September 1, 1996 / Vol. 21, No. 17 / OPTICS LETTERS 1351

Relaxation of guiding center solitons in optical fibers

Michele Midrio, Marco Romagnoli, and Stefan Wabnitz

Fondazione Ugo Bordoni, via B. Castiglione 59, 00142 Roma, Italy

Pierluigi Franco

Pirelli Cavi s.p.a., viale Sarca 222, 20146 Milano, Italy

Received March 25, 1996

We present a perturbative study of the mutual coupling between solitons and dispersive waves in periodicallyamplified links. Our analysis describes the limits of soliton transmissions operating beyond the averagesoliton regime. 1996 Optical Society of America

The existence of optical solitons in optical f iber systemswith lumped amplifiers can be understood from theconcept of the average or guiding center soliton.1 Thisdescription, which is valid as long as the amplifier sep-aration is much shorter than the dispersion distance,as a f irst degree of approximation averages all per-turbations over the amplif ier spacing. On the otherhand, when one is modeling fiber lasers or high-bit-ratetransmissions it is important to consider situations inwhich the amplif ier spacing is comparable with thedispersion distance. The resulting resonance (ornonlinear phase matching) between the periodic am-plification and the solitons leads to the generation ofdispersive waves at a discrete set of sidebands:analytical descriptions of this process were presentedin Refs. 2 and 3. However, these perturbative treat-ments do not quantitatively reproduce the numericalresults in the practical case of strongly perturbedsolitons, e.g., for amplif ier gains of 10 dB (or amplifierspacings of 50 km) or larger.4

We show that soliton perturbation theory can stillbe applied to describe strongly perturbed solitons,provided that the dynamics of the soliton amplitude istaken into account in a self-consistent manner in thecomputation of the generated radiation.

Pulse propagation in a periodically amplified f iberlink obeys, in soliton units, the perturbed nonlinearSchrodinger equation1

i≠V≠Z

112

≠2V≠T2 1 jV j2V

­ 2i

"G 2 G

1Xn­2`

dsZ 2 nZAd

#V , (1)

where ZA denotes the spacing between the amplif iersof gain G ­ expsGZAd 2 1 and G is the f iber loss. Theperiodic amplitude variation of the soliton amplitudecan be removed with the substitution V ­ AsZdU , soEq. (1) reduces to1

i≠U≠Z

112

≠2U≠T 2 1 jU j2U ­ iP ; AsZd jU j2U , (2)

where AsZd averages to zero and is given by

0146-9592/96/171351-03$10.00/0

AsZd ­ repZA

"exps22GZd 2

1 2 exps22GZAd2GZA

#

­Xnfi0

an expsinkAZd .

Here repZAindicates that the function is periodically

repeated with period ZA. Moreover, kA ­ 2pyZA,and the Fourier coefficients read as an ­ 2f1 1

inpysGZAdg21.In the absence of perturbations (i.e., with P ­ 0),

Eq. (2) has the one-soliton solution5 U sT , Zd ­ US ;2h sechs2hT dexps2ih2Zd. Whenever P fi 0, the solu-tion of Eq. (2) can be written as U ­ US 1 UR , whereUR is a radiative field. In the average soliton regime,which is defined by the condition ZA ,, 1, the solitonamplitude (and time width) remain approximately con-stant with Z, and the level of generated radiation re-mains small. On the other hand, whenever ZA ­ O s1dthe simulations show that the soliton amplitude hrelaxes to lower values.1 Physically, through theshedding of energy into the continuum the solitoncontinuously reshapes itself so that the average solitonregime is gradually recovered.

Note that Eq. (2) has a Hamiltonian structure, incontrast with Eq. (1). As a consequence, the totalenergy

C0 ­ CS 1 CR ; 4hsZd 21p

Z 1`

2`

dw logf1 2 jbsw, Zdj2g

(3)

is a conserved quantity of the field U that obeys theperturbed Eq. (2).6 Here CS, R represent the solitonand radiation energies, respectively, and b is the off-diagonal element of the Zakharov–Shabat scatteringmatrix.5 By applying the perturbed inverse scatter-ing transform theory,6 – 8 one obtains that the evolutionof b obeys the simple linear equation

≠bsw, Zd≠Z

­ 2iw2b 2 iAsZd sw2 1 h2dUSp, (4)

where USsw, Zd ­ p sechfpwys2hdgexps2ih2Zd is theFourier transform of US . Note that b is related tothe associated f ield f that was introduced by Gordon2

through the relation f sw, Zd ­ bpsw, Zdyf4sw2 1 h2dg.

