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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 27 — Dec. 17, 2012
  • pp: 28447–28454
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Phase fluctuation compensation for long-term transfer of stable radio frequency over fiber link

Bo Ning, Peng Du, Dong Hou, and Jianye Zhao  »View Author Affiliations


Optics Express, Vol. 20, Issue 27, pp. 28447-28454 (2012)
http://dx.doi.org/10.1364/OE.20.028447


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Abstract

We developed a new radio frequency dissemination system based on an optical fiber link. A 1.55 μm mode-locked fiber laser was used as optical transmitter in the system. To actively reduce the phase fluctuation induced by the fiber length variations with high resolution, we proposed a novel compensation technique. In our technique, we directly control the phase of optical pulses generated by the laser to compensate the fluctuation. The phase-controlling method is based on both pump power modulation and cavity length adjusting. We performed the transfer in a 22-km outdoor fiber link, with a transfer stability of 3.7 × 10−14 at 1 s and 6.6 × 10−18 at 16000 s. The integrated timing jitter in 24 hours was reduced from 14 ps to 35 fs.

© 2012 OSA

1. Introduction

Though the aforementioned research proves that it is very effective to use optical group-delay actuators for compensation, the compensation ranges of the optical devices are limited. Especially in transfers with long fiber links, the phase drift, which may reach tens or hundreds of picoseconds, could be beyond the adjusting range of the devices. To solve this problem, Hou et al. proposed a new compensation system [16

16. D. Hou, P. Li, C. Liu, J. Zhao, and Z. Zhang, “Long-term stable frequency transfer over an urban fiber link using microwave phase stabilization,” Opt. Express 19(2), 506–511 (2011). [CrossRef] [PubMed]

]. Instead of stabilizing the fiber to some reference, they physically controlled the cavity length of the MLL and generated a phase shift to cancel the fluctuation. However, one vital disadvantage of the idea is that accuracy of the compensation is low, because the accuracy of the phase shift provided in this way is strongly limited by the resolutions of the intra-cavity piezo actuator. As a result, the compensated fluctuation is still in picosecond scale (6.3 ps) and cannot reach the requirements of many applications.

In this paper, we propose a new compensation scheme. Instead of physically controlling the cavity length of a MLL, we utilized pump power modulation as well as a phase shifter to achieve the phase shift with much higher resolution. Results show that the phase fluctuation is reduced to sub-100-femtosecond scale.

2. Technique description

In [16

16. D. Hou, P. Li, C. Liu, J. Zhao, and Z. Zhang, “Long-term stable frequency transfer over an urban fiber link using microwave phase stabilization,” Opt. Express 19(2), 506–511 (2011). [CrossRef] [PubMed]

], a commercial piezoelectric transducer (PZT) was used in an Er-doped fiber MLL, of which the repetition rate is 100 MHz. The extreme resolution of the PZT is 0.6 nm (PI, P840.2). Based on the theory of MLL [4

4. J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Kluwer Academic Publishers / Springer, 2004).

], the extreme minimal phase shift provided by controlling the PZT can be calculated as:
[cnL0cn(L0+0.6nm)]τ100MHz300fs,
(1)
where c is the light velocity, n ≈1.5 is the average refraction index of the cavity, and L0 ≈2 m is the original length of the cavity. τ ≈1 ms is operation time of the technique proposed in [16

16. D. Hou, P. Li, C. Liu, J. Zhao, and Z. Zhang, “Long-term stable frequency transfer over an urban fiber link using microwave phase stabilization,” Opt. Express 19(2), 506–511 (2011). [CrossRef] [PubMed]

], and it is very close to the response time of the physical cavity length adjusting. The result implies that the resolution phase shifting is larger than 100 fs and thus it is very difficult to achieve sub-100 fs integrated timing jitter in this way.

Here, a new compensation technique which provides much higher accuracy is proposed. We first introduce a novel method for phase-controlling the MLL, which is crucial in the technique. Then, the detailed setup for the compensation loop is given.

