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  • Editor: Xi-Cheng Zhang
  • Vol. 39, Iss. 9 — May. 1, 2014
  • pp: 2545–2548
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Eavesdropping time and frequency: phase noise cancellation along a time-varying path, such as an optical fiber

Gesine Grosche  »View Author Affiliations


Optics Letters, Vol. 39, Issue 9, pp. 2545-2548 (2014)
http://dx.doi.org/10.1364/OL.39.002545


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Abstract

Single-mode optical fiber is a highly efficient connecting medium used not only for optical telecommunications but also for the dissemination of ultrastable frequencies or timing signals. Ma et al. [Opt. Lett. 19, 1777 (1994)] described a measurement and control system to deliver the same optical frequency at two places, namely the two ends of a fiber, by eliminating the “fiber-induced phase-noise modulation, which corrupts high-precision frequency-based applications.” I present a simple detection and control scheme to deliver the same optical frequency at many places anywhere along a transmission path, or in its vicinity, with a relative instability of 1 part in 1019. The same idea applies to radio frequency and timing signals. This considerably simplifies future efforts to make precise timing or frequency signals available to many users, as required in some large-scale science experiments.

© 2014 Optical Society of America

To date, efforts have focused on long-distance connections [8

8. N. R. Newbury, P. A. Williams, and W. Swann, Opt. Lett. 32, 3056 (2007). [CrossRef]

10

10. K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, Science 336, 441 (2012). [CrossRef]

] between just two points, one “remote” lab and one “local” lab connected by an optical fiber, using methods similar to that proposed in 1994 by Ma et al. [11

11. L. S. Ma, P. Jungner, J. Ye, and J. L. Hall, Opt. Lett. 19, 1777 (1994). [CrossRef]

] to correct phase perturbations between the local and remote end. For example, we have transmitted optical frequencies with a relative accuracy of 1019 over 146 km deployed fiber [12

12. G. Grosche, O. Terra, K. Predehl, R. Holzwarth, B. Lipphardt, F. Vogt, U. Sterr, and H. Schnatz, Opt. Lett. 34, 2270 (2009). [CrossRef]

] and remotely characterized optical clock lasers online with hertz-level resolution [13

13. A. Pape, O. Terra, J. Friebe, M. Riedmann, T. Wübbena, E. M. Rasel, K. Predehl, T. Legero, B. Lipphardt, H. Schnatz, and G. Grosche, Opt. Express 18, 21477 (2010).

]. Significant efforts are now underway to establish national and even international metrology fiber networks.

One important question [14

14. C. Gao, B. Wang, W. L. Chen, Y. Bai, J. Miao, X. Zhu, T. C. Li, and L. J. Wang, Opt. Lett. 37, 4690 (2012). [CrossRef]

19] is how to distribute reference frequencies to many users simultaneously in a cost-effective way. Surprisingly, with one point-to-point connection (such as a long stabilized fiber), we can “tap” this fiber anywhere and locally derive a reference frequency with the same precision as that achieved at the end point [18

18. G. Grosche, “Verfahren zum Bereitstellen einer Referenz-Frequenz,” German patentDE 200810062139 (June24, 2010).

]. We present the patented concept, an experimental setup achieving relative frequency instability of 1019, and several extensions of the idea; these include a branching design and the multipoint dissemination of time using two-way transfer. The methods complement the Ethernet-based “White Rabbit” [19] and enable high-fidelity distribution of frequency or time even for large-area science projects, such as radio telescope arrays [7

7. P. E. Dewdney, P. J. Hall, R. T. Schilizzi, and T. J. L. W. Lazio, Proc. IEEE 97, 1482 (2009). [CrossRef]

] or national metrology networks.

