## Phase noise in fiber-optic parametric amplifiers and converters and its impact on sensing and communication systems |

Optics Express, Vol. 18, Issue 20, pp. 21449-21460 (2010)

http://dx.doi.org/10.1364/OE.18.021449

Acrobat PDF (1802 KB)

### Abstract

We present a theoretical analysis describing the spectral dependence of phase noise in one-pump fiber parametric amplifiers and converters. The analytical theory is experimentally validated and found to have high predictive accuracy. The implications related to phase-coded sensing and communications systems are discussed.

© 2010 OSA

## 1. Introduction

1. S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. **2**(6), 498–513 (2008). [CrossRef]

2. P. Kylemark, M. Karlsson, T. Torounidis, and P. A. Andrekson, “Noise Statistics in Fiber Optical Parametric Amplifiers,” J. Lightwave Technol. **25**(2), 612–620 (2007). [CrossRef]

3. Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express **18**(3), 2884–2893 (2010). [CrossRef] [PubMed]

*not*an

*additive*but rather a

*multiplicative*noise process, making the standard noise figure definition dependent on the input signal power. Consequently, the standard technique used to calculate the equivalent noise figure in links containing a cascade of amplifiers [4] is not applicable in FOPA-amplified systems.

7. S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective Suppression of Idler Spectral Broadening in Two-Pump Parametric Architectures,” IEEE Photon. Technol. Lett. **15**(5), 673–675 (2003). [CrossRef]

8. A. Durécu-Legrand, A. Mussot, C. Simonneau, D. Bayart, T. Sylvestre, E. Lantz, and H. Maillotte, “Impact of pump phase modulation on system performances of fiber optical parametric amplifiers,” Electron. Lett. **41**(6), 350–352 (2005). [CrossRef]

9. P. Kylemark, J. Ren, Y. Myslivets, N. Alic, S. Radic, P. A. Andrekson, and M. Karlsson, “Impact of Pump Phase-Modulation on the Bit-Error Rate in Fiber-Optical Parametric-Amplifier-Based Systems,” IEEE Photon. Technol. Lett. **19**(1), 79–81 (2007). [CrossRef]

10. R. Elschner, C.-A. Bunge, B. Huttl, A. G. i Coca, C. L. Schmidt, R. Ludwig, C. Schubert, and K. Petermann, “Impact of Pump-Phase Modulation on FWM-Based Wavelength Conversion of D(Q)PSK Signals,” IEEE J. Sel. Top. Quantum Electron. **14**(3), 666–673 (2008). [CrossRef]

11. R. Jiang, C.-S. Bres, N. Alic, E. Myslivets, and S. Radic, “Translation of Gbps Phase-Modulated Optical Signal From Near-Infrared to Visible Band,” J. Lightwave Technol. **26**(1), 131–137 (2008). [CrossRef]

12. J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. **15**(23), 1351–1353 (1990). [CrossRef] [PubMed]

15. Y. Kim, S. Kim, Y.-J. Kim, H. Hussein, and S.-W. Kim, “Er-doped fiber frequency comb with mHz relative linewidth,” Opt. Express **17**(14), 11972–11977 (2009). [CrossRef] [PubMed]

16. N. Nishizawa and J. Takayanagi, “Octave spanning high-quality supercontinuum generation in all-fiber system,” J. Opt. Soc. Am. B **24**(8), 1786–1792 (2007). [CrossRef]

17. N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs,” J. Opt. Soc. Am. B **24**(8), 1756–1770 (2007). [CrossRef]

18. M. Sköld, J. Yang, H. Sunnerud, M. Karlsson, S. Oda, and P. A. Andrekson, “Constellation diagram analysis of DPSK signal regeneration in a saturated parametric amplifier,” Opt. Express **16**(9), 5974–5982 (2008). [CrossRef] [PubMed]

20. M. Matsumoto, “Phase noise generation in an amplitude limiter using saturation of a fiber-optic parametric amplifier,” Opt. Lett. **33**(15), 1638–1640 (2008). [CrossRef] [PubMed]

## 2. Theory

### 2.1 Statistics of Nonlinear Phase Noise

*P*and carrier frequency

_{p}*ν*is amplified in a high-power Erbium-doped fiber amplifier (EDFA), thereby accumulating white Gaussian optical noise. The original pump RIN and laser phase noise are considered to be negligible. The optical amplifier noise,

_{p}*n(t) = n*, is a complex white Gaussian random process [22]. The in-phase and quadrature components of the noise have zero mean and variance of

