Improved digital backward propagation for the compensation of inter-channel nonlinear effects in polarization-multiplexed WDM systems
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1 Iproved digital backward propagation for the copensation of inter-channel nonlinear effects in polarization-ultiplexed WDM systes Eduardo F. Mateo, 1,* Xiang Zhou, and Guifang Li 1 1 CREOL, The College of Optics and Photonics, University of Central Florida, 4000 Central Florida Blvd., Orlando, FL, 3816, USA AT&T Labs Research, 00 Laurel Avenue South, Middletown, NJ, 07748, USA *eateo@creol.ucf.edu Abstract: An iproved split-step ethod (SSM) for digital backward propagation (DBP) applicable to wavelength-division ultiplexed (WDM) transission with polarization-division ultiplexing (PDM) is presented. A coupled syste of nonlinear partial differential equations, derived fro the Manakov equations, is used for DBP. The above syste enables the ipleentation of DBP on a channel-by-channel basis, where only the effect of phase-isatched four-wave ixing (FWM) is neglected. A novel forulation of the SSM for PDM-WDM systes is presented where new ters are included in the nonlinear step to account for inter-polarization ixing effects. In addition, the effect of inter-channel walk-off is included. This substantially reduces the coputational load copared to the conventional SSM. 011 Optical Society of Aerica OCIS codes: ( ) Fiber optics counications; ( ) Coherent counications; ( ) Nonlinear optics, fibers. References and links 1. P. P. Mitra, and J. B. Stark, Nonlinear liits to the inforation capacity of optical fibre counications, Nature 411(6841), (001).. M. G. Taylor, Coherent Detection Method using DSP for Deodulation of Signal and Subsequent Equalization of Propagation Ipairents, IEEE Photon. Technol. Lett. 16(), (004). 3. X. Liu and D. A. Fishan, A Fast and Reliable Algorith for Electronic Pre-Equalization of SPM and Chroatic Dispersion, in OFC, (Optical Society of Aerica, 006), paper OThD4. 4. E. Yaazaki, F. Inuzuka, K. Yonenaga, A. Takada, and M. Koga, Copensation of interchannel crosstalk induced by optical fiber nonlinearity in carrier phase-locked WDM syste, IEEE Photon. Technol. Lett. 19(1), 9 11 (007). 5. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Ki, F. Yaan, and G. Li, Electronic post-copensation of WDM transission ipairents using coherent detection and digital signal processing, Opt. Express 16(), (008). 6. E. Ip, and J. M. Kahn, Copensation of Dispersion and Nonlinear Ipairents Using Digital Backpropagation, J. Lightwave Technol. 6(0), (008). 7. G. Goldfarb, M. G. Taylor, and G. Li, Experiental Deonstration of Fiber Ipairent Copensation Using the Split-Step Finite-Ipulse-Response Filtering Method, IEEE Photon. Technol. Lett. 0(), (008). 8. F. Zhang, Y. Gao, Y. Luo, J. Li, L. Zhu, L. Li, Z. Chen, and A. Xu, Experiental Deonstration of Intrachannel Nonlinearity Mitigation in Coherent QPSK Systes with Nonlinear Electrical Equalizer, Electron. Lett. 46(5), (010). 9. R. Waegeans, S. Herbst, L. Holbein, P. Watts, P. Bayvel, C. Fürst, and R. I. Killey, 10.7 Gb/s electronic predistortion transitter using coercial FPGAs and D/A converters ipleenting real-tie DSP for chroatic dispersion and SPM copensation, Opt. Express 17(10), (009). 10. X. Liu, F. Buchali, and R. W. Tkach, Iproving the Nonlinear Tolerance of Polarization-Division-Multiplexed CO-OFDM in Long-Haul Fiber Transission, J. Lightwave Technol. 7(16), (009). 11. F. Yaan and G. Li, Nonlinear Ipairent Copensation for Polarization-Division Multiplexed WDM Transission Using Digital Backward Propagation, IEEE Photonics J. 1(), (009). # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 570
2 1. E. Ip, Nonlinear Copensation Using Backpropagation for Polarization-Multiplexed Transission, J. Lightwave Technol. 8(6), (010). 13. E. F. Mateo, F. Yaan, and G. Li, Efficient copensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transission, Opt. Express 18(14), (010). 14. G. P. Agrawal, Nonlinear fiber optics, (Acadeic Press, 007). 15. O. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, Optiization of the split-step Fourier ethod in odeling optical-fiber counications systes, J. Lightwave Technol. 1(1), (003). 16. E. F. Mateo, and G. Li, Copensation of interchannel nonlinearities using enhanced coupled equations for digital backward propagation, Appl. Opt. 48(5), F6 F10 (009). 