propagation of short soliton pulses through a parabolic index fiber with dispersion decreasing along...

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Optics Communications 281 (2008) 3361–3368

Propagation of short soliton pulses through a parabolic indexfiber with dispersion decreasing along length

Dipankar Ghosh, Mousumi Basu *

Department of Physics, Bengal Engineering and Science University, Shibpur P.O.: Botanic Garden, Howrah 711 103, West Bengal, India

Received 25 September 2007; received in revised form 18 February 2008; accepted 18 February 2008

Abstract

A parabolic index dispersion decreasing fiber (DDF) has been designed and optimized to produce high capacity soliton communica-tion system. Variation of different fiber parameters such as core radius, effective core area and GVD factor along the 25 km of DDFlength has been carried out to optimize a best possible DDF which can sustain the propagation of fundamental soliton. The variationof non-linearity with length along with the conventional power and GVD factor variation has been included in the generalized non-linearSchrodinger equation (NLSE). This NLSE has been solved numerically by split step Fourier method for shorter pulse propagation,incorporating the term for third order dispersion and intrapulse Raman scattering. Stable soliton pulses in transmission system have beenachieved by our simulation, when a correction factor due to Raman induced soliton mean frequency shift is incorporated to the GVDprofile predicted by the fundamental soliton condition. The interaction of neighboring soliton pulse pair through the proposed fiber hasalso been studied.� 2008 Elsevier B.V. All rights reserved.

PACS: 42.65.Tg; 42.81.Dp; 42.65.Wi

Keywords: Dispersion decreasing fiber; GVD factor; Non-linearity; Soliton; Self frequency shift

1. Introduction

From the last two decades, soliton based fiber optic com-munication systems are drawing considerable research inter-ests due to their immense high capacity transmissionpotential for long haul systems. In the context of ideal loss-less optical fibers with anomalous dispersion, the two chirpcontributions coming from group velocity dispersion(GVD) and self phase modulation (SPM) annul each otherresulting a chirp free pulse propagation – called fundamen-tal soliton. Thus to preserve the soliton character, the pulsemust maintain its peak power. Although the recent pro-gresses in fabrication technology resulted in a very lowattenuation �0.2 dB/km [1] of the optical fibers at1550 nm operating wavelength, this amount of loss still

0030-4018/$ - see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2008.02.017

* Corresponding author. Tel.: +91 33 2668 4561/2/3x427.E-mail address: [email protected] (M. Basu).

imposes limitations on the design of optical communicationsystem. Fiber losses are detrimental as they reduce peakpower of solitons exponentially along the fiber length, lead-ing to increase in pulse width with propagation. To maintaina stable soliton transmission the constant anomalous disper-sion fibers need periodically spaced amplifiers so that pulsescan propagate as guiding center or average solitons.

An interesting scheme has been proposed by Tajima thatinstead of using a large number of optical amplifiers, a sol-iton can also survive inside a lossy optical fiber, called dis-persion decreasing fiber (DDF) [2]. Here, the local balancebetween GVD (jb2j) and SPM can be restored by changingdispersion along the fiber length. As soliton peak powerreduces exponentially, the fundamental soliton condition[3,4] can still be maintained at every point if jb2j alsodecreases exponentially along the length [3,4] by decreasingthe core radius continuously along the length. The fabrica-tion of 38 km loss-compensating dispersion-decreasing

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3362 D. Ghosh, M. Basu / Optics Communications 281 (2008) 3361–3368

fiber with exponential dispersion decreasing profile wasreported [5]. The propagation of solitons through fabri-cated DDF and constant-dispersion fiber of same path-averaged dispersion was compared and the output pulsewidth of the DDF was found out to be as short as the inputpulse width, whereas the output pulse width of the con-stant-dispersion fiber was increased by a factor of 3 [6].Different tapered dispersion fibers were fabricated by vary-ing the speed of fiber drawing from a preform [7,8].

Side by side, DDFs are also used for producing ultrashort solitons by pulse compression mechanism [9], wherean enhanced compression of higher order solitons inDDF due to combined effects of negative third order dis-persion and Raman self scattering has been observed. Insome recent work, the non-linear tunneling of optical soli-tons has been investigated through both dispersion andnon-linear barriers by employing the exact solution of thegeneralized non-linear Schrodinger equation (NLSE) withvariable coefficients [10]. In recent days the field of DDFresearch is getting more interested in view of generationof parabolic pulses [11–13] for normally dispersive fiberswhich are numerically as well as experimentally investi-gated by several researchers.

