Abstract

A novel angle-measurement technique based on fringe analysis for phase-measuring profilometry is proposed. A two-dimensional (2-D) angle between two mirror surfaces is determined by least-squares fitting of a plane to the 2-D distribution of the phase difference introduced by the 2-D tilt angle. To evaluate the performance of the proposed technique, numerical simulations that use the Fourier-transform technique and the phase-shift technique for fringe analysis were performed, and the results are compared. A 2-D angle-measurement interferometer based on a Mirau interference microscope was developed that demonstrated the validity of the proposed principle. It is shown by simulation and experiment that the proposed 2-D angle-measurement technique can achieve both a wide measurement range and a high angular resolution simultaneously.

© 2003 Optical Society of America

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References

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  1. J. Rohlin, “An interferometer for precision angle measurement,” Appl. Opt. 2, 762–763 (1963).
    [CrossRef]
  2. D. Malacara, Garries, “Interferometric measurement of angles,” Appl. Opt. 9, 1630–1633 (1970).
    [CrossRef] [PubMed]
  3. G. D. Chapman, “Interferometric angular measurement,” Appl. Opt. 13, 1646–1651 (1974).
    [CrossRef] [PubMed]
  4. P. Shi, E. Stujns, “New optical methods for measuring small-angle rotations,” Appl. Opt. 27, 4342–4344 (1988).
    [CrossRef] [PubMed]
  5. P. R. Yoder, E. R. Schlesinger, J. L. Chickvary, “Active annular-beam laser autocollimator system,” Appl. Opt. 14, 1890–1895 (1975).
    [CrossRef] [PubMed]
  6. F. J. Shuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
    [CrossRef]
  7. A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Prec. Eng. 4, 5–8 (1982).
    [CrossRef]
  8. P. S. Huang, S. Kiyono, O. Kamada, “Angle measurement based on the internal reflection effect,” Appl. Opt. 31, 6047–6055 (1992).
    [CrossRef] [PubMed]
  9. P. S. Huang, J. Ni, “Angle measurement based on the internal reflection effect and the use of right-angle prisms,” Appl. Opt. 34, 4976–4981 (1995).
    [CrossRef] [PubMed]
  10. P. S. Huang, J. Ni, “Angle measurement based on the internal reflection effect and the use of elongated critical-angle prisms,” Appl. Opt. 35, 2239–2241 (1996).
    [CrossRef] [PubMed]
  11. P. S. Huang, Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37, 6636–6642 (1998).
    [CrossRef]
  12. Technical data for the electronic autocollimator Elcomat HR (Moeller-Wedel Optical GmbH, Rosengarten 10-22880 Wedel, Germany).
  13. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  14. C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [CrossRef] [PubMed]
  15. D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe pattern analysis using 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
    [CrossRef]
  16. J. H. Brunining, D. R. Herriott, J. E. Gallager, D. P. Rosenfel, A. D. White, D. J. Brangaccio, “Digital wave front measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef]
  17. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
    [CrossRef] [PubMed]
  18. P. D. Groot, “Phase-shift calibration errors in interferometers with spherical Fizeau cavities,” Appl. Opt. 34, 2856–2863 (1995).
    [CrossRef] [PubMed]
  19. H. Takasaki, “Moiré topography,” Appl. Opt. 9, 1467–1472 (1970).
    [CrossRef] [PubMed]
  20. M. Idesawa, T. Yatagai, T. Soma, “Scanning moiré method and automatic measurement of 3D shape,” Appl. Opt. 16, 2152–2162 (1977).
    [CrossRef] [PubMed]
  21. Z. T. Ge, M. Takeda, “High precision 2-D angle measurement using fringe analysis techniques,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE Vol. 4451, 448–457 (2001).
    [CrossRef]
  22. Z. T. Ge, M. Takeda, “A high precision 2-D angle measurement interferometer,” in Interferometry XI: Applications, W. Osten, ed., Proc. SPIE4778, 277–287 (2002).
    [CrossRef]
  23. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]
  24. S. Kiyono, Z. T. Ge, “Sub-nanometric calibration of a differential interferometer,” Precis. Eng. 19, 187–197 (1996).
    [CrossRef]

1998 (1)

1996 (2)

1995 (2)

1992 (1)

1988 (1)

1987 (2)

1986 (1)

1983 (1)

F. J. Shuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
[CrossRef]

1982 (2)

A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Prec. Eng. 4, 5–8 (1982).
[CrossRef]

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

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]

1977 (1)

1975 (1)

1974 (2)

1970 (2)

1963 (1)

Bachor, H.-A.

