Controllability of Solitons Propagating in Optical Fiber Transmission System

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Copyright © IFAC Large Scale Systems: Theory and Applications, Osaka, Japan, 2004

IFAC PUBLICATIONS www.elsevier.comllocatelifac

CONTROLLABILITY OF SOLITONS PROPAGATING IN OPTICAL FIB ER TRANSMISSION SYSTEM Akihiro Maruta·

• Graduate School of Engineering, Osaka University, 2-1 Yamada-oka, Suita, Osaka, 565-0871 Japan TEL : +81-6-6879-7728, FAX: +81-6-6879-7688 e-mail : [email protected]

Abstract: We review theoretical studies on transmISSIOn control and dispersion management techniques which have introduced in optical soli ton communication systems to increase the bit rate and extend the transmission distance. A systematic procedure to synthesize a control system is presented and the ability to reduce the timing jitter arising from the amplifier noise is compared for several control systems. The dispersion-managed(DM) soli ton solution is also presented. Copyright © 2004 IFAC Keywords: Optical fibers, Optical pulses, Optical nonlinearities, Controllability, Variational analysis

than the ideal soliton affected by the obstacles. In a real system, band pass filters, synchronized modulaters, and/or saturable absorbers as controllers are periodically inserted in the transmission line. In a dispersion managed line , anomalous dispersion fiber and normal one are alternatively con catenated in an amplifier spacing to achive low averaged dispersion and high local dispersion simultaneously. We have fortunately found that, in addition to a single-hump soliton, a family of multi-hump soli tons can propagate in a properly designed dipersion managed line(Maruta et al., 2002) .

1. INTRODUCTION

Optical soliton in a fiber is formed by the balance between the group velocity dispersion and the Kerr nonlinearities(Hasegawa and Kodama, 1995). Since the waveform of the 'ideal' soliton propagating in a transmission system doesn't change along the link, it is t.he remarkable advantage for long-distance transmission. In reality, the Kerr nonlinearity introduces unavoidable obstacles such as interactions between a soli ton and amplifiers' noise or adjacent solitons even in a single channel case. For multichannel case in wavelength division multiplexed (WDM) system, cross phase modulation (X PM) and four wave mixing (FWM) between channels degrade the transmission quality severely and limit the maximum bit rate and link length . To overcome these difficulties, the concept of 'soliton control(Kodama et al., 1994)' and 'dispersion management(Hasegawa et al., 1997) ' have been introduced . The properly designed controller and/or dispersion management makes a soliton robuster

2. OPTICAL FIBER TRANSMISSION SYSTEM The master equation which describes behavior of optical pulse propagation in an optical fiber transmission system is given by the perturbed non linear Schrodinger equation (NLSE) as

.oE

Jh.o 2 E

ITz-2

629

ot 2

2

.

+SIEI E=lgE+cR E

,

(1)

where E(z, t)[v'W], z[m], and t[s] represent the complex envelope of electric field, the propagation distance, and the retarded time , respectively. 112(z)[s2/m], S(z)[l/(mW)], and g(z)[l/m] are fiber's GVD , non linearity, and loss for 9 < 0 or gain for 9 > 0, respectively. eRd v'W /m] expresses the perturbation term representing harmful effects for transmission and/or transmission cont.roller. Without loss of generalit.y, Eq .(l) is transformed into

112' a2u

.au t

az' - ""2 &t 2 + lul

2

u

= eR... ,

(2)

where /J '( ') fJ2

z

112(Z')

= 8(z')a 2 (z')

( u~,t~'

) _ E(z', t) a() z,

=

eR... z'(z)

1

[Wv'W] ,

2

dA _ b2 (Z) A 2C R dZ 2 P + A,

(3)

dp dZ

9«()d(] ,

= u(z', t)v'lJ,

Z

[I/\\,] .

