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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 15 — Jul. 19, 2010
  • pp: 16102–16111
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Brillouin amplification in phase coherent transfer of optical frequencies over 480 km fiber

O. Terra, G. Grosche, and H. Schnatz  »View Author Affiliations


Optics Express, Vol. 18, Issue 15, pp. 16102-16111 (2010)
http://dx.doi.org/10.1364/OE.18.016102


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Abstract

We describe the use of fiber Brillouin amplification (FBA) for the coherent transmission of optical frequencies over a 480 km long optical fiber link. FBA uses the transmission fiber itself for efficient, bi-directional coherent amplification of weak signals with pump powers around 30 mW. In a test setup we measured the gain and the achievable signal-to-noise ratio (SNR) of FBA and compared it to that of the widely used uni-directional Erbium doped fiber amplifiers (EDFA) and to our recently built bi-directional EDFA. We measured also the phase noise introduced by the FBA and used a new and simple technique to stabilize the frequency of the FBA pump laser. We then transferred a stabilized laser frequency over a wide area network with a total fiber length of 480 km using only one intermediate FBA station. After compensating the noise induced by the fiber, the frequency is delivered to the user end with an uncertainty below 2 × 10−18 and an instability σy (τ) = 2 × 10−14 /(τ/s).

© 2010 OSA

1. Introduction

In order to keep the number of intermediate stations as low as possible, we have investigated the use of fiber Brillouin amplification (FBA) [9

9. N. Olsson and J. van der Ziel, “Cancellation of fiber loss by semi-conductor laser pumped Brillouin amplification at 1.5 μm,” Appl. Phys. Lett. 48(20), 1329 (1986). [CrossRef]

11

11. M. Ferreira, J. Rocha, and J. Pinto, “Analysis of the gain and noise characteristics of fibre Brillouin amplifiers,” Opt. Quantum Electron. 26(1), 35–44 (1994). [CrossRef]

] as an alternative technique. This technique enables the amplification of a very small input signal (a few nano Watt) by more than 50 dB in a single gain step, with relatively low pump powers (about 30 mW). Moreover, it enables bi-directional amplification because it uses the fiber itself as the gain medium and different sections of the same fiber for each direction.

2. Fiber Brillouin amplification (FBA)

Stimulated Brillouin scattering (SBS) is a nonlinear process which results from the interaction of light with stimulated acoustic waves. In fused silica single mode fibers, acoustic waves with velocity vA (vA ≈6 × 103 m/s) back-scatter light and shift down its frequency by the Brillouin frequency of (νB = 2 n vA / λ). This shift frequency νB is about 11 GHz for a refractive index n = 1.451, and a wavelength of λ = 1.5 μm. The threshold power for SBS is given by [12

12. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and brillouin scattering,” Appl. Opt. 11(11), 2489–2494 (1972). [CrossRef] [PubMed]

14

14. G. Agrawal, Applications of Nonlinear Fiber Optics, (Academic Press, 2001).

]:
Pcrit=21AγLeff(1+ΔνlaserΔνB)
(1)
where A is the effective mode area of the fiber (1 × 10−10 m2), and γ is the gain coefficient of the nonlinear process (5 × 10−11 m/W). The effective gain length is Leff = (1-e-αL)/α, therefore L eff ≈21 km for an attenuation coefficient of α = 0.2 dB/km. The Brillouin linewidth depends on the life time of the scattered excitation (τB), ΔνB = 1/πτB Hz. For a 148 km long fiber we measured ΔνB ≈10 MHz. We used a laser with linewidth of about Δνlaser = 5 kHz, which is much narrower than ΔνB. Therefore, the SBS threshold power is only 2 mW according to Eq. (1).

