Abstract

We investigate general properties of the interferograms from a frequency comb laser in a non-linear dispersive medium. The focus is on interferograms at large delay distances and in particular on their broadening, the fringe formation and shape. It is observed that at large delay distances the interferograms spread linearly and that its shape is determined by the source spectral profile. It is also shown that each intensity point of the interferogram is formed by the contribution of one dominant stationary frequency. This stationary frequency is seen to vary as a function of the path length difference even within the interferogram. We also show that the contributing stationary frequency remains constant if the evolution of a particular fringe is followed in the successive interferograms found periodically at different path length differences. This can be used to measure very large distances in dispersive media.

© 2011 Optical Society of America

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  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, 635–639 (2000).
    [CrossRef] [PubMed]
  2. R. Holzwarth, Th. Udem, T. W. H¨ansch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
    [CrossRef] [PubMed]
  3. S. T. Cundiff, and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75, 325–342 (2003).
    [CrossRef]
  4. S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of Time and Frequency at the Outset of the 21st Century,” Science 306, 1318–1324 (2004).
    [CrossRef] [PubMed]
  5. Th. Udem, R. Holzwarth, and T. W. H¨ansch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
    [CrossRef] [PubMed]
  6. L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
    [CrossRef]
  7. J. Ye, H. Schnatz, and L. W. Hollberg, “Optical frequency combs: from frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9, 1041–1058 (2003).
    [CrossRef]
  8. J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. 29, 1153–1155 (2004).
    [CrossRef] [PubMed]
  9. M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. van den Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” J. Eur. Opt. Soc. Rapid Publ. 3, 08003 (2008).
    [CrossRef]
  10. Y. Salvade, N. Schuhler, S. Leveque, and S. Le Floch, “High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source,” Appl. Opt. 47, 2715–2720 (2008) (and references therein).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. P. Balling, P. Křen, P. Mašika, and S. A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express 17, 9300–9313 (2009).
    [CrossRef] [PubMed]
  13. M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H. P. Urbach, and J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. 34, 1982–1984 (2009).
    [CrossRef] [PubMed]
  14. I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3, 351–356 (2009).
    [CrossRef]
  15. J. Lee, Y.-J. Kim, K. Lee, S. Lee, and S. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics 4, 716–720 (2010).
    [CrossRef]
  16. M. G. Zeitouny, M. Cui, N. Bhattacharya, S. A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A 82, 023808 (2010).
    [CrossRef]
  17. K. P. Birch, and M. J. Downs, “Correction to the Updated Edl’en Equation for the Refractive Index of Air,” Metrologia 31, 315–316 (1994).
    [CrossRef]
  18. V. A. Borovikov, Uniform Stationary Phase Method, IEE Electromagnetic Wave Series (1994).
  19. K. E. Oughstun, and N. A. Cartwright, “Physical significance of the group velocity in dispersive, ultrashort Gaussian pulse dynamics,” J. Mod. Opt. 52, 1089–1104 (2005) (and references therein).
    [CrossRef]

2010 (2)

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

M. G. Zeitouny, M. Cui, N. Bhattacharya, S. A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A 82, 023808 (2010).
[CrossRef]

2009 (3)

2008 (3)

2005 (1)

K. E. Oughstun, and N. A. Cartwright, “Physical significance of the group velocity in dispersive, ultrashort Gaussian pulse dynamics,” J. Mod. Opt. 52, 1089–1104 (2005) (and references therein).
[CrossRef]

2004 (2)

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

S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of Time and Frequency at the Outset of the 21st Century,” Science 306, 1318–1324 (2004).
[CrossRef] [PubMed]

2003 (2)

S. T. Cundiff, and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75, 325–342 (2003).
[CrossRef]

J. Ye, H. Schnatz, and L. W. Hollberg, “Optical frequency combs: from frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9, 1041–1058 (2003).
[CrossRef]

2002 (1)

Th. Udem, R. Holzwarth, and T. W. H¨ansch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
[CrossRef] [PubMed]

2001 (1)

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[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, 635–639 (2000).
[CrossRef] [PubMed]

R. Holzwarth, Th. Udem, T. W. H¨ansch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef] [PubMed]

1994 (1)

K. P. Birch, and M. J. Downs, “Correction to the Updated Edl’en Equation for the Refractive Index of Air,” Metrologia 31, 315–316 (1994).
[CrossRef]

Balling, P.

Bergquist, J. C.