2.1 Pump power modulation for high-accuracy phase shift

In 2006, Hudson et al. proved that by using an intra-cavity electro-optic modulator (EOM) to change the refractive index inside the cavity, the repetition rate of a MLL can be controlled, and remotely located MLLs can be synchronized with 10-fs accuracy [17

17. D. D. Hudson, S. M. Foreman, S. T. Cundiff, and J. Ye, “Synchronization of mode-locked femtosecond lasers through a fiber link,” Opt. Lett. 31(13), 1951–1953 (2006). [CrossRef] [PubMed]

]. Here, instead of using intra-cavity EOM, we explore the possibility of using pump power modulation to control the repetition rate through refractive index adjusting. In our research, a 1.55-μm Er-doped MLL, in which a 25-cm Er-doped fiber (EDF), is used. The repetition rate frep = 144.5 MHz. We use pump power modulation to generate an accurate phase shift of the laser. The underlying theory is that if the pump light intensity is linearly adjusted, the susceptibility of the gain media in the EDF is affected by nonlinear effects and also changes linearly [18

18. A. Yariv, Quantum Electronics Third Edition (John Wiley & Sons, 1989).

]. This change will lead to a linear shift of the EDF’s refraction index. Besides, due to the nonlinear effect caused by the third-order susceptibility, refraction index of single-mode fiber in cavity also changed. When intra-cavity light intensity increases, very slight linear shift of the refraction index also increases [19

19. A. Maitland and M. H. Dumn, Laser Physics (North-Holland Publishing Company, 1969).

]. As a result, n of the cavity can be adjusted by modulating pump power in a linear way. The frequency/phase of pulses is therefore controlled, as frep 1/n in MLLs [4

4. J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Kluwer Academic Publishers / Springer, 2004).

, 20

20. N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, “Frequency stabilization of mode-locked Erbium fiber lasers using pump power control,” Appl. Phys. B 78(3-4), 321–324 (2004). [CrossRef]

].

To evaluate the feasibility of the theory and corresponding performance, we respectively measured the phase shifts provided by power modulation and cavity length adjusting in 1 ms. A voltage-controlled current supply (Thorlabs, ITC110), of which the input range is 0 V-5 V and the response time is less than 10 μs, was used to modulate the power of a 976-nm laser diode (LD) from ~50 mW to ~500 mW. For the cavity length adjusting, the same PZT in [16

16. D. Hou, P. Li, C. Liu, J. Zhao, and Z. Zhang, “Long-term stable frequency transfer over an urban fiber link using microwave phase stabilization,” Opt. Express 19(2), 506–511 (2011). [CrossRef] [PubMed]

] was used. A travel range of 30 μm is achieved with 0 V-5 V input voltages by using a 150-V PZT driver. The phase shifts were measured with a signal analyzer (Agilent 9010A) by detecting the 40th harmonic of frep through a fast photodiode, and the data is given in Fig. 1
Fig. 1 Phase shift generated by different methods in 1 ms. Blue filled square: physically control the cavity length; red filled circle: modulate pump power.
. We can see that for the cavity length controlling method, the generated phase shift goes up with the input voltage in an inconsistent way. This inaccuracy of the phase-controlling is due to the PZT’s resolution. Based on Eq. (1), the inaccuracy is in hundreds-of-femtoseconds scale. The resolution of phase-controlling is therefore strongly limited. However, by using power modulation, a very smooth phase-controlling is achieved. This result implies that accuracy of the phase-controlling is greatly enhanced. Moreover, the low sensitivity of the phase shifts to input voltage change makes the phase-controlling easy to operate. Considering the PZT’s resolution (0.3 nm) and the current supply’s minimal tuning voltage (15 μV), the corresponding extreme minimal phase shift provided can be calculated and is ~616 fs and ~2.5 fs. The greatly improved accuracy not only provides a high-resolution phase shift, but also ensures that a high harmonic of frep can be easily phase-synchronized with a microwave.

2.2. Phase-controlling MLL in long term

Based on analysis results, we built a phase-locked loop (PLL) to control the laser phase. Schematic of the phase-controlling method is shown in Fig. 2
Fig. 2 Experimental setup of the compensation technique. PD: photodiode; EDFA: erbium-doped fiber amplifier; BPF: band-pass filter; LPF: low-pass filter; RF amp: radio frequency amplifier; LF amp:low frequency amplifier; AGC: automatic gain control; OC: optical circulator; BF: back-reflector; PZT: piezo-electric transducer; CM: collimator; SMF: single-mode fiber; WP: wave plate; EDF: erbium-doped fiber; PBS: polarization beam splitter; WDM: wavelength division multiplexer.
. By tuning the pump power in the PLL, the 21st harmonic of frep is phase-locked to a 3.035-GHz microwave, of which phase is controlled with a resistance-capacitance (R-C) phase shifter. Therefore, by controlling the phase shifter, the laser phase can be shifted by an arbitrary value for an approximately fixed frep. The accuracy of the phase shift is enhanced by the high-frequency harmonic synchronization. Loop bandwidth of the PLL is set to approximately 1 kHz by using a low-pass filter. A fast photodiode and a microwave phase detector are used to detect the phase error between the harmonic and the microwave. Unwanted fluctuations and drifts in the phase error signal are removed by a proportional-integral-derivative regulator before it is used to modulate the pump power.

