Abstract

We propose an in situ aberration measurement technique based on an analytical linear model of through-focus aerial images. The aberrations are retrieved from aerial images of six isolated space patterns, which have the same width but different orientations. The imaging formulas of the space patterns are investigated and simplified, and then an analytical linear relationship between the aerial image intensity distributions and the Zernike coefficients is established. The linear relationship is composed of linear fitting matrices and rotation matrices, which can be calculated numerically in advance and utilized to retrieve Zernike coefficients. Numerical simulations using the lithography simulators PROLITH and Dr.LiTHO demonstrate that the proposed method can measure wavefront aberrations up to Z37. Experiments on a real lithography tool confirm that our method can monitor lens aberration offset with an accuracy of 0.7 nm.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  12. L. Duan, X. Wang, G. Yan, A. Y. Bourov, “Practical application of aerial image by principal component analysis to measure wavefront aberration of lithographic lens,”J. Micro/Nanolith. MEMS MOEMS. 11(2), 023009 (2012).
    [CrossRef]
  13. D. Xu, X. Wang, Y. Bu, L. Duan, G. Yan, J. Yang, A. Y. Bourov, “In situ aberration measurement technique based on multi-illumination settings and principal component analysis of aerial images,” Chin. Opt. Lett. 10, 121202 (2012).
    [CrossRef]
  14. C. A. Mack, “Lithography simulation in semiconductor manufacturing,” Proc. SPIE 5645, 63–83 (2005).
    [CrossRef]
  15. T. Fühner, T. Schnattinger, G. Ardelean, A. Erdmann, “Dr. LiTHO - a development and research lithography simulator,” Proc. SPIE 6520, 65203F (2007).
    [CrossRef]
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  19. S. van Haver, O. T. A. Janssen, J. J. M. Braat, A. J. E. M. Janssen, H. P. Urbach, S. F. Pereira, “General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory,” Proc. SPIE 6924, 69240U (2008).
    [CrossRef]
  20. A. J. E. M. Janssen, “Extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19(5), 849–857 (2002).
    [CrossRef] [PubMed]
  21. J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19(5), 858–870 (2002).
    [CrossRef] [PubMed]
  22. C. van der Avoort, J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer-Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
    [CrossRef]
  23. L. Duan, J. Cheng, G. Sun, Y. Chen, “New 0.75 NA ArF scanning lithographic tool,” Proc. SPIE 7973, 79732D (2011).
    [CrossRef]

2012 (2)

L. Duan, X. Wang, G. Yan, A. Y. Bourov, “Practical application of aerial image by principal component analysis to measure wavefront aberration of lithographic lens,”J. Micro/Nanolith. MEMS MOEMS. 11(2), 023009 (2012).
[CrossRef]

D. Xu, X. Wang, Y. Bu, L. Duan, G. Yan, J. Yang, A. Y. Bourov, “In situ aberration measurement technique based on multi-illumination settings and principal component analysis of aerial images,” Chin. Opt. Lett. 10, 121202 (2012).
[CrossRef]

2011 (2)

2010 (2)

2009 (2)

2008 (1)

S. van Haver, O. T. A. Janssen, J. J. M. Braat, A. J. E. M. Janssen, H. P. Urbach, S. F. Pereira, “General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory,” Proc. SPIE 6924, 69240U (2008).
[CrossRef]

2007 (2)

2006 (2)

2005 (3)

C. van der Avoort, J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer-Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

T. Hagiwara, N. Kondo, I. Hiroshi, K. Suzuki, N. Magome, “Development of aerial image based aberration measurement technique,” Proc. SPIE 5754, 1659–1669 (2005).
[CrossRef]

C. A. Mack, “Lithography simulation in semiconductor manufacturing,” Proc. SPIE 5645, 63–83 (2005).
[CrossRef]

2003 (1)

P. Graeupner, R. Garreis, A. Goehnermeiter, T. Heil, M. Lowisch, D. Flagello, “Impact of wavefront errors on low k1 processes at extreme high NA,” Proc. SPIE 5040, 119–130 (2003).
[CrossRef]

2002 (2)

2001 (1)

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

Ardelean, G.

T. Fühner, T. Schnattinger, G. Ardelean, A. Erdmann, “Dr. LiTHO - a development and research lithography simulator,” Proc. SPIE 6520, 65203F (2007).
[CrossRef]

Bourov, A. Y.