S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of Time and Frequency at the Outset of the 21st Century,” Science 306, 1318–1324 (2004).
[CrossRef] [PubMed]

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[CrossRef]

Bhattacharya, N.

M. G. Zeitouny, M. Cui, N. Bhattacharya, S. A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A 82, 023808 (2010).
[CrossRef]

M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H. P. Urbach, and J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. 34, 1982–1984 (2009).
[CrossRef] [PubMed]

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. van den 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, “Correction to the Updated Edl’en Equation for the Refractive Index of Air,” Metrologia 31, 315–316 (1994).
[CrossRef]

Braat, J. J. M.

Cartwright, N. A.

K. E. Oughstun, and N. A. Cartwright, “Physical significance of the group velocity in dispersive, ultrashort Gaussian pulse dynamics,” J. Mod. Opt. 52, 1089–1104 (2005) (and references therein).
[CrossRef]

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, 351–356 (2009).
[CrossRef]

Cui, M.

M. G. Zeitouny, M. Cui, N. Bhattacharya, S. A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A 82, 023808 (2010).
[CrossRef]

M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H. P. Urbach, and J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. 34, 1982–1984 (2009).
[CrossRef] [PubMed]

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. van den 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.

S. T. Cundiff, and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75, 325–342 (2003).
[CrossRef]

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, 635–639 (2000).
[CrossRef] [PubMed]

Curtis, E. A.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[CrossRef]

Diddams, S. A.

S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of Time and Frequency at the Outset of the 21st Century,” Science 306, 1318–1324 (2004).
[CrossRef] [PubMed]

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[CrossRef]

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, 635–639 (2000).
[CrossRef] [PubMed]

Downs, M. J.

K. P. Birch, and M. J. Downs, “Correction to the Updated Edl’en Equation for the Refractive Index of Air,” Metrologia 31, 315–316 (1994).
[CrossRef]

Drullinger, R. E.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[CrossRef]

H¨ansch, T. W.

Th. Udem, R. Holzwarth, and T. W. H¨ansch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
[CrossRef] [PubMed]

R. Holzwarth, Th. Udem, T. W. H¨ansch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef] [PubMed]

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, 635–639 (2000).
[CrossRef] [PubMed]

Hollberg, L.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[CrossRef]

Hollberg, L. W.

J. Ye, H. Schnatz, and L. W. Hollberg, “Optical frequency combs: from frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9, 1041–1058 (2003).
[CrossRef]

Holzwarth, R.

Th. Udem, R. Holzwarth, and T. W. H¨ansch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
[CrossRef] [PubMed]

R. Holzwarth, Th. Udem, T. W. H¨ansch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef] [PubMed]

Itano, W. M.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[CrossRef]

Ivanov, E. N.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[CrossRef]

Janssen, A. J. E. M.

M. G. Zeitouny, M. Cui, N. Bhattacharya, S. A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A 82, 023808 (2010).
[CrossRef]

Jefferts, S. R.

S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of Time and Frequency at the Outset of the 21st Century,” Science 306, 1318–1324 (2004).
[CrossRef] [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, 635–639 (2000).
[CrossRef] [PubMed]

Joo, K.-N.

Kim, S.

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

Kim, S.-W.

Kim, Y.

Kim, Y.-J.

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

Knight, J. C.

R. Holzwarth, Th. Udem, T. W. H¨ansch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef] [PubMed]

Kren, P.

Le Floch, S.

Lee, J.

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

Lee, K.

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

Lee, S.

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

Leveque, S.

Mašika, P.

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, 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, 351–356 (2009).
[CrossRef]

Oates, C. W.

S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of Time and Frequency at the Outset of the 21st Century,” Science 306, 1318–1324 (2004).
[CrossRef] [PubMed]

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[CrossRef]

Oughstun, K. E.

K. E. Oughstun, and N. A. Cartwright, “Physical significance of the group velocity in dispersive, ultrashort Gaussian pulse dynamics,” J. Mod. Opt. 52, 1089–1104 (2005) (and references therein).
[CrossRef]

Rafac, R. J.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[CrossRef]

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, 635–639 (2000).
[CrossRef] [PubMed]

Robinson, H. G.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[CrossRef]

Russell, P. St. J.

R. Holzwarth, Th. Udem, T. W. H¨ansch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef] [PubMed]

Salvade, Y.

Schnatz, H.

J. Ye, H. Schnatz, and L. W. Hollberg, “Optical frequency combs: from frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9, 1041–1058 (2003).
[CrossRef]

Schouten, R. N.