17. E. F. Mateo, L. Zhu, and G. Li, Ipact of XPM and FWM on the digital ipleentation of ipairent copensation for WDM transission using backward propagation, Opt. Express 16(0), (008). 18. P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, Evaluation of the coputational effort for chroatic dispersion copensation in coherent optical PM-OFDM and PM-QAM systes, Opt. Express 17(3), (009). 19. V. Oppenhei, and R. V. Schafer, Digital Signal Processing, (Prentice-Hall, 1975). 0. J. G. Proakis, Digital Counications, (McGraw-Hill, 001). 1. Introduction There has been and continues to be uch research on high data-rate and spectrally-efficient fiber counication systes. Higher bit-rates per channel involve the deployent of highorder odulation forats, requiring increased SNR and hence higher power per channel. Alternatively, higher spectral efficiency also deands tightly spaced wavelength-division ultiplexed (WDM) channels to optiize the operational bandwidth of optical aplifiers. Together with WDM, polarization-division ultiplexing (PDM) is also deployed to double the spectral efficiency. The above scenario clearly leads to increased nonlinearity in the for of intra- and inter-channel effects as well as inter-polarization effects. Therefore, the itigation or copensation of fiber ipairents which involve Kerr nonlinearity and PDM becoes crucial to increasing transission capacity [1]. Recently, digital backward propagation (DBP) has been proposed for the coprehensive copensation of fiber ipairents. DBP is based first, on the coherent detection of the optical signal [] and second, on the ipleentation of backward propagation in the digital doain. This ipleentation consists on solving the z-reversed propagation equations that describe nonlinear transission in fibers. Provided that the channel characteristics are known, and provided the WDM channels share the sae optical path, any deterinistic effect can be pre/post-copensated at the transitter/receiver. The joint copensation of dispersion and nonlinearity allows increasing the launch power to values beyond the traditional nonlinear liit. Therefore, higher OSNR is achieved and transission reach can be extended. In single polarization systes, pre- and post- copensation via DBP were first proposed in [3 6]. Experiental deonstration of DBP in ulti-channel systes was reported in [7] and single channel experients were carried out in [8,9]. DBP in PDM systes using vectorial backward propagation has been reported in [10] for single channel and in [11,1] for WDM. Despite its proven efficacy in both iproving perforance and extending reach, DBP is still challenging in ters of DSP coplexity and therefore, it is still far fro being deployed in current systes. Recently, an advanced split-step ethod (SSM) was presented in [13] to reduce the coputational load of single-polarization DBP. In [13], the copensation of single-polarization inter-channel effects via DBP was shown to increase the transission reach fro 800 to 000 k. Moreover, the coputational load can be reduced by ore than a factor of 4 with respect to the conventional SSM. In this paper, an advanced SSM ethod is presented for PDM systes. Several aspects are different in the PDM case copared to the single-polarization case studied in [13]. First, a new coupled syste of nonlinear partial differential equations is derived for the backward propagation of PDM signals. Such syste is obtained fro the Manakov equations, instead of the scalar nonlinear Schrodinger equation, which can be used to describe vectorial nonlinear propagation in fiber with randoly varying birefringence [14]. In contrast to the scalar case, # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 571
3 the coupled syste of equations for PDM includes non-conservative ters in the for of phase-atched interaction between the odulated polarization tributaries. When such nonconservative ters are consider in DBP, a new solution (with no counterpart in the singlepolarization case) has to be obtained for the nonlinear step of the SSM. In this paper, we propose a quasi-analytical solution for the coputation of the non-conservative contribution. In addition, the advanced SSM presented in [13] is applied to the PDM case. Here, a walk-off factorization is applied to reduce the coputation coplexity of the SSM. Such advanced- SSM is now extended for the PDM and the factorization of the walk-off is also applied to the PDM non-conservative ters. Fro