Abstract

A pulse-to-pulse alignment method based on interference fringes and the second-order temporal coherence function of optical frequency combs is proposed for absolute distance measurement. The second-order temporal coherence function of the pulse train emitted from optical frequency combs is studied. A numerical model of the function is developed with an assumption of Gaussian pulse and has good agreement with experimental measurements taken by an ordinary Michelson interferometer. The experimental results show an improvement of standard deviation of peak finding results from 27.3 nm to 8.5 nm by the method in ordinary laboratory conditions. The absolute distance measurement with the pulse-to-pulse alignment method is also proposed and experimentally proved.

© 2015 Optical Society of America

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References

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  1. K. Minoshima and H. Matsumoto, “High-accuracy measurement of 240-m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt. 39(30), 5512–5517 (2000).
    [Crossref] [PubMed]
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    [Crossref]
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    [PubMed]
  4. R. Wynands and S. Weyers, “Atomic fountain clocks,” Metrologia 42(3), S64–S79 (2005).
    [Crossref]
  5. I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
    [Crossref]
  6. J. Lee, Y. J. Kim, K. Lee, S. Lee, and S. W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics 4(10), 716–720 (2010).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  9. Y. Liang, J. Huang, M. Ren, B. Feng, X. Chen, E. Wu, G. Wu, and H. Zeng, “1550-nm time-of-flight ranging system employing laser with multiple repetition rates for reducing the range ambiguity,” Opt. Express 22(4), 4662–4670 (2014).
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    [Crossref] [PubMed]
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    [Crossref]
  14. D. Wei, S. Takahashi, K. Takamasu, and H. Matsumoto, “Analysis of the temporal coherence function of a femtosecond optical frequency comb,” Opt. Express 17(9), 7011–7018 (2009).
    [Crossref] [PubMed]
  15. P. Balling, P. Křen, P. Mašika, and S. A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express 17(11), 9300–9313 (2009).
    [Crossref] [PubMed]
  16. N. Chanthawong, S. Takahashi, K. Takamasu, and H. Matsumoto, “A new method for high-accuracy gauge block measurement using 2 GHz repetition mode of a mode-locked fiber laser,” Meas. Sci. Technol. 23(5), 054003 (2012).
  17. G. Wu, M. Takahashi, H. Inaba, and K. Minoshima, “Pulse-to-pulse alignment technique based on synthetic-wavelength interferometry of optical frequency combs for distance measurement,” Opt. Lett. 38(12), 2140–2143 (2013).
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    [Crossref] [PubMed]
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    [Crossref]
  20. K. P. Birch and M. J. Downs, “An updated edlén equation for the refractive index of air,” Metrologia 30(3), 155–162 (1993).
    [Crossref]

2014 (5)

2013 (1)

2012 (2)

S. A. van den Berg, S. T. Persijn, G. J. P. Kok, M. G. Zeitouny, and N. Bhattacharya, “Many-wavelength interferometry with thousands of lasers for absolute distance measurement,” Phys. Rev. Lett. 108(18), 183901 (2012).
[Crossref] [PubMed]

N. Chanthawong, S. Takahashi, K. Takamasu, and H. Matsumoto, “A new method for high-accuracy gauge block measurement using 2 GHz repetition mode of a mode-locked fiber laser,” Meas. Sci. Technol. 23(5), 054003 (2012).

2011 (1)

2010 (1)

J. Lee, Y. J. Kim, K. Lee, S. Lee, and S. W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics 4(10), 716–720 (2010).
[Crossref]

2009 (5)

T. Udem, R. Holzwarth, and T. Hänsch, “Femtosecond optical frequency combs,” Eur. Phys. J. Spec. Top. 172(1), 69–79 (2009).
[Crossref]

D. Wei, S. Takahashi, K. Takamasu, and H. Matsumoto, “Analysis of the temporal coherence function of a femtosecond optical frequency comb,” Opt. Express 17(9), 7011–7018 (2009).
[Crossref] [PubMed]

P. Balling, P. Křen, P. Mašika, and S. A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express 17(11), 9300–9313 (2009).
[Crossref] [PubMed]

S. W. Kim, “Metrology: combs rule,” Nat. Photonics 3(6), 313–314 (2009).
[Crossref]

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

2008 (1)

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” J. Eur. Opt. Soc. Rapid Publ. 3, 08003 (2008).
[Crossref]

2005 (1)

R. Wynands and S. Weyers, “Atomic fountain clocks,” Metrologia 42(3), S64–S79 (2005).
[Crossref]

2000 (2)

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[Crossref] [PubMed]

K. Minoshima and H. Matsumoto, “High-accuracy measurement of 240-m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt. 39(30), 5512–5517 (2000).
[Crossref] [PubMed]

1993 (1)

K. P. Birch and M. J. Downs, “An updated edlén equation for the refractive index of air,” Metrologia 30(3), 155–162 (1993).
[Crossref]

Aketagawa, M.

