Abstract

We introduce a simple and robust scheme for optical frequency transfer of an ultra-stable source light field via an optical frequency comb to a field at a target optical frequency, where highest stability is required, e.g., for the interrogation of an optical clock. The scheme relies on a topology for end-to-end suppression of the influence of optical path-length fluctuations, which is attained by actively phase-stabilized delivery, combined with common-path propagation. This approach provides a robust stability improvement without the need for additional isolation against environmental disturbances such as temperature, pressure or humidity changes. We measure residual frequency transfer instabilities by comparing the frequency transfers carried out with two independent combs simultaneously. Residual fractional frequency instabilities between two systems of 8 × 10−18 at 1 s and 3 × 10−21 at 105 s averaging time are observed. We discuss the individual noise contributions to the residual instability. The presented scheme is technically simple, robust against environmental parameter fluctuations and enables an ultra-stable frequency transfer, e.g., to optical clock lasers or to lasers in gravitational wave detectors.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article

Corrections

Erik Benkler, Burghard Lipphardt, Thomas Puppe, Rafał Wilk, Felix Rohde, and Uwe Sterr, "End-to-end topology for fiber comb based optical frequency transfer at the 10−21 level: erratum," Opt. Express 28, 15023-15024 (2020)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-28-10-15023

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References

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2019 (6)

E. Oelker, R. B. Hutson, C. J. Kennedy, L. Sonderhouse, T. Bothwell, A. Goban, D. Kedar, C. Sanner, J. M. Robinson, G. E. Marti, D. G. Matei, T. Legero, M. Giunta, R. Holzwarth, F. Riehle, U. Sterr, and J. Ye, “Demonstration of 4.8 × 10−17 stability at 1 s for two independent optical clocks,” Nat. Photonics 13(10), 714–719 (2019).
[Crossref]

C. Sanner, N. Huntemann, R. Lange, C. Tamm, E. Peik, M. S. Safronova, and S. G. Porsev, “Optical clock comparison test of Lorentz symmetry,” Nature 567(7747), 204–208 (2019).
[Crossref]

A. Didier, S. Ignatovich, E. Benkler, M. Okhapkin, and T. E. Mehlstäubler, “946 nm Nd:YAG digital-locked laser at 1.1 × 10−16 in 1 s and transfer-locked to a cryogenic silicon cavity,” Opt. Lett. 44(7), 1781–1784 (2019).
[Crossref]

M. Wada, S. Okubo, K. Kashiwagi, F. Hong, K. Hosaka, and H. Inaba, “Evaluation of fiber noise induced in ultrastable environments,” IEEE Trans. Instrum. Meas. 68(6), 2246–2252 (2019).
[Crossref]

A. Liehl, P. Sulzer, D. Fehrenbacher, T. Rybka, D. V. Seletskiy, and A. Leitenstorfer, “Deterministic nonlinear transformations of phase noise in quantum-limited frequency combs,” Phys. Rev. Lett. 122(20), 203902 (2019).
[Crossref]

S. Herbers, S. Dörscher, E. Benkler, and C. Lisdat, “Phase noise of frequency doublers in optical clock lasers,” Opt. Express 27(16), 23262–23273 (2019).
[Crossref]

2018 (5)

A. Tourigny-Plante, V. Michaud-Belleau, N. Bourbeau Hébert, H. Bergeron, J. Genest, and J.-D. Deschênes, “An open and flexible digital phase-locked loop for optical metrology,” Rev. Sci. Instrum. 89(9), 093103 (2018).
[Crossref]

M. A. Norcia, J. R. K. Cline, J. A. Muniz, J. M. Robinson, R. B. Hutson, A. Goban, G. E. Marti, J. Ye, and J. K. Thompson, “Frequency measurements of superradiance from the strontium clock transition,” Phys. Rev. X 8(2), 021036 (2018).
[Crossref]

P. W. Graham and S. Jung, “Localizing gravitational wave sources with single-baseline atom interferometers,” Phys. Rev. D 97(2), 024052 (2018).
[Crossref]

A. Rolland, P. Li, N. Kuse, J. Jiang, M. Cassinerio, C. Langrock, and M. E. Fermann, “Ultra-broadband dual-branch optical frequency comb with 10−18 instability,” Optica 5(9), 1070–1077 (2018).
[Crossref]

K. Kashiwagi, Y. Nakajima, M. Wada, S. Okubo, and H. Inaba, “Multi-branch fiber comb with relative frequency uncertainty at 10−20 using fiber noise difference cancellation,” Opt. Express 26(7), 8831–8840 (2018).
[Crossref]

