Abstract

A method, believed to be new, for the absolute interferometric testing of flat or spherical surfaces is presented. It is based on the classic three-flat test, combined with additional measurements of one test piece in different rotational positions. Full-surface absolute maps for each test piece are determined with a data-processing technique based on the rotationally sheared maps of the rotated surface. An optimized numerical reconstruction algorithm employing linear filtering and superposition of the different rotational shear spectra in the angular frequency domain is used to reconstruct the rotationally sheared data. The technique does not require any assumptions about the surfaces under test; has low error propagation, even in the case of high spatial resolution; and is computationally efficient.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. C. Huang, “Propagation errors in precision Fizeau interferometry,” Appl. Opt. 32, 7016–7021 (1993).
    [CrossRef] [PubMed]
  3. C. J. Evans, “Compensation of errors introduced by nonzero fringe densities in phase-measuring interferometers,” Ann. CIRP 42, 577–580 (1993).
    [CrossRef]
  4. G. Schulz, J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077–1084 (1967).
    [CrossRef] [PubMed]
  5. G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
    [CrossRef]
  6. G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIII.
    [CrossRef]
  7. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).
  8. J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
    [CrossRef]
  9. M. Küchel, “Method and apparatus for absolute interferometric testing of plane surfaces,” U.S. patent5,106,194 (21April1992).
  10. G. Schulz, J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring-error compensation,” Appl. Opt. 31, 3767–3780 (1992).
    [CrossRef] [PubMed]
  11. G. Schulz, “Absolute flatness testing by an extended rotation method using two angles of rotation,” Appl. Opt. 32, 1055–1059 (1993).
    [CrossRef] [PubMed]
  12. C. Ai, J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32, 4698–4705 (1993).
    [CrossRef] [PubMed]
  13. C. J. Evans, R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015–1021 (1996).
    [CrossRef] [PubMed]
  14. C. Ai, J. Wyant, L.-Z. Shao, R. E. Parks, “Method and apparatus for absolute measurement of entire surfaces of flats,” U.S. patent5,502,566 (26March1996).
  15. C. J. Evans, R. J. Hocken, W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and ‘absolute testing,’” Ann. CIRP 45/2, 791–806 (1996).
  16. M. Küchel, “Ein neues Verfahren zur Absolutprüfung von Planflächen,” Presented at the Annual Conference of the DGaO, Kloster Banz, Germany, 21–24 May 1997.
  17. R. Mercier, M. Lamare, P. Picart, J. P. Marioge, “Two-flat method for bi-dimensional measurement of absolute departure from best sphere,” Pure Appl. Opt. 6, 117–126 (1997).
    [CrossRef]
  18. R. P. Bourgeois, J. Magner, H. P. Stahl, “Result of the calibration of interferometer transmission flats for the LIGO pathfinder optics,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 86–94 (1997).
    [CrossRef]
  19. R. E. Parks, L. Shao, C. J. Evans, “Pixel-based absolute topography test of three flats,” Appl. Opt. 37, 5951–5956 (1998).
    [CrossRef]
  20. A. E. Jensen, “Absolute calibration method for laser Twyman–Green wave-front testing interferometers,” J. Opt. Soc. Am. 63, 1313 (1973).
  21. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  22. K.-E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutprüfung von Sphärizitätsnormalen,” Opt. Acta 27, 563–580 (1980).
    [CrossRef]
  23. K.-E. Elssner, R. Burow, J. Grzanna, R. Spolaczyk, “Absolute sphericity measurement,” Appl. Opt. 28, 4649–4661 (1989).
    [CrossRef] [PubMed]
  24. K. Freischlad, “Method and apparatus for absolutely measuring flat and spherical surfaces with high spatial resolution,” U.S. patent6,184,994 (6February2001).
  25. K. R. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  26. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  27. K. R. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
    [CrossRef]
  28. D. J. Whitehouse, “Some theoretical aspects of error separation techniques in surface metrology,” J. Phys. E 9, 531–536 (1976).
    [CrossRef]
  29. C. Elster, I. Weingärtner, “Exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A 16, 2281–2285 (1999).
    [CrossRef]
  30. D. Achilles, Die Fourier-Transformation in der Signalverarbeitung (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  31. Suggested by one of the reviewers.

1999

1998

1997

R. Mercier, M. Lamare, P. Picart, J. P. Marioge, “Two-flat method for bi-dimensional measurement of absolute departure from best sphere,” Pure Appl. Opt. 6, 117–126 (1997).
[CrossRef]

1996

C. J. Evans, R. J. Hocken, W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and ‘absolute testing,’” Ann. CIRP 45/2, 791–806 (1996).

