## Modeling nonlinear phase noise in differentially phase-modulated optical communication systems

Optics Express, Vol. 17, Issue 5, pp. 3226-3241 (2009)

http://dx.doi.org/10.1364/OE.17.003226

Acrobat PDF (283 KB)

### Abstract

Using an alternative approach for evaluating the Bit-Error Rate (BER), we present a numerical and experimental investigation of the performance of phase-modulated optical communication systems in the presence of nonlinear phase noise and dispersion. The numerical method is based on the well known Karhunen-Loève expansion combined with a linearization technique of the Nonlinear Schrödinger Equation (NLSE) to account for the nonlinear interaction between signal and noise. Our numerical results show a good agreement with experiments.

© 2009 Optical Society of America

## 1. Introduction

1. A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol. **23**, 115–130 (2005). [CrossRef]

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

3. H. Kim and A. H. Gnauck, “Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise,” IEEE Photon. Technol. Lett. **15**, 320–322 (2003). [CrossRef]

^{-6}, the Multicanonical Monte Carlo method [4

4. 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]

5. X. Wei, X. Liu, and C. Xu, “Numerical simulation of the SPM penalty in a 10-Gb/s RZ-DPSK system,” IEEE Photon. Technol. Lett. **15**, 1636–1638 (2003). [CrossRef]

6. A. Mecozzi, “Limits to long-haul coherent transmission set by the Kerr nonlinearity and noise of the in-line amplifiers,” J. Lightwave Technol. **12**, 1993–2000 (1994). [CrossRef]

9. K.-P. Ho, “Performance of DPSK Signals With Quadratic Phase Noise,” IEEE Trans. Commun. **53**, 1361–1365 (2005). [CrossRef]

10. J.-A. Huang, K.-P. Ho, H.-K. Chen, S. K. Liaw, and H.-C. Wang, “Impact of nonlinear phase noise to DPSK signals: experimental verification of a simplified theoretical model,” IEEE Photon. Technol. Lett. **17**, 2236–2238 (2005). [CrossRef]

11. K.-P. Ho and H.-C. Wang, “Effect of dispersion on nonlinear phase noise,” Opt. Lett. **31**, 2109–2111 (2006). [CrossRef] [PubMed]

13. A. Demir, “Nonlinear Phase Noise in Optical-Fiber-Communication Systems,” J. Lightwave Technol. **25**, 2002–2032 (2007). [CrossRef]

14. 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. **15**, 1071–1082 (1997). [CrossRef]

*et al*. [19

19. R. Holzlohner, 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]

21. D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. **8**, 1816–1823 (1990). [CrossRef]

20. R. Holzlohner, C. R. Menyuk, W. L. Kath, and V. S. Grigoryan, “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]

22. J.-S. Lee and C.-S. Shim, “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,” J. Lightwave Technol. **12**, 1224–1229 (1994). [CrossRef]

14. 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. **15**, 1071–1082 (1997). [CrossRef]

## 2. Linearization of the NLSE

14. 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. **15**, 1071–1082 (1997). [CrossRef]

**W**(

**f**), which accounts for the nonlinear interaction between signal and noise.

*A*(

*z*= 0,

*t*) =

*s*(

*z*= 0,

*t*)+

*a*(

*z*= 0,

*t*), where

*A*(

*z*,

*t*) is the slowly varying complex envelope,

*s*(

*z*= 0,

*t*) is the signal with peak power

*P*

_{0}and

*a*(

*z*= 0,

*t*) is the ASE of a single optical amplifier in one polarization. The NLSE in one polarization describes the propagation of this signal and noise inside the fiber and is given by

*β*

_{2}and

*β*

_{3}are the second and third order dispersion parameters,

*γ*is the nonlinear parameter and

*α*is the attenuation. If the signal

*s*(

*z*,

*t*) is much stronger than the noise

*a*(

*z*,

*t*), then NLSE can be linearized by neglecting the noise-noise beat term arising from the fiber nonlin-earity, i.e. terms involving

*a*(

^{n}*z*,

*t*), where

*n*> 1, are not taken into account. In the following, we consider a single fiber of length

*L*, then we extend the result for a multi-span system.

### 2.1. Single fiber

*s*(

*z*= 0,

*t*) as a continuous wave with power

*P*

_{0}and neglecting the noise

*a*(

*z*= 0,

*t*), i.e.

