Abstract

Two-wavelength interferometry that is based on a Fourier-transform method has been investigated. A phase profile at a synthetic wavelength has been measured from a two-wavelength interferogram with two spatial carrier frequencies. A phase error caused by the difference between modulation intensities at two wavelengths has been theoretically and numerically analyzed. A phase map without the error can be obtained from a power-spectrum adjustment in the two-wavelength interferogram.

© 1998 Optical Society of America

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  1. A. F. Fercher, H. Z. Hu, U. Vry, “Rough surface interferometry with a two-wavelength heterodyne speckle interferometer,” Appl. Opt. 24, 2181–2188 (1985).
    [CrossRef] [PubMed]
  2. H. J. Tiziani, A. Rothe, N. Maier, “Dual-wavelength heterodyne differential interferometer for high-precision measurements of reflective aspherical surfaces and step heights,” Appl. Opt. 35, 3525–3533 (1996).
    [CrossRef] [PubMed]
  3. V. Gusmeroli, M. Martinelli, “Two-wavelength interferometry by superluminescent source filtering,” Opt. Commun. 94, 309–312 (1992).
    [CrossRef]
  4. G. Beheim, “Fiber-optic interferometer using frequency-modulated laser diodes,” Appl. Opt. 25, 3469–3472 (1986).
    [CrossRef] [PubMed]
  5. K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26, 2810–2816 (1987).
    [CrossRef] [PubMed]
  6. P. J. de Groot, “Three-color laser-diode interferometer,” Appl. Opt. 30, 3612–3616 (1991).
    [CrossRef] [PubMed]
  7. O. Sasaki, H. Sasazaki, T. Suzuki, “Two-wavelength sinusoidal phase/modulating laser-diode interferometer insensitive to external disturbances,” Appl. Opt. 30, 4040–4045 (1991).
    [CrossRef] [PubMed]
  8. W. M. Wang, K. T. V. Grattan, W. J. O. Boyle, A. W. Palmer, “Active optical feedback in a dual-diode laser configuration applied to displacement measurements with a wide dynamic range,” Appl. Opt. 33, 1795–1801 (1994).
    [CrossRef] [PubMed]
  9. R. Onodera, Y. Ishii, “Two-wavelength laser-diode interferometer with fractional fringe techniques,” Appl. Opt. 34, 4740–4746 (1995).
    [CrossRef] [PubMed]
  10. J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt. 10, 2113–2118 (1971).
    [CrossRef] [PubMed]
  11. C. Polhemus, “Two-wavelength interferometry,” Appl. Opt. 12, 2071–2074 (1973).
    [CrossRef] [PubMed]
  12. C. C. Williams, H. K. Wickramasinghe, “Optical ranging by wavelength multiplexed interferometry,” J. Appl. Phys. 60, 1900–1903 (1986).
    [CrossRef]
  13. A. J. den Boef, “Two-wavelength scanning spot interferometer using single-frequency diode lasers,” Appl. Opt. 27, 306–311 (1988).
    [CrossRef]
  14. L. Bartolini, G. Fornetti, M. Ferri De Collibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
    [CrossRef]
  15. P. Y. Chien, R. P. Pan, C. L. Pan, “Double phase modulation approach to an interferometric system,” Opt. Commun. 93, 39–43 (1992).
    [CrossRef]
  16. J.-M. Desse, “Three-color differential interferometry,” Appl. Opt. 36, 7150–7156 (1997).
    [CrossRef]
  17. H. Matsumoto, “Synthetic interferometric distance-measuring system using a CO2 laser,” Appl. Opt. 25, 493–498 (1986).
    [CrossRef]
  18. K. Seta, T. O’ishi, “Distance meter utilizing the intermode beat of a He–Ne laser,” Appl. Opt. 29, 354–359 (1990).
    [CrossRef] [PubMed]
  19. R. Dändliker, R. Thalmann, D. Prongué, “Two-wavelength laser interferometry using superheterodyne detection,” Opt. Lett. 13, 339–341 (1988).
    [CrossRef] [PubMed]
  20. C. L. Wang, Y. H. Chuang, C. L. Pan, “Two-wavelength interferometer based on a two-color laser-diode array and the second-order correlation technique,” Opt. Lett. 20, 1071–1073 (1995).
    [CrossRef] [PubMed]
  21. K. Minoshima, H. Matsumoto, “In situ measurements of shapes and thicknesses of optical parts by femtosecond two-color interferometry,” Opt. Commun. 138, 6–10 (1997).
    [CrossRef]
  22. Y. Ishii, R. Onodera, “Two-wavelength laser-diode interferometry that uses phase-shifting techniques,” Opt. Lett. 16, 1523–1525 (1991).
    [CrossRef] [PubMed]
  23. R. Onodera, Y. Ishii, “Two-wavelength laser-diode heterodyne interferometry with one phase meter,” Opt. Lett. 20, 2502–2504 (1995).
    [CrossRef]
  24. R. Onodera, Y. Ishii, “Fourier description of phase-measuring process in two-wavelength phase-shifting interferometry,” Opt. Commun. 137, 27–30 (1997).
    [CrossRef]
  25. 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]
  26. R. Onodera, Y. Ishii, “Two-wavelength phase-shifting interferometry insensitive to the intensity modulation of dual laser diodes,” Appl. Opt. 33, 5052–5061 (1994).
    [CrossRef] [PubMed]

