## Long distance measurement with femtosecond pulses using a dispersive interferometer |

Optics Express, Vol. 19, Issue 7, pp. 6549-6562 (2011)

http://dx.doi.org/10.1364/OE.19.006549

Acrobat PDF (954 KB)

### 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 *μ*m (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

## 1. Introduction

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]

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]

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]

*μ*m in comparison to a counting laser for a measured distance of 50 m [7

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]

*μ*m (3 × 10

^{−8}), with a statistical averaging of less than 200 nm.

## 2. Measurement principle

*Ê*(

_{r}*ω*)| and

*φ*(

_{r}*ω*) are the electric amplitude and phase of the pulse from the reference arm. |

*Ê*(

_{m}*ω*)| and

*φ*(

_{m}*ω*) are the electric amplitude and phase from the measurement arm. The fundamental property of the interferogram is that it is an intensity measurement that contains the phase difference

*φ*–

_{r}*φ*. This allows the reconstruction of the path-length difference. For a distance propagation in air, the interference term can be written as, with

_{m}*n*and

*c*being the refractive index of air and the speed of light in vacuum, respectively. The distance

*L*is the geometrical pulse separation between the interfering pulses after the beam splitter.

*L*/2 can be viewed as the synthetic wavelength and spatial overlap between the pulses can always be accomplished when the displacement of the measurement arm is around a multiple of the half inter-pulse distance. The displacement of the measurement arm, Δ

_{pp}*l*, can be written as, with

*m*an integer and

*L*the inter-pulse distance, calculated by

_{pp}*L*=

_{pp}*c/n*. Here

_{g}f_{r}*c*is the speed of light in vacuum,

*f*is the repetition frequency and

_{r}*n*is the group refractive index calculated by using the wavelength at the maximum of the power spectral density (PSD) of the pulse. The choice of wavelength for the calculation of

_{g}*n*will be further discussed in Section 3. Δ

_{g}*l*is the total displacement of the measurement arm.

*L*

_{1}and

*L*

_{2}are the pulse separations before and after the movement of the measurement arm respectively. The signs of

*L*

_{1}and

*L*

_{2}are defined as positive when the pulse from the measurement arm is ahead the pulse from the reference arm. The factor two comes from the back and forth propagation. If the jitter of

*f*is ignored, the absolute uncertainty of this method should not increase due to increasing the integer number

_{r}*m*. This indicates that the maximum distance measured by this technique is only limited by the coherence length of the laser source, allowing for low relative uncertainty at long distance. In practice

*f*is locked and stabilized within 1 Hz, indicating 1

_{r}*μ*m uncertainty in 1 km measured distance. Once the intensity of the spectral interference pattern containing the phase information is recorded, a Fourier filter was used to reconstruct the distance information [11

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]

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 Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B **17**, 1795–1802 (2000). [CrossRef]

*ω*is the angular frequency and

*n*is the group refractive index of air. For an unchirped spectral interferogram,

_{g}*L*tends to a constant value as shown in Fig. 3(f). In the above equation,

*dφ*/

*dω*is the derivative of the unwrapped phase, with respect to circular frequency. In order to minimize the uncertainties introduced by Fast Fourier Transform, a zero-padding process and if necessary, a filtering window have to be used to get the best reconstruction.

## 3. Numerical model for analysis

^{14}Hz(816 nm). Using this PSD, the numerical model implemented plane wave propagation at 10

^{6}individual frequencies, separated by the repetition frequency. The refractive index of air is calculated by the updated Edl

*é*n’s equation [19

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]

*ω*. We can compare this with the distance which we input into the program,

*i.e.*2 mm. We observe that, Fig. 4(b) is mainly a constant at 2 mm, with the oscillations on both sides coming coming from the numerical errors of the FFT.