a perforance point of view, the ipact of the PDM phase-atched non-conservative ters is analyzed, for the first tie to our knowledge, in the context of digital backward propagation.. Digital backward propagation for PDM-WDM systes In a PDM-WDM syste with coherent detection, the full reconstruction of the vector optical field can be achieved by using a polarization- and phase-diverse receiver. The reconstructed field will be used as the input for DBP in order to copensate the transission ipairents. Let Ex ( t, z ) and Ey ( t, z ) be the polarization tributaries of the coplex received field for the th-channel where 1,,, N and N is the nuber of channels. The reconstructed total optical field is given by: E xex y Ey, where E ˆ exp( ) E it and f / is the channel spacing. In general, optical counication fibers exhibit residual birefringence responsible for the rando scattering of the state of polarization over a length scale of [14]. Moreover, the typical power values used in counication systes lead to rather long nonlinear lengths. Along the nonlinear length, the state of polarization changes fast and randoly, and the effect of the local state of polarization on the overall nonlinear interaction can be averaged over the entire Poincaré sphere. As a consequence of the above, the vector optical propagation can be described by the so-called Manakov syste [11,14], which is expressed as follows for backward propagation: 3 E i E(, ) (, ) x y E 3 x y 8 E i E 3 E( y, x) E 0, (1) z t 6 t 9 where j represents the jth-order dispersion, is the absorption coefficient, is the nonlinear paraeter and t is the retarded tie frae. The above syste includes both coherent (FWM) and incoherent (SPM, XPM) nonlinear effects between channels and polarization coponents. As explained in [16], the coherent nature of FWM requires: i) the full reconstruction of the entire WDM band, ii) enough up-sapling to avoid aliasing of newly generated FWM products, iii) very short step sizes and iv) phase-locked local oscillators to preserve the relative phase between channels. Alternatively, inter-channel coherent ters can be oitted in backward propagation by introducing the field expressions E ˆ exp( ) E it into Eq. (1), expanding the ( xy, ) E ters and ignoring phase-isatched ters. This leads to the following coupled equations, Eˆ x ˆ ˆ L icx Ex ikey 0, z Eˆ y L ic E ik E z ˆ * ˆ y y x 0, () # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 57
4 where L, C and K represent the linear dispersive operator, the phase-insensitive XPM contribution and a polarization ixing (PolM) ter, respectively, given by: L D D D t t t 3, (3) where, P 8 C Px Py R q (4) 9 q K (5) 8 ˆ* ˆ EyqExq, 9 q Eˆ and R P P( y, x). Clearly, the first two ters on the right hand side of Eq. (4) represent the SPM contribution whereas R includes the XPM contribution. The dispersion paraeters are given by: D - /, 1 3 D i / - 3 / and D3-3/ 6. Equations () neglect any interaction where the relative phase between the WDM channels is relevant. Moreover, when the PolM ter K is included, the relative phase between the polarization coponents of each channel is relevant. Therefore, the relative phase of the polarization tributaries has to be preserved at the receiver. This condition is typically fulfilled in polarization diverse receivers, where each local oscillator is split into orthogonal coponents to receive the PDM tributaries of each channel. The above syste of equations is solved in the digital doain by the well-known Split- Step Method (SSM) [14 16]. This ethod relies on decoupling the linear and nonlinear contributions in Eq. () over a sufficiently short distance. In order for this ethod to be accurate, the step size has to be short enough to ensure: (i) The solution of the linear part fro z to z h is not perturbed by the variations of the optical fields due to nonlinear effects and (ii) The solution of the nonlinear part fro z to z h is not perturbed by the variations of the optical fields due to linear effects. Under these conditions, the step size will be liited by the fastest of the above variations. Typically, the linear step is solved in the frequency doain using efficient algoriths for both the direct and inverse Fourier Transfors. -1 ˆ Eˆ ( t, z h) F F E ( t, z) H (, h), with the following ulti-channel linear transfer function, (6) 3 ( ) ( ) H(, h) exp i i3 h (7) 6 The above approxiation is valid provided that the spectral change induced by nonlinearity is weak over the step length. Fourier doain filtering requires block-by-block coputation which can be efficiently ipleented by the overlap-and-add or the overlap-andsave ethods [18,19]. For the nonlinear step, the linear ter envelopes as follows, E ˆ exp( / ) ( x, x) E( x, x) z, Eq. () becoe, L is neglected. By transforing the optical # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 573
5 Ex z z icxe Ex ike Ey 0, z (8) E y z * z icye Ey ike Ex 0, z The above syste lacks a closed-for solution due to the coupling ter K. This is a FWM-like ter which provides energy transfer between channels and polarization coponents. To solve the above syste, a ulti-step approach will be used. First, by neglecting the coupling ter K, Eq. (8) have the following solution, zh z 0 z E E exp i C e dz. (9) In a second step, the ters C are neglected in Eqs. (8). By taking the average of the coupling ters, Equations (8) becoe, 1 z Ke dz, z (10) Ex i Ey 0, z E y * i Ex 0, z By taking derivates, the above syste can be decoupled and it has the following general solution, where ( xy, ) i z i z, (11) E a e b e (1) a and b ( xy, ) are integration constants. By applying initial conditions, Eq. (1) can be rewritten as, E E cos z ie sin z, x x0 y0 * E E cos z ie sin z, y y0 x0 Finally, by proceeding in a siilar way as with the SSM, we can group Eq. (9) and Eq. (13) as follows, where, iy ix * ix E ( t, z h) E ( t, z) e cos Q ie ( t, z) e Q sinc Q, x x y i y y x y E ( t, z h) E ( t, z) e cos Q ie ( t, z) e Q sinc Q, zh z C e dz, z (13) (14) (15) zh z Q K e dz. (16) z In the conventional split-step ethod, the integrals in Eqs. (15-16) are approxiated by, # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 574
6 ( t, z h) h C ( t, z), (17) eff Q ( t, z h) h K ( t, z), (18) eff where h [exp( h) 1] is the effective step size. In general, within the split-step eff foralis, the above approxiations are valid provided the functions C and K are not significantly odified due to dispersion along the step size. Dispersion plays a dual role in odifying the wavefors. On the one hand, it creates intra-channel pulse broadening at a length scale given by 1 L D B. This pulse broadening is intrinsic to each channel and depends on its baud-rate, B. On the other hand, dispersion creates walk-off effect between channels. The delay between two channels and q occurs in a length scale given by the walk-off length, L 1 wo B [15,17] where q q is the frequency difference. This dual action of dispersion directly translates into the step size requireents. Typically, the nuber of channels is large and in( Lwo ) L D which akes walk-off the liiting effect for the step size [17]. One way to relax the step size requireents for the copensation of inter-channel effects is to separate the effects of pulse broadening and walkoff. To that end, let us rewrite Eqs. (4) and (5) by including the tie delay caused by the dispersive walk-off, 8 C ( t, z) Px ( t, z) Py ( t, z) R q( t dqz, z), (19) 9 q 8 * K( t, z) Eyq( t dqz, z) Exq( t dqz, z), (0) 9 q where d ( ) is the walk-off paraeter between channels and q. By Fourier q q transforing the above expressions, we can write, 8 idqz C (, z) Px (, z) Py (, z) R q(, z)e, (1) 9 q 8 * idqz K(, z) Eyq(, z) Exq(, z)e. () 9 q x( ) F x( t). By taking the above expressions into account, Where we use the notation, Eqs. (15) and (16) can be approxiated as, 8-1 j ( t, z h) - Px ( t, z) Py ( t, z) heff F R q(, z) Wq (, h), 9 " q ¹ (3) where, 8 1 ˆ* (, ) (, ) ˆ Q t z h F Eyq z Exq(, z) Wq (, h), 9 q (4) W q hid z q e 1 (, h). id q (5) # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 575
7 Fro a physical perspective, the filters W q can be viewed as a generalized nonlinear effective length where not only power attenuation but walk-off effects are taken into account to odify the strength of the nonlinear interaction. The walk-off factorization reoves the necessity to follow the dispersive delay within the step. Therefore, the step size becoes now liited by the iniu of the nonlinear or pulse broadening lengths. In WDM systes, the nonlinear length [17] is typically shorter than the pulse broadening length. The reason is the contribution of adjacent channels to the total nonlinear phase shift. Coputationally, the above forulation requires additional direct and inverse Fourier transfors, which gives rise an increased coplexity per step. However, by