If a pump laser with a power (Ppump) is injected in the fiber, it will cause a peak Brillouin gain [9

9. N. Olsson and J. van der Ziel, “Cancellation of fiber loss by semi-conductor laser pumped Brillouin amplification at 1.5 μm,” Appl. Phys. Lett. 48(20), 1329 (1986). [CrossRef]

] of:

g=γLeffPpumpA
(2)

If the pump laser with a frequency νpump is injected in the opposite direction to a signal laser with a frequency νsig, it will be amplified by a process called fiber Brillouin amplification (FBA) [10

10. R. Tkach and A. Chraplyvy, “Fibre Brillouin amplifiers,” Opt. Quantum Electron. 21(1), S105–S112 (1989). [CrossRef]

,11

11. M. Ferreira, J. Rocha, and J. Pinto, “Analysis of the gain and noise characteristics of fibre Brillouin amplifiers,” Opt. Quantum Electron. 26(1), 35–44 (1994). [CrossRef]

], if the pump frequency νpump = νsig + νB. Due to the high gain of the Brillouin process (9 dB /mW) [Eq. (2)], very weak signals can be amplified with a very high gain using only few mW of pump power.

2.1 Amplification and signal-to-noise ratio

For measuring the amplification due to FBA we performed a simple test. We used a 25 km spool of single mode fiber (SMF28) as FBA gain medium and narrow linewidth fiber lasers as pump and signal lasers. These lasers have a wavelength of λ = 1542 nm, a linewidth of about 5 kHz, up to 100 mW output power and a tuning range of about 2 nm. As described in [15

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

], the signal laser is stabilized to our optical frequency reference to reach a linewidth of about 10 Hz.

With our 10 GHz resolution optical spectrum analyzer it is not possible to resolve the amplification caused by FBA, because ΔνB is only about 10 MHz. Therefore, we constructed a Mach-Zehnder interferometer with the 25 km fiber spool in the measurement arm and an AOM in the reference arm, as shown in Fig. 1
Fig. 1 Set-up to measure the gain of FBA or EDFA: VA: variable attenuator, PD: photodetector, signal: signal laser with isolator, pump: pump laser, AOM: acousto-optic modulator, a 25 km fiber is used as gain medium for FBA.
The AOM introduces a frequency shift (55 MHz) in the reference arm to produce heterodyne beat at the photo detector. A variable attenuator is installed in the front of the 25 km arm to adjust the input power for the 25 km fiber without changing the power in the reference arm. The pump laser is injected from the opposite end of the fiber, with a power around 20 mW and with frequency up-shifted by νpump - νsig = νB = 10.972 GHz. We tuned the variable attenuator at the input of the 25 km fiber to inject different signal powers into the FBA, to simulate different fiber lengths. The pump power is optimized to obtain the maximum amplification for every signal power. This allows the gain and the SNR of the device to be measured electrically using RF spectrum analyzer. The measurement is recorded with 100 kHz analyzer filter bandwidth.

For comparison we performed the same measurement for a bi-directional EDFA and a commercial single-pass EDFA and include the result in Fig. 2
Fig. 2 FBA in comparison to EDFAs (uni-directional and bi-directional) for different signal powers received at the output of a 25 km fiber: (a) gain (b) SNR. The spectrum analyzer bandwidth is 100 kHz.
. Note, the latter is not usable in the optical frequency transfer scheme, since bi-directional operation is required to compensate the fiber phase noise.

Figure 2 shows the gain (Fig. 2a) and the SNR (Fig. 2b) for different signal powers for the devices under test. Although both the bi-directional EDFA and FBA achieve almost the same SNR, the gain of FBA is significantly higher than that of the bi-directional EDFA. This is especially true for low signal powers. As an example, the gain of FBA is about 1000 times higher than that of the bi-directional EDFA for signal powers less than 50 nW. For a transparent network, the amplifier gain has to compensate the loss in each section of the network. FBA with 50 dB gain compensates the loss of a 250 km long fiber with a typical attenuation coefficient of 0.2 dB/km. In contrast, a bi-directional EDFA with a gain of 25 dB only allows a span of about 120 km.

