transmission of pulses in a dispersion-managed fiber link with extra nonlinear segments

Optics Communications 245 (2005) 227–236

www.elsevier.com/locate/optcom

Transmission of pulses in a dispersion-managed fiber linkwith extra nonlinear segments

Rodislav Driben a,*, Boris A. Malomed a, P.L. Chu b

a Department of Interdisciplinary Studies, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University,

Tel Aviv 69978, Israelb Optoelectronics Research Centre, Department of Electronic Engineering, City University of Hong Kong, Tat Chee Avenue,

Kowloon, Hong Kong

Received 14 June 2004; received in revised form 18 October 2004; accepted 20 October 2004

Abstract

We introduce an extended version of the dispersion-management (DM) model, which includes an extra nonlinear

element, and consider transmission of return-to-zero pulses in this system (they are not solitons). The pulses feature

self-compression, accompanied by generation of side peaks (in the temporal domain). An optimal transmission dis-

tance, zopt, is identified, up to which the pulse continues to compress itself (the eventual width-compression factor is

.2), while the amplitude of the side peaks remains small enough. The distance zopt virtually does not depend on the

strength S of the DM part of the system in the interval 1.5 < S < 11, but it is sensitive to the nonlinearity strength

in the extra segment. The system provides essentially stronger suppression of the noise-induced jitter of the pulses than

the ordinary DM model. The most important issue is interaction between adjacent pulses, which is a basic difficulty in

the case of DM solitons. In a broad parameter region, the system provides effective isolation between pulses. The min-

imum initial temporal distance between them, necessary for the isolation, is quite small, slightly larger than 1.5 the

pulse�s width. The transmission actually improves the quality of multi-pulse arrays, as it leads to deepening of hiatuses

between originally overlapping pulses.

� 2004 Elsevier B.V. All rights reserved.

PACS: 42.79.�e; 42.79. FzKeywords: Return-to-zero transmission format; Pulse jitter suppression; Inter-symbol interference; Nonlinearity management

0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2004.10.052

* Corresponding author. Tel.: +972 50 7287 645; fax: +972

364 101 89.

E-mail address: [email protected] (R. Driben).

1. Introduction

The potential offered by the technique of disper-sion management (DM) for enhancement of data

transmission in fiber-optic telecommunications in

ed.

mailto:[email protected]

228 R. Driben et al. / Optics Communications 245 (2005) 227–236

the return-to-zero (RZ) format, i.e., by streams of

separated pulses, is well known (see a recent review

[1]). Recently, this potential was realized in the

first commercial long-haul (2900 km) link built in

Australia [2]. A universal characteristic of theRZ-DM regime is the DM strength (see, e.g., [3]),

S ¼ bnLn þ baj jLa

T 2FWHM

; ð1Þ

where Ln and La are lengths of the periodically

alternating segments with the normal and anoma-

lous group-velocity-dispersion (GVD) coefficients,

bn > 0 and ba < 0, respectively, and TFWHM is the

standard temporal width of the pulse.

In most cases, the data-carrying pulses in DMsystems are assumed to be DM solitons [3], i.e.,

wave packets localized in the temporal domain,

which feature strictly periodic oscillations, without

any systematic degradation, even over extremely

large transmission distances. In terms of applica-

tions, the use of the periodic DM solitons is not

a mandatory condition: irreversible evolution of

RZ pulses may be tolerated, provided that it isslow enough, and does not lead to unacceptable

deterioration of the pulse�s characteristics (this

means, for instance, that the pulses do not become

too broad). An example of a beneficial effect due to

systematic evolution of the pulse is a possibility to

enhance the suppression of the Gordon–Haus jit-

ter of the solitons, transmitting them through a

‘‘hyperbolic’’ link, with a slowly varying (�1/z,where z is the propagation distance) local path-av-

erage GVD coefficient, �b ¼ ðbnLn� j ba j LaÞ=ðLn þ LaÞ [4].

A well-known problem, which restricts the use

of the DM solitons to the range of weak DM, with

S[ 2.5, is interaction between the solitons

belonging to the same channel (analysis of the

interaction was elaborated in [5]). The source ofthe problem is that, in the strong-DM regime,

the solitons periodically spread out, which leads,

through their overlapping and formation of

‘‘ghost’’ pulses, to accumulation of mutual distor-

tion induced by the four-wave mixing [6].

