Chaos, Solitons and Fractals 29 (2006) 823–828

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The effect of quintic nonlinearity on thepropagation characteristics of dispersion

managed optical solitons

S. Konar *, Manoj Mishra, S. Jana

Department of Applied Physics, Birla Institute of Technology, Mesra, Ranchi 835 215, India

Accepted 22 April 2005

Abstract

The role of quintic nonlinearity on the propagation characteristics of optical solitons in dispersion managed opticalcommunication systems has been presented in this paper. It has been shown that quintic nonlinearity has only marginalinfluence on single pulse propagation. However, numerical simulation has been undertaken to reveal that quintic non-linearity reduces collision distance between neighbouring pulses of the same channel. It is found that for lower mapstrength the collapse distance between intra channel pulses is very much sensitive to the dispersion map strength.� 2005 Elsevier Ltd. All rights reserved.

1. Introduction

Dispersion managed (DM) optical communication systems have attracted a great deal of attention [1–12] as theyfelicitate very high bit rate of transmission of data over conventional soliton based systems. The advantage of DM sol-itons over conventional one are mainly four folds. DM solitons have larger peak power than the ideal soliton under thesame average dispersion, thereby giving enhanced signal to noise ratio at the receiver end [1]. They are less influenced bythe Gordon-Haus timing jitter [2], collision induced frequency shifts [3] and four wave mixing processes in wavelength-division-multiplexing systems. The last factor could be achieved with considerable success by allowing large local dis-persion [15,16]. Unlike in a constant dispersion system where the pulse propagates maintaining its shape unchanged, inthe dispersion managed system the pulse width oscillates periodically and the shape of the pulse ranges from hyperbolicsecant to Gaussian to flattop, depending on the strength of the dispersion management [17,6].

Though tremendous amount of work [1–18] has been reported in the literature on the propagation characteristics ofreturn-to-zero (RZ) pulses in DM systems, yet many issues remain unanswered. For example, since optical fibers aremade of glass where Kerr nonlinearity arises due to electrostriction, soliton propagation in the DM systems have beenstudied based on Kerr nonlinearity [1–9] only. However, this is true only at low power. Transmission at very high bitrate requires pulses of lower temporal width and higher peak power. At high peak power higher order nonlinearitymight influence the propagation dynamics [20,21] In optical fibers, the most important higher order nonlinearity is

0960-0779/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2005.04.122

* Corresponding author. Tel.: +91 6512276274; fax: +91 6512275401.E-mail address: swakonar@yahoo.com (S. Konar).

824 S. Konar et al. / Chaos, Solitons and Fractals 29 (2006) 823–828

the quintic nonlinearity, which arises as a result of nonlinear saturation in the optical fields. Therefore, effect of quinticnonlinearity, though typically weak, needs to be taken care of seriously in the investigation of DM soliton systems de-signed for very high bit rate of transmission. It should be pointed out that the effect of this nonlinearity on the solitonpropagation in optical fibers with constant dispersion has drawn much attention in recent years [20,21]. In addition,quintic nonlinearity plays a significant role in fiber lasers too [22–24]. Thus, the main aim of this paper is to investigatethe breathing soliton propagation in DM systems under the influence of Kerr and quintic nonlinearity.

2. Governing equations

The dynamics of an optical pulse propagating in a loss less optical fiber possessing Kerr and quintic nonlinearity andperiodically varying dispersion is governed by the modified nonlinear Schrodinger equation [20],

ioEoz� b2ðzÞ

2

o2Eot2þ kn2

n0

a0jEj2E þ kn2

n0

b0jEj4E ¼ 0 ð1Þ

where E is the slowly varying electrical field envelope, b2 is the group velocity dispersion varying periodically with z, t isthe retarded time, k is the wave number. The parameters n0, n2 and n4 are respectively, the linear refractive index, thirdand fifth order nonlinear refractive index coefficient. Two constants a0 and b0 are arising due to the confinement of elec-tric field in the fiber and 0 < a0, b0 < 1 for Gaussian/super Gaussian pulses. At this stage one may introduce followingnormalization: Propagation distance z is normalized with 2.5 times the dispersion map length L in km, time is normal-ized to T0 (in picosecond), the electric field envelope E is normalized with a characteristics power P0 such thatE ¼

ffiffiffiffiffiP 0

pu. With above normalization Eq. (1) takes the form

iouoZþ dðzÞ

2

o2uos2þ juj2uþ cjuj4u ¼ 0 ð2Þ

where the normalized dispersion is written as

dðzÞ ¼ � 2:5bðjÞ2 L

T 20

; ð3Þ

c ¼ n4b0P 0

n2a0, and the characteristic power P 0 ¼ n0

ð2:5Lkn2a0Þ. In optical fibers, quintic nonlinearity is defocusing, thus giving c

negative and generally 0 < jcj � 1. The chosen dispersion map is symmetrical as shown in Fig. 1 and the fiber dispersionis the sum of a locally varying part and an average dispersion hdi and is given by

dðZÞ ¼d1 þ hdi if 0 6 Z 6 L=4

d2 þ hdi if L=4 6 Z 6 3L=4

d1 þ hdi if 3L=4 6 Z 6 L

������� ð4Þ

The map strength of the dispersion map is defined as [9]

