Analytical Modeling of Nonlinear Fiber Propagation for Four Dimensional Symmetric Constellations
Hami Rabbani, Mostafa Ayaz, Lotfollah Beygi, Gabriele Liga, Alex Alvarado, Erik Agrell, Magnus Karlsson
aa r X i v : . [ ee ss . SP ] S e p PREPRINT, SEPTEMBER 23, 2020 1
Analytical Modeling of Nonlinear Fiber Propagationfor Four Dimensional Symmetric Constellations
Hami Rabbani, Mostafa Ayaz, Lotfollah Beygi, Gabriele Liga,
Member , IEEE , Alex Alvarado,
Senior Member , IEEE , Erik Agrell,
Fellow , IEEE , Magnus Karlsson,
Senior Member , IEEE , Fellow , OSA
Abstract —Coherent optical transmission systems naturallylead to a four dimensional (4D) signal space, i.e., two polarizationseach with two quadratures. In this paper we derive an anlayticalmodel to quantify the impact of Kerr nonlinearity on such 4Dspaces, taking the interpolarization dependency into account.This is in contrast to previous models such as the GN andEGN models, which are valid for polarization multiplexed (PM)formats, where the two polarizations are seen as independentchannels on which data is multiplexed. The proposed modelagrees with the EGN model in the special case of independenttwo-dimensional modulation in each polarization. The modelaccounts for the predominant nonlinear terms in a WDM system,namely self-phase modulation and and cross-phase modulation.Numerical results show that the EGN model may inaccuratelyestimate the nonlinear interference of 4D formats. This nonlinearinterference discrepancy between the results of the proposedmodel and the EGN model could be up to 2.8 dB for a systemwith 80 WDM channels. The derived model is validated by split-step Fourier simulations, and it is shown to follow simulationsvery closely.
Index Terms — Coherent transmission, Enhanced Gaussian noisemodel, Four dimensional signals, Gaussian noise model, Kerrnonlinerity, Optical fiber communications . I. I
NTRODUCTION T HE amount of traffic carried on optical backbone net-works continues to grow at a rapid pace, and makesefficient use of available resources indispensable. The Kerrnonlinearity is the overriding factor that leads to signal dis-tortion and limits the capacity of optical fiber transmissionsystems [1]. Studying the ultimate limits of such systems iskey to avoid the capacity crunch. To circumvent the capacitycruch problem, spectrally-efficient modulation formats haveattracted substantial attention.
H. Rabbani, M. Ayaz and L. Beygi are with the the EE Dept. of K.N. Toosi University of Technology. E-mails: [email protected],[email protected] and [email protected]. Liga and A. Alvarado are with the Information and CommunicationTheory Lab, Signal Processing Systems Group, Department of ElectricalEngineering, Eindhoven University of Technology, Eindhoven 5600 MB, TheNetherlands. E-mails: {g.liga,a.alvarado}@tue.nlE. Agrell is with the Dept. of Electrical Engineering, Chalmers Universityof Technology, Sweden. E-mail: [email protected]. Karlsson is with the Dept. of Microtechnology and Nanoscience,Photonics Laboratory, Chalmers University of Technology, Sweden. E-mail:[email protected]. Liga is funded by the EuroTechPostdoc programme under the Eu-ropean Union’s Horizon 2020 research and innovation programme (MarieSkłodowska-Curie grant agreement No. 754462). This work has receivedfunding from the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation programme (grant agreementNo. 757791).
Optimized 2D modulation formats have become increas-ingly popular in optical communications. However, furtheroptimization is possible if the full 4D signal space (whichis inherent in optical coherent detection) is exploited. Theidea of 4D modulation formats was introduced to opticalcommunications as far back in time as the coherent receiverwas explored [2]–[5]. Agrell and Karlsson [6], [7] beganoptimizing modulation formats in a 4D space for coherentoptical communication systems in 2009. A number of 4Dmodulation formats have recently been proposed for purposesof maximizing generalized mutual information, optimizingpower efficiency, and other equally compelling motivations[8]–[11]. 4D coded modulation with bit-wise decoders wasstudied in [12]. Recently, other 4D coded modulation schemeshave been proposed in [13], [14].Although quite a few approximate analytical models fornonlinear fibre propagation are currently available in theliterature [15]–[20], all of these models aim to predict thenonlinear interference (NLI) in polarization multiplexed (PM)systems. What follows is a short description of analyticalmodels proposed for such PM optical systems.To analytically evaluate the quality of transmissions offiber-optic links, many research works have been devoted toextracting channel models both in the time and frequencydomains [16], [20], [21]. The Gaussian noise (GN) modelsin highly dispersive optical communications systems werepresented in [17], [21]–[23]. The 4D GN-type channel modelwas first proposed in [24]. The finite-memory GN model wasintroduced in [25]. Due to the Gaussianity assumption of thesignal, GN model is not able to predict the modulation formatdependence property of NLI.The authors of [16] for the first time addressed amodulation-format-dependent time-domain model, assumingonly the dominant nonlinear terms of cross-channel interfer-ence (XCI), known as cross-phase modulation (XPM) terms.Later, this time-domain model was studied comprehensivelyin [18] and compared with the GN model to address thediscrepancy between these two models. In much the sameway as in [18], the authors of [20] derived a new perturbationmodel (in the frequency domain) dropping the assumption ofGaussianity of the transmitted signal. This model was labelledenhanced Gaussian noise (EGN) model. As its name suggests,the EGN model added a number of correction terms to theGN model formulation, which fully captured the modulationformat dependency of the NLI. Moreover, the frequency-domain approach in [20] allows the model to fully account forall the different contributions of the NLI in a WDM spectrum,