2017 (4)

H. Leopardi, J. Davila-Rodriguez, F. Quinlan, J. Olson, J. A. Sherman, S. Diddams, and T. Fortier, “Single-branch Er:fiber frequency comb for precision optical metrology with 10−18 fractional instability,” Optica 4(8), 879–885 (2017).
[Crossref]

N. Ohmae, N. Kuse, M. E. Fermann, and H. Katori, “All-polarization-maintaining, single-port Er:fiber comb for high-stability comparison of optical lattice clocks,” Appl. Phys. Express 10(6), 062503 (2017).
[Crossref]

A. Liehl, D. Fehrenbacher, P. Sulzer, A. Leitenstorfer, and D. V. Seletskiy, “Ultrabroadband out-of-loop characterization of the carrier-envelope phase noise of an offset-free Er:fiber frequency comb,” Opt. Lett. 42(10), 2050–2053 (2017).
[Crossref]

D. G. Matei, T. Legero, S. Häfner, C. Grebing, R. Weyrich, W. Zhang, L. Sonderhouse, J. M. Robinson, J. Ye, F. Riehle, and U. Sterr, “1.5µm lasers with sub-10 mHz linewidth,” Phys. Rev. Lett. 118(26), 263202 (2017).
[Crossref]

2016 (4)

J. M. Hogan and M. A. Kasevich, “Atom-interferometric gravitational-wave detection using heterodyne laser links,” Phys. Rev. A 94(3), 033632 (2016).
[Crossref]

S. Kolkowitz, I. Pikovski, N. Langellier, M. D. Lukin, R. L. Walsworth, and J. Ye, “Gravitational wave detection with optical lattice atomic clocks,” Phys. Rev. D 94(12), 124043 (2016).
[Crossref]

Y. Yao, Y. Jiang, H. Yu, Z. Bi, and L. Ma, “Optical frequency divider with division uncertainty at the 10−21 level,” Natl. Sci. Rev. 3(4), 463–469 (2016).
[Crossref]

T. Puppe, A. Sell, R. Kliese, N. Hoghooghi, A. Zach, and W. Kaenders, “Characterization of a DFG comb showing quadratic scaling of the phase noise with frequency,” Opt. Lett. 41(8), 1877–1880 (2016).
[Crossref]

2015 (6)

S. Häfner, S. Falke, C. Grebing, S. Vogt, T. Legero, M. Merimaa, C. Lisdat, and U. Sterr, “8 ×10−17 fractional laser frequency instability with a long room-temperature cavity,” Opt. Lett. 40(9), 2112–2115 (2015).
[Crossref]

E. Benkler, C. Lisdat, and U. Sterr, “On the relation between uncertainties of weighted frequency averages and the various types of allan deviations,” Metrologia 52(4), 565–574 (2015).
[Crossref]

L. A. M. Johnson, P. Gill, and H. S. Margolis, “Evaluating the performance of the npl femtosecond frequency combs: Agreement at the 10−21 level,” Metrologia 52(1), 62–71 (2015).
[Crossref]

A. Al-Masoudi, S. Dörscher, S. Häfner, U. Sterr, and C. Lisdat, “Noise and instability of an optical lattice clock,” Phys. Rev. A 92(6), 063814 (2015).
[Crossref]

B. T. R. Christensen, M. R. Henriksen, S. A. Schäffer, P. G. Westergaard, D. Tieri, J. Ye, M. J. Holland, and J. W. Thomsen, “Nonlinear spectroscopy of Sr atoms in an optical cavity for laser stabilization,” Phys. Rev. A 92(5), 053820 (2015).
[Crossref]

S. Cook, T. Rosenband, and D. R. Leibrandt, “Laser frequency stabilization based on steady-state spectral-hole burning in Eu3+:Y2SiO5,” Phys. Rev. Lett. 114(25), 253902 (2015).
[Crossref]

2014 (1)

D. Nicolodi, B. Argence, W. Zhang, R. Le Targat, G. Santarelli, and Y. Le Coq, “Spectral purity transfer between optical wavelengths at the 10−18 level,” Nat. Photonics 8(3), 219–223 (2014).
[Crossref]

2013 (2)

C. Hagemann, C. Grebing, T. Kessler, S. Falke, N. Lemke, C. Lisdat, H. Schnatz, F. Riehle, and U. Sterr, “Providing 10−16 short-term stability of a 1.5µm laser to optical clocks,” IEEE Trans. Instrum. Meas. 62(6), 1556–1562 (2013).
[Crossref]