Thus, there are strong reasons to look for RZ

regimes in the DM links which use pulses that

are different from solitons. This line of the theoret-ical and experimental studies has recently drawn

renewed interest [7,8]. The objective of the present

work is to put forward a promising scheme, in

which non-soliton pulses are transmitted through

a link composed of DM cells including an extra

segment, with strong nonlinearity and negligibleGVD. In fact, the system of this type may be re-

garded as a combination of the standard DM

one and the split-step model (SSM), that was intro-

duced in [9,10]. The SSM is a periodic concatena-

tion of fiber segments which are assumed,

respectively, purely dispersive and purely nonlin-

ear; as well as its DM counterpart, the SSM sup-

ports transmission of its own species of stablesolitons in a broad range of parameters [9,10].

It should be mentioned that, while a majority of

theoretical and experimental studies of the pulse

transmission were dealing with two-segment DM

maps, some works reported that further improve-

ment of the system�s performance can be achieved

with more sophisticated three-segment maps. For

instance, in a recent paper [11] it was demonstratedthat a three-segment map may be especially effi-

cient in suppressing effects of collisions between

pulses (not necessarily solitons) belonging to differ-

ent channels in the WDM (wavelength-division-

multiplied) version of the DM system.

In this work, we demonstrate that a three-step

map including a strongly nonlinear element se-

cures transmission of pulses over long (althoughlimited) distances with improvement of their

shape (gradually making them taller and nar-

rower). The model also provides stronger jitter

suppression than its usual DM counterpart. It is

quite important too that the new model sup-

presses the interaction between the pulses, and

helps to resolve the ISI (inter-symbol interference)

problem, providing for an effective isolation ofadjacent pulses that were overlapping at the

launch point, z = 0. The latter advantage may

be essential, in view of the above-mentioned fun-

damental difficulty encountered in the usual DM

schemes operating with solitons, viz., strong inter-

action effects in the cases when the DM strength

is not small.

The formulation of the model and results ob-tained for the single-pulse propagation in it,

including the jitter suppression, are given in Sec-

tion 2. Section 3 deals with the ISI suppression

R. Driben et al. / Optics Communications 245 (2005) 227–236 229

in two-pulse configurations, and Section 4 con-

cludes the paper.

2. The model and transmission of pulses in it

We consider a periodic single-channel system,

which is modeled by the general nonlinear Schro-

dinger equation for the local amplitude u of the

electromagnetic wave [3,1],

iuz �1

2bðzÞutt þ cðzÞ j uj2u ¼ 0, ð2Þ

where b(z) and c(z) are the local GVD and nonlin-

earity coefficients, and t is the local time variable

(defined so as to absorb the group-velocity term

in the equation, whose coefficient may also be a

function of z). The three-step map assumed in

the present model is composed of segments with

b; cf g ¼b1 ¼ 0; c1f g if 0 < z < L1;

b2; c0f g if L1 < z < L1 þ L2;

b3 ¼ �b2; c0f g if L1 þ L2 < z < L1 þ L2 þ L3:

8><>:

ð3ÞThe three-step cell (3) repeats itself with the period

L ” L1 + L2 + L3. Here, c0 and c1 are, respectively,the nonlinearity of the system fiber, and of the

additional strongly nonlinear segment.

The latter element may be realized in severaldifferent ways. One possibility is the use of a long

segment of a dispersion-shifted fiber, with a usual

value of the nonlinear coefficient and very weak

dispersion. Another realization may be based on

a short (less than a meter) piece of an Erbium-

doped fiber, which, if properly designed and

pumped, may feature the nonlinearity coefficient

larger by a factor of 5 Æ 105 than in the regular fiber[12]. Also quite short may be a segment of a pho-

tonic-crystal fiber, which can provide for very

strong nonlinearity with diverse arrangements of

its microstructure [13]. Lastly, the nonlinear ele-

ment must not necessarily be a piece of a fiber; in-

stead, it may be a compact module, based on a

second-harmonic-generating crystal, in which

strong cubic nonlinearity is induced by the quad-ratic one through the cascading mechanism, while

the module�s GVD may be completely neglected in

view of its small size. Elements of this type were

studied earlier in models of the ‘‘nonlinearity man-

agement’’ and nonlinear pulse shaping [8] (see also

[14]). As concerns the loss and gain terms, they are

not explicitly included in Eq. (2), following the

usual assumption of the local compensation ofthe fiber loss by lumped amplifiers.

The simulations were started by launching an

unchirped Gaussian pulse at the point z = 0,

u0ðtÞ ¼ffiffiffiffiffiP 0

pexp �t2=T 2

� �; ð4Þ

where P0 is the peak power, and the above-

mentioned standard temporal width, which ap-

pears in Eq. (1), is related to T in Eq. (4) as

T 2FWHM ¼ ð2 ln 2ÞT 2 � 1:386T 2. Following the

analogy with the definition of the DM strength

(1), we find it convenient to define a dimensionless

nonlinearity strength of the extra segment,

NS � c1P 0L1 ð5Þ(actually, it is the the nonlinear phase shift at the

center of the pulse passing the nonlinear segment).

