Abstract

A phase-shifting interferometry (PSI) with equal phase steps by use of a frequency-tunable diode laser and a Fabry–Perot cavity is proposed for the Carré algorithm. The measurement accuracy of the Carré algorithm depends on the equality of the phase steps. Using the Fabry–Perot cavity as a highly stable optical frequency reference, a high degree of phase step equality can be realized in PSI with an optical frequency shift. Our experimental scheme realizes an optical frequency step equality higher than 5.1 × 10−5 and a measurement repeatability of λ/800.

© 2005 Optical Society of America

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References

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  1. D. Malacara, M. Servin, Z. Malacara, “Interferogram analysis for optical testing,” in Optical Engineering, B. J. Thompson, ed. (Marcel Dekker, 1998), Vol. 61, pp. 247–278.
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2004 (3)

A. Patil, R. Langoju, P. Rastogi, “An integral approach to phase shifting interferometry using a super-resolution frequency estimation method,” Opt. Express 12, 4681–4697 (2004).
[Crossref] [PubMed]

A. Patil, B. Raphael, P. Rastogi, “Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search,” Opt. Lett. 29, 1381–1383 (2004).
[Crossref] [PubMed]

H. Inaba, T. Ikegami, F.-L. Hong, A. Onae, Y. Koga, T. R. Schibli, K. Minoshima, H. Matsumoto, S. Yamadori, O. Tohyama, S. Yamaguchi, “Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis,” IEEE J. Quantum Electron. 40, 929–936 (2004).
[Crossref]

2003 (2)

N. Kuramoto, K. Fujii, “Interferometric determination of the diameter of a silicon sphere using a direct optical frequency tuning system,” IEEE Trans. Instrum. Meas. 52, 631–635 (2003).
[Crossref]

L. Z. Cai, Q. Liu, X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28, 1808–1810 (2003).
[Crossref] [PubMed]

2000 (2)

Q. Kemao, S. Fangjun, W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11, 1220–1223 (2000).
[Crossref]

X. Chen, M. Gramaglia, J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39, 585–591 (2000).
[Crossref]

1997 (2)

G. Stoilov, T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. Lasers Eng. 28, 61–69 (1997).
[Crossref]

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14, 918–930 (1997).
[Crossref]

1996 (2)

1995 (2)

1994 (2)

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, V. A. Deason, “Multiphase fringe analysis with unknown phase shifts,” Opt. Eng. 33, 2039–2044 (1994).
[Crossref]

G.-S. Han, S.-W. Kim, “Numerical correction of reference phases in phase shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33, 7321–7325 (1994).
[Crossref] [PubMed]

1992 (1)

1991 (1)

1987 (2)

1966 (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[Crossref]

Cai, L. Z.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[Crossref]

Chen, J.

Chen, X.

Ciddor, P. E.

Creath, K.

de Groot, P.

Deason, V. A.

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, V. A. Deason, “Multiphase fringe analysis with unknown phase shifts,” Opt. Eng. 33, 2039–2044 (1994).
[Crossref]

Deck, L. L.

Dragostinov, T.

G. Stoilov, T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. Lasers Eng. 28, 61–69 (1997).
[Crossref]

Eiju, T.

Fangjun, S.

Q. Kemao, S. Fangjun, W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11, 1220–1223 (2000).
[Crossref]

Farrant, D. I.

Fujii, K.

N. Kuramoto, K. Fujii, “Interferometric determination of the diameter of a silicon sphere using a direct optical frequency tuning system,” IEEE Trans. Instrum. Meas. 52, 631–635 (2003).
[Crossref]

Gramaglia, M.

Han, G.-S.

Hariharan, P.

Hibino, K.

Hong, F.-L.

H. Inaba, T. Ikegami, F.-L. Hong, A. Onae, Y. Koga, T. R. Schibli, K. Minoshima, H. Matsumoto, S. Yamadori, O. Tohyama, S. Yamaguchi, “Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis,” IEEE J. Quantum Electron. 40, 929–936 (2004).
[Crossref]

Ikegami, T.

H. Inaba, T. Ikegami, F.-L. Hong, A. Onae, Y. Koga, T. R. Schibli, K. Minoshima, H. Matsumoto, S. Yamadori, O. Tohyama, S. Yamaguchi, “Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis,” IEEE J. Quantum Electron. 40, 929–936 (2004).
[Crossref]

Inaba, H.

