The Gaussian Noise Model in the Presence of Inter-channel Stimulated Raman Scattering

January 9, 2018 1 The Gaussian Noise Model in the Presence of Inter -channel Stimulated Raman Scattering Daniel Semrau, Student Member , IEEE, and Polina Bayvel, F ellow , IEEE, F ellow , OSA Abstract —A Gaussian noise (GN) model is pr esented that properly accounts for an arbitrary frequency dependent signal power profile along the link. This enables the evaluation of the impact of inter -channel stimulated Raman scattering (ISRS) on the optical Kerr nonlinearity . Additionally , the frequency dependent fiber attenuation can be taken into account and transmission systems that use hybrid amplification schemes can be modeled, where distrib uted Raman amplification is partly applied over the optical spectrum. T o include the latter two cases, a set of coupled ordinary differential equations must be numerically solved in order to obtain the signal power profile yielding a semi-analytical model. However for lumped amplification and negligible variation in fiber attenuation, a less complex and fully analytical model is pr esented which is referred to as the ISRS GN model. The deriv ed model is exact to first- order for Gaussian modulated signals and extensively validated by numerical split-step simulations. A maximum deviation of 0 . 1 dB in nonlinear interference power between simulations and the ISRS GN model is found. The model is applied to a transmission system that occupies an optical bandwidth of 10 THz, repr esenting the entire C+L band. At optimum launch power , changes of up to 2 dB in nonlinear interference po wer due to ISRS are reported. Furthermore, comparable models published in the literature are benchmarked against the ISRS GN model. Index T erms —Optical fiber communications, Gaussian noise model, Nonlinear interference, nonlinear distortion, Stimulated Raman Scattering, First-order perturbation, C+L band trans- mission I . I N T R O D U C T I O N A N AL YTICAL models that predict the performance degra- dation in optical fiber communications due to Kerr non- linearity hav e enjoyed significant popularity in recent years. Most approaches analytically solve the nonlinear Schr ¨ odinger equation using a first-order perturbation approach with respect to the nonlinearity coefficient. The resulting expressions offer unique insight into the underlaying parameter dependencies and are key enablers for efficient system design [1], rapid achiev able rate estimations of point-to-point links [2]–[4] and physical layer aware network optimization. The latter is essential for optical network abstraction and virtualization leading to optimal and intelligent techniques to maximize optical network capacity [5]. Analytical models also offer a significant reduction in computational complexity with minor This w ork was supported by a UK EPSRC programme grant UNLOC (EP/J017582/1) and a Doctoral Training Partnership (DTP) studentship for Daniel Semrau. D. Semrau and P . Bayvel are with the Optical Networks Group, Uni- versity College London, London WC1E 7JE, U.K. (e-mail: { uceedfs; p.bayvel } @ucl.ac.uk.) inaccuracies compared to split-step simulations and experi- ments [6]–[11]. The literature of fers a wide range of analytical models vary- ing in accuracy and comple xity [12]–[23]. The first approaches in the context of modern coherent receiv ers and dispersion uncompensated links date back to 1993 and 2002 [12], [13], enabling the computation of the perturbation caused by Kerr nonlinearity . Similar results were independently derived by other groups and the model became widely known as the Gaussian noise (GN) model [14]–[16], [22], [24]. A key assumption of the works is the signal Gaussianity assumption, which is that the signal can be written as a Gaussian process at the fiber input. As a result of this assumption, the GN model relies on large accumulated dispersion [20, Sec. 6] and signals with high cardinality [19, Sec. 4]. T wo conditions that are satisfied in most cases of modern coherent fiber communication. The popularity of the GN model undoubtedly origins in its moderate complexity . Howe ver , as a result it f ails to predict certain properties of nonlinear interference such as modula- tion format dependence [17]–[19], symbol rate dependence [25]–[29], nonlinear phase noise [19], [30], long temporal correlations [18], [19] and the dependence on the memory length of the fiber-optic channel [31]. In order to account for those properties, significantly more complex models have been proposed [17]–[21], [31]. Comprehensive ov erviews can be found in [32], [33]. The impact of the mentioned properties are usually small for lumped, dispersion unmanaged, multi-span systems that use high-order modulation formats and the GN model can be considered sufficiently accurate. Recently , the con ventional GN model was e xperimentally v alidated for the central channel and optical bandwidths up to 7 . 3 THz with a deviation of only 0 . 