Abstract

Phase-demodulation algorithms in interferometry often depend on a sequence of evenly spaced reference phase shifts. These phase shifts must be accurately calibrated and can be distorted by geometric effects, especially when spherical components with high curvature are tested. Here the resulting measurement errors are quantified through mathematical analysis, and it is shown that modern phase-estimation algorithms can be effective in a spherical Fizeau cavity with a numerical aperture as large as 0.95.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 14, pp. 501–598.
  2. R. P. Grosso, R. Crane, “Precise optical evaluation using phase measuring interferometric techniques,” in Interferometry, G. W. Hopkins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.192, 65–74 (1979).
  3. P. Hariharan, Optical Interferometry (Academic, Orlando, Fla., 1985), Chap. 9, pp. 155–159.
  4. K. Kinnstätter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [CrossRef]
  5. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
    [CrossRef]
  6. L. A. Selberg, “Interferometer accuracy and precision,” in Optical Fabrication and Testing, D. R. Campbell, C. W. Johnson, M. Lorenzen, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1400, 24–32 (1990).
  7. P. de Groot, “Predicting the effects of vibration in phase shifting interferometry,” in Optical Fabrication and Testing Workshop, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 189–192.
  8. C. L. Koliopoulos, “Avoiding phase-measuring interferometry's pitfalls,” Photon. Spectra 22, 169–176 (1988).
  9. Y. Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef] [PubMed]
  10. R. C. Moore, F. H. Slaymaker, “Direct measurement of phase in a spherical wave Fizeau interferometer,” Appl. Opt. 19, 2196–2200 (1980).
    [CrossRef] [PubMed]
  11. K. Creath, P. Hariharan, “Phase-shifting errors in interferometric tests with high-numerical-aperture reference surfaces,” Appl. Opt. 33, 24–25 (1994).
    [CrossRef] [PubMed]
  12. P. de Groot, R. Smythe, L. Deck, “Laser diodes map surface features of complex parts,” Laser Focus World, February1994, pages 95–98.
  13. 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]
  14. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  15. G. Schulz, K.-E. Elssner, “Errors in phase-measurement interferometry with high numerical apertures,” Appl. Opt. 30, 4500–4506 (1991).
    [CrossRef] [PubMed]
  16. K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. Soc. Photo-Opt. Instrum. Eng.680, 19–28 (1986).
  17. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. XXVI, Chap. 5, pp. 349–393.
    [CrossRef]
  18. J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” in Optical Fabrication and Testing Workshop, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper PD-4.
  19. J. van Wingerden, H. H. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]
  20. J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
    [CrossRef] [PubMed]
  21. P. de Groot, “Three-color laser-diode interferometer,” Appl. Opt. 30, 3612–3616 (1991).
    [CrossRef] [PubMed]
  22. C. Joenathan, “Phase-measuring interferometry: new methods and error analysis,” Appl. Opt. 33, 4147–4155 (1994).
    [CrossRef] [PubMed]
  23. J. H. Bruning, D. R. Herriott, J. E. Galllagher, 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]
  24. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  25. J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [CrossRef]
  26. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data sampling window,” Appl. Opt.34 (to be published).
    [PubMed]
  27. P. de Groot, L. Deck, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2248 (1994).
  28. C. S. Williams, Designing Digital Filters (Prentice-Hall, Engle-wood Cliffs, N.J., 1986), Chap. 4, p. 113.
  29. F J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]

1994 (3)

1993 (1)

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

1991 (3)

1990 (2)

1989 (1)

1988 (2)

K. Kinnstätter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef]

C. L. Koliopoulos, “Avoiding phase-measuring interferometry's pitfalls,” Photon. Spectra 22, 169–176 (1988).

1987 (1)

1985 (1)

1983 (1)

1980 (1)

1978 (1)

F J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

1974 (1)

Brangaccio, D. J.

Brophy, C. P.

Bruning, J. H.

J. H. Bruning, D. R. Herriott, J. E. Galllagher, 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. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 14, pp. 501–598.

Burow, R.

Cheng, Y. Y.

Crane, R.

R. P. Grosso, R. Crane, “Precise optical evaluation using phase measuring interferometric techniques,” in Interferometry, G. W. Hopkins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.192, 65–74 (1979).

Creath, K.

K. Creath, P. Hariharan, “Phase-shifting errors in interferometric tests with high-numerical-aperture reference surfaces,” Appl. Opt. 33, 24–25 (1994).
[CrossRef] [PubMed]

J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” in Optical Fabrication and Testing Workshop, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper PD-4.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. XXVI, Chap. 5, pp. 349–393.
[CrossRef]

K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. Soc. Photo-Opt. Instrum. Eng.680, 19–28 (1986).

de Groot, P.

P. de Groot, R. Smythe, L. Deck, “Laser diodes map surface features of complex parts,” Laser Focus World, February1994, pages 95–98.

P. de Groot, “Three-color laser-diode interferometer,” Appl. Opt. 30, 3612–3616 (1991).
[CrossRef] [PubMed]

P. de Groot, “Predicting the effects of vibration in phase shifting interferometry,” in Optical Fabrication and Testing Workshop, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 189–192.

