Abstract

High-precision long geometrical distance measurement performs a vital role in large-scale manufacturing and future light detection and ranging (LIDAR) for tight formation. Its high precision, fast measurement rate, and large ambiguity range have traditionally made dual-comb ranging (DCR) a powerful tool for absolute distance measurement. However, DCR experiences the same issues caused by the refractive index of air as other laser-based ranging systems. The conventional method used to compensate refractive index of air is through using empirical equations by monitoring environment parameters. This real-time compensation method relies on precise sensors and cannot be easily applied to long-distance measurement. Thus, a two-color compensation method is proposed that requires only two co-propagating lights at different wavelengths, without specific identification of the refractive index of air. In this paper, the two-color method is combined with a low-noise DCR realized by a digital correction method. Mode resolved and phase-stable comb spectra are available for ranging at both two wavelengths with ~200 THz difference. The experimental result demonstrates 46 nm precision and 2.7 m ambiguity range by two-color DCR (TC-DCR) with 0.1 s coherent averaging at 1 kHz repetition rate difference. This method achieves a precision of the order of ~10−8 and an accuracy of the order of ~10−7, which is comparable to the single-color DCR results by empirical equations with environmental sensing. The proposed two-color DCR demonstrates great potential for long-distance measurement in open air.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref]
  42. Y. Liu, X. Zhao, G. Hu, C. Li, B. Zhao, and Z. Zheng, “Unidirectional, dual-comb lasing under multiple pulse formation mechanisms in a passively mode-locked fiber ring laser,” Opt. Express 24(19), 21392–21398 (2016).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  44. Z. Zhu, K. Ni, Q. Zhou, and G. Wu, “Digital correction method for realizing a phase-stable dual-comb interferometer,” Opt. Express 26(13), 16813–16823 (2018).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]

2018 (8)

Z. Zhu, G. Xu, K. Ni, Q. Zhou, and G. Wu, “Synthetic-wavelength-based dual-comb interferometry for fast and precise absolute distance measurement,” Opt. Express 26(5), 5747–5757 (2018).
[Crossref] [PubMed]

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

C. Wang, Z. Deng, C. Gu, Y. Liu, D. Luo, Z. Zhu, W. Li, and H. Zeng, “Line-scan spectrum-encoded imaging by dual-comb interferometry,” Opt. Lett. 43(7), 1606–1609 (2018).
[Crossref] [PubMed]

E. Hase, T. Minamikawa, T. Mizuno, S. Miyamoto, R. Ichikawa, Y.-D. Hsieh, K. Shibuya, K. Sato, Y. Nakajima, A. Asahara, K. Minoshima, Y. Mizutani, T. Iwata, H. Yamamoto, and T. Yasui, “Scan-less confocal phase imaging based on dual-comb microscopy,” Optica 5(5), 634 (2018).
[Crossref]

Z. Zhu, G. Xu, K. Ni, Q. Zhou, and G. Wu, “Improving the accuracy of a dual-comb interferometer by suppressing the relative linewidth,” Meas. Sci. Technol. 29(4), 45007–45011 (2018).
[Crossref]

R. Liao, Y. Song, W. Liu, H. Shi, L. Chai, and M. Hu, “Dual-comb spectroscopy with a single free-running thulium-doped fiber laser,” Opt. Express 26(8), 11046–11054 (2018).
[Crossref] [PubMed]

Z. Zhu, K. Ni, Q. Zhou, and G. Wu, “Digital correction method for realizing a phase-stable dual-comb interferometer,” Opt. Express 26(13), 16813–16823 (2018).
[Crossref] [PubMed]

2017 (3)

T. Minamikawa, Y.-D. Hsieh, K. Shibuya, E. Hase, Y. Kaneoka, S. Okubo, H. Inaba, Y. Mizutani, H. Yamamoto, T. Iwata, and T. Yasui, “Dual-comb spectroscopic ellipsometry,” Nat. Commun. 8(1), 610–617 (2017).
[Crossref] [PubMed]

K. Shibuya, T. Minamikawa, Y. Mizutani, H. Yamamoto, K. Minoshima, T. Yasui, and T. Iwata, “Scan-less hyperspectral dual-comb single-pixel-imaging in both amplitude and phase,” Opt. Express 25(18), 21947–21957 (2017).
[Crossref] [PubMed]