L. Duan, X. Wang, G. Yan, A. Y. Bourov, “Practical application of aerial image by principal component analysis to measure wavefront aberration of lithographic lens,”J. Micro/Nanolith. MEMS MOEMS. 11(2), 023009 (2012).
[CrossRef]

D. Xu, X. Wang, Y. Bu, L. Duan, G. Yan, J. Yang, A. Y. Bourov, “In situ aberration measurement technique based on multi-illumination settings and principal component analysis of aerial images,” Chin. Opt. Lett. 10, 121202 (2012).
[CrossRef]

L. Duan, X. Wang, A. Y. Bourov, B. Peng, P. Bu, “In situ aberration measurement technique based on principal component analysis of aerial image,” Opt. Express 19(19), 18080–18090 (2011).
[CrossRef] [PubMed]

A. Y. Bourov, L. Li, Z. Yang, F. Wang, L. Duan, “Aerial image model and application to aberration measurement,” Proc. SPIE 7640, 764032 (2010).
[CrossRef]

Braat, J. J. M.

S. van Haver, O. T. A. Janssen, J. J. M. Braat, A. J. E. M. Janssen, H. P. Urbach, S. F. Pereira, “General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory,” Proc. SPIE 6924, 69240U (2008).
[CrossRef]

C. van der Avoort, J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer-Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19(5), 858–870 (2002).
[CrossRef] [PubMed]

Bu, P.

Bu, Y.

Chen, Y.

L. Duan, J. Cheng, G. Sun, Y. Chen, “New 0.75 NA ArF scanning lithographic tool,” Proc. SPIE 7973, 79732D (2011).
[CrossRef]

Cheng, J.

L. Duan, J. Cheng, G. Sun, Y. Chen, “New 0.75 NA ArF scanning lithographic tool,” Proc. SPIE 7973, 79732D (2011).
[CrossRef]

Dierichs, M.

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

Dirksen, P.

C. van der Avoort, J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer-Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19(5), 858–870 (2002).
[CrossRef] [PubMed]

Duan, L.

L. Duan, X. Wang, G. Yan, A. Y. Bourov, “Practical application of aerial image by principal component analysis to measure wavefront aberration of lithographic lens,”J. Micro/Nanolith. MEMS MOEMS. 11(2), 023009 (2012).
[CrossRef]

D. Xu, X. Wang, Y. Bu, L. Duan, G. Yan, J. Yang, A. Y. Bourov, “In situ aberration measurement technique based on multi-illumination settings and principal component analysis of aerial images,” Chin. Opt. Lett. 10, 121202 (2012).
[CrossRef]

L. Duan, X. Wang, A. Y. Bourov, B. Peng, P. Bu, “In situ aberration measurement technique based on principal component analysis of aerial image,” Opt. Express 19(19), 18080–18090 (2011).
[CrossRef] [PubMed]

L. Duan, J. Cheng, G. Sun, Y. Chen, “New 0.75 NA ArF scanning lithographic tool,” Proc. SPIE 7973, 79732D (2011).
[CrossRef]

A. Y. Bourov, L. Li, Z. Yang, F. Wang, L. Duan, “Aerial image model and application to aberration measurement,” Proc. SPIE 7640, 764032 (2010).
[CrossRef]

Erdmann, A.

T. Fühner, T. Schnattinger, G. Ardelean, A. Erdmann, “Dr. LiTHO - a development and research lithography simulator,” Proc. SPIE 6520, 65203F (2007).
[CrossRef]

Flagello, D.

P. Graeupner, R. Garreis, A. Goehnermeiter, T. Heil, M. Lowisch, D. Flagello, “Impact of wavefront errors on low k1 processes at extreme high NA,” Proc. SPIE 5040, 119–130 (2003).
[CrossRef]

Fühner, T.

T. Fühner, T. Schnattinger, G. Ardelean, A. Erdmann, “Dr. LiTHO - a development and research lithography simulator,” Proc. SPIE 6520, 65203F (2007).
[CrossRef]

Garreis, R.

P. Graeupner, R. Garreis, A. Goehnermeiter, T. Heil, M. Lowisch, D. Flagello, “Impact of wavefront errors on low k1 processes at extreme high NA,” Proc. SPIE 5040, 119–130 (2003).
[CrossRef]

Goehnermeiter, A.

