Abstract

A phase-evaluation method of multiple-beam Fizeau patterns that combines two-beam phase-stepping algorithms with the moiré effect was previously reported [Appl. Opt. 34, 3639–3643 (1995)]. The method is based on a multiplicative moiré image-formation process obtained by the direct superposition of high-frequency multiple-beam Fizeau carrier fringes upon a transmission grating (working as a phase modulator). We present a comparison between this multiplicative moiré two-beam phase-stepping method and the well-known Fourier-transform method for the topographic measurement of an undoped silicon wafer. The discrepancy between the two methods yields a rms phase-difference value of the order of (∼2π/90).

© 1998 Optical Society of America

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References

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  1. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, New York, 1992), Chap. 14, pp. 501–598.
  2. M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, D. Robinson, G. T. Reid, eds. (Institute of Physics, Reading, UK, 1993), Chap. 5, pp. 141–193.
  3. D. C. Williams, N. S. Nassar, J. E. Banyard, M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
    [CrossRef]
  4. M. Takeda, H. Ina, S. Kobayashi, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef] [PubMed]
  5. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
    [CrossRef]
  6. A. K. Asundi, K. H. Yung, “Phase-shifting and logical moiré,” J. Opt. Soc. Am. A 8, 1591–1600 (1991).
    [CrossRef]
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    [CrossRef] [PubMed]
  8. S. M. Pandit, N. Jordache, “Interferogram analysis based on the data-dependent systems method for nanometrology applications,” Appl. Opt. 34, 6695–6703 (1995).
    [CrossRef] [PubMed]
  9. O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9, 59–61 (1984).
    [CrossRef] [PubMed]
  10. J. H. Bruning, D. R. Herriot, J. E. Gallager, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
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    [CrossRef]
  12. G. W. Johnson, D. C. Leiner, D. T. Moore, “Phase-locked interferometry,” Opt. Eng. 18, 46–52 (1979).
  13. O. Sasaki, H. Okazaki, “Sinusoidal phase modulating interferometry for surface profile measurement,” Appl. Opt. 25, 3137–3140 (1986).
    [CrossRef] [PubMed]
  14. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, “Semiconductor wafer technical flat planeness testing interferometer,” Appl. Opt. 25, 1117–1121 (1986).
    [CrossRef] [PubMed]
  15. B. V. Dorrío, A. F. Doval, C. López, R. Soto, J. Blanco-García, J. L. Fernández, M. Pérez-Amor, “Fizeau phase-measuring interferometry using the moiré effect,” Appl. Opt. 34, 3639–3643 (1995).
    [CrossRef] [PubMed]
  16. B. V. Dorrío, C. López, A. F. Doval, J. M. Alén, J. Bugarín, A. Fernández, J. Blanco-García, J. L. Fernández, M. Pérez-Amor, “Measurement range analysis in Fizeau phase-stepping interferometry using the moiré effect,” Appl. Opt. 36, 3635–3644 (1997).
    [CrossRef] [PubMed]
  17. R. Józwicki, M. Kujawinska, L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
    [CrossRef]
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  19. B. V. Dorrío, J. Blanco-García, A. F. Doval, C. López, R. Soto, J. L. Fernández, M. Pérez-Amor, “Phase error calculation in a Fizeau interferometer by Fourier expansion of the intensity profile,” Appl. Opt. 35, 61–64 (1996).
    [CrossRef] [PubMed]
  20. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Oreb, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
    [CrossRef]
  21. R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1987).
  22. B. V. Dorrío, J. Blanco-García, A. F. Doval, C. López, R. Soto, J. L. Fernández, M. Pérez-Amor, “Analysis of the measurement range in Fizeau interferometry using the moiré effect,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 204–218 (1995).
    [CrossRef]
  23. T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
    [CrossRef]
  24. J. Gu, F. Chen, “Fast Fourier transform, iteration, and least-squares-fit demodulation image processing for analysis of single-carrier fringe pattern,” J. Opt. Soc. Am. A 12, 2159–2164 (1995).
    [CrossRef]
  25. R. Józwicki, “Influence of aberrations interferometer elements on measurements errors,” Appl. Opt. 30, 3126–3132 (1991).
    [CrossRef]
  26. J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. London 62, 405–417 (1949).
    [CrossRef]

1997

1996

1995

1992

R. Józwicki, M. Kujawinska, L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

1991

1986

1984

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9, 59–61 (1984).
[CrossRef] [PubMed]

1983

1979

G. W. Johnson, D. C. Leiner, D. T. Moore, “Phase-locked interferometry,” Opt. Eng. 18, 46–52 (1979).

1978

1974

1949

J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. London 62, 405–417 (1949).
[CrossRef]

Alén, J. M.

Asundi, A. K.