4 dB in nonlinear interference (NLI) power [6], [34]. An assumption of all abo ve-mentioned works is that every frequency component e xperiences the same power e volution along the link. They are therefore inaccurate in the prediction of ultra-wideband transmission systems where the variation of the fiber attenuation is not ne gligible and for bandwidths where inter-channel stimulated Raman scattering (ISRS) is significant. Inter-channel stimulated Raman scattering (ISRS) is a non- parametric nonlinear process that amplifies low frequency components at the expensi ve of high frequency components within the same optical signal. In modern optical commu- nications that use coherent technology in combination with high dispersive links, ISRS effecti vely introduces a different power profile for each frequency component [35]–[37]. For January 9, 2018 2 C band transmission (approximately 5 THz), as defined from the a vailability of the erbium doped fiber amplifier (EDF A), ISRS is not significant and its impact is negligible in most cases. Howe ver , for systems that occupy the entire C+L band (approximately 10 THz) or beyond, ISRS becomes significant and it must be taken into account. Beyond C+L band trans- mission could be enabled by lumped Raman amplification or the use of other dopants such as bismuth [38] or quantum dots in semiconductor amplifiers [39] [40]. The first approach to include ISRS in the GN model was published in [41]. The authors approximated the signal power profile with an exponential decay using an effecti ve attenuation coefficient that matches the (frequency dependent) ef fective length in the presence of ISRS. Although first conclusions on the impact of ISRS could be drawn, this method exhibits two shortcomings. First, when ISRS is significant, the resulting signal power profile does not resemble an exponential decay , particularly in the beginning of the fiber span, where Kerr nonlinearity prev ails. Second, the work assumes that, during the four-wa ve mixing (FWM) process, ev ery participating frequency component attenuates in the same manner as the channel of interest. In other words, for the induced nonlinear perturbation at frequency f , every frequency component in the triplet ( f , f 1 , f 2 ) attenuates as f during the nonlinear mixing. This does not accurately represent the FWM process, as each frequency component attenuates in a different manner . A more rigorous approach to include ISRS in the GN model was published in [42], [43]. In both works, the channel under test (i.e. frequency component f ) attenuates precisely accord- ing to the signal power profile g ( f ) resulting from ISRS (or any arbitrary profile), lifting the exponential decay assumption in [41]. Ho wever , for an attenuation profile that is linear in frequency (like the one resulting from ISRS), the frequencies in the triplet ( f , f 1 , f 2 ) also attenuate according to f during the FWM process. Therefore, this approach overestimates the impact of ISRS on the K err nonlinearity and the frequency dependent signal power profile is not accurately taken into account. In this paper , a Gaussian noise model is presented that properly accounts for any arbitrary frequency dependent signal power profile. This enables the modeling of nonlinear interfer- ence in ultra-wideband re gimes where ISRS is significant. The model is referred to as the ISRS GN model. Additionally , the variation in fiber attenuation and hybrid amplification schemes can be included. In general, the signal power profile is obtained by numerically solving a set of coupled ordinary dif ferential equations (ODE). This yields a semi-analytical ISRS GN model which relies on a numerical ODE solver . Howe ver , for lumped amplification and negligible variation in fiber attenuation, a fully analytically model is deriv ed based on a linear approximation on the ISRS gain function. This reduces model as well as computational complexity . The analytical ISRS GN model holds for bandwidths up to approximately 15 THz after which the Raman gain function cannot be considered linear anymore. This paper is organized as follows. In Sec. II, the ISRS GN model is presented and its key deri vation steps are briefly outlined. The detailed deri vation can be found in the Appendix A. The model is extensi vely validated by split-step simulations in Sec. III and applied to a C+L band transmission system based on standard single mode fiber (SMF) spans in Sec. IV. In Sec. III and IV, the results in [41]–[43] are benchmark ed against the ISRS GN model. I I . T H E I S R S G N M O D E L In order to maximize the information throughput of an optical communication system, it is vital to ev aluate and maximize the performance of each indi vidual channel that is transmitted. After coherent detection and electronic dispersion compensation, the channel dependent signal-to-noise ratio (SNR) can be calculated as SNR i ≈ P i nP ASE + η n P 3 i , (1) where P i is the launch power of channel i , P ASE is the ampli- fied spontaneous emission (ASE) noise po wer over the channel bandwidth and η n is the nonlinear interference coefficient after n spans. The SNR is a function of the spectral location of the channel within the optical signal as P ASE and η n are frequency dependent quantities. When the channel bandwidth B ch is small