Optimization of Split Transmitter-Receiver Digital Nonlinearity Compensation in Bi-Directional Raman Unrepeatered System

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1 applied sciences Article Optimization of Split Transmitter-Receiver Digital Nonlinearity Compensation in Bi-Directional Raman Unrepeatered System Qiang Zheng 1, Zhilin Yuan 1, Yuan Li 2 and Wei Li 1, * 1 Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan , China; zheng_qiang@foxmail.com (Q.Z.); zhilin_yuan@accelink.com (Z.Y.) 2 Department of Computer Science, Central China Normal University, Wuhan , China; yuanli@mail.ccnu.edu.cn * Correspondence: weilee@hust.edu.cn; Tel.: Received: 7 May 2018; Accepted: 8 June 2018; Published: 13 June 2018 Abstract: A oretical model of nonlinear signal-to-noise interaction (NSNI) in a bi-directional Raman amplified system with receiver-side digital back-propagation (DBP) or split-dbp is given, which is helpful for design of such a system. In proposed model, distributed Raman gain and spontaneous Raman scattering are taken into account. The results of oretical calculation are compared with results of transmission simulations, which indicates that oretical model matches well with results of simulations when pre-compensation length is less than 100 km. For cases of pre-compensation lengths more than 100 km, oretical model has an error of less than 0.1 db compared with simulations. By using oretical model, efficiency of split-dbp is analyzed, and results are compared with transmission simulations. Both results of oretical calculation and simulations show that split-dbp can effectively mitigate NSNI in such a system. By adopting split-dbp, with an appropriate pre-compensation length, signal-to-noise ratio (SNR) of signal increases by about 1 db. In addition, impact of double Rayleigh scattering (DRB) is also analyzed using proposed model, and results show that DRB has little impact on system. Keywords: optical fiber communications; digital back-propagation; Raman amplification; nonlinear signal-to-noise interaction; nonlinearity mitigation 1. Introduction In last decade, digital signal processing (DSP) algorithm has been developed and modern DSP-based coherent receivers can fully compensate for linear channel impairments. Consequently, nonlinear impairments, which include mainly Kerr effects such as self-phase modulation (SPM), cross-phase modulation (XPM), and four-wave mixing (FWM), limit transmission performance [1 3]. The digital back-propagation (DBP) algorithm is now commonly acknowledged as one of most suitable candidates for joint linear and nonlinear impairments compensation [4 6]. In DBP, received electric signals propagate in a simulated virtual fiber that has opposite parameters to practical transmission fiber, and dispersion and nonlinearity of practical transmission fiber are eliminated. However, for traditional DBP algorithms, noise in system, such as Amplified Spontaneous Emission (ASE), double Rayleigh scattering (DRB), and transceiver noise, is not taken into account. In practical optical fiber communication system, such noises in fiber would interact with signal through Kerr effect, which is called nonlinear signal-to-noise interaction (NSNI). As noises are not considered in traditional DBP algorithm, NSNI induced by noises cannot be dealt with. Recently, an algorithm called stochastic DBP was proposed, in which complex probability Appl. Sci. 2018, 8, 972; doi: /app