Y.-S. Jang and S.-W. Kim, “Compensation of the refractive index of air in laser interferometer for distance measurement: A review,” Int. J. Precis. Eng. Manuf. 18(12), 1881–1890 (2017).
[Crossref]

2016 (8)

K. Meiners-Hagen, T. Meyer, G. Prellinger, W. Pöschel, D. Dontsov, and F. Pollinger, “Overcoming the refractivity limit in manufacturing environment,” Opt. Express 24(21), 24092–24104 (2016).
[Crossref] [PubMed]

I. Coddington, N. Newbury, and W. Swann, “Dual-comb spectroscopy,” Optica 3(4), 414–426 (2016).
[Crossref]

H. Wu, F. Zhang, T. Liu, J. Li, and X. Qu, “Absolute distance measurement with correction of air refractive index by using two-color dispersive interferometry,” Opt. Express 24(21), 24361–24376 (2016).
[Crossref] [PubMed]

A. Nishiyama, S. Yoshida, Y. Nakajima, H. Sasada, K. Nakagawa, A. Onae, and K. Minoshima, “Doppler-free dual-comb spectroscopy of Rb using optical-optical double resonance technique,” Opt. Express 24(22), 25894–25904 (2016).
[Crossref] [PubMed]

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

T. Ideguchi, T. Nakamura, Y. Kobayashi, and K. Goda, “Kerr-lens mode-locked bidirectional dual-comb ring laser for broadband dual-comb spectroscopy,” Optica 3(7), 748–753 (2016).
[Crossref]

Y. Liu, X. Zhao, G. Hu, C. Li, B. Zhao, and Z. Zheng, “Unidirectional, dual-comb lasing under multiple pulse formation mechanisms in a passively mode-locked fiber ring laser,” Opt. Express 24(19), 21392–21398 (2016).
[Crossref] [PubMed]

D. Wei, K. Takamasu, and H. Matsumoto, “Is the Two-Color Method Superior to Empirical Equations in Refractive Index Compensation?” Opt. Photonics J. 06(08), 8–13 (2016).
[Crossref]

2015 (1)

2014 (1)

T. Ideguchi, A. Poisson, G. Guelachvili, N. Picqué, and T. W. Hänsch, “Adaptive real-time dual-comb spectroscopy,” Nat. Commun. 5(1), 3375–3382 (2014).
[Crossref] [PubMed]

2013 (2)

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).
[Crossref] [PubMed]

G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Rep. 3(1), 1894 (2013).
[Crossref] [PubMed]

2012 (1)

2010 (3)

J.-D. Deschênes, P. Giaccarri, and J. Genest, “Optical referencing technique with CW lasers as intermediate oscillators for continuous full delay range frequency comb interferometry,” Opt. Express 18(22), 23358–23370 (2010).
[Crossref] [PubMed]

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent dual-comb spectroscopy at high signal-to-noise ratio,” Phys. Rev. A 82(4), 043817 (2010).
[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]

2009 (2)

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]

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent linear optical sampling at 15 bits of resolution,” Opt. Lett. 34(14), 2153–2155 (2009).
[Crossref] [PubMed]

2008 (2)

K.-N. Joo, Y. Kim, and S.-W. Kim, “Distance measurements by combined method based on a femtosecond pulse laser,” Opt. Express 16(24), 19799–19806 (2008).
[Crossref] [PubMed]

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 13902–13905 (2008).
[Crossref] [PubMed]

2006 (1)

2002 (2)

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

W. T. Estler, K. L. Edmundson, G. N. Peggs, and D. H. Parker, “Large-scale metrology - An update,” Cirp Annals-Manufacturing Technology 51(2), 587–609 (2002).
[Crossref]

2001 (1)

C. Sabol, R. Burns, and C. A. Mclaughlin, “Satellite Formation Flying Design and Evolution,” J. Spacecr. Rockets 38(2), 270–278 (2001).
[Crossref]

2000 (1)

1996 (1)

1993 (2)

K. P. Birch and M. J. Downs, “An Updated Edlén Equation for the Refractive Index of Air,” Metrologia 30(3), 155–162 (1993).
[Crossref]