This opens two possibilities: using an additional fiber spool as discrete FBA-module is a suitable solution when operational or safety requirements forbid injecting more than a few mW optical power into the transmission fiber. However, when higher power levels are allowed in the installed transmission fiber, this fiber itself can be used as gain medium. In order to test this option of a distributed amplifier further, we performed measurements using the FBA and the bi-directional EDFA on 148 km and 332 km of installed, commercial fibers. The installed dark fiber is buried and part of a wide-area network connecting PTB with other research institutes in Germany [7

7. H. Schnatz, O. Terra, K. Predehl, T. Feldmann, T. Legero, B. Lipphardt, U. Sterr, G. Grosche, R. Holzwarth, T. Hansch, T. Udem, Z. Lu, L. Wang, W. Ertmer, J. Friebe, A. Pape, E.-M. Rasel, M. Riedmann, and T. Wub, “Phase-coherent frequency comparison of optical clocks using a telecommunication fiber link,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57(1), 175–181 (2010). [CrossRef]

].

2.2 Phase noise

One of the most important aspects in frequency transfer applications is how much additional phase noise is introduced by the amplifier to the signal propagating through the fiber. We used the Mach-Zehnder interferometer shown in Fig. 1 to measure the phase noise of the bi-directional EDFA and FBA. Figure 4a
Fig. 4 (a) Phase noise of the free-running interferometer without (black line) and with a bi-directional EDFA (red ο). (b) Phase noise of 25 km spooled SMF28 fiber without (black line) and with using FBA (blue ο).
shows the measured phase noise for the free-running interferometer without (black line) and with installed EDFA (red circles ο). At low Fourier frequencies (f < 30 Hz) the EDFA possibly adds a small amount of phase noise – this is still less than 0.1 rad2/Hz at f = 1 Hz. We attribute this phase noise to the approximately 6 m fiber used as gain medium inside the amplifier.

2.3 Pump laser stabilization

The maximum amplification is achieved when the pump frequency matches the condition νpump = νsig + νB. Therefore, the pump laser frequency has to be stabilized. If pump and signal lasers are at the same site, a beat between the signal and the pump laser can be detected with a fast photodetector (bandwidth > 11 GHz). The beat signal is then locked to a microwave reference. But this technique needs complex and expensive equipment, which handles GHz frequencies. Furthermore, this technique will not be applicable if the pump laser is located at an intermediate station.

We have developed a simpler method based on the observation that the back-scattered pump power increases when the pump frequency is equal to sig + νB). Figure 5a
Fig. 5 (a) Measurement setup for the scattered and transmitted pump power changes. The pump is injected in the opposite direction to the signal laser. CIR: circulator and OPM: optical power meter. (signal power is about 1 μW after 148 km fiber) (b).The change of the scattered pump power when the pump frequency is swept around sig + νB) and a Gaussian fit.
, shows the system used to measure the scattered and transmitted pump powers when the pump is injected in the opposite direction of the signal in the 148 km fiber. Although the signal power is less than 1 μW at the output of the 148 km fiber the back-scattered pump power increased when the pump frequency is swept around (νsig + νB), see Fig. 5b. The transmitted pump power is decreased also by the same amount. A small modulation of the pump laser frequency and lock-in detection was used to monitor this change in the scattered pump power and stabilize the pump laser frequency at νsig + νB.

To determine the change of the Brillouin frequency (νB) with temperature [16

16. J. Geng, S. Staines, M. Blake, and S. Jiang, “Distributed fiber temperature and strain sensor using coherent radio-frequency detection of spontaneous Brillouin scattering,” Appl. Opt. 46(23), 5928–5932 (2007). [CrossRef] [PubMed]

], we measured the temperature dependence of the 25 km fiber in the range from 16 °C to 25 °C. The Brillouin frequency temperature dependence is found to be 0.4 MHz/K. Typical peak to peak diurnal temperature variations for a buried fiber (depth < 1.5 m) are well below 1 K. Even if we assume an upper limit for temperature variations of about 10 K, the stabilization circuit can easily handle a pump laser frequency shift of about 4 MHz.

