Nonlinear Inverse Synthesis for Optical Links with Distributed Raman Amplification

Transcription

1 1 Noliear Iverse Sythesis for Optical Liks with Distributed Rama Amplificatio So Thai Le, Jaroslaw E. Prilepsky, Paweł Rosa, Jua D. Aia-Castañó ad Sergei K. Turitsy Abstract Noliear Fourier trasform () ad eigevalue commuicatio with the use of oliear sigal spectrum (both discrete ad cotiuous), have bee recetly discussed as a promisig trasmissio method to combat fiber oliearity impairmets. However, because the -based trasmissio method employs the itegrability property of the lossless oliear Schrödiger equatio (NLSE), the origial approach ca oly be applied directly to optical liks with ideal distributed Rama amplificatio. I this paper, we ivestigate i details the impact of a o-ideal Rama gai profile o the performace of the oliear iverse sythesis (NIS) scheme, i which the trasmitted iformatio is ecoded directly oto the cotiuous part of the oliear sigal spectrum. We propose the lossless path-averaged (LPA) model for fiber liks with o-ideal Rama gai profile by takig ito accout the average effect of the Rama gai. We show that the NIS scheme employig the LPA model ca offer a performace gai of 3 db regardless of the Rama gai profiles. Idex Terms Coheret, oliear Fourier trasform, iverse scatterig, orthogoal frequecy divisio multiplexig, oliear sigal processig, oliear optics. C I. INTRODUCTION otiuig demad from the growig umber of badwidth-hugry applicatios ad o-lie services (such as cloud computig, HD video streams, o-lie cotet sharig ad may others) is pushig the required optical commuicatio system capacity close to the theoretical limit of a stadard sigle-mode fiber (SSMF) [1], which is imposed by the iheret fiber oliearity (Kerr effect) []. I the last decade, there have bee extesive efforts i attemptig to surpass the Kerr oliearity limit through various oliearity compesatio techiques, icludig digital back-propagatio (DBP) [3], digital [4] ad optical [5-7] phase cojugatios (OPCs) at the mid-lik or istalled at the trasmitter [8], ad phase-cojugated twi waves [9-11]. However, there are still may limitatios ad challeges to overcome i applyig the aforemetioed oliear compesatio methods, because the trasmissio techologies utilized i optical fiber Research supported by Egieerig ad Physical Scieces Research Coucil (EPSRC) through the project UNLOC (EP/J1758/1). This work was preseted i part at the 15 Europea Coferece o Optical Commuicatio (ECOC), Sep. Valecia, Spai. S. T. Le, J. E. Prilepsky ad S. K. Turitsy are with Asto Istitute of Photoic Techologies (AIPT), Asto Triagle, Birmigham, B4 7ET, UK (correspodig author phoe: +44() ; let1@asto.ac.uk). P. Rosa ad J. Aia-Castañó are with Istituto de Optica, IO-CSIC, CSIC, Madrid, 86, Spai. commuicatio systems were origially developed for liear (radio or ope space) commuicatio chaels. I recet years, there has bee a growig iterest i the alterative approach of desigig fiber optical commuicatio systems i which the oliearity is take ito accout as a essetial elemet rather tha a destructive effect [1]. The core idea behid this approach is based o the fact that without perturbatio the oliear Schrödiger equatio (NLSE), which govers the propagatio of optical sigal i SSMF, belogs to the class of the so-called itegrable oliear systems [13-15]. As a cosequece, the field evolutio over the NLSE chael ca be preseted withi a special basis of oliear ormal modes (oliear sigal spectrum), icludig o-dispersive solito (discrete) ad quasi-liear dispersive radiatio (cotiuous) modes. The evolutio of such special oliear modes i the fiber chael is essetially liear, which meas that the oliearity-iduced cross-talk betwee these modes is effectively abset durig the propagatio (without sigal corruptio due to oise). Therefore, the parameters of oliear modes ca be effectively used to ecode ad trasmit iformatio i fiber chael without sufferig from oliear crosstalk [15-19]. This geeral idea was first itroduced by Hasegawa ad Nyu i [1] ad was termed there as eigevalue commuicatio. There are two mai directios i the commuicatios methodology, which ca be categorized accordig to what part of the oliear spectrum (solitoic discrete part or cotiuous part) is used for the modulatio ad trasmissio. I particular, [19-3] studied the discrete (solitoic) compoets of the oliear spectrum for data commuicatios. This approach is ofte referred as oliear frequecy divisio multiplexig (NFDM) ad some iitial experimetal demostratios have bee reported recetly [1-3]. However, the NFDM method requires cosiderable optimizatio of the pulse shapes for the purpose of maximizig the resultig spectral efficiecy (SE) [4]. The secod approach based o the modulatio of the cotiuous part of the oliear spectrum, has bee proposed i [17] ad assessed i detail i [5, 6] (for ideal Rama amplificatio ad EDFA-based optical liks) ad was termed there as the oliear iverse sythesis (NIS) method. Fially we ote that there already exists a very recet study where both the cotiuous ad discrete parts of the oliear spectrum have bee used simultaeously [7]. However, all trasmissio schemes employig the s are based o the itegrability of the lossless NLSE ad thus ca be effectively applied oly to optical liks with ultra-log fiber laser-based distributed Rama amplificatio providig a flat quasi-lossless gai profile [8].

