Abstract

The phase ambiguity in conventional interferometers can be removed by using two laser diodes of different optical frequencies to generate a synthetic wavelength. However, the stability requirements for a two-color interferometric laser gauge that must provide unambiguous determination of the optical fringe order over a large distance can be severe. We derive upper limits on the optical wavelength uncertainty and express them as a function of optical path difference between the object and reference beams, phase measurement errors, and the synthetic wavelength. A simple stabilization arrangement is proposed, involving simultaneous servo control of both lasers with a single Fabry–Perot étalon. The experimental implementation of the proposed system demonstrates its effectiveness for long-term (16-h) stabilized two-color interferometry over a distance of 250 mm, with a 15-mm synthetic wavelength and a repeatability of 40 nm. For periods of < 1000 s, the repeatability was 8 nm.

© 1991 Optical Society of America

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References

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  1. R. P. Grosso, R. C. 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).
  2. N. A. Massie, R. D. Nelson, S. Holly, “High performance real-time heterodyne interferometry,” Appl. Opt. 18, 1799–1803 (1979).
    [CrossRef]
  3. K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26, 2810–2816 (1987).
    [CrossRef] [PubMed]
  4. J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt. 10, 2113–2118 (1971).
    [CrossRef] [PubMed]
  5. C. Polhemus, “Two-wavelength interferometry,” Appl. Opt. 12, 2071–2074 (1973).
    [CrossRef] [PubMed]
  6. Y. Cheng, J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. 23, 4539–4543 (1984).
    [CrossRef] [PubMed]
  7. K. Creath, Y. Cheng, J. C. Wyant, “Contouring aspheric surfaces using two-wavelength phase-shifting interferometry,” Opt. Acta. 32, 1455–1464 (1985).
    [CrossRef]
  8. N. A. Massie, H. J. Caulfield, “Absolute distance interferometry,” in Interferometric Metrology, N. A. Massie, ed., Proc. Soc. Photo-Opt. Instrum. Eng.816, 149–157 (1987).
  9. C. C. Williams, H. K. Wickramasinghe, “Absolute optical ranging with 200-nm resolution,” Opt. Lett. 14, 542–544 (1989).
    [CrossRef] [PubMed]
  10. C. C. Williams, H. K. Wickramasinghe, “Optical ranging by wavelength-multiplexed interferometry,” J. Appl. Phys. 60, 1900–1903 (1986).
    [CrossRef]
  11. R. Daendliker, R. Thalmann, D. Prongue, “Two-wavelength laser interferometry using superheterodyne detection,” Opt. Lett. 13, 339–341 (1988).
    [CrossRef]
  12. A. J. den Boef, “Two-wavelength scanning spot interferometer using single-frequency diode lasers,” Appl. Opt. 27, 306–311 (1988).
    [CrossRef]
  13. A. F. Fercher, U. Vry, W. Werner, “Two-wavelength speckle interferometry on rough surfaces using a mode hopping diode laser,” Opt. Lasers Eng. 11, 271–279 (1989).
    [CrossRef]
  14. P. de Groot, “Interferometric laser profilometer for rough surfaces,” Opt. Lett. 16, 357–359 (1991).
    [CrossRef] [PubMed]
  15. Y. A. Bykovskii, V. L. Velichanskii, I. G. Goncharov, V. A. Maslov, “Use of a Fabry–Perot resonator for the stabilization of the frequency of an injection laser,” Sov. Phys. Semicond. 4, 580–583 (1970).
  16. J. Schwider, R. Burow, K-E. Elssner, J. Grzanna, R. Spolanczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  17. 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]
  18. Y. C. Chung, T. M. Shay, “Frequency stabilization of a diode laser to a Fabry–Perot interferometer,” Opt. Eng. 27, 424–427 (1988).
    [CrossRef]
  19. C. W. Gillard, N. E. Buholz, “Progress in absolute distance interferometry,” Opt. Eng. 23, 348–353 (1983).
  20. C. R. Tilford, “Analytical procedure for determining lengths from fractional fringes,” Appl. Opt. 16, 1857–1860 (1977).
    [CrossRef] [PubMed]
  21. C. J. Walsh, “Limit to multiwavelength interferometry imposed by frequency instability of the source,” Appl. Opt. 26, 29–31 (1987).
    [CrossRef] [PubMed]
  22. H. Hori, Y. Kitayama, M. Kitano, T. Yabuzaki, T. Ogawa, “Frequency stabilization of GaAlAs laser using a Doppler-free spectrum of the Cs-D2 line,” IEEE J. Quantum Electron. QE-18, 169–174 (1983).
    [CrossRef]
  23. C. Steinmetz, R. Burgoon, J. Herris, “Accuracy analysis and improvements to the Hewlett-Packard laser interferometer system,” in Interferometric Metrology, N. A. Massie, ed., Proc. Soc. Photo-Opt. Instrum. Eng.816, 79–94 (1987).
  24. 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]
  25. U. Vry, “Absolute statistical error in two-wavelength rough-surface interferometry (ROSI),” Opt. Acta. 33, 1221–1225 (1986).
    [CrossRef]

