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1160 J. Opt. Soc. Am. B/Vol. 24, No. 5 /May 2007 S. S. Carroll and J. Atai

Interaction of solitons in a stabilized dispersionmanaged link

Saul S. Carroll and Javid Atai

School of Electrical and Information Engineering, University of Sydney, Sydney, New South Wales 2006, Australia

Received July 24, 2006; revised December 27, 2006; accepted December 31, 2006;posted January 10, 2007 (Doc. ID 73341); published April 17, 2007

We investigate the interactions of dispersion managed (DM) solitons in a stabilized model where the stabili-zation is provided by coupling a lossy core to the system’s main fiber. It is found that interaction of solitons inthis model is quite complex in that they can attract or repel, depending on the initial phase and initial sepa-ration of solitons. An important finding is that bound states of DM solitons exist in this model. It is also foundthat collision distance increases as the net average dispersion increases from normal to anomalous. In the case

of D̄= +0.1, a region in the plane ��0 ,L� has been identified where periodic bound states of two solitons exist.© 2007 Optical Society of America

OCIS codes: 190.0190, 190.4370, 190.3270, 060.0060, 060.4370, 060.5530.

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. INTRODUCTIONispersion managed (DM) solitons have received muchttention due to their capability in radically improvinghe bit rate and quality of information transmission inber-optic telecommunication networks.1–16 A DM line isonstructed by concatenating alternating spans of anoma-ous and normal dispersion fiber so that the path-veraged dispersion is much smaller than the local dis-ersion. This results in suppression of the detrimentalffects of Gordon–Haus timing jitter2,17 and the efficiencyf four-wave mixing. A very interesting feature of DMolitons is that, unlike bright solitons in a standard non-inear Schrödinger equation, they propagate in a stable

anner in anomalous, zero, and normal averageispersions.12,18–20

Although dispersion management alleviates Gordon–aus jitter to some extent, sufficient suppression of this

ffect and interactions between solitons21 (both within ahannel and between the channels in the case of wave-ength division multiplexing) is only possible through these of guiding filters.22,23 Consequently, a considerablemount of research has been directed to study the com-ined effect of different filtering schemes and dispersionanagement on soliton propagation.24–27 An important

onsideration in systems where both DM and filtering areresent is finding regimes for which the solitons remaintable. For example, in the lumped amplification-filteringodel where the filters and amplifiers are considered as

iscrete elements that are inserted periodically in theink, inappropriate placing of filters in DM cells will re-ult in soliton instability.24,25

In Ref. 28, a model was proposed that supports com-letely stable propagation of DM solitons in the presencef filtering and amplification. The model consists of an ad-itional lossy core coupled to the main soliton-bearingore. This lossy core effectively couples out all the low-nergy waves from the main core, thus “cleaning” the

0740-3224/07/051160-6/$15.00 © 2

oliton.29,30 As was noted in Ref. 28, this model can be re-lized by simply inserting short segments of the dual-coreber periodically along the link. Such a configurationmounts to placing an additional core uniformly along theine. Since the coupling length is typically several centi-

eters, the segments’ lengths will not exceed a feweters.It should also be mentioned that this approach, i.e., sta-

ilization of solitons using an extra lossy core, was ini-ially proposed in a model of an ordinary link (constantispersion) with amplification and filtering.29–31 It wasemonstrated, analytically and numerically, that such aodel supports completely stable solitons. Stability and

nteraction of solitons in such a dual-core link have alsoeen investigated.31–33

A major source of performance degradation in long-aul, high-bit-rate dispersion managed links is the inter-ction between neighboring pulses either in the samehannel or in the different channels. Soliton interactionsccur when the separation between two soliton pulsesets close enough together, either in the same channel orifferent channels, that the pulses interact in a nonlinearanner. Such interactions can cause shifting of soliton

ulses within their time slot, which, in turn, may result inheir merger or decay.