2 I this paper, we exted the results firstly preseted i [9] ad discuss i details the impact of the o-ideal Rama gai profile o the performace of NIS-based trasmissio systems. We itroduce a LPA NIS scheme which offers 3 db performace gai regardless of the particular Rama profile. To demostrate the effectiveess of the LPA NIS scheme, without loss of geerality, we cosider here ope-cavity radom distributed feedback (DFB) laser Rama amplificatio, as this scheme provides the best performace amog various other Rama amplificatio schemes [3]. The remaider of the paper is orgaized as follows. The basic of -based trasmissio method icludig system desigs is reviewed i Sectio II. I Sectio III the desig ad characteristics of Radom DFB laser Rama amplifier is preseted. I Sectios IV, the cocept of LPA model ad modified NIS scheme accoutig for o-ideal gai profile are itroduced. The simulatio setup, results ad discussio is preseted i Sectio V. Sectio VI cocludes the paper. II. BASICS OF -BASED TRANSMISSION METHOD As metioed earlier, -based trasmissio employs the oliear sigal spectrum (discrete ad/or cotiuous parts) for the purpose of data modulatio ad trasmissio i fiber optical commuicatio liks. This approach ca be realized i various ways. Herei we recall the basics of s ad differet cocepts of the -based systems. A. Basics of operatio Accordig to the iverse scatterig trasform (IST) theory, the propagatio of a complex sigal q(z, t) i oliear itegrable systems, such as the oe govered by the lossless ormalized NLSE (Eq. 1), ca be decomposed ito the liear propagatio of its oliear spectral data (oliear spectrum) [14]. We write the ormalized NLSE (the aomalous dispersio case) as 1 jqz qtt q q, (1) where z stads for the propagatio distace ad t is the time i the frame co-movig with the group velocity of the evelope. Accordig to the IST theory, the solutio of lossless NLSE (1) with a give iput sigal q(, t) ca be foud through three basic steps: i) Obtaiig the oliear spectral data of the iput sigal through the. ii) Propagatig the oliear spectral data to a desired distace, ad this propagatio is trivial ad liear. iii) Defiig the output sigal through the iverse () give the evolved oliear spectral data at the desired distace z. Basically, the coverts the sigal ito the correspodet oliear spectrum, icludig both cotiuous ad discrete parts. This operatio is achieved by solvig the spectral Zakharov Shabat problem (ZSP) [14], which correspods to a scatterig problem for a o-hermitia Dirac-type system of equatios for two auxiliary fuctios v 1, (t), with the NLSE iput pulse profile, q(,t) q(t), servig as a effective potetial: dv1 dv q( t) v jv1, q( t) v1 jv () dt dt Here, ζ is a (geerally complex) eigevalue, ζ = ξ+jη, q (t) is the complex cojugatio of the potetial q(t), which is assumed to decay as t ± (techically, we simply trucate our q(t) to a fiite duratio). To defie oliear spectral data, for real ζ = ξ we ca select two specific liearly-idepedet solutios of Eq. () as T * ( t, ) 1,, ( t, ), 1, with the coditio at the trailig ed of our iput profile j t, T t e The, the two Jost scatterig coefficiets ( spectral amplitudes) a(ξ) ad b(ξ) are give by: jt a( ) 1 ( t, ) e t, ( ) (, ) j t b t e t After defiig the Jost scatterig coefficiets, the cotiuous part of the sigal oliear spectrum (the left reflectio coefficiet) is defied as: r( ) b( ) / a( ), (3) The solitos (discrete part) correspod to the complex eigevalues ζ = ξ + jη i the upper-half of the complex plae, where a(ζ ) =. Together with the eigevalue ζ, each solitoic degree of freedom is characterized by the secod complex quatity, the so-called ormig costat: C b a (4) ' ( ) / ( ), I geeral, the maps the iitial field, q(,t), oto a set of scatterig data Ʃ = [(r(ξ), ξ is real); (ζ, C )], where the idex rus over all discrete eigevalues of the ZSP (if the