_{r}(t) + jn_{i}(t)*N*, where

_{0}Δν/2*N*is the noise power spectral density in one polarization and

_{0}*Δν*is the optical bandwidth of interest. The optical signal-to-noise ratio of the pump wave (measured within 0.1nm optical bandwidth) is given by

*OSNR*, where

_{0.1nm}= P_{p}/(2N_{0}Δν_{0.1nm})*Δν*is the frequency bandwidth corresponding to 0.1nm at the wavelength of

_{0.1nm}*c/ν*

_{p}.*h*is the optical filter impulse response, and

_{in}(t)*n’(t)*is the complex field of the filtered optical noise. The noisy complex pump field then enters HNLF characterized by fiber length

*L*, nonlinear coefficient

*γ*, and negligible intra-channel dispersion. While the assumption that HNLF has no intra-channel dispersion allows derivation of closed-form expressions, it is also justified in most practical cases as the pump positioning in the proximity of the zero-dispersion HNLF frequency is used to maximize the FOPA gain bandwidth [23]. Next, after the propagation through HNLF, neglecting the depletion and transmission loss, the pump complex field becomes:

*N*to be the total single-polarization filtered noise power and note that the post-filtering variances of the

_{0}’*n*and

_{r}’(t)*n*equal

_{i}’(t)*N*, then the variance of NPN is

_{0}’/2*2γ*

^{2}L^{2}P_{p}N_{0}’.*ϕ*, is non-central χ

_{NL}(t)^{2}-distributed with probability distribution function:

*γL(P*, while the variance is

_{p}+ N_{0}’)*γ*Eq. (4) is an approximation since a closed form analytical solution for the probability density function (PDF) of nonlinear phase exists only for rectangular and Lorentzian optical filter transfer functions [24

^{2}L^{2}(2P_{p}N_{0}’ + N_{0}’^{2}).24. N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical preamplifier,” Opt. Express **13**(12), 4568–4579 (2005). [CrossRef] [PubMed]

### 2.2 Signal/Idler Phase Noise

25. R. H. Stolen and J. E. Bjorkholm, “Parametric Amplification and Frequency Conversion in Optical Fibers,” IEEE J. Quantum Electron. **18**(7), 1062–1072 (1982). [CrossRef]

*γ*is frequency-independent, and (e) Raman scattering is neglected. By incorporating these assumptions, the total phase mismatch is time-dependent and given bywhere the first term represents the linear phase mismatch, and the second term is the nonlinear phase shift due to the noisy pump power. The linear phase mismatch,

*Δβ(t)*, owes its time dependence to the phase modulation of the pump wave, as is often the case in continuous-wave FOPAs. The signal and idler amplitudes are affected by NPN and PMN via modulation of the exponential gain constant:

*ϕ*and

_{s}(t,0)*ϕ*represent the time-dependent phases of the signal and pump lasers before entering HNLF. Since narrow linewidth (<1MHz) lasers are practically available, the initial phase noises tend to be small in comparison to the third and fourth term, which represent NPN and PMN, respectively. The last term represents the AQN contribution to the signal and idler phase noise.

_{p}(t,0)18. M. Sköld, J. Yang, H. Sunnerud, M. Karlsson, S. Oda, and P. A. Andrekson, “Constellation diagram analysis of DPSK signal regeneration in a saturated parametric amplifier,” Opt. Express **16**(9), 5974–5982 (2008). [CrossRef] [PubMed]

*SNR*The simulation parameters used are specified in Table 1 ; they correspond to the constructed FOPA used in the experimental section of this work (Sec. 3). As expected from analysis in Sec. 2.1, the phase is corrupted by increased pump optical noise filter’s bandwidth and reduced pump OSNR. The second term in Eq. (7) shows an additional noise source that is present in the signal phase and absent in the idler phase. The inverse tangent term is responsible for wavelength dependence of signal NPN contribution and will be termed

_{phase}= 1/Var{ϕ_{NL}(t)}.*chromatic NPN*(CNPN) in subsequent discussion. Figure 2(b) depicts the analytically predicted and simulated spectral dependence of signal and idler phase and amplitude fidelity for pump OSNR of 40dB and a fixed optical Gaussian 3-dB bandwidth of 40GHz. The amplitude SNR in this calculation is defined as

*(Mean{A*, where

_{s,i}(t)})^{2}/Var{ A_{s,i}(t)}*A*is the time-varying signal/idler amplitude. All simulations were performed using a commercially available full Generalized Nonlinear Schrodinger Equation (GNLSE) solver (

