Polarization-Division Multiplexing Based on the Nonlinear Fourier Transform
Jan-Willem Goossens, Mansoor I. Yousefi, Yves Jaouën, Hartmut Hafermann
PPolarization-Division Multiplexing Based on theNonlinear Fourier Transform J AN -W ILLEM G OOSSENS , M ANSOOR
I. Y
OUSEFI , Y VES J AOUËN AND H ARTMUT H AFERMANN Mathematical and Algorithmic Sciences Lab, Paris Research Center, Huawei Technologies France Communications and Electronics Department, Telecom ParisTech, Paris, 75013, France * [email protected] Abstract:
Polarization-division multiplexed (PDM) transmission based on the nonlinear Fouriertransform (NFT) is proposed for optical fiber communication. The NFT algorithms are generalizedfrom the scalar nonlinear Schrödinger equation for one polarization to the Manakov systemfor two polarizations. The transmission performance of the PDM nonlinear frequency-divisionmultiplexing (NFDM) and PDM orthogonal frequency-division multiplexing (OFDM) aredetermined. It is shown that the transmission performance in terms of Q-factor is approximatelythe same in PDM-NFDM and single polarization NFDM at twice the data rate and that thepolarization-mode dispersion does not seriously degrade system performance. Compared withPDM-OFDM, PDM-NFDM achieves a Q-factor gain of 6.4 dB. The theory can be generalized tomulti-mode fibers in the strong coupling regime, paving the way for the application of the NFTto address the nonlinear effects in space-division multiplexing. © 2018 Optical Society of America
OCIS codes: (060.2330) Fiber optics communications,(060.4230) Multiplexing, (060.4370) Nonlinear optics, fibers
References and links
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1. Introduction
Nonlinear frequency-division multiplexing (NFDM) is an elegant method to address the nonlineareffects in optical fiber communication. The scheme can be viewed as a generalization ofcommunication using fiber solitons [1] and goes back to the original idea of eigenvaluecommunication [2].In this approach, information is encoded in the nonlinear spectrum of the signal, definedby means of the nonlinear Fourier transform (NFT), [3–5]. The evolution of the nonlinearspectral components in fiber is governed by simple independent equations. As a result, thecombined effects of the dispersion and nonlinearity can be compensated in the digital domain bythe application of an all-pass-like filter. Furthermore, interference-free communication can beachieved in network environments.The nonlinear spectrum consists of a continuous part and, in the case of the anomalous dispersionfiber, also a discrete part (solitonic component). In principle all degrees-of-freedom can bemodulated. Prior work has mostly focused on either continuous spectrum modulation [6–10],r discrete spectrum modulation [6]. Transmission based on NFT has been experimentallydemonstrated for the continuous spectrum [11–13], discrete spectrum [14], as well as the fullspectrum [15].Modern coherent optical fiber systems are based on polarization-division multiplexed (PDM)transmission to improve the achievable rates. However, research on data transmission using theNFT is limited to the scalar nonlinear Schrödinger equation, which does not take into accountpolarization effects (with the exception of [16]).In this paper we overcome this limitation by generalizing the NFDM to the Manakov system,proposing PDM-NFDM. We develop a stable and accurate algorithm to compute the forwardand inverse NFT of a two-dimensional signal. It is shown that the PDM-NFDM based on thecontinuous spectrum modulation is feasible, and that the data rate can be approximately doubledcompared to the single polarization NFDM. Compared to the PDM-OFDM, the PDM-NFDMexhibits a peak Q-factor gain of 6.4 dB, in a system with 25 spans of 80 km standard single-modefiber and 16 QAM.In this paper, we set the discrete spectrum to zero and modulate only the continuous spectrum.Even though not all available degrees of freedom are modulated, the achievable rates reachremarkably close to the upper bound [17, 18].The differential group delay and randomly varying birefringence give rise to temporal pulsebroadening due to polarization-mode dispersion (PMD). The PMD might be compensated inconventional systems, and may even be beneficial in reducing the nonlinear distortions [19]. Onthe other hand, PMD changes the nonlinear interaction between signals in the two polarizations.The impact of PMD on NFDM is not fully investigated yet [20]. In this paper, we show that linearpolarization effects can be equalized in PDM-NFDM at the receiver using standard techniques,and that the PMD does not seriously degrade performance.
