## Abstract

We derive an analytical formula to estimate the variance of nonlinear phase noise caused by the interaction of amplified spontaneous emission (ASE) noise with fiber nonlinearity such as self-phase modulation (SPM), cross-phase modulation (XPM), and four-wave mixing (FWM) in coherent orthogonal frequency division multiplexing (OFDM) systems. The analytical results agree very well with numerical simulations, enabling the study of the nonlinear penalties in long-haul coherent OFDM systems without extensive numerical simulation. Our results show that the nonlinear phase noise induced by FWM is significantly larger than that induced by SPM and XPM, which is in contrast to traditional WDM systems where ASE-FWM interaction is negligible in quasi-linear systems. We also found that fiber chromatic dispersion can reduce the nonlinear phase noise. The variance of the total phase noise increases linearly with the bit rate, and does not depend significantly on the number of subcarriers for systems with moderate fiber chromatic dispersion.

© 2010 OSA

## 1. Introduction

Coherent optical orthogonal frequency division multiplexing (OFDM) has drawn significant attention in optical communications due to its high spectral efficiency using hundreds of subcarriers with higher-order modulation formats and its robustness to fiber chromatic dispersion and polarization mode dispersion [1–5]. However, due to the large number of subcarriers, OFDM is believed to suffer from high peak-to-average power ratio, which makes it less suitable for legacy optical communication systems with periodic inline chromatic dispersion compensation fibers [6]. In Ref [7], a simple formula for estimating the deterministic distortions caused by four-wave mixing (FWM) is developed, and it is found that the nonlinear limit in OFDM systems is independent on the number of OFDM subcarriers in the absence of dispersion. Ref [8]. analytically studied the combined effect of dispersion and FWM in OFDM multi-span systems and concluded that dispersion could significantly reduce the amount of FWM. Recently, significant research effort has been put in nonlinear compensation for coherent OFDM systems [9–16]. Of particular interest is the digital backward propagation [14–16], a technique in which the signal is propagated backwards in distance using digital signal processing (DSP) so that the deterministic linear and nonlinear impairments can be compensated. However, the nonlinear phase noise caused by the interaction between amplified spontaneous emissions (ASE) noise and fiber Kerr nonlinearity, also known as Gordon-Mollenauer effects [17], cannot be compensated using digital backward propagation. Nonlinear phase noise has been studied extensively for single-carrier systems [17–32]; however, to the best of our knowledge, nonlinear phase noise effects have not been investigated for OFDM systems.

In this paper, we derive an analytical formula to calculate the nonlinear phase noise induced by the interaction of ASE with SPM, XPM and FWM in coherent OFDM transmission systems. The analytical model is verified with numerical simulation results, enabling the study of the nonlinear phase noise in coherent OFDM systems without lengthy simulations. With the analytical model, we quantitatively compare the nonlinear phase noise induced by SPM, XPM and FWM, separately, and find that the nonlinear phase noise induced by FWM is dominant compared to that induced by SPM and XPM. This is in contrast to the results of Ref [27]. for WDM systems, in which it is found that ASE-FWM interaction is negligible in quasi-linear systems. This difference is likely due to the sub-carriers of OFDM systems interacting coherently, since they are derived from the same laser source. We also study the effects of fiber chromatic dispersion and the bit rate on the total phase noise in coherent OFDM systems and find that, the total phase noise decreases as fiber chromatic dispersion increases, achieving the limit of linear phase noise. The total phase noise scales up as bit rate increases.

The remainder of the paper is organized as follows. Section 2 describes the mathematical analysis of the nonlinear phase noise. In section 3 we show the validation of the mathematical model with numerical simulation of coherent OFDM systems, and study the impact of fiber chromatic dispersion, number of subcarriers, and bit rate on the variance of the total phase noise. Section 4 gives the conclusion.

## 2. Mathematical analysis for the nonlinear phase noise in coherent OFDM systems

The nonlinear Schrödinger equation governing light propagation in optical fiber is [33]

*γ*is the nonlinear coefficient, $w(z)={\displaystyle {\int}_{0}^{z}\alpha (s)ds}$, and$\alpha (z)$is the fiber loss/amplifier gain profile.

There are large numbers of subcarriers in OFDM systems, making each subcarrier a quasi-cw wave due to low bit rate. The OFDM signal can be described as [8]

*N*is the total number of subcarriers, ${u}_{l}(t,z)$ is the slowly varying field envelope, and ${\omega}_{l}=2\pi l/{T}_{block}$ is the frequency offset from a reference, with ${T}_{block}$ as the OFDM symbol time. First we derive the analytical formula for the variance of nonlinear phase noise including the interaction of ASE noise with SPM and XPM. Second, we include the nonlinear phase noise variance induced by FWM.