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

Schuhler, N.

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, 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, 351–356 (2009).
[CrossRef]

Udem, T.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[CrossRef]

Udem, Th.

Th. Udem, R. Holzwarth, and T. W. H¨ansch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
[CrossRef] [PubMed]

R. Holzwarth, Th. Udem, T. W. H¨ansch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef] [PubMed]

Urbach, H. P.

M. G. Zeitouny, M. Cui, N. Bhattacharya, S. A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A 82, 023808 (2010).
[CrossRef]

M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H. P. Urbach, and J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. 34, 1982–1984 (2009).
[CrossRef] [PubMed]

van den Berg, S. A.

M. G. Zeitouny, M. Cui, N. Bhattacharya, S. A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A 82, 023808 (2010).
[CrossRef]

M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H. P. Urbach, and J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. 34, 1982–1984 (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, 9300–9313 (2009).
[CrossRef] [PubMed]

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

Wadsworth, W. J.

R. Holzwarth, Th. Udem, T. W. H¨ansch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef] [PubMed]

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, 635–639 (2000).
[CrossRef] [PubMed]

Wineland, D. J.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[CrossRef]

Ye, J.

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

J. Ye, H. Schnatz, and L. W. Hollberg, “Optical frequency combs: from frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9, 1041–1058 (2003).
[CrossRef]

S. T. Cundiff, and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75, 325–342 (2003).
[CrossRef]

Zeitouny, M. G.

M. G. Zeitouny, M. Cui, N. Bhattacharya, S. A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A 82, 023808 (2010).
[CrossRef]

M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H. P. Urbach, and J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. 34, 1982–1984 (2009).
[CrossRef] [PubMed]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Ye, H. Schnatz, and L. W. Hollberg, “Optical frequency combs: from frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9, 1041–1058 (2003).
[CrossRef]

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

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

J. Mod. Opt. (1)

K. E. Oughstun, and N. A. Cartwright, “Physical significance of the group velocity in dispersive, ultrashort Gaussian pulse dynamics,” J. Mod. Opt. 52, 1089–1104 (2005) (and references therein).
[CrossRef]

Metrologia (1)

K. P. Birch, and M. J. Downs, “Correction to the Updated Edl’en Equation for the Refractive Index of Air,” Metrologia 31, 315–316 (1994).
[CrossRef]

Nat. Photonics (2)

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

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

Nature (1)

Th. Udem, R. Holzwarth, and T. W. H¨ansch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (1)

M. G. Zeitouny, M. Cui, N. Bhattacharya, S. A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A 82, 023808 (2010).
[CrossRef]

Phys. Rev. Lett. (1)

R. Holzwarth, Th. Udem, T. W. H¨ansch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

S. T. Cundiff, and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75, 325–342 (2003).
[CrossRef]

Science (2)

S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of Time and Frequency at the Outset of the 21st Century,” Science 306, 1318–1324 (2004).
[CrossRef] [PubMed]

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, 635–639 (2000).
[CrossRef] [PubMed]

Other (1)

V. A. Borovikov, Uniform Stationary Phase Method, IEE Electromagnetic Wave Series (1994).

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Figures (4)

Fig. 1
Fig. 1

Behaviour of the cosine of the phase function for various fringes within one correlation. (a) Cross correlation for a propagation distance X = 120 m. The three lines select three different fringes from the cross-correlation pattern. (b,c,d) The corresponding 3-D plots of the cosine of the phase function of the three selected fringes are shown as a function of the wavelength and the scanning time t. (e,f,g) The cosine of the phase function of the three selected fringes as a function of frequency for five intensity values in each fringe.

Fig. 2
Fig. 2

(a) Analysis of the cosine of the phase function at the brightest fringe of the correlation pattern for various delay distances ranging up to 100 m in air. (b) Wavelength (green, continuous) and width (blue, dotted) of brightest fringe as a function of delay distance. (c) Cross-sections from the cosine of the phase function shown in (a) illustrating the phase function at 3 m (red, dotted) and 60 m (blue, continuous).

Fig. 3
Fig. 3

Comparison between exact and asymptotic cross-correlations for 100 and 200 m in air using the first and the second order stationary phase method. (a) Exact simulation (blue, dotted) compared to first order asymptotic calculation (red, continuous) for 100 m propagation in air. (b) Exact simulation (blue, dotted) compared to second order asymptotic calculation (green, continuous) for 100 m propagation in air. (c) Exact simulation (blue, dotted) compared to first order asymptotic calculation (red, continuous) for 200 m propagation in air. (d) Exact simulation (blue, dotted) compared to second order asymptotic calculation (green, continuous) for 200 m propagation in air.