latter are preset) (Fig. 1). More detailed explaatios o ca be foud i [19, 5]. Numerical methods for ca be foud i [5, 31, 3] ad the refereces therei. B. Iverse operatio The iverse maps the scatterig data Ʃ oto the field q(t), Fig. 1: This is achieved via the solutio of Gelfad Levita Marcheko equatios (GLME) for the ukow fuctios A 1, (x, t) [14, 17, 5]: x 1(, ) ( ) (, ) A x t F t y A x y dy (5) A x t F t y A x y dy F x t x t x (, ) ( ) 1(, ) ( ), The quatity F(t) ca cotai cotributios from the solitoic (discrete) ad radiatio (cotiuous) spectrum parts: F( t) F ( t) F ( t ), (6) slo 1 F ( ) j t, ( ) j t slo t j Ce Frad r e d Havig solved the GLME (5), the output of the, i.e. the field profile i the true space-time domai, is give by x t rad q( t) lim A ( t, x ) (7) More detailed explaatios o ca be foud i [19, 5]. Effective umerical methods for (with discrete ad/or cotiuous parts) ca be foud i [5, 3, 33] ad refereces therei. T

3 Digital backpropagatio Tx Phase-shift Rx Trasmissio i NFD Tx Phase-shift Rx 3 qt () r( ), C, r( ) C qt () Fig. 1. Illustratio of ad for a give iput the potetial q(t), which is assumed to decay as t ± C. Evolutio of the oliear spectrum Whe oe is iterested i the solutio q(l, t) at the distace z = L, the quatities r(ξ), C, ζ i Eq. (3-4) are replaced with r(z, ξ), C (z) ζ (z). Uder the oise-free assumptio, the evolutio of oliear spectrum i lossless NLSE chael is trivial ad liear: j z j z r( z, ) r(, ) e, () z, C ( z) C () e (8) So, each discrete eigevalue ζ is a itegral of motio ad does ot chage, while the reflectio coefficiet ad ormig costat obey a simple dyamics, similarly to the evolutio of ordiary Fourier modes i a liear dispersive chael. This remarkable property makes oliear spectrum (discrete ad/or cotiuous parts) ideal iformatio carriers i oliear fiber chaels. D. -based system desigs As the evolutio of oliear spectrum is essetially liear i oliear lossless fiber chael, the oliear spectrum ca be used for data modulatio ad trasmissio. The two basic desigs for -based trasmissio systems are preseted i the Fig.. Optical fiber Optical fiber Fig.. Basic desigs of -based trasmissio systems, icludig trasmissio i the oliear Fourier domai (NFD) ad digital back propagatio i NFD. The first desig (Fig. ) ca be referred as trasmissio i the oliear Fourier domai (NFD) as the trasmitted iformatio is ecoded directly oto the oliear sigal spectrum (discrete ad/or cotiuous parts) via the. Herei, provides three degrees of freedom for data modulatio, amely the discrete spectrum (C ), cotiuous spectrum (r(ξ)) ad discrete eigevalues (ζ ). The modulatios of discrete spectrum, cotiuous spectrum ad discrete eigevalues are ofte cosidered separately due to the umerical complexity of the full - cycle. The resulted trasmissio methods ca be termed as NFDM, NIS ad eigevalue commuicatio, respectively. Herei, we focus o the NIS scheme as it ca be combied effectively with high modulatio format ad traditioal coheret trasmissio techologies, offerig highest flexibility i the system desig [5, 6]. I the secod desig, the s are used to cacel the oliearity distortio i fiber optical commuicatio systems. This scheme ca be uderstood as the digital back propagatio with the use of the operatios, -DBP, Fig. [34]. Here, the sigal ecodig ad modulatio is performed i the space-time domai. As the evolutio of the oliear spectrum is liear, the iterplay of dispersio ad oliearity durig trasmissio ca be removed usig a sigle phase-shift operatio. After removig the liear phase shift iside the NFD, the trasmitted sigal ca be recovered via the. I geeral, both aforemetioed system desigs ca be effectively applied to cacel the determiistic oliear distortios i fiber optical liks. However, from a practical viewpoit, the first desig provides several advatages over the secod -DBP desig. First of all, it admits the