## Abstract

A probability-based model is developed to describe cross phase modulation in multichannel multilevel amplitude/phase modulated coherent systems. Standard deviation of nonlinear phase-shift is evaluated in 16-QAM coherent systems accordingly and by numerical simulation for different values of chromatic dispersion and symbol rate. Furthermore, an error analysis is provided to evaluate the accuracy of the model which demonstrates maximum relative error of 12% in the field of interest.

© 2011 OSA

## 1. Introduction

Although recent advances in digital signal processing (DSP), in conjunction with coherent receiving techniques and Polarization Multiplexing (PolMux), have created a promising opportunity for multilevel phase/amplitude modulation formats in optical communication systems, Cross Phase Modulation (XPM) is known to be the most important limitation [1]. DSP algorithms executed on high-speed electronic processors are capable of estimating and tracking the frequency and phase errors, and they can also compensate for some transmission impairments including Chromatic Dispersion (CD) [2–4]. However impairments from XPM are not easily managed by DSP techniques [5]. A frequency domain DSP-based phase recovery method has been recently proposed to mitigate XPM phase noise for Quadrature Phase Shift Keying (QPSK) systems [6, 7]. While this is potentially applicable to more complex modulation formats, such investigations might benefit from a simple and intuitive understanding of XPM.

Considerable effort has been put into studying and modelling XPM in past decades, resulting in different methods, each focusing on features and requirements of the system under study. In On-Off Keying (OOK) systems where information is modulated onto the intensity of light wave, XPM is explained in two steps; first, intensity fluctuations impose a phase shift on the optical signal through the Kerr effect and second, the nonlinear phase shift is converted into intensity (PM-IM conversion) as a result of Group Velocity Dispersion (GVD) and this unwanted intensity is detected at the receiver [8–10]. Phase modulated systems such as Phase Shift Keying (PSK) and QPSK are more susceptible to fiber nonlinearities since the imposed phase shift directly affects the received signal. Amplified Spontaneous Emission (ASE) from optical amplifiers is known to be the main source of intensity fluctuations in such systems and causes nonlinear phase noise. The ASE accompanying optical signals in the same channel can affect the phase through Self Phase Modulation (SPM) as well as that from neighbouring channels which alters the signal through XPM [11, 12].

With the thriving introduction and application of coherent multilevel phase/amplitude modulation systems, particularly 16-point Quadrature Amplitude Modulation (16-QAM) which is an attractive choice for channel bitrates of 100 Gb/s and beyond [13, 14], nonlinear phase noise is no longer the only source of Kerr impairment. In addition, phase shifts caused by signal power fluctuations in other channels through fiber nonlinearity are of practical interest for such systems. This is similar to mixed format hybrid systems, where intensity fluctuations of OOK channels affect the phase of PSK or QPSK channels directly through fiber nonlinearity [15–17].

In this paper we present a time-domain probability based model to explain nonlinear XPM, and apply it to a 16-QAM lumped amplified dispersion compensated coherent system. In section 2 we provide the theoretical background and needed assumptions to explain the statistical nature of the model. Then in section 3 we apply the model to systems with different parameters and compare the results with the results obtained by simulation.

## 2. Principles and limits of the model

Nonlinear phase shift, Δ*ϕ _{NL}*, prompted by the optical power

*P*of any other channel is equal to 2

_{i}*γP*where

_{i}z*γ*is fiber nonlinear parameter and

*z*is the length of interaction.

*P*changes with time and degrades from its ideal shape because of filters and fiber impairments such as CD. The solution for Δ

_{i}*ϕ*is generally achieved by solving the Coupled Nonlinear Schrödinger Equations (CNLSE) numerically and the bit error rate is calculated according to deviations of nonlinear phase shift. These solutions are complicated and offer little insight. We seek a more intuitive approach.

_{NL}#### 2.1. Assumptions

We make three simplifying assumptions which lead to our probability-based model for nonlinear aberration of the phase of a received multilevel QAM signal. First, we ignore the effect of optical filters on the shape of signals. Second, we consider the interfering channels to be immune to all nonlinearities. Third, we assume zero dispersion within the bandwidths of probe and pump channels individually, which translates into the preservation of signal shape along the fiber. This assumption has proven to be useful in [17] in the analysis of XPM for hybrid optical coherent systems. It should be noticed that we do not neglect the effect of dispersion completely. In fact, as it will be explained in more detail later, the walk-off effect, which is basically a manifestation of different propagation velocities at different channels caused by fiber dispersion, has a vital role in the model. These assumptions guarantee that the pump signal keeps its initial ideal shape and only its power decays as the signal propagates along fiber. Based on the third assumption it is possible to define a single point on probe signal which encounters nonlinearities due to each copropagating pump channel. We then consider the total XPM in a multichannel system to be the sum of contributions from each independent interfering channel. Based on the model, we will also explain how the nonlinear phase shift grows in a multi-span system. We show later that these assumptions provide results that are in close agreement with simulation.