To explain the basic idea, we first consider the existing scheme for point-to-point stabilization. Figure 1 shows a commonly implemented and well-characterized method for phase-stable transmission of an ultrastable optical frequency νlocal from a local point A to a remote point Z [8

8. N. R. Newbury, P. A. Williams, and W. Swann, Opt. Lett. 32, 3056 (2007). [CrossRef]

10

10. K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, Science 336, 441 (2012). [CrossRef]

,12

12. G. Grosche, O. Terra, K. Predehl, R. Holzwarth, B. Lipphardt, F. Vogt, U. Sterr, and H. Schnatz, Opt. Lett. 34, 2270 (2009). [CrossRef]

,20

20. P. A. Williams, W. C. Swann, and N. R. Newbury, J. Opt. Soc. Am. B 25, 1284 (2008). [CrossRef]

]. Analogous, earlier designs enable the phase-stable transmission of radio frequency [21

21. M. Musha, Y. Sato, K. Nakagawa, K. Ueda, A. Ueda, and M. Ishiguro, Appl. Phys. B. 82, 555 (2006).

,22

22. F. Narbonneau, M. Lours, S. Bize, A. Clairon, G. Santarelli, O. Lopez, C. Daussy, A. Amy-Klein, and C. Chardonnet, Rev. Sci. Instrum. 77, 064701 (2006). [CrossRef]

] or pulsed [23

23. G. Marra, H. S. Margolis, and D. J. Richardson, Opt. Express 20, 1775 (2012).

] signals. The method is reminiscent of [11

11. L. S. Ma, P. Jungner, J. Ye, and J. L. Hall, Opt. Lett. 19, 1777 (1994). [CrossRef]

,24

24. J. Ye, J. L. Peng, R. J. Jones, K. W. Holman, J. L. Hall, D. J. Jones, S. A. Diddams, J. Kitching, S. Bize, J. C. Bergquist, L. W. Hollberg, L. Robertsson, and L. S. Ma, J. Opt. Soc. Am. B 20, 1459 (2003). [CrossRef]

] (though different) and can be understood by viewing the entire transmission path from A to the mirror at Z as the long arm of an interferometer.

Fig. 1. Stabilized fiber link connecting the remote point Z with the local point A. For a detailed discussion, see for instance [8,20].

We denote by Δϕij the phase shift experienced by a signal travelling from any point i to point j and assume symmetry of the transmission path: Δϕij=Δϕji. The acousto-optic modulator (AOM) AOM2 provides a fixed frequency shift to distinguish light that has reached the mirror at Z from light reflected anywhere else along the transmission path. At photodetector DetA, the returned light is superimposed with local light traveling through a short reference arm. This yields a beat signal with frequency
fDetA=2(fAOM1+Δϕ˙AZ/2π+fAOM2)
and phase reflecting the momentary phase difference between the two interferometer arms. The beat signal fDetA is compared to a synthesized signal at frequency fsynth with a phase-frequency comparator [21

21. M. Musha, Y. Sato, K. Nakagawa, K. Ueda, A. Ueda, and M. Ishiguro, Appl. Phys. B. 82, 555 (2006).

] or a simple mixer. A servo acts on the phase and frequency of AOM1 to maintain a constant phase difference between the beat signal and the synthesizer signal, resulting in a fixed phase relationship (modulo the frequency offset fsynth/2) between light at point Z and at point A. Thus the frequency delivered to point Z is given by νZ=νlocal+fsynth/2=νA+fsynth/2.

To generate a phase-stable signal at an intermediate point C along the transmission path, we now tap the transmitted signal in both directions. Figures 1 and 2 and the analysis describe in detail one implementation ([18

18. G. Grosche, “Verfahren zum Bereitstellen einer Referenz-Frequenz,” German patentDE 200810062139 (June24, 2010).

], p. 2); similar setups are suitable for periodically modulated or pulsed signals [18

18. G. Grosche, “Verfahren zum Bereitstellen einer Referenz-Frequenz,” German patentDE 200810062139 (June24, 2010).

]. Here the terminology of optical frequency transfer is used.

Fig. 2. Signal generation at point C, from signals tapped in forward and backward directions, with frequencies νf and νb.

The forward propagating light has frequency
νf:=νCf=νA+fAOM1+ΔϕAC/2π,
whereas backward propagating light has frequency
νb:=νCb=νA+fAOM1+Δϕ˙AZ/2π+2fAOM2+Δϕ˙ZC/2π.

We superimpose forward- and backward propagating light on photodetector DetC (here ignoring noise beyond the tapping point) to generate a beat signal at frequency
νbνf=Δϕ˙CZ/2π+2fAOM2+Δϕ˙ZC/2π=2(Δϕ˙CZ/2π+fAOM2).
(1)

The beat signal is amplified and its frequency digitally divided by two. Applying this as a correction frequency fcorr:=(νbνf)/2 to the forward propagating light at point C, for instance using another AOM, we obtain a stable signal at point C:
νC-out=νf+fcorr=νZ=νlocal+fsynth/2.