P. Graeupner, R. Garreis, A. Goehnermeiter, T. Heil, M. Lowisch, D. Flagello, “Impact of wavefront errors on low k1 processes at extreme high NA,” Proc. SPIE 5040, 119–130 (2003).
[CrossRef]

Graeupner, P.

P. Graeupner, R. Garreis, A. Goehnermeiter, T. Heil, M. Lowisch, D. Flagello, “Impact of wavefront errors on low k1 processes at extreme high NA,” Proc. SPIE 5040, 119–130 (2003).
[CrossRef]

Hagiwara, T.

J. K. Tyminski, T. Hagiwara, N. Kondo, H. Irihama, “Aerial image sensor: in-situ scanner aberration monitor,” Proc. SPIE 6152, 61523D (2006).
[CrossRef]

T. Hagiwara, N. Kondo, I. Hiroshi, K. Suzuki, N. Magome, “Development of aerial image based aberration measurement technique,” Proc. SPIE 5754, 1659–1669 (2005).
[CrossRef]

He, L.

Heil, T.

P. Graeupner, R. Garreis, A. Goehnermeiter, T. Heil, M. Lowisch, D. Flagello, “Impact of wavefront errors on low k1 processes at extreme high NA,” Proc. SPIE 5040, 119–130 (2003).
[CrossRef]

Hiroshi, I.

T. Hagiwara, N. Kondo, I. Hiroshi, K. Suzuki, N. Magome, “Development of aerial image based aberration measurement technique,” Proc. SPIE 5754, 1659–1669 (2005).
[CrossRef]

Hu, J.

Irihama, H.

J. K. Tyminski, T. Hagiwara, N. Kondo, H. Irihama, “Aerial image sensor: in-situ scanner aberration monitor,” Proc. SPIE 6152, 61523D (2006).
[CrossRef]

Janssen, A. J. E. M.

S. van Haver, O. T. A. Janssen, J. J. M. Braat, A. J. E. M. Janssen, H. P. Urbach, S. F. Pereira, “General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory,” Proc. SPIE 6924, 69240U (2008).
[CrossRef]

C. van der Avoort, J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer-Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19(5), 858–870 (2002).
[CrossRef] [PubMed]

A. J. E. M. Janssen, “Extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19(5), 849–857 (2002).
[CrossRef] [PubMed]

Janssen, O. T. A.

S. van Haver, O. T. A. Janssen, J. J. M. Braat, A. J. E. M. Janssen, H. P. Urbach, S. F. Pereira, “General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory,” Proc. SPIE 6924, 69240U (2008).
[CrossRef]

Kondo, N.

J. K. Tyminski, T. Hagiwara, N. Kondo, H. Irihama, “Aerial image sensor: in-situ scanner aberration monitor,” Proc. SPIE 6152, 61523D (2006).
[CrossRef]

T. Hagiwara, N. Kondo, I. Hiroshi, K. Suzuki, N. Magome, “Development of aerial image based aberration measurement technique,” Proc. SPIE 5754, 1659–1669 (2005).
[CrossRef]

Li, L.

A. Y. Bourov, L. Li, Z. Yang, F. Wang, L. Duan, “Aerial image model and application to aberration measurement,” Proc. SPIE 7640, 764032 (2010).
[CrossRef]

Liu, S.

Liu, W.

Lowisch, M.

P. Graeupner, R. Garreis, A. Goehnermeiter, T. Heil, M. Lowisch, D. Flagello, “Impact of wavefront errors on low k1 processes at extreme high NA,” Proc. SPIE 5040, 119–130 (2003).
[CrossRef]

Ma, M.

Mack, C. A.

C. A. Mack, “Lithography simulation in semiconductor manufacturing,” Proc. SPIE 5645, 63–83 (2005).
[CrossRef]

Magome, N.

T. Hagiwara, N. Kondo, I. Hiroshi, K. Suzuki, N. Magome, “Development of aerial image based aberration measurement technique,” Proc. SPIE 5754, 1659–1669 (2005).
[CrossRef]

McCoo, E.

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

Peng, B.

Pereira, S. F.

S. van Haver, O. T. A. Janssen, J. J. M. Braat, A. J. E. M. Janssen, H. P. Urbach, S. F. Pereira, “General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory,” Proc. SPIE 6924, 69240U (2008).
[CrossRef]

Pongers, R.

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

Qiu, Z.

Schnattinger, T.