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

1993

1992

1990

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

K. R. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
[CrossRef]

1989

1986

1984

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

1983

1980

K.-E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutprüfung von Sphärizitätsnormalen,” Opt. Acta 27, 563–580 (1980).
[CrossRef]

1976

D. J. Whitehouse, “Some theoretical aspects of error separation techniques in surface metrology,” J. Phys. E 9, 531–536 (1976).
[CrossRef]

1974

1973

A. E. Jensen, “Absolute calibration method for laser Twyman–Green wave-front testing interferometers,” J. Opt. Soc. Am. 63, 1313 (1973).

1967

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

G. Schulz, J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077–1084 (1967).
[CrossRef] [PubMed]

Achilles, D.

D. Achilles, Die Fourier-Transformation in der Signalverarbeitung (Springer-Verlag, Berlin, 1978).
[CrossRef]

Ai, C.

C. Ai, J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32, 4698–4705 (1993).
[CrossRef] [PubMed]

C. Ai, J. Wyant, L.-Z. Shao, R. E. Parks, “Method and apparatus for absolute measurement of entire surfaces of flats,” U.S. patent5,502,566 (26March1996).

Bourgeois, R. P.

R. P. Bourgeois, J. Magner, H. P. Stahl, “Result of the calibration of interferometer transmission flats for the LIGO pathfinder optics,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 86–94 (1997).
[CrossRef]

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Elssner, K.-E.

Elster, C.

Estler, W. T.

C. J. Evans, R. J. Hocken, W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and ‘absolute testing,’” Ann. CIRP 45/2, 791–806 (1996).

Evans, C. J.

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

C. J. Evans, R. J. Hocken, W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and ‘absolute testing,’” Ann. CIRP 45/2, 791–806 (1996).

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

C. J. Evans, “Compensation of errors introduced by nonzero fringe densities in phase-measuring interferometers,” Ann. CIRP 42, 577–580 (1993).
[CrossRef]

Freischlad, K.

K. Freischlad, “Method and apparatus for absolutely measuring flat and spherical surfaces with high spatial resolution,” U.S. patent6,184,994 (6February2001).

Freischlad, K. R.

Fritz, B. S.

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

Gallagher, J. E.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Grzanna, J.

Herriott, D. R.

Hocken, R. J.

C. J. Evans, R. J. Hocken, W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and ‘absolute testing,’” Ann. CIRP 45/2, 791–806 (1996).

Huang, C.

Jensen, A. E.

A. E. Jensen, “Absolute calibration method for laser Twyman–Green wave-front testing interferometers,” J. Opt. Soc. Am. 63, 1313 (1973).

Kestner, R. N.

Koliopoulos, C. L.

Küchel, M.

M. Küchel, “Ein neues Verfahren zur Absolutprüfung von Planflächen,” Presented at the Annual Conference of the DGaO, Kloster Banz, Germany, 21–24 May 1997.

M. Küchel, “Method and apparatus for absolute interferometric testing of plane surfaces,” U.S. patent5,106,194 (21April1992).

Lamare, M.

R. Mercier, M. Lamare, P. Picart, J. P. Marioge, “Two-flat method for bi-dimensional measurement of absolute departure from best sphere,” Pure Appl. Opt. 6, 117–126 (1997).
[CrossRef]

Magner, J.

R. P. Bourgeois, J. Magner, H. P. Stahl, “Result of the calibration of interferometer transmission flats for the LIGO pathfinder optics,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 86–94 (1997).
[CrossRef]

Marioge, J. P.

R. Mercier, M. Lamare, P. Picart, J. P. Marioge, “Two-flat method for bi-dimensional measurement of absolute departure from best sphere,” Pure Appl. Opt. 6, 117–126 (1997).
[CrossRef]

Mercier, R.

R. Mercier, M. Lamare, P. Picart, J. P. Marioge, “Two-flat method for bi-dimensional measurement of absolute departure from best sphere,” Pure Appl. Opt. 6, 117–126 (1997).
[CrossRef]

Merkel, K.

Parks, R. E.

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

C. Ai, J. Wyant, L.-Z. Shao, R. E. Parks, “Method and apparatus for absolute measurement of entire surfaces of flats,” U.S. patent5,502,566 (26March1996).

Picart, P.

R. Mercier, M. Lamare, P. Picart, J. P. Marioge, “Two-flat method for bi-dimensional measurement of absolute departure from best sphere,” Pure Appl. Opt. 6, 117–126 (1997).
[CrossRef]

Rosenfeld, D. P.

Schulz, G.

G. Schulz, “Absolute flatness testing by an extended rotation method using two angles of rotation,” Appl. Opt. 32, 1055–1059 (1993).
[CrossRef] [PubMed]

G. Schulz, J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring-error compensation,” Appl. Opt. 31, 3767–3780 (1992).
[CrossRef] [PubMed]

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

K.-E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutprüfung von Sphärizitätsnormalen,” Opt. Acta 27, 563–580 (1980).
[CrossRef]

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

G. Schulz, J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077–1084 (1967).
[CrossRef] [PubMed]

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIII.
[CrossRef]

Schwider, J.

