ELS EV 1ER Physica D 123 (1998) 267-270

PS'YSICA

Quasi-soliton for ultra-high speed communications A k i r a H a s e g a w a 1

a Department of Electronics and Information Systems, Osaka UniversiO, 2-1, Yamada-oka, Suita, Osaka 565, Japan

Abstract

Properly designed variations of fiber (group velocity) dispersion can propagate quasi-solitons, new soliton-like pulse which allows significantly improved transmission property by reducing much of the intrinsic problems of solitons. Copyright (~) 1998 Elsevier Science B.V. All rights reserved.

Keywords: Optical soliton; Optical communication; Quasi soliton; Dispersion management

1. Introduction

It is almost a quarter of a century since the discov- ery of the optical soliton in fiber and suggestion of its use for high speed communications has taken place [ 1 ]. However, a soliton communication system has not yet been implemented, in part due to conservative atti- tude of communication industries, but also due to the fact that a soliton system has not demonstrated deci- sively its merit over conventional (NRZ) systems. The latter fact originates from the following three intrinsic problems. First, the inefficient use of the frequency bandwidth which originates from the necessity that two adjacent solitons representing 1-1 bits must be placed with a separation of more than six pulse width in order to avoid their interaction. Second, the time jitter caused by amplifier noise (the Gordon-Haus ef- fect [2]) which originates from the fact that a soliton requires a negative dispersion to balance the nonlin- earity. Third, the radiation of non-soliton components and frequency shifts when a soliton goes through dis- continuities such as localized amplifiers or when it

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collides with solitons in other wavelength channels. Attempts have been made to reduce these effects by means of soliton control by filters [3-5] and signif- icant improvements in transmission rate have been achieved but the net results have not yet provided convincing argument to let an operating company to deploy a soliton system. Since much of these prob- lems are the consequences of the ideal soliton proper- ties, they may be solved by de-emphasizing the role of nonlinearity in balancing the dispersion and by re- placing it's role partially by the linear compression mechanism. The quasi-soliton is the terminology used by Kumar and Hasegawa [6] for a nonlinear station- ary pulse that is produced based on this concept. The quasi-soliton is minted upon various approaches de- veloped recently in soliton based communication re- search.

2. Quasi-soliton

Let us consider the propagation of a nonlinear pulse in the anomalous dispersion fiber of length Z i. The complex amplitude q of the light wave in a fiber with variable dispersion d ( Z ) is described by the nonlinear

0167-2789/98/$19.00 Copyright (~) 1998 Elsevier Science B.V. All rights reserved. PII: S0167-2789(98)00126-2

268

Schr6dinger equation

. Oq d ( Z ) 02q +~fq l2 q = - i f ' q , ~-5-~ + ~ - o'1"2

for 0 < Z < Z I , (1)

where ot is the normalized nonlinear coefficient. Oscillations in amplitude due to fiber loss can be eliminated by introducing the reduced amplitude u, with q = a ( Z ) u and a ( Z ) = e x p ( - F Z ) . Here, u satisfies

OU de(Z' ) 32u + otlul2 u = 0, (2) i oz--- S + ~ OT 2

where

Z

z'= faZ(Z) dZ, and d~(Z')- d(Z) aZ(Z ) . (3) 0

Kumar and Hasegawa have constructed a stationary (Z'-independent) localized solution of Eq. (2) by in- troducing a chirped pulse through

u = ~ v ( p ( Z ' ) r , Z') exp i - - ~ r 2 , (4)

where p and C are the real functions of Z t. Substitut- ing Eq. (4) in Eq. (2) and using the coordinate r = p T, we obtain

OV dep 2 02v K1 r 2 p i - - + - - + otplvl2v -- - - v, (5)

OZ r 2 Or e 2

where

+ C2de [~ = - C p d e , K1 - (6) p3

Here we choose the dispersion profile such that KI is a constant. If we further impose,

dep = const. (= 1), (7)

in Eq. (5), we have

3v 132v ( Klz 2 ) i f f -~ + ~-~r 2 + alvf2 2 v = 0, (8)

Z ~ where Z" = f o p(s ) ds. If ot = 0, Eq. (8) represents the wave equation for the linear harmonic oscillator, while if K1 = 0, Eq. (8) gives the ideal NLS equation. Note in Eq. (8) that both the nonlinear and the linear

A. Hasegawa/Physica D 123 (1998) 267-270

2

,---, 1.5

~- 0.5

-10 l0

/-- / ~e" Soliton

...... / ~Quaii-s°l i t°n

-5 0 5 Normalized Time z

Fig. 1. Plot of f(r;ot, Kl) vs. time ~ (solid line) with Ki = 1, K2 = -0.15915 and a = 0.5. The broken line shows the soliton solution of the same pulse width [6].

potentials have attractive (or trapping) sign. Equation (8) was solved by means of IST [7] but the antisym- metric solution obtained there is not appropriate for the present purpose. Kumar and Hasegawa [1] con- structed the stationary solution by applying the gauge transformation,

v = f ( r ) exp( iK2Z") (9)

where f satisfies

l d Z f 4-otf2 K 2 f _ K I ~ 2 dr 2 - -~-r- j = 0. (10)