1997

J.-M. Desse, “Three-color differential interferometry,” Appl. Opt. 36, 7150–7156 (1997).
[CrossRef]

K. Minoshima, H. Matsumoto, “In situ measurements of shapes and thicknesses of optical parts by femtosecond two-color interferometry,” Opt. Commun. 138, 6–10 (1997).
[CrossRef]

R. Onodera, Y. Ishii, “Fourier description of phase-measuring process in two-wavelength phase-shifting interferometry,” Opt. Commun. 137, 27–30 (1997).
[CrossRef]

1996

1995

1994

W. M. Wang, K. T. V. Grattan, W. J. O. Boyle, A. W. Palmer, “Active optical feedback in a dual-diode laser configuration applied to displacement measurements with a wide dynamic range,” Appl. Opt. 33, 1795–1801 (1994).
[CrossRef] [PubMed]

R. Onodera, Y. Ishii, “Two-wavelength phase-shifting interferometry insensitive to the intensity modulation of dual laser diodes,” Appl. Opt. 33, 5052–5061 (1994).
[CrossRef] [PubMed]

1992

V. Gusmeroli, M. Martinelli, “Two-wavelength interferometry by superluminescent source filtering,” Opt. Commun. 94, 309–312 (1992).
[CrossRef]

P. Y. Chien, R. P. Pan, C. L. Pan, “Double phase modulation approach to an interferometric system,” Opt. Commun. 93, 39–43 (1992).
[CrossRef]

1991

1990

L. Bartolini, G. Fornetti, M. Ferri De Collibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

K. Seta, T. O’ishi, “Distance meter utilizing the intermode beat of a He–Ne laser,” Appl. Opt. 29, 354–359 (1990).
[CrossRef] [PubMed]

1988

1987

1986

1985

1982

1973

1971

Bartolini, L.

L. Bartolini, G. Fornetti, M. Ferri De Collibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

Beheim, G.

Boyle, W. J. O.

W. M. Wang, K. T. V. Grattan, W. J. O. Boyle, A. W. Palmer, “Active optical feedback in a dual-diode laser configuration applied to displacement measurements with a wide dynamic range,” Appl. Opt. 33, 1795–1801 (1994).
[CrossRef] [PubMed]

Chien, P. Y.

P. Y. Chien, R. P. Pan, C. L. Pan, “Double phase modulation approach to an interferometric system,” Opt. Commun. 93, 39–43 (1992).
[CrossRef]

Chuang, Y. H.

Creath, K.

Dändliker, R.

de Groot, P. J.

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

den Boef, A. J.

Desse, J.-M.