3. Frequency transfer over a 480 km optical fiber

The 480 km fiber consists of four commercial fibers with overall attenuation of 115 dB, two of them are connecting PTB, Braunschweig to Cörmigk with an attenuation of about 70 dB and the other two are connecting PTB to the Institute of Quantum Optics (IQ), Hanover with attenuation of about 45 dB. In this setup local and user end as well as the intermediate station are located at PTB, which allows for easy testing. The light is sent in the first fiber (166 km) to Cörmigk, where it is returned to the intermediate station at PTB by fiber 2. The light is amplified and sent in the third fiber (74 km) to IQ, where it is connected to a fourth fiber to return the light back to PTB (the user end).

For a full round-trip in the fiber the signal accumulates 230 dB loss before it reaches the phase compensation detector PD1. In order to compensate this loss, we used two FBA amplifiers for pumping four fiber sections. A FBA amplifier is used at the intermediate station to amplify the light in the return direction with a pump power of 45 mW and in the forward direction with a pump power of 30 mW. It is also used at the local end with a pump power of 9 mW to amplify the return light and at the virtual user end with a pump power of 18 mW to amplify the forward light. Pump light is injected into the fiber using simple fiber couplers with a coupling ratio depending on the available pump powers. It compromises between signal and pump power loss. We used one amplifier for each direction because of the 80 MHz frequency difference between the forward and the return light introduced by the AOM. For the forward direction FBA2 is used with frequency νFpump = νsig + νB + 55 MHz and for the return direction FBA1 is used with frequency νRpump = νsig + νB - 25 MHz. Where νB is the Brillouin frequency and νsig is the frequency of the transfer laser (signal). A part of the Brillouin-scattered light is directed by circulators to the DC photodetectors PD (3 and 4), where it is used to stabilize the frequency of the pump laser using the method discussed in section (2.3).

The phase noise of the out-of-loop beat is measured before and after applying the compensation scheme by detecting the phase changes with respect to a reference oscillator using a digital phase detector, after division by a suitable ratio to keep the phase changes below one radian [17

17. F. Walls, A. Clements, C. Felton, M. Lombardi, and M. Vanek, “Extending the Range and Accuracy of Phase Noise Measurements,” National Institute of Standards and Technology (NIST) Technical Note 1337, TN129 (1990).

]. Figure 7
Fig. 7 Phase noise of the out-of-loop signal (OL) before (black dashed), and after (red solid) compensation. The green curve (ο) gives the theoretical compensation limit according to [18].
shows the phase noise in (rad2/Hz) for the out-of-loop signal measured before and after applying the compensation. The phase noise reduction is 39 dB at 1 Hz, which is near to the theoretical limit of 41 dB at 1 Hz predicted by Williams et al. [18

18. W. 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 (2008). [CrossRef]

], SD (f) = (1/3) (2π)2 τ 2 delay f 2 Sφ (f) = 7.5 × 10−5 f 2 Sφ (f), where Sφ (f) is the phase noise of the free-running fiber, τdelay is the time for a one-way trip in the fiber and f is the Fourier frequency. It is close to optimal (green ο in Fig. 7) within the theoretical compensation bandwidth; the phase noise is suppressed up to Fourier frequencies of about 50 Hz.

The mean value of the transmitted frequency is shifted from that of the reference laser by 64 μHz with a statistical uncertainty of 54 μHz. This shift corresponds to fractional frequency deviation of 3 × 10−19. The statistical uncertainty is obtained by dividing the standard deviation of the measurement by N since it is white phase noise and not by N, as discussed in [20

20. W. Lee, D. Yu, C. Park, and J. Mun, “The uncertainty associated with the weigted mean frequency of a phase-stabilized signal with white phase noise,” Metrologia 47(1), 24–32 (2010). [CrossRef]

], where N = 65000 is the total number of data points.

