## Alleviation of additional phase noise in fiber optical parametric amplifier based signal regenerator |

Optics Express, Vol. 20, Issue 24, pp. 27254-27264 (2012)

http://dx.doi.org/10.1364/OE.20.027254

Acrobat PDF (1692 KB)

### Abstract

We theoretically and numerically explain the power saturation and the additional phase noise brought by the fiber optical parametric amplifier (FOPA). An equation to calculate an approximation to the saturated signal output power is presented. We also propose a scheme for alleviating the phase noise brought by the FOPA at the saturated state. In simulation, by controlling the decisive factor dispersion difference term Δ*k* of the FOPA, amplitude-noise and additional phase noise reduction of quadrature phase shift keying (QPSK) based on the saturated FOPA is studied, which can provide promising performance to deal with PSK signals.

© 2012 OSA

## 1. Introduction

1. M. Matsumoto, “Fiber-based all-optical signal regeneration,” IEEE J. Sel. Top. Quantum Electron. **18**(2), 738–752 (2012). [CrossRef]

2. M. Gao, J. Kurumida, and S. Namiki, “Wide range operation of regenerative optical parametric wavelength converter using ASE-degraded 43-Gb/s RZ-DPSK signals,” Opt. Express **19**(23), 23258–23270 (2011). [CrossRef] [PubMed]

4. C. S. Brès, A. O. J. Wiberg, J. Coles, and S. Radic, “160-Gb/s optical time division multiplexing and multicasting in parametric amplifiers,” Opt. Express **16**(21), 16609–16615 (2008). [PubMed]

5. P. O. Hedekvist and P. A. Anderson, “Noise characteristics of fiber-based optical phase conjugators,” J. Lightwave Technol. **17**(1), 74–79 (1999). [CrossRef]

7. S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. **15**(7), 957–959 (2003). [CrossRef]

8. 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]

9. P. Kylemark, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Semi-analytic saturation theory of fiber optical parametric amplifiers,” J. Lightwave Technol. **24**(9), 3471–3479 (2006). [CrossRef]

10. S. Watanabe, F. Futami, R. Okabe, R. Ludwig, C. Schmidt-Langhorst, B. Huettl, C. Schubert, and H. Weber, “An optical parametric amplified fiber switch for optical signal processing and regeneration,” J. Sel. Top. Quantum Electron. **14**(3), 674–680 (2008). [CrossRef]

8. 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]

11. 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]

## 2. Signal output power in a saturated FOPA

9. P. Kylemark, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Semi-analytic saturation theory of fiber optical parametric amplifiers,” J. Lightwave Technol. **24**(9), 3471–3479 (2006). [CrossRef]

*E*

_{0},

*E*

_{1}, and

*E*

_{2}are the electric field complex amplitudes of the pump power, and two sideband amplitudes, respectively. Δ

*k*,

*γ*, and

*z*are the propagation constant mismatch, third order nonlinear parameter, and propagated distance in the fiber, where Δ

*k*=

*k*

_{1}+

*k*

_{2}−2

*k*

_{0}. Since the total power

*P*= |

*E*

_{0}|

^{2}+ |

*E*

_{1}|

^{2}+ |

*E*

_{2}|

^{2}is conserved as we ignore the loss in the fiber, Eq. (1) can be conveniently rewritten in terms of normalized dimensionless amplitude variables. To this end, the normalized pump power

*η*(

*z*)≡|

*E*

_{0}|

^{2}/

*P*and the normalized signal and idler amplitudes

*a*

_{1,2}≡|

*E*

_{1,2}|/

*P*

^{1/2}are introduced. The phase matching condition is defined as

*ϕ*(

*z*) = Δ

*kz*+

*ϕ*

_{1}(

*z*) +

*ϕ*

_{2}(

*z*)−2

*ϕ*

_{0}(

*z*). Kylemark et al. [9

9. P. Kylemark, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Semi-analytic saturation theory of fiber optical parametric amplifiers,” J. Lightwave Technol. **24**(9), 3471–3479 (2006). [CrossRef]

13. G. Cappellini and S. Trillo, “Third-order three-wave mixing in singlemode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B **8**(4), 824–838 (1991). [CrossRef]

*η*,

*a*

_{1}, and

*a*

_{2}are the normalized pump power, and the two normalized sideband amplitudes, respectively.

