Abstract

Testing for flatness of an optical parallel plate in a Fizeau interferometer suffers from problems caused by multiple-beam interference noise. Each internal-reflection component can be separated from the signal by its modulation frequency in a wavelength-scanned interferometer; however, the frequency depends on the thickness and the refractive-index dispersion of the test plate and on the nonlinearity of the scanning source. With a new 19-sample wavelength-scanning algorithm we demonstrate the elimination of the reflection noise, the effect of the dispersion up to the second order of the reflectance of the test plate, and as the nonlinearity of the source. The algorithm permits large tolerance in the air-gap distance, thus making it somewhat independent of the thickness of the test plate. The minimum residual reflection noise with this algorithm for testing a glass plate is ∼λ/600. Experimental results show that the front surface of the test plate was measured within 1 nm rms of its true shape over a 230-mm-diameter aperture.

© 2003 Optical Society of America

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References

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  1. M. V. Mantravadi, “Testing nearly parallel plates,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 22.
  2. 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]
  3. C. Ai, J. C. Wyant, “Testing an optical window of a small wedge angle: effect of multiple reflections,” Appl. Opt. 32, 4904–4912 (1993).
    [CrossRef] [PubMed]
  4. J. Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]
  5. F. Lexer, C. K. Hitzenberger, A. F. Fercher, M. Kulhavy, “Wavelength-tuning interferometry of intraocular distances,” Appl. Opt. 36, 6548–6552 (1997).
    [CrossRef]
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    [CrossRef]
  7. K. Okada, H. Sakuta, T. Ose, J. Tsujiuchi, “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable-source phase shifting interferometry,” Appl. Opt. 29, 3280–3285 (1990).
    [CrossRef] [PubMed]
  8. P. J. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39, 2658–2663 (2000).
    [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]
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    [CrossRef]
  11. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  14. K. Hibino, “Error-compensating phase measuring algorithms in a phase shifting Fizeau interferometer,” Opt. Rev. 6, 529–538 (1999).
    [CrossRef]
  15. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  19. B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, P. S. Fairman, “Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. 39, 5161–5171 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2002 (1)

K. Hibino, T. Takatsuji, “Suppression of multiple-beam interference noise in testing an optical parallel plate by wavelength-scanning interferometry,” Opt. Rev. 9, 60–65 (2002).
[CrossRef]

2000 (2)

1999 (2)

K. Hibino, “Error-compensating phase measuring algorithms in a phase shifting Fizeau interferometer,” Opt. Rev. 6, 529–538 (1999).
[CrossRef]

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

1998 (2)

1997 (2)

1995 (1)

1993 (1)

1992 (1)

1991 (1)

1990 (1)

1987 (1)

1985 (1)

1983 (1)

1974 (1)

Ai, C.

Brangaccio, D. J.

Brohinsky, W. R.

Bruning, J. H.

Burow, R.

Chen, J.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.
[CrossRef]

de Groot, P. J.

Deck, L. L.

L. L. Deck, “Multiple surface phase shifting interferometry,” in Optical Manufacturing and Testing IV, International Symposium on Optical Science, Engineering and Instrumentation, H. P. Stahl, ed., Proc. SPIE4451, 424–431 (2001).
[CrossRef]

Elssner, K.-E.

Fairman, P. S.

B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, P. S. Fairman, “Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. 39, 5161–5171 (2000).
[CrossRef]

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Farrant, D. I.

Fercher, A. F.

Forbes, G.

Frankena, H. J.

Freund, C. H.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Gallagher, J. E.

Gilliand, Y.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Grzanna, J.

Herriott, D. R.

Hibino, K.

Hiratsuka, H.

Hitzenberger, C. K.

Ishii, Y.

Kido, E.

Kulhavy, M.

Larkin, K. G.

Leistner, A. J.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Lexer, F.

Mantravadi, M. V.

M. V. Mantravadi, “Testing nearly parallel plates,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 22.

Merkel, K.

Murata, K.

Okada, K.

Oreb, B. F.

Ose, T.

Rosenfeld, D. P.

Sakuta, H.

Schwider, J.

Seckold, J. A.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Smorenburg, C.

Spolaczyk, R.

Stetson, K. A.

Takatsuji, T.

K. Hibino, T. Takatsuji, “Suppression of multiple-beam interference noise in testing an optical parallel plate by wavelength-scanning interferometry,” Opt. Rev. 9, 60–65 (2002).
[CrossRef]

Tsujiuchi, J.

Walsh, C. J.

B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, P. S. Fairman, “Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. 39, 5161–5171 (2000).
[CrossRef]

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Ward, B. K.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

White, A. D.

Wingerden, J.

Wyant, J. C.

Yoshimura, T.