Bone, D. J.

Brangaccio, D. J.

Brunining, J. H.

Chapman, G. D.

Chickvary, J. L.

Eiju, T.

Ennos, A. E.

A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Prec. Eng. 4, 5–8 (1982).
[CrossRef]

Gallager, J. E.

Garries,

Ge, Z. T.

S. Kiyono, Z. T. Ge, “Sub-nanometric calibration of a differential interferometer,” Precis. Eng. 19, 187–197 (1996).
[CrossRef]

Z. T. Ge, M. Takeda, “High precision 2-D angle measurement using fringe analysis techniques,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE Vol. 4451, 448–457 (2001).
[CrossRef]

Z. T. Ge, M. Takeda, “A high precision 2-D angle measurement interferometer,” in Interferometry XI: Applications, W. Osten, ed., Proc. SPIE4778, 277–287 (2002).
[CrossRef]

Groot, P. D.

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.

Huang, P. S.

Idesawa, M.

Ina, H.

Kamada, O.

Kiyono, S.

S. Kiyono, Z. T. Ge, “Sub-nanometric calibration of a differential interferometer,” Precis. Eng. 19, 187–197 (1996).
[CrossRef]

P. S. Huang, S. Kiyono, O. Kamada, “Angle measurement based on the internal reflection effect,” Appl. Opt. 31, 6047–6055 (1992).
[CrossRef] [PubMed]

Kobayashi, S.

Li, Y.

Malacara, D.

Ni, J.

Oreb, B. F.

Roddier, C.

Roddier, F.

Rohlin, J.

Rosenfel, D. P.

Sandeman, R. J.

Schlesinger, E. R.

Shi, P.

Shuda, F. J.

F. J. Shuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
[CrossRef]

Soma, T.

Stujns, E.

Takasaki, H.

Takeda, M.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

Z. T. Ge, M. Takeda, “A high precision 2-D angle measurement interferometer,” in Interferometry XI: Applications, W. Osten, ed., Proc. SPIE4778, 277–287 (2002).
[CrossRef]

Z. T. Ge, M. Takeda, “High precision 2-D angle measurement using fringe analysis techniques,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE Vol. 4451, 448–457 (2001).
[CrossRef]

Virdee, M. S.

A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Prec. Eng. 4, 5–8 (1982).
[CrossRef]

White, A. D.

Yatagai, T.

Yoder, P. R.

Appl. Opt. (16)

H. Takasaki, “Moiré topography,” Appl. Opt. 9, 1467–1472 (1970).
[CrossRef] [PubMed]

D. Malacara, Garries, “Interferometric measurement of angles,” Appl. Opt. 9, 1630–1633 (1970).
[CrossRef] [PubMed]

G. D. Chapman, “Interferometric angular measurement,” Appl. Opt. 13, 1646–1651 (1974).
[CrossRef] [PubMed]

J. H. Brunining, D. R. Herriott, J. E. Gallager, D. P. Rosenfel, A. D. White, D. J. Brangaccio, “Digital wave front measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef]

P. R. Yoder, E. R. Schlesinger, J. L. Chickvary, “Active annular-beam laser autocollimator system,” Appl. Opt. 14, 1890–1895 (1975).
[CrossRef] [PubMed]

M. Idesawa, T. Yatagai, T. Soma, “Scanning moiré method and automatic measurement of 3D shape,” Appl. Opt. 16, 2152–2162 (1977).
[CrossRef] [PubMed]

D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe pattern analysis using 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
[CrossRef]

P. Shi, E. Stujns, “New optical methods for measuring small-angle rotations,” Appl. Opt. 27, 4342–4344 (1988).
[CrossRef] [PubMed]

P. S. Huang, Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37, 6636–6642 (1998).
[CrossRef]

P. D. Groot, “Phase-shift calibration errors in interferometers with spherical Fizeau cavities,” Appl. Opt. 34, 2856–2863 (1995).
[CrossRef] [PubMed]

P. S. Huang, J. Ni, “Angle measurement based on the internal reflection effect and the use of right-angle prisms,” Appl. Opt. 34, 4976–4981 (1995).
[CrossRef] [PubMed]

P. S. Huang, J. Ni, “Angle measurement based on the internal reflection effect and the use of elongated critical-angle prisms,” Appl. Opt. 35, 2239–2241 (1996).
[CrossRef] [PubMed]