Banyard, J. E.

D. C. Williams, N. S. Nassar, J. E. Banyard, M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Blanco-García, J.

Brangaccio, D. J.

Bruning, J. H.

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

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

Bugarín, J.

Burow, R.

Chen, F.

Dorrío, B. V.

Doval, A. F.

Elssner, K.-E.

Farrant, D. I.

Fernández, A.

Fernández, J. L.

Gallager, J. E.

Gonzalez, R. C.

R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1987).

Greivenkamp, J. E.

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

Grzanna, J.

Gu, J.

Herriot, D. R.

Hibino, K.

Holden, J.

J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. London 62, 405–417 (1949).
[CrossRef]

Holly, S.

Ina, H.

Johnson, G. W.

G. W. Johnson, D. C. Leiner, D. T. Moore, “Phase-locked interferometry,” Opt. Eng. 18, 46–52 (1979).

Jordache, N.

Józwicki, R.

R. Józwicki, M. Kujawinska, L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

R. Józwicki, “Influence of aberrations interferometer elements on measurements errors,” Appl. Opt. 30, 3126–3132 (1991).
[CrossRef]

Kobayashi, S.

Kreis, T.

Kujawinska, M.

R. Józwicki, M. Kujawinska, L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, D. Robinson, G. T. Reid, eds. (Institute of Physics, Reading, UK, 1993), Chap. 5, pp. 141–193.

Kwon, O. Y.

Leiner, D. C.

G. W. Johnson, D. C. Leiner, D. T. Moore, “Phase-locked interferometry,” Opt. Eng. 18, 46–52 (1979).

López, C.

Macy, W. W.

Massie, N. A.

Moore, D. T.

G. W. Johnson, D. C. Leiner, D. T. Moore, “Phase-locked interferometry,” Opt. Eng. 18, 46–52 (1979).

Nassar, N. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Nelson, R. D.

Okazaki, H.

Oreb, B. F.

Oreb, K. G.

Pandit, S. M.

Pérez-Amor, M.

Rosenfeld, D. P.

Salbut, L.

R. Józwicki, M. Kujawinska, L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

Sasaki, O.

Schultz, G.

G. Schultz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (Wiley-Interscience, New York, 1992), pp. 93–167.

Schwider, J.

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, “Semiconductor wafer technical flat planeness testing interferometer,” Appl. Opt. 25, 1117–1121 (1986).
[CrossRef] [PubMed]

G. Schultz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (Wiley-Interscience, New York, 1992), pp. 93–167.

Soto, R.

Spolaczyk, R.

Takeda, M.

Virdee, M. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

White, A. D.

Williams, D. C.

D. C. Williams, N. S. Nassar, J. E. Banyard, M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Wintz, P.

R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1987).

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Yung, K. H.

Appl. Opt.

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

N. A. Massie, R. D. Nelson, S. Holly, “High-performance real-time heterodyne interferometry,” Appl. Opt. 19, 154–160 (1978).
[CrossRef]

W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
[CrossRef] [PubMed]

M. Takeda, H. Ina, S. Kobayashi, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
[CrossRef] [PubMed]

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, “Semiconductor wafer technical flat planeness testing interferometer,” Appl. Opt. 25, 1117–1121 (1986).
[CrossRef] [PubMed]

O. Sasaki, H. Okazaki, “Sinusoidal phase modulating interferometry for surface profile measurement,” Appl. Opt. 25, 3137–3140 (1986).
[CrossRef] [PubMed]

B. V. Dorrío, C. López, A. F. Doval, J. M. Alén, J. Bugarín, A. Fernández, J. Blanco-García, J. L. Fernández, M. Pérez-Amor, “Measurement range analysis in Fizeau phase-stepping interferometry using the moiré effect,” Appl. Opt. 36, 3635–3644 (1997).
[CrossRef] [PubMed]

S. M. Pandit, N. Jordache, “Interferogram analysis based on the data-dependent systems method for nanometrology applications,” Appl. Opt. 34, 6695–6703 (1995).
[CrossRef] [PubMed]

B. V. Dorrío, A. F. Doval, C. López, R. Soto, J. Blanco-García, J. L. Fernández, M. Pérez-Amor, “Fizeau phase-measuring interferometry using the moiré effect,” Appl. Opt. 34, 3639–3643 (1995).
[CrossRef] [PubMed]

B. V. Dorrío, J. Blanco-García, A. F. Doval, C. López, R. Soto, J. L. Fernández, M. Pérez-Amor, “Phase error calculation in a Fizeau interferometer by Fourier expansion of the intensity profile,” Appl. Opt. 35, 61–64 (1996).
[CrossRef] [PubMed]

R. Józwicki, “Influence of aberrations interferometer elements on measurements errors,” Appl. Opt. 30, 3126–3132 (1991).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

G. W. Johnson, D. C. Leiner, D. T. Moore, “Phase-locked interferometry,” Opt. Eng. 18, 46–52 (1979).

R. Józwicki, M. Kujawinska, L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

Opt. Laser Technol.