#### 2.2. Variance of the phase shift

While the approach is general, we consider an implementation of 16-QAM in which three different intensity levels *P*
_{1}, *P*
_{2} and *P*
_{3} are present, as deduced from the constellation shown in Fig. 1(a). A point on a probe signal encounters a random sequence of three power levels which decay due to fiber attenuation, *α*, as shown in Fig. 1(b) and therefore, nonlinear impact of the first colliding symbol is greater than the second one and so on. The nonlinear impact at the end of the fiber is negligible and we would only consider the interaction along the effective length of fiber. Waveform distortion due to CD shown in Fig. 1(b) at different positions is considered a secondary effect in the model and is neglected. In building the phase shift sample space, it is important to notice that the occurrence probability of the three power levels are not the same. Half of the constellation points correspond to the middle power, *P*
_{2}, a quarter of constellation points correspond to high power, *P*
_{3}, and another quarter correspond to low power, *P*
_{1}. Consequently the occurrence probability of *P*
_{1} or *P*
_{3} is 0.25 while probability of *P*
_{2} is 0.5.

An important factor in the extent of XPM, as previously mentioned, is the walk-off effect which is a result of group velocity mismatch between pump and probe channels. As any sample point on a probe signal traverses the effective length of fiber *L _{e}*, it experiences XPM from

*N*symbols of pump signal, which is equal to

_{W}*B*is symbol rate,

_{S}*D*is the fiber dispersion coefficient, Δ

*ν*is channel spacing,

*λ*is optical wavelength and c is the speed of light in vacuum.

We define the normalized nonlinear phase shift due to *m*th walk-off symbol as:

*L*=

_{s}*L*/

_{e}*N*is symbol walk-off length. The nonlinear phase shift caused by

_{W}*m*th symbol,

*δϕ*(

*m*), is

*P*

_{1}Φ(

*m*) with a probability of 1/4,

*P*

_{2}Φ(

*m*) with a probability of 1/2 and

*P*

_{3}Φ(

*m*) with a probability of 1/4. Simple calculations give us the following relationship for variance of the total nonlinear phase shift: where ${P}_{e}^{2}$ is given by:

Linearisation of exponential for large values of *N _{W}* results in the following approximation for variance of nonlinear phase shift:

Figure 2 examines the validity of the approximation given in Eq. (5) where relative error of the approximation is given in percent with respect to *N _{W}*. As the graph shows, the relative error is less than 10% for values of

*N*bigger than 3.

_{W}Since bit streams in different channels are independent, the standard deviation of the total nonlinear phase shift *σ _{ϕt}* in a multichannel system could be calculated as sum of uncorrelated variables:

*σ*is calculated for

_{ϕi}*i*th pump.

In a multi-span system, where dispersion is compensated in each span and optical amplifiers maintain the original launched power into fiber, any point on probe signal experiences the same set of pump power levels in each span. Therefore, nonlinear phase shift is multiplied by the number of spans and so is the standard deviation:

where*N*is number of spans and

_{s}*σ*is standard deviation of phase shift in one span.

_{ϕ}#### 2.3. Accuracy of model

The model assumptions are the sources of an error between the predicted nonlinear phase shift and what would result from CNLSE simulation. Although pump power and therefore XPM are much higher at the beginning of the fiber where signal has its closest form to ideal, we anticipate that ideal non-filtered shape of pump signal (first and second assumptions) imposes more nonlinear phase shift since XPM is higher for steeper intensity edges. Furthermore, the fact that different frequency elements of the probe signal travel at different speeds, which was ignored by third assumption, could be interpreted as another source of error. Lower-frequency elements of a signal coincide with different parts of a pump signal than higher frequency elements. Although higher and lower frequency sides embody less energy and most of signal’s power resides around the center, nonlinear phase shifts caused by higher and lower frequency components induce an error from our model when segments add up after dispersion compensation.

Additionally, the model ignores the issue of synchronization between pump and probe at the beginning of the fiber. In fact, a probe sample point may coincide with any section of a pump symbol when it starts propagating, which would lead to different XPM phase shifts. This is less important for large *N _{W}* since the probe sample point passes over the pump symbol faster, but it could be of more concern for smaller values of

*N*.

