Abstract

Polarization aberrations (PA) can be presented by Jones pupil and can also impact the imaging performance of immersion projection optics significantly. Precise PA measurement is most important for resolution enhancement technology and holistic lithography at 7nm node and below, in order to improve the pattern fidelity and processing stability. However, the current imaging-based measurement method of PA by linear approximation has not taken the coupling effect of the PA coefficients into account. This paper proposes a nonlinear measurement method of PA based on a rigorous nonlinear model to improve the measurement accuracy significantly. In this invention, the new spectrum modulation theory is developed to establish a rigorous quadratic form of PA and aerial image spectrum. A hybrid genetic algorithm is developed to solve the quadratic form inversely to obtain the PA accurately. An overall simulation validates that this method provides a superior quality of PA measurement with very high precision of 10-4λ.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  5. X. Xu, W. Huang, and M. Xu, “Orthonormal polynomials describing polarization aberration for M-fold optical systems,” Opt. Express 24(5), 4906–4912 (2016).
    [Crossref] [PubMed]
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    [Crossref]
  7. N. Yamamoto, J. Kye, and H. J. Levinson, “Polarization aberration analysis using Pauli-Zernike representation,” Proc. SPIE 6520, 65200Y (2007).
    [Crossref]
  8. Y. Li, X. Guo, X. Liu, and L. Liu, “A technique for extracting and analyzing the polarization aberration of hyper-numerical aperture image optics,” Proc. SPIE 9042, 904204 (2013).
    [Crossref]
  9. X. Ma, C. Han, Y. Li, L. Dong, and G. R. Arce, “Pixelated source and mask optimization for immersion lithography,” J. Opt. Soc. Am. A 30(1), 112–123 (2013).
    [Crossref] [PubMed]
  10. J. Li and E. Y. Lam, “Robust source and mask optimization compensating for mask topography effects in computational lithography,” Opt. Express 22(8), 9471–9485 (2014).
    [Crossref] [PubMed]
  11. X. Ma, C. Han, Y. Li, B. Wu, Z. Song, L. Dong, and G. R. Arce, “Hybrid source mask optimization for robust immersion lithography,” Appl. Opt. 52(18), 4200–4211 (2013).
    [Crossref] [PubMed]
  12. X. Ma, J. Gao, C. Han, Y. Li, L. Dong, and L. Liu, “Efficient source polarization optimization for robust optical lithography,” Proc. SPIE 9052, 90520T (2014).
    [Crossref]
  13. X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and R. G. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
    [Crossref]
  14. B. Zhu, X. Wang, S. Li, G. Yan, L. Shen, and L. Duan, “Wavefront aberration measurement method for a hyper-NA lithographic projection lens based on principal component analysis of an aerial image,” Appl. Opt. 55(12), 3192–3198 (2016).
    [Crossref] [PubMed]
  15. H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
    [Crossref]
  16. Z. Qiu, X. Wang, Q. Yuan, and F. Wang, “Coma measurement by use of an alternating phase-shifting mask mark with a specific phase width,” Appl. Opt. 48(2), 261–269 (2009).
    [Crossref] [PubMed]
  17. L. Dong, Y. Li, X. Dai, H. Liu, and K. Liu, “Measuring the polarization aberration of hyper-NA lens from the vector aerial image,” Proc. SPIE 9283, 928313 (2014).
    [Crossref]
  18. S. Liu, S. Xu, X. Wu, and W. Liu, “Iterative method for in situ measurement of lens aberrations in lithographic tools using CTC-based quadratic aberration model,” Opt. Express 20(13), 14272–14283 (2012).
    [Crossref] [PubMed]
  19. Z. Xiang and Y. Li, “Retrieve polarization aberration from image degradation: a new measurement method in DUV lithography,” Proc. SPIE 10460, 84 (2017).
    [Crossref]
  20. X. Ma, Y. Li, and L. Dong, “Mask optimization approaches in optical lithography based on a vector imaging model,” J. Opt. Soc. Am. A 29(7), 1300–1312 (2012).
    [Crossref] [PubMed]
  21. D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
    [Crossref]