This can be viewed as detecting at point C the additional phase shift between points C and Z, and applying this to the signal coupled out at point C, so that its frequency and phase follow those of the signal at point Z. The output at point C is thus ideally as stable as νZ. Since νZ=(νb+νf)/2, applying fcorr to the backward propagating light at point C also yields νZ [15

15. Y. Bai, B. Wang, X. Zhu, C. Gao, J. Miao, and L. J. Wang, Opt. Lett. 38, 3333 (2013). [CrossRef]

,18

18. G. Grosche, “Verfahren zum Bereitstellen einer Referenz-Frequenz,” German patentDE 200810062139 (June24, 2010).

].

Furthermore, high-power ultrastable light can be made available [18

18. G. Grosche, “Verfahren zum Bereitstellen einer Referenz-Frequenz,” German patentDE 200810062139 (June24, 2010).

]. If we wish to preserve optical power in the main link, asymmetric beam splitters or tap couplers extract only a small percentage of the light. The extracted power may be boosted, for example with an erbium-doped fiber amplifier before detector DetC. Alternatively, the weak (μW) extracted signals are first superimposed with light from a laser at frequency νL0 giving two strong heterodyne beat signals νbνL0 and νfνL0. Their difference frequency is again νbνf and is independent of νL0 and its fluctuations. As before, after division by two, it may serve as a correction frequency. Alternatively, if the mean frequency of the two heterodyne beat signals, (νbνL0+νfνL0)/2, is added as a correction to νL0 (e.g., via an AOM), we obtain ultrastable, high-power laser light at frequency νL=νZ.

Open diamonds in Fig. 3 show the frequency instability at point C when implementing the new scheme as shown in Figs. 1 and 2: νC-out reaches a relative instability (ADEV in >10kHz bandwidth) of 1017 after 1000 s. Free-running fiber noise (full green triangles, fcorr) is clearly suppressed.

Fig. 3. Measured frequency instability (ADEV). Green triangles: fcorr, represents free-running fiber (see text); black open circles: νZ, stabilized remote output at Z; blue open diamonds: νC-out, stabilized output at point C, first design; full blue squares: νC-out for the improved design.

Analysis of the fiber paths that contribute to frequency fluctuations at point C, see Fig. 2, shows that νC-out=(νf+νb)/2+(f1+b1)/2+f3+f4. [Notation: we write f1 etc., for both the fiber paths and the frequency fluctuations arising from them.] The total length of exposed, uncompensated fiber is 23m. An improved design, which minimizes uncompensated fiber paths, is shown in Fig. 4.

For this design (full blue squares in Fig. 3), νC-out is as stable as νZ for short times (1017 at 10 s); the excess frequency instability at 1000 s is below 1018. After 10000 s, an instability below 1019 is reached; this is a factor 40 lower than reported in [15

15. Y. Bai, B. Wang, X. Zhu, C. Gao, J. Miao, and L. J. Wang, Opt. Lett. 38, 3333 (2013). [CrossRef]

]. The total uncompensated fiber length (Fig. 4) was below 0.8 m, with f2, f3, and part of f4 being colocated inside a box to minimize air currents and mounted on a 12 mm thick aluminum board as a thermal mass. Further noise reduction is possible by environmental shielding, length-matching fibers so temperature changes enter common mode (f2f3+f4), and/or active temperature stabilization.

Fig. 4. Improved “tentacle” design for detection unit. FRM, Faraday rotator mirror; AOM, acousto-optic modulator; DetC, photodector. Fiber paths f1 and b1 contribute no noise to νC-out.

Fig. 5. Eavesdropping on a TWTFT link between points A and Z, which simultaneously send signals Sforw and Sback. At each extraction point, signal SB_out exits the apparatus at time τAZ/2.

Implementation may use receiver modules of standard TWTFT-modems, as in point-to-point fiber-based time transfer [32

32. M. Rost, D. Piester, W. Yang, T. Feldmann, T. Wübbena, and A. Bauch, Metrologia 49, 772 (2012). [CrossRef]

,33

33. O. Lopez, A. Kanj, P. E. Pottie, D. Rovera, J. Achkar, C. Chardonnet, A. Amy-Klein, and G. Santarelli, Appl. Phys. B 110, 3 (2013). [CrossRef]

] or newly developed electronic delay lines and time signal encoders/decoders [34

34. L. Sliwczynski, P. Krehlik, A. Czubla, L. Buczek, and M. Lipinski, Metrologia 50, 133 (2013). [CrossRef]

].

