Abstract

A simultaneous multi-channel absolute position alignment system is investigated to determine the absolute position of a grating mark. By employing a multi-order grating interferometer and a multi-channel phase extraction method, many equivalent measurement results are generated simultaneously for stable and consistent measurement. By combining measurement results of different orders, low-order signals enabled large unambiguous measurement ranges, and high-order signals enhanced the measurement accuracy. Comparison experiments performed using an incremental HeNe reference interferometer yielded the standard deviations of smaller than 11.48nm under laboratory conditions. The proposed scheme will enable a new class of absolute position alignment system for industrial applications.

© 2016 Optical Society of America

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References

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2015 (2)

2014 (1)

2013 (3)

A. J. Fleming, “A review of nanometer resolution position sensors: Operation and performance,” Sens. Actuators A Phys. 190, 106–126 (2013).
[Crossref]

S. Zhou, C. Xie, Y. Yang, S. Hu, X. Xu, and J. Yang, “Moiré-based phase imaging for sensing and adjustment of in-plane twist angle,” IEEE Photon. Technol. Lett. 25(18), 1847–1850 (2013).
[Crossref]

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[Crossref]

2012 (4)

J. P. Zhu, S. Hu, J. S. Yu, and Y. Tang, “Alignment method based on matched dual-grating moiré fringe for proximity lithography,” Opt. Eng. 51(11), 113603 (2012).
[Crossref]

E. Manske, G. Jäger, T. Hausotte, and R. Füß, “Recent developments and challenges of nanopositioning and nanomeasuring technology,” Meas. Sci. Technol. 23(7), 074001 (2012).
[Crossref]

G. Berkovic and E. Shafir, “Optical methods for distance and displacement measurements,” Adv. Opt. Photonics 4(4), 441–471 (2012).
[Crossref]

M. C. Leibovici, G. M. Burrow, and T. K. Gaylord, “Pattern-integrated interference lithography: prospects for nano- and microelectronics,” Opt. Express 20(21), 23643–23652 (2012).
[Crossref] [PubMed]

2010 (1)

C. Wagner and N. Harned, “EUV lithography: Lithography gets extreme,” Nat. Photonics 4(1), 24–26 (2010).
[Crossref]

2009 (1)

2008 (2)

2005 (3)

2002 (1)

G. Pugh and M. Giorgi, “Evaluation of ASML ATHENA alignment system on Intel front-end processes,” Proc. SPIE 4689, 286–294 (2002).
[Crossref]

2001 (1)

R. Navarro, S. Keij, A. J. den Boef, S. Schets, F. van Bilsen, G. Simons, R. Schuurhuis, and J. Burghoorn, “Extended ATHENA alignment performance and application for the 100-nm technology node,” Proc. SPIE 4344, 682–694 (2001).
[Crossref]

1989 (1)

M. Suzuki and A. Une, “An optical-heterodyne alignment technique for quarter-micron x-ray lithography,” J. Vac. Sci. Technol. B 7(6), 1971–1976 (1989).
[Crossref]

1988 (1)

1986 (1)

X. Yang and C. Yin, “A new method for the design of zero reference marks for grating measurement systems,” J. Phys. E Sci. Instrum. 19(1), 34–37 (1986).
[Crossref]

1979 (1)

G. Bouwhuis and S. Wittekoek, “Automatic alignment system for optical projection printing,” IEEE Trans. Electron Devices ED 26(4), 723–728 (1979).
[Crossref]

1977 (1)

D. C. Flanders, H. I. Smith, and S. Austin, “A new interferometric alignment technique,” Appl. Phys. Lett. 31(7), 426–428 (1977).
[Crossref]

Abou-Zeid, A.

Alonso, J.

Austin, S.

D. C. Flanders, H. I. Smith, and S. Austin, “A new interferometric alignment technique,” Appl. Phys. Lett. 31(7), 426–428 (1977).
[Crossref]

Berkovic, G.