*L*+ 2 mm and 339 ×

_{pp}*L*– 2 mm respectively, where

_{pp}*L*is the inter-pulse distance, calculated by Eq. (5). In our experiments 339 ×

_{pp}*L*corresponds to a distance of around 100 m. In Fig. 5(a) the fringes are more separated in the high frequency side and the other way round in Fig. 5(c), but these chirps can hardly be seen from the two figures. The chirp is clearly visible in the figures showing the extracted distances, that is Fig. 5(b) for 339 ×

_{pp}*L*– 2 mm and Fig. 5(d) for 339 ×

_{pp}*L*+ 2 mm. An increasing or decreasing derivative of the unwrapped phase always means chirp of the spectral interferogram. Here the twin image ambiguity disappears: A decreasing curve in Fig. 5(b) means the path length difference is shorter than 339 ×

_{pp}*L*. This can be understood by considering that, higher frequency waves propagate slower than lower frequency waves in air. When the path length difference between the two arms is shorter than 339 ×

_{pp}*L*, the pulse reflected from the long arm is in front of the pulse reflected from the short arm. The high frequency component of the pulse from the long arm are closer to the pulse from the short arm. We see that the central region of the curves in Fig. 5(b) and Fig. 5(d) are approximately linear. This is mainly due to the fact that for a small frequency range around 3.674×10

_{pp}^{14}Hz the refractive index of air is approximately a linear function of frequency.

^{14}Hz, as labelled with the dot in both curves. This fact implies that, for deriving the distance information, it is not necessary to know the entire distance curve in Fig. 5(b) but only the small region around 3.674×10

^{14}Hz is interesting. The same derived distance of 2 mm in Fig. 5(b) and Fig. 5(d) at 3.674×10

^{14}Hz means the fringe densities in Fig. 5(a) and Fig. 5(c) around the frequency 3.674×10

^{14}Hz is the same, as labelled by the rectangle. These are the regions that we are interested which contain the distance information.

^{14}Hz. Here 3.674×10

^{14}Hz is a special frequency not because it is the maximum of the PSD, but because it is the frequency which we used to calculate

*L*. Actually, the exact distance or path length difference which we measure can be extracted from almost the entire curves shown in Fig. 5(b), except for the small regions at the edges which have the numerical artifact due to the Fourier transform. We can choose an arbitrary frequency component in Fig. 5(b), but remember that

_{pp}*L*should also be calculated from the frequency we choose. To illustrate this more clearly, we give an example. Let us select two points from Fig. 5(b), for example, point 1,

_{pp}*f*

_{1}= 3.658 × 10

^{14}Hz

*L*

_{1}= 2.009 mm and point 2,

*f*

_{2}= 3.688 × 10

^{14}Hz

*L*

_{2}= 1.997 mm. These two frequencies correspond to two different group refractive indices in air and hence two inter-pulse distances. Using Eq. (5), we have

*L*

_{pp1}= 294.063777 mm and

*L*

_{pp2}= 294.063742 mm. The total distance calculated using different frequency components, is the same.

## 4. Experimental setup

## 5. Measurement procedure

### 5.1. A new calibration approach

20. C. Dorrer, “Influence of the calibration of the detector on spectral interferometry,” J. Opt. Soc. Am. B **16**, 1160–1168 (1999). [CrossRef]

### 5.2. Long distance measurement

*x*= 1150. By comparing it with the PSD measured with the Fourier transform spectrometer, we know that this corresponds to the frequency of 3.674 × 10

^{14}Hz (816 nm). The measurement is carried out by first placing the mechanical car with the retro reflector, in the long arm, at the closest possible position to the dichroic mirror. The translation stage in the reference arm is scanned and the spectral interferogram is found and recorded. At this time the path length difference between both arms is around 1.5 m, corresponding to 9 times of the cavity length. Let us label the position of the measurement arm at this position of the retro reflector as A and the distance between the two interfering pulses as

*L*. In Fig. 8(a) we show a typical measured spectrum at this position. Then the modulated spectrum is analysed and the derivative of the unwrapped phase