2017 (1)

Z. Xiang and Y. Li, “Retrieve polarization aberration from image degradation: a new measurement method in DUV lithography,” Proc. SPIE 10460, 84 (2017).
[Crossref]

2016 (2)

2015 (2)

X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and R. G. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
[Crossref]

X. Xu, W. Huang, and M. Xu, “Orthogonal polynomials describing polarization aberration for rotationally symmetric optical systems,” Opt. Express 23(21), 27911–27919 (2015).
[Crossref] [PubMed]

2014 (3)

J. Li and E. Y. Lam, “Robust source and mask optimization compensating for mask topography effects in computational lithography,” Opt. Express 22(8), 9471–9485 (2014).
[Crossref] [PubMed]

X. Ma, J. Gao, C. Han, Y. Li, L. Dong, and L. Liu, “Efficient source polarization optimization for robust optical lithography,” Proc. SPIE 9052, 90520T (2014).
[Crossref]

L. Dong, Y. Li, X. Dai, H. Liu, and K. Liu, “Measuring the polarization aberration of hyper-NA lens from the vector aerial image,” Proc. SPIE 9283, 928313 (2014).
[Crossref]

2013 (3)

2012 (2)

2010 (1)

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

2009 (2)

J. Ruoff and M. Totzeck, “Orientation Zernike polynomials: a useful way to describe the polarizationeffects of optical imaging systems,” J. Microlith., Microfabr. Microsyst. 8, 031404 (2009).

Z. Qiu, X. Wang, Q. Yuan, and F. Wang, “Coma measurement by use of an alternating phase-shifting mask mark with a specific phase width,” Appl. Opt. 48(2), 261–269 (2009).
[Crossref] [PubMed]

2007 (1)

N. Yamamoto, J. Kye, and H. J. Levinson, “Polarization aberration analysis using Pauli-Zernike representation,” Proc. SPIE 6520, 65200Y (2007).
[Crossref]

2003 (1)

J. Haddadnia, M. Ahmadi, and K. Faez, “An efficient feature extraction method with pseudo-Zernike moment in RBF neural network-based human face recognition system,” EURASIP J. Adv. Signal Process. 2003(9), 267692 (2003).
[Crossref]

2001 (1)

H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[Crossref]

1994 (2)

Ahmadi, M.

J. Haddadnia, M. Ahmadi, and K. Faez, “An efficient feature extraction method with pseudo-Zernike moment in RBF neural network-based human face recognition system,” EURASIP J. Adv. Signal Process. 2003(9), 267692 (2003).
[Crossref]

Arce, G. R.

Arce, R. G.

X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and R. G. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
[Crossref]

Chipman, R. A.

Dai, X.

L. Dong, Y. Li, X. Dai, H. Liu, and K. Liu, “Measuring the polarization aberration of hyper-NA lens from the vector aerial image,” Proc. SPIE 9283, 928313 (2014).
[Crossref]

Dam, T.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Dierichs, M.

H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[Crossref]

Dong, L.

X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and R. G. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
[Crossref]

X. Ma, J. Gao, C. Han, Y. Li, L. Dong, and L. Liu, “Efficient source polarization optimization for robust optical lithography,” Proc. SPIE 9052, 90520T (2014).
[Crossref]

L. Dong, Y. Li, X. Dai, H. Liu, and K. Liu, “Measuring the polarization aberration of hyper-NA lens from the vector aerial image,” Proc. SPIE 9283, 928313 (2014).
[Crossref]

X. Ma, C. Han, Y. Li, L. Dong, and G. R. Arce, “Pixelated source and mask optimization for immersion lithography,” J. Opt. Soc. Am. A 30(1), 112–123 (2013).
[Crossref] [PubMed]

X. Ma, C. Han, Y. Li, B. Wu, Z. Song, L. Dong, and G. R. Arce, “Hybrid source mask optimization for robust immersion lithography,” Appl. Opt. 52(18), 4200–4211 (2013).
[Crossref] [PubMed]

X. Ma, Y. Li, and L. Dong, “Mask optimization approaches in optical lithography based on a vector imaging model,” J. Opt. Soc. Am. A 29(7), 1300–1312 (2012).
[Crossref] [PubMed]

Duan, L.