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
[Crossref]

1991 (1)

P. K. C. Wang, “Navigation strategies for multiple autonomous mobile robots moving in formation,” J. Robot. Syst. 8(2), 177–195 (1991).
[Crossref]

1984 (1)

1972 (1)

1967 (1)

K. Earnshaw and J. Owens, “A dual wavelength optical distance measuring instrument which measures air density,” IEEE J. Quantum Electron. 3(6), 257–258 (1967).
[Crossref]

1966 (1)

B. Edlén, “The Refractive Index of Air,” Metrologia 2(2), 71–80 (1966).
[Crossref]

Arai, K.

G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Rep. 3(1), 1894 (2013).
[Crossref] [PubMed]

Asahara, A.

Baumann, E.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

Bendahmane, A.

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[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]

Bobroff, N.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
[Crossref]

Burns, R.

C. Sabol, R. Burns, and C. A. Mclaughlin, “Satellite Formation Flying Design and Evolution,” J. Spacecr. Rockets 38(2), 270–278 (2001).
[Crossref]

Chai, L.

Chun, B. J.

Ciddor, P. E.

Coddington, I.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

I. Coddington, N. Newbury, and W. Swann, “Dual-comb spectroscopy,” Optica 3(4), 414–426 (2016).
[Crossref]

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent dual-comb spectroscopy at high signal-to-noise ratio,” Phys. Rev. A 82(4), 043817 (2010).
[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]

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent linear optical sampling at 15 bits of resolution,” Opt. Lett. 34(14), 2153–2155 (2009).
[Crossref] [PubMed]

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 13902–13905 (2008).
[Crossref] [PubMed]

Deng, Z.

Deschênes, J.-D.

Diddams, S. A.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

Dontsov, D.

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]

Earnshaw, K.

K. Earnshaw and J. Owens, “A dual wavelength optical distance measuring instrument which measures air density,” IEEE J. Quantum Electron. 3(6), 257–258 (1967).
[Crossref]

Earnshaw, K. B.

Edlén, B.

B. Edlén, “The Refractive Index of Air,” Metrologia 2(2), 71–80 (1966).
[Crossref]

Edmundson, K. L.

W. T. Estler, K. L. Edmundson, G. N. Peggs, and D. H. Parker, “Large-scale metrology - An update,” Cirp Annals-Manufacturing Technology 51(2), 587–609 (2002).
[Crossref]

Estler, W. T.

W. T. Estler, K. L. Edmundson, G. N. Peggs, and D. H. Parker, “Large-scale metrology - An update,” Cirp Annals-Manufacturing Technology 51(2), 587–609 (2002).
[Crossref]

Fernald, F. G.

Freude, W.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

Ganin, D.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

Genest, J.

Giaccarri, P.

Giorgetta, F. R.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

Goda, K.

Gu, C.

Guelachvili, G.

T. Ideguchi, A. Poisson, G. Guelachvili, N. Picqué, and T. W. Hänsch, “Adaptive real-time dual-comb spectroscopy,” Nat. Commun. 5(1), 3375–3382 (2014).
[Crossref] [PubMed]

Hänsch, T. W.

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

T. Ideguchi, A. Poisson, G. Guelachvili, N. Picqué, and T. W. Hänsch, “Adaptive real-time dual-comb spectroscopy,” Nat. Commun. 5(1), 3375–3382 (2014).
[Crossref] [PubMed]

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

Hase, E.

E. Hase, T. Minamikawa, T. Mizuno, S. Miyamoto, R. Ichikawa, Y.-D. Hsieh, K. Shibuya, K. Sato, Y. Nakajima, A. Asahara, K. Minoshima, Y. Mizutani, T. Iwata, H. Yamamoto, and T. Yasui, “Scan-less confocal phase imaging based on dual-comb microscopy,” Optica 5(5), 634 (2018).
[Crossref]

T. Minamikawa, Y.-D. Hsieh, K. Shibuya, E. Hase, Y. Kaneoka, S. Okubo, H. Inaba, Y. Mizutani, H. Yamamoto, T. Iwata, and T. Yasui, “Dual-comb spectroscopic ellipsometry,” Nat. Commun. 8(1), 610–617 (2017).
[Crossref] [PubMed]

Herman, D.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

Hernandez, E. N.

