Abstract

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  35. K. Nienhuis and B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part III: General formulae for small aberrations: experimental verification of the theoretical results,” Physica 14, 590-608 (1949).
    [CrossRef]
  36. A. Björck, Numerical Methods for Least Squares Problems (Cambridge University, 1996).
    [CrossRef]
  37. W. H. Press, S. A. Teukolsky, W. Vetterling, and B. P. Flannery, Numerical Recipes in C++ (Cambridge University, 2002).
    [PubMed]
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    [CrossRef] [PubMed]

2007 (5)

2006 (3)

2003 (1)

S. R. Restaino, S. W. Teare, M. DiVittorio, G. C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1 m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491-2495 (2003).
[CrossRef]

2002 (1)

B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. 41, 1565-1569(2002).
[CrossRef]

1997 (2)

1995 (4)

1994 (3)

1993 (1)

1992 (1)

M. Melozzi and L. Pezzati, “Interferometric testing of annular apertures,” Proc. SPIE 1781, 241-248 (1992).
[CrossRef]

1989 (1)

1983 (1)

1982 (1)

1980 (1)

1978 (1)

1976 (1)

1972 (1)

1949 (1)

K. Nienhuis and B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part III: General formulae for small aberrations: experimental verification of the theoretical results,” Physica 14, 590-608 (1949).
[CrossRef]

1947 (1)

B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations,” Physica 13, 605-620 (1947).
[CrossRef]

Bille, J.

Björck, A.

A. Björck, Numerical Methods for Least Squares Problems (Cambridge University, 1996).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Brase, J. M.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 1-11 (2007).

Chen, H.

B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. 41, 1565-1569(2002).
[CrossRef]

Chen, Q.

Chow, W. W.

Combs, R. L.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 1-11 (2007).

Dai, G.-m.

DiVittorio, M.

S. R. Restaino, S. W. Teare, M. DiVittorio, G. C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1 m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491-2495 (2003).
[CrossRef]

Dong, N.

B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. 41, 1565-1569(2002).
[CrossRef]

Evans, C.

Fischer, D. J.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. Vetterling, and B. P. Flannery, Numerical Recipes in C++ (Cambridge University, 2002).
[PubMed]

Fochs, S. N.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 1-11 (2007).

Fox, D. G.

Gilbreath, G. C.

S. R. Restaino, S. W. Teare, M. DiVittorio, G. C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1 m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491-2495 (2003).
[CrossRef]

Goelz, S.

Grimm, B.

Hou, X.

Hurd, R. L.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 1-11 (2007).

Jeong, T. M.

Kim, C.-J.

King, C. M.

Ko, D-K.

LaFortune, K. N.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 1-11 (2007).

Lee, J.

Liang, J.

Lopez, R.

Mahajan, V. N.

Markey, J. K.

Melozzi, M.

M. Melozzi and L. Pezzati, “Interferometric testing of annular apertures,” Proc. SPIE 1781, 241-248 (1992).
[CrossRef]

Mozurkewich, D.

S. R. Restaino, S. W. Teare, M. DiVittorio, G. C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1 m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491-2495 (2003).
[CrossRef]

Nienhuis, K.

K. Nienhuis and B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part III: General formulae for small aberrations: experimental verification of the theoretical results,” Physica 14, 590-608 (1949).
[CrossRef]

Nijboer, B. R. A.

K. Nienhuis and B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part III: General formulae for small aberrations: experimental verification of the theoretical results,” Physica 14, 590-608 (1949).
[CrossRef]

B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations,” Physica 13, 605-620 (1947).
[CrossRef]

Noll, R. J.

O'Bryan, J. T.

Olivier, S. S.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 1-11 (2007).

Parks, R. E.

Pax, P. H.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 1-11 (2007).

Pezzati, L.

M. Melozzi and L. Pezzati, “Interferometric testing of annular apertures,” Proc. SPIE 1781, 241-248 (1992).
[CrossRef]

Powell, I.

Prata, A.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. Vetterling, and B. P. Flannery, Numerical Recipes in C++ (Cambridge University, 2002).
[PubMed]

Qi, B.

B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. 41, 1565-1569(2002).
[CrossRef]

Restaino, S. R.

S. R. Restaino, S. W. Teare, M. DiVittorio, G. C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1 m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491-2495 (2003).
[CrossRef]

Rimmer, M. P.