REPRINT, SEPTEMBER 23, 2020 2 including: the self-channel interference (SCI), and unlike [18],all XCI and multi-channel interference (MCI) terms. It wasshown in [26] that the GN and time-domain model [16], [18]failed to accurately predict the NLI, whilst the EGN modelwas able to capture both the modulation format and the symbolrate dependency of the NLI. The achievable rate in nonlinearWDM systems was evaluated in [27].Recently, [28] proposed a modulation-format-dependentmodel in the presence of stimulated Raman scattering. Theauthors of [28] added a modulation format correction term toXPM, while SCI was computed under a Gaussian assumption.A general nonlinear model in the presence of Kerr nonlinearityand stimulated Raman scattering was proposed in [29], whichaccounts for the modulation-format-dependent SCI, XCI, andMCI terms. A survey of channel models proposed in theliterature up to 2015 was presented in [30].All of the aforementioned works are valid for PM modu-lation formats in which polarizations act as two independentchannels. In this paper, we concentrate on symmetric constel-lations and derive an accurate analytical model that is able topredict the impact of NLI on 4D optical transmission systemswhere data is jointly transmitted on both polarizations. Unlikethe previous models [18], [20], [31], the derived model is builton the fact that the x- and y-polarization are dependent ofone another, making it possible to predict the performanceof optimized 4D modulation formats in the presence of fibernonlinearities. A comprehensive approach to deriving the SCIterm in the frequency domain is currently being developedin [32], thus enabling the computation of the NLI power ofarbitrary zero-mean 4D constellations.The paper computes the SCI and XPM nonlinear terms. Ourmodel is derived following a time-domain approach, as in [18],[33], and does not include other XCI terms apart from XPM,nor does it contain MCI [20, Fig. 7]. Although the derivationof a comprehensive analytical model that can take into accountall terms of NLI (SCI, XCI, and MCI) goes beyond the scopeof this paper, the model in this paper computes the lion’s shareof the NLI in multi-channel WDM systems, i.e., the SCI andXPM terms [26, Fig. 2].The rest of this paper is organized as follows. In Sec. II, wedescribe the electrical field in a 4D space and also review thefirst order solution to Manakov equation. The main result ofthis work is presented in Sec. III. In Sec. IV, we validate theproposed model by split-step Fourier simulations, and comparea wide variety of 4D formats in terms of the experienced NLI.Sec. V concludes the paper. The detailed derivations of themain result of this paper are included in the Appendix.II. PRELIMINARIES
The electric field of the optical wave intrinsically comprisestwo polarizations, each with two quadratures, thus in total four Constellations which are symmetric with respect to the origin, and havethe same power in both polarizations. degrees of freedom, any one of which can be considered as adimension. The electrical field can therefore be written as E = (cid:20) E x E y (cid:21) = (cid:20) E x , r + iE x , i E y , r + iE y , i (cid:21) , (1)where indices x and y stand for polarization states, and r andi the real and imaginary parts, resp., of the electrical field.The propagation of dual-polarized signals in a dispersiveand nonlinear optical fiber is governed by the Manakovequation [34, Ch. 2] ∂∂z E ( t, z ) = − iβ ∂ ∂t E ( t, z )+ i γf ( z ) E † ( t,z ) E ( t,z ) E ( t,z ) , (2)where γ is the nonlinearity coefficient, β is the group ve-locity dispersion, and f ( z ) accounts for the link’s loss/gainprofile. In the case of perfectly distributed amplification f ( z ) = 1 , while in the case of lumped amplification f ( z ) = exp {− α mod ( z, L ) } where α is the loss coefficient, L is thespan length and mod ( z, L ) is the modulo operation and showsthe distance between the point z and the nearest precedingamplifier.We wish to evaluate the variance of SCI (intra-channel inter-ference) and XPM (inter-channel interference) terms based onthe first order perturbation approach, as these terms contributeto the NLI as predominant factors. We consider a channelof interest (COI) whose central frequency is set to zero, andan interfering channel with central frequency Ω . The XPMcontributions of multiple WDM channels sum up incoherently,so there is no need to consider more than one channel pair[18, Sec. 2]. The linear solution of the Manakov equation atdistance z for two channels is [18, Eq. (1)] E ( z, t ) = X k a k g ( t − kT, z )+ e − i Ω t + iβ z X k b k g ( t − kT − β Ω z, z ) , (3)where a k = [ a k, x a k, x ] T and b k = [ b k, x b k, x ] T are col-umn vectors containing two elements, which represent the k -th symbol transmitted by the COI and interfering chan-nel, resp. The dispersed pulse is represented by g ( t, z ) = exp ( − izβ ∂ t / g ( t, [35], where g ( t, is the input pulse,and ∂ t is the time derivative operator. The symbol rate ofchannels is denoted by T − .Without loss of generality, we concentrate on detectingthe zeroth symbol in the COI, i.e., a . The receiver for theCOI is assumed to fully compensate for the linear link’simpairments. The received symbol at the end of the link istherefore expressed as a + ∆ a , where ∆ a is the NLIcontribution. The first order solution to Manakov equationis obtained based on the perturbation approach [35, Eq. (3)], Throughout this paper we use ( · ) x and ( · ) y to represent variables associ-ated to polarizations x and y, resp. Expectations are denoted by E {·} , and twodimensional complex functions are denoted using boldface (e.g., E ) symbolswhose Hermitian conjugate is shown by ( · ) † . REPRINT, SEPTEMBER 23, 2020 3 which gives ∆ a (Ω) = i γ X h,k,l S h,k,l a † k a h a l + i γ X h,k,l X h,k,l (cid:16) b † k b h I + b h b † k (cid:17) a l . (4)In (4) I is the × identity matrix, and S h,k,l and X h,k,l are[35, Eqs. (4) and (5)] S h,k,l = Z L d z Z ∞−∞ d tf ( z ) g ∗ ( t, z ) g ( t − lT, z ) · g ∗ ( t − kT, z ) g ( t − hT, z ) , (5)and X h,k,l = Z L d z Z ∞−∞ d tf ( z ) g ∗ ( t, z ) g ( t − lT, z ) · g ∗ ( t − kT − β Ω z, z ) g ( t − hT − β Ω z, z ) , (6)resp. The first and second terms on the right-hand side of(4) are responsible for estimating the SCI and XPM terms,resp. Using the fact that g ( t, z ) = R d w ˜ g ( w ) exp ( − iwt + iw β z/ / (2 π ) , where ˜ g ( w ) is the Fourier transform of g ( t, (see [35, Appendix] and [35, Eqs. (11) and (12)]), (5)and (6) are expressed in the frequency domain as S h,k,l = Z d w (2 π ) ρ s ( w , w , w ) e i ( w h − w k + w l ) T , (7)and X h,k,l = Z d w (2 π ) ρ xp ( w , w , w ) e i ( w h − w k + w l ) T , (8)resp., where R d w stands for R R/ − R/ R R/ − R/ R R/ − R/ d w d w d w in which R = 2 π/T , and ρ s ( w , w , w ) = ˜ g ∗ ( w − w + w ) · ˜ g ( w )˜ g ∗ ( w )˜ g ( w ) Z L d zf ( z ) e iβ ( w − w )( w − w ) z , (9)and ρ xp ( w , w , w ) = ˜ g ∗ ( w − w + w ) · ˜ g ( w )˜ g ∗ ( w )˜ g ( w ) Z L d zf ( z ) e iβ ( w − w +Ω)( w − w ) z . (10)One may want to take all the NLI terms such as SCI, XCIand MCI into account. In this regard, (3) should be extendedto a general equation, which accounts for N terms, where N is the number of WDM channels occupying the full C-bandspectrum, and as a result, (4) will contain N terms, whichstem from E † ( t, z ) E ( t, z ) E ( t, z ) in (2). Nonlinear analysisof all the NLI terms however falls outside the scope of thepaper and is left for future work.III. T HE KEY RESULT : NLI
VARIANCE
This section is devoted to providing the key result of thiswork, which is the variance of (4). Not only is the key resultable to predict the NLI of most 4D constellations used in practice, it is straightforward enough to be easily calculatedwith even the simplest of computers. The detailed derivation ofthe key result will be given in the Appendix. The key resultis obtained under some simplifying assumptions, which arediscussed below.The first assumption is that the data symbols in the x-and y-polarization are correlated with each other. The secondassumption is that the data symbols in different time slots areindependent of one another. Here, we consider a multi-channelWDM system where channels across the spectrum can havedifferent launch powers and different 4D modulation formats.The probability distribution in each WDM channel is assumeduniform over all constellation points. We further assume thatthe launch power in the x- and y-polarization are the same,meaning that P COI E {| a x | } = E {| a y | } , P INT E {| b x | } = E {| b y | } , (11)where P COI and P INT are the total launch power transmitted inthe COI and interfering channel, resp. It is also assumed that E {| a x | } = E {| a y | } , E {| b x | } = E {| b y | } . (12)The last key assumption is that E { a x } = E { a y } = E { a x } = E { a x a ∗ y } = E {| a x | a x } = E {| a y | a x } = 0 . This assump-tion holds for most zero-mean symmetric constellations withrespect to the origin that have the same power in bothpolarizations. Although we will show the NLI variance forNyquist rectangular spectral shape channels (sinc pulse), theresults can also be used for near rectangular signal spectralshape such as a root raised cosine with small roll off factor.The NLI variance on the n -th channel (COI) caused by (4)is given by σ NLI ,n =Var (X Ω ∆ a (Ω) ) , (13)Since the data symbols in different WDM channels are uncor-related, we can write (13) as σ NLI ,n = σ SCI + N X j =1 ,j = n σ XPM (Ω) , Ω = | j − n | π ∆ f, (14)where ∆ f is the channel spacing. The SCI and XPM variancesgiven in (14) are expressed as σ SCI = σ SCI , x + σ SCI , y , (15)and σ XPM (Ω) = σ XPM , x (Ω) + σ XPM , y (Ω) , (16)resp., in which σ SCI , x and σ XPM , x are the SCI and XPMvariances in the x-polarization, resp. The same is true for they-polarized terms given in (15) and (16). The terms σ SCI , x and σ XPM , x (Ω) , given in (15) and (16), resp., are equal to σ SCI , x = 881 γ P COI (Ψ S + Ψ X + Ψ X + 3 Z ) , (17) REPRINT, SEPTEMBER 23, 2020 4
Table II
NTEGRAL EXPRESSIONS FOR THE TERMS USED IN (17)
AND (18). T
HEFUNCTIONS ρ S ( · ) AND ρ XP ( · ) ARE GIVEN IN (9)
AND (10),
RESP .Term Integral Expression S T R d w (2 π ) d w ′ (2 π ) ρ s ( w , w , w ) ρ ∗ s ( w ′ , w ′ , w + w + w ′ − w − w ′ ) X T R d w (2 π ) d w ′ π ρ s ( w , w , w ) ρ ∗ s ( w , w ′ , w ′ − w + w ) X T R d w (2 π ) d w ′ π ρ s ( w , w , w ) ρ ∗ s ( w ′ , w , w + w − w ′ ) Z T R d w (2 π ) | ρ s ( w , w , w ) | X T R d w (2 π ) d w ′ π ρ xp ( w , w , w ) ρ ∗ xp ( w − w + w ′ , w ′ , w ) Z T R d w (2 π ) | ρ xp ( w , w , w ) | Table IIT
HE TERMS USED IN (17)
AND (18). T
HE VALUES OF ϕ , · · · , ϕ AREGIVEN IN T ABLE
III.Term Expression Ψ ϕ − ϕ + 24 + 2 ϕ + ϕ − ϕ Ψ ϕ −
15 + 5 ϕ Ψ ϕ − ϕ Φ ϕ −
15 + 5 ϕ Table IIIE
XPRESSIONS FOR THE TERMS ϕ , · · · , ϕ USED IN T ABLE
II.Term Expression Term Expression Term Expression ϕ E {| a x | } E {| a x | } ϕ E {| a x | } E {| a x | } ϕ E {| a x | | a y | } E {| a x | } ϕ E {| a y | | a x | } E {| a x | } ϕ E {| a x | | a y | } E {| a x | } ϕ E {| b x | } E {| b x | } ϕ E {| b x | | b y | } E {| b x | } and σ XPM , x (Ω) = 881 γ P COI P INT (Ω) (Φ (Ω) X (Ω) + 6 Z (Ω)) . (18)The terms S , X , X , Z , X (Ω) , and Z (Ω) in Table Idepend on the spectral properties of the signal, in contrastwith Ψ , Ψ , Ψ and Φ , given in Table II, which depend onthe modulation format. The SCI and XPM variances in they-polarization can be obtained from (17) and (18), resp., byswapping x and y in (17), (18) and Table III.In the special case of independent polarizations that thesame format is used in both polarizations, Table III yields ϕ = ϕ = ϕ and ϕ = ϕ = 1 . These values used inTable II give Ψ = ϕ − ϕ +12 , Ψ = 5 ϕ − , Ψ = ϕ − ,and Φ = 5 ϕ − . These values used in combination withthe integral expressions in Table I can be shown to coincidewith the EGN model.IV. N UMERICAL R ESULTS
This section is focused on investigating the NLI of 4Dmodulation formats from the database [36]. A coherent trans-mission link consisting of 100 km spans of a standard single-mode fiber was simulated. The following parameters were used: Dispersion coefficient D = γ = α = T − =
32 Gbaud, and channel spacing ∆ f = η n = σ NLI ,n P , (19)assuming P COI = P INT = P , where σ NLI ,n is defined in (14).The SNR of the COI n is SNR n = P/ ( σ ASE + σ NLI ,n ) , where σ ASE is the variance of the amplified spontaneous emissionnoise (ASE). We first validate the 4D model using the split-step Fourier method (SSFM), and then compare a wide rangeof 4D constellations.
