## Moment-generating function method used to accurately evaluate the impact of the linearized optical noise amplified by EDFAs |

Optics Express, Vol. 22, Issue 6, pp. 6620-6633 (2014)

http://dx.doi.org/10.1364/OE.22.006620

Acrobat PDF (801 KB)

### Abstract

In a nonlinear optical fiber communication (OFC) system with signal power much stronger than noise power, the noise field in the fiber can be described by linearized noise equation (LNE). In this case, the noise impact on the system performance can be evaluated by moment-generating function (MGF) method. Many published MGF calculations were based on the LNE using continuous wave (CW) approximation, where the modulated signal needs to be artificially simplified as an unmodulated signal. Results thus obtained should be treated carefully. Reliable results can be obtained by replacing the CW-based LNE with the accurate LNE proposed by Holzlöhner *et al* in Ref. [

© 2014 Optical Society of America

## 1. Introduction

1. R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, and W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. **20**, 389–400 (2002). [CrossRef]

8. M. Secondini, E. Forestieri, and C. R. Menyuk, “A combined regular-logarithmic perturbation method for signal-noise interaction in amplified optical systems,” J. Lightwave Technol. **27**, 3358–3369 (2009). [CrossRef]

1. R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, and W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. **20**, 389–400 (2002). [CrossRef]

6. A. Demir, “Nonlinear phase noise in optical-fiber-communication systems,” J. Lightwave Technol. **25**, 2002–2032 (2007). [CrossRef]

8. M. Secondini, E. Forestieri, and C. R. Menyuk, “A combined regular-logarithmic perturbation method for signal-noise interaction in amplified optical systems,” J. Lightwave Technol. **27**, 3358–3369 (2009). [CrossRef]

8. M. Secondini, E. Forestieri, and C. R. Menyuk, “A combined regular-logarithmic perturbation method for signal-noise interaction in amplified optical systems,” J. Lightwave Technol. **27**, 3358–3369 (2009). [CrossRef]

1. R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, and W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. **20**, 389–400 (2002). [CrossRef]

**20**, 389–400 (2002). [CrossRef]

**20**, 389–400 (2002). [CrossRef]

10. R. Holzlöhner, C. R. Menyuk, and W. L. Kath, “Efficient and accurate computation of eye diagrams and bit error rates in a single-channel CRZ system,” IEEE Photon. Technol. Lett. **14**, 1079–1081 (2002). [CrossRef]

11. R. Holzlöhner, C. R. Menyuk, and W. L. Kath, “A covariance matrix method to compute bit error rates in a highly nonlinear dispersion-managed soliton system,” IEEE Photon. Technol. Lett. **15**, 688–690 (2003). [CrossRef]

**20**, 389–400 (2002). [CrossRef]

10. R. Holzlöhner, C. R. Menyuk, and W. L. Kath, “Efficient and accurate computation of eye diagrams and bit error rates in a single-channel CRZ system,” IEEE Photon. Technol. Lett. **14**, 1079–1081 (2002). [CrossRef]

11. R. Holzlöhner, C. R. Menyuk, and W. L. Kath, “A covariance matrix method to compute bit error rates in a highly nonlinear dispersion-managed soliton system,” IEEE Photon. Technol. Lett. **15**, 688–690 (2003). [CrossRef]

**20**, 389–400 (2002). [CrossRef]

10. R. Holzlöhner, C. R. Menyuk, and W. L. Kath, “Efficient and accurate computation of eye diagrams and bit error rates in a single-channel CRZ system,” IEEE Photon. Technol. Lett. **14**, 1079–1081 (2002). [CrossRef]

12. R. Holzlöhner and C. R. Menyuk, “Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems,” Opt. Lett. **28**, 1894–1896 (2003). [CrossRef] [PubMed]

**20**, 389–400 (2002). [CrossRef]

13. A. Demir, “Non-Monte Carlo formulations and computational techniques for the stochastic nonlinear Schrodinger equation,” J. Comput. Phys. **201**, 148–171 (2004). [CrossRef]

13. A. Demir, “Non-Monte Carlo formulations and computational techniques for the stochastic nonlinear Schrodinger equation,” J. Comput. Phys. **201**, 148–171 (2004). [CrossRef]

13. A. Demir, “Non-Monte Carlo formulations and computational techniques for the stochastic nonlinear Schrodinger equation,” J. Comput. Phys. **201**, 148–171 (2004). [CrossRef]

