Abstract

We propose a direct and real-time displacement measurement using an optical frequency comb, able to compensate optically for index of refraction variations due to atmospheric parameters. This scheme could be useful for applications requiring stringent precision over a long distance in air, a situation where dispersion becomes the main limitation. The key ingredient is the use of a mode-locked laser as a precise source for multi-wavelength interferometry in a homodyne detection scheme. By shaping temporally the local oscillator, one can directly access the desired parameter (distance variation) while being insensitive to fluctuations induced by parameters of the environment such as pressure, temperature, humidity and CO2 content.

© 2012 OSA

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  2. P. Exertier, P. Bonnefond, F. Deleflie, F. Barlier, M. Kasser, R. Biancale, and Y. Ménard, “Contribution of laser ranging to earth’s sciences,” C. R. Geosci.338, 958–967 (2006).
    [CrossRef]
  3. K. Djerroud, O. Acef, A. Clairon, P. Lemonde, C. N. Man, E. Samain, and P. Wolf, “Coherent optical link through the turbulent atmosphere,” Opt. Lett.35, 1479–1481 (2010).
    [CrossRef] [PubMed]
  4. J. Lee, Y. -J. Kim, K. Lee, S. Lee, and S. -W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics4, 716–720 (2010).
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  6. F. Delplancke, “The prima facility phase-referenced imaging and micro-arcsecond astrometry,” New Astron. Rev.52, 199–207 (2008).
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  8. J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett.29, 1153–1155 (2004).
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  9. K. -N. Joo and S. -W. Kim, “Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser,” Opt. Express14, 5954–5960 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. B. Lamine, C. Fabre, and N. Treps, “Quantum improvement of time transfer between remote clocks,” Phys. Rev. Lett.101, 123601 (2008).
    [CrossRef] [PubMed]
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  13. I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics3, 351–356 (2009).
    [CrossRef]
  14. P. Balling, P. Kren, P. Masika, and S. A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express17, 9300–9313 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
  16. 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, 1084–1086 (1992).
    [CrossRef]
  17. K. Meiners-Hagen and A. Abou-Zeid, “Refractive index determination in length measurement by two-colour interferometry,” Meas. Sci. Technol.19, 084004 (2008).
    [CrossRef]
  18. S. Azouigui, T. Badr, J. -P. Wallerand, M. Himbert, J. Salgado, and P. Juncar, “Transportable distance measurement system based on superheterodyne interferometry using two phase-locked frequency-doubled Nd:YAG lasers,” Rev. Sci. Instrum.81, 053112 (2010).
    [CrossRef] [PubMed]
  19. C. Helstrom, “The minimum variance of estimates in quantum signal detection,” IEEE Trans. Inf. Theory14, 234–242 (1968).
    [CrossRef]
  20. S. Braunstein and C. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett.72, 3439–3443 (1994).
    [CrossRef] [PubMed]
  21. O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense gaussian quantum light: a multimodal approach,” Phys. Rev. A85, 1–4 (2012).
    [CrossRef]
  22. A. N. Golubev and A. M. Chekhovsky, “Three-color optical range finding,” Appl. Opt.33, 7511–7517 (1994).
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    [CrossRef] [PubMed]
  25. M. Hsu, V. Delaubert, P. Lam, and W. Bowen, “Optimal optical measurement of small displacements,” J. Opt. B: Quantum Semiclassical Opt.6, 495–501 (2004).
    [CrossRef]
  26. B. Edlén, “The refractive index of air,” Metrologia2, 71–80 (1966).
    [CrossRef]
  27. P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt.35, 1566–1573 (1996).
    [CrossRef] [PubMed]
  28. G. Bönsch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlén’s formulae,” Metrologia35, 133–139 (1998).
    [CrossRef]
  29. A. Ishida, “Two-wavelength displacement-measuring interferometer using second-harmonic light to eliminate air-turbulence-induced errors,” Jpn. J. Appl. Phys.28, L473–L475 (1989).
    [CrossRef]
  30. O. Pinel, Pu. Jian, R. Medeiros de Araújo, J. Feng, B. Chalopin, C. Fabre, and N. Treps, “Generation and characterization of multimode quantum frequency combs,” Phys. Rev. Lett.108, 083601 (2012).
    [CrossRef] [PubMed]
  31. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science306, 1330–1336 (2004).
    [CrossRef] [PubMed]
  32. Formulas for refractive index of air are available on http://emtoolbox.nist.gov/Wavelength/Documentation.asp
  33. R. Macovez, M. Mariano, S. D. Finizio, and J. Martorell, “Measurement of the dispersion of air and of refractive index anomalies by wavelength-dependent nonlinear interferometry,” Opt. Express17, 13881–13888 (2009).
    [CrossRef] [PubMed]