One thing should be concerned is that due to environmental influences [21

21. W. K. Tan, H. Y. Wong, A. E. Kelly, M. Sorel, J. H. Marsh, and A. C. Bryce, “Temperature behaviour of pulse repetition frequency in passively mode-locked InGaAsP/InP laser diode - Experimental results and simple model,” IEEE J. Sel. Top. Quantum Electron. 13(5), 1209–1214 (2007). [CrossRef]

], cavity length of the laser drifts, and when it happens, stabilizing frep with pump power modulation could be difficult. This is because that the adjusting range of frep provided in this way is limited by the maximum power of LD and the minimum power required for mode-locking. To solve the problem, we brought in cavity length adjusting. After frep is stabilized to the reference frequency by modulating pump power, the control signal for the power modulation is detected by an analysis system. If the signal changes, it means that the cavity length has drifted and the power has been tuned to stabilizing frep. The system then drives PZT so that the control signal returns to near its initial value. The drift is therefore compensated directly. The system filters out most alternating signals. Only smooth voltage signal with slow change is out for driving PZT. By limiting the control signal within a certain small range, the output power of the MLL remains approximately constant, while the phase is accurately controlled.

2.3 Compensation loop

At the local end, the generated microwave corresponds the returned pulses is amplified by a radio frequency amplifier with automatic gain control to maintain stable high power. The amplified microwave is then fed into two frequency mixers to detect the phase differences between it and two local signals: the initial reference signal and the phase-shifted signal. The detections are carried out at the 21st harmonic of frep (3.035 GHz) to attain high detection sensitivity. Assuming the fiber-induced phase fluctuation observed at the remote site is φp, phase of the returned signal, φreturn, is given as:
φreturn=φ0+φc+2φp,
(2)
where φ0 is the initial of the reference signal and φc is the shifted phase provided by the R-C phase shifter. By mixing the signal with the initial reference signal and the phase-shifted reference signal, two phase differences, φir and φpr can be obtained respectively as:
φir=φc+2φpφpr=2φp.
(3)
φir and φpr are monitored by an analysis system, and the system control the phase of the laser through the phase shifter to null φir−(1/2)φpr, with an operation speed of 1000 times/s. Therefore, the following equation is satisfied:
φc=φp.
(4)
Thus, the phase of the received signal at the remote site is φ0 and the fiber-induced phase fluctuation is compensated.

3. Results and analysis

To test the performance of our compensation scheme, stability of the phase is measured. The measurement is performed by phase comparing the 21st harmonic of frep at the receiving end directly with the reference microwave signal by using a frequency mixer. To achieve stable measurement, thermal environment of the measurement setup for the long-term drift is controlled carefully. Most of the components in the measurement (including frequency mixer and photodiode) and the coaxial cables are in stable-temperature condition (< 0.1 K peak-peak), with the help of temperature-stabilized mounts and insulating material. The cables are in small-lengths (< 3 cm) to reduce thermal length drifts. All the components are placed in vibration-isolated condition and packed with aluminum box to stabilize thermal condition. To reduce the long-term influence of air flow and protect the thermal equilibrium, the tests were operated in a closed room with no human activity in it. We logged the output voltage of the frequency mixer by using an analog-digital converter (measurement bandwidth 7 Hz) and converted them into phase changes [23

23. G. Marra, H. S. Margolis, and D. J. Richardson, “Dissemination of an optical frequency comb over fiber with 3 × 10-18 fractional accuracy,” Opt. Express 20(2), 1775–1782 (2012). [CrossRef] [PubMed]

]. The converting rate is 2 times per second. Figure 3(a)
Fig. 3 Phase fluctuation (a) and phase noise (b) of the 21st harmonic of frep at the remote end. a) blue: phase fluctuation not compensated; red: phase fluctuation suppressed; black: background fluctuation measured by applying the same radio-frequency signal . b) blue: phase noise not suppressed; red: phase noise suppressed; grey: measurement noise ñoor, measured when 22-km SMF is replaced by 0.5-m fiber.
shows the timing drifts measured in an out-of-loop condition. When the compensation technique is not used, the integrated root-mean-square (rms) timing drift is ~14 ps in 24 hours, and the fluctuation range exceeds 50 ps. When using the technique, the rms drift is reduced to 35 fs and the suppressed fluctuation is mostly within 170 fs, while the noise floor of our measurement produces fluctuations within 33 fs (peak-peak).