**20**, 389–400 (2002). [CrossRef]

**14**, 1079–1081 (2002). [CrossRef]

11. R. Holzlöhner, C. R. Menyuk, and W. L. Kath, “A covariance matrix method to compute bit error rates in a highly nonlinear dispersion-managed soliton system,” IEEE Photon. Technol. Lett. **15**, 688–690 (2003). [CrossRef]

## 2. Noise propagator obtained from accurate LNE

*L*can be obtained from the accurate LNE [1

**20**, 389–400 (2002). [CrossRef]

### 2.1. Noise propagator in a fiber of length L

*L*can be obtained by extending the RK4IP in Refs. [14

14. J. Hult, “A fourth-order Runge-Kutta in the interaction picture method for simulating supercontinuum generation in optical fibers,” J. Lightwave Technol. **25**, 3770–3775 (2007). [CrossRef]

15. Z. Zhang, L. Chen, and X. Bao, “A fourth-order Runge-Kutta in the interaction picture method for numerically solving the coupled nonlinear Schrödinger equation,” Opt. Express **18**, 8261–8276 (2010). [CrossRef] [PubMed]

**20**, 389–400 (2002). [CrossRef]

*L̂*is associated with CD effect, whereas the nonlinear operator

*N̂*is caused by Kerr nonlinearity.] By introducing

*ã*=

*e*

^{L̂(z−z0)}

*ã*and

^{I}*N̂*=

^{I}*e*

^{−L̂(z−z0)}

*N̂e*

^{L̂(z−z0)}, the accurate LNE in the interaction picture (IP) has the form

*z*

_{0}=

*z*+

_{n}*h*/2 with step size

*h*=

*z*

_{n}_{+1}−

*z*and denoting

_{n}*ã*=

_{n}*ã*(

*z*),

_{n}*ã*

_{n}_{+1}=

*ã*(

*z*

_{n}_{+1}),

14. J. Hult, “A fourth-order Runge-Kutta in the interaction picture method for simulating supercontinuum generation in optical fibers,” J. Lightwave Technol. **25**, 3770–3775 (2007). [CrossRef]

15. Z. Zhang, L. Chen, and X. Bao, “A fourth-order Runge-Kutta in the interaction picture method for numerically solving the coupled nonlinear Schrödinger equation,” Opt. Express **18**, 8261–8276 (2010). [CrossRef] [PubMed]

*h*=

*z*

_{n}_{+1}−

*z*can be calculated as For the fiber of length

_{n}*L*, the noise propagator has the form

15. Z. Zhang, L. Chen, and X. Bao, “A fourth-order Runge-Kutta in the interaction picture method for numerically solving the coupled nonlinear Schrödinger equation,” Opt. Express **18**, 8261–8276 (2010). [CrossRef] [PubMed]

*k̂*(

_{i}*i*= 2, 3, 4) in Eq. (2) needs one dense matrix multiplication. Here

*N*is the number of Fourier components used for signal representation.

_{w}### 2.2. Equivalent noise propagator of a multi-span system

*L*can be written as [1

**20**, 389–400 (2002). [CrossRef]

5. P. Serena, A. Orlandini, and A. Bononi, “Parametric-gain approach to the analysis of single-channel DPSK/DQPSK systems with nonlinear phase noise,” J. Lightwave Technol. **24**, 2026–2037 (2006). [CrossRef]

7. L. D. Coelho, L. Molle, D. Gross, N. Hanik, R. Freund, C. Caspar, E.-D. Schmidt, and B. Spinnler, “Modeling nonlinear phase noise in differentially phase-modulated optical communication systems,” Opt. Express **17**, 3226–3241 (2009). [CrossRef] [PubMed]

**27**, 3358–3369 (2009). [CrossRef]

*I*is a unit matrix and

*p*. In Eq. (6), with

_{n}*G*being the EDFA gain shown in Fig. 1, the variance of the real or imaginary part of input ASE can be expressed as [cf. Ref. [16

16. E. Forestieri, “Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering,” J. Lightwave Technol. **18**, 1493–1503 (2000). [CrossRef]

*K*-span system consisting of (

*K*+ 1) EDFAs, as shown in Fig. 1, its PG has the form

*P*can be obtained either by using Cholesky decomposition or symmetric (square root) decomposition [8

_{n,eq}**27**, 3358–3369 (2009). [CrossRef]

## 3. MGF calculation

*I*(

*t*

_{s}) expressed using Karhunen-Loève series expansion (KLSE).