2012

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense gaussian quantum light: a multimodal approach,” Phys. Rev. A85, 1–4 (2012).
[CrossRef]

O. Pinel, Pu. Jian, R. Medeiros de Araújo, J. Feng, B. Chalopin, C. Fabre, and N. Treps, “Generation and characterization of multimode quantum frequency combs,” Phys. Rev. Lett.108, 083601 (2012).
[CrossRef] [PubMed]

2010

K. Djerroud, O. Acef, A. Clairon, P. Lemonde, C. N. Man, E. Samain, and P. Wolf, “Coherent optical link through the turbulent atmosphere,” Opt. Lett.35, 1479–1481 (2010).
[CrossRef] [PubMed]

J. Lee, Y. -J. Kim, K. Lee, S. Lee, and S. -W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics4, 716–720 (2010).
[CrossRef]

S. Azouigui, T. Badr, J. -P. Wallerand, M. Himbert, J. Salgado, and P. Juncar, “Transportable distance measurement system based on superheterodyne interferometry using two phase-locked frequency-doubled Nd:YAG lasers,” Rev. Sci. Instrum.81, 053112 (2010).
[CrossRef] [PubMed]

2009

2008

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]

B. Lamine, C. Fabre, and N. Treps, “Quantum improvement of time transfer between remote clocks,” Phys. Rev. Lett.101, 123601 (2008).
[CrossRef] [PubMed]

Y. Salvadé, N. Schuhler, S. Lévêque, and S. Le Floch, “High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source,” Appl. Opt.47, 2715–2720 (2008).
[CrossRef] [PubMed]

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

F. Delplancke, “The prima facility phase-referenced imaging and micro-arcsecond astrometry,” New Astron. Rev.52, 199–207 (2008).
[CrossRef]

2006

2004

M. Hsu, V. Delaubert, P. Lam, and W. Bowen, “Optimal optical measurement of small displacements,” J. Opt. B: Quantum Semiclassical Opt.6, 495–501 (2004).
[CrossRef]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science306, 1330–1336 (2004).
[CrossRef] [PubMed]

J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett.29, 1153–1155 (2004).
[CrossRef] [PubMed]

2001

A. Weiss, M. Hennes, and M. Rotach, “Derivation of refractive index and temperature gradients from optical scintillometry to correct atmospherically induced errors for highly precise geodetic measurements,” Surv. Geophys.22, 589–596 (2001).
[CrossRef]

2000

1998

G. Bönsch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlén’s formulae,” Metrologia35, 133–139 (1998).
[CrossRef]

1996

1994

S. Braunstein and C. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett.72, 3439–3443 (1994).
[CrossRef] [PubMed]

A. N. Golubev and A. M. Chekhovsky, “Three-color optical range finding,” Appl. Opt.33, 7511–7517 (1994).
[CrossRef] [PubMed]

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, 1084–1086 (1992).
[CrossRef]

1989

A. Ishida, “Two-wavelength displacement-measuring interferometer using second-harmonic light to eliminate air-turbulence-induced errors,” Jpn. J. Appl. Phys.28, L473–L475 (1989).
[CrossRef]

1988

1968

C. Helstrom, “The minimum variance of estimates in quantum signal detection,” IEEE Trans. Inf. Theory14, 234–242 (1968).
[CrossRef]

1966

B. Edlén, “The refractive index of air,” Metrologia2, 71–80 (1966).
[CrossRef]

1965

P. L. Bender and J. C. Owen, “Correction of optical distance measurement for the fluctuating atmospheric index of refraction,” J. Geophys. Res.70, 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, 084004 (2008).
[CrossRef]

Acef, O.