flexibility of choosig what part of oliear spectrum oe employs for the data modulatio, ad thus, sigificatly reduces the umerical complexity associated with the full through appropriate modulatio techiques. Secod, i the first desig the computatioal load ca be effectively split betwee the trasmitter ad receiver sides. O the other had, the secod desig requires modificatio oly at the receiver side, while modificatios are ecessary at both sides i the first desig. III. RANDOM DFB RAMAN AMPLIFIER As discussed above, the -based methods ca be applied directly oly to lossless or quasi-lossless optical liks. I practice, such coditio is difficult to achieve. As a result, it is importat to ivestigate the impact of o-flat gai profile o the performace of -based systems ad develop appropriate modificatio of the approach. Herei, we take ito accout a ope-cavity radom DFB laser Rama amplificatio scheme, which ca provide various gai profiles by cotrollig the forward pump power (FPP) [3]. Rama Pump 1366m Tx FBG Fig. 3. Schematic of the radom DFB laser Rama amplifier Rx Rama Pump 1366m The schematic desig of the radom DFB laser Rama amplifier that allows achievig d order pumpig with a sigle wavelegth pump is show i Fig. 3. I this scheme, a high reflectivity (99%) fiber Bragg gratig (FBG) cetered at 1455 m with GHz badwidth is deployed at the ed of the trasmissio spa to reflect back-scattered Rayleigh Stokes-shifted light from the backward pump (at 1366 m) ad stimulate radom DFB lasig at 1455m (wavelegth of the FBG). This radom DFB laser acts as the first order pump, together with d order pump, to amplify the sigal at 155 m. The lack of a FBG o the side of the forward pump sigificatly reduces the relative-itesity-oise trasfer from the forward pump to the Stokes-shifted light at 1455 m [35], which ca seriously hider coheret trasmissio [36]. We simulated the sigal ad oise power excursio for differet pump power ratios i Rama amplifiers usig the

4 4 experimetally verified model [8] with a appropriate boudary coditios that shows a high degree match with the OTDR traces. I all cases the cosidered Rama pumps are fully depolarized. The backward pump powers were chose accordigly to provide a et gai of db. The simulated gai ad oise profiles alog 8 km legth SMF spa are show i Fig. 4 for differet FPPs. It ca be see i Fig. 4 that, whe the FPP is icreased, the Rama gai icreases while the oise Gai Profile (db) Noise Power (dbm) FPP= FPP= Distace (km) Fig. 5. AGV ad NLCC as a fuctio of the forward pump power; the spa legth was 8km. power decreases, leadig to a better sigal-to-oise ratio. To characterize the o-flatess of the Rama gai profile, we defie the average gai variatio (AGV) of the Rama gai profile as: AGV L G( z) k dz / LG, k L G( z) dz / L, (9) FPP= FPP=.4 FPP=.8 FPP=3. FPP=.8 FPP=3.6 FPP=.4 FPP= FPP=4 FPP=1.6 FPP=1. FPP=.8 FPP=.4 FPP=3. FPP=3.6 FPP=4 FPP=1. -4 FPP=.8 FPP=.4 FPP= Distace (km) z z FPP Forward pump power i W Fig. 4. Rama gai ad oise profiles alog 8km SMF spa for differet value of the forward pump power (FPP). AGV (a.u) Forward Pump Power (W) NLCC (a.u) where G(z) is the Rama gai ormalized to 1 at the begiig of the spa, L is the spa legth, ad k is termed as the oliear correctio coefficiet (NLCC). The AGV is show i Fig. 5 as a fuctio of the FPP. The FPP = case correspods to the backward-pumpig-oly scheme. From Fig. 5 we see that the AGV ca be effectively reduced by icreasig the FPP (up to its optimum value) at the cost of icreasig the system power cosumptio. The optimum value of FPP was foud to be ~.7 W, givig the AGV.55. By varyig the FPP to vary the AGV we ca effectively ivestigate the impact of Rama gai flatess o the NIS-based trasmissio systems. IV. MODIFIED NIS FOR NON-IDEAL DISTRIBUTED RAMAN AMPLIFICATION As discussed i the previous sectio, for practical Rama amplificatio