Abstract

We experimentally demonstrate long distance measurements with a femtosecond frequency comb laser using dispersive interferometry. The distance is derived from the unwrapped spectral phase of the dispersed interferometer output and the repetition frequency of the laser. For an interferometer length of 50 m this approach has been compared to an independent phase counting laser interferometer. The obtained mutual agreement is better than 1.5 μm (3 × 10−8), with a statistical averaging of less than 200 nm. Our experiments demonstrate that dispersive interferometry with a frequency comb laser is a powerful method for accurate and non-incremental measurement of long distances.

© 2011 Optical Society of America

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  1. R. Dandliker, K. Hug, J. Politch, and E. Zimmermann, “High-accuracy distance measurements with multiplewavelength interferometry,” Opt. Express 34, 2407–2412 (1995).
  2. B. L. Swinkels, N. Bhattacharya, and J. J. M. Braat, “Correcting movement errors in frequencysweeping interferometry,” Opt. Lett. 30, 2242–2244 (2005).
    [CrossRef] [PubMed]
  3. D. 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]
  4. K. Minoshima, and H. Matsumoto, “High-accuracy measurement of 240 m distance in an optical tunnel by using of a compact femtosecond laser,” Appl. Opt. 39, 5512–5517 (2000).
    [CrossRef]
  5. J. Lee, Y.-J. Kim, K. Lee, S. Lee, and S.-W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics 4, 716–720 (2010).
    [CrossRef]
  6. J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. 29, 1153–1155 (2004).
    [CrossRef] [PubMed]
  7. 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]
  8. P. Balling, P. Kren, P. Masika, and S. A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express 17, 9300–9313 (2009).
    [CrossRef] [PubMed]
  9. Y. Yamaoka, K. Minoshima, and H. Matsumoto, “Direct measurement of the group refractive index of air with interferometry between adjacent femtosecond pulses,” Appl. Opt. 41, 4318–4324 (2002).
    [CrossRef] [PubMed]
  10. L. Lepetit, G. Chriaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12(12), 2467–2474 (1995).
    [CrossRef]
  11. K.-N. Joo, and S.-W. Kim, “Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser,” Opt. Express 14, 5954–5960 (2006).
    [CrossRef] [PubMed]
  12. K. N. Joo, Y. Kim, and S. W. Kim, “Distance measurements by combined method based on a femtosecond pulse laser,” Opt. Express 16, 19799–19806 (2008).
    [CrossRef] [PubMed]
  13. 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]
  14. 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).
    [CrossRef] [PubMed]
  15. S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20, 095302 (2009).
    [CrossRef]
  16. U. Schnell, E. Zimmermann, and R. Dandliker, “Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry,” Pure Appl. Opt. 4, 643–651 (1995).
    [CrossRef]
  17. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  18. C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fouriertransform spectral interferometry,” J. Opt. Soc. Am. B 17, 1795–1802 (2000).
    [CrossRef]
  19. K. P. Birch, and M. J. Downs, “Correction to the updated Edlén equation for the refractive index of air,” Metrologia 31, 315–316 (1994).
    [CrossRef]
  20. C. Dorrer, “Influence of the calibration of the detector on spectral interferometry,” J. Opt. Soc. Am. B 16, 1160–1168 (1999).
    [CrossRef]
  21. A. Bartels, D. Heinecke, and S. A. Diddams, “Passively mode-locked 10 GHz femtosecond Ti:sapphire laser,” Opt. Lett. 33, 1905–1907 (2008).
    [CrossRef] [PubMed]
  22. R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier envelope offset noise of mode locked lasers,” Appl. Phys. B 82, 265–273 (2006).
    [CrossRef]
  23. D. W. Rush, P. T. Ho, and G. L. Burdge, “The coherence time of a modelocked pulse train,” Opt. Commun. 52, 41–45 (1984).
    [CrossRef]

2010 (1)

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

2009 (4)

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

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]

S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20, 095302 (2009).
[CrossRef]

2008 (3)

