Abstract

I propose a systematic way to derive efficient, error-compensating algorithms for phase-shifting interferometry by integer approximation of well-known data-sampling windows. The theoretical basis of the approach is the observation that many of the common sources of phase-estimation error can be related to the frequency-domain characteristics of the sampling window. Improving these characteristics can therefore improve the overall performance of the algorithm. Analysis of a seven-frame example algorithm demonstrates an exceptionally good resistance to first- and second-order distortions in the phase shift and a much reduced sensitivity to low-frequency mechanical vibration.

© 1995 Optical Society of America

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References

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  1. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.
  2. P. Hariharan, Optical Interferometry (Academic, Orlando, Fla., 1985).
  3. 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).
  4. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  5. 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]
  6. J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” in Optical Fabrication and Testing Workshop, Vol. 13 of OSA 1994 Technical Digest Series (Optical Society of America, Washington, D.C., 1994), Postdeadline paper PD4.
  7. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  8. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  9. 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]
  10. 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).
  11. P. de Groot, “Phase-shift calibration errors in interferometers with spherical Fizeau cavities,” Appl. Opt. 34, 2856–2863 (1995).
    [CrossRef]
  12. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]
  13. C. S. Williams, Designing Digital Filters (Prentice-Hall, Engle-wood Cliffs, N.J., 1986), Chap. 4.
  14. K. Creath, P. Hariharan, “Phase-shifting errors in interferometric tests with high-numerical aperture reference surfaces,” Appl. Opt. 33, 24–25 (1994).
    [CrossRef] [PubMed]
  15. 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, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2248, 136–140 (1994).
  16. J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]
  17. P. de Groot, “Predicting the effects of vibration in phase shifting interferometry,” in Optical Fabrication and Testing Workshop, Vol. 13 of OSA 1994 Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 189–192.
  18. P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
    [CrossRef]

1995 (2)

1994 (1)

1992 (1)

1991 (1)

1990 (1)

1987 (1)

1983 (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.

Bruning, J. H.

Burow, R.

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]

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).

J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” in Optical Fabrication and Testing Workshop, Vol. 13 of OSA 1994 Technical Digest Series (Optical Society of America, Washington, D.C., 1994), Postdeadline paper PD4.

de Groot, P.

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

P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
[CrossRef]

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

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, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2248, 136–140 (1994).

Deck, L.

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, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2248, 136–140 (1994).

Eiju, T.

Elssner, K.-E.

Frankena, H. J.

Freischlad, K.

Gallagher, 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.

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.

Koliopoulos, C. L.

Larkin, K. G.

Merkel, K.

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 OSA 1994 Technical Digest Series (Optical Society of America, Washington, D.C., 1994), Postdeadline paper PD4.

Schwider, J.

Smorenburg, C.

Spolaczyk, R.

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.

Appl. Opt. (6)

J. Opt. Soc. Am. A (3)

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 (8)

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

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, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2248, 136–140 (1994).

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

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

P. Hariharan, Optical Interferometry (Academic, Orlando, Fla., 1985).

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).

J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” in Optical Fabrication and Testing Workshop, Vol. 13 of OSA 1994 Technical Digest Series (Optical Society of America, Washington, D.C., 1994), Postdeadline paper PD4.

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).

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Figures (7)

Fig. 1
Fig. 1

Comparison of two data-sampling windows. The time scale is normalized to the period of the fundamental phase-shift frequency v0. As the number of data samples increases, it becomes increasingly more practical to design integer-math PSI algorithms that approximate a Von Hann window.

Fig. 2
Fig. 2

Positive-frequency portions of the Fourier transforms of the windows shown in Fig. 1. The frequency-space graph shows rapid variations around v = 2v0 for the rectangular window, which in practice results in high sensitivity to phase-shift errors. The transform of the Von Hann window, on the other hand, is relatively flat in the neighborhood of v = 2v0, resulting in improved performance.

Fig. 3
Fig. 3

Theoretical P–V surface measurement error in PSI that is due to miscalibration of the phase shift. The graph compares the seven-frame algorithm in Table 1 with the Schwider–Hariharan five-frame algorithm for an illumination wavelength of 600 nm. An arbitrary offset of 0.3 nm has been added to both curves.