schemes, the o-flatess level characterized by the AGV, ca be as high as ~.43 (for backward-pumpigoly scheme). This high level of o-flatess may deprive all the oliearity cacellatio beefit of NIS ad of other based trasmissio schemes. As a result, the LPA model for Rama-based optical liks should be developed i a similar maer to EDFA-based optical liks [6] i order to apply NIS ad, potetially, other -based trasmissio schemes. The geeral model of the NLSE for optical liks with Rama amplifiers ca be writte as jqz q / q q jg( z) q, (1) where t is the time i the frame co-movig with the group velocity of the evelope, β < is the dispersio coefficiet, γ is the Kerr oliearity coefficiet ad g(z) is the distributed distace-depedet Rama gai coefficiet. We cosider here the case that the same pumpig scheme is applied to all fiber spas. I this case g(z) is a periodic fuctio with a period equal to the spa legth. By itroducig the stadard chage of variables [37]: z q( z, t) exp g( y) dy A( z, t ), Eq. (1) ca be rewritte as: jaz A / G( z) A A, (11) which is the lossless NLSE with a distace-depedet oliear coefficiet; here G(z) is the istataeous gai z G( z) exp g( y) dy We assume here that the dyamic of the evelope A(z, t) does ot chage sigificatly after each fiber spa. I this case, the distace-depedet oliear coefficiet i (11) ca be replaced by its averaged value over each fiber spa, givig the effective LPA NLSE [37] L ja A / G( z) dz / L A A (1) z From a egieerig poit of view, the LPA NLSE model ca be obtaied from the geeral NLSE model by removig the loss term ad updatig the oliear coefficiet i such way that the oliear phase-shift acquired by the sigal durig propagatio over oe spa is uchaged. Of course, as show i [6] the LPA NLSE is a approximated model so its

5 5 accuracy depeds strogly o the sigal ad system parameters such as badwidth, pulse shape, power ad trasmissio distace. A detailed ivestigatio o the accuracy of LPA NLSE model for optical liks with lumped amplificatios ca be foud i [6]. For the Rama amplificatio, simulatio results (ot show here) idicate very similar depedece of the accuracy of LPA model o sigal s ad system s parameters i compariso to the case of lumped amplificatio. Ecoder Decoder Normalizatio IFFT Phase-shift I Q IQ modulator I Q Coheret Reciver LO Guard Packet 1 time Packet Packet N Fig. 6 Block diagram of NIS-based trasmissio systems, Illustratio of a burst mode trasmissio Based o the obtaied LPA NLSE model, we ca desig the appropriate modified NIS scheme accoutig for the o-ideal Rama gai profile as show i Fig. 6. Firstly, the ormalizatio is performed o the iitial sigal q(t) (modulated with arbitrary pulse shape ad modulatio format) to brig the LPA NLSE model to the stadard ormalized form (Eq. 1) t / T t, z / Z z, q k Z q, (13) S S S where the time ormalizatio T s is a free parameter (e.g., a characteristic time scale or a reciprocal badwidth) ad the associated space scale is Z s =T s / β ; k is NLCC defied as i Eq. (9). The depedece of the NLCC o FPP is show i Fig. 5 (red curve). After the ormalizatio, the liear Fourier spectrum of the ecoded iput waveform (Q(ω)) is mapped oto the cotiuous part of the oliear spectrum of aother sigal (s(t)) to be trasmitted usig the. Let r(ξ) deote the cotiuous part of the oliear spectrum of s(t) the the mappig operatio of the block ca be expressed as: r( ) Q ( ) (14) / This defies the pre-processig of the iitial sigal at the trasmitter ad ca be cosidered as oliearity precompesatio. Geerally, the mappig rule betwee Q(ω) ad the reflectio coefficiet r(ξ), Eq. (3), ca be take arbitrarily. However, the usage of mappig give by Eq. (14), utilized i the series of previous works o the NIS method [17, 5-6], esures the exact covergece of the oliear quatities to the respective liear aalogs i the low power limit, which simplifies the