Abstract

Long-path pulse-to-pulse interferometers of two-color frequency combs are developed using fundamental and second harmonics of a mode-locked fiber laser. Interferometric phase difference between two-color frequency combs was precisely measured by stabilizing the fundamental fringe phase by controlling the repetition frequency of the comb, and a stability of 10−10 for 1000 s was achieved in the measurement of an optical path length difference between two wavelengths. In long-term measurements performed for 10 h, results of phase variation of interferometric measurements were highly consistent with the fluctuations in the calculated difference of refractive indices of air at two wavelengths with an accuracy of 10−10. The difference between the measured optical distances corresponding to two wavelengths and the optical distance corresponding to the fundamental wavelength were used in the two-color method; high-accuracy self-correction of the fluctuation of refractive index of air was performed with an uncertainty of 5 × 10−8 for 10-h measurements when the maximum refractive index change was on the order of 10−6.

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References

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  1. G. Boensch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlen's formulae,” Metrologia 35(2), 133–139 (1998).
    [CrossRef]
  2. P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35(9), 1566–1573 (1996).
    [CrossRef] [PubMed]
  3. P. L. Bender and J. C. Owens, “Correction of optical distance measurements for the fluctuating atmospheric index of refraction,” J. Geophys. Res. 70(10), 2461–2462 (1965).
    [CrossRef]
  4. J. C. Owens, “The use of atmospheric dispersion in optical distance measurement,” Bull. Geod. 89(1), 277–291 (1968).
    [CrossRef]
  5. H. Matsumoto and T. Honda, “High-accuracy length-measuring interferometer using the two-colour method of compensating for the refractive index of air,” Meas. Sci. Technol. 3(11), 1084–1086 (1992).
    [CrossRef]
  6. H. Matsumoto, Y. Zhu, S. Iwasaki, and T. O’ishi, “Measurement of the changes in air refractive index and distance by means of a two-color interferometer,” Appl. Opt. 31(22), 4522–4526 (1992).
    [CrossRef] [PubMed]
  7. 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]
  8. K. Meiners-Hagen and A. Abou-Zeid, “Refractive index determination in length measurement by two-colour interferometry,” Meas. Sci. Technol. 19(8), 084004 (2008).
    [CrossRef]
  9. T. Yasui, K. Minoshima, and H. Matsumoto, “Stabilization of femtosecond mode-locked Ti: Sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum Electron. 37(1), 12–19 (2001).
    [CrossRef]
  10. Y. Yamaoka, K. Minoshima, and H. Matsumoto, “Direct measurement of the group refractive index of air with interferometry between adjacent femtosecond pulses,” Appl. Opt. 41(21), 4318–4324 (2002).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  17. P. Balling, P. Kren, P. Masika, and S. A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express 17(11), 9300–9313 (2009).
    [CrossRef] [PubMed]
  18. 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]
  19. 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]
  20. 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]
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    [CrossRef] [PubMed]
  22. H. Inaba, Y. Daimon, F.-L. Hong, A. Onae, K. Minoshima, T. R. Schibli, H. Matsumoto, M. Hirano, T. Okuno, M. Onishi, and M. Nakazawa, “Long-term measurement of optical frequencies using a simple, robust and low-noise fiber based frequency comb,” Opt. Express 14(12), 5223–5231 (2006).
    [CrossRef] [PubMed]

2011

2010

2009

2008

K. Meiners-Hagen and A. Abou-Zeid, “Refractive index determination in length measurement by two-colour interferometry,” Meas. Sci. Technol. 19(8), 084004 (2008).
[CrossRef]

2006

2005

2004

2002

2001

T. Yasui, K. Minoshima, and H. Matsumoto, “Stabilization of femtosecond mode-locked Ti: Sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum Electron. 37(1), 12–19 (2001).
[CrossRef]

2000

1998

G. Boensch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlen's formulae,” Metrologia 35(2), 133–139 (1998).
[CrossRef]

1996

1992

H. Matsumoto and T. Honda, “High-accuracy length-measuring interferometer using the two-colour method of compensating for the refractive index of air,” Meas. Sci. Technol. 3(11), 1084–1086 (1992).
[CrossRef]

H. Matsumoto, Y. Zhu, S. Iwasaki, and T. O’ishi, “Measurement of the changes in air refractive index and distance by means of a two-color interferometer,” Appl. Opt. 31(22), 4522–4526 (1992).
[CrossRef] [PubMed]

1968

J. C. Owens, “The use of atmospheric dispersion in optical distance measurement,” Bull. Geod. 89(1), 277–291 (1968).
[CrossRef]

1965

P. L. Bender and J. C. Owens, “Correction of optical distance measurements for the fluctuating atmospheric index of refraction,” J. Geophys. Res. 70(10), 2461–2462 (1965).
[CrossRef]

Abou-Zeid, A.