S ¼ bð1Þ2 L1 � bð2Þ2 L2

T 2FWHM

���������� ð5Þ

where TFWHM is the full width at half maximum. Eq. (2) is a modified nonlinear Schrodinger equation with variabledispersion and cannot be solved using inverse scattering technique. Therefore we solve Eq. (2) using variational proce-

L/4 L/4 L/2

L/2 L/2

d(Z

)

L Unit Cell

< d >

Fig. 1. Periodic dispersion map with average anomalous dispersion.

S. Konar et al. / Chaos, Solitons and Fractals 29 (2006) 823–828 825

dure. This procedure has been used successfully by several authors [25–27] to address different nonlinear optical prob-lems involving nonlinear Schrodinger equation and its modified form. We note that Eq. (2) can be derived from thevariation of the Lagrangian density C

dZ 1

�1

Z 1

�1CdsdZ ¼ 0 ð6Þ

with

C ¼ i

2u�

ouoZ� u

ou�

oZ

� �� dðZÞ

2

ouoT

��������2

þ 1

2uj j4 þ 1

3juj6 ð7Þ

In order to analytically estimate the periodic variation of pulse parameter in the dispersion managed system, it is con-venient to postulate a simple ansatz

uðZ; sÞ ¼ AðZÞ expb�p2ðZÞðs� T 0ðZÞÞ2 þ iCðzÞðs� T 0ðZÞÞ � ijðZÞsþ irðZÞc ð8Þ

where A(z), p(Z), T0(Z), C(Z), j(Z) and r(Z) are real amplitude, inverse of the pulse width, temporal pulse position,chirp, frequency and phase of the pulse respectively. Using the pulse profile (8), we define a finite dimensional Lagrang-ian L A; p; T 0;C; j; r; dA

dZ ;dpdZ ;

dT 0

dZ ;dCdZ ;

djdZ ;

drdZ

� �, such that

dZ 1

�1LdZ ¼ 0 ð9Þ

The Lagrangian is given by

L ¼Z 1

�1Cds ¼ �

ffiffiffip2

rA2 1

4p3

dCdZ� T 0

pdjdZþ 1

pdrdZ

�þ A2

2ffiffiffi2p

pþ dðZÞp

2þ dðZÞC2

2p3þ dðzÞj2

2p� cA4

3ffiffiffi3p

p

ð10Þ

By analogy from classical mechanics we define a Hamiltonian from the Lagrangian which is given by

H ¼X

abbZ � L; b ¼ C; j; r ð11Þ

where the momenta ab�s are defined as

ab ¼oL

oðbZÞ; bZ ¼

oboZ

The Hamiltonian turns out to be,

H ¼ �A2p2ffiffiffi2p þ 2dðZÞp4 � 2dðZÞC2 � 2cA4p2

3ffiffiffi3p

� �aC þ

5ffiffiffi2p

A2

8� dðZÞp2 � 4cA4

3ffiffiffi3p � dðZÞj2

2

!ar þ

ffiffiffiffiffiffi2pp

A4

4p

þ 11ffiffiffipp

cA6

12ffiffiffi3p

pð12Þ

where

aC ¼ �ffiffiffipp

A2

8p3; aj ¼

ffiffiffipp

A2T 0

2pand ar ¼ �

ffiffiffipp

A2

2pð13Þ

The above Hamiltonian yields following ordinary differential equations:

dAdZ¼ �AdðZÞC ð14aÞ

dpdZ¼ �pdðZÞC ð14bÞ

dCdZ¼ �A2p2ffiffiffi

2p þ 2dðZÞp4 � 2dðZÞC2 � 2cp2A4

3ffiffiffi3p ð14cÞ

drdZ¼ 5

ffiffiffi2p

8A2 � dðZÞp2 þ 4cA4

3ffiffiffi3p � dðZÞj2

2ð14dÞ

djdZ¼ 0 ð14eÞ

dT 0

dZ¼ �jdðZÞ ð14fÞ

826 S. Konar et al. / Chaos, Solitons and Fractals 29 (2006) 823–828

Additionally, the pulse amplitude and the inverse width p are related through the normalized pulse energy E0 throughE0 = A2/p. The dispersion managed solitons are solution of (14) with the periodic boundary conditions C(0) = C(L) andp(0) = p(L).