*Z̃*〉

*(*

_{i}*i*= 1, ···, 4

*M*+ 2) with formula [16

_{n}16. E. Forestieri, “Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering,” J. Lightwave Technol. **18**, 1493–1503 (2000). [CrossRef]

17. Z. Zhang, L. Chen, and X. Bao, “Accurate BER evaluation for lumped DPSK and OOK systems with PMD and PDL,” Opt. Express **15**, 9418–9433 (2007). [CrossRef] [PubMed]

*I*(

*t*

_{s}) is the filtered photoelectric current at time

*t*

_{s}. It consists of signal-signal beating (

*y*), noise-noise beating (

_{ss}*y*), and signal-noise beating (

_{nn}*y*). In Eq. (11),

_{ns}*b̃*(

_{i}*t*

_{s}) is the

*i*th component of |

*b̃*(

*t*

_{s})〉 detailed at the end of Appendix B, while

*λ̃*is the power of

_{i}*i*th component of the noise in Karhunen-Loève presentation. In this work, we take

*ξ*= 1/2 for polarized noise.

## 4. OSNR at the receiver

*B*(0.1nm) can be calculated as In Eq. (12),

_{r}*P̄*is the time-averaged (noise free) signal power, while

_{s}*P*(

_{ASE}*B*) is the noise power within

_{r}*B*. To obtain

_{r}*P̄*and

_{s}*P*(

_{ASE}*B*), one needs to notice that the measurement bandwidth

_{r}*B*[e.g., the bandwidth of the transfer function of an optical spectrum analyzer (OSA)] may not be the same as

_{m}*B*. Thus the

_{r}*P̄*in Eq. (12) becomes the power of the signal filtered by

_{s}*B*, while

_{m}*P*(

_{ASE}*B*) becomes the ASE filtered by

_{r}*B*and weighted by a factor

_{m}*B*[18].

_{r}/B_{m}*P*(

_{ASE}*B*) is the ASE power within

_{m}*B*and

_{m}*N*

_{0}is given by Eq. (22). In Eq. (13), the filter (

*B*) effect on the ASE has been neglected.

_{m}*O*is the low-pass transfer function of the bandpass filter (bandwidth

_{m}*B*). In Eq. (14),

_{m}## 5. Applications to DPSK systems

### 5.1. Comparison with CMM results

5. P. Serena, A. Orlandini, and A. Bononi, “Parametric-gain approach to the analysis of single-channel DPSK/DQPSK systems with nonlinear phase noise,” J. Lightwave Technol. **24**, 2026–2037 (2006). [CrossRef]

5. P. Serena, A. Orlandini, and A. Bononi, “Parametric-gain approach to the analysis of single-channel DPSK/DQPSK systems with nonlinear phase noise,” J. Lightwave Technol. **24**, 2026–2037 (2006). [CrossRef]

**24**, 2026–2037 (2006). [CrossRef]

*N*in our Fig. 1. Thus the first term of

_{in}**24**, 2026–2037 (2006). [CrossRef]

19. P. Serena, A. Bononi, J. C. Antona, and S. Bigo, “Parametric gain in the strongly nonlinear regime and its impact on 10-Gb/s NRZ systems with forward-error correction,” J. Lightwave Technol. **23**, 2352–2363 (2006). [CrossRef]

*N*= 0 and (

_{in}*K*+ 1) being replaced by

*K*. According to Ref. [5

**24**, 2026–2037 (2006). [CrossRef]

*n*in Eq. (22). As plotted in Fig. 1, each span contains a transmission fiber followed by a dispersion-compensating fiber (DCF). The transmission fiber is

_{sp}*l*=100 km long with its CD parameter

*D*= 8 ps/nm/km. Each span is fully compensated. The nonlinear phase accumulated in the fiber, defined as

_{tx}*P*being the time averaged signal power (at the input of the fiber), is 0.2

_{in}*π*. The bandwidth of the optical (electrical) filter in the receiver is

*B*= 1.8

_{o}*R*(

_{b}*B*= 0.65

_{e}*R*), respectively.

_{b}**24**, 2026–2037 (2006). [CrossRef]

**24**, 2026–2037 (2006). [CrossRef]

*h*=3.5 km. The CPU time for each BER calculation is ∼ 0.5 hr. In fact,

_{tr}*h*ranged within 0.3 ∼ 5 km yields almost the same curve.