compared to the total optical bandwidth B , the power spectral density (PSD) of the NLI can be considered locally flat and η n can be approximated as η n ( f i ) = Z B ch 2 − B ch 2 G ( ν + f i ) P 3 i dν ≈ B ch P 3 i G ( f i ) , (2) where f i is the center frequency of channel i and G ( f ) is the PSD of the nonlinear interference. For later use, we further define the total optical launch po wer as P tot = Z G Tx ( ν ) dν = X ∀ i P i , (3) where G Tx is the input PSD of the entire optical signal. The aim of the next sections is to find an analytical expression for the nonlinear interference PSD G ( f ) , in order to compute the channel dependent SNR or similar performance metrics. A. The nonlinear interference power V arious models hav e been proposed in the past in order to calculated the NLI power , where it is generally assumed that all frequency components attenuate in the same manner along a fiber span [12]–[24]. Howe ver , this assumption is no longer satisfied when transmission systems operate at large optical bandwidths (C+L band and beyond). This is because each fre- quency component under goes a different power evolution dur- ing propagation as a result of ISRS and a frequency dependent attenuation coef ficient. In addition, frequency dependent signal power profiles are present in hybrid amplification schemes, where part of the spectrum is amplified using distributed Raman amplifiers in order to reduce the ASE noise power for longer wav elengths or extend the amplification window beyond conv entional EDF A ’ s [44]–[46]. January 9, 2018 3 For a frequency dependent power e volution, the PSD of the NLI after one span is deriv ed in Appendix A and it is found to be G ( f ) = 16 27 γ 2 Z d f 1 Z d f 2 G Tx ( f 1 ) G Tx ( f 2 ) G Tx ( f 1 + f 2 − f ) ·      Z L 0 dζ s ρ ( ζ , f 1 ) ρ ( ζ , f 2 ) ρ ( ζ , f 1 + f 2 − f ) ρ ( ζ , f ) e j φ ( f 1 ,f 2 ,f ,ζ )      2 , (4) where φ = − 4 π 2 β 2 [( f 1 − f )( f 2 − f ) + π β 3 ( f 1 + f 2 )] ζ , γ is the nonlinearity coefficient and ρ ( z , f ) is the normalized signal po wer profile. F or example, the normalized signal po wer profile of a passi ve fiber with constant attenuation coef ficient α is ρ ( z , f ) = e − αz . For multi-span systems, where each span has identical fiber parameters and signal power profiles, the phased-array term      sin  1 2 nφ ( f 1 , f 2 , f , L )  sin  1 2 φ ( f 1 , f 2 , f , L )       2 (5) must be inserted into the integral in 4. For multi-span systems with non-identical signal power profiles for each span (e.g. when non-ideal gain flattening filters are considered), the phased-array term cannot be used. Instead Eq. (4) must be considered, where L must be reinterpreted as the link length and ρ ( z , f ) as the signal power profile for the entire link. Howe ver , identical spans are considered for the remainder of this work. W e note that (4) is different than the result deriv ed in [42, Eq. (16)] and [43, Eq. (18)], where ρ ( ζ , f 1 + f 2 − f ) and ρ ( ζ , f ) are swapped. 1 Howe ver , in section III, it is shown by split-step simulations that (4) is the correct formula. Eq. (4) can be used to model any arbitrary frequency dependent signal po wer profile and it is therefore suitable to ev aluate the impact of ISRS on the optical K err nonlinearity . B. Inter-c hannel stimulated Raman scattering In the following, the PSD of the NLI in the presence of ISRS is presented, which is hereafter referred to as the ISRS GN model. In modern dispersion uncompensated links, ISRS ef fectively amplifies high wa velength components at the expense of low wav elength components [35]–[37]. The resulting frequency dependent signal power profile can be 1 As a consequence of the different result, the f 1 and f 2 dependence vanishes, for power profiles of the form ρ ( z , f ) = e a ( z ) · f + b ( z ) (as the one resulting from ISRS). This means that in the nonlinear process all three frequencies in the triplet ( f , f 1 , f 2 ) attenuate according to frequency f which overestimates the impact of ISRS. In contrast, in Eq. (4) each frequency ( f 1 , f 2 , f ) correctly attenuates with its respecti ve po wer profile. obtained by solving a set of coupled ordinary differential equations [47, Eq. (3)] ∂ P i ∂ z = − M X j = i +1 1 2 f j f i g r (∆ f ) P j P i | {z } ISRS loss + i − 1 X j =1 1 2 g r (∆ f ) P j P i | {z } ISRS gain − α ( f i ) P i , (6) where M is the total number of WDM channels, g r (∆ f ) is the normalized (by the ef fective core area A eff ) Raman gain spectrum for a frequenc y separation ∆ f = | f i − f j | and α ( f ) is the frequency dependent attenuation coefficient. The index of the channel with the highest center frequency is i = 1 . Eq. (6) can be extended to include distributed Raman amplification using [48, Eq. 1]. Eq. (6) has no general analytical solution and must be solved numerically . The obtained power profile can then (after normalization) be used in (4) to yield a semi-analytical model that accurately accounts for ISRS, a frequency dependent attenuation coefficient and distributed Raman amplification. The variation of the attenuation coefficient does typically not exceed 0 . 