Azouigui, S.

S. Azouigui, T. Badr, J. -P. Wallerand, M. Himbert, J. Salgado, and P. Juncar, “Transportable distance measurement system based on superheterodyne interferometry using two phase-locked frequency-doubled Nd:YAG lasers,” Rev. Sci. Instrum.81, 053112 (2010).
[CrossRef] [PubMed]

Bachor, H.

Badr, T.

S. Azouigui, T. Badr, J. -P. Wallerand, M. Himbert, J. Salgado, and P. Juncar, “Transportable distance measurement system based on superheterodyne interferometry using two phase-locked frequency-doubled Nd:YAG lasers,” Rev. Sci. Instrum.81, 053112 (2010).
[CrossRef] [PubMed]

Balling, P.

Barlier, F.

P. Exertier, P. Bonnefond, F. Deleflie, F. Barlier, M. Kasser, R. Biancale, and Y. Ménard, “Contribution of laser ranging to earth’s sciences,” C. R. Geosci.338, 958–967 (2006).
[CrossRef]

Bender, P. L.

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

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.

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]

Biancale, R.

P. Exertier, P. Bonnefond, F. Deleflie, F. Barlier, M. Kasser, R. Biancale, and Y. Ménard, “Contribution of laser ranging to earth’s sciences,” C. R. Geosci.338, 958–967 (2006).
[CrossRef]

Bonnefond, P.

P. Exertier, P. Bonnefond, F. Deleflie, F. Barlier, M. Kasser, R. Biancale, and Y. Ménard, “Contribution of laser ranging to earth’s sciences,” C. R. Geosci.338, 958–967 (2006).
[CrossRef]

Bönsch, G.

G. Bönsch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlén’s formulae,” Metrologia35, 133–139 (1998).
[CrossRef]

Bowen, W.

M. Hsu, V. Delaubert, P. Lam, and W. Bowen, “Optimal optical measurement of small displacements,” J. Opt. B: Quantum Semiclassical Opt.6, 495–501 (2004).
[CrossRef]

Braun, D.

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense gaussian quantum light: a multimodal approach,” Phys. Rev. A85, 1–4 (2012).
[CrossRef]

Braunstein, S.

S. Braunstein and C. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett.72, 3439–3443 (1994).
[CrossRef] [PubMed]

Caves, C.

S. Braunstein and C. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett.72, 3439–3443 (1994).
[CrossRef] [PubMed]

Chalopin, B.

O. Pinel, Pu. Jian, R. Medeiros de Araújo, J. Feng, B. Chalopin, C. Fabre, and N. Treps, “Generation and characterization of multimode quantum frequency combs,” Phys. Rev. Lett.108, 083601 (2012).
[CrossRef] [PubMed]

Chekhovsky, A. M.

Ciddor, P. E.

Clairon, A.

Coddington, I.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics3, 351–356 (2009).
[CrossRef]

Cui, M.

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]

Dändliker, R.

Delaubert, V.

V. Delaubert, N. Treps, C. Harb, P. Lam, and H. Bachor, “Quantum measurements of spatial conjugate variables: displacement and tilt of a gaussian beam,” Opt. Lett.31, 1537–1539 (2006).
[CrossRef] [PubMed]

M. Hsu, V. Delaubert, P. Lam, and W. Bowen, “Optimal optical measurement of small displacements,” J. Opt. B: Quantum Semiclassical Opt.6, 495–501 (2004).
[CrossRef]

Deleflie, F.

P. Exertier, P. Bonnefond, F. Deleflie, F. Barlier, M. Kasser, R. Biancale, and Y. Ménard, “Contribution of laser ranging to earth’s sciences,” C. R. Geosci.338, 958–967 (2006).
[CrossRef]

Delplancke, F.