*A*(

*z*,

*t*) becomes

*A*(

*z*), the NLSE (1) reduces to

*ϕ*(

*z*) =

*γP*

_{0}∫

_{0}

^{z}

*e*

^{-αz′}

*dz*′. If we add the noise

*a*(

*z*,

*t*) to the solution as a perturbation, we have

*a*(

*z*,

*t*)|

^{2}≪

*P*

_{0}, we obtain in frequency domain

*ω*,

*a*(

*z*,

*t*) =

*a*(

_{p}*z*,

*t*)+

*ja*(

_{q}*z*,

*t*), where

*a*(

_{p}*z*,

*t*) and

*a*(

_{q}*z*,

*t*) are real functions. In frequency domain,

*ã*(

*z*,

*ω*) =

*ã*(

_{p}*z*,

*ω*) +

*jã*(

_{q}*z*,

*ω*), where

*ã*(

_{p}*z*,

*ω*) and

*ã*(

_{q}*z*,

*ω*) may be complex. Using these relations and 2

*ã*(

_{p}*z*,

*ω*) =

*ã*(

*z*,

*ω*)+

*ã*

^{*}(

*z*, -

*ω*), we obtain

**ã**

_{out}=

**M**·

**ã**

_{in}, where

**ã**= [

*ã*(

_{p}*z*,

*ω*)

*ã*(

_{q}*z*,

*ω*)]

*, [·]*

^{T}^{T}is the transpose operator and the transmission matrix

**M**in absence of loss is given by

*α*≠ 0 can be obtained by dividing the fiber into many short pieces of length

*dz*. The total transmission matrix is given by the ordered product of the transmission matrices for each segment of the fiber [14

**15**, 1071–1082 (1997). [CrossRef]

23. A. G. Green, P. P. Mitra, and L. G. L. Wegener, “Effect of chromatic dispersion on nonlinear phase noise,” Opt. Lett. **28**, 2455–2457 (2003). [CrossRef] [PubMed]

*dz*≪

*α*

^{-1}and

*dz*=

*N*

_{sec}∈ ℕ, a mean-field approximation can be applied in each section

*δ*, and

_{i}*γ*for each segment are given by

_{dz}**M**should be multiplied by exp (-

*αL*).

15. 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]

**G**

_{1}=

**G**

_{1}(

*z*,

*ω*), [·]

**is the transpose-conjugate operator,**

^{H}*τ*is a time window and

*E*{·} indicates statistical averaging. The power spectral density of the noise at the fiber output is given by Φ(

*z*,

*ω*) =

*G*+

_{pp}*G*=

_{qq}*E*{|

*ã*(

_{p}*z*,

*ω*)|

^{2}}+

*E*{|

*ã*(

_{q}*z*,

*ω*)|

^{2}} and at the input Φ(0,

*ω*) =

*E*{|

*ã*(0,

_{p}*ω*)|

^{2}} +

*E*{|

*ã*(0,

_{q}*ω*)|

^{2}} = Φ

_{ASE}, where Φ

_{ASE}is the power spectral density of the noise generated by the optical amplifier in one polarization. Thus, the power spectral density at the end of the fiber is given by

### 2.2. Multi-span system

**H**

*is the filter transfer matrix and is given by*

_{i}**ã**

_{out}=

**H**

*·*

_{i}**ã**

_{in}. The power spectral density matrix

**G**

_{N+1}(

*z*,

*ω*) can be calculated by passing the noise vector

**ã**through the devices shown in Fig. 1. Thus, the noise at the input of the receiver is given by

**M**

*=*

^{i}**H**

_{N+1}· ∏

^{N}

_{k=i}

*M*·

_{k}**H**

*and*

_{k}*N*is the number of spans. Note that

**H**

_{N+1}is the receiver optical filter and can be incorporated into the receiver structure or left in the link. Therefore, the power spectral density of the noise can be calculated as follows:

^{i}

_{ASE}is the power spectral density of the additive white Gaussian noise generated by the

*i*-th amplifier. Finally, using (9),

*G*,

_{pp}*G*,

_{pq}*G*and

_{qp}*G*can be evaluated in order to calculate the Φ(

_{qq}*z*,

*ω*) at the receiver input. Note that each term of the above sum has a product of the matrices