_{tr}### 5.2. Comparison with experimental data

7. L. D. Coelho, L. Molle, D. Gross, N. Hanik, R. Freund, C. Caspar, E.-D. Schmidt, and B. Spinnler, “Modeling nonlinear phase noise in differentially phase-modulated optical communication systems,” Opt. Express **17**, 3226–3241 (2009). [CrossRef] [PubMed]

21. E. Ip and J. M. Kahn, “Power spectra of return-to-zero optical signals,” J. Lightwave Technol. **24**, 1610–1618 (2006). [CrossRef]

7. L. D. Coelho, L. Molle, D. Gross, N. Hanik, R. Freund, C. Caspar, E.-D. Schmidt, and B. Spinnler, “Modeling nonlinear phase noise in differentially phase-modulated optical communication systems,” Opt. Express **17**, 3226–3241 (2009). [CrossRef] [PubMed]

*N*is filtered by

_{in}*O*, the OSNR is calculated using Eq. (14) with

_{in}*B*= 0.95, yielding the back-to-back RK4IP curve shown in Fig. 3.

_{m}/B_{r}**17**, 3226–3241 (2009). [CrossRef] [PubMed]

*P*=

_{in}*P*is smaller than

_{SMF}*E*, where

_{b}/T_{b}*E*is the energy per bit before the spectral modification. For example, to get nonlinear phase Φ̄

_{b}*= 0.9 for the 25-span system, the fiber input power*

_{N}*P*=

_{in}*P*should be 1.516 mW, which means

_{SMF}*E*=0.127 mW or

_{b}/T_{b}*G*(

*E*) = 2.316 mW (

_{b}/T_{b}*G*= 18.197). Different from the DPSK receiver shown in Fig. 1, where the delay is

*T*= 1/

_{b}*R*= 50 ps, the delay in the receiver of Ref. [7

_{b}**17**, 3226–3241 (2009). [CrossRef] [PubMed]

*T′*=(24.84 GHz)

_{b}^{−1}=40.26 ps. Thus, the DPSK phase factors given in Eq. (35) should be modified as with

*N′*=

*N*(

*T*),

_{b}/T′_{b}*T′*=

_{b}*T*+ Δ

_{b}*T*. In Eq. (16), Δ is introduced as where <

_{b}*δϕ*>, given by Eq. (41) in Appendix C, is the nonlinear phase difference between noise and noise-free signal. As shown in Fig. 3, all RK4IP curves (

*ϕ*

_{0}= 0.31) agree very well with the experiment results. The ASE power is calculated using Eq. (15) with Δ

*λ*= 2

*B*. In Eq. (17),

_{m}*ϕ*

_{0}is a calibration constant that basically shifts the RK4IP curves in the OSNR direction, while <

*δϕ*> determines the slope of the RK4IP curves. To show this, we plot in Fig. 4 the RK4IP results for the 25-span system with Δ = 0.31− <

*δϕ*> and Δ = 0. Also, we consider the RK4IP curves using Eq. (14) to calculate ASE power. Our results for the 5-span, 10-span, and 25-span systems confirm that there is almost no difference between the curve using Eq. (15) with

*ϕ*

_{0}= 0.31 and the curve using Eq. (14) with

*ϕ*

_{0}= 0.57.

*h*=6.0 km. The CPU time for each BER calculation is ∼ 0.5 hr. The step size

_{tr}*h*within 0.3 ∼ 10 km will result in almost the same curve.

_{tr}## 6. Summary

^{−9}) but also it can provide reliable information for the cases using coherent detection, which is now widely used in modern OFC systems. It is now well recognized that traditional Gaussian fitting Q-factor approximation is accurate for OOK detection, while MGF method is accurate for various linear OFC systems.

**24**, 2026–2037 (2006). [CrossRef]

**24**, 2026–2037 (2006). [CrossRef]

**17**, 3226–3241 (2009). [CrossRef] [PubMed]

## Appendix A: Accurate LNE in the EDFA-based systems

*u*(

*z*,

*t*) in a fiber satisfies where

*α*is the fiber loss and

*β*=

_{ωω}*∂*

^{2}

*β/∂ω*

^{2}relates to the CD parameter

*D*(

*λ*) (ps/nm/km) with

^{8}m/s). The slope parameter

^{2}·km)] can be neglected if bit rate

*R*satisfies

_{b}*R*> |

_{b}*β*| [16

_{ωω}/β_{ωωω}16. E. Forestieri, “Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering,” J. Lightwave Technol. **18**, 1493–1503 (2000). [CrossRef]

*u*(

*z*,

*t*) =

*v*(

*z*,

*t*)

*e*

^{−}

^{αz}^{/2}, Eq. (18) can be reduced as In a

*K*-span system amplified by (

*K*+ 1) EDFAs (cf. Fig. 1), Eq. (19) can be modified as where

*w*(

*z*,

*t*) is the ASE forcing modeled as the complex AWGN with correlation In Eq. (21), the fiber length in each span is assumed to be

