Abstract

Phase-shifting masks (PSM) are resolution enhancement techniques (RET) used extensively in the semiconductor industry to improve the resolution and pattern fidelity of optical lithography. Recently, a set of gradient-based PSM optimization methods have been developed to solve for the inverse lithography problem under coherent illumination. Most practical lithography systems, however, use partially coherent illumination due to non-zero width and off-axis light sources, which introduce partial coherence factors that must be accounted for in the optimization of PSMs. This paper thus focuses on developing a framework for gradient-based PSM optimization methods which account for the inherent nonlinearities of partially coherent illumination. In particular, the singular value decomposition (SVD) is used to expand the partially coherent imaging equation by eigenfunctions into a sum of coherent systems (SOCS). The first order coherent approximation corresponding to the largest eigenvalue is used in the PSM optimization. In order to influence the solution patterns to have more desirable manufacturability properties and higher fidelity, a post-processing of the mask pattern based on the 2D discrete cosine transformation (DCT) is introduced. Furthermore, a photoresist tone reversing technique is exploited in the design of PSMs to project extremely sparse patterns.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  1. A. K. Wong, Resolution enhancement techniques, 1 (SPIE Press, 2001).
    [Crossref]
  2. S. A. Campbell, The science and engineering of microelectronic fabrication, 2nd ed. (Publishing House of Electronics Industry, Beijing, China, 2003).
  3. F. Schellenberg, “Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography,” in 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, “Fast and low-complexity mask design in optical microlithography - An inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
    [Crossref] [PubMed]
  7. M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
    [Crossref]
  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. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” in Proc. SPIE,  6154, 1159–1172 (San Jose, CA, 2006).
  11. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” in Proc. SPIE (San Jose, CA, 2007).
  12. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15, 15,066–15,079 (2007).
    [Crossref]
  13. B. Salik, J. Rosen, and A. Yariv, “Average coherent approximation for partially cohernet optical systems,” J. Opt. Soc. Am. A 13 (1996).
    [Crossref]
  14. E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun.38 (1981).
    [Crossref]
  15. P. S. Davids and S. B. Bollepalli, “Generalized inverse problem for partially coherent projection lithography,” in Proc. SPIE (San Jose, CA, 2008).
    [Crossref]
  16. X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” in Proc. SPIE (Taiwan, 2008).
    [Crossref]
  17. X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A25 (2008).
    [Crossref]
  18. N. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. thesis, University of California at Berkeley (1998).
  19. J. Zhang, W. Xiong, M. Tsai, Y. Wang, and Z. Yu, “Efficient mask design for inverse lithography technology based on 2D discrete cosine transformation (DCT),” in Simulation of Semiconductor Processes and Devices, 12 (2007).
    [Crossref]
  20. 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 (1982).
    [Crossref] [PubMed]
  21. M. Born and E. Wolfe, Principles of optics (Cambridge University Press, United Kingdom, 1999).
  22. R. Wilson, Fourier Series and Optical Transform Techniques in Contemporary Optics (John Wiley and Sons, New York, 1995).
  23. N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for OPC,” in BACUS Symposium on Photomask Technology, Proc. SPIE, 2440, 313–327 (1995).
  24. C. Vogel, Computational methods for inverse problems (SIAM Press, 2002).
    [Crossref]

2007 (2)

A. Poonawala and P. Milanfar, “Fast and low-complexity mask design in 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,” Opt. Express 15, 15,066–15,079 (2007).
[Crossref]

2006 (1)

A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” in Proc. SPIE,  6154, 1159–1172 (San Jose, CA, 2006).

2004 (1)

F. Schellenberg, “Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography,” in 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. Dev. 45, 651–665 (2001).
[Crossref]

1996 (1)

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

1994 (1)

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 (1)

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

Arce, G. R.

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15, 15,066–15,079 (2007).
[Crossref]

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” in Proc. SPIE (San Jose, CA, 2007).

X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” in Proc. SPIE (Taiwan, 2008).
[Crossref]

X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A25 (2008).
[Crossref]

Bollepalli, S. B.

P. S. Davids and S. B. Bollepalli, “Generalized inverse problem for partially coherent projection lithography,” in Proc. SPIE (San Jose, CA, 2008).
[Crossref]

Born, M.

M. Born and E. Wolfe, Principles of optics (Cambridge University Press, United Kingdom, 1999).

Campbell, S. A.