*ξ*=

*zγP*is the normalized propagation length, and

*κ*= Δ

*k*/

*γP*is the normalized phase-matching term. In Ref [13

13. G. Cappellini and S. Trillo, “Third-order three-wave mixing in singlemode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B **8**(4), 824–838 (1991). [CrossRef]

10. S. Watanabe, F. Futami, R. Okabe, R. Ludwig, C. Schmidt-Langhorst, B. Huettl, C. Schubert, and H. Weber, “An optical parametric amplified fiber switch for optical signal processing and regeneration,” J. Sel. Top. Quantum Electron. **14**(3), 674–680 (2008). [CrossRef]

*a*,

*b*,

*c*, and

*d*, are the roots of equation d

*η*/d

*ξ*= 0 (Eq. (2a) in Ref [10

10. S. Watanabe, F. Futami, R. Okabe, R. Ludwig, C. Schmidt-Langhorst, B. Huettl, C. Schubert, and H. Weber, “An optical parametric amplified fiber switch for optical signal processing and regeneration,” J. Sel. Top. Quantum Electron. **14**(3), 674–680 (2008). [CrossRef]

*a > b*>

*η*(

*z*) ≥

*c > d*. From Eq. (3) it can be known that the pump power at the output is a periodic function of the normalized propagated length

*ξ*with periodicity of the sn function. When the total power

*P*is unchanged in the fiber, the power will be transferred between the signal/idler and the pump periodically.

*a*,

*b*,

*c*, and

*d*are too complicated to calculate the saturated signal output power from this equation. However, in many applications of saturated phase-insensitive FOPA as an amplitude limiter, the results can be simplified, where the input pump power is very large compared with the power of the input signal. In this paper, we choose fiber with

*γ*= 12/W/km and dispersion slope d

*D*/d

*λ*= 0.03 ps/nm

^{2}/km. The difference between the pump and the zero-dispersion wave-length is

*λ*

_{p}−

*λ*

_{0}= 3 nm. Pump power

*P*

_{0}is 400 mW. The effect of fiber loss is neglected. Signal frequency is set at the peak position in the unsaturated gain spectra, Δ

*ν*= (−2

*γP*

_{0}/

*k*

_{2})1/2/(2π), where Δ

*ν*and

*k*

_{2}are the frequency separation between the signal and the pump and the group-velocity dispersion (GVD) coefficient at the pump wavelength, respectively.

*α*=

*a*

_{1}

^{2}−

*a*

_{1}

^{2}between the input signal and idler approaches zero, and

*a*,

*b*,

*c*, and

*d*can be simplified to

*H*= 4

*ηa*

_{1}

*a*

_{2}cos

*ϕ*−(

*κ*−1)

*η*−3

*η*

^{2}/2 is the Hamiltonian of the degenerate FWM system, which is determined by the initial conditions. Because the pump power only changes in the range of

*bP*and

*cP*, and

*c*corresponds to the saturated length, the saturated output signal power can be written aswhere

*P*is the system power, Δ

*k*is the dispersion difference, and

*γ*is the third-order nonlinear parameter.

## 3. Nonlinear phase shift and phase noise

*ϕ*is equal to π/2 by the classical theory, and cos

*ϕ*= 0. If the signal is amplified to a power comparable (tens of mWs) with that of the pump, the last term with cos

*ϕ*cannot be ignored. The FWM affects not only the power of signal, but the phase of it. We can rewrite Eq. (6) thus asFrom this equation, we can see that the nonlinear phase shift can be considered as the equivalent of a background phase shift (BPS) with the value 2

*γPL*, and a combination of SPM and FWM.

*ϕ*

_{1}by integrating Eq. (7) is too complicated to be expressed. However, because of the peoriodicity in such a system with perturbation, there is a simple method for us to analysis the phase noise introduced by the power fluctuation from input signals. For example, as shown in Fig. 4(a) , consider the power evolution of the signals with input power 0.35 mW (red solid line) and input power 0.05 mW (green dashed line) in the fiber mentioned above. If we move the dashed line to the left, it can be found that the two lines coincide with each other except the lengths in the right-red and left-green rectangle parts, which are defined as

*L*and

_{phaseG}*L*respectively (shown in Fig. 4(b)). It is obvious that

_{phaseR}*L*=

_{phaseG}*L*. Because the pump power is the same (400 mW) and much larger than the signal (0.05 mW and 0.35 mW), the two FWM processes with the same signal power (the coincided part in Fig. 4(b)) will have nearly same pump power and idler power. During the distance where all the powers (pump, signal, and idler) are the same, the FWM processes will bring the same nonlinear phase shift to the signals. Under such simplification, we can calculate the nonlinear phase difference at the end of the fiber.