_{s,i}(t)*VPItransmissionMaker*). The linear phase mismatch,

^{TM}*Δβ = -λ*[26], is considered to be constant (i.e., no pump phase modulation), so that the CNPN term is only a function of pump amplitude noise,

_{p}^{2}/(2πc)S(λ_{p}-λ_{0})(2πc/λ_{s}-2πc/λ_{p})^{2}*n’(t)*. The signal phase exhibits strong variations at the edge of the parametric gain, significantly impairing the amplifier performance in this spectral region. In continuous-wave-pumped FOPAs, the idler phase is dominated by pump phase modulation and the first term in Eq. (8) dominates the noise properties of the idler phase. However, when pump phase modulation can be avoided, as in the case of pulsed-pump FOPAs, the signal phase still exhibits wavelength dependence, whereas the idler phase possesses purely achromatic properties. The increased stability of the idler’s phase is accompanied by a significant increase in amplitude fluctuations, as can be seen in Fig. 2(b). The noise induced by pump amplitude fluctuations, therefore, is distributed differently (in the two quadratures) for the amplified (signal) and converted (idler) wave. Interestingly, the only spectrally independent quadrature noise component is the phase noise of the idler wave.

### 2.3 Phase Modulation Noise

27. J. Hansryd and P. A. Andrekson, “Broad-Band Continuous-Wave-Pumped Fiber Optical Parametric Amplifier with 49-dB Gain and Wavelength-Conversion Efficiency,” IEEE Photon. Technol. Lett. **13**(3), 194–196 (2001). [CrossRef]

28. J. B. Coles, B. P.-P. Kuo, N. Alic, S. Moro, C.-S. Bres, J. M. C. Boggio, P. A. Andrekson, M. Karlsson, and S. Radic, “Bandwidth-efficient phase modulation techniques for stimulated Brillouin scattering suppression in fiber optic parametric amplifiers,” Opt. Express **18**(17), 18138–18150 (2010). [CrossRef] [PubMed]

*V*), the variance of the driving time-varying voltage,

_{π}*σ*, must equal

^{2}_{V(t)}*V*in order to optimally suppress the optical carrier. Then, the required RF noise power spectral density is given by

_{π}^{2}*S*, where

_{n,rf}= (V_{π}^{2}/R_{L})/Δf_{n,rf}*R*is the load impedance, and

_{L}*Δf*is the electrical bandwidth of RF noise. The pump instantaneous frequency,

_{n,rf}*ν*, is a function of the instantaneous pump phase,

_{p}(t) = c/λ_{p}+ (1/2π)dϕ_{p}(t)/dt*ϕ*, defining the linear phase mismatch as

_{p}(t) = πV(t)/V_{π}*β*,

_{4}*β*, etc.) have purposely been omitted since their contributions to the linear phase mismatch are negligible in the bandwidth of interest (100nm). A distant spectral conversion [29

_{6}29. J. M. Chavez Boggio, S. Moro, B. P.-P. Kuo, N. Alic, B. Stossel, and S. Radic, “Tunable Parametric All-Fiber Short-Wavelength IR Transmitter,” J. Lightwave Technol. **28**(4), 443–447 (2010). [CrossRef]

*n’(t) = 0*) and using parameters in Table 1. Figure 3(b) depicts the signal phase SNR spectrum for a 600MHz electrical noise bandwidth. As expected from Eq. (9), the phase SNR reduces as the pump-signal wavelength separation is increased. At the edge of the gain spectrum, the interaction between the

*Δβ(t)L/2*term and the CNPN term in Eq. (7) results in sharp spectral features. These spectral features are smoothed when some amount of pump amplitude noise (and hence NPN) is present, as is always the case in practical FOPA devices. By comparing Fig. 2(b) and Fig. 3(b), we come to expect the PMN to make at least an order of magnitude smaller contribution to the total phase noise than NPN. The two noise variances become comparable when the pump OSNR exceeds 55dB and pump noise is narrowly (i.e. sub-20GHz) filtered in order to reduce NPN. It is interesting to note that PMN increases with increased HNLF length [see Eqs. (7) and (8)]. Hence, we expect FOPAs employing the combination of longer fiber lengths and reduced pump powers to have a larger contribution of PMN to the total phase noise, thus posing another challenge in devising cost-effective FOPA devices.