2. Channel Model with Two Polarizations
Light propagation in two polarizations in optical fiber is modeled by the coupled nonlinearSchrödinger equation (CNLSE) [21]. The fiber birefringence usually varies rapidly and randomlyalong the fiber in practical systems (on a scale of 0.3 to 100 meters). Under this assumption, theaveraging of the nonlinearity in the CNLSE leads to the Manakov-PMD equation [22–24]: ∂ A ∂ Z = − j ∆ β σ Z A − ∆ β σ Z ∂ A ∂ T − α A + j β ∂ A ∂ T − j γ (cid:13)(cid:13) A (cid:13)(cid:13) A . (1)Here A ≡ A ( Z , T ) is the 2 × A and A of thetwo polarization components, Z denotes the distance along the fiber, T represents time, σ Z is a2 × Z ), and ∆ β , ∆ β , α , β and γ areconstant numbers.The term ∂ A / ∂ T is responsible for the PMD [24], while the second line represents loss,chromatic dispersion and Kerr nonlinearity. The factor 8 / ∂ A ∂ Z = j β ∂ A ∂ T − j γ (cid:13)(cid:13) A (cid:13)(cid:13) A . (2)t is convenient to normalize (2). Let Z = T = (cid:113)(cid:0) | β | Z (cid:1) / , A = (cid:114) (cid:46) (cid:0) γ Z (cid:1) . (3)Introducing the normalized variables z = Z / Z , t = T / T and q = A / A , (2) is simplified to theManakov equation j ∂ q ∂ z = ∂ q ∂ t − s (cid:13)(cid:13) q (cid:13)(cid:13) q , (4)where s = s = −
3. NFT of the Two-Dimensional Signals
In this section, basics of the NFT theory of the two-dimensional signals, as well as numericalalgorithms to compute the forward and inverse NFT, are briefly presented [26].
Equation (4) can be represented by a Lax pair ˆ L and ˆ M . This means that, operators ˆ L and ˆ M canbe found such that (4) is in one-to-one correspondence with the Lax equation ∂ ˆ L / ∂ z = [ ˆ M , ˆ L ] .The Lax pair for the Manakov equation was found by Manakov in 1974 [26, 27]. The operator ˆ L is: ˆ L = j (cid:169)(cid:173)(cid:171) ∂∂ t − q − q sq ∗ − ∂∂ t sq ∗ − ∂∂ t (cid:170)(cid:174)(cid:172) . (5)To simplify the presentation, consider the focusing regime with s = − L v = λ v (6)can be solved for the Jost function v (eigenvector) assuming that the signals vanish at t = ±∞ .This gives rise to six boundary conditions for v , denoted by j ± : j ( )± → e ( ) exp (− j λ t ) , j ( i )± → e ( i ) exp ( j λ t ) , i = , , as t → ±∞ , (7)where e ( k ) are unit vectors, i.e., e ( k ) l = δ kl , k , l = , ,
2. Each of the boundary conditions (7) isbounded when λ ∈ C + or λ ∈ C − . The eigenvalue problem (6) under the boundary conditions(7) can be solved, obtaining six Jost functions { j ( i )± ( t , λ )} i = , , for all t . It can be shown that { j ( i ) + ( t , λ )} i = , , and { j ( i )− ( t , λ )} i = , , each form an orthonormal basis for the solution space of(6). Thus we can expand j ( )− ( t , λ ) in the basis of { j ( i ) + ( t , λ )} i = , , : j ( )− ( t , λ ) = a ( λ ) j ( ) + ( t , λ ) + b ( λ ) j ( ) + ( t , λ ) + b ( λ ) j ( ) + ( t , λ ) , (8)where a ( λ ) , b ( λ ) and b ( λ ) are called nonlinear Fourier coefficients. It can be shown that in thefocusing regime the inner product of two Jost functions corresponding to the same eigenvaluedoes not depend on time. This implies that a ( λ ) and b i ( λ ) do not depend on time (as the notationin (8) suggests). The NFT of the q = [ q , q ] is now defined asNFT ( q )( λ ) = ˆ q i = b i ( λ ) a ( λ ) , λ ∈ R , ˜ q i = b i ( λ j ) a (cid:48) ( λ j ) , λ j ∈ C + , i = , , (9)here λ j , j = , , · · · , N , are the solutions of a ( λ j ) = λ j ∈ C + .An important property of NFT ( q ( t , z ))( λ ) is that it evolves in distance according to theall-pass-like filter H ( λ ) = e js λ L . (10) Remark.