H. Inaba, T. Ikegami, F.-L. Hong, A. Onae, Y. Koga, T. R. Schibli, K. Minoshima, H. Matsumoto, S. Yamadori, O. Tohyama, S. Yamaguchi, “Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis,” IEEE J. Quantum Electron. 40, 929–936 (2004).
[Crossref]

Ishii, Y.

Kemao, Q.

Q. Kemao, S. Fangjun, W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11, 1220–1223 (2000).
[Crossref]

Kim, S.-W.

Koga, Y.

H. Inaba, T. Ikegami, F.-L. Hong, A. Onae, Y. Koga, T. R. Schibli, K. Minoshima, H. Matsumoto, S. Yamadori, O. Tohyama, S. Yamaguchi, “Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis,” IEEE J. Quantum Electron. 40, 929–936 (2004).
[Crossref]

Kuramoto, N.

N. Kuramoto, K. Fujii, “Interferometric determination of the diameter of a silicon sphere using a direct optical frequency tuning system,” IEEE Trans. Instrum. Meas. 52, 631–635 (2003).
[Crossref]

Lai, G.

Langoju, R.

Larkin, K. G.

Lassahn, G. D.

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, V. A. Deason, “Multiphase fringe analysis with unknown phase shifts,” Opt. Eng. 33, 2039–2044 (1994).
[Crossref]

Lassahn, J. K.

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, V. A. Deason, “Multiphase fringe analysis with unknown phase shifts,” Opt. Eng. 33, 2039–2044 (1994).
[Crossref]

Liu, Q.

Malacara, D.

D. Malacara, M. Servin, Z. Malacara, “Interferogram analysis for optical testing,” in Optical Engineering, B. J. Thompson, ed. (Marcel Dekker, 1998), Vol. 61, pp. 247–278.

Malacara, Z.

D. Malacara, M. Servin, Z. Malacara, “Interferogram analysis for optical testing,” in Optical Engineering, B. J. Thompson, ed. (Marcel Dekker, 1998), Vol. 61, pp. 247–278.

Matsumoto, H.

H. Inaba, T. Ikegami, F.-L. Hong, A. Onae, Y. Koga, T. R. Schibli, K. Minoshima, H. Matsumoto, S. Yamadori, O. Tohyama, S. Yamaguchi, “Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis,” IEEE J. Quantum Electron. 40, 929–936 (2004).
[Crossref]

Minoshima, K.

H. Inaba, T. Ikegami, F.-L. Hong, A. Onae, Y. Koga, T. R. Schibli, K. Minoshima, H. Matsumoto, S. Yamadori, O. Tohyama, S. Yamaguchi, “Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis,” IEEE J. Quantum Electron. 40, 929–936 (2004).
[Crossref]

Murata, K.

Onae, A.

H. Inaba, T. Ikegami, F.-L. Hong, A. Onae, Y. Koga, T. R. Schibli, K. Minoshima, H. Matsumoto, S. Yamadori, O. Tohyama, S. Yamaguchi, “Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis,” IEEE J. Quantum Electron. 40, 929–936 (2004).
[Crossref]

Oreb, B. F.

Patil, A.

Raphael, B.

Rastogi, P.

Schibli, T. R.

H. Inaba, T. Ikegami, F.-L. Hong, A. Onae, Y. Koga, T. R. Schibli, K. Minoshima, H. Matsumoto, S. Yamadori, O. Tohyama, S. Yamaguchi, “Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis,” IEEE J. Quantum Electron. 40, 929–936 (2004).
[Crossref]

Schmit, J.

Servin, M.

D. Malacara, M. Servin, Z. Malacara, “Interferogram analysis for optical testing,” in Optical Engineering, B. J. Thompson, ed. (Marcel Dekker, 1998), Vol. 61, pp. 247–278.

Stoilov, G.

G. Stoilov, T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. Lasers Eng. 28, 61–69 (1997).
[Crossref]

Taylor, P. L.

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, V. A. Deason, “Multiphase fringe analysis with unknown phase shifts,” Opt. Eng. 33, 2039–2044 (1994).
[Crossref]

Tohyama, O.