Using a Gaussian ansatz for the pulse, similar

to Eq. (4) (but including chirp), it is possible to de-

scribe changes of parameters of a pulse passing a

system�s cell in an analytical form based on thevariational approximation (VA). This approxima-

tion was applied to both the DM (see, e.g., [15])

and SSM [10] systems. However, as the aim of

the present work is not to find a soliton-like solu-

tion, for which the VA is well adjusted, but rather

to investigate nonsteady propagation regimes, fur-

ther use of the VA would amount to multiple

numerical iterations of the single-cell transfer func-tion. A still more serious problem is that it would

be quite difficult to use a variational ansatz that

would allow us to predict the generation of side

peaks, which is the most essential feature limiting

the use of the scheme, as shown below. Therefore,

direct simulations of the full system are more rele-

vant than an analytical consideration.

Systematic simulations of the model with theinitial condition (4) demonstrate that, in a broad

range of parameter values, the propagation leads

to self-compression of the pulse, with simultaneous

generation of side-lobes attached to it in the tem-

poral domain, a typical example of which is dis-

played in Fig. 1(a). In fact, the pulse propagating

in the system performs nearly periodic shape

Fig. 1. A typical example of comparison of the input Gaussian pulse and the output one produced by the propagation through

z = zopt ” 1800 km, which is the optimal distance for the present case (see the definition in the text). The panels (a) and (b) display,

respectively, the distribution of the power and chirp in the pulses (in the latter subplot, the horizontal line refers to the zero chirp in the

input pulse). Note that the side-lobes of the output pulse contain, in this case, only 1.6% of the total energy.


oscillations, Fig. 1 showing the output pulse takenat a point where it is narrowest. In the DM model

per se, the narrowest pulse has zero chirp. In the

present case, this is not necessarily true, as the sys-tem is more complex, although it is plausible that

the pulse�s chirp is specially small when the width


of the pulse attains its minimum. Low chirp in the

output pulse is also an important quality charac-

teristic in some applications. To specify this fea-

ture, Fig. 1(b) shows the distribution of the local

chirp, /tt [/(t) is the phase of the complex fieldu(t)] in the same output pulse which is displayed

in Fig. 1(a).

In Fig. 1 and in the figures displayed below, ex-

cept for the parts which compare the results to those

in the ordinary DM model [parts of Figs. 4–6(b)],

the system�s parameters, which were selected to dis-

play a generic situation, are L1 = L2 = L3 = 20 km,

c0 = 0.5 (W km)�1, c1 = 4 (W km)�1, and b2 =�b3 = �7 ps2 km�1. As concerns the choice of L1

and c1, the actual control parameter is their product

(L1c1 = 80 W�1, in the present case). In fact, the

above-mentioned physical realizations of the non-

linear segment, using short Erbium-doped or pho-

tonic-crystal fibers, may correspond to a very

small L1 combined with a very large c1.The initial parameters of the pulse (4), which

were also selected to represent a practically rele-

vant generic case, are P0 = 1 mW and T = 6 ps

(the latter corresponds to TFWHM = 7.08 ps). The

Fig. 2. The same as in Fig. 1(a), but after the pulse has passed the dista

of the total energy.

cases intended to compare the findings with the re-

sults for the DM model without the nonlinear seg-

ment will differ by the choice of the lengths: L1 = 0,

L2 = L3 = 30 km, so that the total period remains

the same, L = 60 km.The self-compression of the pulse is quite an

obvious effect of the additional nonlinearity added

to the system. Note that the peak power of the in-

put pulse in the case shown in Fig. 1, 1 mW, is too

small for the formation of a soliton, but large en-

ough to make the nonlinearity effects significant.

Without the nonlinear segment inserted into the

DM map, no systematic reduction of the widthwas observed as a result of the transmission.

The example shown in Fig. 1, as well as many

others, suggest that, up to some value of the

transmission length, the overall quality of the

pulse improves, as its width gets reduced,

roughly, by a factor of 2; after that, although

the self-compression of the central body of the

pulse continues, its overall quality starts to dete-riorate due to the growth of the side peaks. A

trade-off between these two trends defines the

maximum acceptable (optimal) transmission

nce 1.67Lopt � 3000 km. In this case, the side-lobes contain 13%


length zopt, as the distance at which the peak

power in the side-lobes achieves the level of 5%

of the peak power at the center of the main pulse.