**M**

*. In order to go over the sum only one time, it is necessary to calculate the system transmission matrix backwards, i.e., the sum should be calculated from*

_{i}*i*=

*N*to 1. Several properties of the interaction between signal and noise can be evaluated using the power spectral density matrix

**G**

_{N+1}(

*z*,

*ω*), as reported in [24

24. P. Serena, A. Bononi, and A. Orlandini, “Fundamental laws of parametric gain in periodic dispersion-managed optical links,” J. Opt. Soc. Am. B **24**, 773–787 (2007). [CrossRef]

*G*

_{N+1}(

*z*,

*ω*) is a positive semidefinite matrix and, therefore, it can be written as

**G**

_{N+1}(

*z*,

*ω*) = Φ

_{ASE}

^{N+1}· (

**W**̃ ·

**W̃**), where [15

^{H}15. 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]

_{ASE}

^{N+1}·

**I**filtered by

**W**̃. Therefore, the noise at the receiver input is given by

**a**̃

*, was normalized by*

_{i}**a**

_{N+1}and is included in

**W**̃. Note that in

*ã*(

*z*,

*ω*) =

*ã*(

_{p}*z*,

*ω*) +

*jã*(

_{q}*z*,

*ω*),

*ã*(

_{p}*z*,

*ω*) and

*ã*(

_{q}*z*,

*ω*) may be complex functions. In order to deal with system functions in the frequency domain, it is more convenient to define a noise vector

**ñ**(

*z*,

*ω*) = [

*ñ*

_{re}(

*z*,

*ω*)

*ñ*(

_{im}*z*,

*ω*)]

^{T}containing the real and imaginary parts of

*a*̃(

*z*,

*ω*):

**f**= [

*f*

_{-M}

*f*

_{-M+1}… 0…

*f*

_{M-1}

*f*]

_{M}*and using the relation*

^{T}*ω*= 2

*πf*, the vector

**n**̃

_{out}(

*z*,

*f*) can be written as a (4

*M*+2) × 1 column vector

**n**̃

_{out}(

**f**) containing the real and imaginary parts of

*a*̃

_{out}(

*z*,

*f*) at the frequencies

**f**and it is given by

**D**

_{0}is a (2

*M*+ 1) × (2

*M*+ 1) anti-diagonal matrix and

**a**̃

_{N+1}(

**f**) is a (4

*M*+2) + 1 column vector. The terms

**W**̃

*(*

_{mn}**f**) with {

*m*,

*n*} ∈ {1,2} are (2

*M*+ 1) + (2

*M*+ 1) diagonal matrices, where each diagonal entry is the

*W̃*(

_{mn}*f*) element of the matrix

*W*̃. Consequently, the noise at the receiver input can also be written as

**n**̃

_{N+1}(

**f**) =

**B**·

_{2}**a**̃

_{n+1}(

**f**),

**B**=

_{2}**B**and

_{2}**W**(

**f**) is the system transmission matrix given by

**B**and

_{1}**B**are given by

_{2}**I**is the identity matrix. The matrices

**B**and

_{1}**B**are used to calculate the real and imaginary parts of

_{2}*a*̃

*(*

_{N+1}*z*,

*f*) and

*a*̃

_{out}(

*z*,

*f*). Note that

**n**〰

_{out}(

**f**) cannot be defined like

**n**〰

_{N+1}(

**f**) because the matrix

**W**〰(

**f**) may be non-symmetric. In fact, it can be seen from (9) that it is non-symmetric if

*β*

_{3}≠ 0. Moreover, the entries in the matrix

**W**(

**f**) corresponding to the frequency

*f*can be dropped if enough simulation bandwidth is available. The break on the matrix symmetry will not affect the result because no significant linear or nonlinear effect will occur at this frequency. On the other hand, the even number of samples will increase the speed of the Fast-Fourier Transform (FFT) algorithm. In the following sections, the system transmission matrix

_{M}**W**(

**f**) has a reduced size of 4

*M*+ 4

*M*and is used to evaluate the BER accounting for nonlinear phase noise.