G. Berkovic and E. Shafir, “Optical methods for distance and displacement measurements,” Adv. Opt. Photonics 4(4), 441–471 (2012).
[Crossref]

Bernabeu, E.

Bosse, H.

Bouwhuis, G.

G. Bouwhuis and S. Wittekoek, “Automatic alignment system for optical projection printing,” IEEE Trans. Electron Devices ED 26(4), 723–728 (1979).
[Crossref]

Burghoorn, J.

R. Navarro, S. Keij, A. J. den Boef, S. Schets, F. van Bilsen, G. Simons, R. Schuurhuis, and J. Burghoorn, “Extended ATHENA alignment performance and application for the 100-nm technology node,” Proc. SPIE 4344, 682–694 (2001).
[Crossref]

Burrow, G. M.

Chen, W.

Chu, H.

Constant, K.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86(20), 204101 (2005).
[Crossref]

Cui, J.

den Boef, A. J.

R. Navarro, S. Keij, A. J. den Boef, S. Schets, F. van Bilsen, G. Simons, R. Schuurhuis, and J. Burghoorn, “Extended ATHENA alignment performance and application for the 100-nm technology node,” Proc. SPIE 4344, 682–694 (2001).
[Crossref]

Flanders, D. C.

D. C. Flanders, H. I. Smith, and S. Austin, “A new interferometric alignment technique,” Appl. Phys. Lett. 31(7), 426–428 (1977).
[Crossref]

Fleming, A. J.

A. J. Fleming, “A review of nanometer resolution position sensors: Operation and performance,” Sens. Actuators A Phys. 190, 106–126 (2013).
[Crossref]

Fu, Y.

Füß, R.

E. Manske, G. Jäger, T. Hausotte, and R. Füß, “Recent developments and challenges of nanopositioning and nanomeasuring technology,” Meas. Sci. Technol. 23(7), 074001 (2012).
[Crossref]

Gallagher, N. C.

Gaylord, T. K.

Giorgi, M.

G. Pugh and M. Giorgi, “Evaluation of ASML ATHENA alignment system on Intel front-end processes,” Proc. SPIE 4689, 286–294 (2002).
[Crossref]

Harned, N.

C. Wagner and N. Harned, “EUV lithography: Lithography gets extreme,” Nat. Photonics 4(1), 24–26 (2010).
[Crossref]

Hausotte, T.

E. Manske, G. Jäger, T. Hausotte, and R. Füß, “Recent developments and challenges of nanopositioning and nanomeasuring technology,” Meas. Sci. Technol. 23(7), 074001 (2012).
[Crossref]

He, Y.

J. P. Zhu, S. Hu, J. S. Yu, Y. Tang, F. Xu, Y. He, S. L. Zhou, and L. L. Li, “Influence of tilt moiré fringe on alignment accuracy in proximity lithography,” Opt. Lasers Eng. 51(4), 371–381 (2013).
[Crossref]

Ho, K.-M.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86(20), 204101 (2005).
[Crossref]

Hu, S.

F. Xu, S. Zhou, S. Hu, W. Jiang, L. Luo, and H. Chu, “Moiré fringe alignment using composite circular-line gratings for proximity lithography,” Opt. Express 23(16), 20905–20915 (2015).
[Crossref] [PubMed]

S. Zhou, C. Xie, Y. Yang, S. Hu, X. Xu, and J. Yang, “Moiré-based phase imaging for sensing and adjustment of in-plane twist angle,” IEEE Photon. Technol. Lett. 25(18), 1847–1850 (2013).
[Crossref]

J. P. Zhu, S. Hu, J. S. Yu, Y. Tang, F. Xu, Y. He, S. L. Zhou, and L. L. Li, “Influence of tilt moiré fringe on alignment accuracy in proximity lithography,” Opt. Lasers Eng. 51(4), 371–381 (2013).
[Crossref]

J. P. Zhu, S. Hu, J. S. Yu, and Y. Tang, “Alignment method based on matched dual-grating moiré fringe for proximity lithography,” Opt. Eng. 51(11), 113603 (2012).
[Crossref]

S. Zhou, Y. Fu, X. Tang, S. Hu, W. Chen, and Y. Yang, “Fourier-based analysis of moiré fringe patterns of superposed gratings in alignment of nanolithography,” Opt. Express 16(11), 7869–7880 (2008).
[Crossref] [PubMed]

Huang, L.