J.-D. Deschênes and J. Genest, “Heterodyne beats between a continuous-wave laser and a frequency comb beyond the shot-noise limit of a single comb mode,” Phys. Rev. A 87(2), 023802 (2013).
[Crossref]

2012 (1)

A. Yamaguchi, N. Shiga, S. Nagano, Y. Li, H. Ishijima, H. Hachisu, M. Kumagai, and T. Ido, “Stability transfer between two clock lasers operating at different wavelengths for absolute frequency measurement of clock transition in 87Sr,” Appl. Phys. Express 5(2), 022701 (2012).
[Crossref]

2011 (4)

G. Krauss, D. Fehrenbacher, D. Brida, C. Riek, A. Sell, R. Huber, and A. Leitenstorfer, “All-passive phase locking of a compact Er:fiber laser system,” Opt. Lett. 36(4), 540–542 (2011).
[Crossref]

A. Ruehl, M. J. Martin, K. C. Cossel, L. Chen, H. McKay, B. Thomas, C. Benko, L. Dong, J. M. Dudley, M. E. Fermann, I. Hartl, and J. Ye, “Ultra-broadband coherent supercontinuum frequency comb,” Phys. Rev. A 84(1), 011806 (2011).
[Crossref]

J. Taylor, S. Datta, A. Hati, C. Nelson, F. Quinlan, A. Joshi, and S. Diddam, “Characterization of power-to-phase conversion in high-speed P-I-N photodiodes,” IEEE Photonics J. 3(1), 140–151 (2011).
[Crossref]

K. Czuba and D. Sikora, “Temperature stability of coaxial cables,” Acta Phys. Pol., A 119(4), 553–557 (2011).
[Crossref]

2009 (1)

B. Kühn and R. Schadrack, “Thermal expansion of synthetic fused silica as a function of OH content and fictive temperature,” J. Non-Cryst. Solids 355(4-5), 323–326 (2009).
[Crossref]

2008 (2)

P. A. Williams, W. C. Swann, and N. R. Newbury, “High-stability transfer of an optical frequency over long fiber-optic links,” J. Opt. Soc. Am. B 25(8), 1284–1293 (2008).
[Crossref]

G. Grosche, B. Lipphardt, and H. Schnatz, “Optical frequency synthesis and measurement using fibre-based femtosecond lasers,” Eur. Phys. J. D 48(1), 27–33 (2008).
[Crossref]

2006 (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[Crossref]

2005 (2)

E. Rubiola, “On the measurement of frequency and of its sample variance with high-resolution counters,” Rev. Sci. Instrum. 76(5), 054703 (2005).
[Crossref]

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2004 (1)

2003 (1)

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90(11), 113904 (2003).
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2002 (1)

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1994 (2)

1981 (1)

1979 (1)

1966 (1)

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1965 (1)

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Safronova, M. S.

C. Sanner, N. Huntemann, R. Lange, C. Tamm, E. Peik, M. S. Safronova, and S. G. Porsev, “Optical clock comparison test of Lorentz symmetry,” Nature 567(7747), 204–208 (2019).
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Sanner, C.

C. Sanner, N. Huntemann, R. Lange, C. Tamm, E. Peik, M. S. Safronova, and S. G. Porsev, “Optical clock comparison test of Lorentz symmetry,” Nature 567(7747), 204–208 (2019).
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E. Oelker, R. B. Hutson, C. J. Kennedy, L. Sonderhouse, T. Bothwell, A. Goban, D. Kedar, C. Sanner, J. M. Robinson, G. E. Marti, D. G. Matei, T. Legero, M. Giunta, R. Holzwarth, F. Riehle, U. Sterr, and J. Ye, “Demonstration of 4.8 × 10−17 stability at 1 s for two independent optical clocks,” Nat. Photonics 13(10), 714–719 (2019).
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Santarelli, G.