Rotter, M. D.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 1-11 (2007).

Rusch, W. V. T.

Silva, D. E.

Stall, H. P.

Sullivan, P. J.

Swantner, W.

Taylor, J. S.

Teare, S. W.

S. R. Restaino, S. W. Teare, M. DiVittorio, G. C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1 m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491-2495 (2003).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. Vetterling, and B. P. Flannery, Numerical Recipes in C++ (Cambridge University, 2002).
[PubMed]

van Brug, H.

Vetterling, W.

W. H. Press, S. A. Teukolsky, W. Vetterling, and B. P. Flannery, Numerical Recipes in C++ (Cambridge University, 2002).
[PubMed]

Wang, J. Y.

Williams, D. R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Wu, F.

Wu, S.

Wyatt, H. J.

H. J. Wyatt, “The form of the human pupil,” Vision Res. 35, 2021-2036 (1995).
[CrossRef] [PubMed]

Yamamoto, R. M.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 1-11 (2007).

Yang, L.

Appl. Opt. (12)

M. P. Rimmer, C. M. King, and D. G. Fox, “Computer program for the analysis of interferometric test data,” Appl. Opt. 11, 2790-2796 (1972).
[CrossRef] [PubMed]

J. Y. Wang and D. E. Silva, “Wavefront interpretation with Zernike polynomials,” Appl. Opt. 19, 1510-1518 (1980).
[CrossRef] [PubMed]

C.-J. Kim, “Polynomial fit of interferograms,” Appl. Opt. 21, 4521-4525 (1982).
[CrossRef] [PubMed]

A. Prata, Jr., and W. V. T. Rusch, “Algorithm for computation of Zernike polynomials expansion coefficients,” Appl. Opt. 28, 749-754 (1989).
[CrossRef] [PubMed]

D. J. Fischer, J. T. O'Bryan, R. Lopez, and H. P. Stall, “Vector formulation for interferogram surface fitting,” Appl. Opt. 32, 4738-4743 (1993).
[CrossRef] [PubMed]

W. Swantner and W. W. Chow, “Gram-Schmidt orthonormalization of Zernike polynomials for general aperture shapes,” Appl. Opt. 33, 1832-1837 (1994).
[CrossRef] [PubMed]

V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Appl. Opt. 33, 8125-8127 (1994).
[CrossRef] [PubMed]

H. van Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. 36, 2788-2790 (1997).
[CrossRef] [PubMed]

C. Evans, R. E. Parks, P. J. Sullivan, and J. S. Taylor, “Visualization of surface figure by use of Zernike polynomials,” Appl. Opt. 34, 7815-7819 (1995).
[CrossRef] [PubMed]

I. Powell, “Pupil exploration and wave-front-polynomial fitting of optical systems,” Appl. Opt. 34, 7986-7997 (1995).
[CrossRef] [PubMed]

X. Hou, F. Wu, L. Yang, and Q. Chen, “Comparison of annular wavefront interpretation with Zernike circle polynomials and annular polynomials,” Appl. Opt. 45, 8893-8901(2006).
[CrossRef] [PubMed]

X. Hou, F. Wu, L. Yang, S. Wu, and Q. Chen, “Full aperture wavefront reconstruction from annular subaperture interferometric data by use of Zernike annular polynomials and a matrix method for testing large aspheric surfaces,” Appl. Opt. 45, 3442-3454 (2006).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Opt. Eng. (2)

S. R. Restaino, S. W. Teare, M. DiVittorio, G. C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1 m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491-2495 (2003).
[CrossRef]

B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. 41, 1565-1569(2002).
[CrossRef]

Opt. Lett. (3)

Physica (2)

B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations,” Physica 13, 605-620 (1947).
[CrossRef]

K. Nienhuis and B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part III: General formulae for small aberrations: experimental verification of the theoretical results,” Physica 14, 590-608 (1949).
[CrossRef]

Proc. SPIE (2)

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 1-11 (2007).

M. Melozzi and L. Pezzati, “Interferometric testing of annular apertures,” Proc. SPIE 1781, 241-248 (1992).
[CrossRef]

Vision Res. (1)

H. J. Wyatt, “The form of the human pupil,” Vision Res. 35, 2021-2036 (1995).
[CrossRef] [PubMed]

Other (8)

G.-m. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).
[CrossRef]

http://scikits.com/KFacts.html.