Jäger, G.

E. Manske, G. Jäger, T. Hausotte, and R. Füß, “Recent developments and challenges of nanopositioning and nanomeasuring technology,” Meas. Sci. Technol. 23(7), 074001 (2012).
[Crossref]

Jiang, W.

Keij, S.

R. Navarro, S. Keij, A. J. den Boef, S. Schets, F. van Bilsen, G. Simons, R. Schuurhuis, and J. Burghoorn, “Extended ATHENA alignment performance and application for the 100-nm technology node,” Proc. SPIE 4344, 682–694 (2001).
[Crossref]

Kim, C.-H.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86(20), 204101 (2005).
[Crossref]

Kim, Y.-S.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86(20), 204101 (2005).
[Crossref]

Lee, J.-H.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86(20), 204101 (2005).
[Crossref]

Leibovici, M. C.

Leung, W.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86(20), 204101 (2005).
[Crossref]

Li, L. L.

J. P. Zhu, S. Hu, J. S. Yu, Y. Tang, F. Xu, Y. He, S. L. Zhou, and L. L. Li, “Influence of tilt moiré fringe on alignment accuracy in proximity lithography,” Opt. Lasers Eng. 51(4), 371–381 (2013).
[Crossref]

Luo, L.

Manske, E.

E. Manske, G. Jäger, T. Hausotte, and R. Füß, “Recent developments and challenges of nanopositioning and nanomeasuring technology,” Meas. Sci. Technol. 23(7), 074001 (2012).
[Crossref]

Meiners-Hagen, K.

Navarro, R.

R. Navarro, S. Keij, A. J. den Boef, S. Schets, F. van Bilsen, G. Simons, R. Schuurhuis, and J. Burghoorn, “Extended ATHENA alignment performance and application for the 100-nm technology node,” Proc. SPIE 4344, 682–694 (2001).
[Crossref]

O’Brien, D. J.

Oh, C.-H.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86(20), 204101 (2005).
[Crossref]

Pollinger, F.

Prather, D.

Pugh, G.

G. Pugh and M. Giorgi, “Evaluation of ASML ATHENA alignment system on Intel front-end processes,” Proc. SPIE 4689, 286–294 (2002).
[Crossref]

Sáez-Landete, J.

Schets, S.

R. Navarro, S. Keij, A. J. den Boef, S. Schets, F. van Bilsen, G. Simons, R. Schuurhuis, and J. Burghoorn, “Extended ATHENA alignment performance and application for the 100-nm technology node,” Proc. SPIE 4344, 682–694 (2001).
[Crossref]

Schneider, G.

Schuurhuis, R.

R. Navarro, S. Keij, A. J. den Boef, S. Schets, F. van Bilsen, G. Simons, R. Schuurhuis, and J. Burghoorn, “Extended ATHENA alignment performance and application for the 100-nm technology node,” Proc. SPIE 4344, 682–694 (2001).
[Crossref]

Shafir, E.

G. Berkovic and E. Shafir, “Optical methods for distance and displacement measurements,” Adv. Opt. Photonics 4(4), 441–471 (2012).
[Crossref]

Simons, G.

R. Navarro, S. Keij, A. J. den Boef, S. Schets, F. van Bilsen, G. Simons, R. Schuurhuis, and J. Burghoorn, “Extended ATHENA alignment performance and application for the 100-nm technology node,” Proc. SPIE 4344, 682–694 (2001).
[Crossref]

Smith, H. I.

D. C. Flanders, H. I. Smith, and S. Austin, “A new interferometric alignment technique,” Appl. Phys. Lett. 31(7), 426–428 (1977).
[Crossref]

Su, X.