Faez, K.

J. Haddadnia, M. Ahmadi, and K. Faez, “An efficient feature extraction method with pseudo-Zernike moment in RBF neural network-based human face recognition system,” EURASIP J. Adv. Signal Process. 2003(9), 267692 (2003).
[Crossref]

Gao, J.

X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and R. G. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
[Crossref]

X. Ma, J. Gao, C. Han, Y. Li, L. Dong, and L. Liu, “Efficient source polarization optimization for robust optical lithography,” Proc. SPIE 9052, 90520T (2014).
[Crossref]

Guo, X.

Y. Li, X. Guo, X. Liu, and L. Liu, “A technique for extracting and analyzing the polarization aberration of hyper-numerical aperture image optics,” Proc. SPIE 9042, 904204 (2013).
[Crossref]

Haddadnia, J.

J. Haddadnia, M. Ahmadi, and K. Faez, “An efficient feature extraction method with pseudo-Zernike moment in RBF neural network-based human face recognition system,” EURASIP J. Adv. Signal Process. 2003(9), 267692 (2003).
[Crossref]

Han, C.

X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and R. G. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
[Crossref]

X. Ma, J. Gao, C. Han, Y. Li, L. Dong, and L. Liu, “Efficient source polarization optimization for robust optical lithography,” Proc. SPIE 9052, 90520T (2014).
[Crossref]

X. Ma, C. Han, Y. Li, L. Dong, and G. R. Arce, “Pixelated source and mask optimization for immersion lithography,” J. Opt. Soc. Am. A 30(1), 112–123 (2013).
[Crossref] [PubMed]

X. Ma, C. Han, Y. Li, B. Wu, Z. Song, L. Dong, and G. R. Arce, “Hybrid source mask optimization for robust immersion lithography,” Appl. Opt. 52(18), 4200–4211 (2013).
[Crossref] [PubMed]

Hu, P.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Huang, W.

Kye, J.

N. Yamamoto, J. Kye, and H. J. Levinson, “Polarization aberration analysis using Pauli-Zernike representation,” Proc. SPIE 6520, 65200Y (2007).
[Crossref]

Lam, E. Y.

Levinson, H. J.

N. Yamamoto, J. Kye, and H. J. Levinson, “Polarization aberration analysis using Pauli-Zernike representation,” Proc. SPIE 6520, 65200Y (2007).
[Crossref]

Li, J.

Li, S.

Li, Y.

Z. Xiang and Y. Li, “Retrieve polarization aberration from image degradation: a new measurement method in DUV lithography,” Proc. SPIE 10460, 84 (2017).
[Crossref]

X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and R. G. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
[Crossref]

L. Dong, Y. Li, X. Dai, H. Liu, and K. Liu, “Measuring the polarization aberration of hyper-NA lens from the vector aerial image,” Proc. SPIE 9283, 928313 (2014).
[Crossref]

X. Ma, J. Gao, C. Han, Y. Li, L. Dong, and L. Liu, “Efficient source polarization optimization for robust optical lithography,” Proc. SPIE 9052, 90520T (2014).
[Crossref]

Y. Li, X. Guo, X. Liu, and L. Liu, “A technique for extracting and analyzing the polarization aberration of hyper-numerical aperture image optics,” Proc. SPIE 9042, 904204 (2013).
[Crossref]

X. Ma, C. Han, Y. Li, B. Wu, Z. Song, L. Dong, and G. R. Arce, “Hybrid source mask optimization for robust immersion lithography,” Appl. Opt. 52(18), 4200–4211 (2013).
[Crossref] [PubMed]

X. Ma, C. Han, Y. Li, L. Dong, and G. R. Arce, “Pixelated source and mask optimization for immersion lithography,” J. Opt. Soc. Am. A 30(1), 112–123 (2013).
[Crossref] [PubMed]

X. Ma, Y. Li, and L. Dong, “Mask optimization approaches in optical lithography based on a vector imaging model,” J. Opt. Soc. Am. A 29(7), 1300–1312 (2012).
[Crossref] [PubMed]

Liu, H.