In particular, the situation shown in Fig. 1 corre-

sponds exactly to z = zopt. Fig. 2 illustrates thedegradation of the pulse, for the transmission dis-

tance essentially exceeding zopt. In particular, in

this case the share of the pulse�s energy trans-

ferred to the side-lobes is almost ten times larger

than at z = zopt.

Analysis of the numerical data reveals that zoptvirtually does not depend on the DM strength (1),

when the latter takes values in a broad interval,

1:5 < S < 11 ð6Þ(outside this interval, the results are much worse).

However, the optimal propagation length is quite

sensitive to the nonlinearity strength (5) of the

additional segment; a typical example of this

dependence, together with the respective depend-ence of the pulse�s compression factor on the NS,

are shown in Fig. 3. The self-compression of the

pulse (without formation of conspicuous side

peaks) offers an advantage in terms of the suppres-

sion of detrimental effects induced by the interac-

Fig. 3. The dashed curve shows the optimal propagation distance (the

the center of the main pulse) versus the nonlinearity strength NS of th

(5). Shown by the continuous curve is the compression factor for the p

versus the input one (4), as a function of NS.

tion between pulses (see details in the next

section), which, as it was said above, is the basic

factor limiting the use of the DM solitons. Fig. 3

suggests that, in the context of particular applica-

tions, a trade-off should be found between thedesirable transmission distance and the necessity

to make the pulses narrow enough.

For the consideration of the interactions be-

tween adjacent pulses, it is also quite important

to understand how much they spread out in the

course of the propagation. To this end, in Fig. 4

we display plots showing the evolution of the

pulse�s width within one cell (at z close to zopt) inthe present model, and in its DM counterpart

(the one lacking the nonlinear segment, L1 = 0).

For this figure, the integral definition of the

squared temporal half-width is adopted, instead

of the above-mentioned TFWHM:

T 2int �

Rþ1�1 t2juðtÞj2dtRþ1�1 juðtÞj2dt

: ð7Þ

As is evident from Fig. 4, the same initial con-figuration produces a pulse that, on average, is def-

initely narrower in the present system than its

one at which the peak power in the side peaks is 5% of that in

e additional segment of the scheme, which is defined as per Eq.

ower distribution, ju(t)j2, in the output pulse (taken at z = zopt),

Fig. 4. Comparison of the evolution of the squared half-widths of the pulse (generated by the same input) within a system�s cell, in the

present model and its DM counterpart (that does not include the extra nonlinear segment). The integral definition (7) of the squared

half-width is adopted here. The cell is taken close to z = zopt. The two plots are juxtaposed so that the borders between the segments

with the anomalous and normal GVD, where the pulse�s width attains its maximum in the ordinary DM system, coincide. In the plot

corresponding to the present model, the width keeps a small constant value inside the nonlinear segment with negligible GVD.


counterpart in the ordinary DM model. We also

note that, if TFWHM were used instead of the inte-

gral definition (7), the comparison would be still

more favorable to the new model, as the contribu-

tion from the side peaks makes the integral expres-

sion in the numerator of Eq. (7) larger.

One of known assets of the DM technique is

the suppression of the soliton�s jitter induced bythe interaction with optical noise. To explore

the same property in the present model, in Fig.

5 we show a typical example of the mean-square

jitter growth with the propagation distance in the

present model, which is compared to the result

for the same input pulse in the pure-DM counter-

part of the system, that does not include the addi-

tional nonlinear segment. The noise was emulatedin two different ways: either as periodic applica-

tion of the shift Dx to the pulse�s central fre-

quency, with a fixed size of jDxj and randomly

chosen sign, or through direct injection of ran-

dom-field wave packets, with amplitudes scaled

to match the former method. Both ways of emu-

lating the noise produced practically identical

results. In the latter case (direct injection of the

noise), the stronger suppression of the jitter in

the full model, as compared to the DM one,

which is obvious in Fig. 5, may be due the fact

that, inside the additional nonlinear segment,

the phase shift acquired by the pulse is much lar-

ger than that of the small-amplitude noise com-ponents, which makes the interaction between

the pulse and the noise effectively incoherent,

i.e., weak. In all the above-mention interval (6)

of the DM strengths, the results for the jitter sup-

pression are very similar to those displayed in

Fig. 5.