K. Meiners-Hagen and A. Abou-Zeid, “Refractive index determination in length measurement by two-colour interferometry,” Meas. Sci. Technol. 19(8), 084004 (2008).
[CrossRef]

Balling, P.

Bender, P. L.

P. L. Bender and J. C. Owens, “Correction of optical distance measurements for the fluctuating atmospheric index of refraction,” J. Geophys. Res. 70(10), 2461–2462 (1965).
[CrossRef]

Bhattacharya, N.

Bitou, Y.

Boensch, G.

G. Boensch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlen's formulae,” Metrologia 35(2), 133–139 (1998).
[CrossRef]

Ciddor, P. E.

Cui, M.

Daimon, Y.

Dändliker, R.

Hirano, M.

Holzwarth, R.

Honda, T.

H. Matsumoto and T. Honda, “High-accuracy length-measuring interferometer using the two-colour method of compensating for the refractive index of air,” Meas. Sci. Technol. 3(11), 1084–1086 (1992).
[CrossRef]

Hong, F. L.

Hong, F.-L.

Hosaka, K.

Hyun, S.

S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S. W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20(9), 6 (2009).
[CrossRef]

Inaba, H.

Iwasaki, S.

Jin, J.

S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S. W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20(9), 6 (2009).
[CrossRef]

Katsuyama, T.

Kawato, S.

Kim, S. W.

S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S. W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20(9), 6 (2009).
[CrossRef]

Kim, S.-W.

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]

Kim, Y.

S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S. W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20(9), 6 (2009).
[CrossRef]

Kim, Y. J.

S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S. W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20(9), 6 (2009).
[CrossRef]

Kim, Y.-J.

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]

Kobayashi, T.

Kohno, T.

Kren, P.

Lee, J.

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, 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]

Lévêque, S.

Lu, Z. H.

Masika, P.

Matsumoto, H.

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]

T. R. Schibli, K. Minoshima, Y. Bitou, F. L. Hong, H. Inaba, A. Onae, and H. Matsumoto, “Displacement metrology with sub-pm resolution in air based on a fs-comb wavelength synthesizer,” Opt. Express 14(13), 5984–5993 (2006).
[CrossRef] [PubMed]

H. Inaba, Y. Daimon, F.-L. Hong, A. Onae, K. Minoshima, T. R. Schibli, H. Matsumoto, M. Hirano, T. Okuno, M. Onishi, and M. Nakazawa, “Long-term measurement of optical frequencies using a simple, robust and low-noise fiber based frequency comb,” Opt. Express 14(12), 5223–5231 (2006).
[CrossRef] [PubMed]

Y. Yamaoka, K. Minoshima, and H. Matsumoto, “Direct measurement of the group refractive index of air with interferometry between adjacent femtosecond pulses,” Appl. Opt. 41(21), 4318–4324 (2002).
[CrossRef] [PubMed]

T. Yasui, K. Minoshima, and H. Matsumoto, “Stabilization of femtosecond mode-locked Ti: Sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum Electron. 37(1), 12–19 (2001).
[CrossRef]

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]

H. Matsumoto and T. Honda, “High-accuracy length-measuring interferometer using the two-colour method of compensating for the refractive index of air,” Meas. Sci. Technol. 3(11), 1084–1086 (1992).
[CrossRef]

H. Matsumoto, Y. Zhu, S. Iwasaki, and T. O’ishi, “Measurement of the changes in air refractive index and distance by means of a two-color interferometer,” Appl. Opt. 31(22), 4522–4526 (1992).
[CrossRef] [PubMed]

Meiners-Hagen, K.

K. Meiners-Hagen and A. Abou-Zeid, “Refractive index determination in length measurement by two-colour interferometry,” Meas. Sci. Technol. 19(8), 084004 (2008).
[CrossRef]

Minoshima, K.