Suzuki, M.

M. Suzuki and A. Une, “An optical-heterodyne alignment technique for quarter-micron x-ray lithography,” J. Vac. Sci. Technol. B 7(6), 1971–1976 (1989).
[Crossref]

Tan, J.

Tang, X.

Tang, Y.

J. P. Zhu, S. Hu, J. S. Yu, Y. Tang, F. Xu, Y. He, S. L. Zhou, and L. L. Li, “Influence of tilt moiré fringe on alignment accuracy in proximity lithography,” Opt. Lasers Eng. 51(4), 371–381 (2013).
[Crossref]

J. P. Zhu, S. Hu, J. S. Yu, and Y. Tang, “Alignment method based on matched dual-grating moiré fringe for proximity lithography,” Opt. Eng. 51(11), 113603 (2012).
[Crossref]

Tao, Z.

Une, A.

M. Suzuki and A. Une, “An optical-heterodyne alignment technique for quarter-micron x-ray lithography,” J. Vac. Sci. Technol. B 7(6), 1971–1976 (1989).
[Crossref]

van Bilsen, F.

R. Navarro, S. Keij, A. J. den Boef, S. Schets, F. van Bilsen, G. Simons, R. Schuurhuis, and J. Burghoorn, “Extended ATHENA alignment performance and application for the 100-nm technology node,” Proc. SPIE 4344, 682–694 (2001).
[Crossref]

Wagner, C.

C. Wagner and N. Harned, “EUV lithography: Lithography gets extreme,” Nat. Photonics 4(1), 24–26 (2010).
[Crossref]

Wedde, M.

Wetzel, E.

Whang, A. J.

Wittekoek, S.

G. Bouwhuis and S. Wittekoek, “Automatic alignment system for optical projection printing,” IEEE Trans. Electron Devices ED 26(4), 723–728 (1979).
[Crossref]

Xie, C.

S. Zhou, C. Xie, Y. Yang, S. Hu, X. Xu, and J. Yang, “Moiré-based phase imaging for sensing and adjustment of in-plane twist angle,” IEEE Photon. Technol. Lett. 25(18), 1847–1850 (2013).
[Crossref]

Xu, F.

F. Xu, S. Zhou, S. Hu, W. Jiang, L. Luo, and H. Chu, “Moiré fringe alignment using composite circular-line gratings for proximity lithography,” Opt. Express 23(16), 20905–20915 (2015).
[Crossref] [PubMed]

J. P. Zhu, S. Hu, J. S. Yu, Y. Tang, F. Xu, Y. He, S. L. Zhou, and L. L. Li, “Influence of tilt moiré fringe on alignment accuracy in proximity lithography,” Opt. Lasers Eng. 51(4), 371–381 (2013).
[Crossref]

Xu, X.

S. Zhou, C. Xie, Y. Yang, S. Hu, X. Xu, and J. Yang, “Moiré-based phase imaging for sensing and adjustment of in-plane twist angle,” IEEE Photon. Technol. Lett. 25(18), 1847–1850 (2013).
[Crossref]

Yang, J.

S. Zhou, C. Xie, Y. Yang, S. Hu, X. Xu, and J. Yang, “Moiré-based phase imaging for sensing and adjustment of in-plane twist angle,” IEEE Photon. Technol. Lett. 25(18), 1847–1850 (2013).
[Crossref]

Yang, R.

Yang, X.

X. Yang and C. Yin, “A new method for the design of zero reference marks for grating measurement systems,” J. Phys. E Sci. Instrum. 19(1), 34–37 (1986).
[Crossref]

Yang, Y.

S. Zhou, C. Xie, Y. Yang, S. Hu, X. Xu, and J. Yang, “Moiré-based phase imaging for sensing and adjustment of in-plane twist angle,” IEEE Photon. Technol. Lett. 25(18), 1847–1850 (2013).
[Crossref]

S. Zhou, Y. Fu, X. Tang, S. Hu, W. Chen, and Y. Yang, “Fourier-based analysis of moiré fringe patterns of superposed gratings in alignment of nanolithography,” Opt. Express 16(11), 7869–7880 (2008).
[Crossref] [PubMed]

Yao, P.

