## Abstract

An analytical model to calculate the variance of cross-phase modulation (XPM) distortion in a wavelength-division multiplexed (WDM) fiber-optic system is developed. The method is based on the first order perturbation technique and it is applicable for both dispersion managed and dispersion uncompensated systems. For dispersion managed systems, it is shown that the variance of XPM distortion scales as *N ^{x}* where

*N*is the number of spans and

*x*∈ [1, 2] depending on the amount of inline-dispersion compensation. The analytical model is found to be in good agreement with simulations in most of the cases.

© 2014 Optical Society of America

## 1. Introduction

Since different wavelength-division multiplexed (WDM) channels have different group speeds in a dispersive fiber, symbols of a “fast” channel walk off the symbols of a “slow” channel. As the “fast” channel walks off, it induces phase distortion on the “slow” channel (due to cross-phase modulation (XPM)) and vice versa, and group velocity dispersion (GVD) converts the phase distortion into amplitude distortion [1, 2]. XPM effect may be considered as a stochastic process if the symbol pattern in the other channels is unknown and it could lead to performance degradation [3].

Due to the impact of the XPM impairments on the performance of a WDM systems, they have drawn considerable attention [1–13]. In [4], XPM distortion and the effect of channel spacing on XPM are investigated based on numerical simulations. In [1,2,5], XPM distortion in multi-span intensity-modulation direct-detection (IM-DD) is investigated in frequency domain both experimentally and theoretically. The effects of bit rate and dispersion compensation on the system performance are also studied. In [6], XPM and stimulated Raman scattering (SRS) distortions for a IM-DD system are studied analytically and numerically. Effect of dispersion compensation on these distortions are also investigated. In [7], the signal distortion due to self-phase modulation (SPM) and XPM is studied using the first and/or second order perturbation theory.

Recently, the modeling of nonlinear distortion in coherent fiber-optic WDM systems has drawn significant interest [8–13]. In [8], the first order Volterra series is used to estimate the channel capacity of a dispersive nonlinear optical fiber. In [9], an analytical model for XPM phase noise and polarization scattering is obtained based on Volterra analysis, and the variance and autocorrelation of the XPM phase noise and polarization scattering are calculated. In [10], propagation impairment due to nonlinear interaction in coherent orthogonal frequency-division multiplexing (OFDM) systems is studied. In [11, 12], an analytical expression for the power spectral density (PSD) of the nonlinear interference in a dispersion uncompensated WDM system is developed by assuming that the nonlinear distortion can be modeled as Gaussian noise. To evaluate the PSD, it is necessary to carry out a triple numerical integration. In [13], channel capacity of a nonlinear fiber-optic system is estimated using a first order perturbation theory.

In this paper, a time domain approach based on first order perturbation technique is used to find the nonlinear distortion due to XPM. We assume that the pulse is linear to the leading order and treat nonlinearity as a perturbation. Unlike the frequency domain approach [11,12], our time domain approach is applicable to the cases of dispersion managed links as well as dispersion uncompensated links. In the case of dispersion uncompensated links, the XPM distortion may be treated as noise and this noise occurring in different spans are nearly independent. Hence, the XPM variance due to each span can be added to obtain the total XPM variance. In contrast, in the case of dispersion managed links, such an approach fails due to the correlation of XPM distortion among spans. When the inline-dispersion compensation is 100% (resonant map),the XPM variance scales as *N*^{2} where *N* is the number of spans. This is because the XPM field generated in each span is identical and they add up linearly leading to quadratic dependance for the variance or power. In general, the XPM variance scales as *N ^{x}* where

*x*∈ [1, 2] depending on the amount of inline-dispersion compensation,

*x*being close to 1 when there is no inline-dispersion compensation.

This paper is organized as follows. In section 2, we present a first order perturbation theory for XPM effects in fibers. Analytical expressions are obtained for the first order corrections. In section 3, analytical expression for the variance of XPM impairment is obtained. In section 4, the theoretical model is validated by numerical simulations. Also, effect of launch power, fiber dispersion, system reach, and channel spacing on the XPM variance is examined. Finally, in section 5, the contributions of this paper are summarized.