Balling, P.

Berg, S. A.

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” J. Eur. Opt. Soc. Rapid Publ. 3, 08003 (2008).
[Crossref]

Bhattacharya, N.

S. A. van den Berg, S. T. Persijn, G. J. P. Kok, M. G. Zeitouny, and N. Bhattacharya, “Many-wavelength interferometry with thousands of lasers for absolute distance measurement,” Phys. Rev. Lett. 108(18), 183901 (2012).
[Crossref] [PubMed]

M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, and H. P. Urbach, “Long distance measurement with femtosecond pulses using a dispersive interferometer,” Opt. Express 19(7), 6549–6562 (2011).
[Crossref] [PubMed]

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” J. Eur. Opt. Soc. Rapid Publ. 3, 08003 (2008).
[Crossref]

Birch, K. P.

K. P. Birch and M. J. Downs, “An updated edlén equation for the refractive index of air,” Metrologia 30(3), 155–162 (1993).
[Crossref]

Bosse, H.

Cao, S.

Chanthawong, N.

N. Chanthawong, S. Takahashi, K. Takamasu, and H. Matsumoto, “A new method for high-accuracy gauge block measurement using 2 GHz repetition mode of a mode-locked fiber laser,” Meas. Sci. Technol. 23(5), 054003 (2012).

Chen, X.

Coddington, I.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Cui, M.

M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, and H. P. Urbach, “Long distance measurement with femtosecond pulses using a dispersive interferometer,” Opt. Express 19(7), 6549–6562 (2011).
[Crossref] [PubMed]

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” J. Eur. Opt. Soc. Rapid Publ. 3, 08003 (2008).
[Crossref]

Cundiff, S. T.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[Crossref] [PubMed]

Diddams, S. A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[Crossref] [PubMed]

Downs, M. J.

K. P. Birch and M. J. Downs, “An updated edlén equation for the refractive index of air,” Metrologia 30(3), 155–162 (1993).
[Crossref]

Feng, B.

Hall, J. L.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[Crossref] [PubMed]

Han, S.

J. Lee, K. Lee, Y. S. Jang, H. Jang, S. Han, S. H. Lee, K. I. Kang, C. W. Lim, Y. J. Kim, and S. W. Kim, “Testing of a femtosecond pulse laser in outer space,” Sci. Rep. 4, 5134 (2014).
[PubMed]

Hänsch, T.

T. Udem, R. Holzwarth, and T. Hänsch, “Femtosecond optical frequency combs,” Eur. Phys. J. Spec. Top. 172(1), 69–79 (2009).
[Crossref]

Holzwarth, R.

T. Udem, R. Holzwarth, and T. Hänsch, “Femtosecond optical frequency combs,” Eur. Phys. J. Spec. Top. 172(1), 69–79 (2009).
[Crossref]

Huang, J.

Inaba, H.

Jang, H.

J. Lee, K. Lee, Y. S. Jang, H. Jang, S. Han, S. H. Lee, K. I. Kang, C. W. Lim, Y. J. Kim, and S. W. Kim, “Testing of a femtosecond pulse laser in outer space,” Sci. Rep. 4, 5134 (2014).
[PubMed]

Jang, Y. S.

J. Lee, K. Lee, Y. S. Jang, H. Jang, S. Han, S. H. Lee, K. I. Kang, C. W. Lim, Y. J. Kim, and S. W. Kim, “Testing of a femtosecond pulse laser in outer space,” Sci. Rep. 4, 5134 (2014).
[PubMed]

Jones, D. J.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[Crossref] [PubMed]

Kang, K. I.

J. Lee, K. Lee, Y. S. Jang, H. Jang, S. Han, S. H. Lee, K. I. Kang, C. W. Lim, Y. J. Kim, and S. W. Kim, “Testing of a femtosecond pulse laser in outer space,” Sci. Rep. 4, 5134 (2014).
[PubMed]

Kim, S. W.