## 3. Evaluation of the bit-error rate

1. A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol. **23**, 115–130 (2005). [CrossRef]

26. G. Bosco and P. Poggiolini, “On the Q factor inaccuracy in the performance analysis of optical direct-detection DPSK systems,” IEEE Photon. Technol. Lett. **16**, 665–667 (2004). [CrossRef]

1. A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol. **23**, 115–130 (2005). [CrossRef]

22. J.-S. Lee and C.-S. Shim, “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,” J. Lightwave Technol. **12**, 1224–1229 (1994). [CrossRef]

27. M. Kac and A. Siegert “On the Theory of Noise in Radio Receivers with Square Law Detectors,” J. Appl. Phys. **18**, 383–397 (1947). [CrossRef]

28. 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]

29. A. Orlandini, P. Serena, and A. Bononi, “An Alternative Analysis of Nonlinear Phase Noise Impact on DPSK Systems,” in *32nd European Conference on Optical Communication, ECOC 2006* (Cannes, France, 2006). [CrossRef]

15. 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]

30. L. D. Coelho, L. Molle, D. Gross, N. Hanik, R. Freund, C. Caspar, and E.-D. Schmidt, “Numerical and Experimental Investigation of the Effect of Dispersion on Nonlinear Phase Noise in RZ-DPSK Systems,” *in 33rd European Conference on Optical Communication, ECOC 2007* (Berlin, Germany, 2007). [CrossRef]

**23**, 115–130 (2005). [CrossRef]

22. J.-S. Lee and C.-S. Shim, “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,” J. Lightwave Technol. **12**, 1224–1229 (1994). [CrossRef]

**W**(

**f**), derived in the previous section, in order to calculate the BER in presence of nonlinear phase noise. This method can yield accurate BER results, taking into account ASE noise, pulse shaping, optical and electrical filtering, interferometer phase error and nonlinear phase noise. We start our derivation with the standard Karhunen-Loève method, where no interaction between signal and noise is considered, then we extend the result by using the system transmission matrix.

### 3.1. Standard Karhunen-Loève method

*s*̃

_{out}(

*f*) is obtained by solving numerically the NLSE using, for example, the Split-Step Fourier method [12], which accounts for all signal distortions. The interaction between signal and noise is included using the system transmission matrix

**W**(

**f**), where only the power of the signal, noise variances and link parameters are considered. The block “C / V” transforms the complex signal into a vector where its real and imaginary parts are separated. Without interaction between signal and noise,

**n**̃

_{out}(

**f**) is given by the sum of the noise generated by each amplifier in the link. Therefore, the received signal can be written back, for simplicity, in its complex scalar form

*E*̃(

*f*) =

*s*̃

_{out}(

*f*) +

*n*̃

_{out}(

*f*), where

*n*̃

_{out}(

*f*) is AWGN. The decision variable at the output of the balanced receiver

*I*(

*t*) is given by [1

_{k}**23**, 115–130 (2005). [CrossRef]

*t*is the sampling time and

_{k}*H*(

_{e}*f*),

*H*(

_{o}*f*),

*H*

_{1}(

*f*) and

*H*

_{2}(

*f*) are the transfer functions of the electrical filter, optical filter, upper and lower branch of the delay interferometer, respectively [1

**23**, 115–130 (2005). [CrossRef]

*φ*(

*f*) should satisfy the second kind homogeneous Fredholm integral equation [22

**12**, 1224–1229 (1994). [CrossRef]

*λ*of the

_{m}*m*eigenfunction

^{th}*φ*(

_{m}*f*) is a real-valued constant. The eigenvalues are ordered as |

*λ*

_{m-1}|≥ |

*λ*|

_{m}*λ*| ≥ |

_{m}*λ*

_{m+1}|, where

*m*is a positive integer. All the eigenfunctions satisfy the following condition:

*δ*is the Kronecker delta. Equation (29) implies that the eigenfunctions form a complete set of orthonormal basis functions over [-∞, ∞]. With this set,

_{ml}*E*̃(

*f*)

*e*can be expanded as

^{j2πft}*c*(

_{m}*t*) = ∫

^{∞}

_{-∞}

*E*̃(

*f*)

*e*

^{j2πft}+

*φ*

_{m}^{*}(

*f*)

*df*is the time dependent coefficient. Since

*E*̃(

*f*) consists of signal and noise, we can define

*s*(

_{i}*t*) = 𝔉

^{-1}{

*s*̃

_{out}(

*f*) +

*φ*

_{i}^{*}(

*f*)} and

*n*(

_{i}*t*) = 𝔉

^{-1}{

*n*̃

_{out}(

*f*) +

*φ*

_{i}^{*}(

*f*)} as the time dependent signal and noise coefficients, respectively. Therefore, we obtain