2006 (2)

R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier envelope offset noise of mode locked lasers,” Appl. Phys. B 82, 265–273 (2006).
[CrossRef]

K.-N. Joo, and S.-W. Kim, “Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser,” Opt. Express 14, 5954–5960 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

2002 (1)

2000 (3)

1999 (1)

1995 (3)

U. Schnell, E. Zimmermann, and R. Dandliker, “Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry,” Pure Appl. Opt. 4, 643–651 (1995).
[CrossRef]

R. Dandliker, K. Hug, J. Politch, and E. Zimmermann, “High-accuracy distance measurements with multiplewavelength interferometry,” Opt. Express 34, 2407–2412 (1995).

L. Lepetit, G. Chriaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12(12), 2467–2474 (1995).
[CrossRef]

1994 (1)

K. P. Birch, and M. J. Downs, “Correction to the updated Edlén equation for the refractive index of air,” Metrologia 31, 315–316 (1994).
[CrossRef]

1984 (1)

D. W. Rush, P. T. Ho, and G. L. Burdge, “The coherence time of a modelocked pulse train,” Opt. Commun. 52, 41–45 (1984).
[CrossRef]

1982 (1)

Balling, P.

Bartels, A.

Belabas, N.

Bhattacharya, N.

Birch, K. P.

K. P. Birch, and M. J. Downs, “Correction to the updated Edlén equation for the refractive index of air,” Metrologia 31, 315–316 (1994).
[CrossRef]

Braat, J. J. M.

Burdge, G. L.

D. W. Rush, P. T. Ho, and G. L. Burdge, “The coherence time of a modelocked pulse train,” Opt. Commun. 52, 41–45 (1984).
[CrossRef]

Chriaux, G.

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.

Cundiff, S. T.

D. 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]

Dandliker, R.

R. Dandliker, K. Hug, J. Politch, and E. Zimmermann, “High-accuracy distance measurements with multiplewavelength interferometry,” Opt. Express 34, 2407–2412 (1995).

U. Schnell, E. Zimmermann, and R. Dandliker, “Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry,” Pure Appl. Opt. 4, 643–651 (1995).
[CrossRef]

Diddams, S. A.

A. Bartels, D. Heinecke, and S. A. Diddams, “Passively mode-locked 10 GHz femtosecond Ti:sapphire laser,” Opt. Lett. 33, 1905–1907 (2008).
[CrossRef] [PubMed]

D. 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]

Dorrer, C.

Downs, M. J.

K. P. Birch, and M. J. Downs, “Correction to the updated Edlén equation for the refractive index of air,” Metrologia 31, 315–316 (1994).
[CrossRef]

Hall, J. L.

D. 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]

Heinecke, D.

Ho, P. T.

D. W. Rush, P. T. Ho, and G. L. Burdge, “The coherence time of a modelocked pulse train,” Opt. Commun. 52, 41–45 (1984).
[CrossRef]

Hug, K.

R. Dandliker, K. Hug, J. Politch, and E. Zimmermann, “High-accuracy distance measurements with multiplewavelength interferometry,” Opt. Express 34, 2407–2412 (1995).

Hyun, S.

S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20, 095302 (2009).
[CrossRef]

Ina, H.

Jin, J.

S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20, 095302 (2009).
[CrossRef]

Joffre, M.

Jones, D.

D. 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.

Joo, K.-N.

Keller, U.

R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier envelope offset noise of mode locked lasers,” Appl. Phys. B 82, 265–273 (2006).
[CrossRef]

Kim, S. W.

Kim, S.-W.

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

S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20, 095302 (2009).
[CrossRef]

K.-N. Joo, and S.-W. Kim, “Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser,” Opt. Express 14, 5954–5960 (2006).
[CrossRef] [PubMed]

Kim, Y.

S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20, 095302 (2009).
[CrossRef]

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

Kim, Y. J.

S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20, 095302 (2009).
[CrossRef]

Kim, Y.-J.

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

Kobayashi, S.