In this paper, we numerically investigate and charac-erize the interaction of solitons in the same channel inhe model proposed in Ref. 28. It will be demonstratedhat the interaction dynamics of DM solitons in thisodel are quite complex. In particular, unlike the funda-ental nonlinear Schrödinger (NLS) solitons, interaction

f solitons is dependent on the initial values of the phaseifference and the initial separation between pulses. It islso shown that there exists a critical initial separationhat minimizes soliton interaction. In addition, it will behown that repulsion or attraction of the DM solitons doesot depend solely on the phase difference between them.

007 Optical Society of America

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S. S. Carroll and J. Atai Vol. 24, No. 5 /May 2007/J. Opt. Soc. Am. B 1161

n Section 2, the model is briefly reviewed. The interac-ion of solitons are presented and discussed in Section 3.

summary of the results is given in Section 4.

. MODELur starting point is the model introduced in Ref. 28,here it was assumed that the effective gain and filteringre uniformly distributed along the DM link that isoupled to a parallel lossy core:

iuz + � 12D�z� − i�1�u�� + �u�2u − i�0u + �v = 0, �1�

ivz + i�v + �u = 0, �2�

here u�z ,�� and v�z ,�� are the local amplitudes of thelectromagnetic waves in the main and added cores, z andare the propagation distance and the reduced time,34,35

nd �0 and �1 are, respectively, the effective gain and theltering coefficients in the main soliton-bearing core (as-uming a distributed approximation), and � is the loss pa-ameter of the added core. � represents the coupling con-tant between the cores, and is set equal to 1 by means ofn obvious scale transformation. In addition, the nonlin-ar coefficient may be set equal to 1 because there is aingle cubic term in the model. All other effects in theossy core, such as dispersion, nonlinearity, and filteringre negligible.29–31

The model assumes a uniform distribution of the effec-ive gain, filtering, and parallel lossy core along theength of the fiber link. However, as was previouslytated, this does not require a dual core to be present inhe entire length in the physical realization of the model,ather, short segments of dual-core fiber periodically in-talled in the link will suffice to create an effectively ho-ogeneous dual-core fiber link.28

The dispersion coefficient D�z� represents the variationn dispersion along the length of the main core. It shoulde noted that the sign of D�z� is opposite to that of theommonly used dispersion coefficient �2.34,35 We assume aeriodic dispersion map:

D�z� = �D+ � 0, 0 � z � L+,

D− � 0, L+ � z � L+ + L−,� �3�

hich is repeated with period 2L�L++L−. The averageispersion is given by

D̄ ��D+L+ + D−L−�

2L. �4�

The dispersion map is the main factor in understand-ng why DM solitons are beneficial for transmission. Aow-average dispersion helps reduce the Gordon–Hausiming jitter,2 while a high local dispersion helps suppresshannel cross talk.3,36

In Ref. 28 only stability of solitons in Eqs. (1) and (2) inhe case of full dispersion compensation, i.e., D̄=0, wasonsidered. However, in this paper, in addition to full dis-ersion compensation, in order to characterize the inter-ction dynamics of DM solitons in this model, we considerhe cases of anomalous average dispersion (D̄= +0.05 and¯ = +0.1), as well as normal average dispersion (D̄

−0.05 and D̄=−0.1). For all dispersion regimes, and toeduce the number of parameters, we consider the casehere L+=L−�L, and for the case of full dispersion com-ensation, we impose the same normalization conditionss in Ref. 10:

LD+ = L�D−� � 1, �5�

hile for the net anomalous and net normal dispersion re-imes, we only set

LD+ � 1. �6�

To simulate the interaction of solitons, we numericallyolved Eqs. (1) and (2) by means of symmetrized split-stepourier method subject to the following initial conditions:

u��,0� = ��� −�

2,0� + ��� +

�

2,0�exp�i�,

v��,0� = ��� −�

2,0� + ��� +

�

2,0�exp�i�, �7�

here � and are the initial separation and phase dif-erence between the solitons, respectively. ��� ,z� and�� ,z� are the exact solutions for the dual-core model withonstant dispersion given by31

���,z� = Asech����1+i eikz,

���,z� = Bsech����1+i eikz,

here

A2 = 12 �2 − 2� + �3 2�1��2,

B =A

k − i�,

ig. 1. Region of stable transmission of DM solitons in theodel of Eqs. (1) and (2) with D̄=0, L−=L+�L, �1=0.3, and �1.35.