Note that if q or q is set to zero in the Manakov equation and the associated ˆ L operator (5), the equation and the operator are reduced, respectively, to the scalar NLSE andthe corresponding ˆ L operator [17]. Also, the theory in this section can straightforwardly begeneralized to include any number of signals q i , i = , , ... , for instance, LP modes propagatingin a multi-mode fiber in the strong coupling regime. Remark
Since the Jost vectors are orthonormal at all times, (8) implies the unimodularitycondition | a ( λ )| + | b ( λ )| + | b ( λ )| = , (11)which will be used in the algorithm. We develop the NFT algorithms for the continuous spectrum that is considered in this paper. Theforward and inverse NFT algorithms are, respectively, based on the Ablowitz-Ladik and discretelayer peeling (DLP) methods. These algorithms generalize the corresponding ones in [9].We begin by rewriting the eigenvalue problem ˆ L v = λ v in the form ∂ v / ∂ t = P v , where P = (cid:169)(cid:173)(cid:171) − j λ q ( t ) q ( t )− q ∗ ( t ) j λ − q ∗ ( t ) j λ (cid:170)(cid:174)(cid:172) . (12)Here and in the remainder of this section we suppress the dependence on the coordinate z . Wediscretize the time interval [ T , T ] according to t [ k ] = T + k ∆ T , where ∆ T = ( T − T )/ N suchthat t [ ] = T and t [ N ] = T . We set q i [ k ] = q i ( T + k ∆ T ) and similarly for the vector v . TheAblowitz-Ladik method is a discretization of ∂ v / ∂ t = P v as follows v [ k + , λ ] = c k (cid:169)(cid:173)(cid:171) z / Q [ k ] Q [ k ]− Q [ k ] ∗ z − / − Q [ k ] ∗ z − / (cid:170)(cid:174)(cid:172) v [ k , λ ] , (13)where Q i [ k ] = q i [ k ] ∆ T , z : = e − j λ ∆ T and k = , , · · · , N −
1. With this discretization v becomes a periodic function of λ with period π / ∆ T . We introduced a normalization factor c k = / (cid:112) | det P [ k ]| in (13), where | det P [ k ]| = + | Q [ k ]| + | Q [ k ]| , to improve numericalstability.The iterative equation (13) is initialized with v [ , λ ] = j ( )− ( T , λ ) = e ( ) z T / ∆ T . After N iterations, the nonlinear Fourier coefficients are obtained as projections onto theJost-solutions j ( i ) + ( T , λ ) , a [ λ ] = z − N − T ∆ T v [ N , λ ] , (14) b i [ λ ] = z N + T ∆ T v i [ N , λ ] , i = , . (15)The continuous component in the NFT ( q )( λ ) is then computed based on (9).he algorithm can be efficiently implemented in the frequency domain. Let us write (13) as V [ k + , λ ] = c k (cid:169)(cid:173)(cid:171) Q [ k ] z − Q [ k ] z − − Q ∗ [ k ] z − − Q ∗ [ k ] z − (cid:170)(cid:174)(cid:172) V [ k , λ ] , (16)where V [ k , λ ] : = ( A [ k , λ ] , B [ k , λ ] , B [ k , λ ]) T , and A [ k , λ ] = a [ k , λ ] B i [ k , λ ] = z − N − T ∆ T + b i [ k , λ ] , i = , . (17)We discretize λ on the interval [ Λ , Λ ] with Λ = − Λ = π /( ∆ T ) such that λ [ k ] = Λ + k ∆Λ .Let tilde ’ ∼ ’ denote the action of the discrete Fourier transform DFT with respect to λ [ k ] ,e.g., ˜ A [ ., l ] = DFT ( A [ ., λ [ k ]]) , where l = , , , · · · , N − (cid:8) z − B i [ ., λ [ k ]] (cid:9) [ l ] = shift (cid:8) ˜ B i (cid:9) [ l ] , where shift denotes circular right shift of the array byone element. Equation (16) in the frequency domain is˜ A [ k + , l ] = c k ( ˜ A [ k , l ] + Q [ k ] shift (cid:2) ˜ B [ k ] (cid:3) [ l ] + Q [ k ] shift (cid:2) ˜ B [ k ] (cid:3) [ l ]) , ˜ B [ k + , l ] = c k (− Q [ k ] ∗ ˜ A [ k , l ] + shift (cid:2) ˜ B [ k ] (cid:3) [ l ]) , (18)˜ B [ k + , l ] = c k (− Q [ k ] ∗ ˜ A [ k , l ] + shift (cid:2) ˜ B [ k ] (cid:3) [ l ]) . The initial condition is given by ˜ A [ ] = DFT (cid:2) a [ k ] (cid:3) [ ] and ˜ B i [ ] =