I am indebted to Fritz Riehle and Giorgio Santarelli for their indefatigable enthusiasm and openness to new ideas, to the authors of [11

11. L. S. Ma, P. Jungner, J. Ye, and J. L. Hall, Opt. Lett. 19, 1777 (1994). [CrossRef]

] for lucid writing, and to the Deutsche Forschungsgemeinschaft (SFB 407 and QUEST, Centre for Quantum Engineering and Space-Time Research) and the European Space Agency for financial support.

References

1.

C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, Phys. Rev. Lett. 104, 070802 (2010). [CrossRef]

2.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, Nature 506, 71 (2014).

3.

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013). [CrossRef]

4.

S. G. Karshenboim, Can. J. Phys. 83, 767 (2005). [CrossRef]

5.

C. Clivati, D. Calonico, G. A. Costanzo, A. Mura, M. Pizzocaro, and F. Levi, Opt. Lett. 38, 1092 (2013). [CrossRef]

6.

T. C. W. Chou, D. B. Hume, T. Rosenband, and D. J. Wineland, Science 329, 1630 (2010). [CrossRef]

7.

P. E. Dewdney, P. J. Hall, R. T. Schilizzi, and T. J. L. W. Lazio, Proc. IEEE 97, 1482 (2009). [CrossRef]

8.

N. R. Newbury, P. A. Williams, and W. Swann, Opt. Lett. 32, 3056 (2007). [CrossRef]

9.

O. Lopez, A. Haboucha, B. Chanteau, C. Chardonnet, A. Amy-Klein, and G. Santarelli, Opt. Express 20, 23518 (2012). [CrossRef]

10.

K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, Science 336, 441 (2012). [CrossRef]

11.

L. S. Ma, P. Jungner, J. Ye, and J. L. Hall, Opt. Lett. 19, 1777 (1994). [CrossRef]

12.

G. Grosche, O. Terra, K. Predehl, R. Holzwarth, B. Lipphardt, F. Vogt, U. Sterr, and H. Schnatz, Opt. Lett. 34, 2270 (2009). [CrossRef]

13.

A. Pape, O. Terra, J. Friebe, M. Riedmann, T. Wübbena, E. M. Rasel, K. Predehl, T. Legero, B. Lipphardt, H. Schnatz, and G. Grosche, Opt. Express 18, 21477 (2010).

14.

C. Gao, B. Wang, W. L. Chen, Y. Bai, J. Miao, X. Zhu, T. C. Li, and L. J. Wang, Opt. Lett. 37, 4690 (2012). [CrossRef]

15.

Y. Bai, B. Wang, X. Zhu, C. Gao, J. Miao, and L. J. Wang, Opt. Lett. 38, 3333 (2013). [CrossRef]

16.

S. W. Schediwy, D. Gozzard, K. G. H. Baldwin, B. J. Orr, R. B. Warrington, G. Aben, and A. N. Luiten, Opt. Lett. 38, 2893 (2013). [CrossRef]

17.

P. Krehlik, L. Sliwczynski, L. Buczek, and M. Lipinski, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60, 1804 (2013). [CrossRef]

18.

G. Grosche, “Verfahren zum Bereitstellen einer Referenz-Frequenz,” German patentDE 200810062139 (June24, 2010).

19.

http://www.ohwr.org/projects/white-rabbit.

20.

P. A. Williams, W. C. Swann, and N. R. Newbury, J. Opt. Soc. Am. B 25, 1284 (2008). [CrossRef]

21.

M. Musha, Y. Sato, K. Nakagawa, K. Ueda, A. Ueda, and M. Ishiguro, Appl. Phys. B. 82, 555 (2006).

22.

F. Narbonneau, M. Lours, S. Bize, A. Clairon, G. Santarelli, O. Lopez, C. Daussy, A. Amy-Klein, and C. Chardonnet, Rev. Sci. Instrum. 77, 064701 (2006). [CrossRef]

23.

G. Marra, H. S. Margolis, and D. J. Richardson, Opt. Express 20, 1775 (2012).

24.