Appl. Opt. (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]

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]

K. A. Stetson, W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631–3637 (1985).
[CrossRef]

K. Okada, H. Sakuta, T. Ose, J. Tsujiuchi, “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable-source phase shifting interferometry,” Appl. Opt. 29, 3280–3285 (1990).
[CrossRef] [PubMed]

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

C. Ai, J. C. Wyant, “Testing an optical window of a small wedge angle: effect of multiple reflections,” Appl. Opt. 32, 4904–4912 (1993).
[CrossRef] [PubMed]

F. Lexer, C. K. Hitzenberger, A. F. Fercher, M. Kulhavy, “Wavelength-tuning interferometry of intraocular distances,” Appl. Opt. 36, 6548–6552 (1997).
[CrossRef]

P. J. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39, 2658–2663 (2000).
[CrossRef]

B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, P. S. Fairman, “Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. 39, 5161–5171 (2000).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Opt. Lett. (2)

Opt. Rev. (2)

K. Hibino, T. Takatsuji, “Suppression of multiple-beam interference noise in testing an optical parallel plate by wavelength-scanning interferometry,” Opt. Rev. 9, 60–65 (2002).
[CrossRef]

K. Hibino, “Error-compensating phase measuring algorithms in a phase shifting Fizeau interferometer,” Opt. Rev. 6, 529–538 (1999).
[CrossRef]

Other (3)

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.
[CrossRef]

M. V. Mantravadi, “Testing nearly parallel plates,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 22.

L. L. Deck, “Multiple surface phase shifting interferometry,” in Optical Manufacturing and Testing IV, International Symposium on Optical Science, Engineering and Instrumentation, H. P. Stahl, ed., Proc. SPIE4451, 424–431 (2001).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of apparatus for testing a near-parallel plate in a wavelength-scanned Fizeau interferometer: abbreviations defined in text. Grating G and mirror M were rotated by a PZT for the wavelength scan.

Fig. 2
Fig. 2

Sampling functions of (a) the new 19-sample algorithm and (b) a 64-sample Fourier algorithm.

Fig. 3
Fig. 3

PV phase errors (for three algorithms) caused by multiple-beam interference noise as a function of the dispersion of the test plate. It is assumed that R = 0.04 and g = 3, 9.

Fig. 4
Fig. 4

PV phase error caused by multiple-beam interference noise as a function of geometric parameter g.

Fig. 5
Fig. 5

Interference fringes observed for a near-parallel BK7 plate.

Fig. 6
Fig. 6

Surface shape of the BK7 plate measured by the new algorithm over an aperture of 230-mm diameter: (a) phase map of the surface relief, (b) diameter relief profile of the surface along the horizontal (x) direction.

Tables (1)

Tables Icon

Table 1 Frequencies of Interference Signals in a Wavelength-Scanned Fizeau Interferometera

Equations (25)

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

Ix, y, t=s0+s1 cosφ1x, y+ν1t+k=2,3, sk cosφkx, y+νkt,
ν1=4πLλ02dλdt,
g=n0T/L,
nλ0+δλ=n0+dndλ0δλ+12d2ndλ20δλ2+
Ψk=4πλpL+qnλT+δp,q =4πλ0+δλpL+qn0+dndλ0δλ+12d2ndλ20δλ2+T+δp,q,
νk=dΨkdt=ν1p+qg1-λ0n0dndλ0-ν1t8πλ02n0d2ndλ20λ0L.
sk=sk1+εp1ν1t+εp2ν1t2+, k=1, 2, 3,,
dλdt=dλdt1+εw1+εw2ν1tπ+,
ν1ν11+εp1+εw1+εp2+εw2ν1tπ,
νkν1p+qg1-λ0n0dndλ0+p+qgεp1+εw1+p+qgεp2+εw2ν1tπ.
φ=arctanr=1m brIx, y, trr=1m arIx, y, tr,
F1ν=r=1m br exp-iνtr,
F2ν=r=1m ar exp-iνtr.
ddνF1ν+iF2νν=ν1=0.
F1νk=F2νk=0, νk=0, 2ν1, 3ν1,, 10ν1.
ddνF1ννk=ddνF2ννk=0, νk=3ν1, 4ν1.
F1,23.5ν10,
F1,24.5ν10.
ar=a1, a2, a3,, a9, a10, a9,, a3, a2, a1, br=-b1, -b2, -b3,,-b9, 0, b9,, b3, b2, b1.
Sa2, b2=|F13.5ν1|2+|F23.5ν1|2+|F14.5ν1|2 +|F24.5ν1|2.
Sa2=0, Sb2=0.
ar=1483-1, 1-23, -5, -43, -7, -1-23, 1, 43, 12, 83, 12, 43, 1, -1-23, -7, -43, -5, 1-23, -1, br=14833-1, 1, 1, -2, 1-43, -11, -1-73, -12, -43, 0, 43, 12, 1+73, 11, -1+43, 2, -1, -1, -3+1.
ar=132cosπ32r-652, br=132sinπ32r-652, r=1, 2,, 64.
φ=arctan- F1νJν¯dν- F2νJν¯dν,
Δφ/4π=0.018ε2 cos 2φ+odispersion2+oε12,

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