2 Appl. Sci. 2018, 8, 97 of 10 method was adapted to compensate for NSNI caused by Erbium-doped fiber amplifiers (EDFAs) in link [7,8]. Moreover, once DBP algorithm was applied to system, noise would continue interacting with signal during DBP process, thus causing extra NSNI. There has been a study of impact of transceiver noise on performance of DBP, which indicated that transceiver noise would induce extra NSNI in receiver-side DBP, thus limiting performance of DBP [9]. To mitigate extra NSNI, a new approach called split-dbp was proposed and studied, which involves moving a part of DBP from receiver side to transmitter side. By this means, part of NSNI can be avoided and performance is improved [10,11]. In Reference [10], D. Lavery et al. demonstrated that split nonlinear compensation scheme is better than both post- and pre-compensation schemes in EDFA amplified systems through simulation. Now, distributed Raman amplification is regarded as a mature and promising amplification scheme for next-generation fiber optical communication systems [12,13]. Actually, DBP algorithm has been applied in Raman amplified unrepeatered systems [14,15], but NSNI caused by distributed Raman gain and spontaneous Raman scattering is in se cases ignored as well. Unlike in EDFA amplified system, gain in Raman amplified systems is distributed along fiber; consequently, noise, which includes ASE from EDFAs, spontaneous Raman scattering, and double Rayleigh scattering, is amplified along fiber. As a result, NSNI in a Raman amplified system is worse than that in EDFA amplified systems. In this paper, we oretically deduced NSNI in a bi-directional Raman amplified unrepeatered system with receiver-side DBP and split-dbp, considering effects of distributed Raman gain, transceiver noise, DRB, and spontaneous Raman scattering along fiber. The origin of extra NSNI in system was studied. Theoretical analysis indicated that extra NSNI can be effectively mitigated by adopting split-dbp. Then performance of split-dbp was investigated through oretical model and simulations, using a bi-directional Raman amplified system with 32 GBd 16QAM modulation. The impact of DRB on system was also studied. 2. Theoretical Model Consider an unrepeatered system with bi-directional Raman amplification and receiver-side DBP, as shown in Figure 1. Assuming that nonlinear signal-to-signal (S-S) interaction is perfectly compensated by receiver-side DBP, effective receiver signal-to-noise ratio (SNR) can be described as: SNR P s P ASE + P S N, (1) where P s is average power of signal; P ASE is average power of ASE noise, including ASE noise from EDFAs and spontaneous Raman estimation; and P S-N is power of NSNI. In Raman amplified system, spontaneous Raman scattering noise is generated along fiber. Considering spontaneous Raman scattering generated at position z, denoted by E sprs (z), it will propagate along fiber with a length of L-z and interact with signal, where L is length of whole fiber. After virtual transmission in DBP with a length of L-z, interaction of noise and signal in practical fiber is eliminated. However, noise continues to propagate and interact with signal in residual virtual fiber of length z, causing extra NSNI. The field of extra NSNI can be described as a four-wave mixing (FWM) process [16]: E NSNI (z, ω q ) = γρ(2l z, 2L)E s (z, ω i )E s (z, ω j )E sprs (z, ω k) +2γρ(2L z, 2L)E s (z, ω i )E s (z, ω k )E sprs (z, ω j ), (2)

3 Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 10 Appl. Sci. 2018, 8, of 10 where ENSNI (z, ω q) is field of NSNI at angular frequency ωq = ωi + ω j ωk, γ is nonlinear coefficient, E s is signal field, and ρ(2 L z,2 L) is FWM efficiency from where E NSNI (z, ω q ) is field of NSNI at angular frequency ω q = ω i + ω j ω k, γ is nonlinear position 2L-z to 2L. The latter can be written as: coefficient, E s is signal field, and ρ(2l z, 2L) is FWM efficiency from position 2L-z to 2L. The latter can be written as: 2L 2L Δ i β ρ(2 L z,2 L) = ( )e ρ(2l z, 2L) = P ξ dξ P(ξ)e i β (3) dξ, (3) 2L z, 2L z where P( ξ ) is normalized power profile in fiber, i j k g where P(ξ) is normalized power profile in fiber, β = β Δ i + β = β j β + ββ k ββ g is β phase is mismatch, phase and βmismatch, i,j,k,g represents and β i, j, kpropagation, g represents constants propagation for angular constants frequencies for angular ω i,j,k,g frequencies. Note that ω i, in j, k, Equation g. Note (2) FWM that in of Equation two noise(2) frequencies FWM of and two anoise signal frequencies frequencyand is neglected a signal frequency becauseis neglected power of because signal is much greater power of than signal noise. much greater than noise. Figure Figure 1. Nonlinear 1. signal-to-noise interaction (NSNI) (NSNI) in a bi-directional in a bi-directional Raman system Raman with system with receiver-side receiver-side digital digital back-propagation (DBP). (DBP). ω g Integrating Integrating Equation Equation (2) (2) within within band of of signal, we canget ge NSNI NSNI field field at at ω : g : E S N (, (z, z ω g ) = E (, z ω ) dωdω =, (4) E NSNI (z, ω g )dω i dω j, (4) S N g NSNI g i j ω i,j,g [ B 2 BB ωi, j, g [, B 2, ] ] where B is signal bandwidth. Assuming that noise and signal are independent random variables, where B is powers signal of bandwidth. two terms Assuming in Equation that (2) noise canand be handled signal are independently, independent random and we can obtain power of NSNI: variables, powers of two terms in Equation (2) can be handled independently, and we can obtain power of NSNI: P S N (z) = 3γ 2 ρ(2l z, 2L) 2 P s (ω i )P s (ω j )P sprs (ω i + ω j ω g )dω i dω j dω g 2 2 P ω S N( i,j,g z) [ = B 2, B 2 ] 3 γ ρ(2 L z,2 L) Ps( ωi) Ps( ωj) PspRs( ωi+ ωj ωg) dωidωjdω, (5) g B B 3γ 2 η(z)pω 2 i, j, g [, ] s P sprs, (5) s 3 γη( zpp ) sprs where P s,sprs (ω) is power of signal and spontaneous Raman scattering at angular frequency ω, P s,sprs where is average PsspRs, ( ω power ) is of power signal of andsignal spontaneous and spontaneous Raman scattering, Raman scattering and η(z) at is angular nonlinear interaction frequency coefficient, ω, P which s, sprs is canaverage be written power as of [15]: signal and spontaneous Raman scattering, and η () z is nonlinear interaction coefficient, which can be written as 2 [15]: η(2l z, 2L) = ρ(2l z, 2L) dωi dω j, (6) 2 η(2 L z,2 L) = ω i,j,g [ B 2, B 2 ] ρ(2 L z,2 L) dωidωj, (6) BB i, j, g [, ] ω Integrating Equation (5) along fiber, we can obtain total NSNI power at end of DBP: P S N = L 0 3γ 2 η(2l z, 2L)P 2 s (z)p sprs (z)dz, (7) Note that in Equation (7), transceiver noise can be added into P sprs (z) at positions 0 and L, for transmitter noise and receiver noise, respectively.

4 L S N = 3 γη(2,2 ) s ( ) sprs( ) 0 P L z L P z P z dz, (7) Appl. Sci. 2018, 8, 972 sprs 4 of 10 Note that in Equation (7), transceiver noise can be added into at positions 0 and L, for transmitter noise and receiver noise, respectively. Because of amplification of forward Raman amplifier, signal power in front part Because of amplification of forward Raman amplifier, signal power in front part of of fiber is very high, and in DBP high-power region is at end. As in preceding fiber analysis, is very most high, and of in overcompensation DBP high-power parts region are in is at high-power end. As in region. preceding Thus, analysis, most of overcompensation overcompensation parts will parts produce are in intense high-power extra NSNI and region. impair Thus, performance overcompensation of signal. parts will produce Split-DBP intense can effectively extra NSNI reduce and impair extra NSNI performance caused by DBP. of When signal. split-dbp Split-DBP is applied can effectively in reducesystem, extra as shown NSNI in caused Figure by2, DBP. NSNI When includes split-dbp two is parts. applied Part in one is system, same as shown case inof Figure 2, NSNI receiver-side includesdbp, two parts. which Part is caused one isby same overcompensation as case of of receiver-side DBP, DBP. which Part istwo caused by occurs overcompensation because NSNI of in receiver-side fiber near DBP. transmitter Part twoside occurs is not because compensated. NSNI