Abstract

We propose and experimentally demonstrate a stable radio frequency (RF) phase dissemination scheme for a long-haul optical fiber loop link based on frequency mixing. Using a single optical source in both directions of the loop link, additional timing jitter caused by group velocity dispersion (GVD) can be eliminated. Impressive scalability provided by the optical link ensures that arbitrary-access node can obtain an RF signal with a stabilized phase to meet the requirements of multiple users. In our experiment, a 2.4 GHz RF signal is distributed to arbitrary points along a 100 km fiber-optic loop link steadily. Stabilities of the recovered signals from two accessing nodes are recorded. The root-mean-square (RMS) phase jitter of the received signal at either accessing node is reduced from 1.87 rad to no more than 0.027 rad during 1800-second measuring time.

© 2016 Optical Society of America

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References

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  1. S. Huang and R. L. Tjoelker, “Stabilized photonic links for deep space tracking, navigation, and radio science applications,” 43rd Annual Precise Time, and Time Interval (PTTI) Systems, and Applications Meeting. 1–8 (2012).
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    [Crossref]
  4. K. W. Holman, D. D. Hudson, and J. Ye, “Remote transfer of a high-stability and ultralow-jitter timing signal,” Opt. Lett. 30(10), 1225–1227 (2005).
    [Crossref]
  5. L. Zhang, L. Chang, Y. Dong, W. Xie, H. He, and W. Hu, “Phase drift cancellation of remote radio frequency transfer using an optoelectronic delay-locked loop,” Opt. Lett. 36(6), 873–875 (2011).
    [Crossref]
  6. G. Marra, R. Slavík, H. S. Margolis, S. N. Lea, P. Petropoulos, D. J. Richardson, and P. Gill, “High-resolution microwave frequency transfer over an 86-km-long optical fiber network using a mode-locked laser,” Opt. Lett. 36(7), 511–513 (2011).
    [Crossref]
  7. A. Zhang, Y. Dai, F. Yin, T. Ren, K. Xu, J. Li, Y. Ji, J. Lin, and G. Tang, “Stable radio-frequency delivery by λ dispersion-induced optical tunable delay,” Opt. Lett. 38(14), 2419–2421 (2013).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  13. P. Krehlik, Ł. Śliwczyński, Ł. Buczek, and M. lipiński, “Multipoint Dissemination of RF Frequency in Fiber Optic Link With Stabilized Propagation Delay,” IEEE Trans. Ultrason. Ferroeletr. Freq. Control 60(9), 1804–1810 (2013).
    [Crossref]
  14. G. Grosche, O. Terra, K. Predehl, R. Holzwarth, B. Lipphardt, F. Vogt, U. Sterr, and H. Schnatz, “Optical frequency transfer via 146 km fiber link with 10−19 relative accuracy,” Opt. Lett. 34(15), 2270–2272 (2009).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  19. Z. Wu, Y. Dai, F. Yin, K. Xu, J. Li, and J. Lin, “Stable radio frequency phase delivery by rapid and endless post error cancellation,” Opt. Lett. 38(7), 1098–1100 (2013).
    [Crossref]
  20. W. Li, W. Wang, W. Sun, W. Wang, and N. Zhu, “Stable radio-frequency phase distribution over optical fiber by phase-drift auto-cancellation,” Opt. Lett. 39(15), 4294–4296 (2014).
    [Crossref]
  21. L. Yu, R. Wang, L. Lu, Y. Zhu, J. Zheng, C. Wu, B. Zhang, and P. Wang, “WDM-based radio frequency dissemination in a tree-topology fiber optic network,” Opt. Express 23(15), 19783–19792 (2015).
    [Crossref]
  22. Y. Bai, B. Wang, C. Gao, J. Miao, X. Zhu, and L. Wang, “Fiber-based radio frequency dissemination for branching networks with passive phase-noise cancelation,” Chin. Opt. Lett. 13(6), 061201 (2015).
    [Crossref]
  23. C. Gao, B. Wang, W. Chen, Y. Bai, J. Miao, X. Zhu, T. Li, and L. Wang, “Fiber-based multiple-access ultrastable frequency dissemination,” Opt. Lett. 37(22), 4690–4692 (2012).
    [Crossref]

2015 (2)

2014 (6)

2013 (3)

2012 (1)

2011 (4)

2010 (1)

2009 (2)

2008 (1)

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D. 48, 35–41 (2008).
[Crossref]

2005 (1)

Akiyama, T.

