Impulse Response of Cross-Phase Modulation Filters in Multi-span Transmission Systems with Dispersion Compensation

Transcription

1 OPTICAL FIBER TECHNOLOGY 4, ARTICLE NO. OF98065 Impulse Reonse of Cross-Phase Modulation Filters in Multi-an Transmission Systems with Diersion Compensation Alberto Bononi, Cristian Francia, and Giovanni Bellotti Dipartimento di Ingegneria dell Informazione, Uniersita ` di Parma, Parma, Italy Received December 1, 1997 As shown by Chiang et al. Ž 1996., cross-phase modulation on each channel of a wavelength-division-multiplexed transmission system can be viewed as a phase modulation, where the modulating signal is the sum of the input intensities of all the other copropagating channels, each filtered by a low-pass filter, whose bandwidth decreases with the walk-off parameter. We give an explicit expression of the impulse reonse of such filter in the general case of diersion mapped transmission systems. We show by simulation that this filter, in intensity-modulated transmission systems, gives an optimal prediction of the signal phase even when the underlying assumption of negligible envelope distortion upon which its derivation is based is strongly violated. As an application, we show an analytical evaluation of the ectral broadening induced by in multi-an compensated systems Academic Press I. INTRODUCTION In a fundamental paper, Chiang et al. 1 have shown that cross-phase modulation Ž. in wavelength-division-multiplexed Ž WDM. optical transmission systems can be modeled as a phase modulator with inputs from the intensity of copropagating waves. Such inputs are low-pass filtered by the relative channel walk-off induced by chromatic diersion: the larger the walk-off, the narrower the filter bandwidth. This model nicely explains the weak dependence of on the number of WDM channels, which has already been observed experimentally and by simulation in. Chiang et al. 1 have extended such a model to cascades of ideally amplified fiber links with diersion compensation. This paper provides a time-domain derivation of the theoretical model proposed in 1, leading to an explicit expression of the filter impulse reonse, both for a single fiber an and for a cascade of amplified links with diersion compensation $5.00 Copyright 1998 by Academic Press Allrightsofreproductioninanyformreserved.

2 37 BONONI, FRANCIA, AND BELLOTTI We find that the expression of the filter phase in 1 is incorrect because of an inconsistency in the use of Fourier transforms. Having the correct phase expression, we show that such a filter is noncausal when the interfering channel travels faster than the observed channel. In such a case, the phase of the interfering channel at the detector contains information about the future information bits of the observed channel. We give a graphical interpretation of the filter impulse reonse in the case of link-by-link diersion compensation in a cascade of amplified links, so that the periodic forward and backward relative walk-off becomes apparent. For the case of intensity-modulation direct-detection Ž IMDD. systems, we provide simulation evidence that the model surprisingly holds even though the assumption of negligible envelope distortion upon which its derivation is based is strongly violated. In other words, we find that the modulation induced on the optical field phase by is nearly independent of the distortion induced on the intensity of the optical field by group velocity diersion Ž GVD.. This is due to the above-mentioned low-pass filtering effect of chromatic diersion-induced channel walk-off on. An an application of the filter impulse reonse, we give an analytical evaluation of the ectral broadening induced by in multi-an compensated systems. The paper is organized as follows. Section II provides the derivation of the filter impulse reonse for a single link, while Section III extends it to the general case of diersion-compensated cascades of amplified fiber links. Section IV presents simulation results providing evidence that the model holds even in the presence of strong intensity distortion due to group velocity diersion. Section V shows simulation and theoretical results on ectral broadening induced by. Finally, Section VI concludes the paper. II. SINGLE SPAN We consider a WDM system composed of one an of single-mode optical fiber, with N copropagating channels having the same polarization. The nonlinear Schrodinger equation describing the propagation along the z axis of the complex envelope A Ž z, t. of the generic channel s, s 1,...,N, is 3 s AsŽ z, t. 1 AsŽ z, t. AsŽ z, t. z t gs s s p ja Ž z, t. A Ž z, t. A Ž z, t., Ž 1. where gs is the group velocity of channel s, is the attenuation coefficient, is the nonlinearity coefficient and the summation is extended to the set of channels p 1,...,N : p s 4. The two terms on the right-hand side of Ž. 1 give rise to self-phase modulation Ž SPM. and, reectively. The j term in Ž 1. appears because of the engineering definition of Fourier transforms 3. A j term appears when using the physicist notation 4. Ý