Shao, L.

Shao, L.-Z.

C. Ai, J. Wyant, L.-Z. Shao, R. E. Parks, “Method and apparatus for absolute measurement of entire surfaces of flats,” U.S. patent5,502,566 (26March1996).

Spolaczyk, R.

Stahl, H. P.

R. P. Bourgeois, J. Magner, H. P. Stahl, “Result of the calibration of interferometer transmission flats for the LIGO pathfinder optics,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 86–94 (1997).
[CrossRef]

Weingärtner, I.

White, A. D.

Whitehouse, D. J.

D. J. Whitehouse, “Some theoretical aspects of error separation techniques in surface metrology,” J. Phys. E 9, 531–536 (1976).
[CrossRef]

Wyant, J.

C. Ai, J. Wyant, L.-Z. Shao, R. E. Parks, “Method and apparatus for absolute measurement of entire surfaces of flats,” U.S. patent5,502,566 (26March1996).

Wyant, J. C.

Ann. CIRP

C. J. Evans, “Compensation of errors introduced by nonzero fringe densities in phase-measuring interferometers,” Ann. CIRP 42, 577–580 (1993).
[CrossRef]

C. J. Evans, R. J. Hocken, W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and ‘absolute testing,’” Ann. CIRP 45/2, 791–806 (1996).

Appl. Opt.

G. Schulz, J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077–1084 (1967).
[CrossRef] [PubMed]

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

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]

K.-E. Elssner, R. Burow, J. Grzanna, R. Spolaczyk, “Absolute sphericity measurement,” Appl. Opt. 28, 4649–4661 (1989).
[CrossRef] [PubMed]

C. Ai, J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32, 4698–4705 (1993).
[CrossRef] [PubMed]

C. Huang, “Propagation errors in precision Fizeau interferometry,” Appl. Opt. 32, 7016–7021 (1993).
[CrossRef] [PubMed]

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

G. Schulz, J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring-error compensation,” Appl. Opt. 31, 3767–3780 (1992).
[CrossRef] [PubMed]

G. Schulz, “Absolute flatness testing by an extended rotation method using two angles of rotation,” Appl. Opt. 32, 1055–1059 (1993).
[CrossRef] [PubMed]

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

J. Opt. Soc. Am.

A. E. Jensen, “Absolute calibration method for laser Twyman–Green wave-front testing interferometers,” J. Opt. Soc. Am. 63, 1313 (1973).

J. Opt. Soc. Am. A

J. Phys. E

D. J. Whitehouse, “Some theoretical aspects of error separation techniques in surface metrology,” J. Phys. E 9, 531–536 (1976).
[CrossRef]

Opt. Acta

K.-E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutprüfung von Sphärizitätsnormalen,” Opt. Acta 27, 563–580 (1980).
[CrossRef]

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

Opt. Commun.

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

Opt. Eng.

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

Pure Appl. Opt.

R. Mercier, M. Lamare, P. Picart, J. P. Marioge, “Two-flat method for bi-dimensional measurement of absolute departure from best sphere,” Pure Appl. Opt. 6, 117–126 (1997).
[CrossRef]

Other

R. P. Bourgeois, J. Magner, H. P. Stahl, “Result of the calibration of interferometer transmission flats for the LIGO pathfinder optics,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 86–94 (1997).
[CrossRef]

M. Küchel, “Ein neues Verfahren zur Absolutprüfung von Planflächen,” Presented at the Annual Conference of the DGaO, Kloster Banz, Germany, 21–24 May 1997.

M. Küchel, “Method and apparatus for absolute interferometric testing of plane surfaces,” U.S. patent5,106,194 (21April1992).

C. Ai, J. Wyant, L.-Z. Shao, R. E. Parks, “Method and apparatus for absolute measurement of entire surfaces of flats,” U.S. patent5,502,566 (26March1996).

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIII.
[CrossRef]

K. Freischlad, “Method and apparatus for absolutely measuring flat and spherical surfaces with high spatial resolution,” U.S. patent6,184,994 (6February2001).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

D. Achilles, Die Fourier-Transformation in der Signalverarbeitung (Springer-Verlag, Berlin, 1978).
[CrossRef]

Suggested by one of the reviewers.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Configurations for the absolute test of three flats A, B, and C with a total of eight measurements.

Fig. 2
Fig. 2

Data analysis flow for the absolute test with a total of eight measurements. The numbers in parentheses refer to the equations in the text.

Fig. 3
Fig. 3

(a) Differencing transfer function with shear of 1 pixel and 4.4 pixels. (b) Spectral error propagation for shearing reconstruction with shear of 1 pixel and 4.4 pixels. The frequency is normalized by the Nyquist frequency.