Y. Nakajima, H. Inaba, K. Hosaka, K. Minoshima, A. Onae, M. Yasuda, T. Kohno, S. Kawato, T. Kobayashi, T. Katsuyama, and F.-L. Hong, “A multi-branch, fiber-based frequency comb with millihertz-level relative linewidths using an intra-cavity electro-optic modulator,” Opt. Express 18(2), 1667–1676 (2010).
[CrossRef] [PubMed]

T. R. Schibli, K. Minoshima, Y. Bitou, F. L. Hong, H. Inaba, A. Onae, and H. Matsumoto, “Displacement metrology with sub-pm resolution in air based on a fs-comb wavelength synthesizer,” Opt. Express 14(13), 5984–5993 (2006).
[CrossRef] [PubMed]

H. Inaba, Y. Daimon, F.-L. Hong, A. Onae, K. Minoshima, T. R. Schibli, H. Matsumoto, M. Hirano, T. Okuno, M. Onishi, and M. Nakazawa, “Long-term measurement of optical frequencies using a simple, robust and low-noise fiber based frequency comb,” Opt. Express 14(12), 5223–5231 (2006).
[CrossRef] [PubMed]

Y. Yamaoka, K. Minoshima, and H. Matsumoto, “Direct measurement of the group refractive index of air with interferometry between adjacent femtosecond pulses,” Appl. Opt. 41(21), 4318–4324 (2002).
[CrossRef] [PubMed]

T. Yasui, K. Minoshima, and H. Matsumoto, “Stabilization of femtosecond mode-locked Ti: Sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum Electron. 37(1), 12–19 (2001).
[CrossRef]

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]

Nakajima, Y.

Nakazawa, M.

Newbury, N. R.

O’ishi, T.

Okuno, T.

Onae, A.

Onishi, M.

Owens, J. C.

J. C. Owens, “The use of atmospheric dispersion in optical distance measurement,” Bull. Geod. 89(1), 277–291 (1968).
[CrossRef]

P. L. Bender and J. C. Owens, “Correction of optical distance measurements for the fluctuating atmospheric index of refraction,” J. Geophys. Res. 70(10), 2461–2462 (1965).
[CrossRef]

Potulski, E.

G. Boensch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlen's formulae,” Metrologia 35(2), 133–139 (1998).
[CrossRef]

Salvadé, Y.

Schibli, T. R.

Schuhler, N.

Swann, W. C.

Takahashi, S.

Takamasu, K.

Urbach, H. P.

van den Berg, S. A.

Wang, L. J.

Wei, D.

Yamaoka, Y.

Yasuda, M.

Yasui, T.

T. Yasui, K. Minoshima, and H. Matsumoto, “Stabilization of femtosecond mode-locked Ti: Sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum Electron. 37(1), 12–19 (2001).
[CrossRef]

Ye, J.

Zeitouny, M. G.

Zhang, J.

Zhu, Y.

Appl. Opt.

Bull. Geod.

J. C. Owens, “The use of atmospheric dispersion in optical distance measurement,” Bull. Geod. 89(1), 277–291 (1968).
[CrossRef]

IEEE J. Quantum Electron.

T. Yasui, K. Minoshima, and H. Matsumoto, “Stabilization of femtosecond mode-locked Ti: Sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum Electron. 37(1), 12–19 (2001).
[CrossRef]

J. Geophys. Res.

P. L. Bender and J. C. Owens, “Correction of optical distance measurements for the fluctuating atmospheric index of refraction,” J. Geophys. Res. 70(10), 2461–2462 (1965).
[CrossRef]

Meas. Sci. Technol.

H. Matsumoto and T. Honda, “High-accuracy length-measuring interferometer using the two-colour method of compensating for the refractive index of air,” Meas. Sci. Technol. 3(11), 1084–1086 (1992).
[CrossRef]

K. Meiners-Hagen and A. Abou-Zeid, “Refractive index determination in length measurement by two-colour interferometry,” Meas. Sci. Technol. 19(8), 084004 (2008).
[CrossRef]

S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S. W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20(9), 6 (2009).
[CrossRef]

Metrologia

G. Boensch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlen's formulae,” Metrologia 35(2), 133–139 (1998).
[CrossRef]

Nat. Photonics

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]

Opt. Express

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]

Y. Nakajima, H. Inaba, K. Hosaka, K. Minoshima, A. Onae, M. Yasuda, T. Kohno, S. Kawato, T. Kobayashi, T. Katsuyama, and F.-L. Hong, “A multi-branch, fiber-based frequency comb with millihertz-level relative linewidths using an intra-cavity electro-optic modulator,” Opt. Express 18(2), 1667–1676 (2010).
[CrossRef] [PubMed]