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1162 J. Opt. Soc. Am. B/Vol. 24, No. 5 /May 2007 S. S. Carroll and J. Atai

=��9 + 8�2�1�2� − 3

4�1,

�2 =8�0�1

8�12 + 3 − �9 + 32�1

2�1 −�

�0k2 + �2� ,

nd k is determined by solving the following cubic equa-ion:

k�k2 + �2� − k =�9 + 32�1

2

2�1�0k2 − ��1 − ��0�.

ig. 2. Interaction of solitons corresponding to D̄=0, L=1.5, �00.68, �1=0.3, �=1.35, and =�. In (a) since �=6��cr, theolitons initially repel each other until they reach a separation of�cr=6.86 and in (b) they attract since �=8��cr.

Numerical simulations were performed for differentalues of L in the range 0.5�L�4 [and their correspond-ng dispersion values as per Eqs. (5) and (6)] with �0 vary-ng from 0.64 to 0.74. �1 and � were kept constant at 0.3nd 1.35, respectively, to ensure that the necessary condi-ions for stability of solitons are satisfied.28–31 The phaseifference was set at either =0 or =�, and the sepa-ation between the pulses was varied from 4 to 8 (normal-zed units).

. INTERACTION OF DISPERSIONANAGED SOLITONS

n Ref. 28, a stability domain for the DM solitons in Eqs.1) and (2) for the case of D̄=0 in the parametric plane�0 ,L� was identified. This region of stability, which ishown in Fig. 1, was our starting point in the analysis ofnteractions of DM solitons in this model. Simulations

ig. 3. Separation of the pulses versus propagation distance forifferent initial separations. The parameters are D̄=−0.05, L1.0, �0=0.68, �1=0.3, �=1.35. (a) =0; (b) =�. It is inter-sting to note that in (b) the pulses with an initial separation ofeventually evolve to � =6.3.

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S. S. Carroll and J. Atai Vol. 24, No. 5 /May 2007/J. Opt. Soc. Am. B 1163

ere systematically performed for a number of points inhis stability region. The points were selected to provide aariety of values, but also to enable a comparison of thehange in one variable, e.g., L staying the same and �0arying and vice versa. To get a more complete picture,e also investigated, for the same points, the interactionf solitons for the cases of net anomalous (D̄= +0.05, D̄=0.1) and net normal (D̄=−0.05, D̄=−0.1) dispersions.he initial separation of pulses ranged from 4 to 8 and theulses were either in phase or � out of phase.A principal issue here is whether the DM solitonserge, and if so, at what point during the propagation theerger occurs. Apart from the fact that this sheds light onechanism(s) involved in such phenomena, it has impli-

ations for real life networks: The system should be de-igned in such a way that the DM solitons do not merge,ecay, or repel. In addition, we are interested in determin-ng how interactions and their outcomes depend on �,, �0, D̄, and L. In particular, it is important to consider

he effect of these parameters on the collision distance,coll. zcoll is defined as the distance along the link wherehe two DM soliton pulses no longer retain an indepen-ent identity.8

A general characteristic observed in the simulationsas that as a result of attraction or repulsion, the solitons

each a certain critical separation ��cr� and propagate forome distance without noticeable interaction until a par-icular collision distance, zcoll, where they collide and sub-equently either decay or merge into a single soliton. Thevolution of solitons to a fixed separation is indicative ofhe existence of bound states in the system. Indeed, ouresults show that at fixed values of the model’s param-ters and within a certain range of � (see below), theolitons always evolve to the same �cr. This clearly dem-nstrates that stable bound states of two stable station-ry pulses do exist in this model. It should be mentionedhat multipulse bound states have been shown to exist inarious models including a model similar to Eqs. (1) and2), but with constant dispersion37 and in modified nonlin-ar Schrödinger equation with high-order filtering

ig. 4. Separation of the pulses versus propagation distance forifferent initial separations and =�. The parameters are D̄+0.05, L=1.0, �0=0.68, �1=0.3, and �=1.35.

erms.38 In DM systems, formation of bound states haseen studied in a cubic-quintic nonlinearity with a com-lex dispersion map and nonlinear gain39 and in multi-hannel DM links.40 In this paper, we report the existencef bound states of dispersion managed solitons belongingo the same channel in Kerr nonlinear media.