01 dB/km across the C+L band ranging from 1530 nm to 1625 nm and might be negligible depending on accuracy and computational complexity requirements [49]. The impact of a frequency dependent attenuation on the NLI coefficient can be loosely upper bounded by assuming that every frequency component attenuates according to the mini- mum in one case and according to the maximum attenuation coefficient in another case. The resulting maximum deviation in NLI coefficient is then ∆ η 1 [ dB ] <  α min α max  [ dB ] , (7) where ( · ) [ dB ] means conv ersion to decibel and α min and α max is the minimum and maximum attenuation coef fi- cient, respectively . In (7) it was assumed that e − αL  1 , ln  π 2 B 2 | β 2 | /α min  ≈ ln  π 2 B 2 | β 2 | /α max  and [50, Eq. 5] was used. For an attenuation deviation of 0 . 01 dB/km ov er 95 nm, (7) yields a maximum de viation of ∆ η 1 [ dB ] < 0 . 2 dB. This contribution might be deemed negligible and the pre- vailing effect that causes a frequency dependent signal power profile is inter -channel stimulated Raman scattering. Eq. (6) can then be solved analytically when the Raman gain spectrum is assumed to be linear up to approximately 14 THz (i.e. up to the Stokes shift). The normalized signal power profile for a spectral component f is then given by [51, Eq. (7)] ρ ( z , f ) = P tot e − αz − P tot C r L eff f R G Tx ( ν ) e P tot C r L eff ν dν , (8) where C r is the slope of a linear regression of the normalized Raman gain spectrum g r (∆ f ) and L eff = 1 − exp( − αz ) α . The z dependence in L eff is suppressed for notational conv enience. The ISRS gain of a 10 THz signal after 100 km propagation, obtained from numerically solving (6) using the Raman gain spectrum as in [52] and its analytical approximation (8) are January 9, 2018 4 -5 -2.5 0 2.5 5 -12 -10 -8 -6 -4 -2 0 2 4 6 Channel frequency f i [THz] ISRS gain [dB] 19 dBm, Eq. ( 8 ) 21 dBm, Eq. ( 8 ) 23 dBm, Eq. ( 8 ) 24 dBm, Eq. ( 8 ) 25 dBm, Eq. ( 8 ) 26 dBm, Eq. ( 8 ) 27 dBm, Eq. ( 8 ) 28 dBm, Eq. ( 8 ) num. solution, ODE ( 6 ) Fig. 1: The net g ain due to ISRS as a function of channel frequency obtained by solving the set of coupled differential equations (6) shown in dotted lines and its analytical approximation (8) sho wn in solid lines for a variety of total optical launch powers P tot . shown in Fig. 1. The precise functions that have been used can be found in [41, Fig. 1]. For a relatively high optical launch power of 28 dBm, the average de viation between the numerical solution and its approximation is 0 . 18 dB which can be considered ne gligible. Therefore, the analytical solution (8) is suf ficiently accurate for modeling the nonlinear interference power . Substituting (8) in (4) yields the reference formula of the analytical ISRS GN model (9) as G ( f ) = 16 27 γ 2 Z d f 1 Z d f 2 G Tx ( f 1 ) G Tx ( f 2 ) G Tx ( f 1 + f 2 − f ) ·      Z L 0 dζ P tot e − αζ − P 0 C r L eff ( f 1 + f 2 − f ) R G Tx ( ν ) e P tot C r L eff ν dν e j φ ( f 1 ,f 2 ,f ,ζ )      2 . (9) Eq. (9) is a key result of this work which is extensiv ely validated in Sec. III and further applied to a C+L band case study in Sec. IV. It is useful to analyze Eq. (8) in more detail. After trivial algebraic manipulations, we obtain that the power transfer between the spectral edges can be computed as ∆ ρ ( z ) [ dB ] = 4 . 3 · P tot C r L eff B , (10) which is independent of the spectral distribution of the in- put power . For modern fiber parameters L eff = 26 km, C r = 0 . 008 1 /W/km/THz (approximately corresponding to a Corning R  V ascade R  EX2000 fiber with A eff = 111 µ m) and an optimum launch po wer of 40 fW/Hz (corresponding to 2 dBm over 40 GHz) as in [6], an ISRS power transfer of ∆ ρ = 1 dB is present at a bandwidth of 5 . 3 THz and ∆ ρ > 4 dB for bandwidths larger than 10 . 6 THz. It should be further noted that the ISRS gain is independent of the Raman gain slope and the optical launch power as long as their product is kept constant. For a gi ven optical bandwidth, this also holds for the nonlinear interference coefficient. This will be useful for relating the results in Sections III and IV to fibers with different Raman gain slopes. I I I . N U M E R I C A L V A L I DAT I O N In this section, the analytical ISRS GN model (9) is validated by split-step simulations for an optical fiber com- munication system with parameters listed in T able I (a). Numerically solving the Manako v equation for the entire C+L band (approximately 10 THz) is extremely challenging, due to high memory requirements and the e xcessiv e use of very large fast Fourier transforms. Therefore, the validation is carried out ov er a bandwidth of B = 1 THz with an artificially tenfold increased Raman gain slope C r . Using (10), the resulting ISRS gains for 1 THz bandwidth coincide with the ones sho wn in Fig. 1 as the product B · C r is kept constant. The reader can therefore con veniently refer to Fig. 1 for the ISRS gains that are present at a particular total launch power (where the abscissa must be scaled down by 10 ). T ABLE I: System Parameters Parameters (a) (b) for section III IV Loss ( α ) [dB/km] 0.2 Dispersion ( D ) [ps/nm/km] 17 Dispersion slope ( S ) [ps/nm 2 /km] 0 0.092 NL coefficient ( γ ) [1/W/km] 1.2 Raman gain slope ( C r ) [1/W/km/THz] 0.28 0.028 Raman gain ( C r · 14 THz) [1/W/km] 4 0.4 Fiber length ( L ) [km] 100 Noise figure [dB] 5 Symbol rate [GBd] 10 50 Channel spacing ( B ch ) [GHz] 10.001 50.001 Number of channels 101 201 Optical bandwidth ( B ) [THz] 1 . 