J. Ye, J. L. Peng, R. J. Jones, K. W. Holman, J. L. Hall, D. J. Jones, S. A. Diddams, J. Kitching, S. Bize, J. C. Bergquist, L. W. Hollberg, L. Robertsson, and L. S. Ma, J. Opt. Soc. Am. B 20, 1459 (2003). [CrossRef]

25.

S. T. Dawkins, J. J. McFerran, and A. N. Luiten, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54, 918 (2007). [CrossRef]

26.

G. Grosche, B. Lipphardt, and H. Schnatz, Eur. Phys. J. D 48, 27 (2008). [CrossRef]

27.

A. Bercy, S. Guellati-Khelifa, F. Stefani, G. Santarelli, C. Chardonnet, P.-E. Pottie, O. Lopez, and A. Amy-Klein, J. Opt. Soc. Am. B 31, 678 (2014). [CrossRef]

28.

S. M. F. Raupach and G. Grosche, “Chirped frequency transfer with an accuracy of 10−18 and its application to the remote synchronisation of timescales,” arXiv: 1308.6725 (2013).

29.

S. A. Jefferts, M. A. Weiss, J. Levine, S. Dilla, E. W. Bell, and T. E. Parker, IEEE Trans. Instrum. Meas. 46, 209 (1997).

30.

J. M. A. Steele, W. Markowitz, and C. A. Lidback, IEEE Trans. Instrum. Meas. 13, 164 (1964).

31.

D. Kirchner, H. Ressler, P. Grudler, F. Baumont, C. Veillet, W. Lewandowski, W. Hanson, W. Klepczynski, and P. Uhrich, Metrologia 30, 183 (1993). [CrossRef]

32.

M. Rost, D. Piester, W. Yang, T. Feldmann, T. Wübbena, and A. Bauch, Metrologia 49, 772 (2012). [CrossRef]

33.

O. Lopez, A. Kanj, P. E. Pottie, D. Rovera, J. Achkar, C. Chardonnet, A. Amy-Klein, and G. Santarelli, Appl. Phys. B 110, 3 (2013). [CrossRef]

34.

L. Sliwczynski, P. Krehlik, A. Czubla, L. Buczek, and M. Lipinski, Metrologia 50, 133 (2013). [CrossRef]

OCIS Codes
(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems
(120.3940) Instrumentation, measurement, and metrology : Metrology
(060.2840) Fiber optics and optical communications : Heterodyne

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: March 4, 2014
Manuscript Accepted: March 8, 2014
Published: April 17, 2014

Citation
Gesine Grosche, "Eavesdropping time and frequency: phase noise cancellation along a time-varying path, such as an optical fiber," Opt. Lett. 39, 2545-2548 (2014)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-39-9-2545