J. Lee, K. Lee, Y. S. Jang, H. Jang, S. Han, S. H. Lee, K. I. Kang, C. W. Lim, Y. J. Kim, and S. W. Kim, “Testing of a femtosecond pulse laser in outer space,” Sci. Rep. 4, 5134 (2014).
[PubMed]

J. Lee, Y. J. Kim, K. Lee, S. Lee, and S. W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics 4(10), 716–720 (2010).
[Crossref]

S. W. Kim, “Metrology: combs rule,” Nat. Photonics 3(6), 313–314 (2009).
[Crossref]

Kim, Y. J.

J. Lee, K. Lee, Y. S. Jang, H. Jang, S. Han, S. H. Lee, K. I. Kang, C. W. Lim, Y. J. Kim, and S. W. Kim, “Testing of a femtosecond pulse laser in outer space,” Sci. Rep. 4, 5134 (2014).
[PubMed]

J. Lee, Y. J. Kim, K. Lee, S. Lee, and S. W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics 4(10), 716–720 (2010).
[Crossref]

Kok, G. J. P.

S. A. van den Berg, S. T. Persijn, G. J. P. Kok, M. G. Zeitouny, and N. Bhattacharya, “Many-wavelength interferometry with thousands of lasers for absolute distance measurement,” Phys. Rev. Lett. 108(18), 183901 (2012).
[Crossref] [PubMed]

Kren, P.

Lee, J.

J. Lee, K. Lee, Y. S. Jang, H. Jang, S. Han, S. H. Lee, K. I. Kang, C. W. Lim, Y. J. Kim, and S. W. Kim, “Testing of a femtosecond pulse laser in outer space,” Sci. Rep. 4, 5134 (2014).
[PubMed]

J. Lee, Y. J. Kim, K. Lee, S. Lee, and S. W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics 4(10), 716–720 (2010).
[Crossref]

Lee, K.

J. Lee, K. Lee, Y. S. Jang, H. Jang, S. Han, S. H. Lee, K. I. Kang, C. W. Lim, Y. J. Kim, and S. W. Kim, “Testing of a femtosecond pulse laser in outer space,” Sci. Rep. 4, 5134 (2014).
[PubMed]

J. Lee, Y. J. Kim, K. Lee, S. Lee, and S. W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics 4(10), 716–720 (2010).
[Crossref]

Lee, S.

J. Lee, Y. J. Kim, K. Lee, S. Lee, and S. W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics 4(10), 716–720 (2010).
[Crossref]

Lee, S. H.

J. Lee, K. Lee, Y. S. Jang, H. Jang, S. Han, S. H. Lee, K. I. Kang, C. W. Lim, Y. J. Kim, and S. W. Kim, “Testing of a femtosecond pulse laser in outer space,” Sci. Rep. 4, 5134 (2014).
[PubMed]

Liang, Y.

Lim, C. W.

J. Lee, K. Lee, Y. S. Jang, H. Jang, S. Han, S. H. Lee, K. I. Kang, C. W. Lim, Y. J. Kim, and S. W. Kim, “Testing of a femtosecond pulse laser in outer space,” Sci. Rep. 4, 5134 (2014).
[PubMed]

Mašika, P.

Matsumoto, H.

Meiners-Hagen, K.

Minoshima, K.

Nenadovic, L.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Newbury, N. R.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Persijn, S. T.

S. A. van den Berg, S. T. Persijn, G. J. P. Kok, M. G. Zeitouny, and N. Bhattacharya, “Many-wavelength interferometry with thousands of lasers for absolute distance measurement,” Phys. Rev. Lett. 108(18), 183901 (2012).
[Crossref] [PubMed]

Pollinger, F.

Qu, X.

Ranka, J. K.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[Crossref] [PubMed]

Ren, M.

Schouten, R. N.

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” J. Eur. Opt. Soc. Rapid Publ. 3, 08003 (2008).
[Crossref]

Stentz, A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[Crossref] [PubMed]

Swann, W. C.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Takahashi, M.

Takahashi, S.

N. Chanthawong, S. Takahashi, K. Takamasu, and H. Matsumoto, “A new method for high-accuracy gauge block measurement using 2 GHz repetition mode of a mode-locked fiber laser,” Meas. Sci. Technol. 23(5), 054003 (2012).