A. SSFM Simulations
Numerically solving the Manakov equation (2) for the entireC band is a big challenge. Part of the problem is, of course,high memory requirements, in addition to the excessive use ofvery large fast Fourier transforms. For this reason, the SSFMstudy was restricted to a bandwidth of 0.5 THz. To validate η n , given in (19), ASE-noise-free SSFM numerical simulationswere performed. In the absence of other noise sources, η n canbe estimated via the received SNR for each channel n via therelationship η n ≈ SNR est n P . (20)The approximate equality in (20) is due to the fact thatthe SSFM-based SNR est estimates also contain higher orderperturbation terms. The SNR for a constellation with M symbols was estimated viaSNR est n = P Mi =1 | ¯ y i | P Mi =1 E {| Y − ¯ y i | | X = x i } , (21)where X and Y are the random variables representing thetransmitted and received symbols, resp., x i is the i -th con-stellation point, and ¯ y i = E { Y | X = x i } . A total number of30000 symbols were used, of which the first 1500 and the last1500 symbols were removed from the transmitted and receivedsequences. All channels used a flat launch power of P = 0 dBm.A WDM system with N = 10 channels and four modulationformats, namely PM-QPSK, subset optimized PM-QPSK (SO-PM-QPSK) [36], [38], PM-16QAM and a4_256 [36], [39]was simulated. Fig. 1 shows the simulation results for η n in dB ( W − ) = 10 log ( η n · W ) using markers for atransmission distance of 500 km. Fig. 1 (a) indicates thatthe 4D model results for SO-PM-QPSK perfectly followsthe simulations, whereas the EGN model fails to estimatethe NLI of this format. Fig. 1 (b) also illustrates that theresults obtained from the 4D model for a4_256 are in goodagreement with simulations, while the EGN model results REPRINT, SEPTEMBER 23, 2020 5
Table IVT
HE VALUE Φ FOR CONSTELLATIONS CHOSEN FROM [36]
ALONG WITH THREE NEW CONSTELLATIONS PROPOSED IN [8], [11], [37].
Modulation Φ Modulation Φ Modulation Φ Modulation Φ Modulation Φ biortho4_8 − tetra4_9 − . PM-QPSK − SO-PM-QPSK − dicyclic4_24 − − l4_25 − . b4_32 − . w4_40 − . w4_49 − . b4_64 − . − − w4_88 − . − . SP-QAM4_128 − . w4_145 − . w4_152 − . w4_169 − . PM-16QAM − . w4_256 − . w4_313 − . w4_409 − . w4_464 − . cross4_512 − . sphere4_512 − . SP-cross4_512 − . − w4_601 − . w4_656 − . w4_800 − . cross4_2048 − . SP-QAM4_2048 − . PM-64QAM − . n η n [ d B ( W − )] PM-QPSK (4D, EGN)SO-PM-QPSK (4D)SO-PM-QPSK (EGN)Simulations n PM-16QAM (4D, EGN)a4_256 (4D)a4_256 (EGN)Simulations(a) (b)
Figure 1. η n as a function of channel number n after 5 spans. The linkconsisting of 5 spans supports N = 10 WDM channels. depart from simulations. These results show that the EGNmodel is inaccurate for the study of arbitrary 4D constellations,and that the NLI can be underestimated (SO-PM-QPSK) oroverestimated (a4_256). The proposed 4D model instead hasthe capacity to predict the NLI of 4D formats with a goodlevel of accuracy. The discrepancy between simulations andthe results obtained from the 4D model is on average about0.2 dB. For PM-QPSK and PM-16QAM, both the EGN modeland 4D model give the same results that match the simulationresults. In the following section, we attempt to identify thereasons behind an increase or decrease in the NLI estimatedfrom the EGN model and 4D model.
B. Comparing a wide range of constellations
This section investigates the NLI of 4D constellations prop-agated in a C-band system. We assume that the entire spectrumis populated with N = 80 WDM channels, and that the linkcomprises 10 spans. Figs. 2 (a) and (c) compare differentformats in terms of η n , while Figs. 2 (b) and (d) comparethem in terms of SNR . We interpret the first two coordinatesin a coordinate list of [36] as the x polarization and the lasttwo as the y polarization. Figs. 2 (a) and (b) give informationabout PM-QPSK, SO-PM-QPSK and dicyclic4_16 [36], [40].We benchmark the 4D model against the EGN model in thisfigure. As can be seen in Fig. 2 (a), the curves are highest inthe middle of spectrum. The 4D model indicates that over theentire spectrum shown in Fig. 2 (a), SO-PM-QPSK undergoesthe most NLI, while PM-QPSK and dicyclic4_16 experiencethe least. It is also noticeable that PM-QPSK and dicyclic4_16have the same NLI. The difference between the experiencedNLI for SO-PM-QPSK and PM-QPSK is about 1.34 dB. This
20 40 60 80363840 n η n [ d B ( W − )] − − . . Launch Power [dBm] S N R [ d B ] PM-QPSK (4D, EGN) dicyclic4_16 (4D)SO-PM-QPSK (4D) dicyclic4_16 (EGN)SO-PM-QPSK (EGN)
20 40 60 80373839 n η n [ d B ( W − )] PM-16QAM (4D, EGN) a4_256 (4D) a4_256 (EGN) − . Launch Power [dBm] S N R [ d B ] . dB . dB . dB(a) (b)(c) (d) Figure 2. (a) and (c) illustrate η n , defined in (19), as a function of channelnumber n after 10 spans, while (b) and (d) illustrate the SNR of COI,i.e., SNR , as a function of launch power after 10 spans. The full C-bandspectrum can accommodate N = 80 WDM channels. means that SO-PM-QPSK is more vulnerable to the Kerrnonlinearity than PM-QPSK. The SCI and XPM terms areresponsible for this gap. The impact of SCI on the COI is high,but the XPM effects in multi-channel WDM systems are evenhigher, and therefore, the better part of this deviation stemsform the XPM terms. To be more specific, the origin of thisdiscrepancy comes form the fact that Φ , given in Table II,for SO-PM-QPSK ( Φ = − ) is larger than for PM-QPSK( Φ = − ).From the curve with triangles (4D model) to the greencurve (EGN model), there is a 2.8 dB increase in the NLI fordicyclic4_16, with SNR falling by around 1.1 dB to approx-imately 16.1 dB (see Fig. 2 (b)). This implies that the EGNmodel significantly overestimates the NLI for dicyclic4_16.This is because ϕ = 0 for this format according to Table III,whereas the EGN model corresponds to setting ϕ = 1 for anyformat. On the other hand, we can see that the EGN model REPRINT, SEPTEMBER 23, 2020 6 underestimates the NLI of SO-PM-QPSK in comparison withthe 4D model. This is because the term ϕ is lower for theEGN model ( ϕ = 1 ) than for the 4D model ( ϕ = 1 . ).Fig. 2 (c) and (d) compare the PM-16QAM and a4_256formats in terms of η n and SNR, resp. Fig. 2 (c) shows thatPM-16QAM is at a disadvantage compared with a4_256. Thedeviation of the NLI between the PM-16QAM and a4_256formats, as shown in Fig. 2 (c), is about 0.3 dB. This deviationmay be rooted in the value of Φ which is smaller for a4_256( Φ = − . ) than for PM-16QAM ( Φ = − . ). We can alsosee in Fig. 2 (c) that the EGN model overestimates the NLIof a4_256 by about 0.6 dB. It is clear from Fig. 2 (d) that theSNR for a4_256 falls from about 17 dB (4D model) to around16.8 dB (EGN model) at 0 dBm launch power.As mentioned earlier, the experienced amount of NLI isdependent on Ψ , Ψ , Ψ , and Φ given in Table II, noneof which is more important than Φ . Table IV shows thevalue of Φ for constellations selected from [36] as wellas three constellations proposed in [8], [11], [37]. All theconstellations shown in Table IV satisfy the assumptions madein Sec. III. It is clear that SO-PM-QPSK and four-dimensionalorthant-symmetric 128-ary modulation (4D-OS128) [37] gen-erate higher NLI than do other formats. For SO-PM-QPSK and4D-OS128, Φ is around − , which is higher than the others.The lowest amount of NLI belongs to constellations whose Φ equals to − , meaning that these constellations undergoapproximately the same NLI as PM-QPSK.V. C ONCLUSION
A nonlinear model which analytically models the impactof the Kerr nonlinearity on a 4D signal space was proposedand analyzed in detail. The model applies to zero-mean dual-polarization 4D formats which are symmetric with respectto the origin and have equal energy on the two polarizationcomponents. Unlike the GN and EGN models, we consider theinterpolarization dependency so as to derive a 4D nonlinearmodel. The proposed model accounts for the SCI and XPMnonlinear terms. This is because the SCI and XPM are thepredominant nonlinear terms in multi-channel WDM systems.We have compared different 4D modulation formats in termsof the experienced NLI, and showed that the derived model isa powerful tool to find 4D formats which are more resistantto the NLI. A
PPENDIX
Our intention, in this section, is to study the variance ofSCI (the first term on the right-hand side of (4)) and XPM(the second term on the right-hand side of (4)) terms. Weproceed by computing the variance of SCI.