T. Akiyama, H. Matsuzawa, E. Haraguchi, T. Ando, and Y. Hirano, “Phase stabilized RF reference signal dissemination over optical fiber employing instantaneous frequency control by vco,” IEEE MTT-S International Microwave Symposium Digest. 217–219 (2012).

Amy-Klein, A.

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D. 48, 35–41 (2008).
[Crossref]

Ando, T.

T. Akiyama, H. Matsuzawa, E. Haraguchi, T. Ando, and Y. Hirano, “Phase stabilized RF reference signal dissemination over optical fiber employing instantaneous frequency control by vco,” IEEE MTT-S International Microwave Symposium Digest. 217–219 (2012).

Bai, Y.

Ben, D.

Bertacco, E. K.

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Buczek, L.

P. Krehlik, Ł. Śliwczyński, Ł. Buczek, and M. lipiński, “Multipoint Dissemination of RF Frequency in Fiber Optic Link With Stabilized Propagation Delay,” IEEE Trans. Ultrason. Ferroeletr. Freq. Control 60(9), 1804–1810 (2013).
[Crossref]

Ł. Śliwczyński, P. Krehlik, Ł. Buczek, and M. Lipiński, “Active propagation delay stabilization for fiber-optic frequency distribution using controlled electronic delay lines,” IEEE Trans. Instrum. Meas. 60(4), 1480–1488 (2011).
[Crossref]

Calonico, D.

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Calosso, C. E.

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Chang, L.

Chardonnet, C.

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D. 48, 35–41 (2008).
[Crossref]

Chen, W.

Clivati, C.

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Costanzo, G. A.

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Dai, Y.

Daussy, C.

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D. 48, 35–41 (2008).
[Crossref]

Dong, Y.

Frittelli, M.

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Gao, C.

Gill, P.

Godone, A.

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Grosche, G.

Haraguchi, E.

T. Akiyama, H. Matsuzawa, E. Haraguchi, T. Ando, and Y. Hirano, “Phase stabilized RF reference signal dissemination over optical fiber employing instantaneous frequency control by vco,” IEEE MTT-S International Microwave Symposium Digest. 217–219 (2012).

He, H.

Hirano, Y.

T. Akiyama, H. Matsuzawa, E. Haraguchi, T. Ando, and Y. Hirano, “Phase stabilized RF reference signal dissemination over optical fiber employing instantaneous frequency control by vco,” IEEE MTT-S International Microwave Symposium Digest. 217–219 (2012).

Holman, K. W.

Holzwarth, R.

Hou, D.

Hu, W.

Huang, S.

S. Huang and R. L. Tjoelker, “Stabilized photonic links for deep space tracking, navigation, and radio science applications,” 43rd Annual Precise Time, and Time Interval (PTTI) Systems, and Applications Meeting. 1–8 (2012).

Hudson, D. D.

Ji, Y.

Krehlik, P.

P. Krehlik, Ł. Śliwczyński, Ł. Buczek, and M. lipiński, “Multipoint Dissemination of RF Frequency in Fiber Optic Link With Stabilized Propagation Delay,” IEEE Trans. Ultrason. Ferroeletr. Freq. Control 60(9), 1804–1810 (2013).
[Crossref]

Ł. Śliwczyński, P. Krehlik, Ł. Buczek, and M. Lipiński, “Active propagation delay stabilization for fiber-optic frequency distribution using controlled electronic delay lines,” IEEE Trans. Instrum. Meas. 60(4), 1480–1488 (2011).
[Crossref]

Lea, S. N.

Levi, F.

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Li, J.

Li, P.

Li, T.

Li, W.

Lin, J.

lipinski, M.

P. Krehlik, Ł. Śliwczyński, Ł. Buczek, and M. lipiński, “Multipoint Dissemination of RF Frequency in Fiber Optic Link With Stabilized Propagation Delay,” IEEE Trans. Ultrason. Ferroeletr. Freq. Control 60(9), 1804–1810 (2013).
[Crossref]

Ł. Śliwczyński, P. Krehlik, Ł. Buczek, and M. Lipiński, “Active propagation delay stabilization for fiber-optic frequency distribution using controlled electronic delay lines,” IEEE Trans. Instrum. Meas. 60(4), 1480–1488 (2011).
[Crossref]

Lipphardt, B.

Liu, C.

Lopez, O.

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D. 48, 35–41 (2008).
[Crossref]

Lours, M.

