January 15, 2000 / Vol. 25, No. 2 / OPTICS LETTERS 93

Disintegration of a soliton in a dispersion-managed opticalcommunication line with random parameters

F. Kh. Abdullaev and B. B. Baizakov

Physical-Technical Institute of the Uzbek Academcy of Sciences, G. Mavlyanov Street 2-b, 700084 Tashkent, Uzbekistan

Received September 15, 1999

The propagation of dispersion-managed solitons in optical f iber links with a random dispersion map has beenstudied. Two types of randomness are considered: random dispersion magnitudes and random lengths of thespans. By numerical simulations, disintegration of a soliton propagating in such an optical communication lineis shown to occur. It is observed that the stability of the soliton propagation is affected more by modulationsof the dispersion magnitudes of the spans than by modulations of the span lengths. Results of numericalsimulations of the soliton breakup distance confirm theoretical predictions in the averaged dynamics limit. 2000 Optical Society of America

OCIS codes: 060.5530, 260.2030, 000.4430.

Since the discovery of the dispersion-managed (DM)solitons in optical fiber links with alternating anoma-lous and normal group-velocity dispersion,1 therehave been significant advances in understandingthese solitons’ basic properties. Increasing researchefforts have been motivated by enhancements in theperformance characteristics of optical communicationlines when DM solitons are used as information bitcarriers. These enhancements are due to the factthat DM solitons are less inf luenced by Gordon–Haustiming jitter than conventional solitons and can bemore closely packed because of their specific pulseprofile. Among the other advantages of DM solitons isthe fact that four-wave mixing effects are signif icantlysuppressed.2 The advantages of DM solitons comefrom the fact that they possess higher pulse energythan conventional solitons in a uniform fiber with thesame path-averaged dispersion.

An optical transmission line based on the dispersion-compensation technique represents a chain of pe-riodically linked pieces of fibers with alternatinganomalous and normal group-velocity dispersion. It isreasonable to suggest that in practical situations corre-sponding fiber pieces will not be identical with respectto either dispersion magnitude or length. Most likelythere will be a random distribution of these parametersover certain mean values. As a result of breaking ofperiodicity, a DM soliton suffers random perturbationsalong its path and eventually disintegrates after somepropagation distance. The question of randomnessin DM communication systems was addressed for thefirst time to our knowledge in Ref. 3, in which thepropagation of solitons in fiber links with f luctuatinggroup-velocity dispersion of the spans was considered.

Our objective in this study is the theoretical and nu-merical investigation of DM soliton dynamics in opticaltransmission lines with a random dispersion map whenthe dispersion magnitudes of the spans or the spanlengths are randomly distributed over certain meanvalues. The main question to be addressed is the dis-tance that a DM soliton covers before it disintegratesas a result of randomness of the dispersion map. Thepresent study deals with the lossless case.

0146-9592/00/020093-03$15.00/0

The propagation of DM solitons is governed by thenonlinear Schrodinger equation for a dimensionlessenvelope of the electric field:

iuz 1d�z�2

utt 1 juj2u � 0 , (1)

where d�z� is a stepwise function describing the dis-persion map, which in turn can be represented asconsisting of periodic d0�z� and random d1�z� parts,d�z� � d0�z� 1 d1�z�. In the absence of randomnessthis function would be a periodic function, d0�z� �d0�z 1 z1 1 z2�, where z1 and z2 are the fiber segmentlengths. We are concerned with the Gaussian white-noise model for d1�z�:

�d1� � 0 , �d1�z1�d1�z2�� � 2s2d�z1 2 z2� , (2)

where s � dd�d0 and s � dz�z1,2 represent thestandard deviation of the dispersion magnitudes andthe span lengths, respectively.

Employing the variational approach,4 we note thatthe underlying nonlinear Schrodinger equation is re-duced to a system of ordinary differential equations forDM soliton parameters. The trial localized waveformis taken as

u�z, t� � A�z�F∑

ta�z�

∏exp�if�z� 1 ib�z�t2� , (3)

where A, a, b, and f are the complex amplitude, width,chirp parameter, and phase, respectively, and F �t�a�z��is the localized function specifying the pulse profile.Here the conserving quantity is

Z `

2`

juj2dt � E � const. (4)

Performing standard calculations,4 we arrive at thefollowing variational equations:

az � 2d�z�ab , bz �C1d�z�

a4 2C2

a3 2 2d�z�b2, (5)

where

2000 Optical Society of America

94 OPTICS LETTERS / Vol. 25, No. 2 / January 15, 2000

C1 �

R`

2` jF j2dt

2R`

2` t2jF j2dt, C2 �

R`

2` jFtj4dt

4R`

2` t2jF j2dt,

t �ta

. (6)

For the Gaussian ansatz F �t� � exp�2t2�, one obtainsC1 � 2 and C2 � 1�

p2, and for F �t� � sech�t� the cor-

responding values are C1 � C2 � 2�p2.We performed numerical simulations of the pulse

propagation in optical fiber links with a random dis-persion map, using both the full nonlinear Schrodingerequations (1) and the corresponding variational equa-tions (5). The fiber segments are assumed to havegroup-velocity dispersions of d1 � 2.446 ps��nm km�and d2 � 22.258 ps��nm km� and lengths z1 � z2 �50 km. The corresponding path-averaged disper-sion is d � �d2z2 1 d1z1���z2 1 z1� � 0.094 ps��nm km�. Taking ts � 10 ps for the pulse duration, wehave a dispersion distance z0 � 270 km, correspondingto d. Obviously, it is convenient to use normalized pa-rameters D1 ! d1�d, D2 ! d2�d, and z1,2 ! z1,2�z0and a dimensionless distance normalized to z0.