0. At k = N − a and b i arefound by recovering V [ N , λ ] through an inverse DFT and using (17). The inverse NFT algorithm consists of two steps. First, we compute v [ N , λ ] from the continuousspectra ˆ q , ( λ ) and invert the forward iterations (13). Second, at each iteration, we compute Q [ k ] from v [ k , λ ] .Substituting b i ( λ ) = ˆ q i ( λ ) a ( λ ) in the unimodularity condition (11), we can compute | a ( λ )|| a ( λ )| = / (cid:113) + | ˆ q ( λ )| + | ˆ q ( λ )| . In the absence of a discrete spectrum, from (9), a ( λ ) (cid:44) λ . We have Re ( log ( a )) = log (| a |) and Im ( log ( a )) = ∠ a . Since a ( λ ) and therefore log ( a ) are analytic functions of λ we can recoverthe phase of a as ∠ a = H ( log (| a |)) , where H denotes the Hilbert transform.The inverse iterations are obtained by inverting the matrix in (13) and dropping terms of order Q i ∼ ∆ T (to yield the same accuracy as for the forward iterations): v [ k , λ ] = c k (cid:169)(cid:173)(cid:171) z − / − Q [ k ] − Q [ k ] Q ∗ [ k ] z / Q ∗ [ k ] z / (cid:170)(cid:174)(cid:172) v [ k + , λ ] . (19)In practice, as in the forward NFT, it is better to invert the frequency domain iterations (18).The signal Q [ k ] can then be obtained from V [ k + , l ] as follows. Recall that shift (cid:2) ˜ B i [ k ] (cid:3) [ l ] = ˜ B i [ k , l − ] . Using the initial conditions for the forward iterations ˜ A [ , l ] = δ , l and ˜ B i [ , l ] = A [ k , l ] = B i [ k , l ] = l ≥ k >
0. In particular,˜ B i [ k , N − ] = k < N . For the first element of ˜ B i [ k ] , from (18) and shift (cid:2) ˜ B [ k ] (cid:3) [− ] = shift (cid:2) ˜ B [ k ] (cid:3) [ N − ] =
0, we obtain˜ A [ k + , ] = c k ˜ A [ k , ] , ˜ B [ k + , ] = − c k Q [ k ] ∗ ˜ A [ k , ] , ˜ B [ k + , ] = − c k Q [ k ] ∗ ˜ A [ k , ] . (a.u.) -4 -2 0 2 4 6 j ^ q j ( a . u . ) ^ q NFT[INFT(^ q )] ^ q NFT[INFT(^ q )] (a.u.) -4 -2 0 2 4 6 j ^ q j ( a . u . ) ^ q NFT[INFT(^ q )] ^ q NFT[INFT(^ q )] Fig. 1. Comparing ˆ q = [ ˆ q , ˆ q ] and NFT ( INFT ( ˆ q )) , where ˆ q and ˆ q are displaced Gaussians.For fixed sampling rate, the algorithm is less accurate at higher input-power. These equations can be solved for Q ∗ i [ k ] = − ˜ B i [ k + , ]/ ˜ A [ k + , ] . The forward and inverse NFT numerical algorithms are tested as follows. Figure 1 comparesa signal containing two polarization components in the nonlinear Fourier domain with itsreconstruction after successive INFT and NFT operations. We take two displaced Gaussians withstandard deviation σ = √ q i , i = ,
2. Weset time windows to T =
64 and take 2048 samples in time and nonlinear Fourier domain. In thediscrete layer peeling method, the signal is periodic in nonlinear Fourier domain with period π / ∆ T ≈
4. PDM-NFDM Transmitter and Receiver
In this section, we describe the transmitter (TX) and receiver (RX) digital signal processing(DSP) in PDM-NFDM and PDM-OFDM. A schematic diagram of the polarization-divisionmultiplexed NFDM and OFDM transmission systems is shown in Fig. 2. The TX DSP producesdigital signals for the in-phase and quadrature components of both polarization components,which are fed into the IQ-Modulator. The modulated signals for the two polarization componentsare combined in the polarization beam combiner before they enter the transmission line consistingof multiple fiber spans. We consider the practically relevant case of lumped amplification usingErbium-doped fiber amplifiers (EDFAs). After propagation through the fiber the two polarizationcomponents are separated in the polarization beam splitter and fed into the polarization-diversityintradyne