H. Inaba, Y. Daimon, F.-L. Hong, A. Onae, K. Minoshima, T. R. Schibli, H. Matsumoto, M. Hirano, T. Okuno, M. Onishi, and M. Nakazawa, “Long-term measurement of optical frequencies using a simple, robust and low-noise fiber based frequency comb,” Opt. Express 14(12), 5223–5231 (2006).
[CrossRef] [PubMed]

T. R. Schibli, K. Minoshima, Y. Bitou, F. L. Hong, H. Inaba, A. Onae, and H. Matsumoto, “Displacement metrology with sub-pm resolution in air based on a fs-comb wavelength synthesizer,” Opt. Express 14(13), 5984–5993 (2006).
[CrossRef] [PubMed]

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

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]

Opt. Lett.

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Figures (7)

Fig. 1
Fig. 1

Experimental set up for the pulse-to-pulse interferometer using two-color optical combs. The output of the fiber comb is separated into three branches; one of them is used for stabilizing the comb (not shown) and other two are used for interferometric experiments for fundamental (ω) and SHG (2ω). EDFA: Er-doped fiber amplifier; FC: fiber collimator; H: half-wave plate; L: lens; C: SHG crystal; DM: dichroic mirror; BS: beam splitter of 10 mm cube; PD1,2: photodetectors. Path difference between the probe and reference is 5.55 m, which corresponds to the repetition rate of 54.0 MHz. In the interferometer, the pulses were highly chirped by propagation in fibers and a PPLN crystal. The pulse durations estimated by the coherence lengths were about 1 ps and 3 ps for fundamental and SHG, respectively.

Fig. 2
Fig. 2

Interference fringe signals for (a) fundamental and (b) SHG wavelengths. Optical path length was changed by the scanning of the repetition frequency, frep. Cross: measurement results; Line: sine curve fitting for guide for eyes.

Fig. 3
Fig. 3

Environmental variation of interference fringe signals of two-color combs ((a) fundamental, (b) SHG) and (c) their difference. In (a) and (b), the origin of each vertical axis is one of the zero phases. Residual sinusoidal variation in the plot of (c) is due to the imperfect temperature control of the SHG crystal oven.

Fig. 4
Fig. 4

Interferometric measurements of two-color frequency combs. The fundamental interference fringe signal (a, b) is stabilized by feedback to frep. Variations of the fundamental and SHG optical path lengths are shown ((a, b) and (c, d), respectively). The origin of vertical axis is zero phase of the fundamental fringe. Moving average of 25 s was applied in the cases of (b) and (d).

Fig. 5
Fig. 5

Long-term variation in the difference between the refractive indices of air at two color wavelengths n2-n1 (a) obtained by the measured phase difference of the two-color interferometer, (b) calculated using environmental parameters, and (c) their difference. The vertical positions of the plots were shifted for clarity as follows. In (a) and (b), the vertical origins of the plots was chosen so that the averages of all the data in two plots were the same. In (c), the vertical origin was shifted so that the average of data was 2x10−9.

Fig. 6
Fig. 6

Long-term variation in the refractive index of air (a) obtained by the measured frep, (b) calculated using environmental parameters, and (c) their difference. The origin of each plot was arbitrary chosen for clarity in the similar way to Fig. 5. In (c), the vertical origin is shifted so that the average is 0.2 × 10−6.

Fig. 7
Fig. 7

Self-correction of air-refractive index by two-color method. The variations in the relative distances to the target distance D, which are obtained from the corresponding refractive indices, are plotted; (a) one-color distance D1 obtained by measured frep (Fig. 6(a)), (b) difference between optical distances corresponding to two colors, D2 - D1, obtained from the direct measurements of phase difference of the two-color interferometers (Fig. 5(a)), and (c) corrected geometrical distance D obtained using Eq. (3). The origin of each plot was arbitrary chosen for clarity in the similar way to Fig. 5. The vertical origin of the plot (c) was shifted so that the average of data is 0.2 × 10−6.

Equations (4)

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

D= D 1 A( D 2 D 1 ),
A=( n 1 1)/( n 2 n 1 ).
ΔD/D={Δ D 1 AΔ( D 2 D 1 )ΔA( D 2 D 1 )ΔAΔ( D 2 D 1 )}/D,
Δ D 1 = cΔ f rep f rep 2 .

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