T. Fühner, T. Schnattinger, G. Ardelean, A. Erdmann, “Dr. LiTHO - a development and research lithography simulator,” Proc. SPIE 6520, 65203F (2007).
[CrossRef]

Shi, T.

Shi, W.

Stoffels, F.

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

Sun, G.

L. Duan, J. Cheng, G. Sun, Y. Chen, “New 0.75 NA ArF scanning lithographic tool,” Proc. SPIE 7973, 79732D (2011).
[CrossRef]

Suzuki, K.

T. Hagiwara, N. Kondo, I. Hiroshi, K. Suzuki, N. Magome, “Development of aerial image based aberration measurement technique,” Proc. SPIE 5754, 1659–1669 (2005).
[CrossRef]

Tang, Z.

Tyminski, J. K.

J. K. Tyminski, T. Hagiwara, N. Kondo, H. Irihama, “Aerial image sensor: in-situ scanner aberration monitor,” Proc. SPIE 6152, 61523D (2006).
[CrossRef]

Urbach, H. P.

S. van Haver, O. T. A. Janssen, J. J. M. Braat, A. J. E. M. Janssen, H. P. Urbach, S. F. Pereira, “General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory,” Proc. SPIE 6924, 69240U (2008).
[CrossRef]

van der Avoort, C.

C. van der Avoort, J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer-Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

van der Laan, H.

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

van Haver, S.

S. van Haver, O. T. A. Janssen, J. J. M. Braat, A. J. E. M. Janssen, H. P. Urbach, S. F. Pereira, “General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory,” Proc. SPIE 6924, 69240U (2008).
[CrossRef]

Wang, F.

Wang, L.

Wang, X.

Willekers, R.

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

Xu, D.

Yan, G.

D. Xu, X. Wang, Y. Bu, L. Duan, G. Yan, J. Yang, A. Y. Bourov, “In situ aberration measurement technique based on multi-illumination settings and principal component analysis of aerial images,” Chin. Opt. Lett. 10, 121202 (2012).
[CrossRef]

L. Duan, X. Wang, G. Yan, A. Y. Bourov, “Practical application of aerial image by principal component analysis to measure wavefront aberration of lithographic lens,”J. Micro/Nanolith. MEMS MOEMS. 11(2), 023009 (2012).
[CrossRef]

Yang, J.

Yang, Z.

A. Y. Bourov, L. Li, Z. Yang, F. Wang, L. Duan, “Aerial image model and application to aberration measurement,” Proc. SPIE 7640, 764032 (2010).
[CrossRef]

Yuan, Q.

Zhang, D.

Zhou, T.

Appl. Opt. (2)

Chin. Opt. Lett. (1)

J. Micro/Nanolith. MEMS MOEMS. (1)

L. Duan, X. Wang, G. Yan, A. Y. Bourov, “Practical application of aerial image by principal component analysis to measure wavefront aberration of lithographic lens,”J. Micro/Nanolith. MEMS MOEMS. 11(2), 023009 (2012).
[CrossRef]

J. Mod. Opt. (1)

C. van der Avoort, J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer-Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

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

Opt. Express (4)

Proc. SPIE (9)

L. Duan, J. Cheng, G. Sun, Y. Chen, “New 0.75 NA ArF scanning lithographic tool,” Proc. SPIE 7973, 79732D (2011).
[CrossRef]

S. van Haver, O. T. A. Janssen, J. J. M. Braat, A. J. E. M. Janssen, H. P. Urbach, S. F. Pereira, “General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory,” Proc. SPIE 6924, 69240U (2008).
[CrossRef]

A. Y. Bourov, L. Li, Z. Yang, F. Wang, L. Duan, “Aerial image model and application to aberration measurement,” Proc. SPIE 7640, 764032 (2010).
[CrossRef]

P. Graeupner, R. Garreis, A. Goehnermeiter, T. Heil, M. Lowisch, D. Flagello, “Impact of wavefront errors on low k1 processes at extreme high NA,” Proc. SPIE 5040, 119–130 (2003).
[CrossRef]

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

T. Hagiwara, N. Kondo, I. Hiroshi, K. Suzuki, N. Magome, “Development of aerial image based aberration measurement technique,” Proc. SPIE 5754, 1659–1669 (2005).
[CrossRef]

J. K. Tyminski, T. Hagiwara, N. Kondo, H. Irihama, “Aerial image sensor: in-situ scanner aberration monitor,” Proc. SPIE 6152, 61523D (2006).
[CrossRef]

C. A. Mack, “Lithography simulation in semiconductor manufacturing,” Proc. SPIE 5645, 63–83 (2005).
[CrossRef]

T. Fühner, T. Schnattinger, G. Ardelean, A. Erdmann, “Dr. LiTHO - a development and research lithography simulator,” Proc. SPIE 6520, 65203F (2007).
[CrossRef]

Other (3)

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

C. A. Mack, Fundamental Principles of Optical Lithography: The Science of Microfabrication (John Wiley & Sons Ltd, 2007).