H. Inaba, T. Ikegami, F.-L. Hong, A. Onae, Y. Koga, T. R. Schibli, K. Minoshima, H. Matsumoto, S. Yamadori, O. Tohyama, S. Yamaguchi, “Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis,” IEEE J. Quantum Electron. 40, 929–936 (2004).
[Crossref]

Xiaoping, W.

Q. Kemao, S. Fangjun, W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11, 1220–1223 (2000).
[Crossref]

Yamadori, S.

H. Inaba, T. Ikegami, F.-L. Hong, A. Onae, Y. Koga, T. R. Schibli, K. Minoshima, H. Matsumoto, S. Yamadori, O. Tohyama, S. Yamaguchi, “Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis,” IEEE J. Quantum Electron. 40, 929–936 (2004).
[Crossref]

Yamaguchi, S.

H. Inaba, T. Ikegami, F.-L. Hong, A. Onae, Y. Koga, T. R. Schibli, K. Minoshima, H. Matsumoto, S. Yamadori, O. Tohyama, S. Yamaguchi, “Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis,” IEEE J. Quantum Electron. 40, 929–936 (2004).
[Crossref]

Yang, X. L.

Yatagai, T.

Yeazell, J. A.

Appl. Opt. (6)

IEEE J. Quantum Electron. (1)

H. Inaba, T. Ikegami, F.-L. Hong, A. Onae, Y. Koga, T. R. Schibli, K. Minoshima, H. Matsumoto, S. Yamadori, O. Tohyama, S. Yamaguchi, “Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis,” IEEE J. Quantum Electron. 40, 929–936 (2004).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

N. Kuramoto, K. Fujii, “Interferometric determination of the diameter of a silicon sphere using a direct optical frequency tuning system,” IEEE Trans. Instrum. Meas. 52, 631–635 (2003).
[Crossref]

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

Meas. Sci. Technol. (1)

Q. Kemao, S. Fangjun, W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11, 1220–1223 (2000).
[Crossref]

Metrologia (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[Crossref]

Opt. Eng. (1)

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, V. A. Deason, “Multiphase fringe analysis with unknown phase shifts,” Opt. Eng. 33, 2039–2044 (1994).
[Crossref]

Opt. Express (1)

Opt. Lasers Eng. (1)

G. Stoilov, T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. Lasers Eng. 28, 61–69 (1997).
[Crossref]

Opt. Lett. (3)

Other (1)

D. Malacara, M. Servin, Z. Malacara, “Interferogram analysis for optical testing,” in Optical Engineering, B. J. Thompson, ed. (Marcel Dekker, 1998), Vol. 61, pp. 247–278.

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Figures (5)

Fig. 1
Fig. 1

Schematic experimental setup for locking the optical frequencies without optical frequency modulation.

Fig. 2
Fig. 2

Error signal E(f) used in the modulation-free method.

Fig. 3
Fig. 3

Time variation of the beat note signal between the ECLD whose frequency was locked using the FPC and the reference He–Ne laser for an integration time of 0.03 s.

Fig. 4
Fig. 4

Experimental setup of the Fizeau interferometer. L1–L3, lenses; PH, pinhole.

Fig. 5
Fig. 5

Three-dimensional phase maps of the wedged glass substrate. The optical frequencies in the measurements for (a) and (b) were different.

Equations (8)

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

I i = I 0 { 1 + γ ( x ,     y ) cos [ φ ( x ,     y ) + α i ] } ,
tan 2 φ ( x ,     y ) = ( 3 I 2 - 3 I 3 - I 1 + I 4 ) ( I 1 + I 2 - I 3 - I 4 ) ( I 1 - I 2 - I 3 + I 4 ) 2 ,
tan 2 α = 3 I 2 - 3 I 3 - I 1 + I 4 I 1 + I 2 - I 3 - I 4 .
I 2 - I 3 = ( 2 I 0 γ sin α ) sin φ ,
I 2 + I 3 - I 1 - I 4 = ( 8 I 0 γ cos α sin 2 α ) cos φ .
2 α = 2 π n L c Δ f ,
FSR = c 2 l ,
E ( f ) = G [ α 1 g 1 T ( f ) - α 2 g 2 ] I in ( f ) ,

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