S. A. Campbell, The science and engineering of microelectronic fabrication, 2nd ed. (Publishing House of Electronics Industry, Beijing, China, 2003).

Cobb, N.

N. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. thesis, University of California at Berkeley (1998).

N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for OPC,” in BACUS Symposium on Photomask Technology, Proc. SPIE, 2440, 313–327 (1995).

Davids, P. S.

P. S. Davids and S. B. Bollepalli, “Generalized inverse problem for partially coherent projection lithography,” in Proc. SPIE (San Jose, CA, 2008).
[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]

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 ED-29, 1828–1836 (1982).
[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]

Ma, X.

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15, 15,066–15,079 (2007).
[Crossref]

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” in Proc. SPIE (San Jose, CA, 2007).

X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” in Proc. SPIE (Taiwan, 2008).
[Crossref]

X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A25 (2008).
[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, “Fast and low-complexity mask design in 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,” in Proc. SPIE,  6154, 1159–1172 (San Jose, CA, 2006).

Pati, Y. C.

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, 774–788 (2007).
[Crossref] [PubMed]

A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” in Proc. SPIE,  6154, 1159–1172 (San Jose, CA, 2006).

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 (1982).
[Crossref] [PubMed]

Rosen, J.

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

Schellenberg, F.

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

F. Schellenberg, Resolution enhancement techniques in optical lithography (SPIE Press, 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 ED-29, 1828–1836 (1982).
[Crossref]

Tsai, M.

J. Zhang, W. Xiong, M. Tsai, Y. Wang, and Z. Yu, “Efficient mask design for inverse lithography technology based on 2D discrete cosine transformation (DCT),” in Simulation of Semiconductor Processes and Devices, 12 (2007).
[Crossref]

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 ED-29, 1828–1836 (1982).
[Crossref]

Vogel, C.

C. Vogel, Computational methods for inverse problems (SIAM Press, 2002).
[Crossref]

Wang, Y.

J. Zhang, W. Xiong, M. Tsai, Y. Wang, and Z. Yu, “Efficient mask design for inverse lithography technology based on 2D discrete cosine transformation (DCT),” in Simulation of Semiconductor Processes and Devices, 12 (2007).
[Crossref]

Wilson, R.

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

Wolf, E.

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun.38 (1981).
[Crossref]

Wolfe, E.

M. Born and E. Wolfe, Principles of optics (Cambridge University Press, United Kingdom, 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, 1 (SPIE Press, 2001).
[Crossref]

Xiong, W.

J. Zhang, W. Xiong, M. Tsai, Y. Wang, and Z. Yu, “Efficient mask design for inverse lithography technology based on 2D discrete cosine transformation (DCT),” in Simulation of Semiconductor Processes and Devices, 12 (2007).
[Crossref]

Yariv, A.

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

Yu, Z.

J. Zhang, W. Xiong, M. Tsai, Y. Wang, and Z. Yu, “Efficient mask design for inverse lithography technology based on 2D discrete cosine transformation (DCT),” in Simulation of Semiconductor Processes and Devices, 12 (2007).
[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]

N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for OPC,” in BACUS Symposium on Photomask Technology, Proc. SPIE, 2440, 313–327 (1995).

Zhang, J.

J. Zhang, W. Xiong, M. Tsai, Y. Wang, and Z. Yu, “Efficient mask design for inverse lithography technology based on 2D discrete cosine transformation (DCT),” in Simulation of Semiconductor Processes and Devices, 12 (2007).
[Crossref]

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 (1)

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

IEEE Trans. Image Process. (1)

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

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]

in Proc. SPIE (2)

A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” in Proc. SPIE,  6154, 1159–1172 (San Jose, CA, 2006).

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

J. Opt. Soc. Am. A (2)

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

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]

Opt. Express (1)

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15, 15,066–15,079 (2007).
[Crossref]

Other (15)

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun.38 (1981).
[Crossref]

P. S. Davids and S. B. Bollepalli, “Generalized inverse problem for partially coherent projection lithography,” in Proc. SPIE (San Jose, CA, 2008).
[Crossref]

X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” in Proc. SPIE (Taiwan, 2008).
[Crossref]

X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A25 (2008).
[Crossref]

N. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. thesis, University of California at Berkeley (1998).