coherent receiver, which provides the input to the RX DSP.We briefly describe the TX and RX DSP in OFDM and NFDM. We first map the incomingbits to signals taken from a QAM constellation. We then oversample the discrete-time signal inthe time domain (by introducing zeros in the frequency domain outside the support of the signal)and then add guard intervals in the time domain. Increasing the guard interval increases theaccuracy of the INFT. Unless stated otherwise, we do not increase the guard intervals during thecomputation of the INFT as the amplitude is increased, so there is a penalty due to inaccuraciesin the algorithm. In NFDM, these steps are performed in what is called the U -domain; see [9]. x DSP inx I inx Q iny Q iny I IQ Mod.
EDFA
Coh.
Rx Rx DSP
SSMF
Tx DSP NFDM
OFDM data encoder Oversample in time Include guard interval Manakov INFT Unnormalize Combine bursts qU ˆ Tx DSP OFDM
OFDM data encoder Oversample in time Include guard interval IFFT Unnormalize Combine bursts
Rx DSP NFDM
Separate bursts Normalize Manakov NFT One tap channel eq.
Remove guard interval
Training sequence equalization Downsample in time Minimum distance decoder Uq ˆ Rx DSP OFDM
Separate bursts Normalize FFT CDC or DBP Remove guard interval Training sequence equalization Downsample in time Minimum distance decoder spans N outx I outx Q outy Q outy I IQ Mod. inx E iny E outx E outy E Fig. 2. System diagram for the polarization-multiplexed NFDM and OFDM transmissionsystems with processing steps in the transmitter and receiver DSP. Steps highlighted inpurple require joint processing of both polarization components. For further explanation seetext.
The signal is subsequently mapped from the U -domain to nonlinear Fourier domain through thetransformation U i ( λ ) = (cid:113) − log ( − | ˆ q i ( λ )| ) e j ∠ ˆ q i ( λ ) , i = , . (20)This step has no analogue in OFDM. Note that contrary to the single-polarization case herethe energy of the signal in the time domain is only approximately proportional to the energy inthe U -domain. At this point, we perform the inverse NFT in the case of NFDM and an inverseFFT in case of OFDM. Then we obtain the unnormalized signal by introducing units using (3).Finally we combine the OFDM or NFDM bursts to obtain the signal to be transmitted. The outputof the DSP are the in-phase and quadrature components of the signals in the two polarizationcomponents. All steps in the DSP are performed independently for the two polarizations, exceptfor the INFT, which requires joint processing.At the RX we invert the steps of the TX DSP. The signal processing begins with burstseparation and signal normalization using (3), followed by the forward NFT (FFT) in caseof NFDM (OFDM). In NFDM, we subsequently equalize the channel using (10), which is asingle-tap phase compensation. Similarly, in OFDM we either compensate the dispersion withthe phase exp (− j β ω N span L / ) , or perform digital backpropagation (DBP) for a fixed numberof steps per span. We then remove the guard intervals in time, down-sample the signal in the timedomain, and obtain the output symbols. The bit error rate (BER) is calculated using a minimumdistance decoder for symbols.In Fig. 3 we compare the time domain signals of NFDM and OFDM at the TX for one of thepolarization components. We use 112 subcarriers over a burst duration of T = | ˆ q , | is increased in the nonlinear Fourier domain. At a signal power of0.5 dBm, one can see a significant amount of broadening of the NFDM signal in the time domain.This effect makes it difficult to control the time duration of signals in NFDM.