D. Nicolodi, B. Argence, W. Zhang, R. Le Targat, G. Santarelli, and Y. Le Coq, “Spectral purity transfer between optical wavelengths at the 10−18 level,” Nat. Photonics 8(3), 219–223 (2014).
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B. Kühn and R. Schadrack, “Thermal expansion of synthetic fused silica as a function of OH content and fictive temperature,” J. Non-Cryst. Solids 355(4-5), 323–326 (2009).
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B. T. R. Christensen, M. R. Henriksen, S. A. Schäffer, P. G. Westergaard, D. Tieri, J. Ye, M. J. Holland, and J. W. Thomsen, “Nonlinear spectroscopy of Sr atoms in an optical cavity for laser stabilization,” Phys. Rev. A 92(5), 053820 (2015).
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R. Paschotta, B. Rudin, A. Schlatter, G. J. Spöhler, L. Krainer, S. C. Zeller, N. Haverkamp, H. R. Telle, and U. Keller, “Relative timing jitter measurements with an indirect phase comparison method,” Appl. Phys. B: Lasers Opt. 80(2), 185–192 (2005).
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C. Hagemann, C. Grebing, T. Kessler, S. Falke, N. Lemke, C. Lisdat, H. Schnatz, F. Riehle, and U. Sterr, “Providing 10−16 short-term stability of a 1.5µm laser to optical clocks,” IEEE Trans. Instrum. Meas. 62(6), 1556–1562 (2013).
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G. Grosche, B. Lipphardt, and H. Schnatz, “Optical frequency synthesis and measurement using fibre-based femtosecond lasers,” Eur. Phys. J. D 48(1), 27–33 (2008).
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C. Tresp, T. Puppe, A. Seer, P. Thoumany, F. Rohde, and R. Wilk, “Characterization of the CEO phase noise of an erbium fiber frequency comb,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2019), p. STu4L.4.

Seletskiy, D. V.

A. Liehl, P. Sulzer, D. Fehrenbacher, T. Rybka, D. V. Seletskiy, and A. Leitenstorfer, “Deterministic nonlinear transformations of phase noise in quantum-limited frequency combs,” Phys. Rev. Lett. 122(20), 203902 (2019).
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A. Liehl, D. Fehrenbacher, P. Sulzer, A. Leitenstorfer, and D. V. Seletskiy, “Ultrabroadband out-of-loop characterization of the carrier-envelope phase noise of an offset-free Er:fiber frequency comb,” Opt. Lett. 42(10), 2050–2053 (2017).
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Sherman, J. A.

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A. Liehl, P. Sulzer, D. Fehrenbacher, T. Rybka, D. V. Seletskiy, and A. Leitenstorfer, “Deterministic nonlinear transformations of phase noise in quantum-limited frequency combs,” Phys. Rev. Lett. 122(20), 203902 (2019).
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H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, “Carrier-envelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation,” Appl. Phys. B: Lasers Opt. 69(4), 327–332 (1999).
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B. T. R. Christensen, M. R. Henriksen, S. A. Schäffer, P. G. Westergaard, D. Tieri, J. Ye, M. J. Holland, and J. W. Thomsen, “Nonlinear spectroscopy of Sr atoms in an optical cavity for laser stabilization,” Phys. Rev. A 92(5), 053820 (2015).
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C. Tresp, T. Puppe, A. Seer, P. Thoumany, F. Rohde, and R. Wilk, “Characterization of the CEO phase noise of an erbium fiber frequency comb,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2019), p. STu4L.4.

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J. Taylor, S. Datta, A. Hati, C. Nelson, F. Quinlan, A. Joshi, and S. Diddam, “Characterization of power-to-phase conversion in high-speed P-I-N photodiodes,” IEEE Photonics J. 3(1), 140–151 (2011).
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D. Nicolodi, B. Argence, W. Zhang, R. Le Targat, G. Santarelli, and Y. Le Coq, “Spectral purity transfer between optical wavelengths at the 10−18 level,” Nat. Photonics 8(3), 219–223 (2014).
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C. Sanner, N. Huntemann, R. Lange, C. Tamm, E. Peik, M. S. Safronova, and S. G. Porsev, “Optical clock comparison test of Lorentz symmetry,” Nature 567(7747), 204–208 (2019).
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Opt. Express (3)

Opt. Lett. (6)

Optica (2)

Phys. Rev. A (5)

J. M. Hogan and M. A. Kasevich, “Atom-interferometric gravitational-wave detection using heterodyne laser links,” Phys. Rev. A 94(3), 033632 (2016).
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B. T. R. Christensen, M. R. Henriksen, S. A. Schäffer, P. G. Westergaard, D. Tieri, J. Ye, M. J. Holland, and J. W. Thomsen, “Nonlinear spectroscopy of Sr atoms in an optical cavity for laser stabilization,” Phys. Rev. A 92(5), 053820 (2015).
[Crossref]

A. Al-Masoudi, S. Dörscher, S. Häfner, U. Sterr, and C. Lisdat, “Noise and instability of an optical lattice clock,” Phys. Rev. A 92(6), 063814 (2015).
[Crossref]