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 498-546.
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations: Part II. Wave Diffraction Optics (SPIE, 2004).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

W. Swantner, Optical Engineering Services, 433 Live Oak Loop NE, Albuquerque, N. Mex. (personal communication).

A. Björck, Numerical Methods for Least Squares Problems (Cambridge University, 1996).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. Vetterling, and B. P. Flannery, Numerical Recipes in C++ (Cambridge University, 2002).
[PubMed]

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Figures (9)

Fig. 1
Fig. 1

Contour plots of elliptical wavefronts fitted by (a) orthonormal elliptical polynomials and (b) Zernike circle polynomials with J = 4 , 11, and 15. The corresponding coefficients are given in Table 2. The peak-to-valley value of the plots is 13.

Fig. 2
Fig. 2

Contour plots of square wavefronts fitted by (a) orthonormal square polynomials and (b) Zernike polynomials with J = 28 . The corresponding coefficients are given in Table 3. The peak-to-valley value of the plots is 35.

Fig. 3
Fig. 3

Contour plots of annular wavefronts fitted by (a) orthonormal annular polynomials and (b) Zernike polynomials with J = 28 . The obscuration ratio ϵ = 0.63 , and the corresponding coefficients are shown in Table 4. The peak-to-valley value of the plots is 14.

Fig. 4
Fig. 4

(a) Input wavefront map from the orthonormal hexagonal coefficients. (b) Wavefront map using Zernike coefficients obtained as in Subsection 3A. (c) Wavefront map using Zernike coefficients obtained as in Subsection 3B. (d) Wavefront map using Zernike coefficients obtained as in Subsection 3C. The corresponding coefficients are given in Table 7. The peak-to-valley value of the plots is 4.

Fig. 5
Fig. 5

Wavefront fitting error σ Δ as a function of the obscuration ratio ϵ of an annular pupil for various aberration types given in Table 6.

Fig. 6
Fig. 6

Wavefront fitting error σ Δ when each hexagonal coefficient with a value of unity is used.

Fig. 7
Fig. 7

Wavefront fitting error σ Δ as a function of the aspect ratio b of an elliptical pupil for various aberration types given in Table 6.

Fig. 8
Fig. 8

Wavefront fitting error σ Δ as a function of the semiwidth a of a rectangular pupil for various aberration types given in Table 6.

Fig. 9
Fig. 9

Wavefront fitting error σ Δ when each square coefficient with a value of unity is used.

Tables (7)

Tables Icon

Table 1 Nonzero Elements of Vector c Z for Various Pupils a

Tables Icon

Table 2 Elliptical Coefficients and the Corresponding Zernike Coefficients when an Elliptical Wavefront with an Aspect Ratio b = 0.83 Is Fitted with a Different Number of Polynomials J Showing Independence of the Former but Dependence of the Latter on J a

Tables Icon

Table 3 Square Coefficients and the Corresponding Zernike Coefficients to Illustrate the Relation between the Piston Coefficient and Wavefront Mean a

Tables Icon

Table 4 Annular Coefficients and the Corresponding Zernike Coefficients of an Annular Wavefront with an Obscuration Ratio ϵ = 0.63 to Illustrate the Relation between the Coefficients and the Wavefront Variance a

Tables Icon

Table 5 Hexagonal Coefficients and the Corresponding Zernike Coefficients as Calculated from Eq. (13) a

Tables Icon

Table 6 Examples of Realistic Aberration Types and the Coefficients of Orthonormal Polynomials

Tables Icon

Table 7 Hexagonal Coefficients and Zernike Coefficients Calculated by Different Methods a

Equations (34)