L. Dong, Y. Li, X. Dai, H. Liu, and K. Liu, “Measuring the polarization aberration of hyper-NA lens from the vector aerial image,” Proc. SPIE 9283, 928313 (2014).
[Crossref]

Liu, K.

L. Dong, Y. Li, X. Dai, H. Liu, and K. Liu, “Measuring the polarization aberration of hyper-NA lens from the vector aerial image,” Proc. SPIE 9283, 928313 (2014).
[Crossref]

Liu, L.

X. Ma, J. Gao, C. Han, Y. Li, L. Dong, and L. Liu, “Efficient source polarization optimization for robust optical lithography,” Proc. SPIE 9052, 90520T (2014).
[Crossref]

Y. Li, X. Guo, X. Liu, and L. Liu, “A technique for extracting and analyzing the polarization aberration of hyper-numerical aperture image optics,” Proc. SPIE 9042, 904204 (2013).
[Crossref]

Liu, S.

Liu, W.

Liu, X.

Y. Li, X. Guo, X. Liu, and L. Liu, “A technique for extracting and analyzing the polarization aberration of hyper-numerical aperture image optics,” Proc. SPIE 9042, 904204 (2013).
[Crossref]

Ma, X.

McCoo, E.

H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[Crossref]

McGuire, J. P.

Peng, D.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Pongers, R.

H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[Crossref]

Qiu, Z.

Ruoff, J.

J. Ruoff and M. Totzeck, “Orientation Zernike polynomials: a useful way to describe the polarizationeffects of optical imaging systems,” J. Microlith., Microfabr. Microsyst. 8, 031404 (2009).

Shen, L.

Slonaker, S.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Song, Z.

Stoffels, F.

H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[Crossref]

Tolani, V.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Totzeck, M.

J. Ruoff and M. Totzeck, “Orientation Zernike polynomials: a useful way to describe the polarizationeffects of optical imaging systems,” J. Microlith., Microfabr. Microsyst. 8, 031404 (2009).

Tyminski, J.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

van der Laan, H.

H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[Crossref]

van Greevenbroek, H.

H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[Crossref]

Wang, F.

Wang, X.

Willekers, R.

H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[Crossref]

Wu, B.

Wu, X.

Xiang, Z.

Z. Xiang and Y. Li, “Retrieve polarization aberration from image degradation: a new measurement method in DUV lithography,” Proc. SPIE 10460, 84 (2017).
[Crossref]

Xu, M.

Xu, S.

Xu, X.

Yamamoto, N.

N. Yamamoto, J. Kye, and H. J. Levinson, “Polarization aberration analysis using Pauli-Zernike representation,” Proc. SPIE 6520, 65200Y (2007).
[Crossref]

Yan, G.

Yuan, Q.

Zhu, B.

Appl. Opt. (5)

EURASIP J. Adv. Signal Process. (1)

J. Haddadnia, M. Ahmadi, and K. Faez, “An efficient feature extraction method with pseudo-Zernike moment in RBF neural network-based human face recognition system,” EURASIP J. Adv. Signal Process. 2003(9), 267692 (2003).
[Crossref]

J. Micro/Nanolith. MEMS MOEMS (1)

X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and R. G. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015).
[Crossref]

J. Microlith., Microfabr. Microsyst. (1)

J. Ruoff and M. Totzeck, “Orientation Zernike polynomials: a useful way to describe the polarizationeffects of optical imaging systems,” J. Microlith., Microfabr. Microsyst. 8, 031404 (2009).