4. Conclusion

Acknowledgment

The authors would like to thank Katharina Predehl and Thomas Legero for technical assistance, as well as Piet Schmidt and Fritz Riehle for helpful comments. The work was partly supported by DFG through the Centre for Quantum Engineering and Space-Time Research, QUEST. Osama Terra is supported by a scholarship from the Egyptian National Institute of Standards (NIS) and is a member of the Braunschweig International Graduate School of Metrology, IGSM. Osama Terra’s home address is National Institute of Standards (NIS), Tersa St., Haram-Giza, Egypt, P. O. Box: 136 Giza, Postal code:12211; and his home email address is osama.terra@nis.sci.eg.

References and Links

1.

S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of time and frequency at the outset of the 21st century,” Science 306(5700), 1318–1324 (2004). [CrossRef] [PubMed]

2.

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]

3.

O. Lopez, A. Amy-Klein, C. Daussy, Ch. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 1018 for RF frequency transfer,” Eur. Phys. J. D 48(1), 35–41 (2008). [CrossRef]

4.

O. Terra, G. Grosche, K. Predehl, R. Holzwarth, T. Legero, U. Sterr, B. Lipphardt, and H. Schnatz, “Phase-coherent comparison of two optical frequency standards over 146 km using a telecommunication fiber link,” Appl. Phys. B 97(3), 541–551 (2009). [CrossRef]

5.

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] [PubMed]

6.

E. Desurvire, Erbium-doped fiber amplifiers: principle and applications, (Wiley-Interscience publication, 1994).

7.

H. Schnatz, O. Terra, K. Predehl, T. Feldmann, T. Legero, B. Lipphardt, U. Sterr, G. Grosche, R. Holzwarth, T. Hansch, T. Udem, Z. Lu, L. Wang, W. Ertmer, J. Friebe, A. Pape, E.-M. Rasel, M. Riedmann, and T. Wub, “Phase-coherent frequency comparison of optical clocks using a telecommunication fiber link,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57(1), 175–181 (2010). [CrossRef]

8.

K. Predehl, R. Holzwarth, T. Udem, T. W. Hänsch, O. Terra, G. Grosche, B. Lipphardt, and H. Schnatz, “Ultra Precise Frequency Dissemination across Germany - Towards a 900 km Optical Fiber Link from PTB to MPQ,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper CTuS2.

9.

N. Olsson and J. van der Ziel, “Cancellation of fiber loss by semi-conductor laser pumped Brillouin amplification at 1.5 μm,” Appl. Phys. Lett. 48(20), 1329 (1986). [CrossRef]

10.

R. Tkach and A. Chraplyvy, “Fibre Brillouin amplifiers,” Opt. Quantum Electron. 21(1), S105–S112 (1989). [CrossRef]

11.

M. Ferreira, J. Rocha, and J. Pinto, “Analysis of the gain and noise characteristics of fibre Brillouin amplifiers,” Opt. Quantum Electron. 26(1), 35–44 (1994). [CrossRef]

12.

R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and brillouin scattering,” Appl. Opt. 11(11), 2489–2494 (1972). [CrossRef] [PubMed]

13.

E. Ippen and R. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21(11), 539 (1972). [CrossRef]

14.

G. Agrawal, Applications of Nonlinear Fiber Optics, (Academic Press, 2001).

15.

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

16.

J. Geng, S. Staines, M. Blake, and S. Jiang, “Distributed fiber temperature and strain sensor using coherent radio-frequency detection of spontaneous Brillouin scattering,” Appl. Opt. 46(23), 5928–5932 (2007). [CrossRef] [PubMed]

17.

F. Walls, A. Clements, C. Felton, M. Lombardi, and M. Vanek, “Extending the Range and Accuracy of Phase Noise Measurements,” National Institute of Standards and Technology (NIST) Technical Note 1337, TN129 (1990).

18.

W. 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 (2008). [CrossRef]

19.

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

20.