1991 (1)

1989 (2)

A. F. Fercher, U. Vry, W. Werner, “Two-wavelength speckle interferometry on rough surfaces using a mode hopping diode laser,” Opt. Lasers Eng. 11, 271–279 (1989).
[CrossRef]

C. C. Williams, H. K. Wickramasinghe, “Absolute optical ranging with 200-nm resolution,” Opt. Lett. 14, 542–544 (1989).
[CrossRef] [PubMed]

1988 (3)

1987 (3)

1986 (2)

U. Vry, “Absolute statistical error in two-wavelength rough-surface interferometry (ROSI),” Opt. Acta. 33, 1221–1225 (1986).
[CrossRef]

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

1985 (2)

K. Creath, Y. Cheng, J. C. Wyant, “Contouring aspheric surfaces using two-wavelength phase-shifting interferometry,” Opt. Acta. 32, 1455–1464 (1985).
[CrossRef]

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]

1984 (1)

1983 (3)

C. W. Gillard, N. E. Buholz, “Progress in absolute distance interferometry,” Opt. Eng. 23, 348–353 (1983).

J. Schwider, R. Burow, K-E. Elssner, J. Grzanna, R. Spolanczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

H. Hori, Y. Kitayama, M. Kitano, T. Yabuzaki, T. Ogawa, “Frequency stabilization of GaAlAs laser using a Doppler-free spectrum of the Cs-D2 line,” IEEE J. Quantum Electron. QE-18, 169–174 (1983).
[CrossRef]

1979 (1)

N. A. Massie, R. D. Nelson, S. Holly, “High performance real-time heterodyne interferometry,” Appl. Opt. 18, 1799–1803 (1979).
[CrossRef]

1977 (1)

1973 (1)

1971 (1)

1970 (1)

Y. A. Bykovskii, V. L. Velichanskii, I. G. Goncharov, V. A. Maslov, “Use of a Fabry–Perot resonator for the stabilization of the frequency of an injection laser,” Sov. Phys. Semicond. 4, 580–583 (1970).

Buholz, N. E.

C. W. Gillard, N. E. Buholz, “Progress in absolute distance interferometry,” Opt. Eng. 23, 348–353 (1983).

Burgoon, R.

C. Steinmetz, R. Burgoon, J. Herris, “Accuracy analysis and improvements to the Hewlett-Packard laser interferometer system,” in Interferometric Metrology, N. A. Massie, ed., Proc. Soc. Photo-Opt. Instrum. Eng.816, 79–94 (1987).

Burow, R.

Bykovskii, Y. A.

Y. A. Bykovskii, V. L. Velichanskii, I. G. Goncharov, V. A. Maslov, “Use of a Fabry–Perot resonator for the stabilization of the frequency of an injection laser,” Sov. Phys. Semicond. 4, 580–583 (1970).

