Abstract

Fritz’s method [Opt. Eng. 23, 379 (1984)] of using Zernike polynomials to assess the absolute planarity of test plates is revisited. A refinement is described that takes into account the data decorrelation that appears in experiments. An uncertainty balance is defined by propagation of error contributions through the steps of the method. The resultant measuring procedure is demonstrated on a data set from experiments, and a nanometer level of uncertainty is achieved.

© 1999 Optical Society of America

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References

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  1. Lord Rayleigh, “Interference bands and their applications,” Nature (London) 48, 212–214 (1893).
    [CrossRef]
  2. C. Ai, J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32, 4698–4705 (1993).
    [CrossRef] [PubMed]
  3. K.-E. Elssner, A. Vogel, J. Grzanna, G. Schulz, “Establishing a flatness standard,” Appl. Opt. 33, 2437–2446 (1994).
    [CrossRef] [PubMed]
  4. C. J. Evans, R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015–1021 (1996).
    [CrossRef] [PubMed]
  5. J. Chen, D. Song, R. Zhu, Q. Wang, L. Chen, “Large-aperture high-accuracy phase-shifting digital flat interferometer,” Opt. Eng. 35, 1936–1942 (1996).
    [CrossRef]
  6. P. Hariharan, “Interferometric testing of optical surfaces: absolute measurement of flatness,” Opt. Eng. 36, 2478–2481 (1997).
    [CrossRef]
  7. C. J. Evans, “Comment on the paper ‘Interferometric testing of optical surfaces: absolute measurement of flatness,’” Opt. Eng. 37, 1880–1882 (1998).
    [CrossRef]
  8. I. Powell, E. Goulet, “Absolute figure measurements with a liquid-flat reference,” Appl. Opt. 37, 2579–2588 (1998).
    [CrossRef]
  9. R. E. Parks, C. J. Evans, P. O. Sullivan, L.-Z. Shao, B. Loucks, “Measurement of the LIGO pathfinder optics,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
    [CrossRef]
  10. R. E. Parks, L.-Z. Shao, C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951–5956 (1998).
    [CrossRef]
  11. G. Schulz, J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077–1084 (1967).
    [CrossRef] [PubMed]
  12. R. E. Parks, “Removal of test optics errors,” in Advances in Optical Metrology, N. Balasubramanian, J. C. Wyant, eds., Proc. SPIE153, 56–63 (1978).
    [CrossRef]
  13. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).
    [CrossRef]
  14. C. J. Evans, R. E. Parks, P. J. Sullivan, G. S. Taylor, “Visualization of surface figure by the use of Zernike polynomials,” Appl. Opt. 34, 7815–7819 (1995).
    [CrossRef] [PubMed]
  15. E.g., code v (Optical Research Associates, Pasadena, Calif.)
  16. J. S. Loomis, fringe User’s Manual—Version 2 (University of Arizona, Tucson, Ariz., November1976).
  17. C.-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987), Vol. 10, pp. 193–221.
  18. V. Greco, G. Molesini, “Micro-temperature effects on absolute flatness test plates,” Pure Appl. Opt. 7, 1341–1346 (1998).
    [CrossRef]
  19. V. B. Gubin, V. N. Sharonov, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147–148 (1990).
  20. S. Brandt, Statistical and Computational Methods in Data Analysis, 2nd ed. (North-Holland, Amsterdam, 1970), p. 216.
  21. Bureau International des Poids et MesuresInternational Electrotechnical CommissionInternational Federation of Clinical ChemistryInternational Organization for StandardizationInternational Union of Pure and Applied PhysicsInternational Union of Pure and Applied ChemistryInternational Organization of Legal Metrology, Guide to the Expression of Uncertainty in Measurements (International Organization for Standardization, Geneva, 1993).
  22. Mark IVxp (Zygo Corporation, Middlefield, Conn.).
  23. R. Tronconi, “Misura assoluta di planarità con metodi interferometrici,” laurea dissertation (University of Florence, Florence, Italy, April1997).
  24. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  25. W. H. Press, A. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge, U. Press, Cambridge, 1992).
  26. V. B. Gubin, V. N. Sharonov, “Absolute calibration of spherical surfaces,” Sov. J. Opt. Technol. 57, 554–555 (1990).