Yin, C.

X. Yang and C. Yin, “A new method for the design of zero reference marks for grating measurement systems,” J. Phys. E Sci. Instrum. 19(1), 34–37 (1986).
[Crossref]

Yu, J. S.

J. P. Zhu, S. Hu, J. S. Yu, Y. Tang, F. Xu, Y. He, S. L. Zhou, and L. L. Li, “Influence of tilt moiré fringe on alignment accuracy in proximity lithography,” Opt. Lasers Eng. 51(4), 371–381 (2013).
[Crossref]

J. P. Zhu, S. Hu, J. S. Yu, and Y. Tang, “Alignment method based on matched dual-grating moiré fringe for proximity lithography,” Opt. Eng. 51(11), 113603 (2012).
[Crossref]

Zhou, S.

Zhou, S. L.

J. P. Zhu, S. Hu, J. S. Yu, Y. Tang, F. Xu, Y. He, S. L. Zhou, and L. L. Li, “Influence of tilt moiré fringe on alignment accuracy in proximity lithography,” Opt. Lasers Eng. 51(4), 371–381 (2013).
[Crossref]

Zhu, J. P.

J. P. Zhu, S. Hu, J. S. Yu, Y. Tang, F. Xu, Y. He, S. L. Zhou, and L. L. Li, “Influence of tilt moiré fringe on alignment accuracy in proximity lithography,” Opt. Lasers Eng. 51(4), 371–381 (2013).
[Crossref]

J. P. Zhu, S. Hu, J. S. Yu, and Y. Tang, “Alignment method based on matched dual-grating moiré fringe for proximity lithography,” Opt. Eng. 51(11), 113603 (2012).
[Crossref]

Adv. Opt. Photonics (1)

G. Berkovic and E. Shafir, “Optical methods for distance and displacement measurements,” Adv. Opt. Photonics 4(4), 441–471 (2012).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

D. C. Flanders, H. I. Smith, and S. Austin, “A new interferometric alignment technique,” Appl. Phys. Lett. 31(7), 426–428 (1977).
[Crossref]

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86(20), 204101 (2005).
[Crossref]

IEEE Photon. Technol. Lett. (1)

S. Zhou, C. Xie, Y. Yang, S. Hu, X. Xu, and J. Yang, “Moiré-based phase imaging for sensing and adjustment of in-plane twist angle,” IEEE Photon. Technol. Lett. 25(18), 1847–1850 (2013).
[Crossref]

IEEE Trans. Electron Devices ED (1)

G. Bouwhuis and S. Wittekoek, “Automatic alignment system for optical projection printing,” IEEE Trans. Electron Devices ED 26(4), 723–728 (1979).
[Crossref]

J. Phys. E Sci. Instrum. (1)

X. Yang and C. Yin, “A new method for the design of zero reference marks for grating measurement systems,” J. Phys. E Sci. Instrum. 19(1), 34–37 (1986).
[Crossref]

J. Vac. Sci. Technol. B (1)

M. Suzuki and A. Une, “An optical-heterodyne alignment technique for quarter-micron x-ray lithography,” J. Vac. Sci. Technol. B 7(6), 1971–1976 (1989).
[Crossref]

Meas. Sci. Technol. (1)

E. Manske, G. Jäger, T. Hausotte, and R. Füß, “Recent developments and challenges of nanopositioning and nanomeasuring technology,” Meas. Sci. Technol. 23(7), 074001 (2012).
[Crossref]

Nat. Photonics (1)

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Figures (9)

Fig. 1
Fig. 1 Measurement principle of proposed absolute position alignment method.
Fig. 2
Fig. 2 Diagram of combined coarse and fine absolute position alignment principle.
Fig. 3
Fig. 3 Absolute position alignment system employed in this investigation.
Fig. 4
Fig. 4 Diagram of phase information extraction method used for the measured signal.
Fig. 5
Fig. 5 Diagram of scan sampling used for phase extraction detection.
Fig. 6
Fig. 6 Multi-order composite signal spectrum.
Fig. 7
Fig. 7 Offset residuals measured at identical positions for different orders m.
Fig. 8
Fig. 8 Linear relationship between results of proposed system and HeNe interferometer.
Fig. 9
Fig. 9 Residuals measured at various positions.