#### 2.1 SPM and XPM induced nonlinear phase noise

Inserting Eq. (2) into (1) and considering the effects of SPM and XPM only, we have

Within each OFDM block, ${u}_{l}$ is constant; therefore, the first and second order derivative of ${u}_{l}$ with respect of time, appearing in Eq. (3), can be ignored. Now the exact solution of Eq. (3) can be written as

We define ${L}_{s}$as the fiber span length, and the signal is periodically amplified by amplifiers located at $\kappa {L}_{s}$, *κ* = 1, 2, …, *M*, where *M* is the total number of fiber spans. Consider the noise added by the amplifier located at $\kappa {L}_{s}$. Let us expand the noise field as a discrete Fourier transform

Strictly speaking, the noise field should be expressed as a Fourier transform instead of a discrete Fourier transform. In other words, we have approximated the noise as a field with 2*N* degrees of freedom (DOF) instead of infinite degrees of freedom. For a linear system that employs matched filter at the receiver, 2*N* DOFs accurately describe the noise process [32]. In Ref [17]. it is argued that 2 DOFs per carrier (or total 2*N* DOFs) is sufficient to describe the noise field even in a nonlinear system. The total field immediately after the amplifier located at $\kappa {L}_{s}$ is

We assume that the ASE noise is a white noise process with power spectral density ${\rho}_{ASE}$, from which it follows that,

Now treating ${u}_{l}{}^{+}(t,\kappa {L}_{s})$ as the initial field, Eq. (3) is solved to obtain the field at the end of the optical system, located at $z=M{L}_{s}$, as

The linear phase noise is embedded in the term ${u}_{l}+{n}_{l}\text{'}$, and the nonlinear phase noise caused by SPM and XPM by the amplifier located at $z=\kappa {L}_{s}$ is:

Squaring Eq. (13) and making use of Eq. (9), we obtain the variance of the nonlinear phase noise caused by SPM and XPM

Assuming that the number of subcarriers carrying data is ${N}_{e}$ (equivalently the oversampling factor is $N/{N}_{e}$) and each subcarrier has equal power, summing Eq. (14) over all amplifiers, we obtain the nonlinear phase noise variance caused by SPM and XPM

#### 2.2 FWM induced nonlinear phase noise

Substituting Eq. (2) in Eq. (1), and considering only the FWM effect, we obtain the following equation with the quasi-cw assumption

The solution to Eq. (16) for the field ${u}_{l}$ is as follows

Now consider the noise added by the amplifier located at $\kappa {L}_{s}$. The optical field immediately after the amplifier is shown in Eq. (7). Equation (17) is solved using the initial condition of Eq. (7). Replacing ${u}_{l,{z}_{0}}$ in Eq. (17) with ${u}_{l}{}^{+}(\kappa {L}_{s})$, we obtain the optical field at the end of the fiber span as

Ignoring the higher order term of ${n}_{l}$, we have

From Eq. (22), we have

This distortion can be compensated with the digital backward propagation, and thus it has no impact on the nonlinear phase noise. The third term on the RHS of Eq. (23) $\delta {u}_{l}(M,\kappa )$ describes the ASE-FWM interaction and can be expanded as

After the digital backward propagation removes the deterministic distortions, the phase noise of the received field is

#### 2.3 Total phase noise

In summary, the total phase noise for the *l*
^{th} subcarrier in an OFDM system including the linear phase noise and nonlinear phase noise (induced by interaction between ASE and SPM, XPM and FWM) is as follows:

## 3. Results and discussions

In this section, we validate our analytical model for the variance of the total phase noise in OFDM systems given by Eq. (35) with numerical simulation. The following parameters are used throughout the paper unless otherwise specified: the bit rate is 10 Gb/s, the amplifier spacing is 100 km, and the noise figure (NF) is 6 dB. A single type of fiber is used between amplifiers. To separate the deterministic (although bit pattern dependent) distortions due to nonlinear effects from the ASE-induced nonlinear noise effects, we use digital backward propagation [14–16]. Since digital backward propagation compensates for both dispersion and deterministic nonlinear effects, we do not use the cyclic prefix. 2048 OFDM frames are used to get a good Monte Carlo statistics. Each OFDM subcarrier is modulated with binary-phase-shift-keying (BPSK) data. Figure 1 shows the coherent OFDM system structure in our simulation.

For Figs. 2
-3
, we choose a fiber dispersion *D* of 1 ps/nm/km and a total launch power of 0 dBm. Here we use only one subcarrier (*N _{e}* = 1) to carry data while the total number of subcarriers is 8 (8th-folder oversampling), so that the nonlinear phase noise model that includes SPM effects alone can be validated. The subcarrier carrying data is located at the central of the OFDM spectrum, corresponding to the first subcarrier

*U*

_{0}in Fig. 1 due to FFT operation. The signal spectrum before entering into the fiber span is shown in Fig. 2. And in Fig. 3, the solid lines show the analytical linear phase noise and nonlinear phase noise variance induced by SPM only, the dashed line with circles show the numerical simulation results for the variance of linear phase noise and SPM induced nonlinear phase noise, as a function of fiber propagation distance. As can be seen, the agreement is quite good.

In order to validate the nonlinear phase noise model including the ASE interaction with SPM, XPM, and FWM effects in Eq. (35), we use 8 subcarriers to carry data for an OFDM system with 64 subcarriers. The subcarrier carrying data is located at the central of the OFDM spectrum, corresponding to the subcarriers *U _{0}*~

*U*and

_{3}*U*~

_{60}*U*in Fig. 1. Figure 4 shows the OFDM signal spectrum, and Fig. 5 shows the variance of the linear phase noise and nonlinear phase noise for numerical simulation (dashed line with circles) and analytical calculation (solid line), respectively. We see that very good agreement is achieved, which validates our model for the nonlinear phase noise considering SPM, XPM, and FWM effects.In Ref [7], the authors showed that the nonlinear degradation due to FWM effects in OFDM systems is nearly independent of the number of ODFM subcarriers used in the system in the absence of chromatic dispersion, while in Ref [8]. the authors studied the chromatic dispersion effects on the FWM and showed that chromatic dispersion could decrease the FWM effects significantly. However, both of these analyses focused on the deterministic nonlinear effects. In this section, we will study the dependence of the nonlinear phase noise effects on fiber dispersion and bit rate in an OFDM system with digital backward propagation.

_{63}In Figs. 6
-8
, we fix the transmission distance to be 1000 km, the total number of subcarriers is 128 with 64 subcarriers carrying data (two-fold oversampling). Figure 6 shows the OFDM signal spectrum. Figure 7
shows the variance of the total phase noise (linear + nonlinear phase noise) as a function of the launch power, for *D =* 17 ps/nm/km and D = 0 ps/nm/km. Solid lines and circles show the analytical results and the numerical simulation results, respectively. As can be seen from Fig. 7, the nonlinear tolerance increases significantly as the fiber chromatic dispersion parameter increases. It is also shown in Fig. 7 that the total variance of the phase noise initially decreases with launch power since the linear phase noise is dominant at low launch powers. However, as the launch power increases beyond −2 dBm, the variance increases with D = 0 ps/nm/km since nonlinear phase noise becomes dominant at higher powers.

In Fig. 8, we show the impact of the bit rate on the total phase noise for a transmission fiber with *D* = 17 ps/nm/km and *D* = 0 ps/nm/km. The total launch power is −3 dBm. Solid lines show the analytical results, while filled circles show the numerical simulation results. From Fig. 8, we see that the variance of the total phase noise scales linearly with the bit rate. This could be explained by the fact that, with the increase of the bit rate, the OFDM symbol time ${T}_{block}$ decreases, which leads to the increase of the total phase noise as described in Eqs. (15), (33) and (34). The qualitative explanation for the increase in phase noise when the bit rate increases is as follows: as the bit rate increases, OSNR requirement for a given BER increases. This is because the receiver filter bandwidth scales with bit rate, which leads to the increase of the total noise within the receiver bandwidth. The same thing happens with phase noise: the total phase noise within the receiver bandwidth increases as the bit rate increases.

For a BPSK system with coherent detection, from [35], one would obtain the bit error rate *BER* as a function of the phase noise variance ${\sigma}^{2}$ as

In Fig. 9 , we show the impact of the number of subcarriers on the variance of total phase noise, obtained analytically using Eq. (35). Two-folder oversampling is used in the simulation. The total launch power is −3 dBm, the bit rate is 10 Gb/s. Figure 9 shows that, in the absence of dispersion, the variance of total phase noise scales linearly with the number of subcarriers, while with moderate levels of dispersion, the variance of total phase noise is almost constant because the linear phase noise is dominant for such systems.

Finally Fig. 10 shows the variance of the nonlinear phase noise as a function of propagation distance for SPM induced nonlinear phase noise alone (dashed line), XPM induced nonlinear phase noise alone (dashed line with ‘x’), and FWM induced nonlinear phase noise alone for D = 0 ps/nm/km (solid line with circles), D = 10 ps/nm/km (solid line with triangles) and D = 17 ps/nm/km (solid line with diamonds), obtained analytically using Eqs. (15) and (34). From Fig. 10, we see that, for an OFDM system with large number of subcarriers, nonlinear phase noise induced by FWM is significantly larger than that induced by SPM and XPM. This is in contrast to the results of Ref [27]. for WDM systems, in which it is found that ASE-FWM interaction is negligible in quasi-linear systems. This difference is likely due to the fact that the subcarriers of OFDM system are derived from the same laser source and interact coherently. We also see that, with moderate levels of fiber chromatic dispersion, the nonlinear phase noise induced by FWM decreases since the phase matching becomes more difficult.

## 4. Conclusions

In summary, we derive an analytical formula for the variance of the nonlinear phase noise in an OFDM system, taking into account the interaction of ASE noise with SPM, XPM and FWM effects. Our analytical results agree well with the numerical simulation. In addition, we quantitatively compared the nonlinear phase noise induced by SPM, XPM and FWM, and found that the nonlinear phase noise induced by FWM is the dominant nonlinear effect in OFDM systems with digital backward propagation. We also studied the effect of dispersion and bit rate on the nonlinear phase noise. Our results show that nonlinear phase noise can be suppressed with fiber chromatic dispersion, approaching the limit of linear phase noise for large dispersion values. Finally, we show that for OFDM systems, the variance of the total phase noise scales linearly with the channel bit rate.

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