_{W}To quantify the accuracy of the model we introduce the error (in percent) as:

*σ*is given by Eq. (3) and

_{ϕ}*σ*is the standard deviation of nonlinear phase shift given by CNLSE simulation.

_{ϕS}## 3. Results and comparison

In this section we provide and compare results given by the presented model and results from numerical solutions of CNLSE. The symmetric split-step method is used to solve CNLSE with minimum step size 1 m, maximum step width 1000 m, and maximum phase change 0.05 degrees. Number of symbols used to simulate is 512. All CNLSE results were carried out using VPItransmissionMaker* ^{TM}*.

All results are for a 16-QAM coherent system. A 100 km fiber with 0.2 dB/km attenuation, 0.08 ps/(nm^{2}.km) dispersion slope, 2.6×10^{−20}m^{2}/W nonlinear index and effective core area of 80 *μ*m^{2}is used. ASE is neglected, signals are dispersion compensated and no distributed amplification is performed. Channel spacing is set to 50 GHz and the probe is at 1551.66 nm. The bandwidth of the optical filters are twice the symbol rate, and the probe launched power is kept at a low value to minimize the effect of SPM on *σ _{ϕ}*.

Figure 3(a) shows the standard deviation of nonlinear phase shift with launched power for a two channel (one pump - one probe) 10 Gbaud system. Results from both CNLSE simulation and our model are presented for dispersion parameters of 8, 12, 16 and 20 ps/(km.nm). Agreement between results from the model and from CNLSE is good and it appears that the model provides an upper limit on *σ _{ϕ}*. Similar graphs are shown in Fig. 3(b) for 15 Gbaud and in Fig. 3(c) for 20 Gbaud. Graphs of Fig. (3) reveal that for any of the three symbol rates, the difference between results from simulation and the model are smaller for bigger values of dispersion. Also it can be seen that for any value of dispersion, the difference between the results from simulation and the model are smaller for bigger values of symbol rate. These facts are in agreement with the prediction of subsection 2.3 that the model is a better approximation for larger

*N*.

_{W}Figure 4(a) illustrates the dispersion dependence of the nonlinear phase shift. The standard deviation of nonlinear phase shift, *σ _{ϕ}* is given for two sybmol rates (10 and 20 Gbaud) and launched power of 26 mW. The graph shows that agreement between the model and simulation breaks down for very small values of dispersion. The model is no longer valid for small values of

*N*as we previously mentioned, and since the number of walk-off symbols is small for small dispersions the break down is expected. On the other hand, agreement between the model and simulation gets better for larger dispersions since pump/probe synchronization at the beginning of the fiber is less of an issue for larger values of

_{W}*N*.

_{W}Figure 4(b) evaluates the validity of the multiple channel description of the model given in Eq. (6) which depicts the standard deviation of nonlinear phase shift with launched power for two-pump and four-pump systems. Channels are 50 GHz apart and probe is in the middle. Fiber dispersion is 16 ps/(nm.km). There is a good agreement between model and simulation.

Figure 5 shows the standard deviation of the phase with number of spans for fiber dispersion parameters 8 and 20 ps/(nm.km) at 15 Gbaud. It is evident from the graphs that simulation results follow Eq. (7) as well, maintaining the relative error at almost the same value.

Figure 6 shows the values of error between simulation and model, *Err*, given in Eq. (8) with symbol rate and fiber dispersion in a contour diagram. The values of *Err* are less than 12% and can get as samll as 2% in the given range of dispersion parameters and symbol rates. Higher values of dispersion and symbol rate product, or equivalently higher number of walk-off symbols, *N _{W}*, results in

*Err*closers to zero and better match between the model and simulation.

The presented results show a good agreement between the model and solutions of CNLSE. The model works for multi channels systems as well as muslti span systems. It is important to notice that in contrast to the fact that model’s third assumption neglects the effect of CD on the shape of signals, the model is a better approximation for larger values of dispersion. We could consider this as an indication for the validity of the third assumption.

## 4. Conclusion

We presented a mathematical model for nonlinear cross-talk in multilevel QAM coherent systems. The model is based on three assumptions and uses probability methods to describe XPM in such systems. We applied the model to 16-QAM format for different values of dispersion and symbol rate and compared resultant standard deviations of nonlinear phase shift to those given by numerical simulation. This comparison reveals that our model overestimates the phase shifts from XPM by at mose 12% for typical contemporary symbol rates and dispersion parameters bigger than 8 ps/(nm.km). Hence we expect that this simple model provides a convenient tool for estimating XPM impairment from any intensity dependent (e.g. QAM) modulation format and for developing future techniques to mitigate this impairment.

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