1998

C. J. Evans, “Comment on the paper ‘Interferometric testing of optical surfaces: absolute measurement of flatness,’” Opt. Eng. 37, 1880–1882 (1998).
[CrossRef]

V. Greco, G. Molesini, “Micro-temperature effects on absolute flatness test plates,” Pure Appl. Opt. 7, 1341–1346 (1998).
[CrossRef]

I. Powell, E. Goulet, “Absolute figure measurements with a liquid-flat reference,” Appl. Opt. 37, 2579–2588 (1998).
[CrossRef]

R. E. Parks, L.-Z. Shao, C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951–5956 (1998).
[CrossRef]

1997

P. Hariharan, “Interferometric testing of optical surfaces: absolute measurement of flatness,” Opt. Eng. 36, 2478–2481 (1997).
[CrossRef]

1996

C. J. Evans, R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015–1021 (1996).
[CrossRef] [PubMed]

J. Chen, D. Song, R. Zhu, Q. Wang, L. Chen, “Large-aperture high-accuracy phase-shifting digital flat interferometer,” Opt. Eng. 35, 1936–1942 (1996).
[CrossRef]

1995

1994

1993

1990

V. B. Gubin, V. N. Sharonov, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147–148 (1990).

V. B. Gubin, V. N. Sharonov, “Absolute calibration of spherical surfaces,” Sov. J. Opt. Technol. 57, 554–555 (1990).

1984

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).
[CrossRef]

1983

1967

1893

Lord Rayleigh, “Interference bands and their applications,” Nature (London) 48, 212–214 (1893).
[CrossRef]

Ai, C.

Brandt, S.

S. Brandt, Statistical and Computational Methods in Data Analysis, 2nd ed. (North-Holland, Amsterdam, 1970), p. 216.

Burow, R.

Chen, J.

J. Chen, D. Song, R. Zhu, Q. Wang, L. Chen, “Large-aperture high-accuracy phase-shifting digital flat interferometer,” Opt. Eng. 35, 1936–1942 (1996).
[CrossRef]

Chen, L.

J. Chen, D. Song, R. Zhu, Q. Wang, L. Chen, “Large-aperture high-accuracy phase-shifting digital flat interferometer,” Opt. Eng. 35, 1936–1942 (1996).
[CrossRef]

Elssner, K.-E.

Evans, C. J.

R. E. Parks, L.-Z. Shao, C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951–5956 (1998).
[CrossRef]

C. J. Evans, “Comment on the paper ‘Interferometric testing of optical surfaces: absolute measurement of flatness,’” Opt. Eng. 37, 1880–1882 (1998).
[CrossRef]

C. J. Evans, R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015–1021 (1996).
[CrossRef] [PubMed]

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

R. E. Parks, C. J. Evans, P. O. Sullivan, L.-Z. Shao, B. Loucks, “Measurement of the LIGO pathfinder optics,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
[CrossRef]

Flannery, B. P.

W. H. Press, A. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge, U. Press, Cambridge, 1992).

Fritz, B. S.

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).
[CrossRef]

Goulet, E.

Greco, V.

V. Greco, G. Molesini, “Micro-temperature effects on absolute flatness test plates,” Pure Appl. Opt. 7, 1341–1346 (1998).
[CrossRef]

Grzanna, J.

Gubin, V. B.

V. B. Gubin, V. N. Sharonov, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147–148 (1990).

V. B. Gubin, V. N. Sharonov, “Absolute calibration of spherical surfaces,” Sov. J. Opt. Technol. 57, 554–555 (1990).

Hariharan, P.

P. Hariharan, “Interferometric testing of optical surfaces: absolute measurement of flatness,” Opt. Eng. 36, 2478–2481 (1997).
[CrossRef]

Kestner, R. N.

Kim, C.-J.

C.-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987), Vol. 10, pp. 193–221.

Loomis, J. S.

J. S. Loomis, fringe User’s Manual—Version 2 (University of Arizona, Tucson, Ariz., November1976).

Loucks, B.

R. E. Parks, C. J. Evans, P. O. Sullivan, L.-Z. Shao, B. Loucks, “Measurement of the LIGO pathfinder optics,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
[CrossRef]

Merkel, K.