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

Opt. Express (4)

Proc. SPIE (7)

H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[Crossref]

L. Dong, Y. Li, X. Dai, H. Liu, and K. Liu, “Measuring the polarization aberration of hyper-NA lens from the vector aerial image,” Proc. SPIE 9283, 928313 (2014).
[Crossref]

Z. Xiang and Y. Li, “Retrieve polarization aberration from image degradation: a new measurement method in DUV lithography,” Proc. SPIE 10460, 84 (2017).
[Crossref]

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

N. Yamamoto, J. Kye, and H. J. Levinson, “Polarization aberration analysis using Pauli-Zernike representation,” Proc. SPIE 6520, 65200Y (2007).
[Crossref]

Y. Li, X. Guo, X. Liu, and L. Liu, “A technique for extracting and analyzing the polarization aberration of hyper-numerical aperture image optics,” Proc. SPIE 9042, 904204 (2013).
[Crossref]

X. Ma, J. Gao, C. Han, Y. Li, L. Dong, and L. Liu, “Efficient source polarization optimization for robust optical lithography,” Proc. SPIE 9052, 90520T (2014).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 the impact mechanism of the PA on imaging.
Fig. 2
Fig. 2 Process of the hybrid genetic algorithm.
Fig. 3
Fig. 3 process of the simulation experiment.
Fig. 4
Fig. 4 PA of our designed immersion optics extracted from ray tracing by code V.
Fig. 5
Fig. 5 the comparison of the measurements with the true values. (a) is obtained under X-polarized illumination, and (b) is obtained under Y-polarized illumination.
Fig. 6
Fig. 6 the errors between the measurements and the true values.

Tables (2)

Tables Icon

Table 1 simulation parameters.

Tables Icon

Table 2 The RMSE of measured pupil and true pupil.

Equations (31)