factorizing the walk-off effect, the step size can be substantially increased in typical WDM scenarios. In general, the syetric version of the split-step ethod ust be used in order to iprove the algorith efficiency [14,15]. Here, the nonlinear phase shift is calculated by using the value of the optical field in the id-segent. In this case a correction factor has to be added to the filter, W [13]. By perforing a change of variable in Eqs. (15-16), we can rewrite as, q ( idq) h e 1 Wq(, h) e id q ( idq) h/ h e 1 h/ heff e Figure 1 depicts a scheatic for the ipleentation of the split-step ethod, where for siplicity, the XPM odule is configured by using the equivalent forulation C 8 P R / 9 instead of Eq. (4). q q (6) Fig. 1. (A): Block diagra for the ipleentation of one step using the advanced SSM..The XPM and Xpol odules are shown below. (B) Sketch of the XPM odule including the filtering for walk-off factorization. For conventional SSFM, the FFT, IFFT and filters W are reoved. (C) Block of the Xpol odule including the filtering for walk-off factorization. For conventional SSFM, the FFT, IFFT and filters W are reoved. q q # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 576
8 3. Siulation results and discussion A 00 Gb/s per channel (dual polarization) 16-QAM PDM-WDM syste has been siulated using the VPI TransissionMaker. The transission syste consists of ten spans of NZ-DSF fiber with a length of 100 k per span, a dispersion paraeter of D 4.4 ps/k/n and a dispersion slope of Ds ps/k/n. The loss is 0. db/k and the nonlinear coefficient is, 1.46 /W/k. Fiber loss is copensated per span using Erbiu-doped fiber aplifiers with a noise figure of 5 db. A 4 channel WDM syste with channel spacing of 50 GHz has been siulated. 16 QAM has been selected as the odulation forat because it requires higher OSNRs and, hence, it can take ore advantage of nonlinearity copensation techniques. The fiber is odeled as a concatenation of sections with different birefringence axes. The rate of variation of the rando birefringence is odeled by using a Gaussian distribution with a variance inversely proportional to the fiber correlation length, which is assued to be 50. Within each section, propagation is odeled by solving the exact nonlinear propagation equations (see Eqs. (6) in [14]). In our siulation, no tie delay is induced between polarization coponents. Thus, the differential group delay between polarization odes (DGD) is equal to zero. The entire WDM band is autoatically up-sapled by VPI to properly account for third order nonlinear effects. The step-size used by VPI is chosen to keep the nonlinear phase-shift below 0.05 degrees. Raised-cosine filters are used for deultiplexing. After forward propagation, a polarization-diverse coherent receiver is odeled as shown in Fig.. Fig.. Siulated TX-RX configuration with polarization diverse coherent detection. Local oscillators, as well as transitted lasers, are assued to have zero linewidth. LOs frequencies are chosen to atch the transitter lasers. After detection, each polarization tributary is sapled at saples/sybol and backward-propagated using Eqs. (). After DBP, polarization deultiplexing is perfored by applying the inverse Jones atrix of the syste. Then a phase estiation algorith is perfored to copensate for residual constellation rotation due to residual uncopensated nonlinearity. In this work, NZ-DSF is used because of its lower dispersion, which increases the strength of inter-channel nonlinearity with respect to SSFM. In this scenario, DBP becoes ore necessary than in, for instance, systes deploying SSFM, where high local dispersion itigates inter-channel effects ore efficiently. # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 577
9 Five different cases will be investigated depending on the nonlinear effects included in DBP: DBP1: SPM copensation where the ters R and K are neglected. This case assues that no inforation fro adjacent channels is considered for DBP. DBP: Incoherent inter-channel copensation where the ter K is neglected. This case includes incoherent ters only; thus, the relative phase between channels and polarization coponents does not need to be preserved for DBP. All the transitted channels are included for DBP. DBP3: Coherent inter-channel copensation. This case includes both incoherent interchannel coponents R and the coherent polarization ter K. Therefore, the relative phase between