Fig. 4
Fig. 4

(top) Spectral distribution of the stationary frequencies after 200 m propagation in air. (bottom) Frequency distribution inside a cross-correlation after 200 m propagation in air.

Equations (24)

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Γ ( X ) = m = 0 S ( m ω r + ω 0 ) cos [ ( m ω r + ω 0 ) n ( m ω r + ω 0 ) X c ] ,
Γ ( X ) = T r 2 = h X ( n ¯ X c + T r )
t n ¯ X c ( mod T r )
h X ( t ) = 1 π 0 S ( ω + ω 0 ) cos { ( ω + ω 0 ) [ n ( ω + ω 0 ) n ¯ ] X c + ω 0 n ¯ X c + ω t } d ω .
h X ( t ) = R 2 π π 2 α X cos [ γ X f X 2 ( t ) + θ ] d t 0 h X = 0 ( t 0 ) ] cos [ f X ( t ) α X t 0 ] .
ζ ( X ) = τ c 4 α X = τ c 2 c [ 2 d n ( ω ) d ω + ω d 2 n ( ω ) d ω 2 ] ω = ω c X .
𝒟 = 3 2 τ c 2 α 3 .
h X ( t ) = 1 π Re { 0 S ( ω + ω 0 ) exp [ i ( n ¯ X c ξ ( ω + ω 0 ) + ω t ) ] d ω } ,
ξ ( ω + ω 0 ) = ( ω + ω 0 ) [ n ( ω + ω 0 ) n ¯ 1 ]
h X = 0 ( t ) = 1 2 π S ( | ω | + ω 0 ) exp ( i ω t ) d ω
S ( | ω | + ω 0 ) = h X = 0 ( t 1 ) exp ( i ω t 1 ) d t 1 .
h X ( t ) = Re [ 1 π h X = 0 ( t ) g X ( t t 1 ) d t 1 ] ,
g X , I ( s ) = I exp [ i ω s + i n ¯ X c ξ ( ω + ω 0 ) ] d ω
d d ω [ ω s + n ¯ X c ξ ( ω + ω 0 ) ] = s + n ¯ X c ξ ( ω + ω 0 ) = 0 .
h X ( t ) = 1 2 π 0 S ( ω + ω 0 ) { exp [ i ϕ ( ω ) ] + exp [ i ϕ ( ω ) ] } d ω ,
h X ( t ) 1 π { C 1 X 1 / 2 cos [ ϕ ( ω dom ) + σ π 4 ] + C 2 X 3 / 2 cos [ ϕ ( ω dom ) + σ 3 π 4 ] } ,
C 1 = 2 π | ϕ ( 2 ) ( ω 0 ) | S ( ω dom + ω 0 )
C 2 = π / 2 | ϕ ( 2 ) | 3 / 2 [ S ( 2 ) ϕ ( 3 ) S ( 1 ) ϕ ( 2 ) ϕ ( 4 ) S 4 ϕ ( 2 ) + 5 ( ϕ ( 3 ) ) 2 S 12 ( ϕ ( 2 ) ) 2 ] .
h X ( t ) 1 π S ( ω dom + ω 0 ) 2 π X | d 2 k d ω 2 | ω = ω dom cos [ k ( ω dom + ω 0 ) X ω dom n ¯ X c + ω t + σ π 4 ] .
h X ( t ) 1 π { C 1 X 1 / 2 cos [ ϕ ( ω dom ) + σ π 4 ] σ C 2 X 3 / 2 sin [ ϕ ( ω dom ) + σ π 4 ] } .
h X ( t ) 1 π cos [ k ( ω dom + ω 0 ) X ω dom n ¯ X c + ω t + σ π 4 + ϑ ] .
h X = 0 ( t 0 ) exp [ i ( t + β X n ¯ X / c 2 α X ) t 0 ] d t 0 .
{ t 1 X = n ¯ c β + 2 α ω p ( X , t 1 ) t 2 X = n c β + 2 α ω p ( X , t 2 ) t 2 t 1 X = 2 α [ ω p ( X , t 2 ) ω p ( X , t 1 ) ] .
X = | Δ x 2 c α [ ω p ( X , t 2 ) ω p ( X , t 1 ) ] | .

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