*I*(

*t*) as a series summation:

_{k}### 3.2. Discrete analysis

*f*and coincide with the discrete frequency vector of the DFT. The resulting summation is given by

*e*and

_{m}*K*are defined as

_{ml}*e*and

_{S,m}*e*are the signal and noise part of

_{n,m}*e*and

_{m}*f*(

*m,l*) = ((

*m*,

*l*) -

*M*- 1)Δ

*f*. Note that

*I*(

*t*) and

_{k}*e*are random variables and

_{m}*K*is a deterministic variable. Equations (26

_{ml}26. G. Bosco and P. Poggiolini, “On the Q factor inaccuracy in the performance analysis of optical direct-detection DPSK systems,” IEEE Photon. Technol. Lett. **16**, 665–667 (2004). [CrossRef]

*q*=

_{i,m}*φ*(

_{i}*f*)√Δ

_{m}*f*. Equation (39) shows that the second kind homogeneous Fredholm integral becomes a standard eigenvalue problem [31], where the eigenvalues and eigenvectors of the matrix

**K**are used to expand

**e**and also define the MGF. The coefficients of the sum (32) are given by

*s*(

_{i}*t*) =

_{k}**q**

^{H}*·*

_{i}**e**

_{s}and

*n*(

_{i}*t*) =

_{k}**q**

_{i}**+**

^{H}**e**, where

_{n}*n*(

_{i}*t*),

_{k}*i*= 1,2,…, 2

*M*, are zero-mean complex Gaussian random variables with independent real and imaginary components of variance

*σ*

^{2}= Φ

^{out}

_{ASE}Δ

*f*/2. Thus, the decision sample

*I*(

*t*) can be expressed as quadratic form of Gaussian random variables and its MGF can be evaluated in a closed form as follows [1

_{k}**23**, 115–130 (2005). [CrossRef]

28. 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]

*α*, =

_{i}*λ*|

_{i}*s*(

_{i}*t*)|

_{k}^{2},

*β*, = 2

_{i}*λ*

_{i}*σ*

^{2}and

*ξ*= 1 or

*ξ*= 2 for polarized or unpolarized noise, respectively. The mean and variance of

*I*(

*t*) are given by

_{k}**23**, 115–130 (2005). [CrossRef]

28. 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]

### 3.3. Extended Karhunen-Loève method

**e**and

_{W}**K**are a 4

_{W}*M*×1 column vector and a 4

*M*× 4

*M*matrix, respectively, given by

*s*(

_{i,w}*t*) =

_{k}**q**

_{i,w}**H**·

**e**

_{w,s}and

*n*(

_{iw}*t*) =

_{k}**q**

_{i,w}**+**

^{H}**e**, where

_{w,n}*n*(

_{i,w}*t*),

_{k}*i*= 1,2,…, 4

*M*, are zero-mean real Gaussian random variables with variance .

*σ*

^{2}= Φ

^{N+1}

_{ASE}Δ

*f*/2. The MGF of

*I*(

*t*)can be written in the same form as in (40) and the BER can be calculated using the Laplace inverse transform and saddlepoint integration or approximation methods, as described in [1

_{k}**23**, 115–130 (2005). [CrossRef]

**18**, 1493–1503 (2000). [CrossRef]

## 4. Simulation and experimental set-up

32. E. T. Spiller, W. L. Kath, R. O. Moore, and C. J. McKinstrie, “Computing large signal distortions and bit-error ratios in DPSK transmission systems,” IEEE Photon. Technol. Lett. **17**, 1022–1024 (2005). [CrossRef]

*N*, the total input power at MP1 was set such that the accumulated mean nonlinear phase shift of the system set-up (a) and (b),

*P*

_{(*)}is the average fiber input power,

*L*

^{(*)}

_{eff}is the effective length and

*γ*

_{(*)}the nonlinear parameter. Therefore, by fixing the accumulated mean nonlinear phase shift, the amount of nonlinearity in each span is varied, which should give the same performance in absence of dispersion for any number of spans

*N*[7]. In this configuration, only the variance of the nonlinear phase noise is evaluated in terms of BER and the effect of dispersion on nonlinear phase noise can be isolated.

