Intrachannel nonlinear penalties in dispersion-managed transmission systems

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Intrachannel nonlinear penalties in dispersion-managed transmission systems
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  626 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 3, MAY/JUNE 2002 Intrachannel Nonlinear Penalties inDispersion-Managed Transmission Systems Shiva Kumar, John C. Mauro, Srikanth Raghavan  , Member, IEEE  , and Dipak Q. Chowdhury  Invited Paper   Abstract— We have developed a model for intra-channelimpairments in dispersion-managed systems and validated itusing numerical simulations. We have shown that intrachannelimpairments can be reduced using optimal precompensation. Wehave studied the intrachannel penalty for different modulationformats, and results show that the optimum precompensationis independent of modulation format. The intrachannel penaltydecreases as the spectrum of the modulation format broadens.  Index Terms— Cross-phase modulation, four-wave mixing, non-linear optics, optical communications. I. I NTRODUCTION I N STRONGLY pulse-overlapped return-to-zero (RZ) trans-mission systems with bit rate 40 Gb/s, the intrachannelimpairments are more dominant than interchannel ones. In thispaper,weintroduceamodelwehavedevelopedforintrachannelfour-wave mixing (FWM) and intrachannel cross-phase mod-ulation (XPM). The nonlinear mixing of closely spaced pulsepairs generates temporal side bands or ghost pulses due to intra-channel FWM [1]–[4]. The ghost pulses falling on the logical “1” bits induce amplitude jitter, which is one of the dominantpenalties at bit rates 40 Gb/s [3]–[8]. Suppose we have three consecutive bits of “1” centered at , 0, and . Theghost pulse generated bythenonlinear mixingof pulsesat0 andinterferes with the pulse at . This interference modifiesthe amplitude of the pulse at and gives rise to amplitude jitter. Here, we model the interaction in a random bit sequenceoflength consistingofbits“1”and“0.”Weconsiderthepulsetriplet interactions that modify the energy of the center bit. Dueto energy change, the power in the center of the bit slot fluc-tuates. After summing over all possible triplets, we obtain anexpression for the power jitter due to intrachannel FWM.The nonlinear interaction between two adjacent pulses dueto intrachannel XPM has drawn considerable attention in thecontext of dispersion-managed solitons [9]–[11]. The adjacent pulse causes phase modulation of the probe pulse due to in-trachannel XPM, which translates into timing fluctuations dueto dispersion. Although the energy of a pulse does not changeby intrachannel XPM, pulse width and amplitude of the pulsefluctuate giving rise to amplitude jitter. Due to amplitude and Manuscript received January 28, 2002; revised March 13, 2002.The authors are with the Science and Technology Division, Corning Incorpo-rated, Corning, NY 14831 USA.Publisher Item Identifier S 1077-260X(02)05899-9. timingchanges,thepowerinthecenterofbitslotfluctuates.Fora random bit sequence of length , the amplitude and timingshifts due to all the pair interactions involving the probe pulseare summed to obtain the noise variance of power in the centerof the bit slot caused by intrachannel XPM.The pulsewidth and amplitude of the pulse also fluctuate duetoself-phase modulation (SPM). Wecombinethe powerchangeat the center of bit slot due to SPM, intrachannel XPM, andintrachannel FWM. By performing an ensemble average overall realizations of bit streams, total noise variance of power dueto intrachannel impairments is computed.The energy jitter due to intrachannel FWM is sensitive tothe amount of precompensation. References [8] and [12] have shown that splitting the dispersion compensation equally be-tween input and output leads to a considerable improvement inperformance. However, our results indicate that 80% of disper-sion compensation at the input and 20% at the output improvesthe performance for nonresonant dispersion maps (residual dis-persion per amplifier spacing 0 ps/nm km). The difference inresults is due to the fact that our dispersion maps differ fromthatin[8]and[12]significantly.In[8]singlepositivedispersion fiber is used as a transmission fiber over 1600 km, while we usea periodic dispersion map with 80-km period. In [12], disper-sion map and bit rate are such that intrachannel XPM is dom-inant while in our system mostly intrachannel FWM is domi-nant. We found that, in the case of nonresonant map, the energychange due to interaction with different triplets becomes