Caulfield, H. J.

N. A. Massie, H. J. Caulfield, “Absolute distance interferometry,” in Interferometric Metrology, N. A. Massie, ed., Proc. Soc. Photo-Opt. Instrum. Eng.816, 149–157 (1987).

Cheng, Y.

K. Creath, Y. Cheng, J. C. Wyant, “Contouring aspheric surfaces using two-wavelength phase-shifting interferometry,” Opt. Acta. 32, 1455–1464 (1985).
[CrossRef]

Y. Cheng, J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. 23, 4539–4543 (1984).
[CrossRef] [PubMed]

Chung, Y. C.

Y. C. Chung, T. M. Shay, “Frequency stabilization of a diode laser to a Fabry–Perot interferometer,” Opt. Eng. 27, 424–427 (1988).
[CrossRef]

Crane, R. C.

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

K. Creath, Y. Cheng, J. C. Wyant, “Contouring aspheric surfaces using two-wavelength phase-shifting interferometry,” Opt. Acta. 32, 1455–1464 (1985).
[CrossRef]

Daendliker, R.

de Groot, P.

den Boef, A. J.

Eiju, T.

Elssner, K-E.

Fercher, A. F.

A. F. Fercher, U. Vry, W. Werner, “Two-wavelength speckle interferometry on rough surfaces using a mode hopping diode laser,” Opt. Lasers Eng. 11, 271–279 (1989).
[CrossRef]

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]

Gillard, C. W.

C. W. Gillard, N. E. Buholz, “Progress in absolute distance interferometry,” Opt. Eng. 23, 348–353 (1983).

Goncharov, I. G.

Y. A. Bykovskii, V. L. Velichanskii, I. G. Goncharov, V. A. Maslov, “Use of a Fabry–Perot resonator for the stabilization of the frequency of an injection laser,” Sov. Phys. Semicond. 4, 580–583 (1970).

Grosso, R. P.

R. P. Grosso, R. C. 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.

Herris, J.

C. Steinmetz, R. Burgoon, J. Herris, “Accuracy analysis and improvements to the Hewlett-Packard laser interferometer system,” in Interferometric Metrology, N. A. Massie, ed., Proc. Soc. Photo-Opt. Instrum. Eng.816, 79–94 (1987).

Holly, S.

N. A. Massie, R. D. Nelson, S. Holly, “High performance real-time heterodyne interferometry,” Appl. Opt. 18, 1799–1803 (1979).
[CrossRef]

Hori, H.

H. Hori, Y. Kitayama, M. Kitano, T. Yabuzaki, T. Ogawa, “Frequency stabilization of GaAlAs laser using a Doppler-free spectrum of the Cs-D2 line,” IEEE J. Quantum Electron. QE-18, 169–174 (1983).
[CrossRef]

Hu, H. Z.

Kitano, M.

H. Hori, Y. Kitayama, M. Kitano, T. Yabuzaki, T. Ogawa, “Frequency stabilization of GaAlAs laser using a Doppler-free spectrum of the Cs-D2 line,” IEEE J. Quantum Electron. QE-18, 169–174 (1983).
[CrossRef]

Kitayama, Y.

H. Hori, Y. Kitayama, M. Kitano, T. Yabuzaki, T. Ogawa, “Frequency stabilization of GaAlAs laser using a Doppler-free spectrum of the Cs-D2 line,” IEEE J. Quantum Electron. QE-18, 169–174 (1983).
[CrossRef]

Maslov, V. A.

Y. A. Bykovskii, V. L. Velichanskii, I. G. Goncharov, V. A. Maslov, “Use of a Fabry–Perot resonator for the stabilization of the frequency of an injection laser,” Sov. Phys. Semicond. 4, 580–583 (1970).

Massie, N. A.