A. SCI variance
For the sake of brevity, we only focus on the x-polarizedelement of (4) because we can obtain the results for the y-polarized component under the substitution x → y, y → x. Using(4), the x-polarized component of SCI term is given by ∆ a , SCI , x = i γ X h,k,l S h,k,l (cid:16) a h, x a ∗ k, x a l, x + a h, y a ∗ k, y a l, x (cid:17) . (22) The variance of (22) is therefore equal to σ SCI , x = E { ∆ a , SCI , x ∆ a ∗ , SCI , x } − E { ∆ a , SCI , x } E { ∆ a ∗ , SCI , x } , (23)where E { ∆ a , SCI , x } = 0 . This is because under the assump-tions made in Sec. III, we have E { a h, x } = E { a h, x } = E { a h, x a ∗ h, y } = E {| a h, x | a h, x } = E {| a h, y | a h, x } = 0 (see[31, Appendix A]), and as a result, E { a h, x a ∗ k, x a l, x } = E { a h, y a ∗ k, y a l, x } = 0 for all h , k , and l . Substituting (22) into(23) gives σ SCI , x = 6481 γ X h,k,l,h ′ ,k ′ ,l S h,k,l S ∗ h ′ ,k ′ ,l ′ (cid:16) E { a h, x a ∗ k, x a l, x a ∗ h ′ , x a k ′ , x a ∗ l ′ , x } + E { a h, x a ∗ k, x a l, x a ∗ h ′ , y a k ′ , y a ∗ l ′ , x } + E { a h, y a ∗ k, y a l, x a ∗ h ′ , x a k ′ , x a ∗ l ′ , x } + E { a h, y a ∗ k, y a l, x a ∗ h ′ , y a k ′ , y a ∗ l ′ , x } (cid:17) . (24)We can rewrite (24) as σ SCI , x = X i =1 σ SCI , x ,i , (25)where σ SCI , x ,i represents the i -th term in (24). We only give theprocedure of calculating σ SCI , x , in detail, and we can followthe same approach for the others. The contribution of σ SCI , x , was calculated in the first term of Eq. (36) and Eq. (105) of[41], and σ SCI , x , is more challenging to compute than thesecond and third terms, which is why we focus on calculatingthis term. The term σ SCI , x , is given by σ SCI , x , =6481 γ X h,k,l,h ′ ,k ′ ,l S h,k,l S ∗ h ′ ,k ′ ,l ′ E { a h, y a ∗ k, y a l, x a ∗ h ′ , y a k ′ , y a ∗ l ′ , x } , (26)whose expectation term is equal to [31, Eqs. (26) and (27)] E { a h, y a ∗ k, y a l, x a ∗ h ′ , y a k ′ , y a ∗ l ′ , x } = E {| a x | | a y | } , h = k = l = h ′ = k ′ = l ′ , E {| a y | } E {| a x | | a y | } , h = h ′ = l = k = k ′ = l ′ , E {| a y | } E {| a x | | a y | } , h = k = l = h ′ = k ′ = l ′ , E { a y a ∗ x } E {| a y | a ∗ y a x } , h = l ′ = l = k = h ′ = k ′ , E { a ∗ y a x } E {| a y | a y a ∗ x } , k = l = h = h ′ = k ′ = l ′ , E {| a y | } E {| a x | | a y | } , k = k ′ = h = l = h ′ = l ′ , E { a ∗ y a x } E {| a y | a y a ∗ x } , l = h ′ = h = k = k ′ = l ′ , E {| a x | } E {| a y | } , l = l ′ = h = k = h ′ = k ′ , E {| a y | } E {| a x | | a y | } , h ′ = k ′ = h = k = l = l ′ , E { a y a ∗ x } E {| a y | a ∗ y a x } , k ′ = l ′ = h = k = l = h ′ , E {| a y | } E {| a x | } , h = h ′ = k = k ′ = l = l ′ , E {| a y | } E { a x a ∗ y } E { a y a ∗ x } , h = k = l = h ′ = k ′ = l ′ , E {| a y | } E {| a x | } , h = k = l = l ′ = k ′ = h ′ , E {| a y | } E { a x a ∗ y } E { a y a ∗ x } , h = h ′ = k = l = k ′ = l ′ , E {| a y | } E { a x a ∗ y } E { a y a ∗ x } , h = l ′ = k = l = h ′ = k ′ , E {| a y | } E { a x a ∗ y } E { a y a ∗ x } , h = l ′ = k = k ′ = l = h ′ . (27) REPRINT, SEPTEMBER 23, 2020 7
We can hence write (26) as σ SCI , x , = X j =1 σ SCI , x , ,j , (28)where σ SCI , x , ,j stands for the contribution of the j -th case,given in (27), to (26). We first remove from (27) the termswhich involve E { a x a ∗ y } or E { a ∗ x a y } . By substituting (7) into(26), we can write σ SCI , x , , , given in (28), as σ SCI , x , , = 6481 γ E {| a x | | a y | } Z d w (2 π ) d w ′ (2 π ) ρ s ( w , w , w ) · ρ ∗ s ( w ′ , w ′ , w ′ ) X h e i ( w − w + w − w ′ + w ′ − w ′ ) hT . (29)Using the identity [35, Eq. (14)] ∞ X k = −∞ e ikT w = 2 πT ∞ X n = −∞ δ ( w − πnT ) , (30)for the sinc pulse and considering (11), we can write (29) as σ SCI , x , , = 881 γ P COI ϕ S , (31)where S is given in Table I and ϕ is given in Table III.By using (7) once again in (26), and considering (11) theterm σ SCI , x , , , given in (28), is equal to σ SCI , x , , = 881 γ P COI ϕ Z d w (2 π ) d w ′ (2 π ) ρ s ( w , w , w ) · ρ ∗ s ( w ′ , w ′ , w ′ ) X h = l e i ( w − w ′ ) hT + i ( w − w + w ′ − w ′ ) lT , (32)where ϕ is given in Table III. Using the same approach givenin [31, Eq. (29)], we have X h = l e i ( w − w ′ ) hT e i ( w − w + w ′ − w ′ ) lT = X h,l e i ( w − w ′ ) hT · e i ( w − w + w ′ − w ′ ) lT − X h e i ( w − w ′ + w − w + w ′ − w ′ ) hT . (33)Considering (30), we can rewrite (33) for the sinc pulse as X h = l e i ( w − w ′ ) hT e i ( w − w + w ′ − w ′ ) lT = 4 π T δ ( w − w + w ′ − w ′ ) · δ ( w − w ′ ) − πT δ ( w − w ′ + w − w + w ′ − w ′ ) . (34)By inserting (34) into (32), we get σ SCI , x , , = 881 γ P COI ϕ ( X − S ) , (35)where X and S are given in Table I. Considering (7), (11),(26), (28), (33) and (34), we can express σ SCI , x , , , given in(28), as σ SCI , x , , = 881 γ P COI ϕ Z d w (2 π ) d w ′ (2 π ) ρ s ( w , w , w ) · ρ ∗ s ( w ′ , w ′ , w ′ ) (cid:16) π T δ ( w − w ′ + w ′ − w ′ ) δ ( w − w ) − πT δ ( w − w + w − w ′ + w ′ − w ′ ) (cid:17) . (36) The term δ ( w − w ) is a bias term and should be discarded.Bias terms are those for which w = w , w = w , w ′ = w ′ ,or w ′ = w ′ . These terms create a constant phase shift,and thus, irrelevant for the noise variance we would like tocompute (see [16, Sec. VIII, Eqs. (63)–(67)], [18, Sec. 3,Eq. (17)], [20, Appendix A], [42, Sec.IV-B and the text after(63)] and [17, Appendix C]). Eq. (36) is therefore reduced to σ SCI , x , , = − γ P COI ϕ S , (37)and we can express the same formula for σ SCI , x , , . Followingthe same approach, the term σ SCI , x , , , given in (28), con-tributes to (26) as σ SCI , x , , = 881 γ P COI ϕ ( X − S ) , (38)where X is given in Table II.The last step in calculating (28) is to investigate the impactof the last six situations on the NLI variance. We start with σ SCI , x , , , which contributes to (28) as σ SCI , x , , =6481 γ E {| a y | } E {| a x | } Z d w (2 π ) d w ′ (2 π ) ρ s ( w ,w ,w ) · ρ ∗ s ( w ′ , w ′ , w ′ ) X h = k = l e i ( w − w ′ ) hT e − i ( w − w ′ ) kT e i ( w − w ′ ) lT , (39)where the triple summation is expressed as X h = k = l e i ( w − w ′ ) hT − i ( w − w ′ ) kT + i ( w − w ′ ) lT = X h,k,l e i ( w − w ′ ) hT · e − i ( w − w ′ ) kT + i ( w − w ′ ) lT − X h = k = l e i ( w − w ′ − w + w ′ ) hT + i ( w − w ′ ) lT − X h = l = k e i ( w − w ′ + w − w ′ ) hT e − i ( w − w ′ ) kT − X h = k = l e i ( w − w ′ ) hT · e i ( − w + w ′ + w − w ′ ) kT + 2 X h e i ( w − w ′ − w + w ′ + w − w ′ ) hT . (40)Considering (11), (30), (40), and (39), we have σ SCI , x , , = 881 γ P COI Z d w (2 π ) d w ′ (2 π ) ρ s ( w , w , w ) · ρ ∗ s ( w ′ , w ′ , w ′ ) (cid:16) π T δ ( w − w ′ ) δ ( w ′ − w ) δ ( w − w ′ ) − π T δ ( w − w ′ + w ′ − w ) δ ( w − w ′ ) − π T δ ( w ′ − w ) · δ ( w − w ′ + w − w ′ ) − π T δ ( w ′ − w + w − w ′ ) · δ ( w − w ′ ) + 8 πT δ ( w − w ′ + w ′ − w + w − w ′ ) (cid:17) , (41)which can be written as σ SCI , x , , = 881 γ P COI ( Z − X − X +2 S ) , (42)where Z , X , X and S are given in Table I. The sameapproach holds for σ SCI , x , , , but the bias terms should notbe taken into account. Considering (31), (35), (37), (38) and REPRINT, SEPTEMBER 23, 2020 8 (42), and using (11) and (12), we can express (26) as σ SCI , x , = 881 γ P COI [( ϕ − ϕ − ϕ + 4) S +( ϕ + ϕ − X + ( ϕ − X + Z ] , (43)which is called the interpolarization nonlinear effect, and theterm ϕ is given in Table III. This expression is not availablein the literature.The contributions of σ SCI , x , , σ SCI , x , , and σ SCI , x , , given in(26), can be calculated through the same procedure, so theirdetailed derivations will not be repeated here, and we onlygive the final results for them as follows σ SCI , x , = σ SCI , x , = 881 γ P COI [( ϕ − ϕ − ϕ + 4) S + (2 ϕ − X ] , (44) σ SCI , x , = 881 γ P COI [( ϕ − ϕ + 12) S + (4 ϕ − X + ( ϕ − X + 2 Z ] , (45)and we call (45) the intra-polarization nonlinear effect. Byexcluding the bias terms from [41, Eq. (105)], and using itinto the first term of [41, Eq. (36)], we can get (45). Putting(45), (44) and (43) together, we obtain the total variance ofthe SCI nonlinear term (24), which is expressed as (17) withcoefficients from Tables I and III. B. XPM variance
Here we calculate σ XPM , x (Ω) in (18) for a single pair ofchannels with fixed separation Ω . For notational convenience,the dependence on Ω is dropped throughout the section. Asmentioned in Sec. II, the second term of (4) gives rise to theXPM nonlinear term. The x-polarized component of this termis i γ X h,k,l X h,k,l (cid:16) b h, x b ∗ k, x a l, x + b h, y b ∗ k, y a l, x + b h, x b ∗ k, y a l, y (cid:17) , (46)whose variance is equal to σ XPM , x = 6481 γ X h,k,l,h ′ ,k ′ ,l ′ X h,k,l X ∗ h ′ ,k ′ ,l ′ (cid:16) E { b h, x b ∗ k, x b ∗ h ′ , x b k ′ , x }· E { a l, x a ∗ l ′ , x } + 2 E { b h, x b ∗ k, x b ∗ h ′ , y b k ′ , y } E { a l, x a ∗ l ′ , x } + 2 E { b h, y b ∗ k, y b ∗ h ′ , x b k ′ , x } E { a l, x a ∗ l ′ , x } + E { b h, y b ∗ k, y b ∗ h ′ , y b k ′ , y }· E { a l, x a ∗ l ′ , x } + E { b h, x b ∗ k, y b ∗ h ′ , x b k ′ , y } E { a l, y a ∗ l ′ , y } (cid:17) . (47)We now focus on the calculation of the first term of (47),and we can compute the others in a similar way. To evaluatethe fourth order moment given in the first term of (47), thefollowing cases should be taken into account. E { b h, x b ∗ k, x b ∗ h ′ , x b k ′ , x } = E {| b x | } , h = k = h ′ = k ′ E {| b x | } , h = k = h ′ = k ′ E {| b x | } , h = h ′ = k = k ′ . (48) Bias terms in [41, Eq. (105)] are those which involve δ m − n , δ k − n , δ m ′ − n ′ and δ k ′ − n ′ . Using (48) and (8), we can write the contribution of the firstterm of (47) to the NLI, as σ XPM , x , = 6481 γ Z d w (2 π ) d w ′ (2 π ) ρ xp ( w , w , w ) ρ ∗ xp ( w ′ , w ′ , w ′ ) (cid:16) E {| b x | } E {| a x | } X h e i ( w − w − w ′ + w ′ ) hT X l e i ( w − w ′ ) lT +4 E {| b x | } E {| a x | } h X h = h ′ e i ( w − w ) hT − ( w ′ − w ′ ) h ′ T X l e iw lT · e − w ′ lT + X h = k e i ( w − w ′ ) hT − ( w − w ′ ) kT X l e i ( w − w ′ ) lT i(cid:17) . (49)Considering (11), (30) and (33), we can express (49) as σ XPM , x , = 881 γ P COI P INT Z d w (2 π ) d w ′ (2 π ) ρ xp ( w , w , w ) · ρ ∗ xp ( w ′ , w ′ , w ′ ) (cid:16) ϕ π T δ ( w − w − w ′ + w ′ ) · δ ( w − w ′ )+4 h ( 8 π T δ ( w − w ) δ ( w ′ − w ′ ) − π T · δ ( w − w + w ′ − w ′ )) δ ( w − w ′ ) + ( 8 π T δ ( w − w ′ ) · δ ( w − w ′ ) − π T δ ( w − w + w ′ − w ′ )) δ ( w − w ′ ) i(cid:17) . (50)It should be noticed that the term δ ( w − w ) δ ( w ′ − w ′ ) is abias term and should be ignored. By excluding this term from(50), we have σ XPM , x , = 881 γ P COI P INT [( ϕ −
2) 4 X + 4 Z ] , (51)where X and Z are given in Table I. Analogous expressionshold for other terms of (47). We can therefore express (47) as(18). R EFERENCES[1] R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel,“Capacity limits of optical fiber networks,”
J. Lightw. Technol. , vol. 28,no. 4, pp. 662–701, Feb. 2010.[2] S. Betti, F. Curti, G. De Marchis, and E. Iannone, “Exploiting fiber op-tics transmission capacity: 4-quadrature multilevel signalling,”
Electron.Lett. , vol. 26, no. 14, pp. 992–993, July 1990.[3] ——, “A novel multilevel coherent optical system: 4-quadrature signal-ing,”
J. Lightw. Technol. , vol. 9, no. 4, pp. 514–523, Apr. 1991.[4] S. Benedetto and P. Poggiolini, “Theory of polarization shift keyingmodulation,”
IEEE Trans. Commun. , vol. 40, no. 4, pp. 708–721, Apr.1992.[5] R. Cusani, E. Iannone, A. M. Salonico, and M. Todaro, “An efficientmultilevel coherent optical system: M-4Q-QAM,”
J. Lightw. Technol. ,vol. 10, no. 6, pp. 777–786, June 1992.[6] E. Agrell and M. Karlsson, “Power-efficient modulation formats incoherent transmission systems,”
J. Lightw. Technol. , vol. 27, no. 22,pp. 5115–5126, Nov. 2009.[7] M. Karlsson and E. Agrell, “Which is the most power-efficient modula-tion format in optical links?”
Opt. Express , vol. 17, no. 13, pp. 10 814–10 819, June 2009.[8] K. Kojima, T. Yoshida, T. Koike-Akino, D. S. Millar, K. Parsons,M. Pajovic, and V. Arlunno, “Nonlinearity-tolerant four-dimensional2A8PSK family for 5–7 bits/symbol spectral efficiency,”
J. Lightw.Technol. , vol. 35, no. 8, pp. 1383–1391, Feb. 2017.[9] M. Reimer, S. O. Gharan, A. D. Shiner, and M. O’Sullivan, “Optimized4 and 8 dimensional modulation formats for variable capacity in opticalnetworks,” in
Proc. Optical Fiber Communication Conf. , Anaheim, CA,USA, Mar. 2016.
REPRINT, SEPTEMBER 23, 2020 9 [10] T. Nakamura, E. L. T. de Gabory, H. Noguchi, W. Maeda, J. Abe, andK. Fukuchi, “Long haul transmission of four-dimensional 64SP-12QAMsignal based on 16QAM constellation for longer distance at samespectral efficiency as PM-8QAM,” in
Proc. European Conf. OpticalCommunication , Valencia, Spain, Sep. 2015.[11] B. Chen, C. Okonkwo, H. Hafermann, and A. Alvarado, “Polarization-ring-switching for nonlinearity-tolerant geometrically shaped four-dimensional formats maximizing generalized mutual information,”
J.Lightw. Technol. , vol. 37, no. 14, pp. 3579–3591, July 2019.[12] A. Alvarado and E. Agrell, “Four-dimensional coded modulation withbit-wise decoders for future optical communications,”
J. Lightw. Tech-nol. , vol. 33, no. 10, pp. 1993–2003, May 2015.[13] J. Cai, M. V. Mazurczyk, H. G. Batshon, M. Paskov, C. R. Davidson,Y. Hu, O. V. Sinkin, M. A. Bolshtyansky, D. G. Foursa, and A. N.Pilipetskii, “Performance comparison of probabilistically shaped QAMformats and hybrid shaped APSK formats with coded modulation,”
J.Lightw. Technol. , vol. 38, no. 12, pp. 3280–3288, June 2020.[14] F. Frey, S. Stern, J. K. Fischer, and R. F. H. Fischer, “Two-stage codedmodulation for Hurwitz constellations in fiber-optical communications,”
J. Lightw. Technol. , vol. 38, no. 12, pp. 3135–3146, June 2020.[15] A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modelingof the impact of nonlinear propagation effects in uncompensated opticalcoherent transmission links,”
J. Lightw. Technol. , vol. 30, no. 10, pp.1524–1539, May 2012.[16] A. Mecozzi and R. J. Essiambre, “Nonlinear Shannon limit in pseu-dolinear coherent systems,”
J. Lightw. Technol. , vol. 30, no. 12, pp.2011–2024, June 2012.[17] P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinearpropagation in a strongly dispersive optical communication system,”
J.Lightw. Technol. , vol. 31, no. 8, pp. 1273–1282, Apr. 2013.[18] R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinearnoise in long, dispersion-uncompensated fiber links,”
Opt. Express ,vol. 21, no. 22, pp. 25 685–25 699, Nov. 2013.[19] V. Curri, A. Carena, P. Poggiolini, G. Bosco, and F. Forghieri, “Ex-tension and validation of the GN model for non-linear interference touncompensated links using Raman amplification.”