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

I ( x a , y a ) = S ( f s , g s ) k = x , y , z | T ( f , g ) K f ; k ( f , g ) E i ( f s , g s ) e 2 π i { f x + g y + γ z } d f d g | 2 d f s d g s ,
K f ( f , g ) = A ( f , g ) M o ( f , g ) J ( f , g ) ,
I ( x a , y a ) = S ( f s , g s ) k = x , y , z | f , g = ± q i q ( f , g ) K f ; k ( f , g ) E i ( f s , g s ) e 2 π i { f x + g y + γ z } | 2 d f s d g s ,
T ( f , g ) = q 0 δ ( 0 , 0 ) + q 1 δ ( f + 1 , g + 1 ) + q 1 δ ( - f + 1 , - g + 1 ) + + q t δ ( t f + 1 , t g + 1 ) + q t δ ( - t f + 1 , - t g + 1 ) ,
{ t 2 f + 1 2 + t 2 g + 1 2 ( N A / λ 0 ) 2 ( t + 1 ) 2 f + 1 2 + ( t + 1 ) 2 g + 1 2 > ( N A / λ 0 ) 2 , t N + .
I ( x a , y a ) = S ( f s , g s ) k = x , y , z | q 0 K f ; k ( 0 , 0 ) E i ( f s , g s ) + q 1 K f ; k ( f + 1 , g + 1 ) E i ( f s , g s ) e 2 π i { f + 1 x + g + 1 y + γ z } + q 1 K f ; k ( - f + 1 , - g + 1 ) E i ( f s , g s ) e 2 π i { - f + 1 x - g + 1 y + γ z } + + q t K f ; k ( t f + 1 , t g + 1 ) E i ( f s , g s ) e 2 π i { t f + 1 x + t g + 1 y + γ z } + q t K f ; k ( - t f + 1 , - t g + 1 ) E i ( f s , g s ) e 2 π i { - t f + 1 x - t g + 1 y + γ z } | 2 d f s d g s .
S ( f s , g s ) = δ ( f s , g s ) ,
E i ( f s , g s ) = [ E i , x ( f s , g s ) E i , y ( f s , g s ) ] = { [ 1 , 0 ] T , X polarization [ 0 , 1 ] T , Y polarization .
I ( x a , y a ) = k = x , y , z | q 0 K f ; k ( 0 , 0 ) E i ( f s , g s ) + q 1 K f ; k ( f + 1 , g + 1 ) E i ( f s , g s ) e 2 π i { f + 1 x + g + 1 y + γ z } + q 1 K f ; k ( - f + 1 , - g + 1 ) E i ( f s , g s ) e 2 π i { - f + 1 x - g + 1 y + γ z } + + q t K f ; k ( t f + 1 , t g + 1 ) E i ( f s , g s ) e 2 π i { t f + 1 x + t g + 1 y + γ z } + q t K f ; k ( - t f + 1 , - t g + 1 ) E i ( f s , g s ) e 2 π i { - t f + 1 x - t g + 1 y + γ z } | 2 .
I ( x a , y a ) = I 1 + I 2 + I 3 = | C 11 a + C 12 b | 2 + | C 21 a + C 22 b | 2 + | C 31 a + C 32 b | 2 = [ a H , b H ] k = 1 , 2 , 3 [ C k 1 H C k 1 C k 1 H C k 2 C k 2 H C k 1 C k 2 H C k 2 ] [ a b ] = [ a H , b H ] S x p o l [ a b ] ,
{ a = [ a 1 , a 2 , , a j max ] T b = [ b 1 , b 2 , , b j max ] T , j max is the expasion order .
I ( x a , y a ) = I ' 1 + I ' 2 + I ' 3 = | C ' 11 a ' + C ' 12 b ' | 2 + | C ' 21 a ' + C ' 22 b ' | 2 + | C ' 31 a ' + C ' 32 b ' | 2 = [ a ' H , b ' H ] k = 1 , 2 , 3 [ C ' k 1 H C ' k 1 C ' k 1 H C ' k 2 C ' k 2 H C ' k 1 C ' k 2 H C ' k 2 ] [ a ' b ' ] = [ a ' H , b ' H ] S y p o l [ a ' b ' ] .
{ a ' = [ a ' 1 , a ' 2 , , a ' j max ] T b ' = [ b ' 1 , b ' 2 , , b ' j max ] T , j max is the expasion order .
I ( x a , y a ) = p i H S i p o l p i , w h e r e , i = x o r y , p x = [ a b ] , p y = [ a ' b ' ] .
I ˜ ( f , g ) = F { I ( x a , y a ) } = F { p i H S i p o l p i } , w h e r e , i = x o r y , p x = [ a b ] , p y = [ a ' b ' ] . = p i H F { S i p o l } p i = p i H S i - p o l p i
s n , m = s 0 ; n , m δ ( 0 , 0 ) + s + 1 ; n , m δ ( f + 1 , g + 1 ) + s 1 ; n , m δ ( - f + 1 , - g + 1 ) + s + 2 ; n , m δ ( 2 f + 1 , 2 g + 1 ) + s 2 ; n , m δ ( -2 f + 1 , - 2 g + 1 ) + .