01 10 . 05 Roll-off factor [%] 0.01 Number of symbols [ 2 x ] 15 Step size [m] 5 Gaussian modulation was implemented in order to emulate the signal Gaussianity assumption of the GN model. Addi- tionally , a small channel bandwidth was chosen such that B  B ch . A matched root-raised-cosine (RRC) filter was used to obtain the output symbols and the SNR was ideally estimated as the ratio between the variance of the transmitted symbols E [ | X | 2 ] and the variance of the noise σ 2 , where σ 2 = E [ | X − Y | 2 ] and Y represents the receiv ed symbols after digital signal processing. The nonlinear interference coefficient was then estimated via (1) and was compared with the predictions of the ISRS GN model via (2) and (9). A spectrally uniform launch power was assumed yielding G Tx ( f ) = P tot B Π  f B  , (11) for the model calculations, where Π ( x ) denotes the rectan- gular function. In the simulation en vironment, a frequency dependent po wer profile was implemented to emulate the power transfer between channels due to ISRS based on (8). January 9, 2018 5 − 500 − 250 0 250 500 35 36 37 38 39 40 41 42 43 a) Channel frequency f i [GHz] η 1 [dB] no ISRS 19 dBm 23 dBm 25 dBm 25 dBm, model in [42,43] 25 dBm, model in [41] 7 9 11 13 15 17 19 21 23 25 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 -500 GHz -170 GHz 0 GHz 170 GHz 500 GHz b) T otal launch power P tot [dBm] ∆ η n [dB] after 1 span after 10 spans Fig. 2: The nonlinear interference coefficient after 1 span as a function of channel frequency for different total launch powers is shown in a) and the NLI change due to ISRS as a function of total launch power is shown in b). Solid lines represent the analytical ISRS GN model (9) and markers represent results obtained by split-step simulations. For comparison, the model in [41] is shown with a dotted line and the model in [42,43] is shown with a dashed line for 25 dBm launch power in a). 1 2 3 4 5 6 8 10 0 1 2 3 4 5 6 7 8 9 10 11 Span number n NLI accumulation η n [dB] − η 1 [dB] − 500 GHz, no ISRS 500 GHz, no ISRS − 500 GHz, 25 dBm 500 GHz, 25 dBm 7 8 9 10 8 9 10 11 b) − 500 − 250 0 250 500 0.02 0.04 0.06 0.08 0.1 a) Channel frequency f i [GHz] Coherence factor  no ISRS 25 dBm Fig. 3: The accumulation of NLI as a function of span number for the most outer frequencies of the signal with and without ISRS. Lines represent the analytical ISRS GN model (9) and markers represent split-step simulations. The inset a) shows the coherence factor after 10 spans and the inset b) shows a magnified area of the figure. The results for the nonlinear interference coef ficient after one span as a function of the channel frequency f i is sho wn in Fig. 2a) and as a function of total launch po wer in Fig. 2b). The accuracy of the ISRS GN model is remarkable with a maximum de viation of < 0 . 1 dB. The de viation is slightly higher at the exact spectral edges. At the exact spectral edges the NLI PSD varies over the channel bandwidth and the NLI PSD cannot be considered locally flat as in (2). This is not an inherit approximation of the ISRS GN model and it can be lifted by properly integrating over the NLI PSD. As expected, the ISRS GN model con verges to the con ventional GN model for low launch powers. For increasing launch powers, the nonlinear interference PSD begins to tilt. The NLI is decreased for channels that experience net ISRS loss and increased for channels that experience net ISRS gain. Moreov er, as shown in Fig. 2b), the NLI interference coefficient depends exponentially on the launch po wer . Fig. 2b) indicates further that the launch po wer dependence on the NLI coefficient is stronger for an increasing number of spans. For Gaussian modulation, the NLI accumulation in decibel as a function of fiber spans can be written as [14, Sec. IX] ( η n ) [ dB ] − ( η 1 ) [ dB ] = (1 +  ) · ( n ) [ dB ] , (12) where ( · ) [ dB ] denotes con version to decibel scale and  is the coherence factor that is a measure for coherent accumulation of the NLI. As the coherence factor depends on the signal power profile (cf. [14, Fig. 10] and [11, Fig. 3]), it is affected by ISRS. The accumulation of NLI together with the resulting coherence factor obtained from the ISRS GN model and simulation results are sho wn in Fig. 3. Indeed, ISRS introduces a power dependent tilt on the coherent accumulation. This corresponds to the increasing po wer dependence of the NLI coefficient with increasing span number . January 9, 2018 6 -5 -2.5 0 2.5 5 24 25 26 27 28 29 30 31 32 a) Channel frequency f i [THz] η 1 [dB] no ISRS 19 dBm 21 dBm 24 dBm 25 dBm 26 dBm 27 dBm 28 dBm 10 12 14 16 18 20 22 24 26 28 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 -5 THz -2 THz 2 THz 5 THz 0 THz b) T otal launch power P tot [dBm] ∆ η n [dB] after 1 span after 10 spans Fig. 4: The nonlinear interference coefficient after 1 span as a function of channel frequency for different total launch powers is shown in a) and the NLI as a function of total launch power is shown in b) obtained by the analytical ISRS GN model (9). The uniform optimum launch power for the system under test is 24 dBm. -5 -2.5 0 2.5 5 0 . 