J. Zhang, W. Xiong, M. Tsai, Y. Wang, and Z. Yu, “Efficient mask design for inverse lithography technology based on 2D discrete cosine transformation (DCT),” in Simulation of Semiconductor Processes and Devices, 12 (2007).
[Crossref]

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 (1982).
[Crossref] [PubMed]

M. Born and E. Wolfe, Principles of optics (Cambridge University Press, United Kingdom, 1999).

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

N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for OPC,” in BACUS Symposium on Photomask Technology, Proc. SPIE, 2440, 313–327 (1995).

C. Vogel, Computational methods for inverse problems (SIAM Press, 2002).
[Crossref]

F. Schellenberg, Resolution enhancement techniques in optical lithography (SPIE Press, 2004).

A. K. Wong, Resolution enhancement techniques, 1 (SPIE Press, 2001).
[Crossref]

S. A. Campbell, The science and engineering of microelectronic fabrication, 2nd ed. (Publishing House of Electronics Industry, Beijing, China, 2003).

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” in Proc. SPIE (San Jose, CA, 2007).

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 (12)

Fig. 1.
Fig. 1.

Optical lithography system with partially coherent illuminations

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2.

A partially coherent system represented by a SVD decomposition of a sum of coherent systems.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3.

Eigenvalues α k of sum of coherent systems decomposition by SVD.

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4.

Fig. 4. (a) The amplitude of the first equivalent kernel corresponding to the largest eigen-value |ϕ1(x,y)|, (b) second equivalent kernel corresponding to the second largest eigenvalue |ϕ2(x,y)|.

Download Full Size | PPT Slide | PDF

Fig. 5.
Fig. 5.

Approximated forward process model.

Download Full Size | PPT Slide | PDF

Fig. 6.
Fig. 6.

Top row (input masks), left to right: desired pattern, optimized real-valued mask and optimized trinary mask. The bottom row illustrates the corresponding binary output patterns. White, grey and black represent 1, 0 and -1, respectively. σ=0.3.

Download Full Size | PPT Slide | PDF

Fig. 7.
Fig. 7.

Top row (input masks), left to right: desired pattern, optimized real-valued mask and optimized trinary mask. The bottom row illustrates the corresponding binary output patterns. White, grey and black represent 1, 0 and -1, respectively. σ=0.6.

Download Full Size | PPT Slide | PDF

Fig. 8.
Fig. 8.

The relationship between the number of maintained DCT low frequency components and the output pattern errors.

Download Full Size | PPT Slide | PDF

Fig. 9.
Fig. 9.

Left to right: output pattern when the desired pattern is inputted, post-processed trinary mask with the DCT post-processing maintaining 136 low frequency components, and the binary output pattern of the post-processed trinary mask. White, grey and black represent 1, 0 and -1, respectively.

Download Full Size | PPT Slide | PDF

Fig. 10.
Fig. 10.

Left to right: desired pattern, output pattern when the desired pattern is inputted, optimized trinary mask, and the output pattern of the optimized trinary mask. White, grey and black represent 1, 0 and -1, respectively. σ=0.3.

Download Full Size | PPT Slide | PDF

Fig. 11.
Fig. 11.

Left to right: Photoresist distribution, optimized trinary mask using the photoresist tone reversing method, and the output pattern of the optimized trinary mask. White and black represent positive and negative photoresist, respectively in the first figure. White, grey and black represent 1, 0 and -1, respectively in the second and the third figures. σ=0.3.

Download Full Size | PPT Slide | PDF

Fig. 12.
Fig. 12.

Left to right: Photoresist distribution, post-processed trinary mask using the DCT post-processing maintaining 276 low frequency components, and the binary output pattern of the post-processed trinary mask. White and black represent positive and negative photoresist, respectively in the first figure. White, grey and black represent 1, 0 and -1, respectively in the second and the third figures. σ=0.3.