_{A}*dφ*is plotted in Fig. 8(b). This curve is compared to the curves during the calibration, Fig. 7(b) and Fig. 7(d) by using Eq. (9) and the derived pulse separation is shown in Fig. 8(c). The derived separation between the interfered pulses is almost a constant, implying that at short path-length difference the chirp of the pulse can be ignored. Subsequently for each measurement the mechanical car was moved over a long distance which we label as position B. This long distance is around 50 m, corresponding to 339 times the cavity length. At this time the distance between the two interfered pulses is

_{A}/dx*L*and a new spectral interferogram is recorded. One such spectral interferogram is shown in Fig. 9(a). The reflected beam drops considerably in power, leading to a drop in the modulation depth, but the fringes were clearly visible. The derived pulse separation after calibration is shown in Fig. 9(c). At pixel number

_{B}*x*= 1150 the read out of the distance

*L*= 0.9455 mm. The derived pulse separation is decreasing respected to wavelength. This means the path length difference is 339 ×

_{B}*L*+ 0.9455 mm. The turbulence in air causes the fringes in the spectral interferogram to vibrate, which can be seen in the corresponding curve in Fig. 9(c). This leads to an uncertainty of around 2

_{pp}*μ*m. The measured spectral interferogram and the derived pulse separation are also compared with simulation. Fig. 9(b) is the simulated spectral interferogram for this distance. To give a good comparison, the x-axis of Fig. 9(b) was chosen to be the wavelength, and the power of the reflected beam was taken to be only 1/10 of the original. The pulse separation calculated from the simulation is shown in Fig. 9(d). Besides the aliasing on both sides, the central parts of (b) and (d) match quite well, as indicated by the red line in Fig. 9-(c).

*L*. Fig. 10(a) and Fig. 10(c) show the spectral interferogram and the derived pulse separation at

_{pp}*L*= −2.9228 mm. The spectral interferometer and the derived curve of the pulse separation is also compared with the simulations, as shown in Fig. 10(c) and Fig. 10(d). The total displacement of the measurement arm is calculated from Eq. (3), where

_{B}*L*is calculated by

_{pp}*L*=

_{pp}*c/n*and

_{g}f_{r}*n*is the group refractive index of air at 3.674 × 10

_{g}^{14}Hz (816 nm).

### 5.3. Measurement Result

*f*for each group. Within each group 6 independent measurements were performed. In each measurement, we arbitrarily chose

_{r}*L*

_{1}and

*L*

_{2}in Eq. (3). At each position the spectral interferogram is recorded 5 times to statistically minimize the uncertainty. One 50 m measurement took about 10 minutes, mostly occupied by transporting the retroreflector down the 50 m bench. The measured displacement is compared with measurement of the HeNe fringe counting laser interferometer and the result is shown in Fig. 11. The agreement in 5 measurements are all within one wavelength, with the standard deviation of around 1

*μ*m. The average of all measurements shows the agreement within 200 nm on 100 m pulse propagation in air, as shown by the dotted line in Fig. 11. We attribute the residual difference and uncertainty in the comparison measurement to vibrations of the setup and air turbulence. The variation and uncertainty of the environmental parameters, limits the accuracy in measuring the absolute distance. For example, a typical uncertainty of 0.1°C in temperature already leads to an uncertainty of the refractive index of air of about 1×10

^{−7}, corresponding to 10

*μ*m at 100 m propagation. Also, an uncertainty of 1 hPa in pressure leads to an uncertainty of 2.7×10

^{−7}, corresponding to 26

*μ*m at 100 m propagation. Moreover the updated Edlen’s equation itself has an intrinsic uncertainty of 1×10

^{−8}, implying 1

*μ*m over a path length difference of 100 m. The influences of the environmental parameters on the refractive index at the wavelengths of both the He-Ne laser and the frequency comb cancel in first order. For this is the reason the agreement between the experimental results is much better than 10

*μ*m. The agreement between the frequency comb and the He-Ne laser is much better than the achievable absolute accuracy in air, showing that the measurement result is not limited by the chosen method.