In insame fiber way, near transmitter we can determine side is not total compensated. power of NSNI In at same end way, of wereceiver-side can determine DBP by total following: power of NSNI at end of receiver-side DBP byl following: PS N= 3 γη( z, l) P P S N = l s( z) PspRs( z) dz 0 3γ 2 η(z, l)ps 2 (z)p sprs (z)dz, (8) L 0 + L + 3 (2,2 ) ( ) ( ), (8) γη L z L l P 3γ 2 s z P η(2l z, 2L l)ps 2 sprs z dz l (z)p sprs (z)dz l where l is pre-compensation length of transmitter-side DBP. By substituting Equations (7), (8) where l is pre-compensation length of transmitter-side DBP. By substituting to (1), final SNR of signal can be calculated. In practical calculations, by dividing fiber Equations (7), (8) to (1), final SNR of signal can be calculated. In practical calculations, by dividing into several fiberblocks, into several integral blocks, in Equations integral (7) and in Equations (8) can be simplified (7) and (8) to can summation be simplified of to summation divided offiber blocks. divided fiber blocks. P ( z) Figure Figure 2. NSNI 2. NSNI in ain bi-directional a Raman amplified systemwith with split-dbp. 3. Simulation Setup 3. Simulation Setup Numerical simulations were carried out with a bi-directional Raman amplified system, as shown Numerical in Figure 3. simulations In transmitter, were carried a laser with out with a linewidth a bi-directional of 100 khz Raman and an amplified ideal IQ modulator system, as were shown in Figure assumed 3. Into generate transmitter, signal. a laser A 32 with Gbaud a linewidth 16QAM signal of 100was khz generated and an ideal with a IQpseudo-random modulator were assumed binary to generate sequence of length signal A The 32 Gbaud signal was 16QAM sampled signal at eight wassamples/symbol generated withand a pseudo-random filtered by a binarygaussian sequence low-pass of length filter 2 15 for -1. pulse The signal shaping. was Two sampled EDFAs, ateach eight with samples/symbol a noise figure of and 4 db, filtered were by a Gaussian deployed low-pass filter booster for pulse amplifier shaping. and Two pre-amplifier EDFAs, each before wi atransmission noise figure of link 4 db, and were before deployed as coherent booster amplifier receiver. The andtransmission pre-amplifier link before was a 300-km transmission long fiber, link with and attenuation before coefficients coherentof receiver. 0.2 The transmission link was a 300-km long fiber, with attenuation coefficients of 0.2 db/km for Appl. Sci. 2018, 8, x FOR PEER REVIEW signal and 0.26 db/km for pump, dispersion coefficient 17 ps nm 1 km 1 5 of 10 and nonlinear coefficient 1.1 W 1 km db/km 1. The for signal signal propagation and 0.26 db/km in for fiber pump, was dispersion simulatedcoefficient by solving 17 ps nm modified 1 km 1 and nonlinear nonlinear coefficient 1.1 W Schrödinger equation using 1 km split-step 1. The signal propagation in fiber was simulated by solving Fourier method (SSFM) [17]. modified nonlinear Schrödinger equation using split-step Fourier method (SSFM) [17]. Figure 3. Simulation setup: LD: laser diode; IQ mod.: IQ modulator; OBPF: optical band-pass filter; Figure 3. Simulation setup: LD: laser diode; IQ mod.: IQ modulator; OBPF: optical band-pass filter; Co. Rx: coherent receiver. Co. Rx: coherent receiver. The wavelengths of signal and pump were 1550 nm and 1455 nm, and distributed Raman gain was calculated by solving power coupled equations of distributed Raman amplifier ([17], Equations ). In simulation, forward Raman pump power was set at 450 mw, and backward Raman pump power was set at 600 mw. The relative intensity noise (RIN) transfer from Raman pump to signal was simulated by a time variation of distributed Raman gain. The time variation was produced by multiplying RIN of pump laser and RIN transfer function (TF) from pump to signal [18]. The RIN of pump laser was set at 120 dbc Hz 1.