Download Full Size | PPT Slide | PDF

Equations (31)

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 ( x , y ) = ∫ ∫ ∫ ∫ − ∞ + ∞ TCC ( f 1 , g 1 ; f 2 , g 2 ) M ˜ ( f 1 , g 1 ) M ˜ * ( f 2 , g 2 )
× exp { − i 2 π [ ( f 1 − f 2 ) x + ( g 1 − g 2 ) y ] } d f 1 d g 1 d f 2 d g 2 ,
TCC ( f 1 , g 1 ; f 2 , g 2 ) = ∫ ∫ − ∞ + ∞ γ ˜ ( f , g ) h ˜ ( f + f 1 , g + g 1 ) h ˜ * ( f + f 2 , g + g 2 ) d f d g ,
I ( m , n ) = s ˜ H A s ˜ , m , n = 1 , 2 , … , N ,
s ˜ i = M ˜ ( p , q ) exp [ i 2 π ( pm + qn ) ] i = 1 , 2 , … , N 2 ,
I ( m , n ) = ∑ k = 1 N α k ∣ s ˜ T V k ∣ 2 .
h ˜ k ( p , q ) = S − 1 ( V k ) = ( V k , 1 V k , N + 1 … V k , N ( N − 1 ) + 1 V k , 2 V k , N + 2 … V k , N ( N − 1 ) + 2 ⋮ ⋮ ⋱ ⋮ V k , N V k , 2 N … V k , N 2 ) ,
h k ( m , n ) = IFFT { h ˜ k ( p , q ) } , m , n = 1 , 2 , … , N .
I ( m , n ) = ∑ k = 1 N 2 α k ∣ h k ( m , n ) ⊗ M ( m , n ) ∣ 2 .
I ( m , n ) ≈ ∑ k = 1 i α k ∣ h k ( m , n ) ⊗ M ( m , n ) ∣ 2 , i = 1 , 2 , … , N 2 .
γ ˜ ( f , g ) = λ 2 π ( σ NA ) 2 circ ( λ f 2 + g 2 σ NA )
= { λ 2 π ( σ NA ) 2 for f 2 + g 2 ≤ σ NA λ 0 elsewhere ,
h ( r ) = h ( x , y ) = J 1 ( 2 π r NA ⁄ λ ) 2 π r NA ⁄ λ .
h ˜ ( f , g ) = λ 2 π ( NA ) 2 circ ( λ f 2 + g 2 NA ) = { λ 2 π ( NA ) 2 for f 2 + g 2 ≤ NA λ 0 elsewhere .
D = d ( Z ( m , n ) , Z * ( m , n ) ) = d ( T { M ( m , n ) } , Z * ( m , n ) )
M ̂ ( m , n ) = arg min M ( m , n ) ∈ ℜ N × N d ( T { M ( m , n ) } , Z * ( m , n ) ) .
sig ( x ) = 1 1 + exp [ − a ( x − t r ) ] ,
M ̂ = arg min M d ( sig { ∣ H 1 { m ̲ } ∣ 2 } , Z * ) .
z ̲ i = 1 1 + exp [ − a ∣ ∑ i = 1 N 2 h 1 , ij 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 2 − z ̲ i ) 2 ,
( θ ̂ ̲ ) = arg min θ ̲ { F ( θ ̲ ) }
= arg min θ ̲ { ∑ i = 1 N 2 ( z i * − 1 1 + exp [ − a ∣ ∑ j = 1 N 2 h 1 , ij cos ⁡ θ ̲ j ∣ 2 + a t r ] ) 2 } .
∇ F ( θ ̲ ) = d ̲ θ ̲ = 2 a × sin ⁡ θ ̲ ⊙ { ( H 1 * H [ ( z ̲ * − z ̲ ) ⊙ z ̲ ⊙ ( 1 ̲ − z ̲ ) ⊙ ( H 1 * m ̲ ) ] }
+ 2 a × sin ⁡ θ ̲ ⊙ { ( H 1 H [ ( z ̲ * − z ̲ ) ⊙ z ̲ ⊙ ( 1 ̲ − z ̲ ) ⊙ ( H 1 m ̲ ) ] } ,
θ ̲ k + 1 = θ ̲ k − s θ ̲ d ̲ θ ̲ k ,
E = ∑ i = 1 N 2 ∣ z ̲ i * − z ̲ bi ∣ 2 = ∑ i = 1 N 2 ∣ z ̲ i * − T { M tri } ∣ 2 .
m ̂ ̲ = arg min m ̂ ̲ { F ( m ̲ ) + γ R ( m ̲ ) } ,
r D ( m ̲ i ) = − 4.5 m ̲ i 4 + m ̲ i 2 + 3.5 , i = 1 , … , N 2 .
R = k λ NA = 41.5 nm .

Metrics