3 IMPULSE RESPONSE OF CROSS-PHASE MODULATION FILTERS 373 Note that the contributions giving rise to four wave mixing Ž FWM. have been omitted from the right-hand side of the equation. Thus we suppose we are working with high local GVD, as common in diersion mapping 5, 6, to effectively suppress FWM. Note also that on the left-hand side of Ž. 1 we have neglected the higher order time derivatives of A Ž z, t. s, which cause the GVD-induced distortion of the envelope. It is only under this assumption that Eq. Ž. 1 can be solved exactly, giving A z, t A 0, t z e z e j sž z, t., s 1,...,N, Ž. s s gs where the signal phase is 1 z sž z, t. 0 LeffŽ z. A 0, t sž gs / ž / z z z Ý H A p 0, t dz e dz, Ž 3. 0 gs Ž where is the initial phase, and L z 1 e z. 0 eff is the effective length of the fiber. The two terms in the squared brackets represent SPM and, reectively. 1 1 The coefficient d D gs gp c 4 is the walk-off parameter be- tween channels s and p, where Dc is the diersion coefficient of the fiber and is the wavelength distance between the two channels. This is the case studied in 1 in the frequency domain, using sinusoidally modulated signals. Here we proceed instead in the time domain. With the variable change zd, the term related to in Ž. 3 can be rewritten as / gs 1 0 z Ž d. s Ž z, t. Ý H A 0, t e d. Ž 4 pž. d dz The above variable change has a physical meaning: integrating in Ž. 3 in the ace domain is equivalent to integrating in the time domain that part of the A Ž 0, t. p interfering signal which crosses the reference signal A Ž 0, t. s during the propaga- tion over the distance z. The relative walk-off between the s and the p channels is d sm, so that the integral in the time domain has to be extended over a duration d z,asinž. 4. In Ž. 4 we recognize the convolution integral z Ž z, t. P 0, t h Ž t., Ž 5. Ý ž / s p gs

4 374 BONONI, FRANCIA, AND BELLOTTI where P Ž 0, t. A Ž 0, t. is the input power of signal p and h Ž. p p t is the impulse reonse of a linear filter h Ž. t, which, for both d 0 and d 0, is given by / 1 t dz Žt d. hž t. e Ł, Ž 6 ž. d zd t where we have used the rectangular function Ł 1 for t, and zero outside. In Fig. 1 we plot the impulse reonse h Ž. t, for both d 0 and d 0. Note that when d 0, h Ž. t is noncausal. This should not be a surprise, since the p signal travels faster than s, so that during propagation Ž z, t. s is affected by the future bits of the p signal. The peak of h Ž. t near the origin is due to the fact that the p channel is attenuated during propagation, so that the effect is stronger at the beginning of the interaction. When the fiber has no attenuation Ž 0,h. Ž t. reduces to a rectangle. In the case of no channel walk-off Ž d 0., we have h Ž t. L z t, so that we obtain the well-known formula Ž z, t. L Ž z. eff s eff Ý P Ž 0, t z. p s p gs. The Fourier transform of the impulse reonse Ž. 6 is 1 e Ž j d. z j t HŽ. H hž t. e dt. Ž 7. j d Its squared magnitude and phase are, reectively, z 1 4e sin dz eff z 1 Ž d. Ž 1 e. H L z 1 8 ž Ž. / / sin dz d HŽ. arctg arctg. Ž 9 z ž. cos d z e Ž. FIG. 1. Impulse reonse h t in both cases of positive and negative walk-off parameter.