I ˜ ( f l , g l ) = p i H S l ; i p o l p i ; where, i = x o r y , p x = [ a b ] , p y = [ a ' b ' ] , l = 0 , + 1 , 1 , + 2 , 2 .
K f ( f , g ) = A ( f , g ) M o ( f , g ) J ( f , g ) ,
A ( f , g ) = M n 1 1 ( M λ 0 f n 1 ) 2 ( M λ 0 g n 1 ) 2 n 2 1 ( λ 0 f n 2 ) 2 ( λ 0 g n 2 ) 2 ,
M o ( f , g ) = [ 1 α 2 1 + κ α β 1 + κ α β 1 + κ 1 β 2 1 + κ α β ] ,
{ α = λ 0 n 2 f β = λ 0 n 2 g κ = 1 α 2 β 2 ,
J ( f , g ) = { [ J x x ( f , g ) J x y ( f , g ) J y x ( f , g ) J y y ( f , g ) ] , f 2 + g 2 ( N A λ 0 ) 2 [ 0 0 0 0 ] .
P Z n m ( ρ , θ ) = R n | m | ( ρ ) exp { i m θ } .
J ( f , g ) = [ J x x ( f , g ) J x y ( f , g ) J y x ( f , g ) J y y ( f , g ) ] = [ A x x ( ρ , θ ) exp { i Θ x x ( ρ , θ ) } A x y ( ρ , θ ) exp { i Θ x y ( ρ , θ ) } A y x ( ρ , θ ) exp { i Θ y x ( ρ , θ ) } A y y ( ρ , θ ) exp { i Θ y y ( ρ , θ ) } ] = [ m , n a n m P Z n m ( ρ , θ ) m , n b ' n m P Z n m ( ρ , θ ) m , n b n m P Z n m ( ρ , θ ) m , n a ' n m P Z n m ( ρ , θ ) ] ,
{ a i , b i , a ' i , b ' i | i { 1 , 2 , 3 , } } .
{ C 11 ( i ) = A ( 0 , 0 ) δ m 0 q 0 ( -1 ) i 1 + A ( f + 1 , g + 1 ) C 1 ( f + 1 , g + 1 ) q 1 { Z i ( 1 ) E X P ( 1 ) + Z i ( - 1 ) E X P ( - 1 ) } + + A ( t f + 1 , t g + 1 ) C 1 ( t f + 1 , t g + 1 ) q t { Z i ( t ) E X P ( t ) + Z i ( - t ) E X P ( - t ) } C 21 ( i ) = A ( f + 1 , g + 1 ) C 2 ( f + 1 , g + 1 ) q 1 { Z i ( 1 ) E X P ( 1 ) + Z i ( - 1 ) E X P ( - 1 ) } + + A ( t f + 1 , t g + 1 ) C 2 ( t f + 1 , t g + 1 ) q t { Z i ( t ) E X P ( t ) + Z i ( - t ) E X P ( - t ) } C 31 ( i ) = A ( f + 1 , g + 1 ) C 3 ( f + 1 , g + 1 ) q 1 { Z i ( 1 ) E X P ( 1 ) + Z i ( - 1 ) E X P ( - 1 ) } + + A ( t f + 1 , t g + 1 ) C 3 ( t f + 1 , t g + 1 ) q t { Z i ( t ) E X P ( t ) + Z i ( - t ) E X P ( - t ) } ,
{ C 12 ( i ) = A ( f + 1 , g + 1 ) C 2 ( f + 1 , g + 1 ) q 1 { Z i ( 1 ) E X P ( 1 ) + Z i ( - 1 ) E X P ( - 1 ) } + + A ( t f + 1 , t g + 1 ) C 2 ( t f + 1 , t g + 1 ) q t { Z i ( t ) E X P ( t ) + Z i ( - t ) E X P ( - t ) } C 22 ( i ) = A ( 0 , 0 ) δ m 0 q 0 ( -1 ) i 1 + A ( f + 1 , g + 1 ) C 4 ( f + 1 , g + 1 ) q 1 { Z i ( 1 ) E X P ( 1 ) + Z i ( - 1 ) E X P ( - 1 ) } + + A ( t f + 1 , t g + 1 ) C 4 ( t f + 1 , t g + 1 ) q t { Z i ( t ) E X P ( t ) + Z i ( - t ) E X P ( - t ) } C 32 ( i ) = A ( f + 1 , g + 1 ) C 5 ( f + 1 , g + 1 ) q 1 { Z i ( 1 ) E X P ( 1 ) + Z i ( - 1 ) E X P ( - 1 ) } + + A ( t f + 1 , t g + 1 ) C 5 ( t f + 1 , t g + 1 ) q t { Z i ( t ) E X P ( t ) + Z i ( - t ) E X P ( - t ) } ,
i { 1 , 2 , , j max } ,
{ E X P ( t ) = e 2 π i { t f + 1 x + t g + 1 y + γ z } E X P ( - t ) = e 2 π i { - t f + 1 x - t g + 1 y + γ z } , t N + ,
{ Z i ( t ) = P Z n m ( t λ 0 f + 1 N A , t λ 0 g + 1 N A ) Z i ( - t ) = P Z n m ( - t λ 0 f + 1 N A , - t λ 0 g + 1 N A ) , t N + ,
{ C 1 ( f , g ) = 1 α 2 1 + κ C 2 ( f , g ) = α β 1 + κ C 4 ( f , g ) = 1 β 2 1 + κ ; { C 3 ( f , g ) = α C 5 ( f , g ) = β ,

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