Eq. (10) is solved numerically. Fig. 1 shows f ( r ; a, Kl) (solid line) with ot = 0.5 and Kl = 1, and the broken line shows the soliton solution of the same pulse width, i.e. Kl = 0, K2 = r/2/2, and f = 07/4"~)sech(oT). The soliton requires about 2.9 times the power of the quasi-soliton represented by f ( r ; ~ , KI) for K1 = 1. As Kl increases, the width of the potential function decreases and the pulse width becomes shorter for the same amplitude and the function f approaches the Gaussian shape. When KI = 0, the bounded solution of Eq. (10) is given by ideal soliton shape, f = 0//~/-~)sech(r/T) (complete nonlinear regime) and when ~ = 0, the fundamental eigenfunction of the harmonic oscillator is given by f = exp(--v/-K-l-r2/2) (linear regime).

The corresponding dispersion profile can be ob- tained from Eqs. (6) and (7) as

de(Z ' ) -~ cosh(3Z') + C(0_._~) sinh(3Zt),

= v/Kt + {C(0)} 2, (11)

A. Hasegawa/Physica D 123 (1998) 267-270 269

1 N

"~ 0.8

o 0.6

.~ 0.4

~ 0 . 2

I I I I

0.2 0.4 0.6 0.8

Normalized Distance Z

Fig. 2. The dispersion profile for various C(0) and F with KI = 1. Solid line shows the profile with C(0) = -1.34, F = 0.36, broken line with C(0) = -0.4, F = 0.1, and dotted line with C(0) = -0.8, F = 0 [6].

I Iql ~ - ~ - ~ 30

o 5 .~v~ A ' o , ~ ~ ~ s ,~.~," ,-ea Ti,~e ~ 20 " ÷°~

Fig. 3. Quasi-soliton interaction over a length of 2000km. Para- meters: C(0) = -1.34, Z, = 4.15, KI = 1, a = 0.5, g = 0.835, and F = 0.36 [6].

and

d(Z) = de(Z') e x p ( - 2 F Z ) . (12)

Fig. 2 shows the dispersion profile for different values of C(0) and F.

Using Eqs. (6), (7) and (11), the fundamental solu- tion of Eq. (1) is obtained as

3.18× 10 -16 cm2/W, fiber loss = 0.2dB/km, ampli- fier to = 2.83ps, z0 becomes 16km. The pulse width of the quasi-soliton (Fig. 1) is found to be 1.46t0 and the peak power is 12 mW. We see stable transmission over this distance. Due to the reduced interaction, this example gives transmission rate of approximately 100 Gbit/s.

e x p ( - F Z ) . ( T ) q(T, Z) = ~ J Z

[. C(Z') T2 ] x exp L t 2 + i~6(Z') , (13)

where

Z t

f C(Z') = --;3, fl(Z') = K9 p(s) ds, (14) de -

o

with de(Z') givn by Eq. (11). The chirp C as given by the first expression of Eq. (14) reverses its sign at a distance Z = Ze where de = 0. In order to recover the initial chirp at Z = Za, one needs to install a normal linear dispersion element (such as a grating) which reverses the chrip. When this is done, the quasi-soliton achieves its periodic property at Z = NZa.

Fig. 3 shows the simulation result of the quasi- soliton transmission for the following set of para- meters: fiber dispersion (at Z = 0) is - 0 .5 ps2/km, wavelength = 1.55 Ixm, effective cross sectional area Aeff = 501xm 2, nonlinear coefficient n2 :

3. Conclusion

Ultra-high speed optical communication is at- tracting world-wide interests as the 21st century infra-structure for computer based information age. Most agrees that soliton will play major role as the means of transmission, however, it has not yet demonstrated decisive merit over linear transmis- sion scheme because of it's intrinsic problems. The quasi-soliton concept is developed to overcome these difficulties and to demonstrate decisively its merit as the mean of information carrier of expected terabit/ second ultra-high speed network in the coming century.

References

[1] A. Hasegawa, ED. Dappert, Transmission of stationary optical pulses in dispersive dielectric fibers, I. Anomalous dispersion Appl. Phys. Lett. 23 (1973) 142.

[2] J.P. Gordon, H.A, Hans, Random walk of coherently amplified solitons in optical fiber transmission Opt. Lett. 11 (1986) 665.

[3] A. Mecozzi, J.D. Moores, H.A. Hans, Y. Lai, Soliton transmission control, Opt. Lett. 16 (1991) 1941.

270 A. Hasegawa/Physica D 123 (1998) 267-270

[4] Y. Kodama, A. Hasegawa, Generation of asymptotically stable optical solitons and suppression of the Gordon-Haus effect Opt. Lett. 17 (1992) 31.

[5] M. Nakazawa, K. Suzuki, E. Yamada, H. Kubota, Y. Kimura, M. Takaya, Experimental demonstration of soliton data transmission over unlimited distances with soliton control in time and frequency domains Electron. Lett. 29 (1990) 729.

[6] S. Kumar, A. Hasegawa, Quasi-soliton propagation in dispersion managed optical fibers, Opt. Lett. 22 (1997) 372.

[7] R. Balakrishnan, Soliton propagation in nonuniform media Phys. Rev. A 32 (1985) 1144.