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References

  1. C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, Phys. Rev. Lett. 104, 070802 (2010). [CrossRef]
  2. B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, Nature 506, 71 (2014).
  3. N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013). [CrossRef]
  4. S. G. Karshenboim, Can. J. Phys. 83, 767 (2005). [CrossRef]
  5. C. Clivati, D. Calonico, G. A. Costanzo, A. Mura, M. Pizzocaro, and F. Levi, Opt. Lett. 38, 1092 (2013). [CrossRef]
  6. T. C. W. Chou, D. B. Hume, T. Rosenband, and D. J. Wineland, Science 329, 1630 (2010). [CrossRef]
  7. P. E. Dewdney, P. J. Hall, R. T. Schilizzi, and T. J. L. W. Lazio, Proc. IEEE 97, 1482 (2009). [CrossRef]
  8. N. R. Newbury, P. A. Williams, and W. Swann, Opt. Lett. 32, 3056 (2007). [CrossRef]
  9. O. Lopez, A. Haboucha, B. Chanteau, C. Chardonnet, A. Amy-Klein, and G. Santarelli, Opt. Express 20, 23518 (2012). [CrossRef]
  10. K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, Science 336, 441 (2012). [CrossRef]
  11. L. S. Ma, P. Jungner, J. Ye, and J. L. Hall, Opt. Lett. 19, 1777 (1994). [CrossRef]
  12. G. Grosche, O. Terra, K. Predehl, R. Holzwarth, B. Lipphardt, F. Vogt, U. Sterr, and H. Schnatz, Opt. Lett. 34, 2270 (2009). [CrossRef]
  13. A. Pape, O. Terra, J. Friebe, M. Riedmann, T. Wübbena, E. M. Rasel, K. Predehl, T. Legero, B. Lipphardt, H. Schnatz, and G. Grosche, Opt. Express 18, 21477 (2010).
  14. C. Gao, B. Wang, W. L. Chen, Y. Bai, J. Miao, X. Zhu, T. C. Li, and L. J. Wang, Opt. Lett. 37, 4690 (2012). [CrossRef]
  15. Y. Bai, B. Wang, X. Zhu, C. Gao, J. Miao, and L. J. Wang, Opt. Lett. 38, 3333 (2013). [CrossRef]
  16. S. W. Schediwy, D. Gozzard, K. G. H. Baldwin, B. J. Orr, R. B. Warrington, G. Aben, and A. N. Luiten, Opt. Lett. 38, 2893 (2013). [CrossRef]
  17. P. Krehlik, L. Sliwczynski, L. Buczek, and M. Lipinski, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60, 1804 (2013). [CrossRef]
  18. G. Grosche, “Verfahren zum Bereitstellen einer Referenz-Frequenz,” German patentDE 200810062139 (June24, 2010).
  19. http://www.ohwr.org/projects/white-rabbit .
  20. P. A. Williams, W. C. Swann, and N. R. Newbury, J. Opt. Soc. Am. B 25, 1284 (2008). [CrossRef]
  21. M. Musha, Y. Sato, K. Nakagawa, K. Ueda, A. Ueda, and M. Ishiguro, Appl. Phys. B. 82, 555 (2006).
  22. F. Narbonneau, M. Lours, S. Bize, A. Clairon, G. Santarelli, O. Lopez, C. Daussy, A. Amy-Klein, and C. Chardonnet, Rev. Sci. Instrum. 77, 064701 (2006). [CrossRef]
  23. G. Marra, H. S. Margolis, and D. J. Richardson, Opt. Express 20, 1775 (2012).
  24. J. Ye, J. L. Peng, R. J. Jones, K. W. Holman, J. L. Hall, D. J. Jones, S. A. Diddams, J. Kitching, S. Bize, J. C. Bergquist, L. W. Hollberg, L. Robertsson, and L. S. Ma, J. Opt. Soc. Am. B 20, 1459 (2003). [CrossRef]
  25. S. T. Dawkins, J. J. McFerran, and A. N. Luiten, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54, 918 (2007). [CrossRef]
  26. G. Grosche, B. Lipphardt, and H. Schnatz, Eur. Phys. J. D 48, 27 (2008). [CrossRef]
  27. A. Bercy, S. Guellati-Khelifa, F. Stefani, G. Santarelli, C. Chardonnet, P.-E. Pottie, O. Lopez, and A. Amy-Klein, J. Opt. Soc. Am. B 31, 678 (2014). [CrossRef]
  28. S. M. F. Raupach and G. Grosche, “Chirped frequency transfer with an accuracy of 10−18 and its application to the remote synchronisation of timescales,” arXiv: 1308.6725 (2013).
  29. S. A. Jefferts, M. A. Weiss, J. Levine, S. Dilla, E. W. Bell, and T. E. Parker, IEEE Trans. Instrum. Meas. 46, 209 (1997).
  30. J. M. A. Steele, W. Markowitz, and C. A. Lidback, IEEE Trans. Instrum. Meas. 13, 164 (1964).
  31. D. Kirchner, H. Ressler, P. Grudler, F. Baumont, C. Veillet, W. Lewandowski, W. Hanson, W. Klepczynski, and P. Uhrich, Metrologia 30, 183 (1993). [CrossRef]
  32. M. Rost, D. Piester, W. Yang, T. Feldmann, T. Wübbena, and A. Bauch, Metrologia 49, 772 (2012). [CrossRef]
  33. O. Lopez, A. Kanj, P. E. Pottie, D. Rovera, J. Achkar, C. Chardonnet, A. Amy-Klein, and G. Santarelli, Appl. Phys. B 110, 3 (2013). [CrossRef]
  34. L. Sliwczynski, P. Krehlik, A. Czubla, L. Buczek, and M. Lipinski, Metrologia 50, 133 (2013). [CrossRef]

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