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

1 A Σ F j ( ρ , θ ) F j ( ρ , θ ) d 2 ρ = δ j , j ,
A = Σ d 2 ρ .
Φ ( ρ , θ ) = j = 1 J a j F j ( ρ , θ ) ,
a j = 1 A Σ Φ ( ρ , θ ) F j ( ρ , θ ) d 2 ρ ,
a 1 = 1 A Σ Φ ( ρ , θ ) d 2 ρ = Φ .
a j = [ a j 2 A Σ [ F j ( ρ , θ ) ] 2 d 2 ρ ] 1 / 2 = σ j .
σ 2 = 1 A Σ [ Φ ( ρ , θ ) Φ ] 2 d 2 ρ = 1 A Σ [ j = 1 J a j F j a 1 ] 2 d 2 ρ = j = 2 J j = 2 J a j a j 1 A Σ F j ( ρ , θ ) F j ( ρ , θ ) d 2 ρ = j = 2 J j = 2 J a j a j δ j , j = j = 2 J a j 2 .
Φ ^ ( ρ , θ ) = j = 1 J b j Z j ( ρ , θ ) ,
j = 1 J b j Z j = j = 1 J a j F j .
j = 1 J Z j | Z j b j = j = 1 J Z j | F j a j ,
C Z Z b = C Z F a ,
C Z F = C Z Z M T ,
b = ( C Z Z ) 1 C Z F a = M T a .
a = ( M T ) 1 b ,
S ^ = Zb ,
b ^ = Z 1 S ^ ,
b ^ j = 1 A Σ Φ ( ρ , θ ) Z j ( ρ , θ ) d 2 ρ = j = 1 J a j 1 A Σ Z j ( ρ , θ ) F j ( ρ , θ ) d 2 ρ = j = 1 J a j Z j | F j ,
b ^ = C Z F a .
σ Δ 2 = 1 A Σ ( Δ Δ ) 2 d 2 ρ = Δ 2 Δ 2 .
σ Δ 2 = j = 1 J a j 2 2 j = 1 J j = 1 J b j Z j | F j a j + j = 1 J j = 1 J b j Z j | Z j b j Δ 2 .
σ Δ 2 = a T a 2 b T C Z F a + b T C Z Z b Δ 2 .
σ Δ 2 = a T a 2 a T M C Z F a + a T M C Z Z M T a = a T a 2 a T M C Z Z M T a + a T M C Z Z M T a = a T a a T M C Z Z M T a = a T a a T a = 0.
M C Z Z M T = 1.
σ Δ 2 = a T a 2 a T ( C Z F ) T C Z F a + a T ( C Z F ) T C Z Z C Z F a Δ 2 = a T Pa Δ 2 ,
P = 1 ( C Z F ) T ( 2 C Z Z ) C Z F ,
Δ 2 = Φ Φ ^ 2
= a 1 2 2 a 1 ( c Z ) T C Z F a + [ ( c Z ) T C Z F a ] 2 ,
b 1 = M 1 , 1 a 1 + M 4 , 1 a 4 + M 11 , 1 a 11 = a 1 + 5 43 a 4 + 521 1 , 072 , 205 a 11 = 0.032874 .
b 6 = M 6 , 6 a 6 + M 12 , 6 a 12 + M 14 , 6 a 14 = 10 7 a 6 + 225 6 492 , 583 a 12 2525 14 297 , 774 , 543 a 14 = 0.397393 .
σ Δ 2 = 1 A Σ [ j = 1 J ( a j F j a 1 b j Z j + b j Z j ) ] 2 d 2 ρ = 1 A Σ [ j = 2 J a j F j j = 1 J b j Z j + j = 1 J b j Z j ] 2 d 2 ρ = j = 2 J j = 2 J a j F j | F j a j + j = 1 J j = 1 J b j Z j | Z j b j + j = 1 J j = 1 J b j Z j Z j b j + 2 j = 2 J j = 1 J a j F j Z j b j 2 j = 2 J j = 1 J a j F j | Z j b j 2 j = 1 J j = 1 J b j Z j Z j b j = j = 2 J a j 2 + j = 1 J j = 1 J b j Z j | Z j b j 2 j = 2 J j = 1 J a j F j | Z j b j j = 1 J j = 1 J b j Z j Z j b j = j = 1 J a j 2 2 j = 1 J j = 1 J b j Z j | F j a j + j = 1 J j = 1 J b j Z j | Z j b j Δ 2 ,
Δ 2 = a 1 2 2 a 1 j = 1 J b j Z j + j = 1 J j = 1 J b j Z j Z j b j = ( a 1 Φ ^ ) 2 = ( Φ Φ ^ ) 2 .
Δ 2 = a 1 2 2 a 1 j = 1 J j = 1 J Z j Z j | F j a j + j = 1 J Z j k = 1 J Z j | F k a k j = 1 J Z j k = 1 J Z j | F k a k .
Δ 2 = a 1 2 2 a 1 ( c Z ) T C Z F a + [ ( c Z ) T C Z F a ] 2 ,
Z j = 1 ( 1 ϵ 2 ) π ϵ 1 0 2 π Z j ( ρ , θ ) ρ d ρ d θ ,

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