Although pulse compression and supercontinuum gen-eration by parabolic pulses in DDFs remain a topic ofactive research, the use of DDFs for transmission ofshorter soliton pulses over relatively longer lengths are alsoattracting considerable research interests because of theirpotential for high capacity communication system. A con-siderable amount of work has been reported to study theexponential variations of GVD and power along theDDF length. However, since the variation of core radiusalong the fiber length changes the mode effective area[14], the non-linearity along the length is also varied. Fol-lowing the inclusion of this change in mode effective areawith length of a step index DDF, Gupta et al. [14] haveshown that an exact exponential decrease in dispersionwould not be able to support fundamental solitons, ratherthe variation of non-linear coefficient along DDF lengthshould also be included.

To put our work in the perspective of previous works,we have numerically computed the best possible core radiusprofile of a DDF having parabolic index profile, which sup-ports higher cut off frequency and more field confinement[15] than a conventional step index fiber. The optimumcore radius profile leads to a change in V-parameter ofthe fiber, that finally changes the waveguide contributionto dispersion [1,16] with the length of the proposed DDF.At the same time the variation of V-parameter leads tochange in modal spot size [1,16], thereby producing corre-sponding change in the non-linearity along the length.Thus a local balance between the GVD factor and non-lin-earity is maintained by the above variation to sustain thefundamental soliton condition. The well known NLSEhas been numerically solved by split step Fourier method[3] to authenticate the proposed parabolic index DDF.Here, we have studied the propagation of shorter pulses

(1 ps and 0.7 ps) through a parabolic index DDF by incor-porating third order dispersion and intrapulse Raman scat-tering, for the first time to the best of our knowledge. Ouroptimum profile is shown to support fundamental solitontransmission through 25 km of length, when a correctionfactor due to Raman induced soliton self frequency shift[17] in GVD profile is incorporated, which has not beenreported for step index DDF [14]. The interaction of 1 psneighbouring soliton pulse pair of same amplitude but dif-ferent initial phases for several values of normalized spac-ings (2q0) among pulses have also been studied. Inclusionof the correction factor due to Raman self frequency shiftto the GVD profile of the proposed DDF helped us a lotto achieve an improved high capacity transmission of shortsoliton pulses throughout 25 km.

2. Evolution of pulses through a parabolic index fiber with

decreasing dispersion

The pulse propagation equation in presence of higherorder GVD, non-linearities and intrapulse Raman scatter-ing can be described by the generalized NLSE [3,4]

oAozþ a

2Aþ ib2

2

o2A

oT 2� b3

6

o3A

oT 3¼ icjAj2A� icT RA

o

oTjAj2 ð1Þ

where A(z, T) is the slowly varying envelope of the pulse, ais the fiber loss, c is the non-linear coefficient, b2 = d2b/dx2

and b3 = d3b/dx3 are GVD factor and third order disper-sion (TOD), respectively. TR is the Raman resonant timeconstant which is typically 6 fs for ordinary silica fibers[3,9].

It is well known that in the anomalous dispersion regime(b2 < 0) of an optical fiber, the interplay between GVD andSPM leads to propagation of optical solitons. For funda-mental soliton propagation the ratio (N2) of the dispersivelength ðLD ¼ T 2

0=b2Þ and the non-linear length (LNL = 1/cP0) must be unity [3], specifically

N 2 ¼ LD

LNL

¼ T 20cP 0

jb2j¼ 1 ð2Þ

where T0 is the pulse width and P0 is the peak power of theinput pulse [3]. Such soliton propagation is possible for anideal lossless fiber, which is practically impossible toachieve. However, for optical fibers with continuouslydecreasing dispersion in the anomalous region along thelength of the fiber, the GVD factor and SPM can cancelthe effects of each other so that soliton pulse can be realizedeven in presence of attenuation.