*R*= 20 Gbit/s RZ-50% DPSK signal at a fixed wavelength of 1556.555 nm for pseudo random binary sequences (PRBS) of length 2

_{b}^{7}- 1 (simulation) and 2

^{23}- 1 (experiment). In order to increase the accuracy of the simulation results, the signal power spectrum at the transmitter was measured and used to fit the simulated signal power spectrum, as shown in Fig. 5. The receiver optical filter transfer function and fiber parameters were also measured using standard procedures. Table 1 shows the fiber parameters used in the experiment and simulations, where L is the fiber length,

*α*is the attenuation, D is the chromatic dispersion parameter, S the slope and PMD the polarization-mode dispersion parameter. At the transmitter and in the loop, optical bandpass filters (OBPF) with bandwidth of 3 nm and 5 nm, respectively, were used to limit the noise bandwidth. Two polarization scramblers, at the receiver and in the loop, were used to average the effect of polarization dependent loss (PDL). The 3 dB bandwidth of the optical receiver filter was chosen to be 0.3 nm; the free spectral range (FSR) of the Mach-Zehnder delay interferometer was measured to be 24.84 GHz. The electrical filter was modelled as a fifth-order Bessel filter with a bandwidth of 0.75

*R*, which was determined from the back-to-back measurements. In Section 2, the system transmission matrix

_{b}**W**(

**f**) was derived assuming the signal as a continuous wave. In order to take into account the effect of signal modulation, instead of

*P*

_{0}, an effective fiber input power

*P*

_{eff}was used to calculate the system transmission matrix [15

**24**, 2026–2037 (2006). [CrossRef]

*P*is evaluated by filtering

_{eff}*P*(

*t*) = |

*A*(

*t*)|

^{2}with a proper window and sampling. For a RZ-DPSK signal with sinusoidal intensity profile,

*P*

_{eff}can be calculated analytically using the following equations [36 ]:

*L*

_{eff}and

*L*are the effective and dispersion lengths of the transmission fiber, respectively. For the DSF fiber,

_{D}*L*is zero, which results in a effective power equal the peak power. However, due to filtering and PMD a small memory is introduced to the system. This could be verified after comparing the measured and simulated power spectral densities. Therefore, in our simulations an effective fiber input power of 95% (DSF) and 63% (SMF+DCF) of the fiber input peak power was used to calculate

_{D}**W**(

**f**). In the fiber, the signal propagates in two orthogonal polarizations, which causes PMD and nonlinear coupling between both polarizations. The linear distortion due to PMD can be estimated by allowing a maximum ratio of DGD to symbol duration of 10% [16]. Using the parameters in Table 1, the mean DGD is given by DGD

_{DSF}= 0.87, 1.38, 3.1ps for

*N*= 2, 5, 25, respectively, and DGD

_{SMF+DCF}= 0.93, 1.32, 2.1ps for

*N*= 5, 10, 25, respectively. Therefore, the impact of PMD on the performance can be neglected for this set-up. Moreover, the nonlinear coupling between both polarizations in presence of PMD can be simulated using (1) as an averaged propagation equation [37

37. N. Hanik, “Modelling of nonlinear optical wave propagation including linear mode-coupling and birefringence,” Opt. Commun. **214**, 207–230 (2002). [CrossRef]

*γ*is reduced to 91% of its maximal value. Under these assumptions we were able to match simulation and experimental results.

## 5. Results

*N*= 2, 5 and 25 non-dispersive (DSF) spans. In this case, SPM dominates and the BER was accurately calculated using the extended Karhunen-Loève expansion method, as shown in Fig. 6(a). Since the mean nonlinear phase shift was kept constant, the BER is almost equal for different number of spans. After that the signal was transmitted

*N*times over a fully dispersion compensated span (SMF+DCF) and BER curves were measured and simulated again for back-to-back,

*N*= 5, 10 and 25 spans, as shown in Fig. 6(b). An improvement of 1dB from the DSF curve was observed for both low- and high-OSNR, as shown in Fig. 6. Therefore, using a high-dispersive span, the effect of dispersion reduces the impact of SPM.

