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On the effect of dispersion on nonlinear phase noise Keang-Po Ho Institute of Communications Engineering and Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan.∗ Hsi-Cheng Wang Institute of Communications Engineering, National Taiwan University, Taipei 106, Taiwan. (Dated: February 2, 2008) 6 0 The variance of nonlinear phase noise is analyzed by including the effect of 0 intrachannel cross-phase modulation (IXPM)-induced nonlinear phase noise. 2 Consistent with Ho and Wang1 but in contrary to the conclusion of both n Kumar2 and Green et al.3, the variance of nonlinear phase noise does not a decrease much with the increase of chromatic dispersion. The results are con- J sistent with a careful reexamination of both Kumar2 and Green et al.3. 2 c 2008Optical Society of America 2 (cid:13) OCIS codes: 060.2330,190.4370,190.4380,260.2030 ] Keywords: nonlinearphasenoise,intrachannelcross-phasemodulation. s c i t p Recently,phase-modulatedopticalcommunicationsys- phasenoisewascalculated,thenumericalvalueofEq.(9) o tems are found to have wide applications in long-haul ofRef.3dependingontheopticalfilterbandwidth[ ∆of . lightwavetransmissionsystems4,5. Addeddirectlytothe Eq.(9)there]. Forthecaseofhavinganopticalma±tched s c signalphase,nonlinearphasenoiseisthemajordegrada- filter with ∆ 0 for continuous-wave signal, the vari- i tion for phase-modulation signals1. However, the recent ance of nonlin→ear phase noise of Ref. 3 is independent of s y paperbyKumar2andearlypaperbyGreenetal.3 shown theamountofchromaticdispersion[i.e.,Eq.(9)becomes h that,incontrarytoHoandWang1,nonlinearphasenoise Eq.(8) there]. AnotherinterpretationofRef.3 basedon p becomes muchsmaller for highly dispersive transmission optical matched filter may conclude that the variance of [ system than that for a system with no dispersion. The nonlinear phase noise is independent of chromatic dis- 1 main purpose of this paper is to reconcile the discrep- persion. Optical matched filter is used in this letter and v ancy among those three letters1,2,3. Although there is nonlinear phase noise does not decrease that much with 5 no numerical error in both Kumar2 and Green et al.3, chromatic dispersion, largely consistent with the results 7 the conclusion is unfortunately generalized further than of Ref. 3 with same kind of filter. 1 its numerical results for a single pulse2 or continuous- 1 Unfortunately,themethodofRef.1cannotdirectlyap- wave signal3. The methods in both Refs. 2,3 are correct 0 ply to the systemof bothRefs. 2,3 withoutsome modifi- 6 andtheirresultsareconsistentlargelywiththeresultsof cations. SimilartotheanalyticalresultsofbothRefs.2,6, 0 Ref. 1 after a careful reinterpretation. there is an optical matched filter before the receiver. / s Ref. 2 found that the variance of the peak nonlin- Optical matched filter is either implicitly or explicitly c ear phase noise for a single pulse decreases rapidly assumed1,2,6. For example, the finite variance of linear i s with chromatic dispersion. However, a phase-modulated phase noise [see Eq. (25) of Ref. 2] implicitly assumed a y lightwave transmission is typically a chain of optical matchedfilter7 butthenumericalsimulationofRef.2as- h p pulses with different modulated phases. Ref. 2 ignores sumes an ideal band-pass filter with 70-GHz bandwidth. : the nonlinear phase noise induced from adjacent op- When white Gaussian noise with infinite bandwidth is v tical pulses, called intrachannel cross-phase modulation assumed, the noise power approaches infinity. A finite i X (IXPM) phase noise1. When IXPM phase noise is in- signal-to-noise ratio requires some types of optical filter r cluded, even with the model of Ref. 2, nonlinear phase andanopticalmatchedfilterhasthesmallestbandwidth a noise does not decrease much with chromatic dispersion. anddoes not distortthe signal. Opticalmatched filter is As shownlater in this letter, the IXPMphase noise may usedinsomeexperimentalmeasurementstoimprovethe affectanotherpulsethatishundredsofpicosecondsaway receiver sensitivity8,9. from the originated pulse. To make a direct comparison with Refs. 2,6, we con- The variance of nonlinear phase noise in Ref. 3 also sider a transmission system consisting of two segments decreases significantly for a continuous-wave signal with of equal length within an amplified fiber span. The dis- chromatic dispersion. When the variance of nonlinear persion of the first segment is anomalous whereas that of the second segment is equal in magnitude but op- posite in sign. Within each fiber span, the accumu- lated dispersion as a function of distance is given by ∗Electronicaddress: [email protected] S(z) = β2min(z,L