The exponential variation of pulse power along theDDF length (z) is expressed as P(z) = P0exp(�az). Herethe variation of power along with the variation of non-lin-ear factor c(z), can compensate the dispersion [14] at eachpoint along the length of the DDF and soliton condition(cf Eq. (2)) can be maintained. To achieve the above condi-tion, if the core radius (a) is supposed to decrease along thefiber length (z), parameters like P(z), c(z) and b2(z) will alsocontinuously vary with length. Thus taking the above

0 4 50.0

0.2

0.4

0.6

0.8

1.0

1.2

Actual variation Fitted portion

d2 (b

V)V

⎯⎯

d

V2

V-parameter 321 876

Fig. 1. Variation of V d2ðbV ÞdV 2 as a measure of waveguide dispersion with

V-parameter.

D. Ghosh, M. Basu / Optics Communications 281 (2008) 3361–3368 3363

variations into account the generalized NLSE (cf. Eq. (1))for a DDF can be normalized in the following form:

ovoxþ i

2

b2ðxÞjb2ð0Þj

o2vos2� b3

6T 0jb2ð0Þjo3vos3

¼ i expð�aLDxÞ cðxÞcð0Þ v jvj

2 � T R

T 0

o

osjvj2

� �ð3Þ

where x and s represent the normalized distance and timevariables

x ¼ z=LD; s ¼ T =T 0 ð4Þ

andffiffiffiffiffiP 0

p

Nexp � a

2LDx

� �vðx; sÞ ¼ Aðz; T Þ ð5Þ

The initial GVD factor b2(0) is chosen to be fixed at�10 ps2/km and the TOD (b3) parameter is found out tobe �0.1 ps3/km. Here, Eq. (3) has been solved numericallyby split step Fourier method by choosing the input pulse ina hyperbolic secant form as given by

vð0; sÞ ¼ sec hðsÞ ð6Þ

3. Design and optimization of the proposed DDF

In order to design and optimize a DDF with parabolicrefractive index profile (q = 2) [1], the core radius of the fibershould be decreased [14] in such a manner that the local bal-ance between dispersion and non-linearity is maintainedthroughout the length of the fiber. For this proposed fiber,3.1m/o GeO2 doped core material is chosen and with thehelp of well known Sellemeir’s formula [1], its core refractiveindex (n1) at 1.55 lm and corresponding material dispersion(Dm) [1] are estimated to be�1.448 and�21.139 ps/km nm,respectively. The waveguide contribution to dispersion coef-ficient [1,14,16] is estimated from the well known scalarwave equation [1,14,16] inside the core and normalizedpropagation constant ‘b’ [1] is obtained as a function of V-parameter of the fiber. The corresponding variation of the

factor, V d2ðbV ÞdV 2 is plotted in Fig. 1 and an empirical formula

for parabolic index profile is found out (valid for1.85 < V < 4.0).

Thus, for this proposed parabolic index fiber, the corre-sponding waveguide dispersion (Dw) coefficient in ps/km nm can be described by the following approximateequation:

Dw � �n1D3k0

� 107 �0:00296� 42:89762

�27:369� V 4

� �ð7Þ

Since, total dispersion (Dt in ps/km nm) is the sum of Dm

and Dw, the group velocity dispersion factor (b2) is ex-pressed as

b2 ¼ �k2

0

2pcDm �

n1D3k0

� 107ð�0:00296� 42:89762

�27:369� V 4Þ

ð8Þ

For a given operating wavelength (k0) and relative refrac-tive index difference (D), the core radius (a) depends onV-parameter of the fiber in the following way [1]:

a ¼ k0

2pn1

ffiffiffiffiffiffi2Dp V ð9Þ

As the core radius is directly proportional to the V-para-meter, the GVD factor (b2) now changes according to thevariation of core radius (a) along the DDF length. Withproper optimization of different fiber parameters, coreradius or V-parameter is made to have a decreasing naturewith distance. As the core radius or V-parameter decreases,the modal spot size (W) changes according to the followingequation as given by Marcuse [18].

W ¼ k0

2pn1

ffiffiffiffiffiffi2Dp K1

V �1=2þ K2

V 1=2þ K3

V 5

� �ð10Þ

with

K1 ¼ffiffiffi2p

K2 ¼ e0:149 � 1þ 1:478ð1� e�0:154ÞK3 ¼ 3:76þ expð4:19=20:418Þ

9>=>; ð11Þ

Thus the above variation of modal spot sizes in turnchanges the value of effective area (Aeff) [1,3] of the pro-posed fiber, which is given by

Aeff ¼ pW 2 ð12ÞThe non-linear coefficient (c) [1,3] of the fiber depends

on the effective area by the following relation:

c ¼ 2pn02k0Aeff

ð13Þ

where, n02 an intensity independent constant characteristicof the material, is chosen to be 2.6 � 10�20 m2/W [14].Using Eqs. (10)–(13) the following variation of c, withoperating V-value can be obtained.

c ¼ 16p2n21Dn02

k30

K1

V �1=2 þ K2

V 1=2 þ K3

V 5

� �2ð14Þ

Table 1Value of the coefficients used in the expression of core radius along withthe fiber loss for different D values

D a (dB/km) a1 a2 a3

0.0057 0.199 2.7410 0.9824 15.692950.0060 0.204 2.8125 0.9047 16.064020.0063 0.209 2.8633 0.8445 16.17643

Table 2Value of the coefficients used in the fitted expression of GVD coefficient

D b1 t1 b2 t2

0.0057 �1.83128 5.60243 11.28745 12.653390.0060 �1.07369 5.46758 10.64614 14.343700.0063 �0.75824 5.37753 10.41823 15.41706


Taking into account the variation of core radius (cf. Eq.(9)) and c (cf. Eq. (14)) with V-parameter, the transcenden-tal Eq. (2) has to be satisfied at any point (x) in the DDFand can be rewritten as

jb2ðxÞj ¼ cðxÞT 20P 0 expð�aLDxÞ ð15Þ

The above transcendental equation is solved numeri-cally by an iterative procedure over a DDF length of25 km, as the interplay between GVD and non-linearityin DDF is prominent upto the length of 20–25 km. Here,D is fixed at 0.006 with corresponding attenuation coeffi-cient �0.204 dB/km.

4. Solution of the transcendental equation to get the optimal

profile of the proposed DDF

By solving Eq. (15) for an initial GVD factor b2(0) fixedat �10 ps2/km and D = 0.006, the initial (at x = 0 or z = 0)core radius (a0) is found out to be 3.717 lm. The core radiusdecreases to 3.00 lm after propagating through 25 km DDFlength. If D deviates by ±5% of its chosen value (�0.006), a0

comes out to be at the value of 3.708 lm and 3.723 lm,respectively. The variations of core radius with fiber lengthfor these three D values are shown in Fig. 2. The optimumradius profile of the proposed DDF is exactly fitted withan exponentially decaying expression of the form given by

a ¼ a1 þ a2 expð�z=a3Þ ð16ÞThe coefficients involved in the expression for three differ-

ent D values are furnished in Table 1. For the above men-tioned optimal profiles the chosen b2(0) (i.e. �10 ps2/km)decreases to �2.277, �2.39 and �2.084 ps2/km at z = 25km for D = 0.006 and 0.0060 ±5%, respectively. Fig. 3shows the variations of b2 that can be well expressed in afitted form

b2ðzÞ ¼ �10þ b1f1� expð�z=t1Þg þ b2f1� expð�z=t2Þgð17Þ

The coefficients associated with the above equation aregiven in Table 2.

0 10 12 14 16 18 20 22 24 262.8

3.0

3.2

3.4

3.6

3.8

Cor

e ra

dius

a (μ

m)

Propagating distance z (km)

Δ = 0.0057Δ = 0.0060Δ = 0.0063

8642

Fig. 2. Core radius (a) profile with propagating distance (z) for three Dvalues.

Here the exponential variation of power is compensatedby both dispersion and non-linear coefficient. Thus it canbe seen from Fig. 3 that the GVD profile is not exactlyexponential in nature. It can also be seen from the figurethat although b2(0) remains same for all D, jb2(z)j hasslightly smaller values at lower D compared with that forD = 0.006.

Using the equations as mentioned in the previous sectionthe variation of modal spot sizes can be obtained. Fig. 4shows the variation of modal spot size (W) and the corre-sponding non-linear coefficient (c) with the fiber length.The variation of modal spot size with propagating distance(z) shows increasing tendency. It can be seen from Fig. 4that when D = 0.006, the modal spot size increases from4.305 lm at z = 0 to the value of 5.015 lm at z = 25 km.