20 -15 -10 -5 0 5 10 15 20 t (ns) j q j ( p W ) NFDMOFDM -20 -15 -10 -5 0 5 10 15 20 t (ns) j q j ( p W ) NFDMOFDM
Fig. 3. NFDM and OFDM signals at the TX (left panel) and RX (right panel) using the idealmodel (2) at P = . Parameter Value
Length span 80 km
Number of spans 25
Number of carriers 112 Burst duration 2 ns Guard interval 18 ns Burst data rate 448GBit/s Effective data rate 44.8GBit/s Bandwidth 56 GHz Oversampling factor 16 Constellation 16 QAM Noise Figure 6.2 dB Attenuation 0.2 dB/km Dispersion 16.89 ps/km/nm Nonlinear index 2.6◦10⁻²⁰ W/m² Core area 80 μm²
Table 1. The system parameters used in the simulations.
Pulse broadening in the time domain due to the chromatic dispersion can be estimated by [28] ∆ T = π | β | LB , (21)where B is the signal bandwidth. The guard time intervals for minimizing the interaction amongbursts can be approximated using (21). For the parameters in Table 1, we obtain ∆ T ∼ T = T + T guard = ( + ) ns =
5. Simulation results
In this section, we consider the system shown in Fig. 2 and compare the PDM-OFDM andPDM-NFDM via simulations, taking into account loss and PMD. First, we consider the modelwith loss and periodic amplification, setting the PMD to zero (namely, neglecting the terms inthe first line of the Eq. 1). Next, the PMD effects are studied in Sec. 5.3. The system parametersused are summarized in Table 1.
10 15 20
OSNR (dB) -4 -3 -2 -1 B E R B2BLosslessLossyTransf. lossless
Fig. 4. BER as a function of OSNR for NFDM, for back-to-back, lossless, lossy andtransformed-lossless models.
In the lumped amplification scheme, the signal is periodically amplified after each span. Thechannel is described by the model (1) including the attenuation term, which is manifestlynot integrable. The effect of lumped amplification on NFT transmission has been studiedpreviously [8].We illustrate the effect of the attenuation by comparing the BER as a function of the OSNR inFig. 4 for four models: 1) the back-to-back (B2B) configuration; 2) lossless model (2) (there ispropagation but without loss); 3) lossy model with lumped amplification; and 4) a transformed-lossless model that is introduced below. In producing Fig. 4, we artificially introduced AWGN atthe receiver, to exclude the effects of the signal-noise interaction from the comparison. We fix thepower P = − . A ( z , t ) = A ( z , t ) e − α z [30, 31]. This transforms thenon-integrable equation (1) to the integrable equation (2) (thus with zero loss) and a modifiednonlinearity parameter γ eff ( L ) = L ∫ L γ e − α z dz = γ ( − e − α L )/( α L ) . (22)We refer to the resulting model as the transformed-lossless model.Note that the power of the signal A ( z , t ) in the transformed-lossless model is higher than thepower of the A ( z , t ) in the original lossy model. That is because γ eff < γ , which yields higheramplitudes according to (3).