Holzwarth, R.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

Hovhannisyan, T.

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

Hsieh, Y.-D.

E. Hase, T. Minamikawa, T. Mizuno, S. Miyamoto, R. Ichikawa, Y.-D. Hsieh, K. Shibuya, K. Sato, Y. Nakajima, A. Asahara, K. Minoshima, Y. Mizutani, T. Iwata, H. Yamamoto, and T. Yasui, “Scan-less confocal phase imaging based on dual-comb microscopy,” Optica 5(5), 634 (2018).
[Crossref]

T. Minamikawa, Y.-D. Hsieh, K. Shibuya, E. Hase, Y. Kaneoka, S. Okubo, H. Inaba, Y. Mizutani, H. Yamamoto, T. Iwata, and T. Yasui, “Dual-comb spectroscopic ellipsometry,” Nat. Commun. 8(1), 610–617 (2017).
[Crossref] [PubMed]

Hu, G.

Hu, M.

Ichikawa, R.

Ideguchi, T.

T. Ideguchi, T. Nakamura, Y. Kobayashi, and K. Goda, “Kerr-lens mode-locked bidirectional dual-comb ring laser for broadband dual-comb spectroscopy,” Optica 3(7), 748–753 (2016).
[Crossref]

T. Ideguchi, A. Poisson, G. Guelachvili, N. Picqué, and T. W. Hänsch, “Adaptive real-time dual-comb spectroscopy,” Nat. Commun. 5(1), 3375–3382 (2014).
[Crossref] [PubMed]

Inaba, H.

T. Minamikawa, Y.-D. Hsieh, K. Shibuya, E. Hase, Y. Kaneoka, S. Okubo, H. Inaba, Y. Mizutani, H. Yamamoto, T. Iwata, and T. Yasui, “Dual-comb spectroscopic ellipsometry,” Nat. Commun. 8(1), 610–617 (2017).
[Crossref] [PubMed]

G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Rep. 3(1), 1894 (2013).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Iwata, T.

Jang, Y. S.

Jang, Y.-S.

Y.-S. Jang and S.-W. Kim, “Compensation of the refractive index of air in laser interferometer for distance measurement: A review,” Int. J. Precis. Eng. Manuf. 18(12), 1881–1890 (2017).
[Crossref]

Joo, K. N.

Joo, K.-N.

Kaneoka, Y.

T. Minamikawa, Y.-D. Hsieh, K. Shibuya, E. Hase, Y. Kaneoka, S. Okubo, H. Inaba, Y. Mizutani, H. Yamamoto, T. Iwata, and T. Yasui, “Dual-comb spectroscopic ellipsometry,” Nat. Commun. 8(1), 610–617 (2017).
[Crossref] [PubMed]

Kang, H. J.

Karpov, M.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

Kim, S. W.

Kim, S.-W.

Y.-S. Jang and S.-W. Kim, “Compensation of the refractive index of air in laser interferometer for distance measurement: A review,” Int. J. Precis. Eng. Manuf. 18(12), 1881–1890 (2017).
[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]

K.-N. Joo, Y. Kim, and S.-W. Kim, “Distance measurements by combined method based on a femtosecond pulse laser,” Opt. Express 16(24), 19799–19806 (2008).
[Crossref] [PubMed]

Kim, Y.

Kim, Y. J.

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]

Kippenberg, T. J.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

Kobayashi, Y.

Koos, C.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

Kordts, A.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

Krockenberger, J.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

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]

Li, C.

Li, J.

Li, W.

Liao, R.

Liu, T.

Liu, W.

Liu, Y.

Luo, D.

Marin-Palomo, P.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

Matsumoto, H.

D. Wei, K. Takamasu, and H. Matsumoto, “Is the Two-Color Method Superior to Empirical Equations in Refractive Index Compensation?” Opt. Photonics J. 06(08), 8–13 (2016).
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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).
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Mclaughlin, C. A.

C. Sabol, R. Burns, and C. A. Mclaughlin, “Satellite Formation Flying Design and Evolution,” J. Spacecr. Rockets 38(2), 270–278 (2001).
[Crossref]

Meiners-Hagen, K.