5 Appl. Sci. 2018, 8, of 10 The wavelengths of signal and pump were 1550 nm and 1455 nm, and distributed Raman gain was calculated by solving power coupled equations of distributed Raman amplifier ([17], Equations (8.1.2) (8.1.3)). In simulation, forward Raman pump power was set at 450 mw, and backward Raman pump power was set at 600 mw. The relative intensity noise (RIN) transfer from Raman pump to signal was simulated by a time variation of distributed Raman gain. The time variation was produced by multiplying RIN of pump laser and RIN transfer function (TF) from pump to signal [18]. The RIN of pump laser was set at 120 dbc Hz 1. The step size of SSFM was 0.01 km and in each step spontaneous Raman scattering was added to signal. After transmission, an optical band-pass filter (OBPF) was deployed with a central wavelength of 1550 nm and a bandwidth of 100 GHz. In coherent receiver a laser with same parameters as those in transmitter was used as local oscillator. The photodetectors in receiver were modeled with ideal response properties, and rmal noise and shot noise were considered. The rmal noise was modeled as white noise in spectrum with A Hz 1/2. Then received signal was down-sampled to two samples/symbol for subsequent DSP. In receiver, DSP carrier recovery was applied to compensate for phase shift of laser. Two nonlinear compensation methods were applied in simulation: receiver-side DBP and split-dbp. To fairly compare performance of transmitter-side DBP and split-dbp, in split-dbp scheme, signal was processed by transmitter-side DBP at two samples/symbol, and n processed signal was up-sampled to eight samples/symbol for fiber transmission. The electrical SNR of signal was calculated after DSP. 4. Results and Discussion 4.1. Accuracy of Theoretical Model In Section 2 we gave expression of final SNR of signal in a bi-directional Raman amplified system with receiver-side DBP or split-dbp. In this section, we will verify accuracy of model through simulation and demonstrate efficiency of split-dbp. First, to find optimal pre-compensation length, different pre-compensation lengths were applied, and SNRs of signals were calculated using Equations (7) and (8) with different incident powers. Correspondingly, transmission simulations were carried out with same parameters. The SNRs of signals in transmission simulations were calculated using error vector magnitude (EVM). As shown in Figure 4a, after calculating average amplitude of each point of signal constellation and standard deviation of noise, EVM of signal can be calculated, n SNR of signal can be calculated by SNR = 1/EVM 2. The simulation results were compared with oretical results, and both are shown in Figure 4b. Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 10 (a) (b) Figure 4. (a) The calculation method of error vector magnitude (EVM) of signal, (b) signal-to- calculation ratio (SNR) method vs. pre-compensation of error length. vector magnitude (EVM) of signal, (b) signal-to-noise Figure 4. (a) Thenoise ratio (SNR) vs. The pre-compensation length. In Figure 4b, black lines are oretical calculation results, colorful marks are results of simulations, and different curves denote different incident powers. Comparing results of ory and simulations, it can be found that results of oretical calculation fit well with simulations when pre-compensation length was less than 100 km. When precompensation length was more than 100 km, SNR of oretical calculation was larger than simulation, which we attributed to influence of up-sampling and low-pass filter, which brings error to transmitter-side DBP. As we can see, when pre-compensation was 0, representing receiver-side DBP, with incident power increasing from 6 dbm to 8 dbm, SNR of signal decreased. This indicates that

6 where P is power of DRB, ζ is Rayleigh backscattering coefficient, α is fiber Appl. Sci. 2018, 8, of 10 In Figure 4b, black lines are oretical calculation results, colorful marks are results of simulations, and different curves denote different incident powers. Comparing results of ory and simulations, it can be found that results of oretical calculation fit well with simulations when pre-compensation length was less than 100 km. When pre-compensation length was more than 100 km, SNR of oretical calculation was larger than simulation, which we attributed to influence of up-sampling and low-pass filter, which brings error to transmitter-side DBP. As we can see, when pre-compensation was 0, representing receiver-side DBP, with incident power increasing from 6 dbm to 8 dbm, SNR of signal decreased. This indicates that NSNI caused by DBP affects performance of signal. When split-dbp was applied, SNR of signal increased first and n remained at a high level, with pre-compensation length increasing. For different incident powers, optimal pre-compensation lengths were all round 50 km, but had small differences, and trend was that optimal pre-compensation length decreased with an increase