10 -5 0 5
P (dBm) -4-2024681012 Q - f a c t o r ( d B ) NLSEManakov
Fig. 5. Comparison between NFT transmission based on the NLSE (single polarization) andthe Manakov equation (polarization-multiplexed). P denotes the total power of the signal.At low power the offset between the curves is 3dB, as the signal power doubles when twopolarization components propagate. At higher power signal-noise interactions decrease thisgap. Figure 4 shows that the proposed scheme for loss cancellation is indeed very effective. Using γ eff ( L ) in the inverse and forward NFTs, the BER of the lossy and transformed-lossless modelsare almost identical. In order to assess the transmission performance of the PDM-NFDM transmission, we performedsystem simulations as described in Sec. 4. For now we focus on deterministic impairments andneglect PMD. We simulate transmission of 112 subcarrier NFDM and OFDM pulses basedon a 16QAM constellation over N span =
25 spans of 80km of standard single-mode fiber. Weassume ideal flat-gain amplifiers with a noise figure of N F = . Q -factor at peak power is roughly the same in both cases. This implies that data ratescan approximately be doubled using polarization multiplexing. We show in Sec. 5.3 that thisconclusion still holds in presence of polarization effects.In Fig. 6 we compare the Q -factor as a function of launch power in PDM-OFDM andPDM-NFDM transmission. The optimum launch power in OFDM is significantly smaller thanin NFDM. OFDM transmission is limited by nonlinearity, while NFDM is mainly limited bynonlinear signal-noise interaction. The Q -factor at optimum launch power is significantly higherin NFDM than in OFDM. We observe an overall gain of 6 . Q -factor. Figure 7 shows the (dBm) -8 -6 -4 -2 0 2 4 6 Q - f a c t o r ( d B ) NFDMOFDMOFDM DBP 4 step/spanOFDM DBP 10 step/spanOFDM DBP 16 step/span
Fig. 6. Comparison of polarization-multiplexed OFDM to NFDM transmission in an idealizedsetting. We use the Manakov equation without PMD-effects for 25 spans of 80 km for16QAM. NFDM performs as well as OFDM with 10step/span DBP. The full potential ofNFDM is leveraged in a multi-channel scenario, where DBP is less efficient.Fig. 7. Received constellations for OFDM (left panel, without DBP) and NFDM (right panel)at the respective optimal launch power. The noise in NFDM is clearly non-Gaussian. corresponding received constellations at the respective optimal launch power.In Fig. 6 we further report results obtained by digital backpropagation after OFDM transmission.For a fair comparison with NFDM, we use the same sampling rate in the DBP and apply it withoutprior downsampling. OFDM with DBP achieves similar performance to NFDM for around 10DBP steps per span and exceeds it for 16 steps per span. We note that here we considered asingle-user scenario. The full potential of NFDM is leveraged in a network scenario, where DBPis less efficient. Our results are in good qualitative agreement with those reported in Ref. [8] forthe single-polarization case, while achieving twice the data rate.
The fiber model (1) describes light propagation in linearly birefringent fibers [32, 33]. Here ∆ β is the difference in propagation constants for the two polarization states which are induced byfiber imperfections or stress. In real fibers, this so-called modal birefringence varies randomly,resulting in PMD.We simulate PMD with the coarse-step method, in which continuous variations of the taps Q - f a c t o r ( d B ) = p km0.1 ps = p km0.2 ps = p km Fig. 8. The effect of the number of taps on the Q-factor for different values of D PMD . Thepower of the signal in the simulation is 0.5 dBm. birefringence are approximated by a large number of short fiber sections in which the birefringenceis kept constant. PMD is thus emulated in a distributed fashion [34]. We use fixed-length sectionsof length 1km, larger than typical fiber correlation lengths. At the beginning of each section,the polarization is randomly rotated to a new point on the Poincaré sphere. We apply a uniformrandom phase in the new reference frame. The latter accounts for the fact that in reality thebirefringence will vary in the sections where it is assumed constant, which will lead to a randomphase relationship between the two polarization components [32]. The differential group delay(DGD) of each section is selected randomly from a Gaussian distribution. This