the polarization coponents of each individual channel has to be preserved. All the transitted channels are included for DBP. Finally, full copensation using Manakov syste, Eq. (1), and dispersion copensation (DC) only are also perfored for coparison purposes. This sets the upper and lower bounds of perforance respectively. For the solution of the Manakov syste, all the channels and polarizations are cobined and up-sapled to for a joint band (see [16] for details). The high sapling requireents together with extreely short step sizes akes the Manakov solution ipractical for an eventual DSP ipleentation [17]. As an estiate, in [16] it is shown that FWM copensation requires roughly 5 ties ore operations than XPM copensation. In this paper, PMD effects are not considered. The ipact of PMD on DBP is iportant when the nuber of channels included in the XPM copensation is large. For such cases, the channels at the opposite edges cannot retain their relative orientations because of polarization-ode dispersion. For DBP to work properly these changes in the forward propagation have to be onitored and included in backward propagation. This will require dynaic onitoring of the polarization transfer atrix of the transission fiber. With respect to the coherent polarization effects (i.e. DBP3), they are not independent of the phases of the interacting fields. Although the phase that is coon to both polarizations (chroatic dispersion) at the sae wavelength or coon to the channels (birefringence) cancels out, the phase change coing fro PMD ust be accounted for the coherent polarization effects. Figure 3 shows the baseline results after backward propagation when different effects are copensated. These results are obtained for a step size sufficiently short, fro which the Q- factor behaves asyptotically. Values in Fig. 3 are the Q-factors averaged over all WDM channels. Each channel carries QAM sybols per polarization tributary. The Q- factor is obtained fro the constellation by averaging the standard deviations of the 16 constellation clusters. The 16-QAM Q-factor calculation has been tested with direct error counting of transission over an AWGN channel in coparison with a Gaussian odel [0] which predicts the following relation between the Q-factor and the sybol error rate: Q/0 3 SER erfc 10 / 3, (e.g., a Q-factor of 7.6 db for a SER of 10 ). # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 578
10 Fig. 3. DBP perforance results for a PDM 16-QAM WDM syste consisting of 4 channels at 00 Gb/s. DBP1 is SPM copensation. DBP is incoherent inter-channel copensation. DBP3 is coherent inter-channel copensation. Several coents can be ade fro Fig. 3. First, SPM (DBP1) copensation provides a sall iproveent in ters of Q-factor. This is because inter-channel effects are sufficiently strong to odify the optical wavefors through forward transission. Such odifications perturb the initial conditions for DBP aking SPM copensation inefficient. With respect to DBP, a oderate iproveent of.1 db is obtained with respect to dispersion copensation only. Again, the effect of the coupling ter K in forward transission has an ipact on the initial conditions. Hence, the sole copensation of incoherent processes provides an interediate perforance. On the contrary, the copensation of both coherent and incoherent inter-channel effects (DPB3) provides a rearkable iproveent of ore than 4. db which is close to axiu achievable perforance provided by the solution of the Manakov syste. The sall discrepancy between DBP3 and Manakov coes fro the arginal effect of FWM on the initial conditions. In this paper, a WDM syste of 16QAM channels with 50 GHz spacing has been selected for siulation. Fro a conceptual viewpoint, DBP is a universal ethod in the sense that it can be applied to any odulation forat provided that coherent detection is perfored. However, DBP acquires ore iportance for high-order odulation forats since they require higher OSNR and they becoe, hence, ore exposed to nonlinearity. With regard to channel spacing, a channel spacing that is twice the baud-rate has been chosen. Larger channel spacing would increase the walk-off between channels. This results in a ore efficient averaging of the XPM effects and the perforance will be increased due to weaker ASE-seeded nonlinearities [13]. Saller channel spacing increases FWM efficiency and the difference between Manakov and DBP3 is expected to increase. Since Fig. 3 shows the Q-factor averaged over the WDM channels, it is interesting to analyze the behavior of the Q-factor for each WDM channel. Figure 4 shows such result where each DBP case is plotted at the respective optiu power. A rather hoogeneous behavior is obtained for each DBP case at the optiu power. Results obtained at higher powers revealed that central channels perfor worse due to a higher exposure to nonlinear interactions. # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 579