### 2.4 Phase noise due to Amplified Quantum Noise

*linear*optical amplifier), where

*equal*amount of Gaussian-distributed noise is added by the amplifier to both quadrature components of the signal, are well known [30]. In case of high (>10dB) signal-to-noise ratio at the output of the amplifier, the phase variations are dominated by the imaginary part of complex white Gaussian noise and the phase variance is inversely proportional to the SNR [30]. The SNR is given by

*GP*where

_{s}/P_{n},*G*is the amplifier gain,

*P*is the input signal power, and

_{s}*P*is the total noise power in one quadrature and one polarization. For EDFAs,

_{n}*P*, where

_{n}= ½hν_{s}Δνn_{sp}(G-1)*h*is the Planck’s constant,

*ν*is the signal frequency,

_{s}*Δν*is the optical bandwidth, and

*n*is the spontaneous emission factor [4]. Thus, in the limit of high gain (i.e. G>>1), the phase SNR due to ASE becomes

_{sp}*2P*.

_{s}/(hν_{s}Δνn_{sp})*hν/2*in one polarization, as illustrated in Fig. 4 . The vacuum fluctuations at the signal wavelength are amplified by parametric gain

*G*[26]. In addition, the signal is coupled to the AQN associated with the idler, which is amplified by gain of

_{s}= 1 + (γP_{p}/g)^{2}sinh^{2}(gL)*G*. Adding the two noise contributions, the total power of AQN in one polarization and both quadratures is given by

_{s}-131. K. Shimoda, H. Takahasi, and C. H. Townes, “Fluctuations in Amplification of Quanta with Application to Maser Amplifiers,” J. Phys. Soc. Jpn. **12**(6), 686–700 (1957). [CrossRef]

34. A. V. Kozlovskii, “Photodetection of a weak light signal in various quantum states by using an optical amplifier,” Quantum Electron. **36**(3), 280–286 (2006). [CrossRef]

## 3. Experimental Results and Discussion

25. R. H. Stolen and J. E. Bjorkholm, “Parametric Amplification and Frequency Conversion in Optical Fibers,” IEEE J. Quantum Electron. **18**(7), 1062–1072 (1982). [CrossRef]

*2f*). Looking closely at the high-resolution optical spectrum of the signal wave at different pump OSNRs [Fig. 8(b)], we clearly see the spectral contribution of the narrowband PMN and broadband NPN. Since the measured phase noise will be integrated over 16GHz of electrical bandwidth, we expect the NPN-induced phase noise to dominate the signal phase SNR. The two sharp peaks located 170MHz away from the signal center frequency are the laser cavity sidemodes; they are suppressed by approximately 50dB with respect to the carrier.

_{p}→f_{s}+ f_{i}18. M. Sköld, J. Yang, H. Sunnerud, M. Karlsson, S. Oda, and P. A. Andrekson, “Constellation diagram analysis of DPSK signal regeneration in a saturated parametric amplifier,” Opt. Express **16**(9), 5974–5982 (2008). [CrossRef] [PubMed]

3. Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express **18**(3), 2884–2893 (2010). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgements

## References and links

1. | S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. |

2. | P. Kylemark, M. Karlsson, T. Torounidis, and P. A. Andrekson, “Noise Statistics in Fiber Optical Parametric Amplifiers,” J. Lightwave Technol. |

3. | Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express |

4. | P. C. Becker, N. A. Olsson, and J. R. Simpson, |

5. | R. M. Jopson, U.S. Patent 5 386 394 (1994). |

6. | M. Ho, M. E. Marhic, K. Y. K. Wong, and L. Kazovsky, “Narrow linewidth idler generation in fiber four-wave mixing and parametric amplification by dithering two pumps in opposition of phase,” J. Lightwave Technol. |

7. | S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective Suppression of Idler Spectral Broadening in Two-Pump Parametric Architectures,” IEEE Photon. Technol. Lett. |

8. | A. Durécu-Legrand, A. Mussot, C. Simonneau, D. Bayart, T. Sylvestre, E. Lantz, and H. Maillotte, “Impact of pump phase modulation on system performances of fiber optical parametric amplifiers,” Electron. Lett. |

9. | P. Kylemark, J. Ren, Y. Myslivets, N. Alic, S. Radic, P. A. Andrekson, and M. Karlsson, “Impact of Pump Phase-Modulation on the Bit-Error Rate in Fiber-Optical Parametric-Amplifier-Based Systems,” IEEE Photon. Technol. Lett. |

10. | R. Elschner, C.-A. Bunge, B. Huttl, A. G. i Coca, C. L. Schmidt, R. Ludwig, C. Schubert, and K. Petermann, “Impact of Pump-Phase Modulation on FWM-Based Wavelength Conversion of D(Q)PSK Signals,” IEEE J. Sel. Top. Quantum Electron. |