P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data sampling window,” Appl. Opt.34 (to be published).
[PubMed]

P. de Groot, L. Deck, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2248 (1994).

Deck, L.

P. de Groot, R. Smythe, L. Deck, “Laser diodes map surface features of complex parts,” Laser Focus World, February1994, pages 95–98.

P. de Groot, L. Deck, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2248 (1994).

Eiju, T.

Elssner, K.-E.

Falkenstörfer, O.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Frankena, H. H.

Freischlad, K.

Galllagher, J. E.

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 14, pp. 501–598.

Grosso, R. P.

R. P. Grosso, R. Crane, “Precise optical evaluation using phase measuring interferometric techniques,” in Interferometry, G. W. Hopkins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.192, 65–74 (1979).

Grzanna, J.

Hariharan, P.

Harris, F J.

F J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Herriott, D. R.

Joenathan, C.

Kinnstätter, K.

Koliopoulos, C. L.

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

C. L. Koliopoulos, “Avoiding phase-measuring interferometry's pitfalls,” Photon. Spectra 22, 169–176 (1988).

Lohmann, A. W.

Merkel, K.

Moore, R. C.

Oreb, B. F.

Rosenfeld, D. P.

Schmit, J.

J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” in Optical Fabrication and Testing Workshop, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper PD-4.

Schreiber, H.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Schulz, G.

Schwider, J.

Selberg, L. A.

L. A. Selberg, “Interferometer accuracy and precision,” in Optical Fabrication and Testing, D. R. Campbell, C. W. Johnson, M. Lorenzen, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1400, 24–32 (1990).

Slaymaker, F. H.

Smorenburg, C.

Smythe, R.

P. de Groot, R. Smythe, L. Deck, “Laser diodes map surface features of complex parts,” Laser Focus World, February1994, pages 95–98.

Spolaczyk, R.

Streibl, N.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

K. Kinnstätter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef]

van Wingerden, J.

White, A. D.

Williams, C. S.

C. S. Williams, Designing Digital Filters (Prentice-Hall, Engle-wood Cliffs, N.J., 1986), Chap. 4, p. 113.

Wyant, J. C.

Zöller, A.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Appl. Opt. (12)

J. H. Bruning, D. R. Herriott, J. E. Galllagher, 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]

R. C. Moore, F. H. Slaymaker, “Direct measurement of phase in a spherical wave Fizeau interferometer,” Appl. Opt. 19, 2196–2200 (1980).
[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]

Y. Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

K. Kinnstätter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef]

J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
[CrossRef] [PubMed]

J. van Wingerden, H. H. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
[CrossRef] [PubMed]

P. de Groot, “Three-color laser-diode interferometer,” Appl. Opt. 30, 3612–3616 (1991).
[CrossRef] [PubMed]

K. Creath, P. Hariharan, “Phase-shifting errors in interferometric tests with high-numerical-aperture reference surfaces,” Appl. Opt. 33, 24–25 (1994).
[CrossRef] [PubMed]

C. Joenathan, “Phase-measuring interferometry: new methods and error analysis,” Appl. Opt. 33, 4147–4155 (1994).
[CrossRef] [PubMed]

G. Schulz, K.-E. Elssner, “Errors in phase-measurement interferometry with high numerical apertures,” Appl. Opt. 30, 4500–4506 (1991).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

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

Laser Focus World (1)

P. de Groot, R. Smythe, L. Deck, “Laser diodes map surface features of complex parts,” Laser Focus World, February1994, pages 95–98.

Opt. Eng. (1)

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Photon. Spectra (1)

C. L. Koliopoulos, “Avoiding phase-measuring interferometry's pitfalls,” Photon. Spectra 22, 169–176 (1988).

Proc. IEEE (1)

F J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Other (11)

P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data sampling window,” Appl. Opt.34 (to be published).
[PubMed]

P. de Groot, L. Deck, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2248 (1994).

C. S. Williams, Designing Digital Filters (Prentice-Hall, Engle-wood Cliffs, N.J., 1986), Chap. 4, p. 113.

K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. Soc. Photo-Opt. Instrum. Eng.680, 19–28 (1986).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. XXVI, Chap. 5, pp. 349–393.
[CrossRef]

J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” in Optical Fabrication and Testing Workshop, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper PD-4.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 14, pp. 501–598.

R. P. Grosso, R. Crane, “Precise optical evaluation using phase measuring interferometric techniques,” in Interferometry, G. W. Hopkins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.192, 65–74 (1979).

P. Hariharan, Optical Interferometry (Academic, Orlando, Fla., 1985), Chap. 9, pp. 155–159.

L. A. Selberg, “Interferometer accuracy and precision,” in Optical Fabrication and Testing, D. R. Campbell, C. W. Johnson, M. Lorenzen, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1400, 24–32 (1990).

P. de Groot, “Predicting the effects of vibration in phase shifting interferometry,” in Optical Fabrication and Testing Workshop, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 189–192.