J.-D. Deschênes and J. Genest, “Heterodyne beats between a continuous-wave laser and a frequency comb beyond the shot-noise limit of a single comb mode,” Phys. Rev. A 87(2), 023802 (2013).
[Crossref]

A. Ruehl, M. J. Martin, K. C. Cossel, L. Chen, H. McKay, B. Thomas, C. Benko, L. Dong, J. M. Dudley, M. E. Fermann, I. Hartl, and J. Ye, “Ultra-broadband coherent supercontinuum frequency comb,” Phys. Rev. A 84(1), 011806 (2011).
[Crossref]

Phys. Rev. D (2)

S. Kolkowitz, I. Pikovski, N. Langellier, M. D. Lukin, R. L. Walsworth, and J. Ye, “Gravitational wave detection with optical lattice atomic clocks,” Phys. Rev. D 94(12), 124043 (2016).
[Crossref]

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[Crossref]

Phys. Rev. Lett. (4)

D. G. Matei, T. Legero, S. Häfner, C. Grebing, R. Weyrich, W. Zhang, L. Sonderhouse, J. M. Robinson, J. Ye, F. Riehle, and U. Sterr, “1.5µm lasers with sub-10 mHz linewidth,” Phys. Rev. Lett. 118(26), 263202 (2017).
[Crossref]

S. Cook, T. Rosenband, and D. R. Leibrandt, “Laser frequency stabilization based on steady-state spectral-hole burning in Eu3+:Y2SiO5,” Phys. Rev. Lett. 114(25), 253902 (2015).
[Crossref]

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[Crossref]

A. Liehl, P. Sulzer, D. Fehrenbacher, T. Rybka, D. V. Seletskiy, and A. Leitenstorfer, “Deterministic nonlinear transformations of phase noise in quantum-limited frequency combs,” Phys. Rev. Lett. 122(20), 203902 (2019).
[Crossref]

Phys. Rev. X (1)

M. A. Norcia, J. R. K. Cline, J. A. Muniz, J. M. Robinson, R. B. Hutson, A. Goban, G. E. Marti, J. Ye, and J. K. Thompson, “Frequency measurements of superradiance from the strontium clock transition,” Phys. Rev. X 8(2), 021036 (2018).
[Crossref]

Rev. Mod. Phys. (1)

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[Crossref]

Rev. Sci. Instrum. (2)

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[Crossref]

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P. Giaccari, H. G. Limberger, and P. Kronenberg, “Influence of humidity and temperature on polyimide-coated fiber Bragg gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, (Optical Society of America, 2001), p. BFB2.

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C. Tresp, T. Puppe, A. Seer, P. Thoumany, F. Rohde, and R. Wilk, “Characterization of the CEO phase noise of an erbium fiber frequency comb,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2019), p. STu4L.4.

F. Vernotte, C. E. Calosso, and E. Rubiola, “Three-cornered hat versus Allan covariance,” in 2016 IEEE International Frequency Control Symposium (IFCS), (2016), pp. 1–6.

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Figures (5)