When a 5% decrease in D (= 0.0057) has been consid-ered, the optimized modal spot size of the proposedDDF increases and varies from 4.449 lm to 5.497 lm. Atthe same time increase in D by 5% leads to a lesser valueof optimized W, that is �4.18 lm at initial point, whichincreases to �4.685 lm at end point of propagation. Asspot size increases with distance, the modal effective area

0 10 12 14 16 18 20 22 24 26-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Δ = 0.0057Δ = 0.0060Δ = 0.0063

GVD

coe

ffici

ent β

2 (p

s2 /km

)

Propagating distance z (km)8642

Fig. 3. Variation of the GVD coefficient (b2) with propagating distance (z)for three D values.

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

0 4 8 10 12 14 16 18 20 22 24 264.0

4.2

4.4

4.6

4.8

5.0

5.2

5.4

5.6

Non

linea

r coe

ffici

ent γ

(W-1

km

-1)

Δ = 0.0057Δ = 0.0060Δ = 0.0063


Mod

al s

pot s

ize

W (μ

m)

2 6

Fig. 4. Variation of modal spot size (W) (left side) and correspondingnon-linear coefficient (c) (right side), with propagating distance (z) forthree D values.


also increases, thereby reducing the non-linear coefficientalong the length, which has also been reflected in Fig. 4.Thus, not only the power variation with length but alsothe change in c(z) becomes significant enough to compen-sate the dispersion variation. Therefore we should includethis variation of c(z) to get the best possible DDF.

5. Soliton Propagation through the proposed DDF

For the optimal profile as obtained in Section 4, the prop-agation of optical pulses (with T0 = 5.0, 1.0 and 0.7 ps) is dis-cussed in this section. We have solved the generalized NLSEin modified form (cf. Eq. (3)) numerically by the split stepFourier method [3], using MatLab. It can be seen from theresults that the optimized DDF can support propagationof fundamental solitons for the hyperbolic secant form ofinput pulses having above mentioned initial widths. Thepeak powers corresponding to different values of D are tab-ulated in Table 3. For D = 0.006, the attenuation at 1550 nmoperating wavelength is �0.204 dB/km and the input peakpower required to maintain the soliton condition (cf. Eq.(2)) at T0 = 5 ps becomes comparatively smaller �221mW. However, when the shorter pulse propagation has been

Table 3Estimation of peak powers (P0) corresponding to the pulse widths (T0)

T0 (ps) D P0 (W)

5.0 0.0057 0.2360.0060 0.2210.0063 0.208

1.0 0.0057 5.9010.0060 5.5240.0063 5.208

0.7 0.0057 12.0430.0060 11.2740.0063 10.628

considered, the input peak powers have to be increased tolarger values corresponding to P0 � 5.52 W whenT0 = 1 ps and P0 � 11.27 W at T0 = 700 fs. Although toachieve high capacity transmission, the propagation of shortsoliton pulses with smaller initial pulse widths (T0 = 1 ps and0.7 ps in this study) are necessary, a compromise with highpower requirement is unavoidable.

Our numerical simulation by split step Fourier method,shows that even for shorter initial pulses, soliton propaga-tion is possible by preserving the same pulse widththroughout the total length. As mentioned in Eq. (3), forsufficiently short pulses, the Raman term downshifts thesoliton mean frequency (m). This shift in mean frequencyassociated with Raman effect is given by [17,19]

dmdz¼ � 4

15pT R

T 40

jb2ðzÞj ð18Þ

Using Eq. (17) the soliton mean frequency relative tocarrier frequency can be expressed as

dmdz¼ � 4

15pT R

T 40

jð�10þ b1 þ b2Þzþ b1t1fexpð�z=t1Þ � 1g

þ b2t2fexpð�z=t2Þ � 1gj ð19Þ

Now, incorporating the correction factor 2pb3m(z) arisingdue to the above frequency shift, the optimum GVD factorbopt

2 ðzÞ at any z, is related with the TOD parameter b3 by[20]

bopt2 ðzÞ ¼ b2ðzÞ � 2pb3mðzÞ ð20Þ

Although for the initial pulse width �5 ps, the last term inEq. (20) does not have significant effect, for shorter solitonpulses, the optimal GVD profile should be designedaccording to Eqs. (19) and (20). Thus incorporating theabove correction factor, i.e., the change in GVD associatedwith the shift in soliton mean frequency, as given by Eq.(20), fundamental soliton pulse propagation becomes pos-sible over the 25 km length of the proposed DDF.