Molesini, G.

V. Greco, G. Molesini, “Micro-temperature effects on absolute flatness test plates,” Pure Appl. Opt. 7, 1341–1346 (1998).
[CrossRef]

Parks, R. E.

R. E. Parks, L.-Z. Shao, C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951–5956 (1998).
[CrossRef]

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

R. E. Parks, C. J. Evans, P. O. Sullivan, L.-Z. Shao, B. Loucks, “Measurement of the LIGO pathfinder optics,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
[CrossRef]

R. E. Parks, “Removal of test optics errors,” in Advances in Optical Metrology, N. Balasubramanian, J. C. Wyant, eds., Proc. SPIE153, 56–63 (1978).
[CrossRef]

Powell, I.

Press, W. H.

W. H. Press, A. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge, U. Press, Cambridge, 1992).

Rayleigh, Lord

Lord Rayleigh, “Interference bands and their applications,” Nature (London) 48, 212–214 (1893).
[CrossRef]

Schulz, G.

Schwider, J.

Shannon, R. R.

C.-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987), Vol. 10, pp. 193–221.

Shao, L.-Z.

R. E. Parks, L.-Z. Shao, C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951–5956 (1998).
[CrossRef]

R. E. Parks, C. J. Evans, P. O. Sullivan, L.-Z. Shao, B. Loucks, “Measurement of the LIGO pathfinder optics,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
[CrossRef]

Sharonov, V. N.

V. B. Gubin, V. N. Sharonov, “Absolute calibration of spherical surfaces,” Sov. J. Opt. Technol. 57, 554–555 (1990).

V. B. Gubin, V. N. Sharonov, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147–148 (1990).

Song, D.

J. Chen, D. Song, R. Zhu, Q. Wang, L. Chen, “Large-aperture high-accuracy phase-shifting digital flat interferometer,” Opt. Eng. 35, 1936–1942 (1996).
[CrossRef]

Spolaczyk, R.

Sullivan, P. J.

Sullivan, P. O.

R. E. Parks, C. J. Evans, P. O. Sullivan, L.-Z. Shao, B. Loucks, “Measurement of the LIGO pathfinder optics,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
[CrossRef]

Taylor, G. S.

Teukolsky, A. A.

W. H. Press, A. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge, U. Press, Cambridge, 1992).

Tronconi, R.

R. Tronconi, “Misura assoluta di planarità con metodi interferometrici,” laurea dissertation (University of Florence, Florence, Italy, April1997).

Vetterling, W. T.

W. H. Press, A. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge, U. Press, Cambridge, 1992).

Vogel, A.

Wang, Q.

J. Chen, D. Song, R. Zhu, Q. Wang, L. Chen, “Large-aperture high-accuracy phase-shifting digital flat interferometer,” Opt. Eng. 35, 1936–1942 (1996).
[CrossRef]

Wyant, J. C.

Zhu, R.

J. Chen, D. Song, R. Zhu, Q. Wang, L. Chen, “Large-aperture high-accuracy phase-shifting digital flat interferometer,” Opt. Eng. 35, 1936–1942 (1996).
[CrossRef]

Appl. Opt.

Nature (London)

Lord Rayleigh, “Interference bands and their applications,” Nature (London) 48, 212–214 (1893).
[CrossRef]

Opt. Eng.

J. Chen, D. Song, R. Zhu, Q. Wang, L. Chen, “Large-aperture high-accuracy phase-shifting digital flat interferometer,” Opt. Eng. 35, 1936–1942 (1996).
[CrossRef]

P. Hariharan, “Interferometric testing of optical surfaces: absolute measurement of flatness,” Opt. Eng. 36, 2478–2481 (1997).
[CrossRef]

C. J. Evans, “Comment on the paper ‘Interferometric testing of optical surfaces: absolute measurement of flatness,’” Opt. Eng. 37, 1880–1882 (1998).
[CrossRef]

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).
[CrossRef]

Pure Appl. Opt.

V. Greco, G. Molesini, “Micro-temperature effects on absolute flatness test plates,” Pure Appl. Opt. 7, 1341–1346 (1998).
[CrossRef]

Sov. J. Opt. Technol.