Meyer, T.

Millot, G.

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

Minamikawa, T.

Minoshima, K.

Miyamoto, S.

Mizuno, T.

Mizutani, Y.

Nakagawa, K.

Nakajima, Y.

Nakamura, T.

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.

Newbury, N. R.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent dual-comb spectroscopy at high signal-to-noise ratio,” Phys. Rev. A 82(4), 043817 (2010).
[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]

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent linear optical sampling at 15 bits of resolution,” Opt. Lett. 34(14), 2153–2155 (2009).
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I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 13902–13905 (2008).
[Crossref] [PubMed]

Ni, K.

Nishiyama, A.

Okubo, S.

T. Minamikawa, Y.-D. Hsieh, K. Shibuya, E. Hase, Y. Kaneoka, S. Okubo, H. Inaba, Y. Mizutani, H. Yamamoto, T. Iwata, and T. Yasui, “Dual-comb spectroscopic ellipsometry,” Nat. Commun. 8(1), 610–617 (2017).
[Crossref] [PubMed]

Onae, A.

Owens, J.

K. Earnshaw and J. Owens, “A dual wavelength optical distance measuring instrument which measures air density,” IEEE J. Quantum Electron. 3(6), 257–258 (1967).
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Parker, D. H.

W. T. Estler, K. L. Edmundson, G. N. Peggs, and D. H. Parker, “Large-scale metrology - An update,” Cirp Annals-Manufacturing Technology 51(2), 587–609 (2002).
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Peggs, G. N.

W. T. Estler, K. L. Edmundson, G. N. Peggs, and D. H. Parker, “Large-scale metrology - An update,” Cirp Annals-Manufacturing Technology 51(2), 587–609 (2002).
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Pfeiffer, M. H. P.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

Picqué, N.

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

T. Ideguchi, A. Poisson, G. Guelachvili, N. Picqué, and T. W. Hänsch, “Adaptive real-time dual-comb spectroscopy,” Nat. Commun. 5(1), 3375–3382 (2014).
[Crossref] [PubMed]

Pitois, S.

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
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Poisson, A.

T. Ideguchi, A. Poisson, G. Guelachvili, N. Picqué, and T. W. Hänsch, “Adaptive real-time dual-comb spectroscopy,” Nat. Commun. 5(1), 3375–3382 (2014).
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Pollinger, F.

Pöschel, W.

Potvin, S.

Prellinger, G.

Qu, X.

Randel, S.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

Roy, J.

Sabol, C.

C. Sabol, R. Burns, and C. A. Mclaughlin, “Satellite Formation Flying Design and Evolution,” J. Spacecr. Rockets 38(2), 270–278 (2001).
[Crossref]

Sasada, H.

Sato, K.

Shi, H.

Shibuya, K.

Song, Y.

Swann, W.

Swann, W. C.

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent dual-comb spectroscopy at high signal-to-noise ratio,” Phys. Rev. A 82(4), 043817 (2010).
[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]

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent linear optical sampling at 15 bits of resolution,” Opt. Lett. 34(14), 2153–2155 (2009).
[Crossref] [PubMed]

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 13902–13905 (2008).
[Crossref] [PubMed]

Takahashi, M.

G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Rep. 3(1), 1894 (2013).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Takamasu, K.

D. Wei, K. Takamasu, and H. Matsumoto, “Is the Two-Color Method Superior to Empirical Equations in Refractive Index Compensation?” Opt. Photonics J. 06(08), 8–13 (2016).
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Trocha, P.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

Udem, T.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

Wang, C.

Wang, P. K. C.

P. K. C. Wang, “Navigation strategies for multiple autonomous mobile robots moving in formation,” J. Robot. Syst. 8(2), 177–195 (1991).
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Wei, D.

D. Wei, K. Takamasu, and H. Matsumoto, “Is the Two-Color Method Superior to Empirical Equations in Refractive Index Compensation?” Opt. Photonics J. 06(08), 8–13 (2016).
[Crossref]

Weimann, C.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

Wolf, S.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
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Wu, G.

Wu, H.

Xu, G.