Equations (26)

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S( x )=Acos( 2π P signal x+ φ mark )=Acos( 2π P signal ( x+ x mark ) ).
x mark = P signal 2π φ mark .
x mark =N P highorder +f P highorder .
U ob ( x )=R( x ) U ill ( x ),
U im ( x )= m=M M A m e im 2π P 0 ( x+ x mark ) U ill ( x ).
NAm λ P 0 NA.
U im inter ( x )= 1 2 [ U im ( x ) U im ( x ) ].
S( x )= D U im inter ( x, ξ 0 ) U im inter ( x, ξ 0 ) ¯ dσ.
S( x )= 1 2 I ill m=1 M { A m A m ¯ + A m A m ¯ ( A m A m ¯ + A m A m ¯ )cos[ 4mπ P 0 ( x+ x mark ) ] } .
S( x ) m=1 M { R m + T m cos[ k m ( x+ x mark ) ] }
S( n )= m=1 M n=0 N1 { T m + R m cos[ k m ( nΔ+ x mark ) ] } .
A j ( n )=cos( k j nΔ )
B j ( n )=sin( k j nΔ ),
R Aj ( n )=S( n ) A j ( n ) = m=0 M n=0 N1 { R m cos( k j nΔ )+ T m cos[ k m ( x mark +nΔ ) ]cos( k j nΔ ) } = 1 2 cos[ 1 2 ( N1 )( k m k j )Δ+ k m x mark ] sin[ ( k m k j ) 1 2 NΔ ] sin[ ( k m k j ) 1 2 Δ ] + 1 2 cos[ 1 2 ( N1 )( k m + k j )Δ+ k m x mark ] sin[ ( k m + k j ) 1 2 NΔ ] sin[ ( k m + k j ) 1 2 Δ ]
R Bj ( n )=S( n ) B j ( n ) = m=0 M n=0 N1 { R m sin( k j nΔ )+ T m cos[ k m ( x mark +nΔ ) ]sin( k j nΔ ) } = 1 2 sin[ 1 2 ( N1 )( k m k j )Δ+ k m x mark ] sin[ ( k m k j ) 1 2 NΔ ] sin[ ( k m k j ) 1 2 Δ ] + 1 2 sin[ 1 2 ( N1 )( k m + k j )Δ+ k m x mark ] sin[ ( k m + k j ) 1 2 NΔ ] sin[ ( k m + k j ) 1 2 Δ ]
L=NΔ=q P 0 ,
R Aj ( n )= 1 2 T j cos( k j x mark )N
R Bj ( n )= 1 2 T j sin( k j x mark )N.
x mark j th order = P 0 j4π arctan[ S( n ) B j ( n ) S( n ) A j ( n ) ].
x mark = x mark loworder P highorder x mark highorder P highorder + 1 2 P highorder + x mark highorder ,
N( x )= m=0 a m sin( k m x )+ b m cos( k m x ) .
S real ( x )=S( x )+N( x ).
Erro r m ( x )= 1 k m arctan( a m T m + b m ).
V( Erro r m ( x ) )= ( Erro r m ( x ) a m ) 2 V( a m )+ ( Erro r m ( x ) b m ) 2 V( b m ) +2( Erro r m ( x ) a m )( Erro r m ( x ) b m )cov( a m , b m ).
V( Erro r m ( x ) ) ( 1 k m T m ) 2 V( a m )
U( Erro r m ( x ) )= V( Erro r m ( x ) ) P 0 4πm U( a m ) T m .

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