*L*(km) long, According to Wiener-Khintchine theorem [22],

*N*

_{0}

*in Eq. (21) is the ASE PSD (in one polarization direction) at the output of the*

_{k}*k*th EDFA. Suppose each EDFA has the same gain

*G*and spontaneous-emission parameter

*n*, we have [16

_{sp}**18**, 1493–1503 (2000). [CrossRef]

*v*(

*z*,

*t*) in the fiber into noise-free field

*v*

_{0}(

*z*,

*t*) and its perturbation

*δv*(

*z*,

*t*) [i.e.,

*v*(

*z*,

*t*) =

*v*

_{0}(

*z*,

*t*) +

*δv*(

*z*,

*t*)] and assuming that |

*v*

_{0}| >> |

*δv*| (so that the nonlinear terms of

*δv*can be neglected), Eq. (20) can be decomposed as [1

**20**, 389–400 (2002). [CrossRef]

**20**, 389–400 (2002). [CrossRef]

*a*=

_{l}*a*(

*ω*) = ∫

_{l}*δve*

^{−jωlt}

*dt*and the circulant matrices [

*M*]

_{ν}*=*

_{lm}*ν*

_{l}_{−}

*, [*

_{m}*M*]

_{μ}*=*

_{lm}*μ*

_{l}_{+}

*with in frequency domain, Eq. (24) has the form where*

_{m}*W*=

_{l}*W*(

*z*,

*ω*) is the Fourier component of the forcing term

_{l}*w*(

*z*,

*t*) in Eq. (20). As indicated in Ref. [1

**20**, 389–400 (2002). [CrossRef]

*v*

_{0}|

^{2}in (25) is real,

*M*in (26) is Hermitian, or, its real part

_{ν}*M*]

_{μ}*=*

_{km}*μ*

_{k}_{+}

*, both the real (*

_{m}*M*are symmetric.

_{μ}*ã*= (

*a*,

_{R}*a*)

_{I}*(for*

^{T}*a*=

*a*+

_{R}*ja*) and

_{I}*W̃*= (

*W*−

_{I},*W*)

_{R}*(for −*

^{T}*jW*=

*W*−

_{I}*jW*), Eq. (27) is equivalent to with

_{R}*ν̂*(

*μ̂*) in Eq. (30) is antisymmetric (symmetric), respectively. Calculation of the Kerr term (

*ν̂*+

*μ̂*) according to Eq. (30) has the computational complexity much less than

*N*is the number of Fourier components used for signal representation. In fact, the computational cost of this way is basically determined by the FFTs in Eq. (25), which has the computational complexity of

_{W}*O*(

*N*

_{W}*logN*).

_{W}*T*

_{0}= 1/Δ

*f*and

*M*are given by Eq. (36). Eq. (31) means that Eq. (29) can be equivalently replaced by with boundary condition [23

_{n}23. E. Forestieri and G. Prati, “Exact analytical evaluation of second-order PMD impact on the outage probability for a compensated system,” J. Lightwave Technol. **22**, 988–996 (2004). [CrossRef]

*I*is a (4

*M*+ 2) × (4

_{n}*M*+ 2) unit matrix.

_{n}## Appendix B: Filtered photoelectric current expressed using KLSE

**18**, 1493–1503 (2000). [CrossRef]

17. Z. Zhang, L. Chen, and X. Bao, “Accurate BER evaluation for lumped DPSK and OOK systems with PMD and PDL,” Opt. Express **15**, 9418–9433 (2007). [CrossRef] [PubMed]

*I*(

*t*) can be expressed in the form of

*I*(

*t*) = [〈

*s*(

^{o}*t*+

*T*) +

_{b}*n*(

^{o}*t*+

*T*)|

_{b}*s*(

^{o}*t*) +

*n*(

^{o}*t*)〉 +

*c.c.*]/2. Here

*s*(

^{o}*t*) (

*n*(

^{o}*t*)) represents the signal (noise) field at the input of the optical filter. Dirac bra 〈

*x*| is the conjugate transpose (or Hermitian transpose) of Dirac ket |

*x*〉 [

*x*=

*s*(

^{o}*t*),

*n*(

^{o}*t*),

*s*(

^{o}*t*) +

*n*(

^{o}*t*),

*etc.*]. The Dirac ket differs from usual complex vector in that the

*i*th element of the latter is just the

*i*th Fourier coefficient of the (signal or noise) field, while the

*i*th element of the former is the product of the

*i*th Fourier coefficient and its base function (cf. Eq. (17) in Ref. [17

17. Z. Zhang, L. Chen, and X. Bao, “Accurate BER evaluation for lumped DPSK and OOK systems with PMD and PDL,” Opt. Express **15**, 9418–9433 (2007). [CrossRef] [PubMed]