## 6. Conclusion

*μ*m. If the spectrometer is calibrated, an arbitrary frequency within the frequency comb is needed and the distance can be calculated by using the group refractive index at this particular frequency. In our experiment, we have chosen the frequency at the peak of the PSD. This is not an arbitrary because instead of the spectrum, we calibrated a short displacement. Here, the measured PSD by the CCD line is not calibrated and we have to find a position in the PSD where we know both the actual frequency and the pixel index on the CCD line.

*L*/2. This is because in principle our grating spectrometer has the resolution of around 0.04 nm. At this resolution the fringes become hard to distinguish when the distance between the interfered pulses is more than 5 mm. This distance is much shorter than the inter-pulse distance of our femtosecond laser. In order to measure an arbitrary distance, one easy way is to displace the reference arm. There are other solutions for this problem. For example, in 2006, Joo and coworkers suggested the use of an Fabry-Perot Etalon (FPE) before the spectrometer [11

_{pp}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]

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]

## References and links

1. | R. Dandliker, K. Hug, J. Politch, and E. Zimmermann, “High-accuracy distance measurements with multiple-wavelength interferometry,” Opt. Express |

2. | B. L. Swinkels, N Bhattacharya, and J. J. M. Braat, “Correcting movement errors in frequencysweeping interferometry,” Opt. Lett. |

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 |

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. |

5. | J. Lee, Y.-J. Kim, K. Lee, S. Lee, and S.-W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics , |

6. | J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. |

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,” |

8. | P. Balling, P. Kren, P. Masika, and S. A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express |

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. |

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 |

11. | K.-N. Joo and S.-W. Kim, “Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser,” Opt. Express |

12. | K. N. Joo, Y. Kim, and S. W. Kim, “Distance measurements by combined method based on a femtosecond pulse laser,” Opt. Express |

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

14. | Y. Salvade, N. Schuhler, S. Leveque, and S. Le Floch, “High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source,” Appl. Opt. |

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. |

16. | U. Schnell, E. Zimmermann, and R. Dandliker, “Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry,” Pure Appl. Opt. |

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. |

18. | C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B |

19. | K. P. Birch and M. J. Downs, “Correction to the updated Edlén equation for the refractive index of air,” Metrologia |

20. | C. Dorrer, “Influence of the calibration of the detector on spectral interferometry,” J. Opt. Soc. Am. B |

21. | A. Bartels, D. Heinecke, and S. A. Diddams, “Passively mode-locked 10 GHz femtosecond Ti:sapphire laser,” Opt. Lett. |

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 |

23. | D. W. Rush, P. T. Ho, and G. L. Burdge, “The coherence time of a modelocked pulse train,” Opt. Commun. |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(320.7100) Ultrafast optics : Ultrafast measurements

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: November 23, 2010

Revised Manuscript: December 24, 2010

Manuscript Accepted: January 18, 2011

Published: March 23, 2011

**Citation**

M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, and H. P. Urbach, "Long distance measurement with femtosecond pulses using a dispersive interferometer," Opt. Express **19**, 6549-6562 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6549

Sort: Year | Journal | Reset

### References

- R. Dandliker, K. Hug, J. Politch, and E. Zimmermann, “High-accuracy distance measurements with multiplewavelength interferometry,” Opt. Express 34, 2407–2412 (1995).
- B. L. Swinkels, N. Bhattacharya, and J. J. M. Braat, “Correcting movement errors in frequencysweeping interferometry,” Opt. Lett. 30, 2242–2244 (2005). [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]
- 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]
- 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]
- J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. 29, 1153–1155 (2004). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- C. Dorrer, “Influence of the calibration of the detector on spectral interferometry,” J. Opt. Soc. Am. B 16, 1160–1168 (1999). [CrossRef]
- 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]
- 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]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.