Fig. 4
Fig. 4

Spectral error propagation of the integration of K shear maps for K = 1, … , 5 and N = 2048 pixels. The frequency is normalized by the Nyquist frequency.

Fig. 5
Fig. 5

Total error-propagation factor f tc,BC for K = 2, … , 5 (surfaces B and C) as a function of N.

Fig. 6
Fig. 6

Transfer functions of spline fits of orders 1, 3, and 7. The frequency is normalized by the Nyquist frequency (after Ref. 30).

Fig. 7
Fig. 7

Absolute surface map of a flat with diameter of 150 mm. peak-to-valley spread (p.v.) = 22.70 nm. rms = 6.50 nm. Map resolution 480 × 480 points.

Fig. 8
Fig. 8

Difference map between two different absolute measurements of the flat of Fig. 7. p.v. = 4.54 nm, rms = 0.80 nm. Map resolution 480 × 480 points.

Tables (2)

Tables Icon

Table 1 Absolute Process Example

Tables Icon

Table 2 Difference Map Results

Equations (55)

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

ϕ=4πλhreference+htest+alignment terms,
mx, y=hreference-x, y+htestx, y,
m1x, y=hAxx, y+hCx, y,
m2x, y=hBxx, y+hCx, y,
m3x, y=hBxx, y+hAx, y,
hxx, y=h-x, y.
c1x, y=½m1x, y-m2x, y+m3x, y,
c2x, y=½m1x, y-m2x, y-m3x, y,
c3x, y=½m1x, y+m2x, y-m3x, y,
hAx, y=hA,oddx, y+c1x, y,
hBx, y=hA,oddx, y-c2xx, y,
hCx, y=hA,oddx, y+c3x, y,
hA,oddx, y=½hAx, y-hAxx, y.
djx, y=m3+jx, y-m3x, y.
djx, y=hAαjx, y-hAx, y,
djx, ysjr, θ.
sjr, θ=hAr, θ+αj-hAr, θ.
Sjr, fθ=FTsjr, θ,
Sjr, fθ=αj, fθHr, fθ,
αj, fθ=exp2πiαjfθ-1.
HjPr, fθ=1αj, fθ Sjr, fθ for αj, fθ0,
Lr, fθ=j=1M-3 wjfθSjr, fθαj, fθ,
wjfθ=ρjfθnfθ.
nfθ=j=1M-3 ρjfθ.
ρjfθ=|αj, fθ|2.
lr, θ=FT-1L(r, fθ,
lr, θhx, y.
hA,oddx, y=½hx, y-hxx, y.
rmaxΔαΔx,
Run 1: A=F1, B=F2, C=F3; absolute result for F1: h1,A. Run 2: A=F3, B=F4, C=F1; absolute result for F1: h1,C.
sk,n=hnΔ+αk-hnΔ+qk,n.
qk,nqk,n=σq2δk,kδn,n,
Hp=k=0K-1 wk,pSk,pk,p+k=0K-1 wk,pQk,pk,p,
k,p=exp2πiP αk,p-1,
|Ep|2=σq2k=0K-1|wk,p|2|k,p|2.
|Ep|2=σq2|p|2,
|e|2=1Np=0N-1 |Ep|2.
fmd=1Np=0N-1k=0K-1|wk,p|2|k,p|2.
0=ddρk,p fmd=1Np=0N-1k=0K-11|k,p|2ddρk,pρk,pk=0K-1 ρk,p2.
ρk,p=|k,p|21k=0K-1 ρk,pk=0K-1ρ2k,p|k,p|2
ρk,p=|k,p|2
mk,n=hk,n+rk,n,
rk,nrk,n=σr2δk,kδn,n.
sk,n=mk,n-m0,n+rk,n-r0,n.
Hp=k=1K wk,pSk,pk,p+k=1K wk,pRk,pk,p-k=1K wk,pR0,pk,p.
Ep=k=1K Rk,pwk,pk,p-R0,pk=1Kwk,pk,p,
|Ep|2=σr2k=1K|wk,p|2|k,p|2+k=1Kwk,pk,p2.
fcd=1Np=0N-1k=1K|wk,p|2|k,p|2+k=1Kwk,pk,p2.
σc2=¾σr2
ftu=¾+½fcd.
ftc,A=¾+½fcd+eA,oddr3
ftc,B,C=¾+½fcd-eA,oddr3
eA,oddr3=N-1N1414
ftc, A=1+12Np=0N-1k=1K|wk,p|2|k,p|2+k=1Kwk,pk,p2,
ftc, B,C=12+12Np=0N-1k=1K|wk,p|2|k,p|2+k=1Kwk,pk,p2.

Metrics