (2004).
    [CrossRef]
  4. F. Schellenberg, Resolution Enhancement Techniques in Optical Lithography (SPIE Press, 2004).
  5. 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]
  6. A. Poonawala and P. Milanfar, “Mask design for optical microlithography--An inverse imaging problem,” IEEE Trans. Image Process. 16, 774-788 (2007).
    [CrossRef] [PubMed]
  7. S. Sherif, B. Saleh, and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Trans. Image Process. 4, 1252-1257 (1995).
    [CrossRef] [PubMed]
  8. Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Trans. Semicond. Manuf. 5, 138-152 (1992).
    [CrossRef]
  9. 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]
  10. 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]
  11. 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]
  12. Y. Granik, “Illuminator optimization methods in microlithography,” Proc. SPIE 5524, 217-229 (2004).
    [CrossRef]
  13. Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlithogr., Microfabr., Microsyst. 5, 043002 (2006).
    [CrossRef]
  14. A. Poonawala and P. Milanfar, “Opc and psm design using inverse lithography: A non-linear optimization approach,” Proc. SPIE 6154, 1159-1172 (2006).
  15. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Proc. SPIE 6520, 65200U (2007).
    [CrossRef]
  16. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15, 15066-15079 (2007).
    [CrossRef] [PubMed]
  17. B. Salik, J. Rosen, and A. Yariv, “Average coherent approximation for partially coherent optical systems,” J. Opt. Soc. Am. A 13, 2086-2090 (1996).
    [CrossRef]
  18. A. K. Wong, Optical Imaging in Projection Microlithography (SPIE Press, 2005).
    [CrossRef]
  19. 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]
  20. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  21. R. Wilson, Fourier Series and Optical Transform Techniques in Contemporary Optics (Wiley, 1995).
  22. N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for opc,” Proc. SPIE 2440, 313-327 (1995).
    [CrossRef]
  23. C. Vogel, Computational Methods for Inverse Problems (SIAM Press, 2002).
    [CrossRef]

2007

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

Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlithogr., Microfabr., 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).

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]

2004

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,” Proc. SPIE 5377, 1-20 (2004).
[CrossRef]

2001

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

1995

N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for opc,” Proc. SPIE 2440, 313-327 (1995).
[CrossRef]

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

1992

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

1982

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]

Arce, G. R.

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]

Born, M.

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

Campbell, S. A.

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

Cobb, N.

N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for opc,” Proc. SPIE 2440, 313-327 (1995).
[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, 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]

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]

Granik, Y.

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

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]

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]

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]

Ma, X.

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

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

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]

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

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.

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

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,” 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]

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]

Vogel, C.

C. Vogel, Computational Methods for Inverse Problems (SIAM Press, 2002).
[CrossRef]

Wilson, R.

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

Wolf, E.

M. Born and E. Wolf, 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]

Wong, A. K.

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

A. K. Wong, Optical Imaging in Projection Microlithography (SPIE Press, 2005).
[CrossRef]

Yariv, A.

Zakhor, A.

N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for opc,” Proc. SPIE 2440, 313-327 (1995).
[CrossRef]

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.

IBM J. Res. Dev.

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. Image Process.

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

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

IEEE Trans. Semicond. Manuf.

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.

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

J. Opt. Soc. Am. A

Opt. Express

Proc. SPIE

N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for opc,” Proc. SPIE 2440, 313-327 (1995).
[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]

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

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

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

Other

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

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

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

A. K. Wong, Optical Imaging in Projection Microlithography (SPIE Press, 2005).
[CrossRef]

C. Vogel, Computational Methods for Inverse Problems (SIAM Press, 2002).
[CrossRef]

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

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Optical lithography system with partially coherent illumination.

Fig. 2
Fig. 2

Annular illumination with large, medium, and small partial coherence factors. The dashed circles represent the dimension of the pupil.

Fig. 3
Fig. 3

(Top left) Target mask pattern containing 45 nm features with pitch p = 90 nm is indicated by dashed lines. Aerial images formed by annular illuminations with large (top right: σ inner = 0.8 , σ outer = 0.975 ), medium (bottom left: σ inner = 0.5 , σ outer = 0.6 ), and small (bottom right: σ inner = 0.3 , σ outer = 0.4 ) partial coherence factors. Here NA = 1.25 and k = 0.29 .