Our simulation shows that for wider pulses (�5 ps) asusual fundamental soliton transmission has faced no prob-lem. This is due to the fact that decrease in dispersion coef-ficient along the length of the DDF with optimum coreradius profile, can compensate the local non-linearity c ateach point of the lossy fiber. But, when short pulses aretaken into consideration, GVD profile as predicted by therelation (17) is not capable of maintaining constant pulsewidth due to soliton mean frequency shift in presence ofthird order dispersion and intrapulse Raman scattering.Fig. 5 shows the variation of pulse widths with distanceof propagation along the DDF. It can be seen from the fig-ure that initially the pulse width has been slightly changedfrom T0 = 1.0 to 1.018 ps and T0 = 0.7 to 0.751 ps, whereasafter incorporation of the above correction factor (cf. Eq.(20)), a dramatically improved transmission characteristicsof short solitons has been achieved.

It should also be mentioned here that the soliton spec-trum corresponding to 1 ps pulse when propagated

0 8 10 12 14 16 18 20 22 24 260.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

(b)

without correction factor with correction factor

Puls

e W

idth

T 0 (p

s)

Propagating distance, z (km)

(a)

642

Fig. 5. Variation of pulse width along propagating distance (z) with initialpulse width (a) T0 = 1.0 ps and (b) T0 = 0.7 ps.

0 4 8 10 12 14 16 18 20 22 24 26-0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

(b)T0= 0.7 ps


Wav

elen

gth

shift

(λ−λ

0) in

nm

(a)T0=1.0 ps

62

Fig. 7. Plot of soliton wavelength shift (k � k0) with distance for initialpulse width (a) T0 = 1.0 ps and (b) T0 = 0.7 ps.


through 25 km, has experienced a very small amount of selffrequency shift �0.067 THz, as predicted by the analyticalexpression given by Eq. (18). The pulse evolutions of theshort solitons both in time domain as well as frequencydomain are studied by solving NLSE numerically includingthird order dispersion and Raman terms. The pulse spec-trum of soliton pulse with width T0 = 1.0 ps propagatingthrough 25 km of DDF has been obtained by numericalsimulation and is compared with the input spectrum forthe fundamental soliton (N = 1) in Fig. 6.

The most remarkable feature of Fig. 6 is a very lesseramount of red shift of soliton spectrum that is in goodagreement with the value obtained by the analytical for-mula (cf. Eq. (18)). This red shifted spectral peak corre-sponds to the intense soliton shifting towards right sidein time domain. Thus it can be said that the 1 ps pulse expe-rienced an insignificant amount of positive wavelengthshift, Dk�0.53 nm which is 0.034% of the operating wave-length of 1550 nm, after 25 km length of propagation.Rather, the shorter soliton pulse with T0 = 700 fs, experi-ences a little more red shift towards longer wavelengthregion (i.e., �1552.2 nm) at z = 25 km, which has beenreflected in Fig. 7. Here the variations of soliton wave-

Fig. 6. Input and output pulse spectrum through the proposed DDF.

length shift for pulse widths T0 = 1 ps and 0.7 ps, havebeen plotted as a function of propagating distance.

6. Interaction of neighbouring solitons in the proposed DDF

The time interval between two neighboring pulses deter-mines the bit rate of the communication system and isgiven by

B ¼ 1

2q0T 0

ð21Þ

where 2q0 is the normalized spacing between adjacent soli-tons [4]. The implication of soliton interaction in our pro-posed DDF has been understood by solving the NLSE (cf.Eq. (3)) numerically. Here we take the input pulse corre-sponding to a soliton pair with same amplitude of the fol-lowing form [3,4]:

vð0; sÞ ¼ sec hðs� q0Þ þ sec hðsþ q0Þeiu ð22Þwhere / is the relative phase between the two neighboringsolitons. The simulation for propagation of equal amplitudesoliton pairs with pulse width of 1 ps, at different values of q0

has been performed. Fig. 8 shows the variation of normal-ized relative spacing, i.e. q(z)/q0 between the adjacent inphase (/ = 0) pulses as a function of propagation distance.At distance z = 0, q/q0 is unity for different initial q0 valuesstarting from 3.5 to 6.5. It can be seen that for lower initialseparation of adjacent pulses, i.e., q0 = 3.5, the pulses aregetting closer to each other as they propagate such thatthe normalized separation becomes�0.536, which is almosthalf of the initial separation, at z � 2.76 km. After that valueof z, solitons repel each other more strongly. Unlike the con-stant-dispersion fibers [3], it is seen that even at smaller valueof q0 = 3.5, there is no possibility of direct collision and col-lapse of the pair of pulses at a particular length of our pro-posed DDF. However there is a large change in relative peakpower. The same nature of variation of normalized relativespacing with fiber length can be seen when q0 = 4.0, forwhich minimum normalized separation becomes �0.7 at