Based on the above results, one may conjecture that thenteraction of solitons will be virtually nonexistent if theyre initially separated by �cr. Our simulations have con-rmed this conjecture. This result has significant implica-ions for applications particularly because it shows thatolitons may propagate without any noticeable interac-ion for many DM map periods (�100 DM map periods).

Another important feature of interactions is that, un-ike the NLS-type solitons and ordinary DM solitons, thenitial attraction or repulsion is dependent upon both nd �. In particular, when =�, if initial separation ofolitons is greater than �cr, the solitons attract, and if���cr, the solitons repel. A typical example of the in-

eraction of solitons demonstrating this behavior for thease of D̄=0 is displayed in Fig. 2. On the other hand,

ig. 5. Dependence of zcoll on the average dispersion for (a) �, L=2.5, �0=0.725, �1=0.3, and �=1.35; (b) =0, L=0.8, �00.65, � =0.3, and �=1.35.

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1164 J. Opt. Soc. Am. B/Vol. 24, No. 5 /May 2007 S. S. Carroll and J. Atai

hen =0, the opposite occurs, i.e., if initially ���cr,he solitons repel each other, and they attract if 4��

�cr. For ��4, the solitons interact strongly beforeeaching zcoll=20, leading to either their decay (most com-on for =�) or merger (most common for =0).The interaction dynamics described above is shown in

ig. 3 for D̄=−0.05. It is worth noting that in the case ofonstant normal dispersion, it has been shown that in theodel of coupled Ginzburg–Landau (GL) equations,-out-of-phase solitons attract.32 However, Fig. 3 shows

hat in the presence of dispersion management, both re-ulsion and attraction may occur depending on the initialhase and separation. To the best of our knowledge, suchnteraction dynamics have never been reported before forLS- or GL-type models.We have also considered the dependence of �cr on the

alue of net average dispersion D̄. Our results show thathis dependence is weak. For example, the value of �cr

or D̄= +0.05 is 6.2 (Fig. 4), and that for D̄=−0.05 is 6.3Fig. 3(b)]. In a number of cases, �cr was found to be theame for different values of D̄.

As shown in Fig. 3, varying the initial separation ofolitons results in a different zcoll. We have conducted ex-ensive simulations to determine how various parametersffect zcoll. We have found that as the net dispersion in-reases from D̄=−0.1 to D̄= +0.1, zcoll also increases. Fig-re 5 displays this trend for two different points in thetability region for various values of � and . This re-ult will be useful in selecting a dispersion regime, whichesults in the largest propagation distance without inter-ction, owing to the fact that �cr is weakly dependent onhe value of net average dispersion D̄.

Finally, we have also identified a region in the paramet-ic plane ��0 ,L� for D̄= +0.1, where solitons periodicallyttract and repel each other but never merge or decay (seeig. 6). Changing the initial results only in changinghe initial repulsion to an attraction, and vice versa, with-ut any change in the closest ��min� and farthest ��max�eparations. Changing the initial separation, providedhat it is in the range �min����max, does not affecthe period. This clearly indicates the existence of periodicound states in the model.