W. Lee, D. Yu, C. Park, and J. Mun, “The uncertainty associated with the weigted mean frequency of a phase-stabilized signal with white phase noise,” Metrologia 47(1), 24–32 (2010). [CrossRef]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(140.0140) Lasers and laser optics : Lasers and laser optics

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: May 18, 2010
Revised Manuscript: June 25, 2010
Manuscript Accepted: June 28, 2010
Published: July 15, 2010

Citation
O. Terra, G. Grosche, and H. Schnatz, "Brillouin amplification in phase coherent transfer of optical frequencies over 480 km fiber," Opt. Express 18, 16102-16111 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-16102


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References

  1. S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of time and frequency at the outset of the 21st century,” Science 306(5700), 1318–1324 (2004). [CrossRef] [PubMed]
  2. 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]
  3. O. Lopez, A. Amy-Klein, C. Daussy, Ch. 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(1), 35–41 (2008). [CrossRef]
  4. O. Terra, G. Grosche, K. Predehl, R. Holzwarth, T. Legero, U. Sterr, B. Lipphardt, and H. Schnatz, “Phase-coherent comparison of two optical frequency standards over 146 km using a telecommunication fiber link,” Appl. Phys. B 97(3), 541–551 (2009). [CrossRef]
  5. 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] [PubMed]
  6. E. Desurvire, Erbium-doped fiber amplifiers: principle and applications, (Wiley-Interscience publication, 1994).
  7. H. Schnatz, O. Terra, K. Predehl, T. Feldmann, T. Legero, B. Lipphardt, U. Sterr, G. Grosche, R. Holzwarth, T. Hansch, T. Udem, Z. Lu, L. Wang, W. Ertmer, J. Friebe, A. Pape, E.-M. Rasel, M. Riedmann, and T. Wub, “Phase-coherent frequency comparison of optical clocks using a telecommunication fiber link,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57(1), 175–181 (2010). [CrossRef]
  8. K. Predehl, R. Holzwarth, T. Udem, T. W. Hänsch, O. Terra, G. Grosche, B. Lipphardt, and H. Schnatz, “Ultra Precise Frequency Dissemination across Germany - Towards a 900 km Optical Fiber Link from PTB to MPQ,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper CTuS2.
  9. N. Olsson and J. van der Ziel, “Cancellation of fiber loss by semi-conductor laser pumped Brillouin amplification at 1.5 μm,” Appl. Phys. Lett. 48(20), 1329 (1986). [CrossRef]
  10. R. Tkach and A. Chraplyvy, “Fibre Brillouin amplifiers,” Opt. Quantum Electron. 21(1), S105–S112 (1989). [CrossRef]
  11. M. Ferreira, J. Rocha, and J. Pinto, “Analysis of the gain and noise characteristics of fibre Brillouin amplifiers,” Opt. Quantum Electron. 26(1), 35–44 (1994). [CrossRef]
  12. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and brillouin scattering,” Appl. Opt. 11(11), 2489–2494 (1972). [CrossRef] [PubMed]
  13. E. Ippen and R. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21(11), 539 (1972). [CrossRef]
  14. G. Agrawal, Applications of Nonlinear Fiber Optics, (Academic Press, 2001).
  15. G. Grosche, B. Lipphardt, and H. Schnatz, “Optical frequency synthesis and measurement using fiber-based femtosecond lasers,” Eur. Phys. J. D 48(1), 27–33 (2008). [CrossRef]
  16. J. Geng, S. Staines, M. Blake, and S. Jiang, “Distributed fiber temperature and strain sensor using coherent radio-frequency detection of spontaneous Brillouin scattering,” Appl. Opt. 46(23), 5928–5932 (2007). [CrossRef] [PubMed]
  17. F. Walls, A. Clements, C. Felton, M. Lombardi, and M. Vanek, “Extending the Range and Accuracy of Phase Noise Measurements,” National Institute of Standards and Technology (NIST) Technical Note 1337, TN129 (1990).
  18. W. 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 (2008). [CrossRef]
  19. E. Rubiola, “On the measurement of frequency and of its sample variance with high-resolution counters,” Rev. Sci. Instrum. 76(5), 054703 (2005). [CrossRef]
  20. W. Lee, D. Yu, C. Park, and J. Mun, “The uncertainty associated with the weigted mean frequency of a phase-stabilized signal with white phase noise,” Metrologia 47(1), 24–32 (2010). [CrossRef]

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