= I"z'~

t

T = -

to

.au

b2 (Z)

a2 u

8Z - -2-8'J'2

=

2

+ IUI u = eRa

dZ clK dZ

= RI<

dTo dZ

= b2 (Z)K + RTo

le

- -INA 2Ie

2+ Rc ,

,

K2) 2

-

5IN 2 +A + Ra ,

4h

1:

h T2)f dT ,

-00

Rp

=

Rc

=I1I A 100 {hRe[e]U + 2rf.) L e

h:e A

Im[e] {le - hT2)f dr ,

-00

-2C Im[e]Ue - hr2)f} dr , . (5)

100 {Re[e] f. + C Im[e]rf} dr , RI< = -00 2 100 Im[e]rI dr , = TA P -00 R9 = - 21 1A 100 {p R.e[e] (31 + 2rI.) 2p -I A L

RTo

,

(9)

,

1I =I 100 Im[e] ( 3Ie 2 L e

RA

(4)

Substituting Eqs.(5) into Eq .(2) , we obtain the following dimension less equation, t

C+Rp,

with

With using the period, L', and the pulse's minimum full width at half maximum (FWHM) in a period, tors], we introduce new dimensionless parameters to normalize Eq.(2) defined by U(Z , T)

3

dO 2 -dZ = b2 (Z) (ID -p h

L

S«()a 2 «()d(

= b2(Z)p

-dC = -b 2 (Z)p 2(ID - + C 2)

Z

Here we consider a system in which 112(Z), 8( z ), and g(z) in Eq .(I) are periodic functions of z with their common period of L[m] . This means that 112'(Z') in Eq.(2) is a periodic function of z' with its period of

= 10

T(Z, T)

, (8)

where f( T) is a bell-shape, even and real function of T and the parameters, A(Z) , p(Z), C(Z) , K(Z), To(Z), and O(Z) , represent the amplitude, the reciprocal of pulse width , the chirp, the center frequency, the t.emporal center position, and the phase of the pulse, respectively. By applying the variational approach to Eq.(6) with the ansatz of Eq .(7), we obt.ain the following coupled differential equations of the pulse parameters(Sugahara et al., 1999) .

and ao is a dimensionless constant which is determined below.

L'

(7)

= C(Z)T2/2 - K(Z)Tjp(Z) + O(Z) = p(Z){T - To(Z)} ,

4>(Z, T) {

[I/\\,] ,

S«()a «()d(

[l

f(T) exp(i4» ,

with

z

a(z) = ao exp

= A(Z)

[v'W] ,

eRE S( z')a 2 (z')

=

U(Z, T)

(6)

L

=

where b2(Z) 112'(Z)L' /t'6 and eRu L'..Ji}eR.s. Equation (6) is equivalent to Eq.(I) and the variations of l12(z), 8(z), and g(z) can be represented by a single variable, b2 ( Z), which represents the fiber's dimension less effective dispersion.

L

P

-00

+4K Im[e]r f} dT ,

where R.e[e] and Im[e] represent the real and the imaginary part of e!lue-iq" respectively. Ii , (l L, D, C, N) is defined in Table 2 and

=

I.

3. VARIATIONAL APPROACH

= dl/dr.

In the non-dissipative case in which the even part of Im[e] 0, the pulse's dimensionless energy Eo which is defined by

=

We here assume a solution of Eq .(6) to have the following form with linear chirp,

630

( 10)

Eo

=

oo?

£'

A2

to

P

IUI- dT = -eo = - h .

/ -00

is conserved. Here

eo =

I:

lul 2

where

(11)

(15)

dt [J] represents

Here 6(Z) is t.he delta function and J1.,., is a constant which represents the magnitude of ASE noise.

the pulse's energy averaged in a period when ao in Eq.(3) is determined by ao

£

=

Let us start to synthesize some transmission controllers . Equation (6) is a nonlinear partial differential equat.ion which degree of freedom is infinite, whereas the degree of freedom is reduced to only four in Eq .(9) . Therefore it is convenient to synthesize a transmission control system based on Eq.(9) with control elements listed in Table l.