Kren, P.

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. Photonics 4, 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, 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, 716–720 (2010).
[CrossRef]

Lepetit, L.

Leveque, S.

Likforman, J. P.

Masika, P.

Matsumoto, H.

Minoshima, K.

Nenadovic, L.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3, 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]

Paschotta, R.

R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier envelope offset noise of mode locked lasers,” Appl. Phys. B 82, 265–273 (2006).
[CrossRef]

Politch, J.

R. Dandliker, K. Hug, J. Politch, and E. Zimmermann, “High-accuracy distance measurements with multiplewavelength interferometry,” Opt. Express 34, 2407–2412 (1995).

Ranka, J. K.

D. 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]

Rush, D. W.

D. W. Rush, P. T. Ho, and G. L. Burdge, “The coherence time of a modelocked pulse train,” Opt. Commun. 52, 41–45 (1984).
[CrossRef]

Salvade, Y.

Schlatter, A.

R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier envelope offset noise of mode locked lasers,” Appl. Phys. B 82, 265–273 (2006).
[CrossRef]

Schnell, U.

U. Schnell, E. Zimmermann, and R. Dandliker, “Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry,” Pure Appl. Opt. 4, 643–651 (1995).
[CrossRef]

Schuhler, N.

Stentz, A.

D. 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]

Swinkels, B. L.

Takeda, M.

Telle, H. R.

R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier envelope offset noise of mode locked lasers,” Appl. Phys. B 82, 265–273 (2006).
[CrossRef]

Urbach, H. P.

van den Berg, S. A.

Windeler, R. S.

D. 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]

Yamaoka, Y.

Ye, J.

Zeitouny, M. G.

Zeller, S. C.

R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier envelope offset noise of mode locked lasers,” Appl. Phys. B 82, 265–273 (2006).
[CrossRef]

Zimmermann, E.

R. Dandliker, K. Hug, J. Politch, and E. Zimmermann, “High-accuracy distance measurements with multiplewavelength interferometry,” Opt. Express 34, 2407–2412 (1995).

U. Schnell, E. Zimmermann, and R. Dandliker, “Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry,” Pure Appl. Opt. 4, 643–651 (1995).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. B (1)

R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier envelope offset noise of mode locked lasers,” Appl. Phys. B 82, 265–273 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (3)

Meas. Sci. Technol. (1)

S. Hyun, Y. J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20, 095302 (2009).
[CrossRef]

Metrologia (1)

K. P. Birch, and M. J. Downs, “Correction to the updated Edlén 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.-W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics 4, 716–720 (2010).
[CrossRef]

Opt. Commun. (1)

D. W. Rush, P. T. Ho, and G. L. Burdge, “The coherence time of a modelocked pulse train,” Opt. Commun. 52, 41–45 (1984).
[CrossRef]

Opt. Express (4)

Opt. Lett. (4)

Pure Appl. Opt. (1)

U. Schnell, E. Zimmermann, and R. Dandliker, “Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry,” Pure Appl. Opt. 4, 643–651 (1995).
[CrossRef]

Science (1)

D. 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]

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Figures (11)

Fig. 1
Fig. 1

A schematic to show the measurement principle. The beam from the femtosecond pulse laser is collimated into a Michelson interferometer. The output light from both arms is combined and reflected by a diffraction grating. The diffracted beam is then focused by a lens onto a line CCD.

Fig. 2
Fig. 2

Spectral interferograms recorded for different pulse separations. (a) Two pulses at a short separation. (b) Two pulses at a large separation. (c) After one cavity length, the fringes re-appear, where one pulse is interfering to its next.

Fig. 3
Fig. 3

Data processing procedure for measurement of L. (a) The spectral interferogram i.e. dispersed interference intensity captured by the CCD line. (b) Fourier transform of the measured spectral interferogram. (c) The DC peak and one AC peak are band pass filtered. (d) The wrapped phase. (e) The unwrapped phase. (f) The pulse separation obtained from the derivative of the unwrapped phase.