in incident power. Then, to determine efficiency of split-dbp, pre-compensation length was set to 0 and 50 km, and SNRs of signals were calculated with different incident powers, as well as simulations. The results are shown in Figure 5. Similarly, in this figure black lines denote results of oretical calculation and colorful marks denote results of simulations. Because pre-compensation length was less than 100 km, results of oretical calculation matched well with simulations. And we can see, by adopting split-dbp, optimal incident power increased from 6 dbm to 8 dbm, and maximum SNR increased by about 1 db. This demonstrates efficiency of NSNI mitigation of split-dbp in bi-directional Raman amplified system. However, we can see that when incident power was over 8 dbm, SNR of signal still decreased quickly, Appl. Sci. 2018, 8, x FOR PEER REVIEW 7 of 10 which indicates re are or complicated nonlinear effects that impact signal. Figure SNR vs. vs. incident power Impact Impact of of Double Double Rayleigh Rayleigh Backscattering Backscattering In In oretical oretical analysis analysis and and simulation simulation above, above, DRB DRB in in system system was neglected. was neglected. In this In section this section we take we take DRB DRB into consideration, into consideration, and analyze and analyze impact impact of of DRB DRB on on system. system. In In proposed proposed oretical oretical model, model, we calculate we calculate NSNI caused NSNI by caused transceiver by noise transceiver and spontaneous noise and Raman spontaneous scattering. Raman These scattering. two kinds These of noise two can kinds beof treated noise as can dot be treated noise, as which dot means noise, that which means noise isthat generated noise within is generated a very small within distance. a very small However, distance. DRB However, in system DRB cannot in be system treated cannot as be same treated way. as It issame obvious way. that It is obvious DRB atthat a point DRB is relevant at a point to is relevant Raman gain to and Raman signal gain and power insignal whole power following whole fiber. following To include fiber. To DRB include to proposed DRB to model, proposed DRB model, noise should DRB noise be should be converted into dot noise. For simplicity, we firstly calculate average power distribution of DRB along fiber [19]: L z 2 PDRB( z) = Ps ( z ) ζ exp [ αs + grpp ( z )] dz dz z z (9)

7 Appl. Sci. 2018, 8, of 10 converted into dot noise. For simplicity, we firstly calculate average power distribution of DRB along fiber [19]: P DRB (z) = L z z P s (z )ζ 2 exp [ α s + g R P p (z )]dz dz (9) z where P DRB is power of DRB, ζ is Rayleigh backscattering coefficient, α s is fiber attenuation at wavelength of signal, g R is Raman gain coefficient, and P p is Raman pump power. Note that in Equation (9) consumption of pump due to DRB is ignored. Then DRB noise can be generated at each point in fiber according to its statistical distribution. In oretical model and simulation DRB is treated as Gaussian additive noise. Although exact statistical characteristic of DRB is not very clear yet, according to central limit orem, Gaussian approximation of DRB in a small section of fiber is acceptable. After obtaining power profiles of signal and pumps, power profile of DRB can be calculated using Equation (9). With same parameters as simulation above, we calculated power profile of DRB, which is shown in Figure 6. For comparison, power profiles of signal and spontaneous Raman scattering are plotted in Figure 6 as well. As we can see from figure, compared to spontaneous Raman scattering, DRB has higher power in near-transmitter section, but lower power in near-receiver section of fiber. Because signal power in near-transmitter section of fiber is much higher than that in near-receiver section, noise in near-receiver side section has more severe impact on optical signal-to-noise ratio (OSNR) of signal. That means that spontaneous Raman scattering is a more important noise source than DRB. Figure 7 plots OSNR degradation of signal along fiber. When only spontaneous Raman was considered, in near-transmitter section OSNR of signal decreased slowly, but in near-receiver section of fiber OSNR of signal exhibited a sharp decrease. On or hand, Appl. Sci. in2018, case 8, x FOR with PEER only REVIEW DRB, OSNR of signal exhibited a small drop in near-transmitter 8 of 10 section and decreased slowly in rest section of fiber. And at end of fiber, OSNR of fiber. And case at in which end only of spontaneous fiber, OSNR Raman of scattering case in was which considered only spontaneous is lower than Raman scattering OSNR of was case considered with only is lower DRB. This than indicates OSNR that of spontaneous case with Raman only DRB. scattering This indicates has higher that impact spontaneous system Raman than scattering DRB. has higher impact on system than DRB. Figure 6. Power profiles of signal, spontaneous Raman scattering (sprs), and DRB.