z) where β2 is the group-velocity − 2 dispersioncoefficientandListhelengthofthefiberspan. where denotestheimaginarypartofacomplexnum- ℑ{} In the first order, the temporal distribution of nonlinear ber. Equivalently, the variance of φ (0) is found as the nl phase noise is independent to the number of fiber spans peak variance of nonlinear phase noise in Ref. 2. The if all fiber spans has the same configuration. variance of φ (t) was obtained in Ref. 1 for some dis- nl For a Gaussian pulse launching with an 1/e-pulse cretepoints. Here inthis letter, the whole temporalpro- width of T , at the distance of 0 z L, the pulse file of the variance of φ (t) is derived and calculated. If 0 nl ≤ ≤ becomes thetemporalprofileofφ (t)isconcentratedaround T nl 0 ± that is the 1/e-pulse width of u(0,t)= u(L,t), the peak A T t2 u(z,t)= 0 0 exp (1) nonlinear phase noise from Ref. 2 is more than sufficient [T2 jS(z)]1/2 (cid:26)−2[T2 jS(z)](cid:27) toevaluatethesystemperformance. However,ifthetem- 0 − 0 − poral profile of φ (t) is far wider than T , conclusion nl 0 with a pulse width of τ(z) = [T02+S(z)2/T02]1/2, where derived from the peak nonlinear phase n±oise is not suffi- A0 is the peak amplitude. The fiber loss is first ignored cient to characterize the system performance. Although here but includes afterward. From Refs. 1,5 and using themethodhereissimilartoRefs.1,5,thetemporalpro- a model similar to Ref. 10, the nonlinear phase noise is file for the nonlinear phase noise is never shown and the mainly induced by the nonlinear force of discrepancy between Refs. 1,2,3 is never reconciled. L ∆un(t)=2jγ u(z,t)2n(z,t) h−z(t)e−αzdz, (2) Z | | ⊗ From the model of both Refs. 1,2, with prefect span- 0 (cid:2) (cid:3) by-span dispersion compensation, the temporal profile where γ is the nonlinear fiber coefficient, n(z,t) is the for nonlinear phase noise is independent of the number amplified-spontaneous emission (ASE) noise, h−z(t) is of fiber spans. The temporal profile for a single-span the dispersion from z to L with an overall dispersion of system is derived by the assumption by first-order per- S(z), and α is the fiber attenuation coefficient. The turbation. With the first-order perturbation, the tem- − variance of (2) as a function of time is calculated in poral distribution is also independent to the launched Ref. 1. For system without chromatic dispersion, the power of the signal. For an optical matched filter with variance of (2) is equal to infinity as n(z,t) is commonly h (t) = u(0,t) = u(L,t), we obtain s(0) = √πA2T as o 0 0 assumed as white Gaussian noise. The temporal profile the energy of each optical pulse. of ζ(t) = ∆u (t) h (t) is calculated here, where h (t) n o o ⊗ is the impulse responseofthe opticalfilter precedingthe receiver. As the received signal is s(t) = u(L,t) h (t), ⊗ o Using the property that both ∆un(t) and ζ(t) are cir- the nonlinear phase noise is approximately equal to cular symmetric complex Gaussian random variable, after some algebra and followed Refs. 1,5, the variance of ∆u (t) h (t) n o φ (t)= ℑ{ ⊗ }, (3) nonlinear phase noise as a function of time is nl s(0) 2 σ2(t)=E φ (t)2 = 4γ2A20T02σn2 +∞(cid:12)(cid:12) L exp(cid:26)−t2−jτ(z)2ωt+τS(z(z)2)2−ω22j+S(21zT)0+2ω22T[02τ(z)2+2jS(z)] −αz(cid:27)dz(cid:12)(cid:12) dω, (4) nl (cid:8) nl (cid:9) π Z−∞ (cid:12)(cid:12)Z0 τ(z)2−2jS(z)+2T02 (cid:12)(cid:12) (cid:12) p (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) where2σ2 thepowerspectraldensityofASEnoiseatthe x-axis of the curve in Fig. 1 having a larger dispersion n launching location of z =0. coefficient is scaled down by a factor of 2 than the one Figure 1 shows the temporal profile of the standard with smaller dispersion coefficient. Similarly, the y-axis deviation (STD) of nonlinear phase noise of σ (t) as a ofthecurveinFig.1havingalargerdispersioncoefficient nl functionoftime. They-axisofFig.1isinarbitrarylinear is scaled up by the same factor of 2 than the one with unit. Figure 1 is calculated for a 80-km fiber link with smallerdispersioncoefficient. Afterthescaling,allcurves dispersion coefficient of D = 0,3.5,8, and 17 ps/km/nm in Fig. 1 have more or less the same height and width. and an initial launched pulse of T0 =5 ps. As explained Figure 1 confirms the conclusion of Ref. 2 that the earlier,thetemporaldistributionbyitselfisindependent peak nonlinear phase noise decreases rapidly with chro- ofthe number offiber spans. The number offiber spans, matic dispersion. In term of STD, the peak nonlin- signal-to-noiseratio,andlaunchedpowerscalethewhole ear phase noise with a chromatic dispersion of D = 17 curves of Fig. 1 up or down. ps/km/nm is about 7 times less than the dispersionless Thex-axisofFig.1doesnothavethesamescale. The caseofD =0. However,thetemporaldistributionofthe 3 Ref. 2 also included a minor second-order term [see Eq. (16) there]. The model here includes the correla- 1/4 tionofE n(z ,t+τ)n(z ,t) withapowerspectralden- 1 2 1/2 D= 17 ps/km/nm sity of 2σ{2exp j[S(z ) S(}z )]ω2/2 . The correlation n 1 − 2 1 8 of ASE noise is(cid:8)ignored in Ref. 2. T(cid:9)he temporal pro- 2 3.5 1/8 file here is asymmetric but the temporal profile of Ref. 2 D= 0 1/4 [given by hr(t) in Eq. (16) there] is symmetrical. The 1/2 temporalprofileh(t)inRef.2[Eq.