3. Result and discussion

The set of ODE�s (14) has been solved to find out allowed values of p and C of periodic DM solitons for a given pulseenergy E0. It has been confirmed that like Kerr medium, quintic medium too supports DM solitons. However, in case ofsingle soliton propagation, it has been observed that the p–C phase diagram is only marginally modified with the intro-duction of finite c. Moreover, the modification in p–C values reduces for higher d(Z). Therefore, equal energy DM sol-itons will have similar p–C relationship in Kerr and quintic media. Thus, quintic nonlinearity only marginally modifiesthe p–C relationship still admitting periodic stationary soliton. To further investigate the properties of DM solitons,numerical simulation of Eq. (2) has been performed using split step Fourier method [19]. Initial values of p and C

are chosen from the allowed periodic solution of ODE�s as predicted by the variational method. A single pulse has beenallowed to propagate over 250 DM cells and the pulse sample has been taken after each tenth cell. Fig. 2 shows the pulsedynamics for different strength of quintic nonlinearity parameter c. It is obvious that as predicted by the variationalformalism, the stable pulses are possible in fibers with quintic nonlinearity. The difference in the features of the pulseswith different c is hardly noticeable. This is in conformity with the prediction of the variational theory, that the role ofquintic nonlinearity is only to influence marginally the pulse width and amplitude of pulse with equal energy.

Though quintic nonlinearity has only marginal influence on the single pulse propagation, it may affect the interac-tion of neighbouring pulses of the same channel. Thus, interaction of neighbouring solitons in the same channel in aDM system is considered incorporating the influence of quintic nonlinearity. For simplicity one may consider two adja-cent in phase solitons of equal amplitude with the following initial pulse waveform

Fig.

uð0; sÞ ¼ A expb�p2ðZÞðs� T 0ðZÞÞ2c þ A expb�p2ðZÞðsþ T 0ðZÞÞ2c ð15Þ

where A is determined through the energy width- relationship. The normalized pulse separation 2T0 = 5.0. The contourplot of pulse interaction in the DM system is shown in Fig. 3 for different c values. As conjectured earlier, the pulseinteraction is influenced by the presence of quintic nonlinearity. An important feature of pulse interaction is the reduc-tion of collision length (Lc) with the increase in the value of the strength of the quintic nonlinearity jcj. The variation ofcollision length with the strength of quintic nonlinearity has been depicted in Fig. 4 for different dispersion map strengthS. The variation of collision length Lc with map strength S has been depicted in Fig. 5. It is obvious that the collisionlength decreases with the increase in the value of jcj for S < 2.0. DM system possessing smaller map strength is more

2. Shape of solitons while propagating through 250 DM cells. S = 2. (a) c = 0.0, (b) c = �0.02, (c) c = �0.05, (d) c = �0.1.

–5 0 50

50

100

150

200

Z

–5 0 50

50

100

150

200

τ–5 0 5

0

50

100

150

200

(a) (b) (c)

γ = 0.0 γ = –0.05 γ = –0.1

Fig. 3. Contour plot of soliton interaction in the DM system with quintic nonlinearity. d1 = 3.02, d2 = �2.52, hdi = 0.25, (a) c = 0.0,(b) c = �0.05, (c) c = �0.1.

1 2 3 40

50

100

150

200

Map strength S

Col

lisio

n Le

ngth

Lc

|γ | = 0.00

|γ | = 0.05

|γ | = 0.10

Fig. 5. Variation of collision length Lc with dispersion map strength S for different strength jcj of quintic nonlinearity.

0 0.02 0.04 0.06 0.08 0.150

100

150

200

250

300

350

400

450

Col

lisio

n Le

ngth

Lc

S=0.6

S=1.4

S=2.0

S=2.8

|γ|

Fig. 4. Variation of collision length Lc with the strength jcj of quintic nonlinearity for different dispersion map strength S.

S. Konar et al. / Chaos, Solitons and Fractals 29 (2006) 823–828 827

828 S. Konar et al. / Chaos, Solitons and Fractals 29 (2006) 823–828

influenced by the quintic nonlinearity. As the map strength increases, the influence of quintic nonlinearity diminishes.Influence of quintic nonlinearity is negligible for a DM system with map strength S > 2.0.

4. Conclusion

In conclusion, soliton transmission through dispersion managed optical communication systems taking into accountof quintic nonlinearity have been presented. Following a variational analysis a set of ordinary differential equations wasderived. These equations have been solved for a symmetric map. It has been noticed that quintic nonlinearity has onlymarginal influence on the overall propagation characteristics of a single isolated soliton. Full numerical simulation re-veals that the quintic nonlinearity influences significantly the collapse distance between neighbouring pulses of the samechannel. It has been found that the collision length decreases with the increase in the strength of quintic nonlinearity forS < 2.0. DM system possessing small map strength is more influenced by the quintic nonlinearity.

Acknowledgements

This work is supported by the Department of Science and Technology (DST), Government of India through theR&D grant SP/S2/L-21/99. Manoj Mishra and S. Jana, would like to thank DST for providing Junior ResearchFellowship.

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