## 2. SPM and XPM analytical model

The pulse propagation in optical fiber is described by the nonlinear Schrodinger equation (NLS),

*q*is the electric field envelope,

*β*

_{2}(

*z*) is the dispersion profile,

*γ*

_{0}is nonlinear coefficient and

*α*(

*z*) is the fiber loss/gain profile. Using the transformation, where $w(z)={\int}_{0}^{z}\alpha (s)ds$, Eq. (1) can be rewritten in the lossless form as

*γ*(

*z*) =

*γ*

_{0}exp[−

*w*(

*z*)]. We consider the interaction between two channels of a WDM system. We spilt the field

*u*into two parts, where

*u*is the field of channel

_{j}*j*,

*j*= 1, 2. Substituting Eq. (4) into Eq. (3) and ignoring the four wave mixing terms, we obtain

*n*shows the symbol location in time domain,

*T*is the symbol interval,

_{s}*P*is the power, Ω is the channel separation in radians,

*p*(0,

*T*) is Gaussian pulse shape at

*z*= 0,

*T*

_{0}is the half-width at 1/e- intensity point [14] and

*x*and

_{n}*y*are random variables that take values ±1, ±3,...,±(

_{n}*X*− 1) and ±1, ±3,...,±(

*Y*− 1) (

*X*and

*Y*are the number of amplitude levels of the in-phase and quadrature components, respectively.) with equal probability, respectively. Using the perturbation technique [7, 15], the field envelope in channel

*k*can be expanded as,

*γ*

_{0}is a small parameter and ${u}_{k}^{(m)}$ denotes the

*m*th order solution. Here ${u}_{k}^{(0)}$ represents the 0

*order solution which satisfies*

^{th}*S*(

*z*) is the accumulated dispersion given by Our reference frame is fixed to channel 1. The pulse in channel 2 moves with an inverse group speed of

*β*

_{2}Ω relative to channel 1.

Substituting Eq. (10) into Eq. (5) and collecting all the terms that are proportional to *γ*_{0}, we obtain

*F*(

*z*,

*T*) is of the form

*u*

_{1}due to XPM term, ${\left|{u}_{2}^{(0)}\right|}^{2}{u}_{1}^{(0)}$ in Eq. (18), we make use of the result given by Eq. (21). The forcing function

*F*(

*z*,

*T*) in this case is

*u*

_{1}due to XPM is given by where

*mT*and

_{s}*nT*.

_{s}The first order correction for *u*_{1} due to SPM term,
${\left|{u}_{1}^{(0)}\right|}^{2}{u}_{1}^{(0)}$, can be easily obtained from Eq. (31) by setting the angular frequency Ω to 0, *τ _{m}* and

*τ*to 0 and by replacing the XPM factor 2 with SPM factor 1, i.e.,

_{n}*u*

_{1}is obtained by adding the SPM and XPM contributions,

## 3. Variance calculation

The distortion due to nonlinearity is $\delta {u}_{1}={\gamma}_{0}{u}_{1}^{(1)}$. The variance is obtained by

*E*{.} denotes the ensemble average. In this paper, we mainly focus on the variance due to XPM and we ignore the first term in Eq. (35). So, we have Using Eq. (31), the mean XPM distortion can be found as

*δ*is Kronecker delta function. Using Eq. (39), Eq. (38) becomes

**Case 1:***m*=*n*and*m′*=*n′*This case corresponds to the XPM distortion of the channel 1 due to pulses of channel 2 located at

where So, we have*mT*and_{s}*m′T*. In this case_{s}**Case 4:***m*≠*n*and*m′*=*n′*The same logic as in the previous case, $\text{E}\left\{{a}_{m}{a}_{n}^{*}{a}_{{m}^{\prime}}^{*}{a}_{{n}^{\prime}}\right\}=0$.

Now, total mean of absolute square can be obtained by adding Eqs. (47) and (49)

*K*

_{1}=

*K*

_{2}= 1. So, for QPSK, Eq. (53) reduces to

## 4. Results and discussion

To validate the analytical model, we carried out numerical simulations of the dispersion uncompensated fiber-optic system shown in Fig. 1. Two WDM channels with QPSK modulation and Gaussian pulse shape are assumed. The wavelength division multiplexer (Mux) combines the two channels and the output of the multiplexer is launched to a multi-span fiber-optic system. The fiber loss is exactly compensated with an ideal noiseless amplifier. Amplified spontaneous emission (ASE) noise, laser phase noise, polarization effects, and the coherent receiver imperfections are ignored since the primary focus is to validate our analytical model for XPM impairments. After multi-span transmission, the channels are demultiplexed using an ideal bandpass filter whose bandwidth equals the channel spacing and passed to the optical receiver. The following parameters are used throughout the paper unless otherwise specified: amplifier spacing *L* = 80 km, channel spacing = 50 GHz, symbol rate = 10 Gbaud, pulse width *T _{FWHM}* (full width at half maximum) = 50 ps, QPSK modulation, nonlinear coefficient and loss of transmission fiber are 1.1 W

^{−1}km

^{−1}and 0.2 dB/km, respectively. A random bit sequence of 100 pulses is used for the pump channel (channel 2) and a single pulse is launched to channel 1. Number of Monte-Carlo simulation is 2000.