negli-gibly small immediately after the point of reversal of accumu-lated dispersion sign. Therefore, length of the precompensationfibercanbesochosenthataccumulateddispersionreversessigncloser to the receiver.We have also studied the intrachannel impairments for dif-ferent modulation formats and found that the transmission per-formance improves as the spectrum of the modulation formatincreases. The amount of precompensation required to have op-timum -factor is independent of the modulation format.II. I NTRACHANNEL  FWM M ODEL Let the pulses in a channel be represented by [7], [13] (1)(2) 1077-260X/02$17.00 © 2002 IEEE  KUMAR  et al. : INTRACHANNEL NONLINEAR PENALTIES IN DISPERSION-MANAGED TRANSMISSION SYSTEMS 627 where(3)with , , is the bit periodand is the probability of having a bit “1” in slot , which istaken as 1/2.Consider a pulse in slot zero. The nonlinear interaction of thepulse in slot zero with its neighbors is given by [7](4)where is the dispersion, is the nonlinear coefficient(5)between amplifiers, and is the fiber loss. In the absenceof perturbations given by the right-hand side (RHS) of (4),and are constants. Due to the nonlinear interaction with theadjacentpulses, becomesafunctionofdistance.Multiplying(4) by , subtracting its complex conjugate and integrating, wehave(6)where(7)(8)When corresponding to SPM, or andcorresponding to XPM, the RHS of (6) becomes zero.This implies that the energy of the pulse does not change bySPM and XPM. However, in the next section, we consider thevariation in pulsewidth and amplitude due to SPM and XPMwhich gives rise to power variations.With undepleted pump approximation, the peak power of thepulse at bit slot zero after a propagation distance is given by(9)where(10)and input peak power. With the help of (9), the standarddeviationofpowerchangeduetointrachannelfourwavemixingcan be calculated as follows:(11)(12)whereif if any indices are equal (13)The standard deviation of power change due to intrachannelFWM is given by(14)From (12) and (14), we see that standard deviation is pro-portional to square of input peak power . We calculate thechange in power of slot zero using (9) due to its interactionwith different pulse triplets. The following parameters are usedthroughout this paper unless otherwise specified: nonlinearcoefficient 0.0025 W Km , bit rate 40 Gb/s, dispersionof the precompensating and postcompensating fiber 100ps/nm/km, and amplifier spacing 80 km. Gaussian pulseswith full-width at half-maximum (FWHM) 12.5 ps arelaunched at the input. Dispersion management is achievedusing 40 km of standard single-mode fiber withps/nm/km followed by a negative dispersion fiber of the samelength. The fiber loss is 0.2 and 0.25 dB/km for the positivedispersion fiber and negative dispersion fiber, respectively.The dispersion of the negative dispersion fiber is chosen tobe different in different examples. The postcompensation ischosen such that the accumulated dispersion at the receiver iszero.Fig. 1(a) shows the power change of the pulse in slot zerodue to interaction with different pulse triplets as a function of propagation distance. In this case, the dispersion of the nega-tive dispersion fiber is 14.5 ps/nm km and the net accumu-lated dispersion per amplifier spacing is 100 ps/nm. A 5-kmprecompensating fiber with an accumulated dispersion of 500ps/nm is placed immediately after the transmitter. The rest of the dispersion of the transmission fibers is compensated fully atthe receiver. In Fig. 1(a), each curve corresponds to the changein power of the pulse in slot zero due to its interaction with dif-ferent pulse triplets. For example, the top curve corresponds tothe change in power due to the interaction among the pulse inslot zero and those in slots 1, 1, and 2. All of the curvesfocus around 450 km and at this point, the power change is min-imal. The location of minimum power change occurs shortlyafter the location of reversal of accumulated dispersion, whichoccurs in this example at 400 km. In Fig. 1(b), a 7-km precom-pensating fiber with an accumulated dispersion of 700 ps/nmis used and the point of accumulated dispersion reversal occursat 560 km. In this case, the point of minimum power change isaround 700 km. By changing the amount of precompensation,  628 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 3, MAY/JUNE 2002 Fig. 1. Power change of the pulse in slot zero due to its interaction withdifferent pulse triplets. Average launch power    0 dBm, number of amplifiers   10, amplifier spacing    80 km, accumulated dispersion per amplifierspacing    100 ps/nm. Duty cycle    50%. (a) Precompensation length    5 km.