0 4 8 10 12 14 16 18 20 22 24 260.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5qo= 6.5qo= 6.0qo= 5.5qo= 5.0qo= 4.5qo= 4.0qo= 3.5

Nor

mal

ised

rela

tive

spac

ing

(q/q

0)

Distance (km)62

Fig. 8. Plot of Normalized relative spacing (q/qo) as a function ofpropagation distance for different initial q0 and same / (= 0).

-100

1020

3040

0

5

10

15

20

25

00.5

1

Normalised Time

Distance (km

)

Pow

er

-10 -5 0 5 10 15 20 25 30 35 40

0

5

10

15

20

250

0.51

Normalised Time

Distance (km)

Pow

er

a

b

Fig. 9. (a) Soliton pulse evolution when q0 = 3.5. (b) Soliton pulseevolution when q0 = 6.0.

0 8 10 12 14 16 18 20 22 24 26

9.2

9.4

9.6

9.8

10.0

10.2

10.4

10.6

10.8

Rel

ativ

e Sp

acin

g 2q

(z)

Distance (km)

φ = 90o

φ = 60o

.........φ = 53o

φ = 50o

φ = 45o

φ = 30o

φ = 0o

642

Fig. 10. Relative Spacing 2q(z) between two neighboring solitons as afunction of propagating distance (z in km) for several values of initialphase difference, / (in degree) with initial 2q0 = 10.


z = 4.5 km. The studies on all in phase pair of solitons at dif-ferent initial separation show that when q0 = 5, the normal-ized relative spacing has a small decreasing tendency upto alength of 10 km and after that it is roughly constant. Whenq0 > 5, as the normalized spacing decreases insignificantlywith propagating length so that very small mutual interac-tion among the pair of solitons can be considered. Theabove change in relative spacing with distance can bewell understood from the evolutions of the in phase solitonpairs, shown in Fig. 9a and b when q0 = 3.5 and q0 = 6.0,respectively.

Large change in normalized spacing of the neighboringsolitons is undesirable to minimize the mutual interaction.Hence choosing a value of initial separation of 2q0 = 10,where little amount of interaction may be present, 2q(z)of adjacent pulses have been plotted with distance at differ-ent values of initial phase (/) values in Fig. 10. It can beseen from this figure that the pulses initially at 90� out ofphase, repel each other strongly when propagatingthroughout the fiber. The study shows that when / = 50�between a pulse pair is considered, then over a range offiber length (Zopt) from 15.6 to 19.7 km, the normalizedspacing between the pulses becomes exactly the same asthe initial spacing, i.e., 10, even though 2q(z) outside therange of Zopt becomes 61.3% of the initial values. It wasnot possible to obtain such cases of reasonably same sepa-ration of pulse pair at any particular range of length, forother phase values. At the same time it should also benoticed that at / = 53�, the average value of 2q(z) staysvery close to the initial value of 10 all through the propa-gation length and it comes out exactly at 2q(z) = 10 atthe end of fiber length of 25 km. Thus for an initial phasedifference of / = 53� among the soliton pulse pair can alsobe chosen by fiber optic designers so that the normalizedspacing between soliton pairs can be exactly recovered atthe fiber end, thereby preserving the soliton condition whentransmitted through the DDF, even when q0 = 5.0. Thesoliton pulse evolution at comparatively higher bit rate,

i.e. at the rate of 6100 Gb/s, through the DDF, at q0 =5.0 and / = 53� has been shown in Fig. 11, which makesno significant change when compared with the pulse

-10 -5 0 5 10 15 20 25 30 35 40

0

5

10

15

20

250

0.51

Normalised Time

Distance (km)

Pow

er

Fig. 11. Evolution of soliton pulses with initial phase difference / � 53�and q0 = 5.0.


progression as given in Fig. 9b for higher q0 value. Hence itcan be understood from the above study that more or lessstable soliton pulse propagation can be obtained for a 1 pshyperbolic secant pulse pair at normalized spacing of 10.