V. B. Gubin, V. N. Sharonov, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147–148 (1990).

V. B. Gubin, V. N. Sharonov, “Absolute calibration of spherical surfaces,” Sov. J. Opt. Technol. 57, 554–555 (1990).

Other

W. H. Press, A. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge, U. Press, Cambridge, 1992).

E.g., code v (Optical Research Associates, Pasadena, Calif.)

J. S. Loomis, fringe User’s Manual—Version 2 (University of Arizona, Tucson, Ariz., November1976).

C.-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987), Vol. 10, pp. 193–221.

S. Brandt, Statistical and Computational Methods in Data Analysis, 2nd ed. (North-Holland, Amsterdam, 1970), p. 216.

Bureau International des Poids et MesuresInternational Electrotechnical CommissionInternational Federation of Clinical ChemistryInternational Organization for StandardizationInternational Union of Pure and Applied PhysicsInternational Union of Pure and Applied ChemistryInternational Organization of Legal Metrology, Guide to the Expression of Uncertainty in Measurements (International Organization for Standardization, Geneva, 1993).

Mark IVxp (Zygo Corporation, Middlefield, Conn.).

R. Tronconi, “Misura assoluta di planarità con metodi interferometrici,” laurea dissertation (University of Florence, Florence, Italy, April1997).

R. E. Parks, C. J. Evans, P. O. Sullivan, L.-Z. Shao, B. Loucks, “Measurement of the LIGO pathfinder optics,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
[CrossRef]

R. E. Parks, “Removal of test optics errors,” in Advances in Optical Metrology, N. Balasubramanian, J. C. Wyant, eds., Proc. SPIE153, 56–63 (1978).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the uncertainty evaluation. ⊕ Indicates quadratic combination.

Fig. 2
Fig. 2

Examples of data processing: (a) interferometric data, (b) Zernike fitting, (c) residuals.

Fig. 3
Fig. 3

Examples of computed surface: (a) end surface L, (b) combined uncertainty.

Fig. 4
Fig. 4

Profile of surface L along the x = 0 diameter by (dotted curve) the classic-three-flat method and (solid curve) the modified Fritz method. The extended uncertainty boundaries are marked by dashed curves.

Tables (5)

Tables Icon

Table 1 Zernike Polynomials

Tables Icon

Table 2 Interferometric System Specificationsa

Tables Icon

Table 3 Partial Derivatives Computed from Eqs. (5)

Tables Icon

Table 4 Partial Derivatives Computed from Eqs. (6)

Tables Icon

Table 5 Partial Derivatives Computed from Eqs. (7)

Equations (35)