8 Appl. Sci. 2018, Figure 8, Power profiles of signal, spontaneous Raman scattering (sprs), and DRB. 8 of 10 Figure 7. OSNR vs. distance, with withonly only spontaneous Raman Raman scattering (sprs) (sprs) and and with with both both sprs sprs and and DRB. DRB. Next, we take NSNI into consideration. According analysis above, when receiverside DBP is applied, noise in near-receiver section causes more severe NSNI, because noise Next, we take NSNI into consideration. According analysis above, when receiver-side DBP is applied, noise in near-receiver section causes more severe NSNI, because noise in in near-receiver section is amplified in DBP process. Thus, NSNI caused by spontaneous near-receiver section is amplified in DBP process. Thus, NSNI caused by spontaneous Raman Raman scattering is also higher than NSNI caused by DRB. To demonstrate this point, scattering is also higher than NSNI caused by DRB. To demonstrate this point, SNRs of SNRs of signals were calculated using proposed oretical model with DRB included, signals were calculated using proposed oretical model with DRB included, and results and results were compared with case in which only spontaneous Raman scattering was were compared with case in which only spontaneous Raman scattering was considered, as shown considered, as shown in Figure 8. From Figure 8, we can see that when DRB was considered, in Figure 8. From Figure 8, we can see that when DRB was considered, proposed model had proposed model had good accuracy. When pre-compensation length was less than 60 km, two good accuracy. When pre-compensation length was less than 60 km, two curves were almost curves were almost coincident. With pre-compensation length increasing over 60 km, a difference coincident. With pre-compensation length increasing over 60 km, a difference appeared and appeared Appl. Sci. and 2018, 8, x SNR FOR PEER of REVIEW signal including DRB was lower than SNR of signal 9 of without 10 SNR of signal including DRB was lower than SNR of signal without DRB. This is because NSNI DRB. caused This is because by DRB NSNI in caused near-transmitter by DRB in section near-transmitter increased. section Still, increased. difference Still, is small, whichdifference illustrates is small, that which DRBillustrates has a much that smaller DRB has influence a much on smaller performance influence on of performance system than spontaneous of system Raman than scattering. spontaneous Raman scattering. 5. Conclusions Figure Figure 8. SNR 8. SNR vs. vs. The pre-compensation length, with withand and without without DRB. DRB. In this paper, we oretically deduced NSNI in a bi-directional Raman amplified unrepeatered system with receiver-side DBP and split-dbp, considering effects of distributed Raman gain, spontaneous Raman scattering, and DRB. The results of oretical calculation were compared with results of transmission simulations, which indicated that oretical model has good accuracy when pre-compensation length is less than 100 km. Both results of