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D. 48, 35–41 (2008).
[Crossref]

Lu, L.

Margolis, H. S.

Marra, G.

Matsuzawa, H.

T. Akiyama, H. Matsuzawa, E. Haraguchi, T. Ando, and Y. Hirano, “Phase stabilized RF reference signal dissemination over optical fiber employing instantaneous frequency control by vco,” IEEE MTT-S International Microwave Symposium Digest. 217–219 (2012).

Miao, J.

Mura, A.

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Narbonneau, F.

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D. 48, 35–41 (2008).
[Crossref]

Pan, S.

Petropoulos, P.

Poli, N.

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Predehl, K.

Ren, T.

Richardson, D. J.

Santarelli, G.

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D. 48, 35–41 (2008).
[Crossref]

Schnatz, H.

Slavík, R.

Sliwczynski, L.

P. Krehlik, Ł. Śliwczyński, Ł. Buczek, and M. lipiński, “Multipoint Dissemination of RF Frequency in Fiber Optic Link With Stabilized Propagation Delay,” IEEE Trans. Ultrason. Ferroeletr. Freq. Control 60(9), 1804–1810 (2013).
[Crossref]

Ł. Śliwczyński, P. Krehlik, Ł. Buczek, and M. Lipiński, “Active propagation delay stabilization for fiber-optic frequency distribution using controlled electronic delay lines,” IEEE Trans. Instrum. Meas. 60(4), 1480–1488 (2011).
[Crossref]

Sterr, U.

Sun, W.

Sutyrin, D. V.

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Tang, G.

Terra, O.

Tino, G.

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Tjoelker, R. L.

S. Huang and R. L. Tjoelker, “Stabilized photonic links for deep space tracking, navigation, and radio science applications,” 43rd Annual Precise Time, and Time Interval (PTTI) Systems, and Applications Meeting. 1–8 (2012).

Vogt, F.

Wang, B.

Wang, L.

Wang, P.

Wang, R.

Wang, W.

Wei, J.

Wu, C.

Wu, Z.

Xie, W.

Xu, K.

Ye, J.

Yin, F.

Yu, L.

Zhang, A.

Zhang, B.

Zhang, F.

Zhang, L.

Zhang, Z.

Zhao, J.

Zheng, J.

Zhou, Y.

Zhu, N.

Zhu, X.

Zhu, Y.

Zucco, M. E.

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Appl. Phys. B (1)

D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, ”High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014).
[Crossref]

Chin. Opt. Lett. (1)

Eur. Phys. J. D. (1)

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D. 48, 35–41 (2008).
[Crossref]

IEEE Trans. Aerosp. Electron. Syst. (1)

W. Wang, “GPS-Based Time & Phase Synchronization Processing for Distributed SAR,” IEEE Trans. Aerosp. Electron. Syst. 45(3), 1040–1051 (2009).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

Ł. Śliwczyński, P. Krehlik, Ł. Buczek, and M. Lipiński, “Active propagation delay stabilization for fiber-optic frequency distribution using controlled electronic delay lines,” IEEE Trans. Instrum. Meas. 60(4), 1480–1488 (2011).
[Crossref]

IEEE Trans. Ultrason. Ferroeletr. Freq. Control (1)

P. Krehlik, Ł. Śliwczyński, Ł. Buczek, and M. lipiński, “Multipoint Dissemination of RF Frequency in Fiber Optic Link With Stabilized Propagation Delay,” IEEE Trans. Ultrason. Ferroeletr. Freq. Control 60(9), 1804–1810 (2013).
[Crossref]

Opt. Express (4)

Opt. Lett. (11)

L. Yu, R. Wang, L. Lu, Y. Zhu, C. Wu, B. Zhang, and P. Wang, “Stable radio frequency dissemination by simple hybrid frequency modulation scheme,” Opt. Lett. 39(18), 5255–5258 (2014).
[Crossref]

J. Wei, F. Zhang, Y. Zhou, D. Ben, and S. Pan, “Stable fiber delivery of radio-frequency signal based on passive phase correction,” Opt. Lett. 39(11), 3360–3362 (2014).
[Crossref]

W. Li, W. Wang, W. Sun, W. Wang, and N. Zhu, “Stable radio-frequency phase distribution over optical fiber by phase-drift auto-cancellation,” Opt. Lett. 39(15), 4294–4296 (2014).
[Crossref]