F. Delplancke, “The prima facility phase-referenced imaging and micro-arcsecond astrometry,” New Astron. Rev.52, 199–207 (2008).
[CrossRef]

Djerroud, K.

Edlén, B.

B. Edlén, “The refractive index of air,” Metrologia2, 71–80 (1966).
[CrossRef]

Exertier, P.

P. Exertier, P. Bonnefond, F. Deleflie, F. Barlier, M. Kasser, R. Biancale, and Y. Ménard, “Contribution of laser ranging to earth’s sciences,” C. R. Geosci.338, 958–967 (2006).
[CrossRef]

Fabre, C.

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense gaussian quantum light: a multimodal approach,” Phys. Rev. A85, 1–4 (2012).
[CrossRef]

O. Pinel, Pu. Jian, R. Medeiros de Araújo, J. Feng, B. Chalopin, C. Fabre, and N. Treps, “Generation and characterization of multimode quantum frequency combs,” Phys. Rev. Lett.108, 083601 (2012).
[CrossRef] [PubMed]

B. Lamine, C. Fabre, and N. Treps, “Quantum improvement of time transfer between remote clocks,” Phys. Rev. Lett.101, 123601 (2008).
[CrossRef] [PubMed]

Fade, J.

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense gaussian quantum light: a multimodal approach,” Phys. Rev. A85, 1–4 (2012).
[CrossRef]

Feng, J.

O. Pinel, Pu. Jian, R. Medeiros de Araújo, J. Feng, B. Chalopin, C. Fabre, and N. Treps, “Generation and characterization of multimode quantum frequency combs,” Phys. Rev. Lett.108, 083601 (2012).
[CrossRef] [PubMed]

Finizio, S. D.

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science306, 1330–1336 (2004).
[CrossRef] [PubMed]

Golubev, A. N.

Harb, C.

Helstrom, C.

C. Helstrom, “The minimum variance of estimates in quantum signal detection,” IEEE Trans. Inf. Theory14, 234–242 (1968).
[CrossRef]

Hennes, M.

A. Weiss, M. Hennes, and M. Rotach, “Derivation of refractive index and temperature gradients from optical scintillometry to correct atmospherically induced errors for highly precise geodetic measurements,” Surv. Geophys.22, 589–596 (2001).
[CrossRef]

Himbert, M.

S. Azouigui, T. Badr, J. -P. Wallerand, M. Himbert, J. Salgado, and P. Juncar, “Transportable distance measurement system based on superheterodyne interferometry using two phase-locked frequency-doubled Nd:YAG lasers,” Rev. Sci. Instrum.81, 053112 (2010).
[CrossRef] [PubMed]

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, 1084–1086 (1992).
[CrossRef]

Hsu, M.

M. Hsu, V. Delaubert, P. Lam, and W. Bowen, “Optimal optical measurement of small displacements,” J. Opt. B: Quantum Semiclassical Opt.6, 495–501 (2004).
[CrossRef]

Ishida, A.

A. Ishida, “Two-wavelength displacement-measuring interferometer using second-harmonic light to eliminate air-turbulence-induced errors,” Jpn. J. Appl. Phys.28, L473–L475 (1989).
[CrossRef]

Jian, P.

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense gaussian quantum light: a multimodal approach,” Phys. Rev. A85, 1–4 (2012).
[CrossRef]

Jian, Pu.

O. Pinel, Pu. Jian, R. Medeiros de Araújo, J. Feng, B. Chalopin, C. Fabre, and N. Treps, “Generation and characterization of multimode quantum frequency combs,” Phys. Rev. Lett.108, 083601 (2012).
[CrossRef] [PubMed]

Joo, K. -N.

Juncar, P.

S. Azouigui, T. Badr, J. -P. Wallerand, M. Himbert, J. Salgado, and P. Juncar, “Transportable distance measurement system based on superheterodyne interferometry using two phase-locked frequency-doubled Nd:YAG lasers,” Rev. Sci. Instrum.81, 053112 (2010).
[CrossRef] [PubMed]

Kasser, M.