9 Appl. Sci. 2018, 8, of Conclusions In this paper, we oretically deduced NSNI in a bi-directional Raman amplified unrepeatered system with receiver-side DBP and split-dbp, considering effects of distributed Raman gain, spontaneous Raman scattering, and DRB. The results of oretical calculation were compared with results of transmission simulations, which indicated that oretical model has good accuracy when pre-compensation length is less than 100 km. Both results of oretical calculation and simulations showed that split-dbp can effectively mitigate NSNI in such a system. By adopting split-dbp, with an appropriate pre-compensation length, SNR of signal increased by about 1 db. According to oretical analysis, DRB has much a smaller influence on system than spontaneous Raman scattering. The oretical model given in this paper is helpful in design of a bi-directional Raman amplified system. Author Contributions: Q.Z. finished writing whole manuscript and conducted simulations. Z.Y., Y.L., and W.L. presented idea. Q.Z. and Z.Y. are co-first authors. Funding: This research received no external funding. Acknowledgments: This work is supported by Open Foundation of China Sourn Power Grin and State Key Laboratory of Optical Communication Technologies and Networks (W.R.I.) and Accelink Technologies Company Ltd. The help of Shaohua Yu and Zhixue He from Wuhan Research Institute of Posts and Telecommunications is also acknowledged. Conflicts of Interest: The authors declare no conflict of interest. References 1. Mohamed, A.; AliKarar, M.; Landolosi, T. DSP-based dispersion compensation: Survey and simulation. In Proceedings of 2017 International Conference on Communication, Control, Computing and Electronics Engineering (ICCCCEE), Khartoum, Sudan, January Erdogan, A.T.; Demir, A.; Oktem, T.M. Automatic PMD compensation by unsupervised polarization diversity combining coherent receivers. J. Light. Technol. 2008, 26, [CrossRef] 3. Essiambre, R.J.; Kramer, G.; Winzer, P.J.; Foschini, G.J.; Goebel, B. Capacity limits of optical fiber networks. J. Light. Technol. 2010, 28, [CrossRef] 4. Ip, E.; Kahn, J.M. Compensation of dispersion and nonlinear impairments using digital backpropagation. J. Light. Technol. 2008, 26, [CrossRef] 5. Ip, E. Nonlinear compensation using backpropagation for polarization-multiplexed transmission. J. Light. Technol. 2010, 28, [CrossRef] 6. Maher, R.; Xu, T.; Galdino, L.; Sato, M.; Alvarado, A.; Shi, K.; Savory, S.J.; Thomsen, B.C.; Killey, R.I.; Bayvel, P. Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation. Sci. Rep. 2015, 5, [CrossRef] [PubMed] 7. Irukulapati, N.V.; Wymeersch, H.; Johannisson, P.; Agrell, E. Stochastic digital backpropagation. IEEE Trans. Commun. 2014, 62, [CrossRef] 8. Irukulapati, N.V.; Marsella, D.; Johannisson, P.; Agrell, E. Stochastic digital backpropagation with residual memory compensation. J. Light. Technol. 2016, 34, [CrossRef] 9. Galdino, L.; Semrau, D.; Lavery, D.; Saavedra, G.; Czegledi, C.B.; Agrell, E.; Killey, R.I.; Bayvel, P. On limits of digital back-propagation in presence of transceiver noise. Opt. Express 2017, 25, [CrossRef] [PubMed] 10. Lavery, D.; Ives, D.; Liga, G.; Alvarado, A.; Savory, S.J.; Bayvel, P. The benefit of split nonlinearity compensation for single channel optical fiber communications. IEEE Photon. Technol. Lett. 2016, 28, [CrossRef] 11. Lavery, D.; Maher, R.; Liga, G.; Semrau, D.; Galdino, L.; Bayvel, P. On bandwidth dependent performance of split transmitter-receiver optical fiber nonlinearity compensation. Opt. Express 2017, 25, [CrossRef] [PubMed] 12. Bromage, J. Raman amplification for fiber communications systems. J. Light. Technol. 2004, 22, 79. [CrossRef]

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