G. Marra, R. Slavík, H. S. Margolis, S. N. Lea, P. Petropoulos, D. J. Richardson, and P. Gill, “High-resolution microwave frequency transfer over an 86-km-long optical fiber network using a mode-locked laser,” Opt. Lett. 36(7), 511–513 (2011).
[Crossref]

L. Zhang, L. Chang, Y. Dong, W. Xie, H. He, and W. Hu, “Phase drift cancellation of remote radio frequency transfer using an optoelectronic delay-locked loop,” Opt. Lett. 36(6), 873–875 (2011).
[Crossref]

C. Gao, B. Wang, W. Chen, Y. Bai, J. Miao, X. Zhu, T. Li, and L. Wang, “Fiber-based multiple-access ultrastable frequency dissemination,” Opt. Lett. 37(22), 4690–4692 (2012).
[Crossref]

Z. Wu, Y. Dai, F. Yin, K. Xu, J. Li, and J. Lin, “Stable radio frequency phase delivery by rapid and endless post error cancellation,” Opt. Lett. 38(7), 1098–1100 (2013).
[Crossref]

A. Zhang, Y. Dai, F. Yin, T. Ren, K. Xu, J. Li, Y. Ji, J. Lin, and G. Tang, “Stable radio-frequency delivery by λ dispersion-induced optical tunable delay,” Opt. Lett. 38(14), 2419–2421 (2013).
[Crossref]

K. W. Holman, D. D. Hudson, and J. Ye, “Remote transfer of a high-stability and ultralow-jitter timing signal,” Opt. Lett. 30(10), 1225–1227 (2005).
[Crossref]

G. Grosche, O. Terra, K. Predehl, R. Holzwarth, B. Lipphardt, F. Vogt, U. Sterr, and H. Schnatz, “Optical frequency transfer via 146 km fiber link with 10−19 relative accuracy,” Opt. Lett. 34(15), 2270–2272 (2009).
[Crossref]

G. Marra, H. S. Margolis, S. N. Lea, and P. Gill, “High-stability microwave frequency transfer by propagation of an optical frequency comb over 50 km of optical fiber,” Opt. Lett. 35(7), 1025–1027 (2010).
[Crossref]

Other (2)

S. Huang and R. L. Tjoelker, “Stabilized photonic links for deep space tracking, navigation, and radio science applications,” 43rd Annual Precise Time, and Time Interval (PTTI) Systems, and Applications Meeting. 1–8 (2012).

T. Akiyama, H. Matsuzawa, E. Haraguchi, T. Ando, and Y. Hirano, “Phase stabilized RF reference signal dissemination over optical fiber employing instantaneous frequency control by vco,” IEEE MTT-S International Microwave Symposium Digest. 217–219 (2012).

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Figures (5)

Fig. 1
Fig. 1 Diagram of the circle topology including basic optical loop link(Black line) and the expanding link(red dashed line).
Fig. 2
Fig. 2 Schematic diagram of the arbitrary-access stable phase dissemination scheme. LD: laser diode. MZM: Mach-Zehnder modulator. RF: radio frequency. PD: photo-detector. EC: electrical coupler. OC: optical coupler. EBPF: electrical bandpass filter. SMF: single-mode fiber. FM: frequency multiplier. FD: frequency divider.
Fig. 3
Fig. 3 Experimental setup of the proposed phase fluctuation cancellation scheme for arbitrary-access loop link. The node between a 90/50km and a 10/50km standard single-mode fiber spools. OSC: oscilloscope. EDFA: erbium-doped fiber amplifier. EA: electrical amplifier.
Fig. 4
Fig. 4 Measured phase differences between the recovered signal and the standard RF signal at the connection of a 90-km and a 10-km standard single-mode fiber spools (a), connection of a 50-km and a 50-km standard single-mode fiber spools (b) (red, green: compensated, blue: uncompensated;).
Fig. 5
Fig. 5 Measured waveforms of the recovered 2.4GHz RF signal at the connection of a 90-km and a 10-km standard single-mode fiber spools without (A)–(C), and with (D)–(F) phase-drift cancellation.

Equations (6)

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V s = cos ( ω s t + φ s ) ,
V 1 = cos ( ω s t + φ s + φ p ) ,
V 2 = cos ( 2 ω s t + 2 φ s φ p ) .
V 3 = cos ( 2 ω s t + 2 φ s φ p + 2 φ a ) ,
V 4 = cos ( 2 ω s t + 2 φ s φ p + 2 φ b ) ,
V 5 = cos ( ω s t + φ s ) .

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