11. | R. Jiang, C.-S. Bres, N. Alic, E. Myslivets, and S. Radic, “Translation of Gbps Phase-Modulated Optical Signal From Near-Infrared to Visible Band,” J. Lightwave Technol. |

12. | J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. |

13. | K.-P. Ho, |

14. | S. J. McNaught, J. E. Rothenberg, P. A. Thielen, M. G. Wickham, M. E. Weber, and G. D. Goodno, “Coherent Combining of a 1.26-kW Fiber Amplifier,” Advanced in Solid-State Photonics, paper AMA2 (2010). |

15. | Y. Kim, S. Kim, Y.-J. Kim, H. Hussein, and S.-W. Kim, “Er-doped fiber frequency comb with mHz relative linewidth,” Opt. Express |

16. | N. Nishizawa and J. Takayanagi, “Octave spanning high-quality supercontinuum generation in all-fiber system,” J. Opt. Soc. Am. B |

17. | N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs,” J. Opt. Soc. Am. B |

18. | M. Sköld, J. Yang, H. Sunnerud, M. Karlsson, S. Oda, and P. A. Andrekson, “Constellation diagram analysis of DPSK signal regeneration in a saturated parametric amplifier,” Opt. Express |

19. | M. Skold, M. Karlsson, S. Oda, H. Sunnerud, and P. A. Andrekson, “Constellation diagram measurements of induced phase noise in a regenerating parametric amplifier,” Optical Fiber Communications Conference, paper OML4 (2008). |

20. | M. Matsumoto, “Phase noise generation in an amplitude limiter using saturation of a fiber-optic parametric amplifier,” Opt. Lett. |

21. | R. Elschner, and L. Petermann, “Impact of Pump-Induced Nonlinear Phase Noise on Parametric Amplification and Wavelength Conversion of Phase-Modulated Signals,” European Conference in Optical Communications, paper 3.3.4 (2009). |

22. | E. Desurvire, |

23. | M. E. Marhic, |

24. | N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical preamplifier,” Opt. Express |

25. | R. H. Stolen and J. E. Bjorkholm, “Parametric Amplification and Frequency Conversion in Optical Fibers,” IEEE J. Quantum Electron. |

26. | G. P. Agrawal, |

27. | J. Hansryd and P. A. Andrekson, “Broad-Band Continuous-Wave-Pumped Fiber Optical Parametric Amplifier with 49-dB Gain and Wavelength-Conversion Efficiency,” IEEE Photon. Technol. Lett. |

28. | J. B. Coles, B. P.-P. Kuo, N. Alic, S. Moro, C.-S. Bres, J. M. C. Boggio, P. A. Andrekson, M. Karlsson, and S. Radic, “Bandwidth-efficient phase modulation techniques for stimulated Brillouin scattering suppression in fiber optic parametric amplifiers,” Opt. Express |

29. | J. M. Chavez Boggio, S. Moro, B. P.-P. Kuo, N. Alic, B. Stossel, and S. Radic, “Tunable Parametric All-Fiber Short-Wavelength IR Transmitter,” J. Lightwave Technol. |

30. | J. W. Goodman, |

31. | K. Shimoda, H. Takahasi, and C. H. Townes, “Fluctuations in Amplification of Quanta with Application to Maser Amplifiers,” J. Phys. Soc. Jpn. |

32. | J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. |

33. | C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express |

34. | A. V. Kozlovskii, “Photodetection of a weak light signal in various quantum states by using an optical amplifier,” Quantum Electron. |

**OCIS Codes**

(060.5060) Fiber optics and optical communications : Phase modulation

(190.4970) Nonlinear optics : Parametric oscillators and amplifiers

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 28, 2010

Revised Manuscript: September 5, 2010

Manuscript Accepted: September 17, 2010

Published: September 24, 2010

**Citation**

Slaven Moro, Ana Peric, Nikola Alic, Bryan Stossel, and Stojan Radic, "Phase noise in fiber-optic parametric amplifiers and converters and its impact on sensing and communication systems," Opt. Express **18**, 21449-21460 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-21449