P. Exertier, P. Bonnefond, F. Deleflie, F. Barlier, M. Kasser, R. Biancale, and Y. Ménard, “Contribution of laser ranging to earth’s sciences,” C. R. Geosci.338, 958–967 (2006).
[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. Photonics4, 716–720 (2010).
[CrossRef]

K. -N. Joo and S. -W. Kim, “Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser,” Opt. Express14, 5954–5960 (2006).
[CrossRef] [PubMed]

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. Photonics4, 716–720 (2010).
[CrossRef]

Kren, P.

Lam, P.

V. Delaubert, N. Treps, C. Harb, P. Lam, and H. Bachor, “Quantum measurements of spatial conjugate variables: displacement and tilt of a gaussian beam,” Opt. Lett.31, 1537–1539 (2006).
[CrossRef] [PubMed]

M. Hsu, V. Delaubert, P. Lam, and W. Bowen, “Optimal optical measurement of small displacements,” J. Opt. B: Quantum Semiclassical Opt.6, 495–501 (2004).
[CrossRef]

Lamine, B.

B. Lamine, C. Fabre, and N. Treps, “Quantum improvement of time transfer between remote clocks,” Phys. Rev. Lett.101, 123601 (2008).
[CrossRef] [PubMed]

Le Floch, S.

Lee, J.

J. Lee, Y. -J. Kim, K. Lee, S. Lee, and S. -W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics4, 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. Photonics4, 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. Photonics4, 716–720 (2010).
[CrossRef]

Lemonde, P.

Lévêque, S.

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science306, 1330–1336 (2004).
[CrossRef] [PubMed]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science306, 1330–1336 (2004).
[CrossRef] [PubMed]

Macovez, R.

Man, C. N.

Mariano, M.

Martorell, J.

Masika, P.

Matsumoto, H.

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, 5512–5517 (2000).
[CrossRef]

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, 1084–1086 (1992).
[CrossRef]

Medeiros de Araújo, R.

O. Pinel, Pu. Jian, R. Medeiros de Araújo, J. Feng, B. Chalopin, C. Fabre, and N. Treps, “Generation and characterization of multimode quantum frequency combs,” Phys. Rev. Lett.108, 083601 (2012).
[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, 084004 (2008).
[CrossRef]

Ménard, Y.

P. Exertier, P. Bonnefond, F. Deleflie, F. Barlier, M. Kasser, R. Biancale, and Y. Ménard, “Contribution of laser ranging to earth’s sciences,” C. R. Geosci.338, 958–967 (2006).
[CrossRef]

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. Photonics3, 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. Photonics3, 351–356 (2009).
[CrossRef]

Owen, J. C.

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

Pinel, O.

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense gaussian quantum light: a multimodal approach,” Phys. Rev. A85, 1–4 (2012).
[CrossRef]

O. Pinel, Pu. Jian, R. Medeiros de Araújo, J. Feng, B. Chalopin, C. Fabre, and N. Treps, “Generation and characterization of multimode quantum frequency combs,” Phys. Rev. Lett.108, 083601 (2012).
[CrossRef] [PubMed]

Potulski, E.

G. Bönsch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlén’s formulae,” Metrologia35, 133–139 (1998).
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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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S. Azouigui, T. Badr, J. -P. Wallerand, M. Himbert, J. Salgado, and P. Juncar, “Transportable distance measurement system based on superheterodyne interferometry using two phase-locked frequency-doubled Nd:YAG lasers,” Rev. Sci. Instrum.81, 053112 (2010).
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[CrossRef]

Other

Formulas for refractive index of air are available on http://emtoolbox.nist.gov/Wavelength/Documentation.asp

P. Réfrégier, Noise Theory and Application to Physics: From Fluctuations to Information (Springer Verlag, 2004).

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Figures (5)

Fig. 1
Fig. 1

General detection scheme for measuring pi at the Cramér-Rao bound. A specific example of a full experimental setup is given in Fig.3

Fig. 2
Fig. 2

Relation between the different LO modes in the vector space {v0, v1, v2}. The modes w ϕ p, wϕ, wGVD and w GVD p lie in the same plane.