J.-M. Desse, “Three-color differential interferometry,” Appl. Opt. 36, 7150–7156 (1997).
[CrossRef]

Fercher, A. F.

Ferri De Collibus, M.

L. Bartolini, G. Fornetti, M. Ferri De Collibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

Fornetti, G.

L. Bartolini, G. Fornetti, M. Ferri De Collibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

Grattan, K. T. V.

W. M. Wang, K. T. V. Grattan, W. J. O. Boyle, A. W. Palmer, “Active optical feedback in a dual-diode laser configuration applied to displacement measurements with a wide dynamic range,” Appl. Opt. 33, 1795–1801 (1994).
[CrossRef] [PubMed]

Gusmeroli, V.

V. Gusmeroli, M. Martinelli, “Two-wavelength interferometry by superluminescent source filtering,” Opt. Commun. 94, 309–312 (1992).
[CrossRef]

Hu, H. Z.

Ina, H.

Ishii, Y.

Kobayashi, S.

Maier, N.

Martinelli, M.

V. Gusmeroli, M. Martinelli, “Two-wavelength interferometry by superluminescent source filtering,” Opt. Commun. 94, 309–312 (1992).
[CrossRef]

Matsumoto, H.

K. Minoshima, H. Matsumoto, “In situ measurements of shapes and thicknesses of optical parts by femtosecond two-color interferometry,” Opt. Commun. 138, 6–10 (1997).
[CrossRef]

H. Matsumoto, “Synthetic interferometric distance-measuring system using a CO2 laser,” Appl. Opt. 25, 493–498 (1986).
[CrossRef]

Minoshima, K.

K. Minoshima, H. Matsumoto, “In situ measurements of shapes and thicknesses of optical parts by femtosecond two-color interferometry,” Opt. Commun. 138, 6–10 (1997).
[CrossRef]

O’ishi, T.

Occhionero, G.

L. Bartolini, G. Fornetti, M. Ferri De Collibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

Onodera, R.

Palmer, A. W.

W. M. Wang, K. T. V. Grattan, W. J. O. Boyle, A. W. Palmer, “Active optical feedback in a dual-diode laser configuration applied to displacement measurements with a wide dynamic range,” Appl. Opt. 33, 1795–1801 (1994).
[CrossRef] [PubMed]

Pan, C. L.

Pan, R. P.

P. Y. Chien, R. P. Pan, C. L. Pan, “Double phase modulation approach to an interferometric system,” Opt. Commun. 93, 39–43 (1992).
[CrossRef]

Papetti, F.

L. Bartolini, G. Fornetti, M. Ferri De Collibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

Polhemus, C.

Prongué, D.

Rothe, A.

Sasaki, O.

Sasazaki, H.

Seta, K.

Suzuki, T.

Takeda, M.

Thalmann, R.

Tiziani, H. J.

Vry, U.

Wang, C. L.

Wang, W. M.

W. M. Wang, K. T. V. Grattan, W. J. O. Boyle, A. W. Palmer, “Active optical feedback in a dual-diode laser configuration applied to displacement measurements with a wide dynamic range,” Appl. Opt. 33, 1795–1801 (1994).
[CrossRef] [PubMed]

Wickramasinghe, H. K.

C. C. Williams, H. K. Wickramasinghe, “Optical ranging by wavelength multiplexed interferometry,” J. Appl. Phys. 60, 1900–1903 (1986).
[CrossRef]

Williams, C. C.

C. C. Williams, H. K. Wickramasinghe, “Optical ranging by wavelength multiplexed interferometry,” J. Appl. Phys. 60, 1900–1903 (1986).
[CrossRef]

Wyant, J. C.

Appl. Opt.

W. M. Wang, K. T. V. Grattan, W. J. O. Boyle, A. W. Palmer, “Active optical feedback in a dual-diode laser configuration applied to displacement measurements with a wide dynamic range,” Appl. Opt. 33, 1795–1801 (1994).
[CrossRef] [PubMed]

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

J.-M. Desse, “Three-color differential interferometry,” Appl. Opt. 36, 7150–7156 (1997).
[CrossRef]

Appl. Opt.