Fig. 4
Fig. 4

(a) Numerically simulated interferogram with path length difference equals to 0×Lpp±2 mm. (b) The derived pulse separation as a function of frequency.

Fig. 5
Fig. 5

(a) Numerically simulated interferogram with path length difference equals to 339×Lpp – 2 mm. (b) The derived pulse separation from the spectral interferogram shown in (a). (c) Numerically simulated interferogram with path length difference equals to 339×Lpp + 2 mm. (d) The derived pulse separation from the spectral interferogram shown in (c).

Fig. 6
Fig. 6

Schematic of the experimental setup used to measure distances in the 50 m corridor using spectral interferograms. The mode locked Ti:Sapphire laser generates 40 femtosecond pulses with repetition rate of around 1 GHz. The reference arm consists of a hollow corner cube mounted on the translation stage (TS2). The measurement arm consists of two dispersion compensated mirrors and a gold coated retroreflector mounted on a mechanical stage of the 50 m long measurement bench. The HeNe fringe counting laser interferometer was used for a simultaneous comparison measurement. The returning beams were overlapped and focused on a grating spectrometer. A third gold coated retro reflector on a small translation stage (TS1) was added at the beginning of the long arm of the interferometer. A second HeNe fringe counting interferometer was also added to the setup to enable the calibration measurements.

Fig. 7
Fig. 7

The spectral interferograms and the derivatives of the unwrapped phases at position C1 and C2 in the calibration measurement. (a) The measured interferogram when the measurement arm is located at position “C1”. (b) The derivative of the unwrapped phase c1/dx obtained from the interferogram in (a), with respect to the pixel number x. (c) The measured interferogram when the measurement arm is located at position “C2”. (d) The derivative of the unwrapped phase c2/dx obtained from the interferogram in (c), with respect to the pixel number x.

Fig. 8
Fig. 8

(a) A typical measured spectral interferogram at the beginning of the bench and the derived distance as a function of pixel number. (b) The derivative of the unwrapped phase. (c) The derived pulse separation after calibration.

Fig. 9
Fig. 9

(a) A typical measured interferogram with path length difference longer than 339 × Lpp. (b) The simulated spectral interferogram using L = 0.9455 mm. (c) The derived pulse separation from the measurement data in (a). At pixel number x = 1150 the read out of the distance L = 0.9455 mm. The decreasing slope of the curve indicates a positive sign. (d) The pulse separation calculated from the simulation using L = 0.9455 mm.

Fig. 10
Fig. 10

(a) A typical measured interferogram with path length difference shorter than 339×Lpp. (b) The simulated spectral interferogram using L = −2.9228 mm. (c) The derived pulse separation from the measurement data in (a). At pixel number x = 1150 the read out of the distance L = 2.9228 mm. The increasing slope of the curve indicates a negative sign. (d) The pulse separation calculated from the simulation using L = −2.9228 mm.

Fig. 11
Fig. 11

Comparison measurement of displacements of around 50 m. The error bars indicate the standard uncertainty, derived from measurement reproducibility. The average of all measurements is shown by the dotted line.

Equations (9)

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

S ( ω ) = 2 | E ^ r ( ω ) E ^ m ( ω ) | [ 1 + cos ( φ r ( ω ) φ m ( ω ) ) ] ,
φ ( ω ) = φ r ( ω ) φ m ( ω ) = n ( ω ) ω L / c ,
Δ l = m L p p / 2 + L 1 / 2 L 2 / 2 ,
L ( ω ) = c n g ( d φ d ω ) .
339 × L p p 1 L 1 = 339 × L p p 2 L 2 = 99685.611 mm ,
x = Ω ( ω ) ,
L = c n g d φ d x Ω ( ω ) ,
2 Δ l c = c n g c ( d φ c 1 d x Ω ( ω ) d φ c 2 d x Ω ( ω ) ) ,
L 2 Δ l c = n g c n g d φ d x / ( d φ 1 c d x d φ 2 c d x ) .

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