11 Fig. 4. Q-factor per channel for the different DBP cases. Each case is shown at the respective optiu power. The coputational efficiency of the above backward propagation schees is now analyzed. Figure 5 shows the Q-factors as functions of the step size for the conventional and advanced SSM ipleentations (recall that advanced SSM stands for the walk-off factorization). Fig. 5. Q-factors as functions of step size for the inter-channel copensation cases DBP and DBP3. The results correspond to the respective optiu powers obtained fro Fig. 3 whereas vertical arkers indicate the operational (optiu) step size. This value is obtained by cubic interpolation of the siulation results and by choosing the step size value corresponding to a Q-value penalty of 0.1 db with respect to the plateau value. The operational step size is chosen as a coproise between perforance and coputational load. The advantage of the walk-off factorization is clear in ters of step size. When coparing the advanced- and conventional-ssm, the step size is increased by a factor of 16 for the DBP3 and a factor of 6 for DBP. Since the dispersive walk-off iposes no restrictions on the step size for the advanced- SSM, the latter becoes liited by the nonlinear phase-shift per step, which in turn, depends on the power. This explains the difference between DBP and DBP3 in ters of step size when the advanced SSM is applied. The optiu operation for DBP3 happens at a power # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 580
12 value 3 db higher than the one for DBP. In addition, DBP3 includes additional nonlinear ters which also contribute to reduce the nonlinear length. The ipact of the channel spacing as well as of the nuber of channels on the step size can be extrapolated fro the results obtained in [13] for the scalar case. With respect of fiber dispersion, a different behavior can be extrapolated for the conventional and advanced algoriths. For the conventional, the stepsize is proportional to the walk-off length and hence, it is reduced as the dispersion paraeter of the fiber increases. Alternatively, the advanced SSFM follows the nonlinear length and it is insensitive to dispersion paraeter. Together with the step size, it is iportant to copare the coputation requireents for each ethod. For siplicity only the nuber of coplex ultiplications will be considered, neglecting the nuber of additions. Furtherore, considerations regarding the nueric representation (fixed point/floating point) will be ignored. By recalling the scheatic diagra in Fig. 1, the following nuber of operations is involved in backward propagation for a blocklength of M saples: Intensity operator: M Filtering: ( M P)log ( M P) ( M P ) Exponential operator ( 4 th order Taylor expansion): 6M Cosine operator ( th order Taylor expansion): 4M Sine Cardinal ( 3 th order Taylor expansion): 7M The nuber of ultiplications for the exponential, cosine and sine operators is obtained by saving the square of the arguent in eory [13]. These operators are soeties ipleented using look-up tables. However, look-up tables require large eory presenting a trade-off between eory and speed. In general, the filter ipleentation in the frequency doain is done by the overlap-andadd ethod. This is done using data blocks of M saples with an additional overhead of P saples. Such overhead has to be larger than the filter length in taps [18]. Moreover FFT/IFFT algoriths operate ore efficiently if M P is a power of. Two filter operations are ipleented depending on the case, that is, walk-off filtering with W (, h) and dispersion filtering with H (, h). Their respective group delays are given by, ( N 1) f h, W Bh. H By assuing a sapling rate S, the following overhead values are required for each filter operation, P ( N 1) f h S, W P B h S. H By taking into account the blocks depicted in Fig. 1 the following expressions are given for the total nuber of ultiplications per saple in the conventional SSM, q (7) (8) OP DBP c nsteps 4( M PH )log ( M PH ) ( M PH ) 18 M / M, (9) OP DBP3 c nsteps 4( M PH )log ( M PH ) ( M PH ) 36 M / M. (30) For the advanced SSM, each step involves two filter operations. For siplicity, we assue that the sae overhead is used for both of the. The nuber of ultiplications is now given by, # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 581