*N*= 2, 5 and 25 non-dispersive (DSF) spans. The BER can now be calculated using the extended Karhunen-Loève expansion method. The SPM induced nonlinear phase noise generates an additional penalty comparing with the receiver noise loading scheme. The small difference between the simulated and measured curves for

*N*= 2 and 5 is interpreted as a consequence of a small deviation of the system zero dispersion together with in-line optical filtering and ASE noise accumulation. For

*N*= 25 the accumulated nonlinear phase noise seems to be so strong that the linearization procedure does not work anymore. This problem was also observed by Holzlöhner

*et al*. [19

19. R. Holzlohner, 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. R. Holzlohner, C. R. Menyuk, W. L. Kath, and V. S. Grigoryan, “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]

*N*= 5, 10 and 25 times over a fully dispersion compensated span. In Fig. 7(b), the system performance is almost the same for 5, 10 and 25 spans, showing that the BER depends solely on the mean nonlinear phase shift, which so far was known only for systems without dispersion. Comparing Fig. 7(a) and Fig. 7(b), an improvement of 3dB and 2dB from the DSF curve was observed for low- and high-OSNR, respectively. Therefore, if enough dispersion or filtering is present in the link, which is the case of most WDM systems, the extended Karhunen-Loève expansion method agrees very well with the experimental results, as shown in Fig. 7.

## 6. Conclusion

**W**(

**f**). Additionally, we showed that dispersion can improve the system performance and the gain is found to be larger for low-OSNR than for high-OSNR. It suggests that a system can have the same performance either employing compensation of nonlinear phase noise or FEC. We also verified that the performance of a fully dispersion compensated span is solely given by the mean nonlinear phase shift. This result gives the basic relation between reach-distance and power for a given system performance.

## Acknowledgment

## References and links

1. | A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol. |

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

3. | H. Kim and A. H. Gnauck, “Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise,” IEEE Photon. Technol. Lett. |

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

5. | X. Wei, X. Liu, and C. Xu, “Numerical simulation of the SPM penalty in a 10-Gb/s RZ-DPSK system,” IEEE Photon. Technol. Lett. |

6. | A. Mecozzi, “Limits to long-haul coherent transmission set by the Kerr nonlinearity and noise of the in-line amplifiers,” J. Lightwave Technol. |

7. | K.-P. Ho, |

8. | M. P. Dlubek, A. J. Phillips, and E. C. Larkins, “Nonlinear Evolution of Gaussian ASE Noise in ZMNL Fiber,” J. Lightwave Technol. |

9. | K.-P. Ho, “Performance of DPSK Signals With Quadratic Phase Noise,” IEEE Trans. Commun. |

10. | J.-A. Huang, K.-P. Ho, H.-K. Chen, S. K. Liaw, and H.-C. Wang, “Impact of nonlinear phase noise to DPSK signals: experimental verification of a simplified theoretical model,” IEEE Photon. Technol. Lett. |

11. | K.-P. Ho and H.-C. Wang, “Effect of dispersion on nonlinear phase noise,” Opt. Lett. |

12. | G. P. Agrawal, |

13. | A. Demir, “Nonlinear Phase Noise in Optical-Fiber-Communication Systems,” J. Lightwave Technol. |

14. | 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. |

15. | 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. |

16. | G. P. Agrawal, |

17. | A. V. T. Cartaxo, B. Wedding, and W. Idler, “Influence of fiber nonlinearity on the phase noise to intensity noise conversion in fiber transmission: theoretical and experimental analysis,” J. Lightwave Technol. |

18. | A. V. T. Cartaxo, B. Wedding, and W. Idler, “Influence of fiber nonlinearity on the fiber transfer function: theoretical and experimental analysis,” J. Lightwave Technol. |

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

21. | D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. |

22. | J.-S. Lee and C.-S. Shim, “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,” J. Lightwave Technol. |

23. | A. G. Green, P. P. Mitra, and L. G. L. Wegener, “Effect of chromatic dispersion on nonlinear phase noise,” Opt. Lett. |

24. | P. Serena, A. Bononi, and A. Orlandini, “Fundamental laws of parametric gain in periodic dispersion-managed optical links,” J. Opt. Soc. Am. B |

25. | G. Strang, |

26. | G. Bosco and P. Poggiolini, “On the Q factor inaccuracy in the performance analysis of optical direct-detection DPSK systems,” IEEE Photon. Technol. Lett. |