Z. Zhu, G. Xu, K. Ni, Q. Zhou, and G. Wu, “Improving the accuracy of a dual-comb interferometer by suppressing the relative linewidth,” Meas. Sci. Technol. 29(4), 45007–45011 (2018).
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Z. Zhu, G. Xu, K. Ni, Q. Zhou, and G. Wu, “Synthetic-wavelength-based dual-comb interferometry for fast and precise absolute distance measurement,” Opt. Express 26(5), 5747–5757 (2018).
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Yamamoto, H.

Yan, M.

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

Yasui, T.

Ycas, G.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

Yoshida, S.

Zeng, H.

Zhang, F.

Zhao, B.

Zhao, X.

Zheng, Z.

Zhou, Q.

Zhu, Z.

Appl. Opt. (4)

Cirp Annals-Manufacturing Technology (1)

W. T. Estler, K. L. Edmundson, G. N. Peggs, and D. H. Parker, “Large-scale metrology - An update,” Cirp Annals-Manufacturing Technology 51(2), 587–609 (2002).
[Crossref]

IEEE J. Quantum Electron. (1)

K. Earnshaw and J. Owens, “A dual wavelength optical distance measuring instrument which measures air density,” IEEE J. Quantum Electron. 3(6), 257–258 (1967).
[Crossref]

Int. J. Precis. Eng. Manuf. (1)

Y.-S. Jang and S.-W. Kim, “Compensation of the refractive index of air in laser interferometer for distance measurement: A review,” Int. J. Precis. Eng. Manuf. 18(12), 1881–1890 (2017).
[Crossref]

J. Robot. Syst. (1)

P. K. C. Wang, “Navigation strategies for multiple autonomous mobile robots moving in formation,” J. Robot. Syst. 8(2), 177–195 (1991).
[Crossref]

J. Spacecr. Rockets (1)

C. Sabol, R. Burns, and C. A. Mclaughlin, “Satellite Formation Flying Design and Evolution,” J. Spacecr. Rockets 38(2), 270–278 (2001).
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N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
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Z. Zhu, G. Xu, K. Ni, Q. Zhou, and G. Wu, “Improving the accuracy of a dual-comb interferometer by suppressing the relative linewidth,” Meas. Sci. Technol. 29(4), 45007–45011 (2018).
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B. Edlén, “The Refractive Index of Air,” Metrologia 2(2), 71–80 (1966).
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K. P. Birch and M. J. Downs, “An Updated Edlén Equation for the Refractive Index of Air,” Metrologia 30(3), 155–162 (1993).
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Nat. Commun. (2)

T. Minamikawa, Y.-D. Hsieh, K. Shibuya, E. Hase, Y. Kaneoka, S. Okubo, H. Inaba, Y. Mizutani, H. Yamamoto, T. Iwata, and T. Yasui, “Dual-comb spectroscopic ellipsometry,” Nat. Commun. 8(1), 610–617 (2017).
[Crossref] [PubMed]

T. Ideguchi, A. Poisson, G. Guelachvili, N. Picqué, and T. W. Hänsch, “Adaptive real-time dual-comb spectroscopy,” Nat. Commun. 5(1), 3375–3382 (2014).
[Crossref] [PubMed]

Nat. Photonics (4)

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[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]

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]

Nature (1)

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

Opt. Express (13)

K. N. Joo and S. W. Kim, “Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser,” Opt. Express 14(13), 5954–5960 (2006).
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K.-N. Joo, Y. Kim, and S.-W. Kim, “Distance measurements by combined method based on a femtosecond pulse laser,” Opt. Express 16(24), 19799–19806 (2008).
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J.-D. Deschênes, P. Giaccarri, and J. Genest, “Optical referencing technique with CW lasers as intermediate oscillators for continuous full delay range frequency comb interferometry,” Opt. Express 18(22), 23358–23370 (2010).
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J. Roy, J.-D. Deschênes, S. Potvin, and J. Genest, “Continuous real-time correction and averaging for frequency comb interferometry,” Opt. Express 20(20), 21932–21939 (2012).
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H. J. Kang, B. J. Chun, Y. S. Jang, Y. J. Kim, and S. W. Kim, “Real-time compensation of the refractive index of air in distance measurement,” Opt. Express 23(20), 26377–26385 (2015).
[Crossref] [PubMed]