00 0 . 01 0 . 02 0 . 03 0 . 04 0 . 05 0 . 06 0 . 07 Channel frequency f i [THz] Coherence factor  no ISRS 24 dBm 28 dBm Fig. 5: The coherence factor as a function of channel frequency for a v ariety of total launch powers obtained by the analytical ISRS GN model (9). T o benchmark our results against previous works, the results in [42, Eq. (16)] and [43, Eq. (18)] are shown in dashed in Fig. 2a) using the same frequency dependent po wer profile (8). The dif ference originates for the reason described in Sec. II-A, effecti vely assuming that all frequencies in the nonlinear process attenuate as the one that is ev aluated (i.e. f i ). The de viation therefore increases with increasing ISRS gain tow ards the spectral edges. The model in [41] is shown in dotted lines. The model implements the con ventional GN model, where an effecti ve attenuation coef ficient is used, that matches the ef fective length of the ev aluated channel f i . Consequently , the frequency dependent attenuation within the nonlinear process is not properly accounted for , mainly resulting in an underestimation of the ISRS impact. Bases on the numerical validation carried out in this sec- tion, it is concluded that the ISRS GN model (9) accurately predicts the nonlinear interference resulting from inter-channel stimulated Raman scattering. I V . C + L B A N D T R A N S M I S S I O N In this section, the ISRS GN model is used to ev aluate the impact of ISRS on a C+L band transmission system covering 10 THz of optical bandwidth with parameters listed in T able I (b). The NLI coef ficient as a function of channel frequency is shown in Fig. 4a). For a particular launch power , the corresponding ISRS gains can be found in Fig. 1. Moreov er, the results can be con verted to a different Raman slopes coefficient by subtracting the deviation in decibel from the total launch po wer (cf. Eq. (10)). For example, for halving the Raman gain slope and a total launch po wer of 25 dBm, one can find the resulting NLI coefficients and ISRS gains that indicate 22 dBm. The tilt in NLI coefficient in the absence of ISRS is due to the dispersion slope S (or β 3 ), where lower frequencies experience a higher amount of dispersion and therefore ex- perience reduced NLI. As the launch po wer is increased, the effect of ISRS starts to balance the effect of the dispersion slope in terms of NLI. At a launch po wer of 22 dBm, the NLI PSD is almost flat showing that ISRS and the dispersion slope are some what complementing each other in flatting the NLI spectrum. The system under test exhibits an optimum launch po wer of 24 dBm, which was calculated for a flat input PSD and only considering the center channel. The resulting ISRS po wer transfer ranges from − 3 . 7 dB to 2 . 9 dB while the resulting NLI deviation due to ISRS ranges from − 1 . 7 dB to 2 dB. The slope of the NLI spectrum is − 0 . 24 dB THz ov er its linear-lik e part from − 4 THz to 4 THz. The de viation of the NLI coefficient as a function of total launch power is shown in Fig. 4b). The NLI PSD depends exponentially on the total launch power like the ISRS gain itself and already discussed in Sec. III. The de viation at the spectral edges of the signal is 0 . 5 dB at 18 dBm launch power . In Sec. III it was shown that the coherence factor is changed as a result of ISRS. The same ef fect is seen in Fig. 4b), where the deviation of the NLI is stronger for an increased number January 9, 2018 7 -5 -2.5 0 2.5 5 − 0 . 8 − 0 . 6 − 0 . 4 − 0 . 2 0 Channel frequency f i [THz] ∆ η 1 [dB] no ISRS 21 dBm 24 dBm 26 dBm 27 dBm 28 dBm Fig. 6: Deviation of the NLI coefficient after one span between the analytical ISRS GN model (9) and [41]. The validity of the ISRS GN model is shown in Sec. III. of spans. At 24 dBm the additional de viation of NLI is 0 . 1 dB at the spectral edges after 10 spans. The coherence factor as a function of channel frequency is shown in Fig. 5. The coherence factor is relati vely small (  < 0 . 07 ) due to the large bandwidth. In the absence of ISRS the average coherence factor is 0 . 027 . Using (12) the average coherent NLI accumulation  · ( n ) [ dB ] is 0 . 3 dB and 0 . 5 dB after 10 and 50 spans, respecti vely . F or 24 dBm (optimum) launch po wer, the maximum deviation in coherence factor is found to be 0 . 013 at the spectral edges. This corresponds to a deviation in coherent NLI accumulation of 0 . 1 dB and 0 . 2 dB after 10 and 50 spans, respectiv ely . The change in coherence factor due to ISRS might be deemed negligible depending on the accuracy requirements of the application. T o relate our work to previously published results, the NLI coefficient after one span obtained by the ISRS GN model is compared to the works [41] and [42], [43]. The signal power profile as in (8) was used for all comparisons. The deviation of the NLI coef ficient between the ISRS GN model and [41] is shown in Fig. 6. For optimum launch power ( 24 dBm), the deviation stays belo w 0 . 