3. Transmission of pulse pairs

As it was explained above, a key problem ham-

pering the use of strong-DM schemes in the soli-

ton regime is the interaction between solitons. To

understand if the present model alleviates this

Fig. 5. The growth of the mean-squared temporal displacement of the pulse�s center (jitter) with the propagation distance, under the

action of random noise. The noise is emulated by the shift Dx = ± 0.5 THz of the central frequency, which is added, with a randomly

chosen sign, to the pulse each 60 km. For comparison, the inset shows the same in the ordinary DM model (note the difference in the

scales of the vertical axis: the maximum value on the axis is 6 ps2 in the main plot, and 250 ps2 in the inset, while the maximum value of

the transmission distance is 1200 km in both plots).


well-known difficulty, it is necessary to explore the

transmission of multi-pulse trains. The first step,which we report in this work, is systematic simu-

lations of a pair of two pulses, created with a tem-

poral delay Dt. The objective is to find the

minimum value of Dt that provides stable co-

propagation of the pulses (in particular, without

conspicuous shifts of their centers due to the

interaction).

Simulating the co-propagation of the pulsepairs, we have found that, in the interval (6), the

present model gives rise to the minimum

separation

Dtð Þmin ¼ 1:57T FWHM ð8Þif the propagation distance is chosen to be equal to

the optimal one zopt, that was defined above

for the single-pulse transmission. If Dt exceeds

(Dt)min, the pulses feature virtually no interac-

tion-induced shift of their centers; in the opposite

case, Dt < (Dt)min, the pulses merge into a single

one, within the propagation distance z < zopt.

The small value of (Dt)min is quite promising for

the applications, making it possible to realize ahigh bit rate (per channel). For instance, for the

pulse width TFWHM = 7.08 ps, that was used in

the above examples, Eq. (8) yields (Dt)min =

1.57TFWHM = 11.12 ps, which implies the maxi-

mum bit rate as high as 89 Gb s�1 per channel.

In fact, the system not only prevents the merger

of the pulse pair with Dt > (Dt)min, but also im-

proves the quality of the double-pulse configura-tion, showing a trend to clear the space between

them, which means suppression of the ISI (a de-

tailed discussion of the ISI in a related context can

be found in [8]). These results are illustrated by typ-

ical examples in Fig. 6, through comparison

between the input and output shapes of the

two-pulse configurations, in the case of Dt =1.69TFWHM, which is close to the minimum neces-sary separation given by Eq. (8). In the figure, the

comparison is given, in parallel, for the full model

and its DM counterpart, the effect of the ISI sup-

pression being obvious.

Fig. 6. The comparison between the input and output two-pulse configurations in the full system (a) and its DM counterpart (b), that

does not include the extra nonlinear segment. The output is generated by the transmission of the pair through the distance zopt =

1800 km, which is defined as the optimum distance for the single-pulse transmission.


4. Conclusion

In this work, we have considered an extendedversion of the usual DM model, adding an extra

nonlinear element to it. The additional element

may be realized in a number of ways: as a long seg-

ment of the dispersion-shifted fiber, or as a shortpiece of an Erbium-doped or photonic-crystal


fiber, or, alternatively, as a small module employ-

ing the nonlinear phase shift induced by the sec-

ond-harmonic generation. The subject of the

consideration was transmission of non-soliton

(weakly nonlinear) pulses through the system. Inaccordance with the expectation that the extra

nonlinear element must induce self-focusing of

pulses, we have concluded that the pulse features

self-compression, which is accompanied by

generation of side peaks in the temporal domain.

As a result, it is possible to identify the opti-

mal transmission distance, zopt, up to which the

pulse continues to compress itself, while keep-ing the intrinsic chirp and side-lobe�s amplitude

within acceptable limits. While zopt virtually does

not depend on the DM strength in the broad inter-

val (6), it is quite sensitive to the nonlinearity

strength in the extra segment. The system also

provides for a stronger suppression of the noise-

induced jitter of the pulses than its DM

counterpart.The central issue is suppression of the inter-

action between pulses, which is the main prob-

lem impeding the use of the DM solitons

outside the region of weak DM. We demon-

strate that, in a broad region of parameters,

the system provides for effective isolation be-

tween co-propagating pulses, the minimum dis-

tance necessary for the isolation being quitesmall, �1.57TFWHM. Moreover, the transmission

actually improves the quality of the multi-pulse

pattern, clearing the spaces between pulses, if

they were overlapping originally. Thus, the com-

bined system, which includes the ordinary DM

part and the extra nonlinear element, offers a

potential for improvement of the transmission

of pulse streams in moderately long fiber-opticlinks.

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transmission of pulses in a dispersion-managed fiber link with extra nonlinear segments

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