(b) Precompensation length    7 km. we have seen that the focal point always occurs after the pointof reversal of accumulated dispersion. By pushing the point of minimum power change toward the receiver, we can maximizethe quality of transmission performance.Fig. 2(a) corresponds to the resonant case, i.e., the net dis-persion per amplifier spacing is 0 ps/nm and there is no prec-ompensation. This can be achieved by choosing the dispersionof negative dispersion fiber 17 ps/nm km. Unlike the non-resonant case, there is no focal point and power change due tointrachannel FWM grows linearly with distance. Fig. 2(b) cor-responds to the resonant case with optimum precompensationof 340 ps/nm. We see that intrachannel FWM can be consid-erably suppressed by optimum precompensation. In this case, apostcompensation of 340 ps/nm is used at the receiver.Figs. 3 and 4 show the standard deviation of intrachannelFWM normalized to the mean power using (14) for thecase of nonresonant and resonant dispersion maps, respectively.In the case of nonresonant maps, the standard deviation is closeto zero at the point of minimum power transfer which can bemoved by changing the amount of precompensation. Fig. 2. Same as Fig. 1 except that accumulated dispersion per amplifierspacing    0 ps/nm, i.e., resonant map. (a) No precompensation. (b)Precompensation length    3.4 km. III. I NTRACHANNEL  C ROSS -P HASE  M ODULATION  M ODEL The mean position and amplitude of the reference pulse inslot zero changes due to its interaction with any nearby pulse.The nearby pulse causes phase modulation of the pulse in slotzerowhichtranslatesintoamplitudeandtimingfluctuationsdueto dispersion. The timing and pulsewidth changes due to intra-channel XPM interaction between the pulses inslots and , aregiven by [9](15)(16)(17)(18)where , , and are inverse pulsewidth, chirp, angularfrequency, and pulse center of the pulse in slot , respectively.  KUMAR  et al. : INTRACHANNEL NONLINEAR PENALTIES IN DISPERSION-MANAGED TRANSMISSION SYSTEMS 629 Fig.3. VariationofnormalizedstandarddeviationduetointrachannelFWMina nonresonant map. Diamonds and triangles show the case of precompensationlength of 7 and 5 km, respectively. The parameters are same as in Fig. 1.Fig. 4. Variation of normalized standard deviation due to intrachannel FWMin a resonant map. Diamonds and triangles show the case of precompensationlength of 0 and 3.4 km, respectively. The parameters are same as in Fig. 2.Fig. 5. Variation of the pulse peak position with propagation distancein a nonresonant map of Fig. 1. Diamonds and triangles show the case of precompensation length of 7 and 5 km, respectively. The parameters are sameas in Fig. 1. Using (15)–(18), pulsewidth and timing change of the pulse inslot zero is calculated by summing over all pulse pair interac-tions involving the pulse in slot zero.Fig. 5 shows the variations of the center of a pulse in bit slotzero due to its interaction with an adjacent pulse separated bya bit period of 25 ps using (15)–(18). The periodic change indispersion gives rise to oscillations in the pulse peak position.In this example, the nonresonant dispersion map of Fig. 1 isused. Since the accumulated dispersion at the receiver is zero,the timing shift is quite small after propagation of 800 km. Thereduction in nonlinear interaction is because the pulses overlapnearly completely [1]. Following the explanation in [1], XPM- induced frequency shift is proportional to the time derivative of the intensity of the interacting pulse. When the adjacent pulsescompletely overlap, the time derivative is reduced due to pulsebroadening. Fig. 6. Normalized standard deviation due to intrachannel XPM in anonresonant map. Diamonds and triangles show the case of precompensationlength of 7 and 5 km, respectively. The parameters are same as in Fig. 1.Fig. 7. Normalized standard deviation due to intrachannel XPM and FWM asa function of dispersion of positive dispersion fiber. The other parameters aresame as in Fig. 2(a). By taking a random bit sequence of 3 bits on both sides of the pulse in slot zero, and using (15)–(18) for each of the pulsepair interaction, the variations of peak position and width of thepulse in slot zero is calculated. We assume that the decision onwhether the received bit is “0” or “1” is done at the center of bitslot.Whenthepulsepeakpositionshiftsawayfromthecenterof bit slot due to intrachannel XPM, the power at the center of