Fig. 4
Fig. 4

Theoretical P–V surface measurement error in PSI that is due to quadratic nonlinearities in the phase shift. The graph compares the seven- and five-frame algorithms for an illumination wavelength of 600 nm.

Fig. 5
Fig. 5

Theoretical rms surface measurement error in PSI that is due to mechanical vibration during the measurement. The graph compares the sensitivity of the seven- and five-frame algorithms as a function of vibrational frequency. An arbitrary offset of 0.11 nm has been added to both curves.

Fig. 6
Fig. 6

Experimental P–V surface measurement error in PSI that is due to a deliberate miscalibration of the phase shift. These data were obtained with a white-light interferometric microscope and a mean illumination wavelength of 600 nm. Compare with Fig. 3.

Fig. 7
Fig. 7

Experimental surface maps of residual errors when an interferometric microscope was subjected to a 1-Hz vibration having an amplitude equivalent to one-quarter fringe. There were approximately three fringes in the field of view, and the camera frame rate was 25 Hz. The seven-frame algorithm has a greater resistance to low-frequency vibrations than the five-frame algorithm.

Tables (1)

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Table 1 Examples of Algorithms Derived from an Integer Approximation to the Von Hann Window

Equations (34)

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g ( θ , t ) = Q { 1 + V cos [ θ + ϕ ( t ) ] } ,
G ( θ , v ) = - g ( θ , t ) w ( t ) exp ( - i 2 π v t ) d t .
ϕ ( t ) 2 π v 0 t .
G ( θ , v ) = Q { W ( v ) + ½ V [ W ( v - v 0 ) exp ( i θ ) + W ( v + v 0 ) exp ( - i θ ) ] } ,
W ( v 0 ) = 0 ,
W ( 2 v 0 ) = 0 ,
θ = tan - 1 ( T ) + const ,
T = Im { G ( θ , v 0 ) } Re { G ( θ , v 0 ) } ,
w ( θ , t ) = j w j δ [ t - ( ϕ j / 2 π v 0 ) ] ,
G ( θ , v 0 ) = j g j w j exp ( - i ϕ j ) .
T = j s j g j / j c j g 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 ) ] .
j s j = 0 ,             j c j = 0 .
j s j sin ( - ϕ j ) = j c j cos ( ϕ j ) ,
j s j cos ( ϕ j ) = - j c j sin ( - ϕ j ) .
s j = w j sin ( - ϕ j ) ,
c j = w j cos ( ϕ j ) .
w j = 0.5 + 0.5 cos [ 2 π P ( j - P 2 ) ] ,
P = N 2 π α ,
s j = round [ X w j sin ( - ϕ j ) ] ,
c j = round [ X w j cos ( ϕ j ) ] ,
ϕ j = ( j - P - 1 2 ) α ,
w ˜ j = w j + w j - 1 ,
T = 3 g 1 - 4 g 3 + g 5 - g 0 + 4 g 2 - 3 g 4 .
θ = tan - 1 ( T ) ,
T = 7 ( g 2 - g 4 ) - ( g 0 - g 6 ) - 4 ( g 1 + g 5 ) + 8 g 3 .
φ j = ( j - 3 ) ( π / 2 ) ,
φ j = [ π 2 - + ( j - 3 ) γ ] ( j - 3 ) ,
Δ θ seven = 2 γ - ( 3 γ 2 / 4 ) sin ( 2 θ ) + ( 4 / 16 ) sin ( 2 θ ) + .
Δ θ five = 3 γ / 2 - ( γ / 2 ) cos ( 2 θ ) + ( 2 / 4 ) sin ( 2 θ ) + .
E seven = A 0 | cos ( v π / 2 ) sin 2 ( v π / 4 ) 2 + cos ( 3 v π / 2 ) - cos ( v π / 2 ) 32 | ,
E five = A 0 | cos ( v π / 2 ) sin 2 ( v π / 4 ) 2 | .

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