N. A. Massie, R. D. Nelson, S. Holly, “High performance real-time heterodyne interferometry,” Appl. Opt. 18, 1799–1803 (1979).
[CrossRef]

N. A. Massie, H. J. Caulfield, “Absolute distance interferometry,” in Interferometric Metrology, N. A. Massie, ed., Proc. Soc. Photo-Opt. Instrum. Eng.816, 149–157 (1987).

Merkel, K.

Nelson, R. D.

N. A. Massie, R. D. Nelson, S. Holly, “High performance real-time heterodyne interferometry,” Appl. Opt. 18, 1799–1803 (1979).
[CrossRef]

Ogawa, T.

H. Hori, Y. Kitayama, M. Kitano, T. Yabuzaki, T. Ogawa, “Frequency stabilization of GaAlAs laser using a Doppler-free spectrum of the Cs-D2 line,” IEEE J. Quantum Electron. QE-18, 169–174 (1983).
[CrossRef]

Oreb, B. F.

Polhemus, C.

Prongue, D.

Schwider, J.

Shay, T. M.

Y. C. Chung, T. M. Shay, “Frequency stabilization of a diode laser to a Fabry–Perot interferometer,” Opt. Eng. 27, 424–427 (1988).
[CrossRef]

Spolanczyk, R.

Steinmetz, C.

C. Steinmetz, R. Burgoon, J. Herris, “Accuracy analysis and improvements to the Hewlett-Packard laser interferometer system,” in Interferometric Metrology, N. A. Massie, ed., Proc. Soc. Photo-Opt. Instrum. Eng.816, 79–94 (1987).

Thalmann, R.

Tilford, C. R.

Velichanskii, V. L.

Y. A. Bykovskii, V. L. Velichanskii, I. G. Goncharov, V. A. Maslov, “Use of a Fabry–Perot resonator for the stabilization of the frequency of an injection laser,” Sov. Phys. Semicond. 4, 580–583 (1970).

Vry, U.

A. F. Fercher, U. Vry, W. Werner, “Two-wavelength speckle interferometry on rough surfaces using a mode hopping diode laser,” Opt. Lasers Eng. 11, 271–279 (1989).
[CrossRef]

U. Vry, “Absolute statistical error in two-wavelength rough-surface interferometry (ROSI),” Opt. Acta. 33, 1221–1225 (1986).
[CrossRef]

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]

Walsh, C. J.

Werner, W.

A. F. Fercher, U. Vry, W. Werner, “Two-wavelength speckle interferometry on rough surfaces using a mode hopping diode laser,” Opt. Lasers Eng. 11, 271–279 (1989).
[CrossRef]

Wickramasinghe, H. K.

C. C. Williams, H. K. Wickramasinghe, “Absolute optical ranging with 200-nm resolution,” Opt. Lett. 14, 542–544 (1989).
[CrossRef] [PubMed]

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, “Absolute optical ranging with 200-nm resolution,” Opt. Lett. 14, 542–544 (1989).
[CrossRef] [PubMed]

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

Wyant, J. C.

Yabuzaki, T.

H. Hori, Y. Kitayama, M. Kitano, T. Yabuzaki, T. Ogawa, “Frequency stabilization of GaAlAs laser using a Doppler-free spectrum of the Cs-D2 line,” IEEE J. Quantum Electron. QE-18, 169–174 (1983).
[CrossRef]

Appl. Opt. (11)

N. A. Massie, R. D. Nelson, S. Holly, “High performance real-time heterodyne interferometry,” Appl. Opt. 18, 1799–1803 (1979).
[CrossRef]

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

Y. Cheng, J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. 23, 4539–4543 (1984).
[CrossRef] [PubMed]

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

J. Schwider, R. Burow, K-E. Elssner, J. Grzanna, R. Spolanczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

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]

C. R. Tilford, “Analytical procedure for determining lengths from fractional fringes,” Appl. Opt. 16, 1857–1860 (1977).
[CrossRef] [PubMed]