We also measured the single-sideband phase noise between 1 Hz and 100 kHz at the remote end with a spectrum analyzer. The measured data is plotted in Fig. 3(b). We can see that when the phase noise suppression is activated, the ðber-induced phase noise is reduced by up to 30 dB. The reduced phase noise is −90 dBc/Hz at 1 Hz offset, which is very close to the noise measured when the 22-km SMF is replaced by a 0.5-m ðber while the other parts of the suppression system are still operating. This result implies that the suppression performance is still strongly limited by the measurement noise floor. To obtain signals with lower phase noises, reference microwave signals with higher purities and EDFAs with lower noises could be utilized. By integrating the phase noise between 1 Hz and 100 kHz, we calculate the short-term timing jitter to be less than 28.5 fs.

The focus of the research is to distribute the radio-frequency signal frep. To distribute frep and the carrier-envelope offset frequency (fceo) [4

4. J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Kluwer Academic Publishers / Springer, 2004).

] together, it is possible to use intra-cavity prims [4

4. J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Kluwer Academic Publishers / Springer, 2004).

] to compensate the pump-modulation involved fceo change [4

4. J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Kluwer Academic Publishers / Springer, 2004).

] while maintain a stabilized frep. However, the transfer performance could be degraded as the system is very complex.

4. Conclusions

To test the performance of technique, we transferred a microwave over a 22-km outdoor single-mode fiber. Results show that the integrated timing jitter is reduced to 35 fs in 24 hours, and the frequency stability reaches 3.7 × 10−14 at 1 s and 6.6 × 10−18 at 16000 s. Compared with the results in [16

16. D. Hou, P. Li, C. Liu, J. Zhao, and Z. Zhang, “Long-term stable frequency transfer over an urban fiber link using microwave phase stabilization,” Opt. Express 19(2), 506–511 (2011). [CrossRef] [PubMed]

], the stability provided by us is greatly improved by more than 3 orders of magnitude and is high enough for most of the applications in which microwave transfer using mode-locked laser is needed. Compensating phase fluctuation with tens of picoseconds continuously in 24 hours was also achieved successfully. The results make it possible to use the proposed technique to transfer highly stable and accurate radio frequency from laboratories to other distantly located sites in long term. The technique is very easily implemented with very low cost since no optical group-delay actuators are used. The compensated system could potentially be used over much longer distances (>100 km) by adding bidirectional optical amplifiers with low noises.

Acknowledgment

The authors would like to thank Professor Zhigang Zhang and Professor Zhong Wang for helpful discussions and experimental assistants. This work was supported in part by the Nature Science Foundations of China (No. 11027404), 973 program (2010CB328201) and (2012CB315606).

References and Links

1.

J. Levine, “A review of time and frequency transfer methods,” Metrologia 45(6), S162–S174 (2008). [CrossRef]

2.

S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum. 78(2), 021101 (2007). [CrossRef] [PubMed]

3.

K. Minoshima and H. Matsumoto, “In-situ measurements of shapes and thickness of optical parts by femtosecond two-color interferometry,” Opt. Commun. 138(1-3), 6–10 (1997). [CrossRef]

4.

J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Kluwer Academic Publishers / Springer, 2004).

5.

J. F. Cliche and B. Shillue, “Precision timing control for radioastronomy: maintaining femtosecond synchronization in the Atacama Large Millimeter Array,” IEEE Contr. Syst. Mag. 26(1), 19–26 (2006). [CrossRef]

6.

Y. F. Chen, J. Jiang, and D. J. Jones, “Remote distribution of a mode-locked pulse train with sub 40-as jitter,” Opt. Express 14(25), 12134–12144 (2006). [CrossRef] [PubMed]

7.

E. Allaria, C. Callegari, D. Cocco, W. M. Fawley, M. Kiskinova, C. Masciovecchio, and F. Parmigiani, “The FERMI@Elettra free-electron-laser source for coherent x-ray physics: photon properties, beam transport system and applications,” New J. Phys. 12(7), 075002 (2010). [CrossRef]

8.