Fig. 4
Fig. 4

ACAA gives more accurate aerial imaging for sharper amplitude impulse response. (a) NA = 1.35 , corresponding to a sharper amplitude impulse response, SNR = 18.7 . (b) NA = 0.15 , corresponding to a smoother amplitude impulse response, SNR = 10.2 .

Fig. 5
Fig. 5

Approximated forward process model.

Fig. 6
Fig. 6

Binary mask optimization using the Fourier series expansion model and wavelet penalty. Left to right: the output pattern when target pattern is used as input, the binary optimized mask, and the output pattern of binary optimized mask. Top row illustrates the simulations using the annular illumination with large partial coherence factor ( σ inner = 0.8 , σ outer = 0.975 ); middle row, with medium partial coherence factor ( σ inner = 0.5 , σ outer = 0.6 ); bottom row, with small partial coherence factor ( σ inner = 0.3 , σ outer = 0.4 ).

Fig. 7
Fig. 7

Binary mask optimization using the ACAA and wavelet penalty. Left to right: the output pattern when target pattern is used as input, the binary optimized mask, and the output pattern of binary optimized mask. Top row illustrates the simulations using the annular illumination with large partial coherence factor ( σ inner = 0.8 , σ outer = 0.975 ); middle row, with medium partial coherence factor ( σ inner = 0.5 , σ outer = 0.6 ); bottom row, with small partial coherence factor ( σ inner = 0.3 , σ outer = 0.4 ).

Equations (51)