(12)

4. ANALYSIS OF PERTURBED SOLITON

First, we consider a control system combined BLA with NLG that includes the parameters (00,02,1'1. 1'2) . The following is a synthesis pr
The bahavior of an optical soliton in a fiber where variolls controller .,re installed can be analyzed by using Eqs .(9) . The perturbat.ion term, t:Ru consists of harmful terms for transmission and control terms. The former includes the higher order effects of the optical fiber such as the intrapulse Raman scattering(IRS), the third order dispersion(TOD), and the nonlinear dispersion (NLD) which causes an amplitude-dependent shift of the soliton's temporal position, interactions between neighboring pulses, interactions between neighboring channels in a WDM system, and amplifier's spontaneolls emission (ASE) noise. The latter is the bandwidth-limited amplification (BLA) in combination with the nonlinear gain(NLG), the sliding frequencH'F) efft:,~t, the synchronous amplitude modulator(AM) or the synchronous phase modulator(PM) . For these perturbative terms, ~,(i= A,p,C,K,To,B) in Eq .(9) is summarized in Table 1.

(i) Determine the relation between parameters which satisfies d1//dZ d,,/dZ 0 at (1/,11:) (1,0) in Eqs.(9) . This means that (1/, K) (1,0) becomes a singular point.

=

o < 02 < 2( 30 0 + 1'd

(17)

Eqs.(16) and (17) are t.he necessary condition for (1,0) to be an asymptotically stable point (ASP) in 1/ - " phase space. Table 3 summarizes necessary condition to have an ASP for typical control systems.

(1/, K)

=

Let us consider the timing jitter arising from amplifier noise to compare the ability of the above mentioned transmission control systems. The linearized ODEs of the deviation from the ASP of soliton parameters (~1/, ~'" ~To, ~9) including 8i (Z) are the coupled Langevin equations. The variance of I, he deviation of soli ton parameters can be derived by solving the Langevin equations. The variance of the center position, ~To(Z) for Z ~ 1 is as follows.

=

2A

=

(ii) Linearize Eqs .(9) in the vicinity of the singular point and derive the nece.ssary condition that the trajectory in 1/ - K phase space asymptotically approaches to the singular point.

We here focuse on the conventional soliton case which corresponds to b2 (Z) -1 in Eq.(6) and I( T) sech T in Eq.(7) . In the case, it is convenient to introduce a new variable, 1/(Z), which simultaneously represents the pulse's energy, the pulse amplitude, and the reciprocal of pulse width . Since 1/ is related to A and p through 1/ = A 2 /p,

d1/ _ 2A dA A 2 dp dZ - P dZ - p2 dZ

=

(16)

5. SYNTHESIS OF TRANSMISSION CONTROL SYSTEM

=

=

A2

(~TJ(Z)) '" 2J1.1C Z3 /3 2(J1.IC/A 2 + J1.To)Z 2[(4J1.,., 8 2 + 9/J"A2) /(3A 2 - 28 2 )2 + J1.To]Z J1.,,/{AD(A + D)} + J1. To/ D {J1." + (E2 + F)J1.To }/(EF)

= pRA - --p;:Rp . (13)

=

The pulse's chirp C 0 for the conventional soliton case. Therefore the coupled ODEs of pulse parameters can be reduced to four equations of 1/(Z), K(Z), To(Z), and B(Z). The aut
=

=

=

: no control

: NLG+BLA (18)

: SF + BLA : AM+BLA : PM+BLA

where A 40:2/3, 8 40:21/5/3, C = 40:21/011:0 , D 2maYo cosech 110 (110 coth Yo - 1), E 40:0,

=

(14)

631

=

Table 1. Effects of perturbations on evolution of the pulse paramet.ers. perturbation ASE noise

I

Re

~Ru

TOD NT"O BLA(Oth) BLA(lst)

S(T,Z) u(IUI"jrU ib3UTTT/6 -i8(1UI"Ujr iOroU ()l UT

BLA(2nd)

i02UTT

SF NLG (1st) NLG (2nd) AM

{oTU ;"'IIUI'U

im" c06(OT)U

PM

-mpcos(OT)U

TRS

OroA -()l",A -02A[P" {3Ie(lD +C'le ) +h(h - Id - C2 h)} +2",2 hJe]/(21rJe) 0 'YIA3(3IeIN - hIe) /(21IJe) 'Y2A~(3Iclf - h I g) /(21L I e) m"A(3IeJ" - I L.lb) X cos(OTo)/(2lL 'e) 0

i'Y2IUI U

I

"Ru

R..