To compute the distortion due to SPM, we turned off channel 2 in both analytical model and numerical simulations. The SPM distortion is obtained by subtracting the output field of channel 1 (when channel 2 is turned off) from its linear output obtained by assuming *γ*_{0} = 0. The XPM distortion is obtained by subtracting the SPM distortion from the output field of channel 1 when channel 2 is present.

To find the mean and variance numerically, Monte-Carlo simulation is carried out. Figure 2(a) shows the numerical and analytical first moments of SPM and XPM distortions. Figure 2(b) shows the numerical and analytical second moments of SPM and XPM. Finally, Fig. 3 shows the numerical and analytical XPM variances. The the following parameters are used for Figs. 2 and 3: peak power *P _{peak}* = 0 dBm, fiber dispersion

*β*

_{2}= −10 ps

^{2}/km, and number of spans

*N*= 10. As can be seen in the Figs. 2 and 3, the discrepancy between the analytical and numerical is more at the center of the pulse (higher power) than at the edges of the pulse (lower power). It is due to the fact that the first order perturbation technique is less accurate at higher powers. The maximum discrepancy between the analytical model and numerical simulations for the variance is less than 10%.

As can be seen in Fig. 3, the variance of XPM distortion varies as a function of the temporal position within the symbol interval. So we define the mean variance as

Figure 4(a) shows the mean XPM variance versus peak power. When the launch peak power is 0 dBm, the discrepancy between the analytical model and numerical simulation is less than 10%, whereas it increases to 20% when the launch peak power is 8 dBm which is due to the inaccuracy of first order perturbation technique. To get a better accuracy, higher order perturbation technique can be used [7].

Figure 4(b) shows the dependence of mean XPM variance on the fiber dispersion. From Eq. (54), we find that the variance is caused by the pulse overlapping. So, when the dispersion is low, pulses do not overlap significantly and hence, the variance is quite small. However, the variance grows quickly and beyond 10 ps^{2}/km, it decays slowly. When the dispersion is very large, the XPM effect is reduced since pulses walk-off quickly. The maximum discrepancy between the model and the simulation is less than 10% in this example.

Figure 5(a) shows the mean XPM variance versus number of spans. As can be seen in the Fig. 5(a), the variance of XPM distortion increases almost linearly with the number of spans. This indicates that the total XPM variance can be obtained by adding the XPM variance due to each span. In other words, the XPM distortions in each span may be considered to be independent. However, in the case of dispersion-managed link (see section 4.1), we find that there is correlation among the spans and variances can not be simply added. The quick and rough estimate of the XPM penalty can be done by using the slope of the curve in Fig. 5(a) after one or two spans and it can be extrapolated for a fiber-optic system with larger number of spans. The discrepancy between the model and simulations is less than 10% when the number of spans is 10 which increases to 13% when the number of spans is 20. The discrepancy increases as the number of spans increases which is due to the inaccuracy of first order perturbation technique for long reach and/or large power.

Figure 5(b) shows the XPM variance versus channel spacing. As can be seen, XPM variance decreases significantly as channel spacing increases. This is because as the channel spacing increases, the “fast” channel walks off quickly the “slow” channel and hence, the nonlinear interaction time is reduced.

Next we consider a 28 GBaud WDM system with a channel spacing of 50 GHz. The standard single-mode fiber (SSMF) with the following parameters is used: *β*_{2} = −22 ps^{2}/km, *γ* = 1.1 W^{−1}km^{−1} and loss *α* = 0.2 dB/km. The pulse width *T _{FWHM}* = 35.71 ps. The rest of the parameters are the same as before. Figs. 6(a) and 6(b) show the variances for 2-channel and 5-channel WDM system, respectively. For the 5-channel case, the variance of XPM distortion of the central channel due to other channels is calculated analytically and numerically. Analytical modeling is done by adding the variances of XPM distortion of the central channel due to each channel. This approach ignores the four wave mixing (FWM) among channels, but the numerical simulation includes the FWM impairments. Due to the good agreement between analytical and numerical simulations, we conclude that the FWM is negligible in this example.