D. Wei, S. Takahashi, K. Takamasu, and H. Matsumoto, “Analysis of the temporal coherence function of a femtosecond optical frequency comb,” Opt. Express 17(9), 7011–7018 (2009).
[Crossref] [PubMed]

Takamasu, K.

N. Chanthawong, S. Takahashi, K. Takamasu, and H. Matsumoto, “A new method for high-accuracy gauge block measurement using 2 GHz repetition mode of a mode-locked fiber laser,” Meas. Sci. Technol. 23(5), 054003 (2012).

D. Wei, S. Takahashi, K. Takamasu, and H. Matsumoto, “Analysis of the temporal coherence function of a femtosecond optical frequency comb,” Opt. Express 17(9), 7011–7018 (2009).
[Crossref] [PubMed]

Tan, J.

Udem, T.

T. Udem, R. Holzwarth, and T. Hänsch, “Femtosecond optical frequency combs,” Eur. Phys. J. Spec. Top. 172(1), 69–79 (2009).
[Crossref]

Urbach, H. P.

van den Berg, S. A.

Wei, D.

Weyers, S.

R. Wynands and S. Weyers, “Atomic fountain clocks,” Metrologia 42(3), S64–S79 (2005).
[Crossref]

Windeler, R. S.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[Crossref] [PubMed]

Wu, E.

Wu, G.

Wu, H.

Wynands, R.

R. Wynands and S. Weyers, “Atomic fountain clocks,” Metrologia 42(3), S64–S79 (2005).
[Crossref]

Xing, S.

Yang, R.

Zeitouny, M. G.

S. A. van den Berg, S. T. Persijn, G. J. P. Kok, M. G. Zeitouny, and N. Bhattacharya, “Many-wavelength interferometry with thousands of lasers for absolute distance measurement,” Phys. Rev. Lett. 108(18), 183901 (2012).
[Crossref] [PubMed]

M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, and H. P. Urbach, “Long distance measurement with femtosecond pulses using a dispersive interferometer,” Opt. Express 19(7), 6549–6562 (2011).
[Crossref] [PubMed]

Zeng, H.

Zhang, F.

Appl. Opt. (1)

Eur. Phys. J. Spec. Top. (1)

T. Udem, R. Holzwarth, and T. Hänsch, “Femtosecond optical frequency combs,” Eur. Phys. J. Spec. Top. 172(1), 69–79 (2009).
[Crossref]

J. Eur. Opt. Soc. Rapid Publ. (1)

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” J. Eur. Opt. Soc. Rapid Publ. 3, 08003 (2008).
[Crossref]

Meas. Sci. Technol. (1)

N. Chanthawong, S. Takahashi, K. Takamasu, and H. Matsumoto, “A new method for high-accuracy gauge block measurement using 2 GHz repetition mode of a mode-locked fiber laser,” Meas. Sci. Technol. 23(5), 054003 (2012).

Metrologia (2)

R. Wynands and S. Weyers, “Atomic fountain clocks,” Metrologia 42(3), S64–S79 (2005).
[Crossref]

K. P. Birch and M. J. Downs, “An updated edlén equation for the refractive index of air,” Metrologia 30(3), 155–162 (1993).
[Crossref]

Nat. Photonics (3)

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

J. Lee, Y. J. Kim, K. Lee, S. Lee, and S. W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics 4(10), 716–720 (2010).
[Crossref]

S. W. Kim, “Metrology: combs rule,” Nat. Photonics 3(6), 313–314 (2009).
[Crossref]

Opt. Express (6)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

S. A. van den Berg, S. T. Persijn, G. J. P. Kok, M. G. Zeitouny, and N. Bhattacharya, “Many-wavelength interferometry with thousands of lasers for absolute distance measurement,” Phys. Rev. Lett. 108(18), 183901 (2012).
[Crossref] [PubMed]

Sci. Rep. (1)

J. Lee, K. Lee, Y. S. Jang, H. Jang, S. Han, S. H. Lee, K. I. Kang, C. W. Lim, Y. J. Kim, and S. W. Kim, “Testing of a femtosecond pulse laser in outer space,” Sci. Rep. 4, 5134 (2014).
[PubMed]

Science (1)

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[Crossref] [PubMed]

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Figures (11)

Fig. 1
Fig. 1

Schematic of experimental setup.

Fig. 2
Fig. 2

Illustration of agreement between modeled shape and experimental interferogram. (a) Modeled shape. (b) Experimental interferogram.