. CONCLUSIONe have investigated the interaction of dispersion man-

ged solitons in a stabilized system where stabilization ischieved by periodically coupling the system’s main coreo short segments of a parallel lossy core. It is found thathe interaction dynamics of DM solitons in this model areery complex. In particular, the interactions depend onhe initial phase and separation of solitons. We have nu-erically demonstrated that bound states of DM solitons

xist in this model. It is also found that separation ofulses when the bound state is formed is weakly depen-ent on the value of net dispersion. On the other hand,he collision distance increases as the net average disper-ion varies from normal to anomalous. In the case of D̄+0.1, we have found a region in the parametric plane,

� ,L�, where the solitons exhibit periodic repulsion and

ig. 6. (a) Example of periodic attraction and repulsion of soli-ons for �=7, =0, D̄= +0.1, L=2.5, �0=0.69, �1=0.3, and �1.35; (b) separation of the pulses versus the propagation dis-

ance. (c) shows the region in the plane ��0 ,L� where such peri-dic behavior has been observed. Other parameters in (c) are

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S. S. Carroll and J. Atai Vol. 24, No. 5 /May 2007/J. Opt. Soc. Am. B 1165

ttraction without merging or decaying. This shows thathe model supports periodic bound states.

The authors can be reached via e-mail ataulc@ee.usyd.edu.au and atai@ee.usyd.edu.au.

EFERENCES1. B. A. Malomed, Soliton Management in Periodic Systems

(Springer, 2006).2. M. Suzuki, I. Morita, N. Edagawa, S. Yamamoto, H. Haga,

and S. Akiba, “Reduction of Gordon-Haus timing jitter byperiodic dispersion compensation in soliton transmission,”Electron. Lett. 31, 2027–2029 (1995).

3. N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I.Bennion, “Enhanced power solitons in optical fibres withperiodic dispersion management,” Electron. Lett. 32, 54–55(1996).

4. G. M. Carter, J. M. Jacob, C. R. Menyuk, E. A.Golovchenko, and A. N. Pilipetskii, “Timing-jitter reductionfor a dispersion-managed soliton system: experimentalevidence,” Opt. Lett. 22, 513–515 (1997).

5. M. Matsumoto, “Effects of guiding filters on stretched-pulsetransmission in dispersion-managed fibres,” Electron. Lett.33, 1718–1720 (1997).

6. I. Gabitov, E. G. Shapiro, and S. K. Turitsyn, “Optical pulsedynamics in fibre links with dispersion compensation,” Opt.Commun. 134, 317–329 (1997).

7. Y. Chen and J. Atai, “Average dark soliton dynamics inperiodically dispersion compensated fibre transmissionsystems,” IEEE Photon. Technol. Lett. 10, 1280–1282(1998).

8. T. Yu, E. A. Golovchenko, A. N. Pilipetskii, and C. R.Menyuk, “Dispersion-managed soliton interactions inoptical fibers,” Opt. Lett. 22, 793–795 (1997).

9. T. Georges, “Soliton interaction in dispersion-managedlinks,” J. Opt. Soc. Am. B 15, 1553–1560 (1998).

0. T. I. Lakoba, J. Yang, D. K. Kaup, and B. A. Malomed,“Conditions for stationary pulse propagation in the strongdispersion management regime,” Opt. Commun. 149,366–375 (1998).

1. A. Berntson, N. J. Doran, W. Forysiak, and J. H. B. Nijhof,“Power dependence of dispersion-managed solitons foranomalous, zero, and normal path-average dispersion,”Opt. Lett. 23, 900–902 (1998).

2. J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox,“Stable soliton-like propagation in dispersion managedsystems with net anomalous, zero and normal dispersion,”Electron. Lett. 33, 1726–1727 (1997).

3. D. S. Govan, W. Forysiak, and N. J. Doran, “Long-distance40-Gbit/s soliton transmission over standard fibre by use ofdispersion management,” Opt. Lett. 23, 1523–1525 (1998).

4. D. J. Kaup, B. A. Malomed, and J. Yang, “Interchannelpulse collision in a wavelength-division-multiplexed systemwith strong dispersion management,” Opt. Lett. 23,1600–1602 (1998).

5. A. N. Niculae, W. Forysiak, A. J. Gloag, J. H. B. Nijhof, andN. J. Doran, “Soliton collisions with wavelength-divisionmultiplexed systems with strong dispersion management,”Opt. Lett. 23, 1354–1356 (1998).

6. L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Solitonpropagation in long fibres with periodically compensatedloss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).