19 dB. Even for high ISRS gains at 28 dBm launch power , the maximum deviation is 0 . 8 dB. The deviation of the NLI coefficient between the ISRS GN model and [42, Eq. (13) and (16)] [43, Eq. (18)] is sho wn in Fig. 7. The maximum deviation is 0 . 56 dB and 2 . 1 dB for 24 dBm and for 28 dBm launch power , respectiv ely . The reader is referred to sections II-A and III for the origin of the discrepancy . Based on the case study in this section, it is concluded that the impact of ISRS on the Kerr nonlinearity is significant in C+L band systems. This is strongly depending on launch power and the Raman gain slope due its exponential relation- ship to the nonlinear interference po wer . It should be noted that idealized gain flattening filters (GFF) were considered to compensate the ISRS power transfer at the end of each span. When realistic GFF’ s are considered, the ISRS gain accumulates over distance and the impact on the nonlinear interference po wer is more significant than as shown in this -5 -2.5 0 2.5 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 Channel frequency f i [THz] ∆ η 1 [dB] no ISRS 21 dBm 24 dBm 26 dBm 27 dBm 28 dBm Fig. 7: Deviation of the NLI coefficient after one span between the analytical ISRS GN model (9) and [42, Eq. (13) and (16)] [43, Eq. (18)]. The validity of the ISRS GN model is shown in Sec. III. section. V . C O N C L U S I O N The ISRS GN model was introduced and presented which analytically models the impact of inter-channel stimulated Raman scattering on the nonlinear perturbation caused by Kerr nonlinearity . Its accuracy was compared to split-step simulations and a maximum deviation of 0 . 1 dB in nonlin- ear interference power was found. The model can further account for the frequency dependent fiber attenuation, optical bandwidths beyond the Stokes shift (approximately 14 THz) and hybrid-amplified transmission systems at the expense of greater computational complexity using a semi-analytical approach. It was shown that ISRS changes the nonlinear interference power by up to 2 dB at optimum launch power for the studied C+L band transmission system. For such optical bandwidths and beyond, ISRS must be addressed in order to maximize system performance. From a physical perspectiv e, possible solutions include the use of gain flattening filters, optimized launch power distributions or tailored fiber designs. All of which can be modeled and analyzed using the results in this paper . The deri ved ISRS GN model is therefore a po werful tool for efficient design, optimization, capacity calculations and physical-layer abstractions of ultra-wideband transmission sys- tems that operate over the entire C+L band and be yond. A P P E N D I X A D E R I V A T I O N O F T H E I S R S G N M O D E L In this section, Eq. (4) is derived for one fiber span based on the nonlinear Schr ¨ odinger equation (NLSE) and a first- order regular perturbation approach. The result for one span can then be extended to multiple spans using the phased-array term (5) or reinterpreting a span as the entire link length with an according signal power profile. Instead of a constant attenuation coefficient α , a generic frequency and distance dependent gain coefficient g ( z , f ) is used to model the effect January 9, 2018 8 of inter-channel stimulated Raman scattering. For the sake of brevity , only the key deriv ation steps are outlined. W e begin with the NLSE in the frequency domain which is giv en by [53, Ch. 2] ∂ ∂ z E ( z , f ) = e Γ( z , f ) E ( z , f ) + j γ E ( z , f ) ∗ E ∗ ( z , − f ) ∗ E ( z , f ) , (13) with e Γ( z , f ) = g ( z,f ) 2 + j 2 π 2 β 2 f 2 + j 4 3 π 3 β 3 f 3 and u ( x ) ∗ v ( x ) denoting the con volution operation. The complex en velope of the electric field E ( z , f ) is expanded in a regular perturbation series with respect to the nonlinearity coef ficient γ . The series is then truncated to first-order and we have E ( z , f ) = E (0) ( z , f ) + γ E (1) ( z , f ) . (14) Inserting (14) in (13), we obtain E (0) ( z , f ) = E (0 , f ) · e Γ( z ,f ) , (15) with Γ ( z , f ) = R z 0 e Γ ( ζ , f ) dζ as the solution for the zeroth- order terms and a linear ordinary differential equation for the first-order terms as ∂ ∂ z E (1) ( z , f ) = e Γ( z , f ) E (1) ( z , f ) + Q ( z , f ) , (16) with Q ( z , f ) = j E (0) ( z , f ) ∗ E (0) ∗ ( z , − f ) ∗ E (0) ( z , f ) . The initial condition for the first-order solution is E (1) (0 , f ) = 0 and we obtain E (1) ( z , f ) = e Γ( z ,f ) Z z 0 Q ( ζ , f ) e Γ( ζ ,f ) dζ , (17) as the solution of (16). In order to compute Q ( z , f ) , we assume that the input signal can be modeled as a periodic Gaussian process, a key assumption of the GN model, which is [24, Eq. 13] E (0 , f ) = p f 0 G Tx ( f ) ∞ X n = −∞ ξ n δ ( f − nf 0 ) , (18) where G Tx ( f ) is the power spectral density of the input signal, ξ n is a complex circular Gaussian distributed random variable, T 0 = f − 1 0 is the period of the signal and δ ( x ) denotes the Dirac delta function. For