_{phaseR}*P*

_{1}= Δ

*P*in the input signal power, the phase shift difference Δ

*ϕ*

_{1}can be expressed asBecause of the deep saturation over the length

*L*, the energy transfer direction will change at this area. The phase-matching term

_{phaseR}*ϕ*is near zero, so cos

*ϕ*≈1,

*A*

_{2}≈

*A*

_{1}. In the

*L*part, the phase is matched, so cos

_{phaseG}*ϕ*≈0; and compared with

*P*,

*A*

_{1}≈0 and

*A*

_{2}≈0. Using these approximations, the Δ

*ϕ*

_{1}can be rewritten as

*A*

_{1}and

*A*

_{0}can be further simplified as

*A*

_{1}(

*L*) and

*A*

_{0}(

*L*) further, where

*L*is the length of the fiber. Here, it is easy to see that the nonlinear phase noise is still composed of a background phase noise, and the noise from SPM and FWM.

## 4. Phase noise alleviation

*k*= −2

*γP*

_{0}≈−0.096 /m is perfect. The nonlinear phase shift is defined by the difference between the phase of the signal after and before the FOPA (assuming that the input signal phase is zero). Under this parametric gain saturation condition, the FOPA works as a limiter amplifier, suppressing the amplitude noise of the signal. However, as the red dashed line shown in Fig. 5 , the nonlinear phase shift from the FOPA increases with the input power.

*ϕ*

_{1}should approach 0. By the analysis in Section 2 and 3, wihout integration, we try to find the condition under which Δ

*ϕ*

_{1}= 0.

*ϕ*

_{1}= 0, the term 3

*κ*+ 5 should be larger than zero,

*κ*< −5/3. Furthermore, from Eq. (4c), there is a ouput power of the signal only if

*κ*> −4.

*κ*. If the phase noise can be alleviated,

*κ*should be in a interval that

*k*, we can find a specific condition that alleviating the phase noise while limiting the amplitude at the same time.

*k*≈−0.0175 /m.

## 5. Simulation

*λ*

_{S}. The signal and pump were input into the HNLF via an optical coupler with their state of polarization (SOP) properly controlled by automatic polarization controllers. The HNLF has a zero-dispersion wavelength at

*λ*

_{0}= 1559 nm, dispersion slope d

*D*/d

*λ*= 0.03ps/nm

^{2}/km. and a nonlinear coefficient of

*γ*= 12/W/km. The wavelength of the pump is set to

*λ*

_{P}=

*λ*

_{0}+ 3nm, while a 100 GSymbols/s QPSK signal has wavelength

*λ*

_{S}= 1550 nm. The signal passes through an optical signal-to-noise ratio (OSNR) controller comprising an optical attenuator and an EDFA. Then, the signal enters the FOPA together with the pump, which is a continuous wave. (We can also use the 100 GHz single polarization pulse train as the pump, which is synchronized with the signal). The output signal of the FOPA was transmitted over a 1nm filter and is detected by the coherent receiver. In this paper, the simulation is based on the fully numerical solution of the nonlinear Schrödinger equation (NLSE). The numerical calculation assumes a non-return-to-zero (NRZ) PSK format for the input signal, and white Gaussian noise is added to the complex amplitudes of both signal and pump.

*M*-PSK signal, especially for the signal with low SNR. We compare our scheme with the theory in Ref [6

6. P. Kylemark, P. O. Hedekvist, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Noise characteristics of fiber optical parametric amplifiers,” J. Lightwave Technol. **22**(2), 409–416 (2004). [CrossRef]

## 5. Conclusion

*k*is one decisive factor for the saturated FOPA. The FOPA can be optimized by controlling the dispersion relation between the pump, signal and idler, and this provides the means to deal with PSK signals.