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

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 ,
I ( r ) = M ( r ) h ( r ) 2 ,
I ( r ) = M ( r ) 2 h ( 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 cu ) 2 π r 2 D cu D cu 2 D cl 2 J 1 ( 2 π r 2 D cl ) 2 π r 2 D cl ,
Γ m = { 4 D cu 2 D cl 2 π D 2 ( D cl 2 D cu 2 ) for D 2 D cl m D 2 D cu 0 elsewhere } ,
h ( r ) = J 1 ( 2 π r N A λ ) 2 π r N A λ .
T π [ ( D 2 D cu ) 2 ( D 2 D cl ) 2 ] C N 2 ,
I ( r ) = M ( r 1 ) M ( r 2 ) γ ( r 1 r 2 ) h * ( r r 2 ) h ( r r 2 ) d r 1 d r 2 M ( r ) h C ( r , r ) d r 2 + M ( r ) 2 h I ( r , r ) 2 d r ,
h C ( r , r ) = f ( r , r ) 1 2 h ( r , r ) ,
h I ( r , r ) = [ 1 f ( r , r ) ] 1 2 h ( r , r ) ,
f ( r , r ) = h ( r , r ̂ ) 2 μ ( r , r ̃ ) d r ̃ h ( r , r ̂ ) 2 d r ,
μ ( r , r ̂ ) = γ ( r , r ̂ ) [ γ ( r , r ) γ ( r ̂ , r ̂ ) ] 1 2 .
f ( r , r ) = 1 [ J 1 ( π r d ) ] ( π r d ) 2 d r [ J 1 ( π r ̂ d ) ] ( π r ̂ d ) 2 × ( [ J 1 ( π ( r ̂ r ̇ ) a u ) ] [ π ( r ̂ r ̇ ) a u ] a u 2 a l 2 [ J 1 ( π ( r ̂ r ̇ ) a l ) ] [ π ( r ̂ r ̇ ) a l ] ) d r ̂ ,
f ( r , r ) = 1 [ J 1 ( π r d ) ] ( π r d ) 2 d r IFFT [ FFT ( h ( r ) 2 ) FFT ( γ ( r ) ) ] ,
Γ m = { 4 D c 2 π D 2 for m D 2 D c 0 elsewhere } ,
D = d [ Z ( x , y ) , Z * ( x , y ) ] = d [ T { M ( x , y ) } , Z * ( x , y ) ]
M ̂ ( x , y ) = arg min M ( x , y ) R 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 M d { 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 ̱ ) = d ̱ θ ̱ = a × sin θ ̱ { m Γ m ( H m ) * T [ ( z ̱ * z ̱ ) z ̱ ( 1 ̱ z ̱ ) ( H m ) * ( m ̱ ) ] } + a × sin θ ̱ { m Γ m ( H m ) T [ ( z ̱ * z ̱ ) z ̱ ( 1 ̱ z ̱ ) ( H m ) ( m ̱ ) ] } ,
θ ̱ k + 1 = θ ̱ k s θ ̱ d ̱ θ ̱ k ,
E = i = 1 N 2 z ̱ i * z ̱ b i = i = 1 N 2 z ̱ i * U ( m Γ m H m m ̱ b i 2 t r ) .
F ( θ ̱ ) = d ̱ θ ̱ = a × sin θ ̱ { H T [ ( z ̱ * z ̱ ) z ̱ ( 1 ̱ z ̱ ) H ( m ̱ ) ] } ,
z ̱ i = 1 1 + exp [ a ( j = 1 N 2 h C i j m ̱ j 2 + j = 1 N 2 h I i j 2 m ̱ j 2 ) + a t r ]
i = 1 , , N 2 ,
F ( θ ̱ ) = z ̱ * z ̱ 2 2 = i = 1 N 2 ( z i * z i ) 2 = i = 1 N 2 { z i * 1 1 + exp [ a ( j = 1 N 2 h C i j m ̱ j 2 + j = 1 N 2 h I i j 2 m ̱ j 2 ) + a t r ] } 2 ,
F ( θ ̱ ) = d ̱ θ ̱ = 2 a × sin θ ̱ { H C T [ ( z ̱ * z ̱ ) z ̱ ( 1 ̱ z ̱ ) ( H C ( m ̱ ) ] } + 2 a × sin θ ̱ { H I 2 T [ ( z ̱ * z ̱ ) z ̱ ( 1 ̱ z ̱ ) ( m ̱ ) ] } ,
m ̱ ̂ = arg min m ̱ ̂ [ F ( m ̱ ) + γ R ( m ̱ ) ] ,
R Q ( m ̱ ) = 4 m ̱ T ( 1 ̱ m ̱ ) .
r ( m ̱ i ) = 1 ( 2 m ̱ i 1 ) 2 i = 1 , , N 2 .
J ( m ̱ ) = F ( m ̱ ) + γ Q R Q ( m ̱ ) .
R W = 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 ] n [ 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 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 ] } ,
J ( m ̱ ) = F ( m ̱ ) + γ Q R Q ( m ̱ ) + γ W R W ( m ̱ ) .
F ( θ ̱ ) θ ̱ k = 2 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 ) ] 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 exp ( a m Γ m j = 1 N 2 h i j m 1 + cos θ ̱ j 2 2 + a t r ) × ( a ) m Γ m [ ( j = 1 N 2 h i j m 1 + cos θ ̱ j 2 ) h i k m * ( sin θ ̱ m 2 ) + ( j = 1 N 2 h i j m * 1 + cos θ ̱ j 2 ) h i k m ( sin θ ̱ m 2 ) ] .
F ( θ ̱ ) = d ̱ θ ̱ = a × sin θ ̱ { m Γ m ( H m ) * T [ ( z ̱ * z ̱ ) z ̱ ( 1 ̱ z ̱ ) ( H m ) * ( m ̱ ) ] } + a × sin θ ̱ { m Γ m ( H m ) T [ ( z ̱ * z ̱ ) z ̱ ( 1 ̱ z ̱ ) ( H m ) ( m ̱ ) ] } .
F ( θ ̱ ) θ ̱ k = 2 i = 1 N 2 { z i * 1 1 + exp [ a ( j = 1 N 2 h C i j m ̱ j 2 + j = 1 N 2 h I i j 2 m ̱ j 2 ) + a t r ] } 1 { 1 + exp [ a ( j = 1 N 2 h C i j m ̱ j 2 + j = 1 N 2 h I i j 2 m ̱ j 2 ) + a t r ] } 2 exp [ a ( j = 1 N 2 h C i j m ̱ j 2 + j = 1 N 2 h I i j 2 m ̱ j 2 ) + a t r ] × ( a ) [ ( j = 1 N 2 2 h C i j 1 + cos θ ̱ j 2 ) h C i k ( sin θ ̱ m 2 ) + 2 ( 1 + cos θ ̱ k 2 ) h I i k 2 ( sin θ ̱ m 2 ) ] .
F ( θ ̱ ) = d ̱ θ ̱ = 2 a × sin θ ̱ { H C T [ ( z ̱ * z ̱ ) z ̱ ( 1 ̱ z ̱ ) H C ( m ̱ ) ] } + 2 a × sin θ ̱ { H I 2 T [ ( z ̱ * z ̱ ) z ̱ ( 1 ̱ z ̱ ) ( m ̱ ) ] } .

Metrics