(16)there]isverysim- 1 -200 00 200 400 ilar to (4) here. If IXPM phase noise is included to the -100 00 100 200 numerical method of Ref. 2, different conclusion should -50 00 50 100 be arrived. -25 0 25 The optical matched filter for continuous-waveoptical signal is a very narrow-band optical filter. Using a very Fig.1. ThetemporaldistributionoftheSTDofnonlinear narrow-band filter in the model of Ref. 3, the nonlinear phase noise σ (t). The x-axis is time in picosecond and phasenoiseisindependentofchromaticdispersionthere. nl they-axisisσ (t)inarbitrarylinearunit. Notethatthe Iftheoptimalopticalfilterisusedtodetectasignal,both nl x- and y-axes with difference dispersion do not have the Refs. 1,3 should arrive with similar results. same unit. This letter finds that all three letters of Refs. 1,2,3 should provide consistent results if IXPM phase noise is included in Ref. 2 and optical matched filter is used in nonlinear phase noise also broadens rapidly with chro- Ref. 3. Nonlinear phase noise does not decrease much matic dispersion. With a range from 200 to +400 ps, − with the chromatic dispersion in a practical lightwave thetemporaldistributionofσ (t)forD =17ps/km/nm nl transmission system. With optical matched filter and is about20times wider thanthe case withoutdispersion accordingtobothRefs.3and1,thenonlinearphasenoise of D =0 of within 15 ps. ± for system with large dispersion is approximately equal IftheeffectofchromaticdispersiontoASEnoiseisig- to an equivalent dispersionless continuous-wave system nored (the model of Ref. 2), IXPM phase noise from ad- having the same power. jacentpulsestothesamepulseis100%correlated. Ifthe If the correlation of ASE noise due to chromatic dis- effect of chromatic dispersion to ASE noise is included, persion is included to the model, the temporal profile of thecorrelationbetweenIXPMphasenoisefromadjacent the STD of nonlinear phase noise is asymmetrical with pulsesdecreasesslightlybutthetailofthetemporalpro- respectto its peak. The time-domainasymmetricprofile file increases. For a qualitative understanding without is first observed for nonlinear phase noise here. repeating the calculations in Refs. 1,5, we can assume thatthenonlinearphasenoiseinducedbyadjacentpulses to the same pulse is highly correlated. For highly corre- References latednoise,thecombinednoisehasaSTDapproximately equal to the sum of the individual STD, approximately 1. K.-P. Ho and H.-C. Wang, IEEE Photon. Technol. thesameastheareaofthecurvesofFig.1. Becausemain Lett. 17, 1426 (2005). partsofthefourcurvesinFig.1havethesamepeaksand 2. S. Kumar, Opt. Lett. 30, 3278 (2005). width after scalingupin heightanddownin time by the 3. A. G. Green, P. P. Mitra, and L. G. L. Wegener, samefactor,nonlinearphasenoisedoesnotdecreasethat Opt. Lett. 28, 2455 (2003). much with the increase of chromatic dispersion1,5. 4. A.H. GnauckandP.J.Winzer, J.LightwaveTech- Other than the dispersionless case with D = 0, the nol. 23, 115 (2005). temporal profile of nonlinear phase noise is asymmetric 5. K.-P.Ho, Phase-Modulated Optical Communication with respect to the original center of the pulse of t = 0. Systems (Springer, New York, 2005). In Ref. 3, a shift was observed in frequency domain but 6. J. P. Gordon and L. F. Mollenauer, Opt. Lett. 15, the shift in time domainofFig.1is firstobservedinthis 1351 (1990). letter. The peak of nonlinear phase noise is located ap- 7. J. G. Proakis, Digital Communications (McGraw proximatelyatthecenterofthepulsebutshiftedslightly Hill, New York, 2000), 4th ed. topositivetime. The asymmetrictemporalprofileisdue 8. D.O.CaplanandW.A.Atia,inOpticalFiberCom- totheinclusionofthedispersiveeffectstotheASEnoise. mun. Conf. (Optical Society of America, Washing- Without the inclusionof the dispersive ASE,the tempo- ton, D.C., 2001), paper MM2. ral profile of σ (t) is symmetrical with respect to the 9. W.A.AtiaandR.S.Bondurant,inProc. LEOS’99 nl pulse center of t=0. (IEEE, New York, NY, 1999), paper TuM3. The model here is very similar to the model of 10. A. Mecozzi, C. B. Clausen, and M. Shtaif, IEEE Ref. 2. Only the first-order term is used here but Photon. Technol. Lett. 12, 1633 (2000).