system desig ad checks. The -geerated complex sigal, s(t), is the fed ito the IQ modulator for direct up-covertig ito the optical domai ad lauched ito the fiber. At the receiver, the real ad imagiary parts of the trasmitted sigal are detected with a coheret receiver. The oliear spectrum of the received sigal is the obtaied by usig the. As the evolutio of the sigal oliear spectrum is liear ad trivial withi the LPA NLSE model, Eq. (8), the liear Fourier spectrum of the iitial ecoded complex sigal ca be recovered by applyig a sigle step liear phase-shift removal j L Q( ) r( L, ) e / (15) The, havig urolled the dispersio-iduced phase shift, the iitial ecoded waveform q(t) ca be recovered usig the IFFT operatio ad, fially, it ca be fed ito the stadard decoder for data detectio. I geeral, the DSP at the receiver of a NIS-based system ivolves a sigle operatio ad a sigle liear compesatio step to remove the oliear impairmets without reverse propagatio, idepedetly of the trasmissio distace. This is a sigificat advatage of the NIS method over the other oliear compesatio techiques. V. SIMULATION SETUP AND RESULTS As discussed i [5, 6], the NIS trasmissio method ca be combied with ay modulatio formats ad trasmissio schemes. A compariso of OFDM ad sigle carrier with Nyquist pulse shapig (or orthogoal time divisio multiplexig) for NIS-based systems was provided i [5], revealig that the OFDM is a more suitable modulatio format because it provides a smaller L1-orm, thus dimiishig the processig sigal cut-off error. This ca be explaied by the fact that the L1 orm of the Fourier trasform is always lower or equal to the L1-orm of the time-domai sigal. As a result, i this paper we cosider oly the OFDM scheme. We desig here the 16QAM 56-Gbaud OFDM NIS-based systems i the burst mode regime (Fig. 6), as the operatios have to be performed o retur-to-zero sigals. I this scheme, the eighbourig packets are separated by a guard time, which is % loger tha the dispersio iduced memory. The dispersio iduced memory of the lik ca be estimated as: T BL, (16) where B is the sigal s badwidth ad L is the lik distace. For simplicity, we assume that each packet data cotais oly oe OFDM symbol. To geerate the OFDM sigals, the IFFT size of 14 was used, where 11 subcarriers were filled with data (with Gray-codig) while the remaiig subcarriers were set to zero for oversamplig purpose. The useful OFDM symbol duratio is s. No cyclic prefix was added to the sigal. The et data rate, after removig 7% overhead due to the FEC, was Gb/s (cosiderig oly the burst s bit-rate). Herei, we aim to show that DSP techiques based o model (1) ca be applied effectively eve i the log-haul optical commuicatio systems with as large a badwidth as 56 GHz. The propagatio of sigal i fiber lik was simulated usig the split-step Fourier method with a step size of 1 km, usig the gai profile show i Fig. 4. The Rama oise was modelled as a Gaussia oise, which was added to the sigal

6 6 Q-factor (db) NIS Modified NIS w/o NIS FPP=W Burst Power (dbm) Q-factor (db) NIS w/o NIS Modified NIS FPP=.7W Z=4km Burst Power (dbm) Fig. 8. Performace compariso of OFDM systems with ad without NIS, ad the modified NIS scheme for FPP =.7 W. The trasmissio distace is 4 km 13 (c) (d) Fig. 7. Performace compariso of OFDM systems with ad without NIS, ad the modified NIS scheme for FPP = W (backward pumpig oly case), -(d) costellatios at the optimum lauch powers for system without NIS, with NIS (c) ad with the modified NIS (d) schemes at FPP=W, respectively. after each step (1 km), followig the simulated oise profiles show i Fig. 4. The system performace was evaluated through the EVM ad the estimated BER was the coverted to the Q-factor [38]. The performace of OFDM systems with ad without NIS, ad with the proposed modified NIS scheme is compared i Fig. 7 for the