Fig. 3
Fig. 3

Direct distance measurement with an appropriately shaped LO.

Fig. 4
Fig. 4

Spectral profile of modes wL and w L p for 3 fs FWHM Gaussian pulses.

Fig. 5
Fig. 5

Spectral profile of Hermite-Gaussian modes v0, v1 and v2.

Tables (1)

Tables Icon

Table 1 Summary of the different LO modes and the sensitivities.

Equations (57)

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L ϕ 1 = n ϕ ( λ 1 ) L and L ϕ 2 = n ϕ ( λ 2 ) L .
L = L ϕ 1 + α ( L ϕ 1 L ϕ 2 ) with α = K ( λ 1 ) K ( λ 2 ) K ( λ 1 ) ,
( δ L ) 2 W I shot 3 × 10 14 m .
L = L ϕ 1 + β ( L ϕ 2 L ϕ 1 ) + γ ( L ϕ 3 L ϕ 1 ) .
( t , p ) 0 u ( t , p ) ,
( ω , p ) ( t , p ) e i ω t d t , u ( ω , p ) u ( t , p ) e i ω t d t .
ω 0 = ω | u ( ω , p ) | 2 d ω , Δ ω 2 = ( ω ω 0 ) 2 | u ( ω , p ) | 2 d ω .
( ω , p ) = i ( ω ) e i k ( ω , p ) L ( p ) , k ( ω , p ) = n ϕ ( ω , p ) ω c .
( ω , p ) i ( ω ) exp [ i ( ω 0 t ϕ ( p ) + ( ω ω 0 ) t g ( p ) + ( ω ω 0 ) 2 ω 0 t GVD ( p ) ) ] ,
t ϕ ( p ) = n ϕ ( ω 0 , p ) L ( p ) c ,
t g ( p ) = n g ( ω 0 , p ) L ( p ) c = ( n ϕ ( ω 0 , p ) + ω 0 n ϕ ( ω 0 , p ) ) L ( p ) c ,
t GVD ( p ) = ω 0 ( n ϕ ( ω 0 , p ) + ω 0 2 n ϕ ( ω 0 , p ) ) L ( p ) c .
( ω , p ) ( ω , 0 ) + p p ( ω , p = 0 ) = 0 ( u ( ω , 0 ) + i p i K i w i ( ω ) ) ,
S [ w i ] = 1 K i Re [ u ( p ) , w i ] = p j i = 0 p i .
( p i ) min = 1 2 N K i .
w ϕ ( ω ) = i u ( ω ) = v 0 ( ω )
w g ( ω ) = i ω ω 0 Δ ω u ( ω ) = v 1 ( ω )
w GVD ( ω ) = i 1 3 ( ω ω 0 ) 2 Δ ω 2 u ( ω ) = 1 3 v 0 ( ω ) + 2 3 v 2 ( ω )
S [ w ϕ ] = p ϕ + Δ ω 2 ω 0 2 p GVD
S [ w g ] = p g
S [ w GVD ] = 1 3 ω 0 2 Δ ω 2 p ϕ + p GVD .
w ϕ p ( ω ) = 2 3 v 0 ( ω ) 1 3 v 2 ( ω ) .
S [ w ϕ p ] = p ϕ .
( p ϕ ) min p = 1 2 N 2 3 ω 0 .
w GVD p ( ω ) = v 2 ( ω ) with K GVD p = 2 Δ ω 2 ω 0 .
w L ( ω ) = 1 c K L ( ω 0 v 0 ( ω ) + Δ ω v 1 ( ω ) )
S [ w L ] = p L + K X K L w L , w X p X + K P w K L w L , w P w p P w .
w L p ( ω ) w L ( ω ) w L , w P w w X , w P w w L , w X 1 w X , w P w 2 w P w ( ω )
w L , w X w X , w P w w L , w P w 1 w X , w P w 2 w X ( ω ) .