**18**, 1493–1503 (2000). [CrossRef]

**15**, 9418–9433 (2007). [CrossRef] [PubMed]

*I*(

*t*) =

*y*+

_{ss}*y*+

_{nn}*y*with (

_{ns}*l*= −

*L*, ···

_{s}*L*;

_{s}*m*= −

*M*···

_{n}*M*)

_{n}*Z*〉 represents the decoupled Gaussian random variables with zero mean and real part and imaginary part variance of

*σ*

^{2}. The effects of the optical and electrical filters in the receiver are represented by matrices with their elements being

*L*(±

_{s}*M*) can be neglected. Here [16

_{n}**18**, 1493–1503 (2000). [CrossRef]

## Appendix C: Nonlinearity induced phase difference between noise and noise-free signal

## I: The phase difference caused by N_{in}

_{in}) is much larger than the ASE noise from the EDFAs, which is true for the experiments discussed in Ref. [7

**17**, 3226–3241 (2009). [CrossRef] [PubMed]

_{in}and ignores the effect of N

_{0}

*(*

_{k}*k*= 1, ···,

*K*).

*P̄*, its nonlinear phase accumulated at the fiber output is Φ̄

*=*

_{NL}*P̄γKL*, where

*L*is the fiber length of each span, as denoted in Fig. 1. Due to the optical power fluctuation

*δP̄*, the actual nonlinear phase becomes Relative to the noise-free signal, the noise-induced phase change,

*ϕ*− Φ̄

_{NL}*, varies randomly. The average variance of such phase noise can be calculated as For the experiments in Ref. [7*

_{NL}**17**, 3226–3241 (2009). [CrossRef] [PubMed]

_{in}is filtered with bandwidth of

*B*=3nm. Thus, we have <

_{in}*δP̄*>=

*G*N

_{in}

*B*. Eq. (40) yields where

_{in}*OSNR*≈

*P̄/*(

*G*N

_{in}

*B*) is the input OSNR.

_{in}*δP̄*≥ 0, we have

*ϕ*− Φ̄

_{NL}*=*

_{NL}*δP̄γKL*> 0. This means

*ϕ*rotates faster than Φ̄

_{NL}*and there is a phase difference between the actual optical field and the noise-free signal. As part of the actual field, the noise field also has the same phase shift relative to the noise-free signal. Note that this phase shift will not affect signal-signal and noise-noise beatings. But it will affect the signal-noise beating. In fact, when calculating the signal-noise beating, the noise and signal should be treated consistently. Or, they should be considered within the same coordinate system. Thus the phase shift between noise field and and noise-free signal should be taken into account. In this work, <*

_{NL}*δϕ*> given by Eq. (41) is approximated as the average of such phase shift. Our numerical results plotted in Figs. 3 and 4 confirm the validity of this approximation.

**17**, 3226–3241 (2009). [CrossRef] [PubMed]

*= 0.9. In this case, Eq. (41) yields where arctan(*

_{NL}*x*) + arctan(1/

*x*) =

*π*/2 and arctan(

*x*) ≈

*x*(for

*x*→ 0) have been used. Obviously, <

*δϕ*> in Eq. (42) relates

*ϕ*in Eq. (23) of Ref. [24] with <

_{GM}*δϕ*> +

*ϕ*=

_{GM}*π*/2. Also,

*ϕ*

_{0}in Ref. [24] now becomes

*ϕ*

_{0}+

*π*/2 →

*ϕ*

_{0}in Eq. (17), while Δ

*in Ref. [24] is named as Δ in this work.*

_{GM}## II: The phase difference caused by N_{in} and N_{0}_{k} (*k* = 1, ···, *K*)

_{k}

*K*-span system of Fig. 1 with

*N*≠ 0 and

_{in}*K*not being large enough and with the ASE from each EDFA being filtered by

*O*and the ASE injected at the transmitter being filtered by

_{lk}*O*, Eq. (40) can be generalized as where

_{in}*GN*>>

_{in}B_{in}*KN*

_{0}

*B*[7

_{lk}**17**, 3226–3241 (2009). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, and W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. |

2. | K. Kikuchi, “Enhancement of optical-amplifier noise by nonlinear refractive index and group-velocity dispersion of optical fibers,” IEEE Photon. Technol. Lett. |