backward pumpig oly scheme (FPP= W). For the case of backward pumpig scheme (FPP = W), because of the high o-flatess level of the Rama gai profile (AGV ~.43), applyig directly the NIS method worses the system performace by ~ db. This result clearly idicates that the o-ideal Rama gai profile has a sigificat impact o the NIS-based systems: Whe the AGV is high (AGV~.43 if FPP=W), the NIS method caot produce ay advatage due to the wrog power estimate. However, if the modified NIS scheme is employed, a Q-factor improvemet of ~3 db is observed. This effectively meas that the performace of the NIS scheme is ehaced by 5 db by simply employig the NLCC that takes ito accout the oideal gai profile alog the spa. The received costellatios at optimum lauch powers for three systems uder ivestigatio are show i Figs. 7-(d). A similar performace compariso result is plotted i the Fig. 8 whe FPP =.7 W, which provides the smallest level of oflatess of the Rama gai profile. It should be oted here that by icreasig the FPP, the Rama oise figure is reduced, which ca be referred from the performace at the low power level. It ca be see i Fig. 8 that the NIS method gives aroud db performace gai if FPP =.7W. I this case, the AGV is relatively small (.55), ad the NIS scheme still offers a meaigful performace gai. However, if oe uses the modified NIS method based o Eq. (1), a extra ~1 db gai ca be achieved, givig a total performace gai of ~3 db. Optimum Q-factor (db) w/o NIS NIS 7 Modified NIS Z=4km Forward Pump Power (W) Fig. 9. Optimum Q-factor as a fuctio of FPP for OFDM systems with ad without NIS, ad the modified NIS scheme. The trasmissio distace was 4km The optimum Q-factors i systems with ad without NIS ad with the modified NIS schemes are preseted i Fig. 9 as fuctios of the FPP. As expected, whe the FPP is icreased from W to.7 W, the achievable performace of NIS-based system icreases dramatically as a result of the decrease i the AGV. However, icreasig further the FPP, which icreases the AGV accordigly (if FPP >.7 W), does ot decrease the performace of NIS-based system. We attribute this pheomeo to the reductio of the amplifier oise figure whe the FPP is icreased. O the other had, the modified NIS scheme offers aroud 3 db gai i Q-factor, idepedetly of the FPP. This idicates that if the modified NIS method is applied, the forward pump is ot ecessary, which offers a sigificat reductio i the cost ad power cosumptio whe desigig the NIS-based systems with Rama amplificatios. VI. CONCLUSION We have show that the o-ideal Rama gai has a sigificat impact o the performace of NIS-based systems. As a result, a appropriate modificatio is required i order to apply NIS techiques i optical liks with practical Rama amplificatio schemes. Based o the LPA NLSE model, which

7 7 takes ito accout the average effect of the Rama gai profile, we have proposed a modified NIS scheme for optical liks with Rama amplificatios. By cosiderig various desigs of radom DFB laser Rama amplifier we have show that the modified NIS scheme ca offer a performace gai of 3dB, idepedetly of the Rama gai profile. We strogly believe that the LPA NLSE model ca be effectively applied i differet other -based trasmissio systems. Moreover, it seems possible to attai the further improvemet of the LPA NLSE model performace, which would take ito accout higher order correctios to Eq. (1), similarly to the higherorder guidig-ceter models proposed i [13] for the lumped EDFA amplificatio method. ACKNOWLEDGMENT This work was supported by the UK EPSRC Grat UNLOC (EP/J1758/1). P. Rosa kidly ackowledges the support of the EU Marie Skłodowska-Curie IF CHAOS (65898) project. REFERENCES [1] A. D. Ellis, Z. Jia, ad D. Cotter, "Approachig the No-Liear Shao Limit," JLT, IEEE, vol. 8, pp , 1. [] R. 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