Fig. 1.
Fig. 1. Schematic experimental setup for the evaluation of the frequency transfer instability: CP configuration, free of uncompensated paths in both the DUT system (upper half) and reference system (lower half). The DUT comb generation system (blue box) generates a CEO-free optical frequency comb, while the reference comb generation system (yellow box) provides a CEO frequency measurement based on a common-path $f-2f$ interferometer (green box). NLF: Highly nonlinear fiber, EDFA: Erbium-doped fiber amplifier. See text for further details of the comb generation systems. The source cw field (brown lines) is emitted by a laser stabilized to an ultra-stable Si cavity at 1542 nm, and the target cw field (purple lines) is emitted by a Sr lattice clock laser pre-stabilized to a 48 cm long cavity at 698 nm [37]. By active path length stabilization, the phases at the planes defined by the near-end reference mirrors $\mathrm {RM_{source}}$ and $\mathrm {RM_{target}}$ have a fixed relation to the phases at the far-end planes $\mathrm {CRM_{ref/DUT}}$ and $\mathrm {RM_{user}}$, respectively. The user target plane $\mathrm {RM_{user}}$ could e.g. be located close to the atoms of a Sr lattice clock to transfer the stability of the source laser to the location of the clock interrogation. RF signal paths are indicated by dotted lines. The beat signals between cw fields and combs are detected by photodiodes at RF frequencies $f_\mathrm {source}$, $f_\mathrm {target}$ and $\nu _\mathrm {CEO}$ (ref system only), pre-filtered using PLL tracking filters, and counted with a dead-time-free frequency counter in $\Lambda$-mode [38], such that the data can be evaluated in a post-processing step.
Fig. 2.
Fig. 2. Schematic experimental setup for the evaluation of the frequency transfer instability: NCP configuration, containing uncompensated path segments in the DUT system, as indicated by dashed lines. Compensated optical paths are shown as solid lines. In contrast to the CP case, the superposition of the source field with the DUT comb is implemented completely as fiber optics, and using a separate branch. As in the CP case, the reference system does not comprise uncompensated paths.
Fig. 3.
Fig. 3. Instability plot. Blue: Measured combined instability in the NCP configuration. Orange: Measured combined instability in the CP configuration. The shaded regions indicate $1\sigma$ confidence intervals estimated by the method of Greenhall and Riley [41]. As a measure for the instability, the modified Allan deviation was chosen. For comparison, the green dashed line shows the modified Allan deviation $\mathrm {mod}\,\sigma _y(\tau )= 4.8\times 10^{-17}/\sqrt {2\tau /\mathrm {s}}$ of the most stable optical clocks to date [5] and the red dashed line shows a modified Allan deviation of $\mathrm {mod}\,\sigma _y(\tau )=\sqrt {2}\times 10^{-18}/\sqrt {\tau /\mathrm {s}}$. This corresponds to white frequency noise with an Allan deviation of $\sigma _y(\tau )=2\times 10^{-18}/\sqrt {\tau /\mathrm {s}}$, which is a factor of 24 better than the best optical clocks. For measurement times $\tau\;> \;100$ s, the statistical uncertainty of a frequency comparison between clocks with this performance would not be limited by the transfer via the comb. The $y_\mathrm {DUT-ref}$ time series data can be downloaded from PTB’s open data repository [42] for further analysis, such as estimation of the power spectral density or Allan deviation.
Fig. 4.
Fig. 4. Temporal phase evolution $\varphi _\mathrm {DUT-ref}(t)$ during time intervals with continuous data. The CP (orange, right scale) and NCP (blue, left scale) measurements were not performed simultaneously.
Fig. 5.
Fig. 5. Estimated contributions from various noise processes. Open square symbols: Contributions from free space air paths; Filled circle symbols: Fiber paths. Blue: Humidity, green: Pressure, orange: Temperature. (a) NCP case, i.e. including non-compensated paths. (b) CP case, i.e. end-to-end path length compensated. See text for details.

Tables (3)

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Table 1. Fiber parameters used for the calculation of path length fluctuations.

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Table 2. Geometrical path lengths used for the estimation of the various instability contributions. They are chosen as in the experiment and their locations are indicated in Figs. 1 and 2.

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Table 3. Order-of-magnitude estimates for the various fractional frequency ratio offsets y o f f s due to linear drifts between DUT and reference system during a measurement time of 6 days. As in the case of the instabilities, the offset resulting from humidity in fibers is probably over-estimated using the humidity coefficient in Table 1. For this reason, this coefficient has been reduced by a correction factor of 10.

Equations (9)

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φ m ( t ) = φ C E O ( t ) + m φ r e p ( t ) .
y D U T r e f ( t ) = ν t a r g e t ν s o u r c e | D U T ( t ) ν t a r g e t ν s o u r c e | r e f ( t ) ν t a r g e t ν s o u r c e | n o m ,
y = Δ ν ν 0 = 1 c ( L d n d t + n d L d t ) ,
y ζ = 1 c ( L n ζ + n L ζ ) d ζ d t .
y ζ ( λ 1 ) y ζ ( λ 2 ) = 1 c [ n ( λ 2 ) n ( λ 1 ) ] L ζ d ζ d t + L c [ n ( λ 2 ) ζ n ( λ 1 ) ζ ] d ζ d t ,
n ( λ 2 ) ζ = n ( λ 2 ) 1 n ( λ 1 ) 1 n ( λ 1 ) ζ ,
y ζ ( λ 1 ) y ζ ( λ 2 ) = 1 c [ n ( λ 2 ) n ( λ 1 ) ] L ζ d ζ d t + L c n ( λ 2 ) n ( λ 1 ) n ( λ 1 ) 1 n ζ d ζ d t .
m o d σ y ( τ ) = 3 / 2 S φ 2 π ν 0 τ 3 / 2 10 19 ( τ / s ) 3 / 2 .
k = 2 f h f s

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