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

L-x, y+Kx, y=Dx, y, L-x, y+Mx, y=Ex, y, L-x, y+Mϕx, y=Fx, y, K-x, y+Mx, y=Gx, y,
χ2=i=1NWxi, yi,-j=1J wjAjxi, yi,σi2,
Dρ, θ=j=137 djZjρ, θ+residuals, Eρ, θ=j=137 ejZjρ, θ+residuals, Fρ, θ=j=137 fjZjρ, θ+residuals, Gρ, θ=j=137 gjZjρ, θ+residuals.
Kρ, θ=j=137 kjZjρ, θ, Lρ, θ=j=137 ljZjρ, θ, Mρ, θ=j=137 mjZjρ, θ.
kj=gj-ej+dj2, lj=ej+dj-gj2, mj=ej-dj+gj2;
kj=1sin mϕfj+1-ej+1+dj+1-gj+12-ej+1-dj+1+gj+12cos mϕ-gj, lj=1sin mϕfj+1-ej+1+dj+1-gj+12-ej+1-dj+1+gj+12cos mϕ-ej, mj=1sin mϕfj+1-ej+1+dj+1-gj+12-ej+1-dj+1+gj+12cos mϕ,
kj=1sin mϕej-1+dj-1-gj-12+ej-1-dj-1+gj-12cos mϕ-fj-1-gj, lj=1sin mϕej-1+dj-1-gj-12+ej-1-dj-1+gj-12cos mϕ-fj-1-ej, mj=1sin mϕej-1+dj-1-gj-12+ej-1-dj-1+gj-12cos mϕ-fj-1,
fj=ej
dj=ej-gj, fj=γmdj+1-gj+1-fj+1+ej,
dj=ej-gj, fj=γmgj-1-dj-1+fj-1+ej,
Wn, ρ, θ=Dρ, θψ1n+Eρ, θψ2n+Fρ, θψ3n+Gρ, θψ4n,
ψin=1n=i n=1,, 40otherwise.
Wn, ρ, θ=j=137 Zjρ, θdjψ1n+ejψ2n+fjψ3n+gjψ4n+residuals.
Wn, ρ, θ=s=1111 wsΓsn, ρ, θ+residuals,
σdj2=djej2σej2+djgj2σgj2=σej2+σgj2.
Dρ, θ-j=137 djZjρ, θ=j=137 d˜jZjρ, θ+RDρ, θ, Eρ, θ-j=137 ejZjρ, θ=j=137 e˜jZjρ, θ+REρ, θ, Fρ, θ-j=137 fjZjρ, θ=j=137 f˜jZjρ, θ+RFρ, θ, Gρ, θ-j=137 gjZjρ, θ=j=137 g˜jZjρ, θ+RGρ, θ.
RL-x, y+RKx, y=RDx, y, RL-x, y+RMx, y=REx, y, RK-x, y+RMx, y=RGx, y,
RDx, y=ReDx, y+RoDx, y,
ReDx, y=12RDx, y+RD-x, y, RoDx, y=12RDx, y-RD-x, y.
ReKx, y=12ReGx, y-ReEx, y+ReDx, y, ReLx, y=12ReEx, y+ReDx, y-ReGx, y, ReMx, y=12ReEx, y-ReDx, y+ReGx, y.
rmsRoKx, y=rmsReKx, y, rmsRoLx, y=rmsReLx, y, rmsRoMx, y=rmsReMx, y.
σeoK=rms2ReKx, y+rms2RoKx, y1/2=2 rmsReKx, y, σeoL=rms2ReLx, y+rms2RoLx, y1/2=2 rmsReLx, y, σeoM=rms2ReMx, y+rms2RoMx, y1/2=2 rmsReMx, y.
σK2ρ, θ=σZK2ρ, θ+σeoK2,
Kρ, θ=j=137 kjZjρ, θ±2σKρ, θ,
Wn, ρ, θ=j=137 Wjρ, θ+residuals,
Wjn, ρ, θ=djZjρ, θψ1n+ejZjρ, θψ2n+ψ3n+gjZjρ, θψ4n,
Wjn, ρ, θ+Wj+1n, ρ, θ=ejψ1n+ψ2n+ψ3nZjρ, θ+gjψ4n-ψ1nZjρ, θ+dj+1γmZjρ, θψ3n+Zj+1ρ, θψ1n+ej+1Zj+1ρ, θψ2n+fj+1-γmZjρ, θ+Zj+1ρ, θψ3n+gj+1-γmZjρ, θψ3n+Zj+1ρ, θψ4n,
Wj-1n, ρ, θ+Wjn, ρ, θ=dj-1Zj-1ρ, θψ1n-γmZjρ, θψ3n+ej-1Zj-1ρ, θψ2n+fj-1Zj-1ρ, θ+γmZjρ, θψ3n+gj-1Zj-1ρ, θψ4n+γmZjρ, θψ3n+ejψ1n+ψ2n+ψ3nZjρ, θ+gjψ4n-ψ1nZj(ρ, θ,
Kρ, θ=j=137 kjd1,, d37, e1,, e37, f1,, f37,×g1,, g37, ϕZjρ, θ.
σZK2ρ, θ=i=137Kρ, θdi2σdi2+i=137Kρ, θei2σei2+i=137Kρ, θfi2σfi2+i=137Kρ, θgi2σgi2+Kρ, θϕ2σϕ2,
Kρ, θdi=j=137kjdi Zjρ, θ, Kρ, θei=j=137kjei Zjρ, θ, Kρ, θfi=j=137kjfi Zjρ, θ, Kρ, θgi=j=137kjgi Zjρ, θ, Kρ, θϕ=j=137kjϕ Zjρ, θ.
di=j
12
12
-12

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