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### References

- S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. 2(6), 498–513 (2008). [CrossRef]
- P. Kylemark, M. Karlsson, T. Torounidis, and P. A. Andrekson, “Noise Statistics in Fiber Optical Parametric Amplifiers,” J. Lightwave Technol. 25(2), 612–620 (2007). [CrossRef]
- Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18(3), 2884–2893 (2010). [CrossRef] [PubMed]
- P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers (Academic Press, 1999).
- R. M. Jopson, U.S. Patent 5 386 394 (1994).
- M. Ho, M. E. Marhic, K. Y. K. Wong, and L. Kazovsky, “Narrow linewidth idler generation in fiber four-wave mixing and parametric amplification by dithering two pumps in opposition of phase,” J. Lightwave Technol. 20(3), 469–476 (2002). [CrossRef]
- S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective Suppression of Idler Spectral Broadening in Two-Pump Parametric Architectures,” IEEE Photon. Technol. Lett. 15(5), 673–675 (2003). [CrossRef]
- A. Durécu-Legrand, A. Mussot, C. Simonneau, D. Bayart, T. Sylvestre, E. Lantz, and H. Maillotte, “Impact of pump phase modulation on system performances of fiber optical parametric amplifiers,” Electron. Lett. 41(6), 350–352 (2005). [CrossRef]
- P. Kylemark, J. Ren, Y. Myslivets, N. Alic, S. Radic, P. A. Andrekson, and M. Karlsson, “Impact of Pump Phase-Modulation on the Bit-Error Rate in Fiber-Optical Parametric-Amplifier-Based Systems,” IEEE Photon. Technol. Lett. 19(1), 79–81 (2007). [CrossRef]
- R. Elschner, C.-A. Bunge, B. Huttl, A. G. i Coca, C. L. Schmidt, R. Ludwig, C. Schubert, and K. Petermann, “Impact of Pump-Phase Modulation on FWM-Based Wavelength Conversion of D(Q)PSK Signals,” IEEE J. Sel. Top. Quantum Electron. 14(3), 666–673 (2008). [CrossRef]
- R. Jiang, C.-S. Bres, N. Alic, E. Myslivets, and S. Radic, “Translation of Gbps Phase-Modulated Optical Signal From Near-Infrared to Visible Band,” J. Lightwave Technol. 26(1), 131–137 (2008). [CrossRef]
- J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990). [CrossRef] [PubMed]
- K.-P. Ho, Phase-Modulated Optical Communication Systems (Springer, 2005), Chap. 5.
- S. J. McNaught, J. E. Rothenberg, P. A. Thielen, M. G. Wickham, M. E. Weber, and G. D. Goodno, “Coherent Combining of a 1.26-kW Fiber Amplifier,” Advanced in Solid-State Photonics, paper AMA2 (2010).
- Y. Kim, S. Kim, Y.-J. Kim, H. Hussein, and S.-W. Kim, “Er-doped fiber frequency comb with mHz relative linewidth,” Opt. Express 17(14), 11972–11977 (2009). [CrossRef] [PubMed]
- N. Nishizawa and J. Takayanagi, “Octave spanning high-quality supercontinuum generation in all-fiber system,” J. Opt. Soc. Am. B 24(8), 1786–1792 (2007). [CrossRef]
- N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs,” J. Opt. Soc. Am. B 24(8), 1756–1770 (2007). [CrossRef]
- M. Sköld, J. Yang, H. Sunnerud, M. Karlsson, S. Oda, and P. A. Andrekson, “Constellation diagram analysis of DPSK signal regeneration in a saturated parametric amplifier,” Opt. Express 16(9), 5974–5982 (2008). [CrossRef] [PubMed]
- M. Skold, M. Karlsson, S. Oda, H. Sunnerud, and P. A. Andrekson, “Constellation diagram measurements of induced phase noise in a regenerating parametric amplifier,” Optical Fiber Communications Conference, paper OML4 (2008).
- M. Matsumoto, “Phase noise generation in an amplitude limiter using saturation of a fiber-optic parametric amplifier,” Opt. Lett. 33(15), 1638–1640 (2008). [CrossRef] [PubMed]
- R. Elschner, and L. Petermann, “Impact of Pump-Induced Nonlinear Phase Noise on Parametric Amplification and Wavelength Conversion of Phase-Modulated Signals,” European Conference in Optical Communications, paper 3.3.4 (2009).
- E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley-Interscience, 2002).
- M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators, and Related Devices (Cambridge University Press, 2008).
- N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical preamplifier,” Opt. Express 13(12), 4568–4579 (2005). [CrossRef] [PubMed]
- R. H. Stolen and J. E. Bjorkholm, “Parametric Amplification and Frequency Conversion in Optical Fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, 2007), Chap. 10.
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