5 IMPULSE RESPONSE OF CROSS-PHASE MODULATION FILTERS 375 Equation Ž. 8 agrees with the result in 1. However, Ž. 9 gives the correct expression for the phase, while expression Ž 10. of 1 is incorrect. First, it is has a typo and should read argž i d. argžž 1 1 L. L e cos d L ie sinž d L.., which agrees with Ž here. Second, there is an inconsistency in 1 : the authors start with a version of Eq. Ž. 1 derived according to the physicists notation for Fourier transforms 4, and later they use 1, Eq. Ž 16. the engineer s notation, as we do here. This is why our expression for in Ž. 9 matches with their. However, in the physicists notation, the correct expression is. From Ž. 8 we immediately get the 3-dB bandwidth Bw when the fiber length z is long. In such case the sin Ž. can be neglected and by inection we see that B d w. In the limit d only the DC power component is passed by the filter, so that no is observed. Figure plots the 3-dB bandwidth B w, as a function of, for two typical values of diersion. Since Bw quickly decreases with, more distant channels are more filtered and give a lower contribution to. This justifies the experimental observation that does not increase linearly with the number of channels N. III. CASCADE OF AMPLIFIED LINKS In this section we analyze in the general case of a system composed of M 1 amplifiers, and M links, as in Fig. 3. We suppose that the mth amplifier has gain Gp Ž m. at the pth wavelength, so that the channel power, after the mth Ž m l m. Ž m. amplifier, is P L, t P L, te G, where and l are the attenp m p m1 p i i FIG.. 3-dB bandwidth B D versus channel acing, for 0.1 dbkm and w c diersion D and 17 psnmkm. c

6 376 BONONI, FRANCIA, AND BELLOTTI FIG. 3. Top: M concatenated amplified fiber links. Bottom: impulse reonse in the absence of compensation. uation coefficient and the length of the ith link, reectively, and Lm Ý m i1 l i. The total fiber length is L L M. Since, as seen in Ž. 3, is given as an integral over the link an, by the linearity of the integral the total at the end of the link can be written as M s L, t Ý s l i, t, 10 i1 whre Ž l, t. s i is the contribution over the ith link. In this section it is convenient to write Ž. 5 in a time reference moving with velocity gs, Ý Ž l, t. P Ž 0, t. h Ži. Ž t., Ž 11. s i p where the superscript Ž. i indicates the dependence of the impulse reonse h Ž. t, Ž. Ži. defined in 6, on the length, walk-off, and attenuation parameters l i, d, i of the ith link. Ž1. The at the end of the first link is l, t Ý P 0, t h Ž. s 1 p s p t. The contribution of the second link is Ý Ž l, t. P Ž l, t. h Ž. Ž t. s p 1 Ý p p 1 e 1l 1G Ž1. P 0, t ld Ž1. h Ž. Ž t. Ý P p 0, t Ž. Ž. Ž1. C p h t ld 1, 1 where C Ž. e 1l 1G Ž1. p p is the power gain at the beginning of the second link. From Ž 1. we see that the contribution of the second link can be seen again as a linear filtering of the same input function P Ž 0, t. p. The linear filter is as Ž. Ž1. Ž. in 6, but translated in time by ld and multiplied by the constant C. 1 p

7 IMPULSE RESPONSE OF CROSS-PHASE MODULATION FILTERS 377 In general, the contribution of the mth link m 1,...,M is Ý Ž l, t. P Ž L, t. h Ž m. Ž t. s m p m1 Ý ž / m1 Ž m. Žm. Ži. p p Ý i i1 P Ž 0, t. C h t ld, Ž 13. where the effective gain at the beginning of the mth link is C Ž m. Ł m 1 e i l p i1 ig p Ži., and Cp Ž1. 1. By 10, the total at the end of the Mth link is then Ž L, t. s Ý P Ž 0, t. ˆh Ž t., where the overall filter impulse reonse is p s p ž / ½ 5 M j1 Ž j. Žj. Ži. Ý p Ý i j1 i1 ĥ Ž t. C h t ld. Ž 14. In Fig. 4 we show a block diagram implementing Ž 14.. The equivalent filter for channel p is indeed a parallel bank of filters. For example, consider the simplified case Cp Ž j. 1, j, i.e., the gain Gp Ž j. of the jth amplifier exactly compensates the loss of the preceding Ž j 1. st fiber link. Suppose furthermore that the links have the same length, attenuation, and walk-off l j l, j, j, j, d Ž j. d, j so that we drop the dependence of h on j. In this case, the total overall impulse reonse simplifies to M1 ½ Ý Ž. 5 k0 ĥ Ž t. h t kld Ž 15. FIG. 4. Graphical interpretation of Eq. 14.

8 378 BONONI, FRANCIA, AND BELLOTTI and taking Fourier transforms M1 5 k0 Ĥ Ž. H Ž. Ý e j kld ½ Ž. / sin M ld j ld Ž M1. HŽ. e Ž 16. ž sin ld the same result obtained in 1. Equation Ž 14. is extremely general. It can be used for any type of multistage system, with any amplifier gain profile, with any kind of compensating scheme. In Fig. 3 we show an example of ˆh Ž. t in the case of a noncompensated system, composed of M identical links, when the amplifiers exactly compensate the fiber loss. The overall impulse reonse is just the sum of the M shifted copies of the basic filter Ž. 6. In Fig. 5 we show the case of a compensated system with imperfect compensation, composed of M 4 identical ans of transmission fiber, each followed by a an of compensating fiber of opposite diersion and an amplifier to exactly recover the an loss. The figure shows the individual contributions of each fiber segment to the overall impulse reonse. As opposed to Fig. 3, note that here the Ž identical. contributions of the transmission fibers are partially overlapping, because of the backward relative walk-off achieved by the compensating fibers, which brings the s and p channels back aligned, although not exactly. In this example the reonses of the compensating fibers are noncausal. The compensating fibers, when placed at the end of each transmission an, give typically only a small contribution to the overall filter reonse ˆh Ž. t and thus can be safely neglected. In the case of a perfectly compensated M-an system, neglecting the contribution ˆ Ž. Ž1. of the compensating fibers at each link, we obtain h t Mh Ž. t, so that the effect is M times larger than that of a single link. FIG. 5. Individual fiber contributions to the overall impulse reonse of an imperfectly compensated system. The overall reonse is the sum of those shown.

9 IMPULSE RESPONSE OF CROSS-PHASE MODULATION FILTERS 379 IV. SIMULATIONS We wanted to check how sensitive the results of the present model are when the underlying assumption of undistorted field envelopes used to derive the model is violated. In Figs. 6 and 7 we show the results of computer simulations, performed by the lit-step Fourier method 4. Simulations are carried out for a multi-an WDM system, with 10 ans, channels, and 3 mw peak power per channel, of the compensated type depicted in Fig. 5. One of the channels Ž pump. is onoff keying Ž OOK. modulated with raised cosine pulses with roll-off 0.5, at a bit rate R 10 Gbs, while the other Ž probe. is unmodulated. Figure 6 refers to a system in which at each an the transmission fiber is a nonzero diersion fiber Ž NZDF., with diersion D psnmkm and length l 85 km, and a standard single-mode fiber Ž SMF. with D 17 psnmkm and l 10 km is used for an compensation. Other parameters common to the two fibers are diersion slope D 0.07 pskmnm, nonlinear coefficient.34 W 1 km 1, attenuation 0.1 dbkm. FIG. 6. Simulations of a NZDF compensated system, R 10 Gbs, M 10 ans. Left column: 0.1 nm. Right column: 1 nm. Top row: Pump power Ž mw.. Bottom row: phase of probe signal Ž rad.. Time is normalized to the bit time 1R.

10 380 BONONI, FRANCIA, AND BELLOTTI FIG. 7. Simulations of a SMF compensated system, R 10 Gbs, M 10 ans. Left column: 0.1 nm. Right column: 1 nm. Top row: Pump power Ž mw.. Bottom row: phase of probe signal Ž rad.. Time is normalized to the bit time 1R. Figure 7 refers to a system in which the transmission fiber is SMF, with l 76 km, D 17 psnmkm, D 0.07 pskmnm,.35 W 1 km 1, 0.1 dbkm, and a diersion compensating fiber Ž DCF. with l 13.6 km, D 95 psnmkm, D 0.07 pskmnm, 5W 1 km 1, 0.6 dbkm, is used for compensation. In both figures, the wavelength of exact compensation is halfway between channels. The figures show both the power of the pump Ž top. and the phase of the probe Ž bottom. at the end of the system. The pattern is visible in the pump intensity. In the phase plots, the theoretical predictions of the model, obtained by Ž 14., are shown in bold dashed lines. The right column refers to a system with channel acing 1 nm, while the left column refers to a system with 0.1 nm. In both cases, the pump intensity distortion induced by GVD is quite evident. For the 1 nm acing case, the match between theory and simulations is excellent. This is due to the low-pass filtering effect connected to GVD-induced channel walk-off. For example, in Fig. 6 for 1-nm acing the filter bandwidth, as seen from Fig., is Bw 3.85 GHz, which is enough to mildly smooth out in the probe phase the original 10 Gbs NRZ pump modulation and to substantially filter away from

11 IMPULSE RESPONSE OF CROSS-PHASE MODULATION FILTERS 381 the modulating intensity the GVD-induced high-eed phase-to-intensity Ž PMIM. fluctuations. ' 1 c In fact, the peak frequencies of the PMIM ectrum are 7 fp n ' LD, where c is the eed of light and n 1,,.... In our case, the lowest peak is at fp 19 GHz. The 0.1-nm acing case was selected to show how small the walk-off should be for the low-pass filtering not to be effective, although our simulations take into account only SPM and, and not FWM, which in this case may be the dominant impairment. For example, in Fig. 6 for 0.1-nm acing the filter bandwidth is 10 times larger than before, Bw 38.5 GHz, so that the NRZ probe modulation is passed with little distortion, while the high-eed PMIM fluctuations are a little less suppressed than in the previous case. In Fig. 7 we first note that the shape of the phase is reversed, since in the SMF-compensated case the sign of diersion and thus of walk-off is reversed. The conclusions we reach about the effectiveness of filtering are similar, although the discrepancy between model and simulation is larger. This seems at first sight surprising, since the walk-off in the transmission fiber is larger with a diersion of 17 psnmkm. However, although the filtering is better, the intensity fluctuations due to PMIM conversion at the time points perturbated by and SPM have substantial frequency components below 10 Gbs. In this case in fact we get the lowest PMIM peak at fp 7 GHz. Such components are not filtered by walk-off and account for the larger error in the model. V. APPLICATIONS The explicit expression Ž 14. of the impulse reonse ˆh Ž t. is fundamental to analytically evaluating the transmission impairments induced by in multi-an systems. Here, we use Ž 14. to give an evaluation of the ectral broadening due to in multi-an compensated systems. Suppose A Ž 0, t.ž p 1,...,N. p is an OOKNRZ modulated signal, of the form A Ž 0, t. P 0 B Ž 0, t., where P Ž 0. p ' s s s is the signal mark power at the transmitter, and B 0, t Ý a Ž k. pž t kt. s k s s is the OOK modulating signal at the fiber input, where a Ž k. s is a sequence of independent random variables taking value 1 for mark and 0 for ace with equal probability, pt Ž. is the supporting pulse, T is the bit time, and s is a random delay uniformly distributed between T, T. It is shown in 8 that the autocorrelation of the output stationary process A Ž z, t. in Eq. Ž. is s R Ž. P Ž z. R Ž. R Ž., Ž 17. As s s where P z is the signal mark power at the receiver, s 1 R Ž. 1 Ž T s ž /

12 38 BONONI, FRANCIA, AND BELLOTTI is the contribution of the OOKNRZ modulation, being the triangular function 9, and R Ž. is the contribution. In the case of a large number N of input channels, the distribution of the random phase process Ž z, t. s becomes approximately Gaussian, so that we can write T R e Ž., 19 where Ž. is the variance of the process Ý B Ž 0, t. B Ž p s p p 0, t.p Ž. 0 ˆh Ž. p t. This variance can be analytically calculated and has a rather complicated but closed form 10. This computation requires explicit knowledge of the impulse reonse Ž 14.. The power ectrum, i.e., the Fourier transform of Ž 17., has been compared with computer simulations. Figure 8 shows the results for an 8-channel NZDF-compensated system with M 5 ans. The system parameters are those of section IV. The only differences are the bit rate R.5 Gbs, and the channel acing 0.4 nm. In the figure both the simulated and the analytically calculated power ectrum of the 4th channel are plotted, together with the input OOK ectrum. We can note a very good match between the simulation results and the analytical ones. VI. CONCLUSIONS In future high-eed IMDD WDM transmission systems with appropriate diersion management, after FWM has been properly suppressed, the main cause of eye closure and thus of system degradation will be the two remaining nonlinear Kerr effects, SPM and. It is thus important to be able to accurately predict, in order to estimate its local perturbation of the compensation of diersion, which gives rise to the residual intensity distortion and thus eye closure. This paper has given a time-domain derivation of the very important model for presented in 1. A simple and very general expression for the impulse reonse of the filters in the model has been presented. FIG. 8. Simulations of a NZDF compensated system, N 8, R.5 Gbs, M 5 ans, 0.4 nm.

13 IMPULSE RESPONSE OF CROSS-PHASE MODULATION FILTERS 383 While such filters are obtained in the assumption of undistorted field intensities on all channels, simulation results show that the model still accurately predicts the even in the presence of strong intensity distortion. This unobvious result implies that the modulation induced on the optical field phase by is nearly independent of the distortion induced on the intensity of the optical field by group velocity diersion. This is true in the cases shown here, bit rate 10 Gbs and wavelength acing larger than 0.1 nm, M 10 ans of about 90 km each, for both an NZDF and an SMF compensated system. The model clearly holds for higher bit rates, as the filter bandwidth is independent of the bit rate. For the same systems at a lower rate, we have to increase the number of ans to much larger M to observe a comparable distortion on the intensity. However the phase error between theory and simulation should remain small as in the 10 Gbs case, since the rapid fluctuations due to PMIM conversion get filtered as effectively as in the 10 Gbs case. Finally, as an application of the theory, we have shown that the ectral broadening induced by can be accurately predicted. ACKNOWLEDGMENT This work was supported by the European Community under INCO-DC Project DAWRON. REFERENCES 1 T. Chiang, N. Kagi, M. E. Marhic, and L. Kazovsky, Cross-phase modulation in fiber links with multiple optical amplifiers and diersion compensators, IEEE J. Lightwae Technol., vol. 14, 49 Ž D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, Dependence of cross-phase modulation on channel number in fiber WDM systems, IEEE J. Lightwae Technol., vol. 1, 885 Ž D. Marcuse, Theory of Dielectric Optical Waeguides, nd ed., chap. 9, Academic Press, San Diego, G. W. Agrawal, Nonlinear Fiber Optics, nd ed., chap. 7, Academic Press, San Diego, C. Kurtzke, Suppression of fiber nonlinearities by appropriate diersion management, IEEE Photon. Technol. Lett., vol. 5, 150 Ž A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, R. M. Derosier, C. R. Giles, B. M. Nyman, G. A. Ferguson, J. W. Sulhoff, and J. L. Zyskind, One-third terabits transmission through 150 km of diersion-managed fiber, IEEE Photon. Technol. Lett., vol. 7, 89 Ž J. Wang and K. Petermann, Small signal analysis for diersive optical fiber communication systems, IEEE J. Lightwae Technol., vol. 10, 96 Ž G. Bellotti, C. Francia, and A. Bononi, Spectral broadening due to cross-phase modulation Ž. in OOK WDM transmission systems, in Proceedings, LEOS 97 10th Annual Meeting, San Francisco, CA, pp. 67, paper WAA3, B. Carlson, Communication Systems, 3rd ed., p. 668, McGrawHill, New York, G. Bellotti, Laurea Thesis, Dip. Ing. dell Informazione, Universita ` di Parma, 1997.