27. | M. Kac and A. Siegert “On the Theory of Noise in Radio Receivers with Square Law Detectors,” J. Appl. Phys. |

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

29. | A. Orlandini, P. Serena, and A. Bononi, “An Alternative Analysis of Nonlinear Phase Noise Impact on DPSK Systems,” in |

30. | L. D. Coelho, L. Molle, D. Gross, N. Hanik, R. Freund, C. Caspar, and E.-D. Schmidt, “Numerical and Experimental Investigation of the Effect of Dispersion on Nonlinear Phase Noise in RZ-DPSK Systems,” |

31. | W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, |

32. | E. T. Spiller, W. L. Kath, R. O. Moore, and C. J. McKinstrie, “Computing large signal distortions and bit-error ratios in DPSK transmission systems,” IEEE Photon. Technol. Lett. |

33. | R. O. Moore, G. Biondini, and W. L. Kath, “A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schrödinger Solitons,” SIAM Review |

34. | S. Kumar and L. Liu, “Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems,” Opt. Express |

35. | M. Ohm, R. J. Essiambre, and P. J. Winzer, “Nonlinear phase noise and distortion in 42.7-Gbit/s RZ-DPSK systems,” in |

36. | P. Serena, A. Orlandini, and A. Bononi, “A parametric gain approach to DPSK performance evaluation in presence of nonlinear phase noise,” in |

37. | N. Hanik, “Modelling of nonlinear optical wave propagation including linear mode-coupling and birefringence,” Opt. Commun. |

**OCIS Codes**

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

(060.5060) Fiber optics and optical communications : Phase modulation

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 11, 2008

Revised Manuscript: February 2, 2009

Manuscript Accepted: February 11, 2009

Published: February 17, 2009

**Citation**

Leonardo D. Coelho, Lutz Molle, Dirk Gross, Norbert Hanik, Ronald Freund, Christoph Caspar, Ernst-Dieter Schmidt, and Bernhard Spinnler, "Modeling nonlinear phase noise in differentially phase-modulated optical
communication systems," Opt. Express **17**, 3226-3241 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3226

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### References

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- J. P. Gordon and L. F. Mollenauer, "Phase noise in photonic communications systems using linear amplifiers," Opt. Lett. 15, 1351-1353 (1990). [CrossRef] [PubMed]
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- K.-P. Ho, "Performance of DPSK Signals With Quadratic Phase Noise," IEEE Trans. Commun. 53, 1361-1365 (2005). [CrossRef]
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- R. Holzlohner, 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]
- R. Holzlohner, C. R. Menyuk, W. L. Kath, and V. S. Grigoryan, "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]
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- L. D. Coelho, L. Molle, D. Gross, N. Hanik, R. Freund, C. Caspar, and E.-D. Schmidt, "Numerical and Experimental Investigation of the Effect of Dispersion on Nonlinear Phase Noise in RZ-DPSK Systems," in 33rd European Conference on Optical Communication, ECOC 2007 (Berlin, Germany, 2007). [CrossRef]
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- E. T. Spiller, W. L. Kath, R. O. Moore, and C. J. McKinstrie, "Computing large signal distortions and bit-error ratios in DPSK transmission systems," IEEE Photon. Technol. Lett. 17, 1022-1024 (2005). [CrossRef]
- R. O. Moore, G. Biondini, and W. L. Kath, "A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schr¨odinger Solitons," SIAM Review 50, 523-549 (2008). [CrossRef]
- S. Kumar and L. Liu, "Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems," Opt. Express 15, 2166-2177 (2007). [CrossRef] [PubMed]
- M. Ohm, R. J. Essiambre, and P. J. Winzer, "Nonlinear phase noise and distortion in 42.7-Gbit/s RZ-DPSK systems," in 31st European Conference on Optical Communication, ECOC 2005 (Glasgow, Scotland, 2005). [CrossRef]
- P. Serena, A. Orlandini, and A. Bononi, "A parametric gain approach to DPSK performance evaluation in presence of nonlinear phase noise," in 30th European Conference on Optical Communication, ECOC 2004 (Stockholm, Sweden, 2004).
- N. Hanik, "Modelling of nonlinear optical wave propagation including linear mode-coupling and birefringence," Opt. Commun. 214, 207-230 (2002). [CrossRef]

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