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems: Physical Image Formation (Wiley-VCH Verlag GmbH & Co. KGaA, 2008).

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Figures (9)

Fig. 1
Fig. 1

(a) Schematic diagram of (a) a lithography imaging system; (b) test marks.

Fig. 2
Fig. 2

Schematic diagram of (a) pupil sampling in the pupil plane and (b) aerial image scanning.

Fig. 3
Fig. 3

Distributions of the linear fitting matrices for the typical Zernike coefficients.

Fig. 4
Fig. 4

(a) Values of (S′S); (b) common logarithms of absolute values of (S′S).

Fig. 5
Fig. 5

(a) Aerial image simulated by the simulator; (b) aerial image calculated using the linear model in our method; (c) the differences between aerial images in (a) and (b).

Fig. 6
Fig. 6

Input and retrieved Zernike coefficients.

Fig. 7
Fig. 7

Comparison between the input and rebuilt wavefronts.

Fig. 8
Fig. 8

Measurement accuracy of the Zernike coefficients.

Fig. 9
Fig. 9

Z8 offset measurement across the exposure FPs.

Equations (17)

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I φ ( x , z ) = + J ( f , g ) I coh φ ( f , g ; x , z ) d f d g ,
I coh φ ( f , g ; x , z ) = | E coh φ ( f , g ; x , z ) | 2 .
E coh φ ( f , g ; x , z ) = 1 f 1 f O ( f ' ) exp { i k W φ ( f + f ' , g ) } exp { i 2 π k z p z } exp ( i 2 π f ' x ) d f ' ,
W φ ( f , g ) = j = 1 37 Z j F j ( ρ , θ + φ ) ,
F j ( ρ , θ ) = R n m ( ρ ) Φ j m ( θ ) ,
W φ = j = 1 37 Z j F j ( ρ , θ + φ ) = c s 1 R n 0 ( ρ ) + c t 2 R n m ( ρ ) cos ( m ( θ + φ ) ) + c t 3 R n m ( ρ ) sin ( m ( θ + φ ) ) .
W φ = c s 1 R n 0 ( ρ ) + [ c t 2 cos ( m φ ) + c t 3 sin ( m φ ) ] R n m ( ρ ) cos ( m θ ) + [ c t 3 cos ( m φ ) c t 2 sin ( m φ ) ] R n m ( ρ ) sin ( m θ ) = j = 1 37 Z j ' F j ( ρ , θ ) .
Z ' = Q Z ,
exp ( i k W ) 1 + i k W = 1 + i k j = 1 37 Z j F j ( ρ , θ ) .
E coh ( f , g ; x , z ) E c + i j Z j E j ,
{ E c = 1 f 1 f O ( f ' ) exp { i k 1 [ ( f + f ' ) 2 + g 2 ] N A 2 z } exp ( i 2 π f ' x ) d f ' E j = k 1 f 1 f F j ( ρ , θ ) O ( f ' ) exp { i k 1 [ ( f + f ' ) 2 + g 2 ] N A 2 z } exp { i 2 π f ' x } d f ' ,
I coh ( f , g ; x , z ) | E c | 2 + j = 1 37 Z j 2 Re { i E c * E j } .
I ( x , z ) I 0 ( x , z ) + j = 1 37 Z j T j ( x , z ) ,
{ I 0 ( x , z ) = + J ( f , g ) | E c | 2 d f d g T j ( x , z ) = + J ( f , g ) 2 Re { i E c * E j } d f d g .
b = [ I 1 ( : ) I 0 ( : ) I 2 ( : ) I 0 ( : ) I 6 ( : ) I 0 ( : ) ] , S = [ T Q 1 T Q 2 T Q 6 ] ,
b = S Z .
Z = ( S ' S ) 1 ( S ' b ) .

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