7. J. P. Gordon and H. A. Haus, “Random walk of coherentlyamplified solitons in optical fiber transmission,” Opt. Lett.11, 665–667 (1986).

8. S. K. Turitsyn and E. G. Shapiro, “Dispersion-managedsoliton in optical amplifier transmission systems with zeroaverage dispersion,” Opt. Lett. 23, 682–684 (1998).

9. J. N. Kutz and S. G. Evangelides, Jr., “Dispersion managedbreathers with average normal dispersion,” Opt. Lett. 23,685–687 (1998).

0. V. S. Grigoryan and C. R. Menyuk, “Dispersion-managedsolitons at normal average dispersion,” Opt. Lett. 23,609–611 (1998).

1. B. A. Malomed, “Suppression of soliton jitter andinteractions by means of dispersion management,” Opt.Commun. 147, 157–162 (1998).

2. H. A. Haus, Y. Lai, A. Mecozzi, and J. D. Moores, “Solitontransmission control,” Opt. Lett. 16, 1841–1843 (1991).

3. Y. Kodama and A. Hasegawa, “Generation ofasymptotically stable optical solitons and suppression ofthe Gordon-Haus effect,” Opt. Lett. 17, 31–33 (1992).

4. M. Matsumoto, “Analysis of filter control of dispersion-managed soliton transmission,” J. Opt. Soc. Am. B 15,2831–2837 (1998).

5. M. Matsumoto, “Instability of dispersion managed solitonsin a system with filtering,” Opt. Lett. 23, 1901–1903 (1998).

6. L. F. Mollenauer, P. V. Mamyshev, and J. P. Gordon, “Effectof guiding filters on the behavior of dispersion-managedsolitons,” Opt. Lett. 24, 220–222 (1999).

7. A. Bernston and B. A. Malomed, “Dispersion managementwith filtering,” Opt. Lett. 24, 507–509 (1999).

8. J. Atai and B. A. Malomed, “Stabilized scheme fordispersion management,” J. Opt. Soc. Am. B 17, 1134–1139(2000).

9. B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368(1996).

0. J. Atai and B. A. Malomed, “Stability and interactions ofsolitons in two-component active systems,” Phys. Rev. E 54,4371–4374 (1996).

1. J. Atai and B. A. Malomed, “Exact stable pulses inasymmetric linearly coupled Ginzburg-Landau equations,”Phys. Lett. A 246, 412–422 (1998).

2. E. Efremidis, K. Hizanidis, B. A. Malomed, H. E.Nistazakis, and D. J. Frantzeskakis, “Stable transmissionof solitons in the region of normal dispersion,” J. Opt. Soc.Am. B 17, 952–958 (2000).

3. H. E. Nistazakis, D. J. Frantzeskakis, J. Atai, B. A.Malomed, E. Efremidis, and K. Hizanidis, “Multichannelpulse dynamics in a stabilized Ginzburg-Landau system,”Phys. Rev. E 65, 036605 (2002).

4. G. P. Agrawal, Nonlinear Fibre Optics (Academic, 1995).5. A. Hasegawa and Y. Kodama, Solitons in Optical

Communications (Oxford U. Press, 1995).6. C. Kurtze, “Suppression of fiber nonlinearities by

appropriate dispersion management,” IEEE Photon.Technol. Lett. 5, 1250–1253 (1993).

7. J. Atai and B. A. Malomed, “Bound states of solitary pulsesin linearly couples Ginzburg-Landau equations,” Phys.Lett. A 244, 551–556 (1998).

8. S. Wabnitz and E. Westin, “Optical fiber soliton boundstates and interaction suppression with high-orderfiltering,” Opt. Lett. 21, 1235–1237 (1996).

9. N. Akhmediev, F. Zen, and P. L. Chu, “Pulse-pulseinteraction in dispersion managed fiber systems withnonlinear amplifiers,” Opt. Commun. 201, 217–221 (2002).

0. B. Feng and B. A. Malomed, “Bound states of solitonsbetween close wavelength-separated channels in adispersion-managed fiber-optic link,” Opt. Commun. 219,143–156 (2003).