notational conv enience, we write nf 0 as f n and P ∞ n = ∞ as P ∀ n for the remainder of this deri vation. Using (18), Q ( z , f ) can be written as Q ( z , f ) = j f 3 2 0 X ∀ m X ∀ n X ∀ k p G Tx ( f m ) G Tx ( f n ) G Tx ( f k ) ξ m ξ ∗ n ξ k δ ( f − f m + f n − f k ) e Γ( z ,f m )+Γ ∗ ( z ,f n )+Γ( z ,f k ) . (19) T o first order, it can be shown that only non-degenerate frequency triplets in (19) contrib ute to the nonlinear interfer- ence power . Degenerate frequency triplets merely introduce a constant phase shift of the first-order solution E (1) ( z , f ) , which cancels out when the PSD of E (1) ( z , f ) is computed. For more details, the reader is referred to [54, Ch. IV.B and IV.D]. Therefore, we neglect degenerate frequency triplets in order to keep the deri vation concise. Similar to [54], we define the triplets of non-degenerate frequency components as A i = { ( m, n, k ) : [ m − n + k ] = i and [ m 6 = n or k 6 = n ] } , (20) and rewrite (19) as Q ( z , f ) = j f 3 2 0 X ∀ i δ ( f − f i ) X ∀ ( m,n,k ) ∈ A i ξ m ξ ∗ n ξ k p G Tx ( f m ) G Tx ( f n ) G Tx ( f k ) e Γ( z ,f m )+Γ ∗ ( z ,f n )+Γ( z ,f k ) . (21) Inserting (21) in (17) yields the first-order solution as E (1) ( z , f ) = j f 3 2 0 e Γ( z ,f ) X ∀ i δ ( f − f i ) X ∀ ( m,n,k ) ∈ A i ξ m ξ ∗ n ξ k p G Tx ( f m ) G Tx ( f n ) G Tx ( f k ) Z z 0 dζ e Γ( ζ ,f m )+Γ ∗ ( ζ ,f n )+Γ( ζ ,f k ) − Γ( ζ ,f m − f n + f k ) . (22) In order to obtain the nonlinear interference po wer, we com- pute the a verage PSD of the first-order solution γ E (1) ( z , f ) . Similar to [54, Ch. IV.D], the av erage PSD of (22) multiplied by γ is G NLI ( z , f ) = 2 γ 2 f 3 0 e 2 Re [Γ( z ,f )] X ∀ i δ ( f − f i ) X ∀ ( m,n,k ) ∈ A i G Tx ( f m ) G Tx ( f n ) G Tx ( f k )     Z z 0 dζ e Γ( ζ ,f m )+Γ ∗ ( ζ ,f n )+Γ( ζ ,f k ) − Γ( ζ ,f m − f n + f k )     2 . (23) In the follo wing, we transform the inner summation appearing in (23) into a summation over two independent variables. For the non-degenerate set A i , we have that f m − f n + f k = f i and for a gi ven frequenc y triplet ( f i , f m , f k ) it follo ws that f n = f m + f k − f i . Therefore, (23) can be written as G ( z , f ) = 2 γ 2 f 3 0 e 2 Re [Γ( z ,f )] X ∀ i δ ( f − f i ) X ∀ m X ∀ k G Tx ( f m ) G Tx ( f k ) G Tx ( f m + f k − f )     Z z 0 dζ e Γ( ζ ,f m )+Γ ∗ ( ζ ,f m + f k − f )+Γ( ζ ,f k ) − Γ( ζ ,f )     2 . (24) Finally , we define the normalized signal power profile of a frequency component as ρ ( z , f ) = e R z 0 g ( ζ,f ) dζ and rewrite (24) as an integral expression by letting f 0 → 0 G ( z , f ) = 2 γ 2 ρ ( z , f ) Z d f 1 Z d f 2 G Tx ( f 1 ) G Tx ( f 2 ) G Tx ( f 1 + f 2 − f )      Z z 0 dζ s ρ ( ζ , f 1 ) ρ ( ζ , f 2 ) ρ ( ζ , f 1 + f 2 − f ) ρ ( ζ , f ) e j φ ( f 1 ,f 2 ,f ,ζ )      2 . (25) As (25) was derived for single polarization, 2 γ 2 must be re- placed by 16 27 γ 2 to obtain the nonlinear interference power for dual polarization. Furthermore, the term ρ ( z , f ) outside of the integral can be removed when the span loss is compensated. The result is Eq. (4). 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In 2015 he joined the Optical Networks Group at Univ ersity College London (UCL) where he is currently working towards a Ph.D. degree. His research interests are mainly focused on channel modeling, nonlinear com- pensation techniques and ultra-wideband transmission for long-haul coherent optical communications. Polina Bayvel received the B.Sc. (Eng.) and Ph.D. degrees in electronic and electrical engineering from Uni versity College London (UCL), London, U.K., in 1986 and 1990, respecti vely . Her Ph.D. research focused on nonlinear fiber optics and their applications. In 1990, she was with the Fiber Optics Laboratory , General Physics Institute, Mosco w (Russian Academy of Sci- ences), under the Royal Society Postdoctoral Exchange Fellowship. She was a Principal Systems Engineer with STC Submarine Systems, Ltd., London, U.K., and Nortel Networks (Harlow , U.K., and Ottawa, ON, Canada), where she was inv olved in the design and planning of optical fiber transmission networks. During 1994–2004, she held a Ro yal Society University Research Fellowship at UCL, and, in 2002, she became a Chair in Optical Commu- nications and Networks. She is currently the Head of the Optical Networks Group, UCL. She has authored or co-authored more than 290 refereed journal and conference papers. Her research interests include optical networks, high- speed optical transmission, and the study and mitigation of fiber nonlinearities. Prof. Bayvel is a Fellow of the Royal Academy of Engineering (F .R.Eng.), the Optical Society of America, the U.K. Institute of Physics, and the Institute of Engineering and T echnology . She is a member of the T echnical Program Committee (TPC) of a number of conferences, including Proc. ECOC and Co-Chair of the TPC for ECOC 2005. She was the 2002 recipient of the Institute of Physics Paterson Prize and Medal for contributions to research on the fundamental aspects of nonlinear optics and their applications in optical communications systems. In 2007, she was the recipient of the Royal Society W olfson Research Merit A ward.