C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
[CrossRef] [PubMed]

P. S. Huang, S. Kiyono, O. Kamada, “Angle measurement based on the internal reflection effect,” Appl. Opt. 31, 6047–6055 (1992).
[CrossRef] [PubMed]

J. Rohlin, “An interferometer for precision angle measurement,” Appl. Opt. 2, 762–763 (1963).
[CrossRef]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

Prec. Eng. (1)

A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Prec. Eng. 4, 5–8 (1982).
[CrossRef]

Precis. Eng. (1)

S. Kiyono, Z. T. Ge, “Sub-nanometric calibration of a differential interferometer,” Precis. Eng. 19, 187–197 (1996).
[CrossRef]

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]

Rev. Sci. Instrum. (1)

F. J. Shuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
[CrossRef]

Other (3)

Technical data for the electronic autocollimator Elcomat HR (Moeller-Wedel Optical GmbH, Rosengarten 10-22880 Wedel, Germany).

Z. T. Ge, M. Takeda, “High precision 2-D angle measurement using fringe analysis techniques,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE Vol. 4451, 448–457 (2001).
[CrossRef]

Z. T. Ge, M. Takeda, “A high precision 2-D angle measurement interferometer,” in Interferometry XI: Applications, W. Osten, ed., Proc. SPIE4778, 277–287 (2002).
[CrossRef]

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Figures (18)

Fig. 1
Fig. 1

Simulated interferogram of the plane.

Fig. 2
Fig. 2

Fitting errors of the FFT method (a) without and (b) with a window function.

Fig. 3
Fig. 3

Nonlinearity of the FFT method when the number of fringes varies from (a) 20 to 24 and (b) 7.5 to 225.

Fig. 4
Fig. 4

Fitting error of the PSI method with accurate phase shifts.

Fig. 5
Fig. 5

Nonlinearity of the PSI method with accurate phase shifts.

Fig. 6
Fig. 6

Fitting error of the PSI method with inaccurate phase shifts.

Fig. 7
Fig. 7

Nonlinearity of the PSI method with inaccurate phase shifts.

Fig. 8
Fig. 8

Schematic of the optical setup of the interferometer.

Fig. 9
Fig. 9

Interferogram in a small-measurement range.

Fig. 10
Fig. 10

Calibration curve of the interferometric angle sensor.

Fig. 11
Fig. 11

Comparison of the PSI and FFT methods.

Fig. 12
Fig. 12

Interferogram in a large-measurement range.

Fig. 13
Fig. 13

Calibration curve in a large measurement range.

Fig. 14
Fig. 14

Calibration curve obtained by use of a PZT actuator.

Fig. 15
Fig. 15

Drift of the interferometric angle sensor.

Fig. 16
Fig. 16

Simulated interferogram of the aspheric wave front.

Fig. 17
Fig. 17

Fit of the aspheric wave front obtained in this study.

Fig. 18
Fig. 18

Nonlinearity when the aspheric wave front was measured.

Equations (15)

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

gx, y=ax, y+bx, ycosφx, y,
φx, y=px+qy+rx, y=2πfxx+2πfyy+rx, y,
p=2πfx=2π tan θxλ,q=2πfy=2π tan θyλ.
gx, y=ax, y+cx, y+c*x, y,
cx, y=½bx, yexp i2πfxx+2πfyy+rx, y,
Gη, ζ=Aη, ζ+Cη, ζ+C*-η, -ζ,
φx, y=tan-1Recx, yImcx, y,
Δφx, y=φx, y-φ0x, y=p-p0x+q-q0y+rx, y-r0x, y,
Δφx, y=2πλtan θx-tan θx0x+tan θy-tan θy0y.
Δφx, y=tan-1Imcx, yc0x, yRecx, yc0x, y.
Δθx=θx-θx0=tan θx-tan θx0=λ2π p,Δθy=θy-θy0=tan θy-tan θy0=λ2π q,
gjx, y=ax, y+bx, ycosφx, y+δjj=0, 1,4,
δj=jπ/2 j=0, 1,, 4.
φx, y=tan-12g1-g32g2-g0+g4.
Δφx, y=φx, y-φ0x, y=p-p0x+q-q0y =tan-12g1-g32g20-g00+g40-2g10-g302g2-g0+g42g20-g00+g402g2-g0+g4+4g1-g3g10-g30.

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