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 (5)

Fig. 1
Fig. 1

Spherical Fizeau interferometer cavity. The PZT assembly translates the reference surface axially, thereby shifting the mean phase of the interference pattern. The amount of shift varies with angle φ between radius vector r ̂ and translation vector p ̂ .

Fig. 2
Fig. 2

Phase-shift error ε as a function of normalized field coordinate x/X in a spherical Fizeau interferometer cavity.

Fig. 3
Fig. 3

Maximum N.A. for a spherical Fizeau cavity as a function of maximum allowable calibration error εmax The maximum calibration error is characteristic of the PSI algorithm.

Fig. 4
Fig. 4

P−V excursion of periodic phase errors as a function of calibration error ε for four different PSI algorithms. All of these example algorithms have a nominal a0 = π/2 phase shift. Actual phase shift α is equal to (1 − ε)α0.

Fig. 5
Fig. 5

Variation in effective fringe contrast for four different PSI algorithms as a function of calibration error ε.

Tables (1)

Tables Icon

Table 1 Summary of Results

Equations (40)

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

g j = Q [ 1 + V cos ( θ + ϕ j ) ] ,
ϕ j = ( j j 0 ) α 0 .
θ = tan 1 ( s j g j c j g j ) + const ,
θ = tan 1 [ 2 ( g 1 g 3 ) 2 g 2 ( g 0 + g 4 ) ] .
α = ( 1 ε ) α 0 ,
ϕ j = ( α / α 0 ) ϕ j .
γ ( x , y ) = γ 0 + x γ x + y γ y .
ε = 1 γ ( x , y ) p ̂ · r ̂ .
r ̂ = x R n ̂ x + y R n ̂ y + R R n ̂ z ,
R = [ R 2 ( x 2 + y 2 ) ] 1 / 2 ,
p ̂ = ξ x n ̂ x + ξ y n ̂ y + ρ n ̂ z ,
ρ = [ 1 ( ξ x 2 + ξ y 2 ) ] 1 / 2 .
ε = 1 ( γ 0 + x γ x + y γ y ) R ρ x ξ x y ξ y R .
γ 0 = 2 2 ε 1 ,
N . A . = X / R .
N . A . ( ε max ) = ( 1 1 ε max 1 + ε max ) 1 / 2 .
E ( π ε ) 2 8 .
θ = tan 1 [ 2 ( g 1 g 3 ) 2 g 2 ( g 0 + g 4 ) sin ( α ) ] .
θ = arg ( G ) + const ,
G = j g j w j exp ( i ϕ j ) .
s j = Im { w j exp ( i ϕ j ) } ,
c j = Re { w j exp ( i ϕ j ) } ,
w j = [ s j sin ( ϕ j ) + c j cos ( ϕ j ) ] + i [ s j cos ( ϕ j ) c j sin ( ϕ j ) ] .
G ( α , θ ) = Q j { 1 + V cos [ θ + ( α α 0 ) ϕ j ] } w j exp ( i ϕ j ) ,
G ( α , θ ) = Q { W ( α 0 ) + ½ V [ W ( α 0 α ) exp ( i θ ) + W ( α 0 + α ) exp ( i θ ) ] } ,
W ( α ) = j w j exp ( i ϕ j α α 0 )
θ = arg [ G ( α , θ ) ] Δ θ ( α , θ ) + const ,
Δ θ ( α , θ ) = arg [ W ( α 0 α ) + W ( α 0 + α ) exp ( 2 i θ ) W ( 0 ) ] .
Δ θ ( ε , θ ) = arg { W 0 ( ε ) + W 2 ( ε ) exp [ 2 i θ + i ζ ( ε ) ] } + ξ ( ε ) ,
W 0 ( ε ) = | W ( ε α 0 ) W ( 0 ) | ,
W 2 ( ε ) = | W [ ( 2 ε ) α 0 ] W ( 0 ) | ,
ξ ( ε ) = arg [ W ( ε α 0 ) W ( 0 ) ] ,
ζ ( ε ) = arg { W [ ( 2 ε ) α 0 ] W ( 0 ) } ξ ( ε ) .
cos [ 2 θ + ζ ( ε ) ] = W 2 ( ε ) W 0 ( ε ) .
E ( ε ) = 2 tan 1 { W 2 ( ε ) [ W 0 ( ε ) 2 W 2 ( ε ) 2 ] 1 / 2 } .
E ( ε ) 2 W 2 ( ε ) .
V ( ε ) = V W 0 ( ε ) .
θ = tan 1 ( g 0 g 2 g 1 g 3 ) .
θ = tan 1 [ ( g 0 g 6 ) 7 ( g 2 g 4 ) 4 ( g 1 + g 5 ) 8 g 3 ] .
θ = tan 1 [ 0.006 ( g 0 g 14 ) + 0.166 ( g 2 g 12 ) 0.87 ( g 4 g 10 ) + 1.828 ( g 6 g 8 ) 0.043 ( g 1 + g 13 ) + 0.435 ( g 3 + g 11 ) 1.392 ( g 5 + g 9 ) + 2 g 7 ] .

Metrics