way artifacts inthe wavelength domain caused by a fixed delay for all sections are avoided [35].Within each scattering section, the equation (1) without PMD terms is solved using standardsplit-step Fourier integration. To speed up the simulations, we employ a CUDA/C++ basedimplementation with a MEX interface to Matlab.The resulting DGD of the fiber is Maxwell distributed [36], p (cid:104) ∆ t (cid:105) ( ∆ t ) = π ∆ t (cid:104) ∆ t (cid:105) exp (cid:18) − ∆ t π (cid:104) ∆ t (cid:105) (cid:19) . (23)For the Maxwell distribution, the mean is related to the root-mean-square (RMS) delay by (cid:104) ∆ t (cid:105) (cid:112) π / = (cid:112) (cid:104) ∆ t (cid:105) . The average DGD varies with the square root of the fiber length [28], (cid:104) ∆ t (cid:105) ∼ (cid:112) (cid:104) ∆ t (cid:105) = D PMD √ L , (24)where D PMD is the PMD parameter. Typical PMD values for fibers used in telecommunicationsrange from 0.05 in modern fibers to 0.5 ps/ √ km. In this section, we consider polarization effects on the NFT transmission. Since the nonlinearterm in the Manakov equation is invariant under polarization rotations, the equalization can bedone in in the nonlinear Fourier domain (cf. Fig. 2). We use a simple training sequence basedequalization algorithm to compensate linear polarization effects. The samples of the first NFDMsymbol are used as the training sequence. The filter taps of the equalizer are determined using (dBm) -10 -8 -6 -4 -2 0 2 4 6 8 10 Q - f a c t o r ( d B ) = p km0.2 ps = p km0.5 ps = p kmNo birefringence Fig. 9. The effect of PMD on PDM-NFDM. We used 13, 25 and 61 taps in the equalizer for D PMD = , . . /√ km, respectively. least-squares estimation. To obtain the Q -factor we average the BER over 120 random realizationsof PMD for each data point.Due to its statistical nature, the effects of PMD are often quantified in terms of outageprobabilities. In practice, the system is designed to tolerate a certain amount of PMD, in this caseby fixing the number of taps in the equalizer. When the DGD exceeds this margin, the system issaid to be in outage. We first determine the number of taps to achieve a given outage probability.Fig. 8 shows the Q -factor as a function of the number of taps for different values of the PMDparameter. For D PMD = D PMD = . /√ km this interval corresponds to 12 . Q -factor as a function of launch power with and without PMDeffects. Here we use the number of taps determined as described above. We observe a penalty ofroughly 0.3dB at peak power for D PMD = . /√ km relative to the case without PMD ( D PMD = /√ km). The penalty is therefore not serious for typical fibers used in telecommunication.Compared to the case without PMD effects and random birefringence (labeled “no birefringence"),e find a penalty of roughly 1.2dB at peak power for the case of zero PMD due to the equalization.
6. Conclusions
In this paper, we have proposed polarization-division multiplexing based on the nonlinear Fouriertransform. NFT algorithms are developed based on the Manakov equations. Our simulationsdemonstrate feasibility of polarization multiplexed NFDM transmission over standard single-modefiber. The results show that data rates can approximately be doubled in polarization-multiplexedtransmission compared to the single-polarization case. This is an important step to achieve datarates that can exceed those of conventional linear technology.Numerical simulations of polarization multiplexed transmission over a realistic fiber modelincluding randomly varying birefringence and polarization mode dispersion have shown thatpenalties due to PMD in real fibers do not seriously impact system performance. Our fibersimulations are based on the Manakov equations. As a next step, which is beyond the scope ofthis paper, the results should be verified experimentally.The equation governing light propagation of N modes in multi-mode fibers in the strongcoupling regime can be written in the form [37] ∂ A ∂ Z = j ¯ β ∂ A ∂ T − j γκ (cid:13)(cid:13) A (cid:13)(cid:13) A , (25)where A = ( A , . . . , A N ) T and ¯ β denotes the average group velocity dispersion. The NFT, aswell as the algorithms presented here, generalize to this equation in a straightforward manner.Our approach therefore paves the way to combining the nonlinear Fourier transform withspace-division multiplexing.
7. Acknowledgements7. Acknowledgements