## References and links

1. | M. Matsumoto, “Fiber-based all-optical signal regeneration,” IEEE J. Sel. Top. Quantum Electron. |

2. | M. Gao, J. Kurumida, and S. Namiki, “Wide range operation of regenerative optical parametric wavelength converter using ASE-degraded 43-Gb/s RZ-DPSK signals,” Opt. Express |

3. | G. K. P. Lei, C. Shu, and H. K. Tsang, “Amplitude noise reduction, pulse format conversion, and wavelength multicast of PSK signal in a fiber optical parametric amplifier,” National Fiber Optics Engineers Conference (NFOEC), JW2A.79, Mar. 2012. |

4. | C. S. Brès, A. O. J. Wiberg, J. Coles, and S. Radic, “160-Gb/s optical time division multiplexing and multicasting in parametric amplifiers,” Opt. Express |

5. | P. O. Hedekvist and P. A. Anderson, “Noise characteristics of fiber-based optical phase conjugators,” J. Lightwave Technol. |

6. | P. Kylemark, P. O. Hedekvist, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Noise characteristics of fiber optical parametric amplifiers,” J. Lightwave Technol. |

7. | S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. |

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

9. | P. Kylemark, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Semi-analytic saturation theory of fiber optical parametric amplifiers,” J. Lightwave Technol. |

10. | S. Watanabe, F. Futami, R. Okabe, R. Ludwig, C. Schmidt-Langhorst, B. Huettl, C. Schubert, and H. Weber, “An optical parametric amplified fiber switch for optical signal processing and regeneration,” J. Sel. Top. Quantum Electron. |

11. | 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 |

12. | G. Agrawal, |

13. | G. Cappellini and S. Trillo, “Third-order three-wave mixing in singlemode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B |

14. | R. Elschner and K. Petermann, “Impact of pump-induced nonlinear phase noise on parametric amplification and wavelength conversion of phase modulated signals,” in Proc. Eur. Conf. Opt. Commun. (ECOC), Sep. 2009, Paper. |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(200.6015) Optics in computing : Signal regeneration

**ToC Category:**

Fiber Optical Parametric Amplifiers

**History**

Original Manuscript: July 11, 2012

Revised Manuscript: August 31, 2012

Manuscript Accepted: September 1, 2012

Published: November 19, 2012

**Virtual Issues**

Nonlinear Photonics (2012) *Optics Express*

**Citation**

Lei Jin, Bo Xu, and Shinji Yamashita, "Alleviation of additional phase noise in fiber optical parametric amplifier based signal regenerator," Opt. Express **20**, 27254-27264 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-27254

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### References

- M. Matsumoto, “Fiber-based all-optical signal regeneration,” IEEE J. Sel. Top. Quantum Electron.18(2), 738–752 (2012). [CrossRef]
- M. Gao, J. Kurumida, and S. Namiki, “Wide range operation of regenerative optical parametric wavelength converter using ASE-degraded 43-Gb/s RZ-DPSK signals,” Opt. Express19(23), 23258–23270 (2011). [CrossRef] [PubMed]
- G. K. P. Lei, C. Shu, and H. K. Tsang, “Amplitude noise reduction, pulse format conversion, and wavelength multicast of PSK signal in a fiber optical parametric amplifier,” National Fiber Optics Engineers Conference (NFOEC), JW2A.79, Mar. 2012.
- C. S. Brès, A. O. J. Wiberg, J. Coles, and S. Radic, “160-Gb/s optical time division multiplexing and multicasting in parametric amplifiers,” Opt. Express16(21), 16609–16615 (2008). [PubMed]
- P. O. Hedekvist and P. A. Anderson, “Noise characteristics of fiber-based optical phase conjugators,” J. Lightwave Technol.17(1), 74–79 (1999). [CrossRef]
- P. Kylemark, P. O. Hedekvist, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Noise characteristics of fiber optical parametric amplifiers,” J. Lightwave Technol.22(2), 409–416 (2004). [CrossRef]
- S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett.15(7), 957–959 (2003). [CrossRef]
- 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]
- P. Kylemark, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Semi-analytic saturation theory of fiber optical parametric amplifiers,” J. Lightwave Technol.24(9), 3471–3479 (2006). [CrossRef]
- S. Watanabe, F. Futami, R. Okabe, R. Ludwig, C. Schmidt-Langhorst, B. Huettl, C. Schubert, and H. Weber, “An optical parametric amplified fiber switch for optical signal processing and regeneration,” J. Sel. Top. Quantum Electron.14(3), 674–680 (2008). [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. Express16(9), 5974–5982 (2008). [CrossRef] [PubMed]
- G. Agrawal, Nonlinear Fiber Optics, 4th ed.(Aademic Press, 2007) Chap. 10.
- G. Cappellini and S. Trillo, “Third-order three-wave mixing in singlemode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B8(4), 824–838 (1991). [CrossRef]
- R. Elschner and K. Petermann, “Impact of pump-induced nonlinear phase noise on parametric amplification and wavelength conversion of phase modulated signals,” in Proc. Eur. Conf. Opt. Commun. (ECOC), Sep. 2009, Paper.

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