Fig. 3
Fig. 3

Comparison between fitting results of Gaussian fitting and modified double Gaussian fitting. Interference fringes (red dotted line), Gaussian fitting result (blue line), modified double Gaussian fitting result (pink line).

Fig. 4
Fig. 4

The difference between the adjacent interval length and the center wavelength with the pulse width of 20 fs (only 7 fringes), 30 fs, 40 fs, 50 fs and 100 fs.

Fig. 5
Fig. 5

The interferogram and the difference between the adjacent interval length and the center wavelength with the addition phase of 0, π/2 , π and 3π/2 . (a) The interferogram corresponding to the different addition phase. (b) The difference between the adjacent interval length and the center wavelength corresponding to the different addition phase.

Fig. 6
Fig. 6

Schematic of distance measurement system.

Fig. 7
Fig. 7

Interference fringes of the cw laser and the OFC.

Fig. 8
Fig. 8

Comparison between the intensity normalization and the adjacent interval length normalization of fifteen interferograms. The error bars indicate the standard deviation of each normalization result. (a) The intensity normalization results. (b) The adjacent interval length normalization results.

Fig. 9
Fig. 9

Comparison of peak-finding method.

Fig. 10
Fig. 10

The experiment interference fringes.

Fig. 11
Fig. 11

Comparison of the experimental results.

Equations (13)

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E train (t)=A(t)exp(i ω c t+i( φ 0 +Δ φ ce t)) m= + δ(tm T R )
E total (t)= E train1 (t)+ E train2 (t)
I= m ( | E total (t) | 2 ) 2
I(τ)= m ( | E train1 (tτ)+ E train2 (t) | 2 ) 2 = m T d {[ E train1 (tτ)+ E train2 (t)][ E train1 * (tτ)+ E train2 * (t)]} 2 dt
I(τ)= m [ E train1 2 (tτ) E train1 *2 (tτ)+ E train2 2 (t) E train2 *2 (t)+ E train1 *2 (tτ) E train2 2 (t) + E train1 2 (tτ) E train2 *2 (t)+4 E train1 (tτ) E train1 * (tτ) E train2 (t) E train2 * (t) +2 E train1 (tτ) E train1 *2 (tτ) E train2 (t)+2 E train1 2 (tτ) E train1 * (tτ) E train2 * (t) +2 E train1 * (tτ) E train2 2 (t) E train2 * (t)+2 E train1 (tτ) E train2 (t) E train2 *2 (t)]dt
I(τ)= m ( I 1 + I 2 + I 3 + I 4 )
I 1 = [ E train1 2 (tτ) E train1 *2 (tτ)+ E train2 2 (t) E train2 *2 (t)]dt = [ I train1 2 (tτ)+ I train2 2 (t)]dt = A 4 π 16a
I 2 =4 [ E train1 (tτ) E train1 * (tτ) E train2 (t) E train2 * (t)]dt =4 I train1 (tτ) I train2 (t)dt = A 4 π 8a exp( a 2 τ 2 )
I 3 =Re [ E train1 *2 (tτ) E train2 2 (t)+ E train1 2 (tτ) E train2 *2 (t)]dt =2Re A (tτ) 2 A (t) 2 16 exp(2(i ω c τ+iNΔ φ ce ))dt = A 4 π 16a exp( a 2 τ 2 )cos(2 ω c τ+2NΔ φ ce )
I 4 =2Re [ E train1 (tτ) E train1 *2 (tτ) E train2 (t)+ E train1 2 (tτ) E train1 * (tτ) E train2 * (t) + E train1 * (tτ) E train2 2 (t) E train2 * (t)+ E train1 (tτ) E train2 (t) E train2 *2 (t)]dt =4Re ( A (tτ) 3 A(t) 16 + A(tτ)A (t) 3 16 )exp(i ω c τ+iNΔ φ ce )dt = A 4 π 4a exp( 3 a 2 τ 2 4 )cos( ω c τ+NΔ φ ce )
S(τ)= A 4 π 16a + 3 A 4 π 16a exp( a 2 τ 2 )+ A 4 π 4a exp( 3 a 2 τ 2 4 )
I(x)=3kexp( ( xb c 1 ) 2 )+4kexp( ( xb c 2 ) 2 )+d
L=(N× C n × T R ±( d 0 d 1 + d 2 ))/2

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