1996 Optical Society of America

8818(t )

1352 OPTICS LETTERS / Vol. 21, No. 17 / September 1, 1996

By assuming a slowly varying h (in fact, h isapproximately constant over ZA), one readily obtainsfrom Eqs. (3) and (4) the variation of h and b from thelth to the sl 1 1dst amplif ier as

hfsl 1 1dZAg ­ hslZAd 21

4p

3Z

log

8<: 1 2 jbsw, lZAdj2

1 2 jbfw, sl 1 1dZAgj2

9=;dw ,

(5)

bfw, sl 1 1dZAg

­ bsw, lZAd 1 sw2 1 h2djU j exps2iw2Zd

3X

mfi0am exps2iDmlZAd

exps2iDmZAd 2 1Dm

, (6)

where Dm ­ 2sw2 1 h2d 2 mkA. The iterative solutionof Eqs. (5) and (6) with hs0d ­ 1y2, bsw, 0d ­ 0 yieldsthe desired evolution of the soliton amplitude.

An analytical expression for h can be derivedas follows. Let us assume, first, that h re-mains approximately constant between Z ­ 0 andZm ­ mZA: Eq. (4) yields

jbsw, Zmdj2 > sw2 1 h2d2p2 sech2

√pw2h

!Zm

2

3X

n[VR

janj2 sinc

√DnZm

2

!2

, (7)

where VR denotes the set of integers n that label thesolutions wn ­ 6snkAy2 2 h2d1/2 of the equation Dn ­0. Note from relation (7) that, as Zm grows larger, b issharply peaked around the resonance frequencies wn.

By further assuming that jbj2 ,, 1, we can simplifythe logarithm in Eq. (3). Moreover, whenever Zm .. 1one obtains sinc2sDnZmy2d ! pys2wnZmddsw 2 wnd.These steps lead to

hsZmd 2 hs0d ­ hsZmd 2 1y2

­ 2Zm

4

Xn[VR

√nkA

2

!2janj2

jwnshdjjU shdj2 . (8)

By assuming in Eq. (8) that h varies slowly withdistance Zm, and expressing Eq. (8) in terms of thecontinuous distance Z ­ mZA, one obtains

dh

dZ­ 2

14

Xn[VR

µnkA

2

∂2 janj2

jwnshdjjU shdj2. (9)

We can find an approximate closed-form solution ofEq. (9) by expanding Eq. (9) in Taylor series aroundhs0d ­ 1y2. We obtain

hsZd ­12

2

XAnXBn

"1 2 exp

√2

XBnZ

!#, (10)

with

An ­janj2

Wn

√pkAn

4

!2

sech2spWnd ,

Bn ­An

Wn

"1

2Wn1 2nkAp tanhspWnd

#,

where the index n [ VR and Wn ­ snkAy2 2 1y4d1/2.Note from Eqs. (9) and (10) that the soliton amplitudedecreases with the propagation distance and the rateof loss depends on the spectral intensity of the solitonrelative to the resonance frequencies wn.2

To verify the accuracy of the above perturbativeresults we can compare their predictions with the nu-merical solutions of Eq. (1). Let us consider a f iberloss coefficient a ­ 0.2 dBykm and an amplif ier spac-ing zA ­ 50 km, so the distributed f iber loss is compen-sated by 10 dB of amplifier gain. Figure 1 comparesthe perturbative and numerical results for the growthof the radiation energy CR ­ CS sZ ­ 0d 2 CS sZd ­2 2 4hsZd as a function of the real distance z for dif-ferent values of ZA. The continuous curves were ob-tained from the solution of the map [Eqs. (5) and (6)],whereas the discrete points were obtained from the nu-merical solution of Eq. (1). We used a time-periodicsplit-step integration method. We included absorbersat the edges of the computational time window toprevent the radiation emitted by the guiding centersoliton from reentering through the boundaries. Wecomputed the radiation energy by calculating hsZdthrough the numerical discrete spectral transformmethod of Ref. 9. In Fig. 1 the amplifier spacinggrows from ZA ­ 1 to ZA ­ 4. As can be seen fromFig. 1, perturbative and simulation results agree well,except for at initial stage of relaxation for large valuesof ZA.

This agreement is generally maintained if the contin-uous description of the soliton decay, as in Eqs. (9) and(10), is employed. In fact, Fig. 2 compares the numeri-cal results with these perturbative evolutions of thedispersive wave energy. In this figure the continu-ous curves refer to perturbative equations, whereas

Fig. 1. Perturbative (continuous curves) and numerical(discrete points) evolutions of the dispersive wave energyCR .

September 1, 1996 / Vol. 21, No. 17 / OPTICS LETTERS 1353

Fig. 2. Same as Fig. 1 with analytical (continuous curves)results from Eq. (10) for the case ZA ­ 1 and from Eq. (9)for the others. The curves marked with crosses are fromperturbation theory with a constant-amplitude soliton.

Fig. 3. Soliton amplitude 2h and the corresponding valueof the effective amplif ier spacing ZA

eff ; 4h2ZA at a fixeddistance of 1000 km as a function of the amplifier spacingZA. Continuous curves, from Eq. (9); discrete points, fromnumerical simulation.

the discrete points show the results of the simu-lations. In the case with ZA ­ 1 the solid curvewas obtained for analytical expression (10), whereasfor ZA ­ 2, 3, 4 Eq. (10) is less accurate, and thetheoretical continuous curves were obtained fromEq. (9). In this figure we also show the radiationenergy (see the pair of curves marked with crosses)predicted by Eq. (8), where the soliton reshapingis not taken into account in the computation of the

generated energy [i.e., h ; hs0d on the right-handside of Eq. (8)]. As can be seen, in this case theperturbative results highly overestimate the genera-ted radiation for ZA . 1.

Figure 3 compares numerical (squares) and per-turbative [the solid curve is obtained from Eq. (9)]results for the dependence of the guiding center soli-ton amplitude (and width) 2hszd on the amplifierspacing ZA, where z ­ 1000 km. The circles and thedashed curve show the corresponding effective am-plifier spacing ZA

eff ; 4hszd2ZA. As can be seen,the reshaping of perturbed guiding center solitonsprevents effective amplifier spacings larger than 2.Whenever a dispersion-shifted (step-index) fiber withgroup-velocity dispersion be ­ 21 s220d ps2ykm is con-sidered, ZA ­ 4 corresponds to the case of an initialsoliton with a FWHM value of 6 (27) ps. At 1000 km,these solitons are reshaped through continuum energyemission into 9- (41-) ps solitons. On the other hand,Fig. 3 shows that the initial pulse shape is maintainedalmost unchanged for ZA # 1, which is well above thelimit of validity of the average soliton theory.

We thank T. Georges for having sent us a preprintcontaining similar results obtained with a different ap-proach.10 The research of M. Midrio, M. Romagnoli,and S. Wabnitz was carried out in the framework ofthe agreement between the Fondazione Ugo Bordoniand the Italian Post and Telecommunication Adminis-tration and under the ESTHER program of Euro-pean Economic Community/Advanced CommunicationsTechnologies and Services contracts.

References

1. A. Hasegawa and Y. Kodama, Opt. Lett. 15, 1443(1990).

2. J. P. Gordon, J. Opt. Soc. Am. B 9, 91 (1992).3. S. M. J. Kelly, Electron. Lett. 28, 806 (1992).4. J. N. Elgin and S. M. J. Kelly, Opt. Lett. 18, 787 (1993).5. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34,

62 (1972).6. A. Hasegawa and Y. Kodama, Solitons in Optical

Communications (Oxford U. Press, Oxford, 1995),p. 123.

7. D. J. Kaup and A. C. Newell, Proc. R. Soc. London Ser.A 361, 413 (1978).

8. J. N. Elgin, Phys. Rev. A 47, 331 (1993).9. G. Boffetta and A. R. Osborne, J. Comput. Phys. 102,

252 (1992).10. T. Georges and B. Charbonnier, Opt. Lett. 21, 1232

(1996).