D. C. Williams, N. S. Nassar, J. E. Banyard, M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Opt. Lett.

Proc. Phys. Soc. London

J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. London 62, 405–417 (1949).
[CrossRef]

Other

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

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, D. Robinson, G. T. Reid, eds. (Institute of Physics, Reading, UK, 1993), Chap. 5, pp. 141–193.

G. Schultz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (Wiley-Interscience, New York, 1992), pp. 93–167.

R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1987).

B. V. Dorrío, J. Blanco-García, A. F. Doval, C. López, R. Soto, J. L. Fernández, M. Pérez-Amor, “Analysis of the measurement range in Fizeau interferometry using the moiré effect,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 204–218 (1995).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Power spectrum of S(ξ, η). A constant average intensity i avg(x, y) and an ideal amplitude grating h(x, y) = 0 are considered.

Fig. 2
Fig. 2

Schematic sequence of comparison of the phase-evaluation process for the MMPS and the FTM.

Fig. 3
Fig. 3

Schematic of the experimental setup.

Fig. 4
Fig. 4

(a) High-frequency Fizeau pattern, (b) the principal phase values obtained with the FTM.

Fig. 5
Fig. 5

(a) Null-field multiplicative moiré pattern, (b) the principal phase values obtained with the MMPS.

Fig. 6
Fig. 6

(a) Finite-field moiré pattern and the principal phase values obtained with (b) the FTM and (c) the MMPS.

Fig. 7
Fig. 7

Results obtained for the horizontal central line on the test surface with the FTM and the MMPS employing (a) a high-frequency Fizeau pattern ϕ a , (b) null-field moiré patterns ϕ b , and (c), (d) finite-field moiré patterns ϕ c and ϕ d .

Fig. 8
Fig. 8

Differences between the phases obtained with the FTM and MMPS, ϕ p - ϕ q , for (pq) for Figs. 7(a)7(d).

Equations (15)

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

i x ,   y = k = - +   i k x ,   y exp j 2 π k ξ i x + η i y ,
i k x ,   y = i avg x ,   y a k exp j 2 π k ϕ x ,   y ,
g x ,   y = l = - +   d l δ ,   γ exp j 2 π l ξ g x + h x ,   y ,
d l δ ,   γ = b l exp j 2 π l δ ξ g cos   γ
s x ,   y = k = - l = -   c k l x ,   y d l δ ,   γ × exp j 2 π k ξ i + l ξ g x + k η i y ,
c k l x ,   y = i k x ,   y exp j 2 π lh x ,   y .
S ξ ,   η = k = - l = -   C k l ξ - k ξ i + l ξ g ,   η - k η i d l δ ,   γ
ρ k l = 1 2 π k   ϕ x ,   y x + l   h x ,   y x max 2 + k   ϕ x ,   y y + l   h x ,   y y max 2 1 / 2 ,
m x ,   y = c 0 0 x ,   y d 0 δ ,   γ + ( c 1 - 1 x ,   y d - 1 δ ,   γ × exp j 2 π ξ i - ξ g x + η i y + c - 1 1 x ,   y d 1 δ ,   γ × exp j 2 π - ξ i + ξ g x - η i y ) ,
M ξ ,   η = C 0 0 ξ ,   η d 0 δ ,   γ + C 1 - 1 ξ - ξ i - ξ g , η - η i d - 1 δ ,   γ + C - 1 1 ξ - - ξ i + ξ g , η + η i d 1 δ ,   γ
MMPS ϕ x ,   y - MMPS h x ,   y + MMPS lt x ,   y = tan - 1 q = 1 N 3   μ q m q x ,   y q = 1 N 3 ν q m q x ,   y ,
I ξ ,   η = k = -   C k 0 ξ - k ξ i ,   η - k η i .
C 1 0 ξ - ξ i ,   η - η i = I ξ ,   η H ξ - ξ i ,   η - η i .
FTM ϕ x ,   y + FTM lt x ,   y = tan - 1 Im i 1 x ,   y exp j 2 π ξ i x + η i y Re i 1 x ,   y exp j 2 π ξ i x + η i y ,
FTM ϕ x ,   y - FTM h x ,   y + FTM lt x ,   y = tan - 1 Im ( c 1 - 1 x ,   y d - 1 δ ,   γ exp j 2 π ξ i - ξ g x + η i y ) Re ( c 1 - 1 x ,   y d - 1 δ ,   γ exp j 2 π ξ i - ξ g x + η i y ) ,

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