bitslotreduces.WehavecalculatedthestandarddeviationofpowervariationduetointrachannelXPMusing(15)–(18).Fig.6showsthe standard deviation normalized to the mean power atthe center of bit slot. Comparing Figs. 3 and 6, after a propa-gation of 800 km, normalized standard deviation due to intra-channel XPM is smaller than that due to intrachannel FWM.This is mainly because pulses completely overlap during prop-agation for the dispersion map in Figs. 3 and 6.Fig. 7 shows the normalized standard deviation due to intra-channel XPM and FWM as a function of dispersion of positivedispersion fiber. In this case, the absolute dispersion of positivedispersion fiber is equal to that of negative dispersion fiber andno precompensation is used. At lower dispersions, intrachannelXPM is a dominant penalty, which is consistent with the ex-planation in [1] since pulses partially overlap. As dispersion in-creases, strong overlap of pulses gives rise to ghost pulses thatenhance intrachannel FWM while reducing XPM.IV. R ESULTS AND  D ISCUSSION To calculate the noise variance due to all intrachannel im-pairments, we consider all possible bit patterns of length threeon both sides of slot zero containing a bit “1.” For the given  630 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 3, MAY/JUNE 2002 Fig.8. Validationofthemodel.Thestandarddeviationofpowerisnormalizedto its mean. The parameters are same as in Fig. 1.Fig. 9. Scaling with launch peak power. The standard deviation of power is inabsolute units. Precompensation length    8 km. The other parameters are sameas in Fig. 1. bit pattern, the energy change due to all three pulse interactions(FWM) involving the pulse in slot zero is calculated using (9).Similarly, pulsewidth (or amplitude) and timing shifts due to allpulsepairinteractions(XPM)iscalculated.Using(15)–(18)andignoring the XPM terms, the pulsewidth change due to SPM isalso calculated. Combining all these effects and performing anensemble average over all realizations of bit streams, the noisevarianceofpoweratthecenterofthebitslotforslotzeroiscom-puted. In our parameter space, by increasing the length of bitpattern to four or more, we found that there is very little changein the noise standard deviation.Fig. 8 shows the standard deviation of the power in level “1”normalized to its mean, due to combined (XPM, FWM, andSPM) intrachannel impairments. In Figs. 8–12, total propaga-tion distance is 800 km. As can be seen from Fig. 8, when theprecompensation fiber length is 8 km, the standard deviation isminimum. We expected this result because the energy transferdue to intrachannel four wave mixing is minimal at this level of precompensation. This is because the accumulated dispersionreverses sign around 640 km and the focal point of Fig. 1 movescloser to the receiver.The numerical simulation of thenonlinearSchrödinger equation is carried out using a psuedorandom bitsequence (PRBS) consisting of 1024 bits. The simulation band-width is 1 THz. An eye diagram for optical power is obtained atthe end of transmission link. In the middle of the eye,mean, andstandard deviation of level “1” is computed. Diamonds in Fig. 8show the numerical results which are in good agreement withthe model. Fig. 9 shows the scaling of standard deviation withlaunch power. According to (12) and (14), the standard devia-tion due to intrachannel FWM is proportional to square of inputpeakpower.Thenumericalsimulationsshowthesamebehavior. Fig.10. Variationsofnormalizedstandarddeviationof“1”duetointrachannelXPM and FWM with duty cycle. The other parameters are same as in Fig. 2(a).Fig. 11. Normalized standard deviation of power obtained from numericalsimulations for different modulation formats. The other parameters are sameas in Fig. 1.Fig. 12. Normalized standard deviation of “1” due to intrachannelimpairments as a function of Raman gain in hybrid amplification scheme.Precompensation length    6 km. Raman pump loss    0.35 dB/km. The otherparameters are same as in Fig. 1. Fig.10shows thenormalizedstandard deviationof thepowerdue to intrachannel FWM using (12) and (14) and intrachannelXPM using (15)–(18). The standard deviation increases withdutycycle,aresultwe confirmbyperformingaseriesofnumer-ical simulations of the nonlinear Schrödinger equation. Fig. 11showsthestandarddeviationof“1”fordifferentmodulationfor-mats: NRZ, RZ (50%), i.e., RZ with 50% duty cycle, RZ (33%)and carrier-suppressed RZ (66%) (CS-RZ). The interesting factisthattheoptimumprecompensationisindependentofthemod-ulation format. For the given duty cycle, the performance of 
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