P. R. Bolton, “Noninvasive laser probing of ultrashort single electron bunches for accelerator and light source development,” Int. J. Mod. Phys. B 21(03n04), 527–539 (2007). [CrossRef]

9.

L. S. Ma, P. Jungner, J. Ye, and J. L. Hall, “Delivering the same optical frequency at two places: accurate cancellation of phase noise introduced by an optical fiber or other time-varying path,” Opt. Lett. 19(21), 1777–1779 (1994). [CrossRef] [PubMed]

10.

K. W. Holman, D. J. Jones, D. D. Hudson, and J. Ye, “Precise frequency transfer through a fiber network by use of 1.5-microm mode-locked sources,” Opt. Lett. 29(13), 1554–1556 (2004). [CrossRef] [PubMed]

11.

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

12.

J. Kim, J. Chen, Z. Zhang, F. N. C. Wong, F. X. Kärtner, F. Loehl, and H. Schlarb, “Long-term femtosecond timing link stabilization using a single-crystal balanced cross correlator,” Opt. Lett. 32(9), 1044–1046 (2007). [CrossRef] [PubMed]

13.

J. Kim, J. A. Cox, J. Chen, and F. X. Kartner, “Drift-free femtosecond timing synchronization of remote optical and microwave sources,” Nat. Photonics 2(12), 733–736 (2008). [CrossRef]

14.

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] [PubMed]

15.

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(4), 511–513 (2011). [CrossRef] [PubMed]

16.

D. Hou, P. Li, C. Liu, J. Zhao, and Z. Zhang, “Long-term stable frequency transfer over an urban fiber link using microwave phase stabilization,” Opt. Express 19(2), 506–511 (2011). [CrossRef] [PubMed]

17.

D. D. Hudson, S. M. Foreman, S. T. Cundiff, and J. Ye, “Synchronization of mode-locked femtosecond lasers through a fiber link,” Opt. Lett. 31(13), 1951–1953 (2006). [CrossRef] [PubMed]

18.

A. Yariv, Quantum Electronics Third Edition (John Wiley & Sons, 1989).

19.

A. Maitland and M. H. Dumn, Laser Physics (North-Holland Publishing Company, 1969).

20.

N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, “Frequency stabilization of mode-locked Erbium fiber lasers using pump power control,” Appl. Phys. B 78(3-4), 321–324 (2004). [CrossRef]

21.

W. K. Tan, H. Y. Wong, A. E. Kelly, M. Sorel, J. H. Marsh, and A. C. Bryce, “Temperature behaviour of pulse repetition frequency in passively mode-locked InGaAsP/InP laser diode - Experimental results and simple model,” IEEE J. Sel. Top. Quantum Electron. 13(5), 1209–1214 (2007). [CrossRef]

22.

E. N. Ivanov, S. A. Diddams, and L. Hollberg, “Analysis of noise mechanisms limiting the frequency stability of microwave signals generated with a femtosecond laser,” IEEE J. Sel. Top. Quantum Electron. 9(4), 1059–1065 (2003). [CrossRef]

23.

G. Marra, H. S. Margolis, and D. J. Richardson, “Dissemination of an optical frequency comb over fiber with 3 × 10-18 fractional accuracy,” Opt. Express 20(2), 1775–1782 (2012). [CrossRef] [PubMed]

OCIS Codes
(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems
(140.4050) Lasers and laser optics : Mode-locked lasers
(350.4010) Other areas of optics : Microwaves

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: September 26, 2012
Revised Manuscript: November 15, 2012
Manuscript Accepted: November 25, 2012
Published: December 7, 2012

Citation
Bo Ning, Peng Du, Dong Hou, and Jianye Zhao, "Phase fluctuation compensation for long-term transfer of stable radio frequency over fiber link," Opt. Express 20, 28447-28454 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-28447