Y. Liu, X. Zhao, G. Hu, C. Li, B. Zhao, and Z. Zheng, “Unidirectional, dual-comb lasing under multiple pulse formation mechanisms in a passively mode-locked fiber ring laser,” Opt. Express 24(19), 21392–21398 (2016).
[Crossref] [PubMed]

K. Meiners-Hagen, T. Meyer, G. Prellinger, W. Pöschel, D. Dontsov, and F. Pollinger, “Overcoming the refractivity limit in manufacturing environment,” Opt. Express 24(21), 24092–24104 (2016).
[Crossref] [PubMed]

H. Wu, F. Zhang, T. Liu, J. Li, and X. Qu, “Absolute distance measurement with correction of air refractive index by using two-color dispersive interferometry,” Opt. Express 24(21), 24361–24376 (2016).
[Crossref] [PubMed]

A. Nishiyama, S. Yoshida, Y. Nakajima, H. Sasada, K. Nakagawa, A. Onae, and K. Minoshima, “Doppler-free dual-comb spectroscopy of Rb using optical-optical double resonance technique,” Opt. Express 24(22), 25894–25904 (2016).
[Crossref] [PubMed]

K. Shibuya, T. Minamikawa, Y. Mizutani, H. Yamamoto, K. Minoshima, T. Yasui, and T. Iwata, “Scan-less hyperspectral dual-comb single-pixel-imaging in both amplitude and phase,” Opt. Express 25(18), 21947–21957 (2017).
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Z. Zhu, G. Xu, K. Ni, Q. Zhou, and G. Wu, “Synthetic-wavelength-based dual-comb interferometry for fast and precise absolute distance measurement,” Opt. Express 26(5), 5747–5757 (2018).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

Opt. Lett. (3)

Opt. Photonics J. (1)

D. Wei, K. Takamasu, and H. Matsumoto, “Is the Two-Color Method Superior to Empirical Equations in Refractive Index Compensation?” Opt. Photonics J. 06(08), 8–13 (2016).
[Crossref]

Optica (3)

Phys. Rev. A (1)

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent dual-comb spectroscopy at high signal-to-noise ratio,” Phys. Rev. A 82(4), 043817 (2010).
[Crossref]

Phys. Rev. Lett. (1)

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G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Rep. 3(1), 1894 (2013).
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Science (1)

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref] [PubMed]

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Y.-S. Jang and S.-W. Kim, “Distance Measurements Using Mode-Locked Lasers: A Review,” Nanomanufacturing and Metrology, 1–17 (2018).
[Crossref]

Z. Chen, M. Yan, T. W. Hänsch, and N. Picqué, “A phase-stable dual-comb interferometer,” arXiv:1705.04214 (2017).

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Figures (5)

Fig. 1
Fig. 1 Experimental setup and principle of TC-DCR. (a) Two erbium-doped fiber combs (Comb 1 and Comb 2) are fully stabilized to RF standards: fo1 = fo2 = 10.56MHz, fr1 = 56.090 MHz, fr2 = 56.091 MHz, Δfr = fr2 –fr1 = 1 kHz. The clock rate of the digitizer is equal to fr2. The measured distance between measurement (M) mirror and reference (R) mirror is D, and a commercial interferometer is supplied as comparison. Two optical bandpass filters (OBPF) with ~3 nm FWHM (full width at half maximum) bandwidth are used to extract the two color interferograms at 1560 nm and 780 nm. A CW laser with ~kHz linewidth and center frequencies of fCW = 191.500245THz is used as optical intermediary to generate a beat signal frequency between the two comb modes and CW laser (fCW-1 = fCW − f1, fCW-2 = fCW – f2). The relative beat signal between the two comb modes serves as a compensating signal, and is generated from the electronic mixing process as fb = fCW-2 − fCW-1 = f1 − f2; (b) The frequency-domain principle of the TC-DCR in the multiheterodyne version. Two combs with slight repetition difference generate a sub comb in RF domain at both two colors; (c) Description of the sub comb with frequency spacing Δfr; (d) Phase information calculated from Fourier transform of interferograms.