#### 4.1. Dispersion-managed system

In this section, accuracy of the model is verified for the dispersion-managed system. Figure 7 shows a typical fiber-optic system with pre-, inline-, and post-compensation. Parameters of the transmission fiber (TF) are as follows: fiber loss *α _{TF}* = 0.2 dB/km, fiber nonlinearity

*γ*= 1.1 W

_{TF}^{−1}km

^{−1}, fiber dispersion

*β*

_{2,}

*= −10 ps*

_{TF}^{2}/km, and fiber length

*L*= 80 km. Dispersion compensating fiber (DCF) is used for inline compensation. The parameters of DCF are as follows: fiber loss

_{TF}*α*= 0.5 dB/km, fiber nonlinearity

_{DCF}*γ*= 4.4 W

_{DCF}^{−1}km

^{−1}, fiber dispersion

*β*

_{2,}

*= 150 ps*

_{DCF}^{2}/km. The virtual DCFs are used in digital domain for pre- and post- compensation. The parameters of the virtual DCF are the same as the real DCF except that its nonlinear coefficient is set to zero. Pre- and post-compensation fibers’ lengths are calculated based on the pre-compensation ratio defined as,

*L*and

_{pre}*L*are pre- and post-compensation fiber lengths, respectively. The pre-, inline-, and post-compensation fiber lengths are chosen so that the total accumulated dispersion at the receiver is zero.

_{post}Figure 8 shows the XPM variance versus the pre-compensation ratio after after 10 spans of transmission. The variance of XPM distortion is nearly independent of the pre-compensation ratio. In contrast, the variance of intrachannel impairments shows a strong dependance on the pre-compensation ratio [16]. From Fig. 8, we find that the maximum discrepancy between the model and simulations is 6%.

Next, we define inline-compensation ratio as

*η*when the pre-compensation ratio is 50%. When

*η*= 100%, it corresponds to a resonant dispersion map in which the residual dispersion per span is zero. As can be seen, this is the worst case since the XPM distortion in each span is identical (up to the first order) and they add up coherently. When

*η*= 0%, dispersion is compensated equally at the transmitter and receiver with no inline-compensation, which is the same as the dispersion uncompensated case except for the fact that pre- and post-compensating fibers have nonlinearity. From Fig. 9, we find that

*η*= 70% is the optimum value which corresponds to a residual dispersion

*D*of 185 ps/nm per span. The maximum discrepancy between the model and simulations in this example is 10%.

_{res}Figure 10 shows the XPM variance as a function of the number of spans *N*. When *η* = 100%, a curve fitting to the simulation data shows that variance scales as *N*^{1.95} while it scales as *N*^{1.75} when *η* = 90% and *N*^{1.54} when *η* = 80%. Typically, it varies as *N ^{x}* where

*x*∈ [1, 2]. It can be explained as follows. For a resonant map (

*η*= 100%), XPM distortion in each span is identical and the coherent addition of XPM fields leads to

*N*

^{2}dependence. For a non-resonant map (

*η*= 0%), the XPM distortion in different spans are nearly independent and hence, the variances due to each span can be added which leads to a linear scaling of XPM variance with

*N*. For the other values of

*η*, there is some correlation between the XPM distortions arising from different spans and hence, the variance scales as

*N*where

^{x}*x*∈ (1, 2). It may be noted that variance builds up slowly for the cases of

*η*= 80% and

*η*= 90% as compared to the case of

*η*= 100%. The maximum discrepancy between the model and simulations in this example is less than 10%.

## 5. Conclusions

A time domain approach based on the first order perturbation theory to calculate the variance of XPM distortion in a WDM fiber-optic system is developed. The analysis is applicable to dispersion uncompensated as well as dispersion managed systems. Validity of this approach is examined by comparing it with Monte-Carlo simulations for a range of launch power, dispersion coefficient, number of spans and channel spacing. In the case of dispersion uncompensated system, it is found that the variance scales almost linearly with the number of spans whereas dispersion managed system, the variance scales as *N ^{x}* where

*x*∈ [1, 2] depending on the dispersion map. In a dispersion managed system, the XPM variance is independent of pre-compensation ratio, but it is sensitive to the inline-compensation ratio. The optimum inline-compensation ratio is 70% for the parameter space of this paper.

## 6. Appendix: Differential equation solution

Taking the Fourier transform of Eq. (19), we have

*f̃*(

*z*,

*ω*) is the Fourier transform of

*f*(

*z*,

*T*) and

*f̃*(0,

*ω*) = 0, is

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