Rr.c

S(T,Z) u(lUI')TU ib3UTTT/6

S .. (Z) -4uA'p'Io /h 0

NLD BLA(Oth) BLA(lst) BLA(2nd)

-is(IUI"Ujr

iOroU C>1 UT iOr2UTT

-!J/NA"p"C/h 0 -2C>tP'(lD + C'Ie)/ h -4Or21
STn(Z) 0 hJ{P«lD + C'lIe)/h +",2}/2 3sINA21
SF NLG(lst) NLG(2nd) AM PM

EoTU i')tIUI'U i'Y2IUI'U im" c08(OT)U -mp cos(OT)U

Eo 0 0 2m" sin(OTo )pCJcI 1£ mp ";n(OTo)O.Jo / h-

0 0 0 -2m" sin(OTo)JcI(pILl 0

perturbation ASE noise JRS

TOO

Tahle 2. lntegral formula of Symbol

I Integrand F(.-r) I

J - exp( _.,.2 /2)

J" (M" "'"J" J4 2 J UTY

.../1< .../'/(/2 .../'/(/2

h ID

le IN la

"'
..1'/(/2/4

.,.2 J4 ~

Ig

.,.2J6

.1" .Ib .le

r

c08(2x.,.)

.,." J" cos(2x.,.) .,. P 9in(2x.,.)

~f2/(2p).)

,,'/6 4/3

Id

Ro So(Z) 0 b3",{p'UD - C'le )/(2h) _",2/3} U .. INA'",/(4h) 0 - 2Or t l cl
2 2/3

4/15 7'/('/120 '/('/18+ 2/3 1<2/9 - 2/3

h l~

F(r)dr . (z == f2/(2p), Y ==

..1'/(/2 ..1'/(/2/4 3.../'/(/4 3../'/(/4

If

1:

Sc(Z) 0 b3",p'(lD/Ie +C') .• IN",A'/(21e ) 0 0 Or2P'C (2Ie(lD + C'Ie) -h(6Id - 311. + 2C2 h)} /(lLle) 0 -htA'CUcIN rL I.) /(hle) 2'Y2A'C(lc l f JrJ g ) /(hle) -2m"C(leJ" - hJb) X c.oR(OTo)/(TI.!e) -m p cos(OTo )OJc/(ple)

Sp(Z) 0 b3"'p3C 0 0 0 - 02p3{Je(lD + C
S,dZ) 0 bJ ",Ap·C/2 0

..1'/(/3

16/15

../1
41<2/45 - 2/3 2y cosech 11

1r' coeech y {cothy - y (coth' y - 1/2)} 1< cosech y (y coth y - 1 )

Table 3. Necessary condition to have an ASP for control systems. Yo == 1rf2/2 and the modulation frequency f2 is determined by the hit ;ate. control system NLG + BLA SF + BLA AM + BLA PM

+ BLA

I

parameters Qo,o'2,"'Yl,')'2

il'O,02,CO

I

ASP

necessary condition

('1,"') - (1,0) ('1,,,,) - ('lO,Ko)

Eqs.(5) &. (6) OrO/0r2 - '10/ 3 + KQ, EO/ Or 2 - 4'15Ko/3, 0< 3V6IEol/(4Tjg) < Or2 02 3(00 + ma!ID cosech !ID), 0< (3ma!ID/2)C08ech !ID (!ID coth !ID - 1) < Or2 02 3ar o , 0 < (O/2)VmpYo c08ech Yo < 00

,O

('1,I<,To)

= (1,0,0)

oQ,02,mp,O

('1,1< , To)

= (1,0,0)

ao,o2,m Q

=

=

632

and F m p S1 2yo cosech yO. Eqautions (18) shows that all control systems can reduce the timing jitter and the synchronous modulators are more effective controller than the simple BLA.

=

N

dp dZ { dC dZ

= b2 (Z)p'3 C 2

IN

R=().9 1(J2

~ 10-' ~

=0

. ~ 10" III

-~

,

2(ID 2) = -b (Z)p le + C -

2leh Eop .

10" 1O·1lI _LS-~_.lJl.l.~----7o-~-"'ill---~---'s

Tune T

(19)

Fig.