Opt. Express , vol. 21,no. 3, pp. 3308–17, Feb. 2013.[20] A. Carena, G. Bosco, V. Curri, Y. Jiang, P. Poggiolini, and F. Forghieri,“EGN model of non-linear fiber propagation,”
Opt. Express , vol. 22,no. 13, pp. 16 335–16 362, June 2014.[21] A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modelingof the impact of nonlinear propagation effects in uncompensated opticalcoherent transmission links,”
J. Lightw. Technol. , vol. 30, no. 10, pp.1524–1539, May 2012.[22] P. Poggiolini, “The GN model of non-linear propagation in uncompen-sated coherent optical systems,”
J. Lightw. Technol. , vol. 30, no. 24, pp.3857–3879, Dec. 2012.[23] P. Serena and A. Bononi, “An alternative approach to the Gaussian noisemodel and its system implications,”
J. Lightw. Technol. , vol. 31, no. 22,pp. 3489–3499, Nov. 2013.[24] L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch,“A discrete-time model for uncompensated single-channel fiber-opticallinks,”
IEEE Trans. Commun. , vol. 60, no. 11, pp. 3440–3450, Nov.2012.[25] E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity ofa nonlinear optical channel with finite memory,”
J. Lightw. Technol. ,vol. 32, no. 16, pp. 2862–2876, Aug. 2014.[26] P. Poggiolini, A. Nespola, Y. Jiang, G. Bosco, A. Carena, L. Bertignono,S. M. Bilal, S. Abrate, and F. Forghieri, “Analytical and experimentalresults on system maximum reach increase through symbol rate op-timization,”
J. Lightw. Technol. , vol. 34, no. 8, pp. 1872–1885, Apr.2016.[27] M. Secondini, E. Forestieri, and G. Prati, “Achievable information rate innonlinear WDM fiber-optic systems with arbitrary modulation formatsand dispersion maps,”
J. Lightw. Technol. , vol. 31, no. 23, pp. 3839–3852, Dec. 2013.[28] D. Semrau, E. Sillekens, R. I. Killey, and P. Bayvel, “A modulationformat correction formula for the Gaussian noise model in the presenceof inter-channel stimulated Raman scattering,”
J. Lightw. Technol. ,vol. 37, no. 19, pp. 5122–5131, Oct. 2019.[29] H. Rabbani, G. Liga, V. Oliari, L. Beygi, E. Agrell, M. Karlsson, andA. Alvarado, “A general analytical model of nonlinear fiber propagationin the presence of Kerr nonlinearity and stimulated Raman scattering,” arXiv , 2020. [Online]. Available: http://arxiv.org/abs/1909.08714v2.[30] E. Agrell, G. Durisi, and P. Johannisson, “Information-theory-friendlymodels for fiber-optic channels: A primer,” in
IEEE Information TheoryWorkshop (ITW) , Jerusalem, Israel, Apr.-May 2015. [31] O. Golani, R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Modeling thebit-error-rate performance of nonlinear fiber-optic systems,”
J. Lightw.Technol. , vol. 34, no. 15, pp. 3482–3489, Aug. 2016.[32] G. Liga, A. Barreiro, H. Rabbani, and A. Alvarado, “Extending fibrenonlinear interference power modelling to account for general dual-polarisation 4D modulation formats,” arXiv , Aug, 2020. [Online]. Avail-able: http://arxiv.org/abs/2008.11243.[33] A. Mecozzi and R.-J. Essiambre, “Nonlinear Shannon limit in pseudo-linear coherent systems,”
J. Lightw. Technol. , vol. 30, pp. 2011–2024,Jun. 2012.[34] G. P. Agrawal,
Fiber-Optic Communication Systems , 3rd ed. Wiley,2002.[35] R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Inter-channel nonlinearinterference noise in WDM systems: Modeling and mitigation,”
J.Lightw. Technol. , vol. 33, no. 5, pp. 1044–1053, Mar. 2015.[36] E. Agrell, “Database of sphere packings,” 2014–2020. [Online]. Avail-able: http://codes.se/packings/.[37] B. Chen, A. Alvarado, S. van der Heide, M. van den Hout, H. Hafer-mann, and C. Okonkwo, “Analysis and experimental demonstra-tion of orthant-symmetric four-dimensional 7 bit/4D-sym modulationfor optical fiber communication,” arXiv , 2020. [Online]. Available:http://arxiv.org/abs/2003.12712.[38] M. Sjödin, E. Agrell, and M. Karlsson, “Subset-optimized polarization-multiplexed PSK for fiber-optic communications,”
IEEE Commun. Lett. ,vol. 17, no. 5, pp. 838–840, May 2013.[39] T. A. Eriksson, S. Alreesh, C. Schmidt-Langhorst, F. Frey, P. W.Berenguer, C. Schubert, J. K. Fischer, P. A. Andrekson, M. Karlsson,and E. Agrell, “Experimental investigation of a four-dimensional 256-arylattice-based modulation format,” in
Proc. Optical Fiber CommunicationConf. , Los Angeles, CA, USA, Mar. 2015.[40] L. Zetterberg and H. Brändström, “Codes for combined phase andamplitude modulated signals in a four-dimensional space,”
IEEE Trans.Commun. , vol. 25, no. 9, pp. 943–950, Sep. 1977.[41] A. Carena, G. Bosco, V. Curri, Y. Jiang, P. Poggiolini, andF. Forghieri, “On the accuracy of the GN-model and on analyticalcorrection terms to improve it,” arXiv , 2014. [Online]. Available:http://arxiv.org/abs/1401.6946.[42] P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri,“A detailed analytical derivation of the GN model of non-linear interfer-ence in coherent optical transmission systems,” arXivarXiv