C. J. Walsh, “Limit to multiwavelength interferometry imposed by frequency instability of the source,” Appl. Opt. 26, 29–31 (1987).
[CrossRef] [PubMed]

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]

IEEE J. Quantum Electron. (1)

H. Hori, Y. Kitayama, M. Kitano, T. Yabuzaki, T. Ogawa, “Frequency stabilization of GaAlAs laser using a Doppler-free spectrum of the Cs-D2 line,” IEEE J. Quantum Electron. QE-18, 169–174 (1983).
[CrossRef]

J. Appl. Phys. (1)

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

Opt. Acta. (2)

K. Creath, Y. Cheng, J. C. Wyant, “Contouring aspheric surfaces using two-wavelength phase-shifting interferometry,” Opt. Acta. 32, 1455–1464 (1985).
[CrossRef]

U. Vry, “Absolute statistical error in two-wavelength rough-surface interferometry (ROSI),” Opt. Acta. 33, 1221–1225 (1986).
[CrossRef]

Opt. Eng. (2)

Y. C. Chung, T. M. Shay, “Frequency stabilization of a diode laser to a Fabry–Perot interferometer,” Opt. Eng. 27, 424–427 (1988).
[CrossRef]

C. W. Gillard, N. E. Buholz, “Progress in absolute distance interferometry,” Opt. Eng. 23, 348–353 (1983).

Opt. Lasers Eng. (1)

A. F. Fercher, U. Vry, W. Werner, “Two-wavelength speckle interferometry on rough surfaces using a mode hopping diode laser,” Opt. Lasers Eng. 11, 271–279 (1989).
[CrossRef]

Opt. Lett. (3)

Sov. Phys. Semicond. (1)

Y. A. Bykovskii, V. L. Velichanskii, I. G. Goncharov, V. A. Maslov, “Use of a Fabry–Perot resonator for the stabilization of the frequency of an injection laser,” Sov. Phys. Semicond. 4, 580–583 (1970).

Other (3)

R. P. Grosso, R. C. 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).

N. A. Massie, H. J. Caulfield, “Absolute distance interferometry,” in Interferometric Metrology, N. A. Massie, ed., Proc. Soc. Photo-Opt. Instrum. Eng.816, 149–157 (1987).

C. Steinmetz, R. Burgoon, J. Herris, “Accuracy analysis and improvements to the Hewlett-Packard laser interferometer system,” in Interferometric Metrology, N. A. Massie, ed., Proc. Soc. Photo-Opt. Instrum. Eng.816, 79–94 (1987).

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Figures (5)

Fig. 1
Fig. 1

Upper limits on wavelength instability in a two-color interferometer as a function of synthetic wavelength, for an optical path length (one way) of L = 250 mm between the reference and object mirrors. Four different curves, for uncertainties Δm of 2%, 1%, 0.5% and 0.1% in the optical phase measurement, are shown. Wavelength stability is determined by the requirement that the error in the synthetic wavelength measurement be small enough to permit unambiguous determination of the integer part of the optical fringe order. These curves were calculated by using inequality (18) in the text.

Fig. 2
Fig. 2

Simplified drawing of the two-color interferometer implementation that uses laser diodes LD1 and LD2 and common-cavity stabilization of the synthetic wavelength. Interference filters F1 and F2 are used to separate the two wavelengths λ1 and λ2 and permit a common-path geometry. The phase measurement uses a PZT and a five-point algorithm for 0.5% accuracy. Not shown is the usual arrangement of two quarter-wave plates and analyzer to improve the light efficiency of the interferometer.

Fig. 3
Fig. 3

Feedback current from the Fabry–Perot interferometer for the 785-nm laser diode as a function of time. This plot shows the corrections applied to the diode in response to temperature instabilities that would otherwise have, altered the diode wavelength. The oscillations corresponded to the ±0.002°C resolution of the thermoelectric cooler. The correction for long-term drift over several hours was 0.06°C (0.5 mA). The feedback circuit reduced the uncorrelated relative errors between the optical and synthetic wavelengths to 0.01 ppm.