Equation (11) is Ilsed to derive the second equation of (19) . Soh ·ng Eqs.(19) numerically for an appropriate initial values, p(O) and C(O), and a given constant Eo, we can determine an approximation of single DM soli ton 's energy Eo with which P(Z) and C(Z) are periodic functions with their period of 1. Only for simplicity, we consider a system in which b2 (Z) b1 for IZ - nl < f1/2 « 1/2) and b2 (Z) b2 for 11/2 < IZ - nl« 1/2) and n is an integer. We introduce the following three system paramet.ers, the path-averaged dispersion B == bl /1 + b2 (1 - fd , the map strength S == Ibtlf 1 +lb2 1(1 - fd, and the ratio of accumulated nonlinearity in the fiber of b2 (Z) bl to the Latal accumulated nonlilltrity /( == fl' which completely characterize a single DM soliton(Maruta et al., 2002) .

=

P

ci

6. DM SOUTON SOLUTION In the unperturbed case in which t:Ru Eq .(6), Eq .(9) can be reduced as

I~r-~--.---~-r--~~~-=-Z-_~O-' 8=-0.1 .... 'Z-0.45 WO S=2.2 - -Z-0.5

=

=

Figure 1 shows the DM soliton's waveform obtained by the numerical averaging method(Cautaerts et al., 2000) for (B, S, R) (-0.1,2.2, 0.9). In the averaging method, the initial pulse and the pulse after propagating for a DM period are numerically averaged after adjusting their phases. Repeating the averaging processes successively, one can dig up a periodically stationary solution if it exists. The Gaussin ansatz, f( T) exp( _T 2 /2), is simple, but it fits on the main-lobe of DM soliton well . It can be then used to calculate the pulse energy of the initial input in the numerical averaging method.

=

=

7. INTERACTION BETWEEN NEIGHBORING DM SOLITONS The intrachannel interactions between neighboring DM solitons is one of the remaining obstacle to achieve a transoceanic high speed system . They cause timing and power jitters and the DM solitons finally collide. The variational approach by the use of Eq.( 1~) can be restrictedly applied to the interaction problem(Inoue et al., 2000). This is

1. Waveform of OM (B, S, R) = (-0.1,2.2,0.9).

soliton

for

because the int.eractions are caused by the overlap of pulse's tails. The tail can not be represented by the Gaussian ansatz . We study the nonlinear interactions between inphase neighboring DM soli tons with their initial spacing of T. by numerically solving Eq.(6) in which e:Ru = O. At the input, a couple of chirpfree Gaussian pulses are launched into the DM line. Each initial pulse 's energy is a single OM soliton's approximat.e energy obtained by solving Eq.(19) and pulses' widths are the same. We firstly calculate the interaction distance of a couple of in-phase DM solitons. The definition of the interaction dist.ance Z/ is the shortest propagation distance at which the pulse-to-pulse spacing increases or decreases more than the pulse's FWHM(Inoue et al., 2000) . Figure 2 shows the interaction distance Z/ for various Sand R with B = -0.1, T. = 3. One can see that a couple of pulses can propagate over more than 1000 periods with slightly few time position shift at specific parameters around 2 < S < 2.7 and 0.75 < R < 1. In the parameter range, we may expect the existence of a periodically stable solution which consists of a couple of DM soli tons. To verify the presumption, we apply the numerical averaging method for digging out the fine structure of the periodically stable solution 's waveform. Two adjacent Gaussian pulses are used for the initial input. Figure 3 shows the obtained waveform of a periodically stable solution which consists of a couple of DM solitons for (B, S, R) (-0.1,2.2,0.9). We have named the new kind of DM soli ton as "bi-soliton"(Maruta et al., 2002). Figure 4 shows the waveform of anti-phase bi-soliton for (B, S, R) (-0.1,2.2,0.9).

=

=

633

• •

0.8 0.6

R

El

.. . . .

o

0.4

Z.>IOOO I oo
also shown a DM soliton solution as a periodically stat.ionary pulse propagating in a OM line. In addition, we have introduced a new kind of DM soliton which is named bi-soliton.

0.2

RF-FF-RF-NCES

0 I

Cautaerts, V., A. Marul.a and Y. Kodama (2000). On t.he dispersion-managed soliton. