## Abstract

There has been a trend of migration to high spectral efficiency transmission in optical fiber communications for which the frequency guard band between neighboring wavelength channels continues to shrink. In this paper, we derive closed-form analytical expressions for nonlinear system performance of densely spaced coherent optical OFDM (CO-OFDM) systems. The closed-form solutions include the results for the achievable Q factor, optimum launch power density, nonlinear threshold of launch power density, and information spectral efficiency limit. These analytical results clearly identify their dependence on system parameters including fiber dispersion, number of spans, dispersion compensation ratio, and overall bandwidth. The closed-form solution is further substantiated by numerical simulations using distributed nonlinear Schrödinger equation.

©2010 Optical Society of America

## 1. Introduction

High spectral efficiency transmission can be readily achieved with the concept of coherent optical OFDM (CO-OFDM) [1]. In such systems, the CO-OFDM wavelength channels can be either continuously spaced without frequency guard band [2–5], or densely spaced with extremely small frequency guard band [6,7]. These densely spaced systems present the ultimate limit of achieving high spectral efficiency by allowing very little or no frequency guard band. Most recently, nonlinear transmission performance of CO-OFDM systems have attracted much attention [8–12]. In particular, analytical results are shown in [8] for single-channel transmission without consideration of chromatic dispersion; complete analytical expressions are presented in [9] involving summation of a large number of terms for practical OFDM systems; system performances via numerical simulation is reported in [10–12]. It would be of great interests to derive concise closed-form solutions that capture the dependence of the nonlinear performance on some major system parameters such as chromatic dispersion and dispersion compensation ratio. Similar analytical work was pioneered in [13] where nonlinear launch power and information capacity are derived in closed-form. However, there are two limitations for the report [13]: (i) it only includes the cross phase modulation (XPM) as the dominant effect ignoring four-wave-mixing (FWM) and self-phase-modulation (SPM). This only applies to sparsely spaced WDM systems and would not apply to the densely spaced CO-OFDM systems where XPM, FWM, and SPM are all important, and very often indistinguishable [8–11]; (ii) it assumes that nonlinear phase noise is generated independently in different spans, ignoring an important phase array effect of the FWM products that accounts for the interference among multiple spans [9]. In this paper, we derive closed-form analytical expressions for nonlinear system performance of densely spaced CO-OFDM systems. The closed-form solution entails the results for achievable Q factor, optimum launch power density, nonlinear threshold of launch power density, and information spectral efficiency limit. These analytical results clearly identify the nonlinear performance dependence on system parameters including fiber dispersion, number of spans, dispersion compensation ratio, and overall bandwidth. The closed-form solutions are further substantiated by numerical simulations using distributed nonlinear Schrödinger equation.

## 2. Theoretical derivation and analysis

The class of transmission systems, namely, densely spaced OFDM (DS-OFDM) systems that is treated in this paper is depicted in Fig. 1(a)
where each wavelength channel is OFDM modulated with subcarrier frequency spacing of $\Delta f$ and bandwidth of *W*, separated with neighboring channel by a frequency guard band of $\Delta B$. We define the term of ‘densely spaced’ as the condition of $\Delta B<<\text{\hspace{0.17em}}W$. As such, the frequency guard band can be omitted in the remainder of the investigation. With such an assumption, we study the continuous ‘single-band’ like multi-carrier systems with $\Delta f$subcarrier spacing and total bandwidth of *B* as shown in Fig. 1(b). In such DS-OFDM systems, all the nonlinear effects such as XPM, FWM, and SPM can be considered as FWM between all the subcarriers if we treat multiple densely spaced wavelength channels as an effective big ‘single-band’ OFDM channel that encompasses all the subcarriers. FWM is a third-order nonlinearity effect and its impact on optical fiber communications has been extensively studied [14,15]. Due to the FWM, the interaction of subcarriers at the frequencies of${f}_{i}$, ${f}_{j}$,and${f}_{k}$produces a mixing product at the frequency of ${f}_{g}={f}_{i}+{f}_{j}-{f}_{k}$. The magnitude of the FWM product for *N _{s}* spans of the fiber link is given by [14,15]

*α*and

*L*are respectively the loss coefficient and length of the fiber per span,

*γ*is the third-order nonlinearity coefficient of the fiber, ${\eta}^{\prime}$ is the FWM coefficient which has a strong dependence on the relative frequency spacing between the FWM components given by

In Eq. (2) the overall FWM efficiency is decomposed into two separate contributions: (i) ${\eta}_{1}^{\prime}$, the FWM efficiency coefficient for single span (for simplicity, the contribution from dispersion compensation fiber is omitted), and (ii) ${\eta}_{2}^{\prime}$, the interference effect between *N _{s}* spans of FWM products, also known as phase array effect [9,14]. $\Delta {\beta}_{ijk}\equiv {\beta}_{i}+{\beta}_{j}-{\beta}_{k}-{\beta}_{g}$is the phase mismatch in the transmission fiber. In Eq. (3), we have assumed that the span loss ${e}^{\alpha L}$is much larger than 1, ${e}^{-\alpha L}{e}^{-j\Delta {\beta}_{ijk}L}$is removed from the nominator. In Eq. (4), the subscript ‘1’ stand for the parameters associated with the dispersion compensation fiber (DCF)). Assuming

*m*th subcarrier frequency has the form of ${f}_{m}=m\cdot \Delta f$, the phase mismatch terms $\Delta {\beta}_{ijk}$ and $\Delta {\tilde{\beta}}_{ijk}$in Eq. (4) and Eq. (5) can be rewritten as

*D*(or${D}_{1}$) is the chromatic dispersion of the transmission fiber (or DCF),

*ρ*(or

*ζ*) is the dispersion compensation (or residual dispersion) ratio, ${D}_{r}$is the residual chromatic dispersion per span accounting for both transmission fiber and DCF. At the end of each span, the FWM product ${{P}^{\prime}}_{g}$along with the signal will be amplified by a gain of G equal to the loss of each span ${e}^{-\alpha L}$and the FWM product becomes

We adopt the approach used in [13] where the nonlinear effect is considered as the multiplicative noise to the signal. In essence, we will consider *i*th subcarrier as the reference frequency, and *j* and *k* frequencies as the interferers, namely, frequency *j* and frequency *k* generates a beating frequency component at $\left({f}_{j}-{f}_{k}\right)$, which in turn modulates the subcarrier *i*, creating fourth components of *f _{g}*. It has been shown for large number of subcarriers, the non-degenerate FWM is the dominate effects [9], and

*D*is set to 6 in Eq. (7). Consequently, the nonlinearity impinging on subcarrier

_{x}*i*, ${P}_{{}_{NL}}^{i}$is given as

A factor of one half is added in Eq. (7) because of the double counting in the dual summation. Equation (8) can be understood as the number of photons or amount of energy scattered off the subcarrier *i,* and should be equivalent to the photons scattered into this subcarrier *i* with large bandwidth assumption which we will clarify later. From now on, we drop index *i* and set it to zero, or equivalently, we are investigating the performance of center wavelength channels in broad bandwidth DS-OFDM systems. We also assume all the subcarriers have the same power of *P* for the sake of simplicity. The FWM power at the center subcarriers becomes

*N*of 4000 subcarriers. This implies that in order to compute the FWM effect, it requires a summation in the order of 16 millions (4000x4000) of FWM terms in Eq. (10). Aside from the apparent mathematical complexity, the physical interpretation of FWM dependence on various key system parameters is difficult to ascertain. It is highly desirable to have a concise closed-form solution to the nonlinearity products in Eq. (10).

_{s}Although corroborating by the numerical simulation using distributed Schrödinger equation is the ultimate validation of the closed-form solutions, we would like to go through the derivation step by step below stating our assumptions and intermediate procedures towards final analytical solutions, in order to ensure certain degree of mathematical rigorousness is enforced. In each step we first summarize the main task, lay out the assumption and its justification, and then present the operations as a result of the simplification.

#### 2.1 Conversion from discrete summation to integration

We observe that the FWM coefficient ${\eta}_{1}$ represents the phase array effects and the major contribution of the summation takes place where $jm\cdot \Delta {f}^{2}/{f}_{PA}^{2}\le 1$. This implies that dominate contribution is coming from the terms where $j=m=\mathrm{int}\left({f}_{PA}/\Delta f\right)$ where ‘int’ is the integer round off function. In this work, we assume that phase array effect bandwidth ${f}_{PA}$is much larger than $\Delta f$, namely

which is generally true in CO-OFDM systems. It can be easily shown around $j=m=\mathrm{int}\left({f}_{PA}/\Delta f\right)$, the phase $jm\Delta {f}^{2}/{f}_{PA}^{2}$ inside*sine*function is changing slowly each time when

*j*or

*m*is changed by 1, therefore the conversion from discrete summation is justified as far as the ${\eta}_{1}$is concerned. Similarly if we assume the walk-off bandwidth ${f}_{W}$is much larger than the subcarrier spacing, namelyconversion from discrete to integration can be also justified in relation to ${\eta}_{2}$. We call the conditions of Eqs. (11) and (12) as ‘dense subcarrier’ assumption. Under the assumptions of Eqs. (11) and (12), substituting the continuous integral variable

*f*for $m\cdot \Delta f$, ${f}_{1}$ for $j\cdot \Delta f$, the FWM power is transformed into

According to the definition of *m* in Eq. (10), the variable *f* represents the frequency of the multiplicative noise impairing the channel. We now introduce more convenient and also fundamentally more important terms, power (spectral) densities given by

*I*are respectively FWM noise power (spectral) density and launch power (spectral) density. Substituting Eq. (14) into Eq. (13), we arrive at the FWM noise density

#### 2.2 Conversion of the integration range to more manageable forms

The proof is given in the Appendix A. The FWM power density becomes

#### 2.3 Closed-form expressions for nonlinear noise density

Since *f* is the nonlinearity noise frequency, the integration over *f _{1}* in Eq. (16) would generate the nonlinear noise spectral density. We rewrite Eq. (16) in terms of the one-sided nonlinear multiplicative noise spectral density ${i}_{{}_{NL}}^{}\left(f\right)$ given by

The nonlinear noise spectral density ${i}_{NL}\left(f\right)$has the unit of dBc/Hz similar to phase noise or relative intensity noise (RIN). ${i}_{NL}\left(f\right)$can be integrated into a close-form, the derivation of which is shown in Appendix B. The result of the integration gives

*1/f*noise. This finding makes the authors deduce that our derivation may help explain one class of the flicker noise, namely, third order nonlinearity and dispersion may be one type of mechanisms to produce of

*1/f*noise and the lower bound of the

*1/f*noise signature, ${B}_{0}$ inversely proportional to the bandwidth of the participating noise,

*B*as shown in Eq. (17). Substituting Eq. (20) into Eq. (19), we finally arrive at the closed form expression for the nonlinear noise power density ${I}_{{}_{NL}}^{}$

#### 2.4 Signal-to-noise ratio and spectral efficiency limit in the presence of nonlinearity

The signal power in presence of the nonlinear interference can be expressed as [13]

The noise can be considered as the summation of the white optical amplified-spontaneous-noise (ASE), ${n}_{0}$ and the FWM noise, and is shown given by [13]

*NF,*

*h*is the Planck constant, and

*υ*is the light frequency. The signal-to-noise is thus given by

For the SNR larger than 10, Eq. (26) can be approximated as

The simplification is generally valid for the case of interests where the signal power density is much smaller than ${I}_{0}$.

We have verified through our simulation under ‘dense subcarrier’ and ‘large bandwidth’ assumptions of Eqs. (11), (12), and (18), the FWM noise is of Gaussian distribution. This is also verified previously in [16]. Under the assumption of Gaussian noise distribution, the information spectral efficiency (defined as the maximum information capacity *C* normalized to bandwidth *B*) for single-polarization is readily given by [17]

From Eq. (28), the maximum spectral efficiency ${S}_{opt}$ in the presence of fiber nonlinearity can be easily shown as

#### 2.5 Optimal launch power density, maximum Q, and nonlinear threshold of launch power density

In Eq. (28), the ultimate spectral efficiency is obtained. However in practice, the performance is always lower because of the practical implementation of modulation and coding. We therefore derive a few important parameters that are commonly used in the optical communications community. The first one is the maximum achievable Q factor. Under the Gaussian noise assumption and QPSK modulation, the Q factor is equal to the SNR given by

The optimum launch power density is another important parameter and is defined as the launch power density where the maximum Q takes place. By simply differentiating Q of Eq. (30) over *I*, and setting it to zero, we obtain the optimum launch power density and the optimal Q given by

One of the inconveniences of using the optimum launch power expression in Eq. (31) is that it is dependent on the amplifier noise figure. Another commonly used term is nonlinear threshold launch power density that is defined as the maximum launch power density at which the BER due to the nonlinear noise can no longer be corrected by a certain type of forward-error-correction (FEC) code. For standard Reed-Solomon code RS(255, 239), the threshold *Q* is 9.8 (dB), or linear *q _{0}* of 3.09. In Eq. (30), setting

*n*to zero and

_{0}*Q to*${q}_{0}^{2}$, we arrive at the nonlinear threshold of power density

*q*is the correctable linear Q for a specific FEC.

_{0}The closed-form expressions for nonlinear noise spectral density ${i}_{NL}\left(f\right)$ in Eq. (20), nonlinear noise power density ${I}_{{}_{NL}}^{}$in Eq. (21), nonlinear multi-span noise enhancement factor ${h}_{e}$in Eq. (22), nonlinear characteristic power density ${I}_{0}^{}$in Eq. (23), information spectral efficiency *S* in Eq. (28), system Q factor and its optimal value in Eq. (30) and Eq. (32), optimal launch power density in Eq. (31), and nonlinear threshold of launch power density in Eq. (33) comprise the major findings in this work.

## 3. Corroboration of the theories with numerical simulation

During the derivation in Section 2, there are quite many assumptions and approximations made in order to arrive at concise closed-form expressions. The concern is that whether several approximations may accumulate and make the closed-form solutions inaccurate. In this section, we intend to corroborate the theoretical results with the numerical simulations. Among all the analytical results, the FWM noise density ${I}_{NL}$of Eq. (21) is the most fundamental one, and other expressions can be considered as the derivatives of the ${I}_{NL}$. We have conducted simulation to validate the expression for ${I}_{NL}$.

The parameters for the simulated systems are as follows: 16 wavelength channels, each covering 31-GHz bandwidth, giving total bandwidth B of 496 GHz; OFDM subcarrier frequency spacing of 85 MHz; QPSK modulation for each subcarrier; no frequency guard band between wavelength channels; 10-span of 100 km fiber link; Fiber loss *α* of 0.2 dB/km; nonlinear coefficient$\gamma =1.22{\text{W}}^{-1}k{m}^{-1}$; noise figure of the amplifier of 6 dB; The FWM noise density is simulated by using an perfect optical notch filter to notch out 100 MHz gap at the center of the input signal spectrum, and the power density is measured at the output after 1000-km transmission. Figure 2(a)
shows the simulated nonlinear noise density compared with the computed nonlinear density using the closed-form expression of Eq. (21). Three transmission systems are investigated: (i) SSMF type system with CD of 16 ps/nm/km without any dispersion compensation, abbreviated as ‘system I’, (ii) CD of 16 ps/nm/km, but with dispersion 95% compensated per span, abbreviated as system II, and (iii) non-zero dispersion-shifted type fiber with CD of 4 ps/nm/km, abbreviated as ‘system III’. For systems I, II, and III, the average difference of FWM density is 14%, 12%, and 17%. This shows excellent match between the closed-form formula and simulation, considering the extreme sensitivity of the FWM density as a function of launch power density (cubic dependence). We also perform the simulation of the system Q factors with the above-described three systems, the results of which is shown in Fig. 2(b). We can see a good match between theoretical expressions based on Eq. (23) and Eq. (30) and simulation results. For instance, the difference between the optimal Q from theory and simulation is within 0.15 dB for all simulated dispersion maps. The difference of launch power between the simulation and closed-form theory for the same Q factor is less than 0.4 dB for wide range of launch power density of −28 to −16 dBm/GHz. All these confirm the excellent match between the simulation and the closed-form expression of Q factors in (30).

#### 3.1 Discussion about the closed-form expression

Because the concise closed-form expressions are available, we are ready to quickly identify their dependence on system parameters including fiber dispersion, number of spans, dispersion compensation ratio, and overall bandwidth. In this section, we will discuss in detail the achieved system Q factor, optimum launch power, information spectral efficiency, and multi-span noise enhancement factor

#### 3.2 System Q factor and optimum launch power

The immediate benefits of having closed-form formulas of Eq. (31) and Eq. (32) for system Q factor and optimum launch power density are their scaling over the underlying parameters. From Eq. (31) and Eq. (32), it follows that for every 3 dB increase in fiber dispersion, there is 1 dB increase in the optimal launch power density and the achievable Q; for every 3 dB increase in fiber nonlinear coefficient *γ*, there is 2 dB increase in the optimal launch power and achievable Q. We can quickly generate the optimum launch power and achievable Q for variety of dispersion maps. We use the three systems, systems I, II and III studied in Figs. 2 and Fig. 3
as an example. As shown in Fig. 3(a), system I has the best performance due to large local dispersion and no per-span dispersion compensation. The advantage of system I over system II increases with the increase of the number of spans, for instance from 0 dB to 2.4 dB when the reach increases from single-span to 10 spans. The advantage of system I over system III is maintained at 1.7 dB independent of the number of spans. However, Fig. 3(b) shows the optimal launch power versus number of spans. The optimum launch powers for non-compensated systems, systems I and III are constant. This is because both the linear and nonlinear noises increase linearly with the number of the spans that leads to the optimum power independent of the number of spans. However, for the dispersion compensated system II, the optimum launch power density decreases with number of spans due to the multi-span noise enhancement effect. Another interesting observation from Eq. (31) and Eq. (32) is that both the optimal Q factor and launch power has very week dependence on the overall system bandwidth: proportional to 1/3 power of logarithm of the overall bandwidth. It can be easily shown that for both systems I and II, the Q is decreased by only about 0.7 dB with the 10-fold increase of the overall system bandwidth from 400 to 4000 GHz whereas system II incurs a larger decrease of the Q factor of 0.84 dB with the same bandwidth increase.

#### 3.3 Information spectral efficiency

The information spectral efficiency is important as it represents the ultimate bound of what we can achieve by employing all possible modulations (of course not limited to QPSK) and codes. For large SNR, we simplify Eq. (28) into

*γ*be decreased by a factor of 2.8, or number of spans be reduced by a factor of 2, all of which are difficult to achieve. In a nutshell, it is of diminishing return to improve the spectral efficiency by modifying the optical fiber system parameters. The only effective method to substantially improve the spectral efficiency is to add more dimensions such as resorting to polarization multiplexing that leads to a factor of 2 improvement, or fiber mode multiplexing by at least a factor of two or more dependent on the capability of achievable digital signal processing (DSP). Figure 4 shows the achievable spectral efficiency for the three systems studied in Section 3.1. The only modification is that we assume 40 nm or 5 THz for the total bandwidth. The spectral efficiency for the systems I, II, and III are respectively 5.17, 4.40, and 4.52 b/s/Hz. This shows a total capacity of 25 Tb/s can be achieved for 10x100 SSMF uncompensated EDFA-only single-polarization systems within C-band.

#### 3.4 Multi-span noise enhancement factor

The multi-span noise enhancement factor ${h}_{e}$of Eq. (22) is one of the most important findings in this report. This noise enhancement effect is ignored in the prior analytical results [13]. This multi-span interference effect can be understood as the phase array effect that has been discussed in report [9]. The noise enhancement is referred to the important fact that the overall nonlinear FWM noise of multi-span systems is enhanced by a factor of ${h}_{e}$over the scenario for which the nonlinear FWM noise originated in each span is assumed independent without interference with each other. We note the expression of ${h}_{e}$in Eq. (22) is the first concise close-from result where the multi-span interference effects of all the FWM products are accounted for. From Eq. (22), we conclude that as long as the factor $\alpha L\zeta $is much larger than 1, ${h}_{e}$approaches 1, namely, the nonlinear noise generated in each span can be treated independently in this regime. However, even when the fiber loss $\alpha L$is large but *ζ* is small, or dispersion compensation ratio is large, ${h}_{e}$can be significantly high. Figure 5
shows the noise enhancement factor ${h}_{e}$as a function of dispersion compensation ratio $\rho =1-\zeta $for span losses of 10 and 20 dB. It can be seen that for a dispersion compensation ratio of 95%, the nonlinear noise is enhanced by 8.5 and 7.3 dB for span losses of 10 and 20 dB respectively. It shows that multi-span noise enhancement cannot be ignored even when the span loss is as large as 20 dB if the compensation ratio is higher than 50%.

## 4. Conclusion

In this paper, we have derived closed-form analytical expressions for nonlinear system performance for densely spaced CO-OFDM systems. The closed-form solutions include the results for the achievable Q factor, optimum launch power density, and nonlinear threshold of launch power density, and information spectral efficiency limit. These analytical results clearly identify their dependence on system parameters including fiber dispersion, number of spans, dispersion compensation ratio, and overall bandwidth. The closed-form solution is further substantiated by numerical simulations using distributed nonlinear Schrödinger equation.

## Appendix A: Change of the integration range.

We first argue that we can move the integration boundary for *f* from [-*B*/2-*f*
_{1}, *B*/2-*f*
_{1}] to more tidy form of [-*B*/2, *B*/2]. This is reasonable simplification as the major contribution of the integral is when *f* is around${f}_{W}$. In essence, this fringe effect of subtracting *f _{1}* can be ignored when

*B >>*${f}_{W}$. This corresponds to the ‘large bandwidth’ assumption in Eq. (18). Under the large bandwidth assumption, Eq. (15) can be rewritten as

*f*and

_{1}*f*, Eq. (35) can be expressed as

We now partition ${I}_{{}_{NL}}^{1}$into three regions of integration as follows:

where

We will show below that *A _{1}* is dominant over

*A*and

_{2}*A*, and therefore the contribution of

_{3}*A*and

_{2}*A*can be omitted as a good approximation.

_{3}We first study the special case of *N _{s}* of 1. It is one of the most important cases as it corresponds to the system scenarios for which nonlinear noises originated from different spans are uncorrelated, and the overall noise is simply

*N*times of the noise of one span. As we show later, an uncompensated fiber system is almost a perfect example of independent span noises. Substituting

_{s}*N*

_{s}= 1 into Eq. (38), it can be easily shown that ${A}_{1}$ can be completely integrated in a closed-form as

We now carry out the integration for *A _{2}* over

*f*, we have

_{1}In Eq. (40), we have used the inequality of $x>\mathrm{arg}\mathrm{tan}(x)$ for $x>0$, and $B>>{B}_{0}$, which can be considered as part of the ‘large bandwidth’ assumption. It can be seen that *A _{2}* is upper bounded by $2/{f}_{W}^{2}$, the ratio between

*A*/

_{1}*A*is upper bounded by $\frac{\pi}{4}\mathrm{ln}\left(B/{B}_{0}\right)$. Under large bandwidth assumption of $B>>{f}_{W}$, we consider this is a large number and thus

_{2}*A*in general can be ignored. The upper bound for the term

_{2}*A*can be found as follows

_{3}Therefore the ratio between *A _{1}*/

*A*is upper bounded by $\pi \mathrm{ln}\left(B/{B}_{0}\right)$. Consequently under the large bandwidth assumption,

_{3}*A*is the dominant contribution to ${I}_{{}_{NL}}^{1}$, and can be used as an good approximation to ${I}_{{}_{NL}}^{1}$.

_{1}We now move to the multi-span scenario for which there is an additional factor of ${\eta}_{1}$in the integrand of Eq. (38). We show that by introducing this factor, the conclusion that *A _{1}* being a dominate component in ${P}_{{}_{NL}}^{1}$still stands. ${\eta}_{1}$generates a interference pattern with a main lobe in the low frequency region. When considering the ratio between

*A*and

_{1}*A*, the weight will shift toward

_{2}*A*as

_{1}*A*is integrated at the low frequency of

_{1}*f*, [0,

_{1}*B*/2], and

*A*is integrated at the high frequency of

_{2}*f*, [

_{1}*B*/2,

*∞*]. Therefore the ratio of

*A*/

_{1}*A*should be no less than $\frac{\pi}{4}\mathrm{ln}\left(B/{B}_{0}\right)$ we derived for ${\eta}_{1}$= 1 for the single-span case, and subsequently

_{2}*A*can be ignored to a good degree of approximation. Regarding the ratio of

_{2}*A*/

_{1}*A*, the integration variable ‘

_{3}*f*’ is over low frequency region of [0,

*B*

_{0}/2] for ‘

*A*whereas over [

_{3}’*B*

_{0}/2,

*B*/2] for

*A*. So the existence of the phase array pattern ${\eta}_{1}$will shift weight toward

_{1}*A*. However, we will show the ratio of

_{3}*A*/

_{1}*A*is still upper bounder by a factor in the order of $\pi \mathrm{ln}\left(B/{B}_{0}\right)$ to a good approximation. The phase array effects dependent on the phase difference between each span largely determined by ${f}_{PA}$. When ${f}_{PA}$>>${f}_{W}$, for instance, the dispersion is almost completely compensated, ${\eta}_{1}$remains a constant over a large range frequency range. This is closed to the scenario for the single-span case for which we have shown

_{3}*A*can be ignored. Therefore we only need to study the ratio of

_{3}*A*/

_{1}*A*when ${f}_{PA}$<<${f}_{W}$ as this is the scenario for which the main lob shifts close to the low frequency, favoring

_{3}*A*. We would like to derive the ratio of

_{3}*A*/

_{1}*A*when${f}_{PA}<<{f}_{W}$. ${\eta}_{1}$can be also expressed as

_{3}Substituting Eq. (42) into Eq. (38), we have

As the upper boundary of integration for variable *f* in Eq. (43), $n{f}_{W}^{2}/\left(4{f}_{PA}^{2}\right)$ is much larger than 1 under the assumption of${f}_{PA}<<{f}_{W}$, and can be considered approximated as infinity. Using formula${\int}_{0}^{\infty}\mathrm{sin}\left(f\right)/f\text{\hspace{0.17em}}}df=\pi /2$ and after some simple arrangement, Eq. (43) can be transformed into

In Eq. (44), we have used the assumption of ${f}_{PA}^{}$<< ${f}_{W}^{}$ to justify omitting the second term. It can be also shown that using the closed-form developed for *A _{1}* in Appendix B, under assumption of ${f}_{PA}^{}$<< ${f}_{W}^{}$,

*A*can be approximated as

_{1}Combing Eq. (44) and Eq. (45), we conclude that *A*
_{1}/*A*
_{3} is also upper bounded in the order of $\pi \mathrm{ln}\left(B/{B}_{0}\right)$. In summary, we conclude that *A*
_{1} can be used as an approximation for ${I}_{{}_{NL}}^{1}$ for a wide range of the system parameters. Unless specifically mentioned in this paper, we consider

Using the definition of ${i}_{NL}\left(f\right)$in Eq. (19), we have

There are two other important factors that make Eq. (46) a very good approximation: First, in Eq. (37), not only are *A*
_{2} and *A*
_{3} small compared to *A*
_{1}, but they also have opposite signs and their combined effects tend to cancel each other, which further improves the accuracy of using *A*
_{1} as an approximation to ${I}_{{}_{NL}}^{1}$. Second, the phase array effect will tend to deemphasize the contribution of *A*
_{2} and add more weight to *A*
_{3}. However, because *A*
_{3} is integrated in the low frequency of *f*, in practical application, the low frequency noise including DC components can be estimated and removed. Therefore the lower integration boundary for *f*, *B*
_{0}/2 in Eq. (46) can be also interpreted as that the low frequency noise from [0, *B*
_{0}/2] is being removed due to the phase estimation. Therefore, to be complete, we redefine the B_{0} as

where ${B}_{PE}$is the phase estimation bandwidth which is equal to half of the subcarrier channel spacing $\Delta f$in CO-OFDM systems, or the phase locked loop (PLL) bandwidth in single-carrier systems. This is to accommodate the scenario for which the phase estimation bandwidth is high enough such that the lower integration boundary for *f* in Eq. (46) should be ${B}_{PE}$ instead of ${f}_{W}^{2}/B$.

## Appendix B: Closed-form solution for FWM noise spectral density

In deriving close-form for ${I}_{{}_{NL}}^{1}$ we will repeatedly employ a useful formula of complex functional analysis as follows:

where$f\left(x\right)$is an analytical function over the upper half of the complex plane. Substituting Eq. (42) into Eq. (47), we have

where ‘Re’ stands for operation of extracting the real component. We have exchanged variable ${f}_{1}f$ to ${f}_{2}$ in Eq. (50). Applying formula of Eq. (49) into Eq. (50), we obtain

Substituting the definitions for ${f}_{W}$and ${f}_{PA}$in Eq. (9) into Eq. (51), we have

Substituting Eq. (52) into Eq. (47), we arrive at the FWM noise spectral density

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