K L p = K L 1 w L , w P w 2 + w L , w X 2 2 w L , w P w w L , w X w X , w P w 1 w X , w P w 2 .
( δ L ) 1 HD shot = c 2 N K L p = 2 × 10 11 m .
n ϕ ( σ , T , P , x , P w ) 1 = K ( σ ) X ( T , P , x ) g ( σ ) P w
K ( σ ) = 10 8 ( A + B 130 σ 2 + C 38.9 σ 2 ) ,
X ( T , P , x ) = P D 1 + 10 8 ( E F T ) P 1 + G T [ 1 + H ( x 0.04 % ) ] ,
g ( σ ) = 10 10 ( I J σ 2 ) ,
A = 8091.37 , B = 2333983 , C = 15518 ,
D = 932164.60 , E = 0.5953 , F = 0.009876 , G = 0.0036610 ,
H = 0.5327 ,
I = 3.802 , J = 0.0384 .
n g ( σ , T , P , x , P w ) 1 = ( K ( σ ) + σ K ( σ ) ) X ( T , P , x ) ( g ( σ ) + σ g ( σ ) ) P w ,
u ( ω ) = 1 Δ ω 1 ( 2 π ) 1 / 4 e ( ω ω 0 ) 2 4 Δ ω 2 ,
v n ( ω ) = i 1 2 n n ! H n ( ω ω 0 2 Δ ω ) u ( ω ) .
v 0 ( ω ) = i u ( ω )
v 1 ( ω ) = i ω ω 0 Δ ω u ( ω )
v 2 ( ω ) = i 1 2 ( ( ω ω 0 ) 2 Δ ω 2 1 ) u ( ω ) .
w L ( ω ) = 1 c K L ( ω 0 v 0 ( ω ) + Δ ω v 1 ( ω ) ) w X ( ω ) = L K ( ω 0 ) c K X [ ( ω 0 + Δ ω 2 ω 0 ( δ 1 + δ 2 ) ) v 0 ( ω ) + Δ ω ( 1 + δ 1 ) v 1 ( ω ) + 2 Δ ω 2 ω 0 ( δ 1 + δ 2 ) v 2 ( ω ) ] w P w ( ω ) = L g ( ω 0 ) c K P w [ ( ω 0 + Δ ω 2 ω 0 ( η 1 + η 2 ) ) v 0 ( ω ) + Δ ω ( 1 + η 1 ) v 1 ( ω ) + 2 Δ ω 2 ω 0 ( η 1 + η 2 ) v 2 ( ω ) ]
δ 1 = ω 0 K ( ω 0 ) K ( ω 0 ) , δ 2 = ω 0 2 2 K ( ω 0 ) K ( ω 0 ) ,
η 1 = ω 0 g ( ω 0 ) g ( ω 0 ) , η 2 = ω 0 2 2 g ( ω 0 ) g ( ω 0 ) .
K L = 1 c ω 0 2 + Δ ω 2
K X = K ( ω 0 ) L c ( ω 0 + Δ ω 2 ω 0 ( δ 1 + δ 2 ) ) 2 + Δ ω 2 ( 1 + δ 1 ) 2 + 2 Δ ω 4 ω 0 2 ( δ 1 + δ 2 ) 2
K P w = g ( ω 0 ) L c ( ω 0 + Δ ω 2 ω 0 ( η 1 + η 2 ) ) 2 + Δ ω 2 ( 1 + η 1 ) 2 + 2 Δ ω 4 ω 0 2 ( η 1 + η 2 ) 2
M [ w L ] = p L + K X K L w L , w X p X + K P w K L w L , w P w p P w
M [ w X ] = K L K X w L , w X p L + p X + K P w K X w L , w P w p P w
M [ w P w ] = K L K P w w L , w P w p L + K X K P w w X , w P w p X + p P w
w P w i ( ω ) = 1 1 w X , w P w 2 ( w P w ( ω ) w X , w P w w X )
w L p ( ω ) w L ( ω ) w L , w X w X ( ω ) w L , w P w i w P w i
1 w L , w P w 2 + w L , w X 2 2 w L , w P w w L , w X w X , w P w 1 w X , w P w 2

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