A. F. Fercher, H. Z. Hu, U. Vry, “Rough surface interferometry with a two-wavelength heterodyne speckle interferometer,” Appl. Opt. 24, 2181–2188 (1985).
[CrossRef] [PubMed]

H. Matsumoto, “Synthetic interferometric distance-measuring system using a CO2 laser,” Appl. Opt. 25, 493–498 (1986).
[CrossRef]

G. Beheim, “Fiber-optic interferometer using frequency-modulated laser diodes,” Appl. Opt. 25, 3469–3472 (1986).
[CrossRef] [PubMed]

K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26, 2810–2816 (1987).
[CrossRef] [PubMed]

A. J. den Boef, “Two-wavelength scanning spot interferometer using single-frequency diode lasers,” Appl. Opt. 27, 306–311 (1988).
[CrossRef]

K. Seta, T. O’ishi, “Distance meter utilizing the intermode beat of a He–Ne laser,” Appl. Opt. 29, 354–359 (1990).
[CrossRef] [PubMed]

O. Sasaki, H. Sasazaki, T. Suzuki, “Two-wavelength sinusoidal phase/modulating laser-diode interferometer insensitive to external disturbances,” Appl. Opt. 30, 4040–4045 (1991).
[CrossRef] [PubMed]

R. Onodera, Y. Ishii, “Two-wavelength phase-shifting interferometry insensitive to the intensity modulation of dual laser diodes,” Appl. Opt. 33, 5052–5061 (1994).
[CrossRef] [PubMed]

R. Onodera, Y. Ishii, “Two-wavelength laser-diode interferometer with fractional fringe techniques,” Appl. Opt. 34, 4740–4746 (1995).
[CrossRef] [PubMed]

H. J. Tiziani, A. Rothe, N. Maier, “Dual-wavelength heterodyne differential interferometer for high-precision measurements of reflective aspherical surfaces and step heights,” Appl. Opt. 35, 3525–3533 (1996).
[CrossRef] [PubMed]

J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt. 10, 2113–2118 (1971).
[CrossRef] [PubMed]

C. Polhemus, “Two-wavelength interferometry,” Appl. Opt. 12, 2071–2074 (1973).
[CrossRef] [PubMed]

J. Appl. Phys.

C. C. Williams, H. K. Wickramasinghe, “Optical ranging by wavelength multiplexed interferometry,” J. Appl. Phys. 60, 1900–1903 (1986).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

K. Minoshima, H. Matsumoto, “In situ measurements of shapes and thicknesses of optical parts by femtosecond two-color interferometry,” Opt. Commun. 138, 6–10 (1997).
[CrossRef]

Opt. Commun.

R. Onodera, Y. Ishii, “Fourier description of phase-measuring process in two-wavelength phase-shifting interferometry,” Opt. Commun. 137, 27–30 (1997).
[CrossRef]

V. Gusmeroli, M. Martinelli, “Two-wavelength interferometry by superluminescent source filtering,” Opt. Commun. 94, 309–312 (1992).
[CrossRef]

P. Y. Chien, R. P. Pan, C. L. Pan, “Double phase modulation approach to an interferometric system,” Opt. Commun. 93, 39–43 (1992).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

L. Bartolini, G. Fornetti, M. Ferri De Collibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Phase error Φ′ - Φ as a function of object height w for the modulation-intensity ratio, R = 1.01. A periodic error is caused by the difference between the modulation intensities.

Fig. 2
Fig. 2

Numerical calculation of rms error as a function of the ratio of the modulation intensity. The lower area of the dashed line satisfies a measurement accuracy of Λ/10.

Fig. 3
Fig. 3

Numerical simulation of a two-wavelength interferogram with different modulation intensities, b 2/b 1 = 1.25.