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Fig. 2
Fig. 2 Waveforms and spectra of two-color IGMs. Red color represents IGMs1 and blue color represents IGMs2. (a) Twenty IGMs of both colors are given after 20 ms moving coherent averaging. The echo signals should be cut out to avoid phase disturbance when applying Fourier transform; (b) The spectra of 1-s IGMs1 and 1-s IGMs2. The corresponding orders of the optical modes are shown in the above figure. The parts of the two spectra are provided in (c). Resolution bandwidth (RBW) is 1 Hz.

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Fig. 3
Fig. 3 Two-color phase measurement of TC-DCR. Curve ‘i’, ‘ii’, and ‘iii’ represents Δφ1, Δφ2/2, and Δφ1 −Δφ2/2 respectively. (a) The phase information with two periods of time (0-11 s and 11-21 s with ~10 minutes’ delay) are given to reflect the optical path length drift. The datum during 11-21 s are upshifted 2 π. The Δφ1 − Δφ2/2 drifts 0.11 rad, indicating that D2 – D1 is stable while D1 or D2 drifts 1.12 μm. The dark curves represent phase of IGMs with 20 ms coherent averaging; (b) The phase information between 18.8 and 19.0 s; (c) The phase precision (Allan deviation) of Δφ1, Δφ1 −Δφ2/2, and Δφ1 −Δφ2/2 through 20 ms coherent averaging (Curve ‘iv’) in Fig. 3(a). These results indicate the two-color method can compensate the long-term drift of optical path length caused by the refractive index of air.

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Fig. 4
Fig. 4 Comparison measurement between the TC-DCR and a commercial interferometer. (a) Ranging results of one-color and two-color method with 0.1 s coherent averaging versus results from a commercial interferometer. Note that the refractive index of air is compensated when computing one-color results and the commercial interferometer’s results. The results of the single color are upshifted by 0.1 m to avoid overlap; (b) The residuals among two single color results and the commercial interferometer’s results. The standard deviation of single-color residuals is about 0.10 μm, and the standard deviation of the difference between two single-color results is 3.4 nm; (c) The residuals (with 0.55 μm standard deviation) between two-color results and the commercial interferometer’s results.

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Fig. 5
Fig. 5 Comparison measurement between the TC-DCR and a commercial interferometer. (a) Ranging results of one-color and two-color method with 0.1 s coherent averaging versus results from a commercial interferometer. Note that we compensate the refractive index of air when we compute one-color results and the commercial interferometer’s results. The results of single color are upshifted by 100 μm to avoid overlap. (b) The residuals among two single color results and commercial interferometer’s results. The standard deviation of single-color residuals is about 30 nm, the standard deviation of the difference between two single-color results is 2.7 nm. (c) The residuals (with 0.42 μm standard deviation) between two-color results and commercial interferometer’s results.

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Equations (8)

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D = D 1 − A ( D 2 − D 1 ) ,
f RF ( k ) = m 2 ⋅ f r2 − m 1 ⋅ f r 1 + f o2 − f o1 = m 2 ⋅ Δ f r + Δ m ⋅ f r1 + Δ f o ,
D i = ( N i + Δ φ i / 2 π ) λ i 2 , i = 1 , 2.
δ f RF ( k ) = m 2 ⋅ δ Δ f r + Δ m ⋅ δ f r1 + δ Δ f o ≈ m 2 ⋅ δ Δ f r + δ Δ f o ,
I 1 ( t )= I 0 ( t )exp [ − i ⋅ δ φ ( k c ) ] ,
I 2 ( t )= I 1 [ t − T jitter ( t ) ] exp [ − i 2 π f RF ( k c ) T jitter ( t ) ] .
δ TC = D 1 − A ( D 2 − D 1 ) − D 0 = D 1 − A [ ( D 2 − n 2 D 0 ) − ( D 1 − n 1 D 0 ) + ( n 2 − n 1 ) D 0 ] − D 0 = n 1 δ 1 − A ( n 2 δ 2 − n 1 δ 1 ) ≈ δ 1 − A ( δ 2 − δ 1 ) ,
σ ( δ TC ) = σ 2 ( δ 1 ) + A 2 σ 2 ( δ 2 − δ 1 ) .

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