Perturbations experienced by a soliton propagatingalong fiber links with randomly varying dispersionmagnitudes of the spans or the span lengths will resultin continuous broadening of the soliton. The meansquare of the soliton’s width (at the starting point ofeach unit cell), normalized to its initial value, is shownin Fig. 1. One may concluded from this figure thata DM soliton can propagate more than 100 unit cellswithout noticeable distortion, even with large enoughrandom deviations of dispersion magnitudes of spanss � Dav . This conclusion is consistent with the resultreported in Ref. 3.

To compare the relative weights of two kinds ofdisorder in DM systems, randomness of dispersionmagnitudes of the spans and the span lengths in thedisintegration of a soliton, we calculated the growthrate for the energy, H � D�z� �h2�z� 1 b2�z��h2�z�� 22h�z�, h � 1�a, which is a normalized constant ofthe motion of variational equations in the absence ofrandomness.5 It should be pointed out that the initialvalues of the soliton amplitude h0 � 1.225 and chirpb0 � 0 are taken to correspond to stationary points onthe phase plane of the system of Eqs. (5). Figure 2shows that randomness of dispersion magnitudes ismore detrimental to the stability of a DM soliton thanis randomness of span lengths at a standard devia-tion s � 0.06 for both cases.

It is of interest to compare the robustness of DMand conventional solitons against random f luctuationsof the fiber dispersion. Figure 3 illustrates this issueand suggests that DM solitons are notably more stable.

When the fiber span lengths are small comparedwith the dispersion magnitudes (both in dimensionlessunits), the averaged dynamics approach is valid.5 Inthis approach a real map with fiber segments of alter-nating anomalous and normal dispersion is replacedwith a uniform fiber with path-averaged dispersionDav . In this formulation the problem can be consid-ered a particular case of the random Kepler prob-lem in the context of optical solitons, as recently

reported.6 We note that the Kepler problem and itsrelation to optical soliton systems has already beenexplored.7,8 In Ref. 6, by use of the action-angle vari-ables, an explicit analytical expression was derivedfor the expected distance Ld that a soliton propagatesalong a fiber with randomly varying dispersion beforeit disintegrates:

Ld �1

s2E4

Z `

0

JA�J�

dJ 1.68s2E4

,

A�J� �J�pJ 1 4�

8p3�pJ 1 2�3�128 1 448pJ 1 448p2J2

1 168p3J3 1 21p4J4� , (7)

Fig. 1. Relative mean-squared width of the pulse propa-gating in a DM line with randomly varying dispersionmagnitudes of spans. The dispersion map parameters areD1 � 26, D2 � 224, and z1 � z2 � 0.14. The curves,obtained from Eqs. (5), are averaged over 400 realizations.

Fig. 2. Growth of the energy �H � when the dispersionmagnitudes of the spans and the span lengths are randomlymodulated with standard deviation s � 0.06. The disper-sion map parameters are D1 � 1, D2 � 21, z1 � 1.0, andz2 � 0.1. The curves, obtained from Eqs. (5), are averagedover 400 realizations.

January 15, 2000 / Vol. 25, No. 2 / OPTICS LETTERS 95

Fig. 3. Decay of the normalized pulse amplitude as aresult of randomness of the fiber dispersion, according toEqs. (5). In the DM case, fiber spans are assumed to haveD1 � 1, D2 � 21, z1 � 1.0, z2 � 0.1, and D1,2, randomlymodulated with standard deviation s � 0.06. For theconventional case the fiber dispersion is taken as randomlyvarying around the corresponding path-averaged value Davwith the same value of s � 0.06. Stochastic equations (5)are averaged over 400 realizations.

Fig. 4. Mean distance at which a soliton disintegrates, cal-culated in the averaged dynamics limit and E � 1, for dif-ferent strengths of random modulation of fiber dispersion(according to Eqs. (5), averaged over 400 realizations).

where J is the action variable.6 So for high-energypulses the decay length is strongly reduced. A com-parison of our numerical simulation results with theprediction of the above-described theory6 is shown inFig. 4. The criterion H � 0 (when the separatrix tra-jectory is reached starting from the value H , 0, corre-sponding to the stationary point h0, b0) is accepted forcomplete disintegration of a soliton.

Typical behavior of a soliton propagating along theDM line with randomly varying dispersion magnitudes

Fig. 5. Disintegration of a soliton propagating in a DMline with randomly varying dispersion magnitudes ofspans. This particular realization is for Eq. (1) with aGaussian pulse u�0, t� � 2.45 exp�20.427t2�. The disper-sion map parameters are D1 � 26, D2 � 224, z1 � z2 �0.14, and s � 0.06.

of the spans obtained by numerical solution of Eq. (1)is shown in Fig. 5. Good agreement with the resultsof variational equations (Fig. 4) with regard to thesoliton breakup distance, Ld � 7500 km, at s � 0.06can be seen from this figure. It should be pointedout that the agreement between predictions based onthe variational approach and numerical solution ofthe full nonlinear Schrodinger equation becomes poorwhen emission of linear waves by a soliton is involved.This poor agreement corresponds to near-separatrixtrajectories of variational equations.

In conclusion, we have performed numerical simu-lations of DM soliton propagation in fiber links withrandomly varying map parameters. Soliton disinte-gration owing to randomness is shown to occur in suchan optical communication line. However, disintegra-tion within reasonable distances ��100z0� occurs onlywith large enough deviations �s . 0.04� of the map pa-rameters from the corresponding mean values for D1,2and z1,2. Good agreement between the results of nu-merical simulations of the soliton breakup distance andthe corresponding theoretical prediction in the aver-aged dynamics limit has been demonstrated.

B. Baizakov’s e-mail address is baizakov@physic.uzsci.net.

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