13 OP DBP a nsteps 8( M PW )log ( M PW ) (N 6)( M PW ) 16 M / M, (31) OP DBP3 a nsteps 10( M PW )log ( M PW ) (3N 5)( M PW ) 33 M / M. (3) In general, the overlap-and-add ethod is optiized by choosing the block size that iniizes the nuber of ultiplications. This optiu block size depends on the overhead size. In the SSM, the step size can be relatively sall which causes sall group delays. This eventually yields overheads of only several saples for practical sapling rates. In such scenario, the theoretical optiu block-size that iniizes the nuber of ultiplications is very short. Moreover, it can be deonstrated that the block size has to be large enough to preserve nuerical accuracy. Otherwise, the concatenation of steps leads to draatic error propagation. By taking the above considerations into account, the following rules are applied to choose the block size, 1) The total block-size for filtering ( M P) is a power of two. ) The iniu block size for M Pis chosen to be 6 3) For overhead values greater than, the total block-size ( M P) is chosen to iniize the nuber of ultiplications. By sapling at a frequency of S 50 GHz, Table 1 suarizes the nuber of operations including SPM copensation. Note that the nuber of operations for SPM is also given by Eq. (9) when additions are neglected. 6. Table 1. Suary of results for a 00 Gb/s PDM WDM syste consisting in 4 channels with a 50 GHz channel spacing. Transission is k. Letter -a stands for advanced SSFM and -c stands for conventional SSFM. DC SPM DBP-a DBP-c DBP3-a DBP3-c Q-iproveent h [k] n [steps/span] P [saples] M [saples] P M 6 OP [ 100] Table 1 shows the nuber of ultiplications required for each ethod and each syste. For coparison purposes, the nuber of operations for dispersion copensation is also shown. A factor of 4.3 in coputation savings is obtained for DBP3 when the advanced SSF is used. Alternatively, a factor of 6.8 is obtained for DBP. As explained before, this difference is due to the ipact of the optiu power on the step size. Despite the increased coplexity of the advanced SSM, the large step sizes allowed by the walk-off factorization substantially reduce the overall coputational load. 3. Conclusion An iproved schee for digital backward propagation (DBP) applied to PDM-WDM systes has been introduced. This new schee is based on two new aspects. First, the backward propagation proble has been reforulated by deriving a coupled syste of nonlinear partial differential equations. Such syste, allows the ipleentation of DBP in a channel-bychannel basis by including the ost relevant inter-channel and inter-polarization nonlinear ters. This forulation also allows the selective copensation of different nonlinear effects, # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 58
14 which has an ipact on the axiu achievable Q-factor and the corresponding coputational load. Second, an iproved split-step ethod has been used for the DSP ipleentation of DBP. The split-step foralis is extended to cope with new polarization-ixing ters. Likewise, the effect of dispersive walk-off is factorized by including the relative delay of the WDM channels in the coputation of the nonlinear step. A 00 Gb/s PDM WDM syste, consisting in 4 channels with 50 GHz channel spacing, has been siulated to assess the efficiency of the new DBP schee. Results show that the new syste of coupled nonlinear equations reoves alost all the contribution of deterinistic ipairents. Moreover, the walk-off factorization allowed increasing the SSM step size in a substantial aount. A rigorous coputation of the required nuber of operations showed that the iproved SSM reduces the coputational load by a factor of 4.3. This nuber is increased to alost 7 when polarization ters are not included in the backward propagation equations. # $15.00 USD Received 3 Nov 010; revised 1 Dec 010; accepted 10 Dec 010; published 4 Jan 011 (C) 011 OSA 17 January 011 / Vol. 19, No. / OPTICS EXPRESS 583
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