3. | A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “New analytical results on fiber parametric gain and its effects on ASE noise,” IEEE Photon. Technol. Lett. |

4. | R. Hui, M. O’Sullivan, A. Robinson, and M. Taylor, “Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments,” J. Lightwave Technol. |

5. | P. Serena, A. Orlandini, and A. Bononi, “Parametric-gain approach to the analysis of single-channel DPSK/DQPSK systems with nonlinear phase noise,” J. Lightwave Technol. |

6. | A. Demir, “Nonlinear phase noise in optical-fiber-communication systems,” J. Lightwave Technol. |

7. | L. D. Coelho, L. Molle, D. Gross, N. Hanik, R. Freund, C. Caspar, E.-D. Schmidt, and B. Spinnler, “Modeling nonlinear phase noise in differentially phase-modulated optical communication systems,” Opt. Express |

8. | M. Secondini, E. Forestieri, and C. R. Menyuk, “A combined regular-logarithmic perturbation method for signal-noise interaction in amplified optical systems,” J. Lightwave Technol. |

9. | R. Holzlöhner, “A covariance matrix method for the computation of bit errors in optical transmission systems,” Ph.D. thesis, University of Maryland Baltimore County. Baltimore, Maryland, USA (2003). |

10. | R. Holzlöhner, C. R. Menyuk, and W. L. Kath, “Efficient and accurate computation of eye diagrams and bit error rates in a single-channel CRZ system,” IEEE Photon. Technol. Lett. |

11. | R. Holzlöhner, C. R. Menyuk, and W. L. Kath, “A covariance matrix method to compute bit error rates in a highly nonlinear dispersion-managed soliton system,” IEEE Photon. Technol. Lett. |

12. | R. Holzlöhner and C. R. Menyuk, “Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems,” Opt. Lett. |

13. | A. Demir, “Non-Monte Carlo formulations and computational techniques for the stochastic nonlinear Schrodinger equation,” J. Comput. Phys. |

14. | J. Hult, “A fourth-order Runge-Kutta in the interaction picture method for simulating supercontinuum generation in optical fibers,” J. Lightwave Technol. |

15. | Z. Zhang, L. Chen, and X. Bao, “A fourth-order Runge-Kutta in the interaction picture method for numerically solving the coupled nonlinear Schrödinger equation,” Opt. Express |

16. | E. Forestieri, “Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering,” J. Lightwave Technol. |

17. | Z. Zhang, L. Chen, and X. Bao, “Accurate BER evaluation for lumped DPSK and OOK systems with PMD and PDL,” Opt. Express |

18. | D. Gariepy and G. He, “Measuring OSNR in WDM systemsEffects of resolution bandwidth and optical rejection ratio,” White paper, EXFO Inc. (2009). |

19. | P. Serena, A. Bononi, J. C. Antona, and S. Bigo, “Parametric gain in the strongly nonlinear regime and its impact on 10-Gb/s NRZ systems with forward-error correction,” J. Lightwave Technol. |

20. | A. Bononi, P. Serena, and A. Orlandini, “A unified design framework for single-channel dispersion-managed terrestrial systems,” J. Lightwave Technol. |

21. | E. Ip and J. M. Kahn, “Power spectra of return-to-zero optical signals,” J. Lightwave Technol. |

22. | A. Papoulis, |

23. | E. Forestieri and G. Prati, “Exact analytical evaluation of second-order PMD impact on the outage probability for a compensated system,” J. Lightwave Technol. |

24. | Z. Zhang, L. Chen, and X. Bao, “The noise propagator in an optical system using EDFAs and its effect on system performance: accurate evaluation based on linear perturbation,” arXiv:physics.optics/1207.3362v1. |

**OCIS Codes**

(060.2330) Fiber optics and optical communications : Fiber optics communications

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(060.5060) Fiber optics and optical communications : Phase modulation

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

**ToC Category:**

Optical Communications

**History**

Original Manuscript: November 20, 2012

Manuscript Accepted: February 27, 2014

Published: March 14, 2014

**Citation**

Zhongxi Zhang, Liang Chen, and Xiaoyi Bao, "Moment-generating function method used to accurately evaluate the impact of the linearized optical noise amplified by EDFAs," Opt. Express **22**, 6620-6633 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6620