Fig. 4
Fig. 4

Comparison of the optical wavelength measurement Lλ and the synthetic wavelength result LΛ over a period of 16 h, by using the apparatus in Fig. 3 and common-cavity stabilization. The nominal one-way optical separation of the reference and object mirrors was 250 mm. The room temperature plot indicates that the variations in optical path were due to thermal expansion of the mirror mounts and the optical bench. Despite the temperature fluctuations, the synthetic wavelength measurement remained within λ/4 of the optical wavelength measurement, as required by inequality (10).

Fig. 5
Fig. 5

Results of the two-color measurement of the displacement of the object mirror (see Fig. 2). Measurements were made every 3 s, and the mirror was moved in discrete 2-μm steps every 15 s by using a motorized micrometer. These steps are 10 times larger than the allowed displacement of λ/4 of a single-color interferometer. The synthetic wavelength data were used to remove the phase ambiguities in the optical wavelength data and achieve interferometrically accurate measurement without ambiguity.

Equations (32)

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

L λ = m λ / 2 ,
δ L λ = λ 2 δ m + m 2 δ λ .
Δ L λ = λ 2 Δ m + m 2 Δ λ ,
Δ λ = | δ λ | max ,
Δ m = | δ m | max .
Δ λ λ < λ L ( α Δ m 2 ) .
M = m 1 m 2
1 Λ = 1 λ 1 1 λ 2 .
L Λ = M Λ 2 .
| δ L Λ δ L λ | < λ / 4 .
δ L Λ = Λ 2 δ M + M 2 δ Λ ,
δ L Λ = Λ 2 ( δ m 1 δ m 2 ) + Λ L ( δ λ 1 λ 1 2 δ λ 2 λ 2 2 ) .
L λ = 1 2 ( λ 1 m 1 2 + λ 2 m 2 2 ) .
δ L λ = ( λ 1 4 δ m 1 + λ 2 4 δ m 2 ) + L 2 ( δ λ 1 λ 1 + δ λ 2 λ 2 ) .
| δ m 1 ( Λ 2 λ 1 4 ) δ m 2 ( Λ 2 + λ 2 4 ) + δ λ 1 λ 1 L ( Λ λ 1 1 2 ) σ λ 2 λ 2 L ( Λ λ 2 + 1 2 ) | max < λ 4 .
| δ m 1 | max = Δ m , | δ m 2 | max = Δ m , | δ λ 1 | max = Δ λ , | δ λ 2 | max = Δ λ .
Δ m Λ + Δ λ λ 2 Λ L λ < λ 4 ,
Δ λ λ < Λ L ( λ 8 Λ Δ m 2 ) .
δ λ 1 λ 1 = δ l 1 l 1 ,
Δ l l < λ L ( λ 8 Λ Δ m 2 ) ,
Δ T < 1 χ λ L ( λ 8 Λ Δ m 2 ) ,
δ λ 1 λ 1 = δ l l + { δ λ 1 λ 1 } unc ,
δ λ 2 λ 2 = δ l l + { δ λ 2 λ 2 } unc ,
{ Δ λ λ } unc < λ L ( λ 8 Λ Δ m 2 ) .
α = L λ [ Δ l l + { Δ λ λ } unc ] + Δ m 2 .
θ = tan 1 ( X Y sin β ) ,
X = 2 ( I 4 I 2 ) ,
Y = 2 I 3 I 5 I 1 .
M = f ( M ) + I [ M ( t 1 ) f ( M ) ] ,
m 1 = f ( m 1 ) + m 1 ( 0 ) + I [ M Λ λ 1 f ( m 1 ) m 1 ( 0 ) ] ,
m 2 = f ( m 2 ) + m 2 ( 0 ) + I [ M Λ λ 2 f ( m 2 ) m 2 ( 0 ) ] ,
L λ = 1 2 ( λ 1 m 1 2 + λ 2 m 2 2 ) .

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