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References

  1. J. Levine, “A review of time and frequency transfer methods,” Metrologia45(6), S162–S174 (2008). [CrossRef]
  2. S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum.78(2), 021101 (2007). [CrossRef] [PubMed]
  3. K. Minoshima and H. Matsumoto, “In-situ measurements of shapes and thickness of optical parts by femtosecond two-color interferometry,” Opt. Commun.138(1-3), 6–10 (1997). [CrossRef]
  4. J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Kluwer Academic Publishers / Springer, 2004).
  5. J. F. Cliche and B. Shillue, “Precision timing control for radioastronomy: maintaining femtosecond synchronization in the Atacama Large Millimeter Array,” IEEE Contr. Syst. Mag.26(1), 19–26 (2006). [CrossRef]
  6. Y. F. Chen, J. Jiang, and D. J. Jones, “Remote distribution of a mode-locked pulse train with sub 40-as jitter,” Opt. Express14(25), 12134–12144 (2006). [CrossRef] [PubMed]
  7. E. Allaria, C. Callegari, D. Cocco, W. M. Fawley, M. Kiskinova, C. Masciovecchio, and F. Parmigiani, “The FERMI@Elettra free-electron-laser source for coherent x-ray physics: photon properties, beam transport system and applications,” New J. Phys.12(7), 075002 (2010). [CrossRef]
  8. P. R. Bolton, “Noninvasive laser probing of ultrashort single electron bunches for accelerator and light source development,” Int. J. Mod. Phys. B21(03n04), 527–539 (2007). [CrossRef]
  9. L. S. Ma, P. Jungner, J. Ye, and J. L. Hall, “Delivering the same optical frequency at two places: accurate cancellation of phase noise introduced by an optical fiber or other time-varying path,” Opt. Lett.19(21), 1777–1779 (1994). [CrossRef] [PubMed]
  10. K. W. Holman, D. J. Jones, D. D. Hudson, and J. Ye, “Precise frequency transfer through a fiber network by use of 1.5-microm mode-locked sources,” Opt. Lett.29(13), 1554–1556 (2004). [CrossRef] [PubMed]
  11. K. W. Holman, D. D. Hudson, J. Ye, and D. J. Jones, “Remote transfer of a high-stability and ultralow-jitter timing signal,” Opt. Lett.30(10), 1225–1227 (2005). [CrossRef] [PubMed]
  12. J. Kim, J. Chen, Z. Zhang, F. N. C. Wong, F. X. Kärtner, F. Loehl, and H. Schlarb, “Long-term femtosecond timing link stabilization using a single-crystal balanced cross correlator,” Opt. Lett.32(9), 1044–1046 (2007). [CrossRef] [PubMed]
  13. J. Kim, J. A. Cox, J. Chen, and F. X. Kartner, “Drift-free femtosecond timing synchronization of remote optical and microwave sources,” Nat. Photonics2(12), 733–736 (2008). [CrossRef]
  14. 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] [PubMed]
  15. 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(4), 511–513 (2011). [CrossRef] [PubMed]
  16. D. Hou, P. Li, C. Liu, J. Zhao, and Z. Zhang, “Long-term stable frequency transfer over an urban fiber link using microwave phase stabilization,” Opt. Express19(2), 506–511 (2011). [CrossRef] [PubMed]
  17. D. D. Hudson, S. M. Foreman, S. T. Cundiff, and J. Ye, “Synchronization of mode-locked femtosecond lasers through a fiber link,” Opt. Lett.31(13), 1951–1953 (2006). [CrossRef] [PubMed]
  18. A. Yariv, Quantum Electronics Third Edition (John Wiley & Sons, 1989).
  19. A. Maitland and M. H. Dumn, Laser Physics (North-Holland Publishing Company, 1969).
  20. N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, “Frequency stabilization of mode-locked Erbium fiber lasers using pump power control,” Appl. Phys. B78(3-4), 321–324 (2004). [CrossRef]
  21. W. K. Tan, H. Y. Wong, A. E. Kelly, M. Sorel, J. H. Marsh, and A. C. Bryce, “Temperature behaviour of pulse repetition frequency in passively mode-locked InGaAsP/InP laser diode - Experimental results and simple model,” IEEE J. Sel. Top. Quantum Electron.13(5), 1209–1214 (2007). [CrossRef]
  22. E. N. Ivanov, S. A. Diddams, and L. Hollberg, “Analysis of noise mechanisms limiting the frequency stability of microwave signals generated with a femtosecond laser,” IEEE J. Sel. Top. Quantum Electron.9(4), 1059–1065 (2003). [CrossRef]
  23. G. Marra, H. S. Margolis, and D. J. Richardson, “Dissemination of an optical frequency comb over fiber with 3 × 10-18 fractional accuracy,” Opt. Express20(2), 1775–1782 (2012). [CrossRef] [PubMed]

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