Fig. 4
Fig. 4

(a) Phase distribution at a synthetic wavelength measured from unprocessed spectra. (b) Cross-sectional profile taken from the phase map of (a). The solid curve indicates predetermined object heights. The dotted curve shows a calculated phase with the error in Eq. (10).

Fig. 5
Fig. 5

(a) Frequency spectra of the two-wavelength interferogram shown in Fig. 3. Five spectra correspond to five terms of G(f x , f y ) in Eq. (5). (b) Selected spectrum C 1 and a normalized spectrum Nor.C 2*. The amplitudes of these spectra are equalized.

Fig. 6
Fig. 6

(a) Phase distribution at a synthetic wavelength measured from processed spectra. (b) Cross-sectional profile taken from the phase map of (a). Good agreement between the measured result and the predetermined profile is shown.

Fig. 7
Fig. 7

Experimental configuration for a two-wavelength interferometer that is based on the Fourier-transform method.

Fig. 8
Fig. 8

Experimental two-wavelength interferogram at the synthetic wavelength, Λ = 3.3 μm. The lower part shows a cross-sectional profile of the intensity pattern of the moiré fringe.

Fig. 9
Fig. 9

(a) Phase map at the synthetic wavelength obtained from the two-wavelength interferogram shown in Fig. 8. A phase profile of character G is measured. (b) Cross-sectional profile of the phase map of (a).

Equations (22)

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

g x ,   y = a x ,   y + b 1 x ,   y cos 2 π f 1 x + ϕ 1 x ,   y + b 2 x ,   y cos 2 π f 2 x + ϕ 2 x ,   y ,
ϕ k x ,   y = 2 π w x ,   y / λ k ,
g x ,   y = a x ,   y + c 1 x ,   y exp j 2 π f 1 x + c 1 * x ,   y exp - j 2 π f 1 x + c 2 x ,   y exp j 2 π f 2 x + c 2 * x ,   y exp - j 2 π f 2 x ,
c k x ,   y = b k x ,   y exp j ϕ k x ,   y / 2
G f x ,   f y = A f x ,   f y + C 1 f x - f 1 ,   f y + C 1 * f x + f 1 ,   f y + C 2 f x - f 2 ,   f y + C 2 * f x + f 2 ,   f y ,
- 1 C 1 f x ,   f y + C 2 * f x ,   f y = b 1 x ,   y exp j ϕ 1 x ,   y / 2 + b 2 x ,   y exp - j ϕ 2 x ,   y / 2 = b 1 x ,   y cos Ψ x ,   y exp j Φ x ,   y ,
Ψ x ,   y = π w x ,   y / Γ ,     Φ x ,   y = π w x ,   y / Λ ,
b 1 x ,   y = b 2 x ,   y .
- 1 C 1 f x - 1 ,   f y + C 2 * f x ,   f y = b 1 x ,   y × cos Ψ x ,   y + π x exp j Φ x ,   y + j π x .
- 1 C 1 f x ,   f y + C 2 * f x ,   f y = b R exp j Φ ,
b R = b 1 R - 1 2 2 + R   cos 2 Ψ 1 / 2 ,
tan   Φ = sin   ϕ 1 - R   sin   ϕ 2 cos   ϕ 1 + R   cos   ϕ 2 .
R = b 2 / b 1 ,
tan   Φ tan   Φ - tan   Ψ 2   cos 2   Φ R - 1 ,
sin Φ - Φ = - R - 1 cos   Φ 2   cos   Φ tan   Ψ .
Φ - Φ 1 - R 2 tan   Ψ ,
SUM 1 = f x f y   | C 1 f x ,   f y | 2 ,
SUM 2 = f x f y   | C 2 * f x ,   f y | 2 .
Nor . C 2 * f x ,   f y = SUM 1 / SUM 2 1 / 2 C 2 * f x ,   f y .
f x f y   | C 1 f x ,   f y | 2 = f x f y   | Nor . C 2 * f x ,   f y | 2 .
x y   | c 1 x ,   y | 2 = x y   | Nor . c 2 * x ,   y | 2 .
b 1 x ,   y = Nor . b 2 x ,   y .

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