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### References

- R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. 20, 389–400 (2002). [CrossRef]
- K. Kikuchi, “Enhancement of optical-amplifier noise by nonlinear refractive index and group-velocity dispersion of optical fibers,” IEEE Photon. Technol. Lett. 5, 1079–1081 (1993).
- A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, “New analytical results on fiber parametric gain and its effects on ASE noise,” IEEE Photon. Technol. Lett. 9, 535–537 (1997). [CrossRef]
- R. Hui, M. O’Sullivan, A. Robinson, M. Taylor, “Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments,” J. Lightwave Technol. 15, 1071–1082 (1997). [CrossRef]
- P. Serena, A. Orlandini, A. Bononi, “Parametric-gain approach to the analysis of single-channel DPSK/DQPSK systems with nonlinear phase noise,” J. Lightwave Technol. 24, 2026–2037 (2006). [CrossRef]
- A. Demir, “Nonlinear phase noise in optical-fiber-communication systems,” J. Lightwave Technol. 25, 2002–2032 (2007). [CrossRef]
- L. D. Coelho, L. Molle, D. Gross, N. Hanik, R. Freund, C. Caspar, E.-D. Schmidt, B. Spinnler, “Modeling nonlinear phase noise in differentially phase-modulated optical communication systems,” Opt. Express 17, 3226–3241 (2009). [CrossRef] [PubMed]
- M. Secondini, E. Forestieri, C. R. Menyuk, “A combined regular-logarithmic perturbation method for signal-noise interaction in amplified optical systems,” J. Lightwave Technol. 27, 3358–3369 (2009). [CrossRef]
- R. Holzlöhner, “A covariance matrix method for the computation of bit errors in optical transmission systems,” Ph.D. thesis, University of Maryland Baltimore County. Baltimore, Maryland, USA (2003).
- R. Holzlöhner, C. R. Menyuk, W. L. Kath, “Efficient and accurate computation of eye diagrams and bit error rates in a single-channel CRZ system,” IEEE Photon. Technol. Lett. 14, 1079–1081 (2002). [CrossRef]
- R. Holzlöhner, C. R. Menyuk, W. L. Kath, “A covariance matrix method to compute bit error rates in a highly nonlinear dispersion-managed soliton system,” IEEE Photon. Technol. Lett. 15, 688–690 (2003). [CrossRef]
- R. Holzlöhner, C. R. Menyuk, “Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems,” Opt. Lett. 28, 1894–1896 (2003). [CrossRef] [PubMed]
- A. Demir, “Non-Monte Carlo formulations and computational techniques for the stochastic nonlinear Schrodinger equation,” J. Comput. Phys. 201, 148–171 (2004). [CrossRef]
- J. Hult, “A fourth-order Runge-Kutta in the interaction picture method for simulating supercontinuum generation in optical fibers,” J. Lightwave Technol. 25, 3770–3775 (2007). [CrossRef]
- Z. Zhang, L. Chen, X. Bao, “A fourth-order Runge-Kutta in the interaction picture method for numerically solving the coupled nonlinear Schrödinger equation,” Opt. Express 18, 8261–8276 (2010). [CrossRef] [PubMed]
- E. Forestieri, “Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering,” J. Lightwave Technol. 18, 1493–1503 (2000). [CrossRef]
- Z. Zhang, L. Chen, X. Bao, “Accurate BER evaluation for lumped DPSK and OOK systems with PMD and PDL,” Opt. Express 15, 9418–9433 (2007). [CrossRef] [PubMed]
- D. Gariepy, G. He, “Measuring OSNR in WDM systemsEffects of resolution bandwidth and optical rejection ratio,” White paper, EXFO Inc. (2009).
- P. Serena, A. Bononi, J. C. Antona, S. Bigo, “Parametric gain in the strongly nonlinear regime and its impact on 10-Gb/s NRZ systems with forward-error correction,” J. Lightwave Technol. 23, 2352–2363 (2006). [CrossRef]
- A. Bononi, P. Serena, A. Orlandini, “A unified design framework for single-channel dispersion-managed terrestrial systems,” J. Lightwave Technol. 26, 3617–3631 (2008). [CrossRef]
- E. Ip, J. M. Kahn, “Power spectra of return-to-zero optical signals,” J. Lightwave Technol. 24, 1610–1618 (2006). [CrossRef]
- A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).
- E. Forestieri, G. Prati, “Exact analytical evaluation of second-order PMD impact on the outage probability for a compensated system,” J. Lightwave Technol. 22, 988–996 (2004). [CrossRef]
- Z. Zhang, L. Chen, X. Bao, “The noise propagator in an optical system using EDFAs and its effect on system performance: accurate evaluation based on linear perturbation,” arXiv:physics.optics/1207.3362v1.

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