7. Conclusion

We have designed and optimized a parabolic index DDFhaving D = 0.006 with 5% tolerance. An empirical formulaof dispersion characteristics for this type of profile is esti-mated. In order to maintain the fundamental soliton condi-tion, the non-linear coefficient is also varied apart from thevariation of power and GVD coefficient. The core radiusprofiles are found out by suitable fitting with the numericalresults. The modified NLSE is solved by split step Fouriermethod and 25 km soliton pulse propagation becomes pos-sible. To upgrade it for high capacity transmission system,short pulses are used where inclusion of third order disper-sion and intrapulse Raman scattering lead to an optimumGVD profile. By incorporating a correction factor due tosoliton mean frequency shift to the optimum GVD profile,solitons with constant pulse widths are realized throughoutthe propagation length. From the interaction of neighbor-ing soliton pulses, a nearly stable propagation of input sol-iton pulse pair with same amplitude and initial separationof 10, but a relative phase from 50� to 53�, has been

achieved for the proposed DDF. As a whole, our designshows an improved high bit rate transmission of shorterpulses by suitable tailoring of GVD profile in presence ofRaman induced self frequency shift. Such design will behelpful for computer assisted fiber manufacturing in thecontext of latest technological development.

Acknowledgements

This work has been financially supported by Depart-ment of Science and Technology, Government of India.The authors are thankful to Prof. S.N. Sarkar, Departmentof Electronic Science, University of Calcutta, India, for hisconstructive suggestions.

References

[1] A.K. Ghatak, K. Thyagarajan, Introduction to Fiber Optics, Cam-bridge University Press, UK, 2002.

[2] K. Tajima, Opt. Lett. 12 (1987) 54.[3] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, UK, 2001.[4] G.P. Agrawal, Applications of Nonlinear Fiber Optics, Academic

Press, New York, 2006.[5] D.J. Richardson, R.P. Chamberlin, L. Dong, D.N. Payne, Electron.

Lett. 31 (1995) 1681.[6] A.J. Stentz, R.W. Boyd, A.F. Evans, Opt. Lett. 20 (1995) 1770.[7] K.R. Tamura, M. Nakazawa, Opt. Lett. 26 (2001) 762.[8] V.A. Bogatyrev, M.M. Bubnov, E.M. Dianov, A.S. Kurkov, P.V.

Mamyshev, A.M. Prokhorov, S.D. Rumyantsev, V.A. Semenov, S.L.Semenov, A.A. Sysoliatin, S.V. Chernikov, A.N. Gur’yanov, G.G.Devyatykh, S.I. Miroshnichenko, J. Lightwave Technol. 9 (1991) 561.

[9] Z. Shumin, L. Fuyun, X. Wencheng, Y. Shiping, W. Jian, D. Xiaoyi,Opt. Commun. 237 (2004) 1.

[10] G. Yang, R. Hao, L. Li, Z. Li, G. Zhou, Opt. Commun. 260 (2006)282.

[11] T. Hirooka, M. Nakazawa, Opt. Lett. 29 (2004) 498.[12] A. Latkin, S.K. Turitsyn, A.A. Sysoliatin, Opt. Lett. 32 (2007) 331.[13] C. Finot, L. Provost, P. Petropoulos, D.J. Richardson, Opt. Express

15 (2007) 852.[14] D. Gupta, G. Kumar, K. Thyagarajan, Opt. Commun. 237 (2004)

309.[15] A. Carnevale, U. Paek, G.E. Peterson, United States Patent

4,412,722, 1983.[16] E.G. Neumann, Single Mode Fibers: Fundamentals, Springer,

Heldelberg, 1988.[17] J.P. Gordon, Opt. Lett. 11 (1986) 662.[18] D. Marcuse, J. Opt. Soc. Am. 68 (1978) 103.[19] E.L. Buckland, R.W. Boyd, A.F. Evans, Opt. Lett. 22 (1997) 454.[20] A. Mostofi, H.H. Hanza, P.L. Chu, IEEE J. Quantum Electron. 33

(1997) 620.

propagation of short soliton pulses through a parabolic index fiber with dispersion decreasing along...

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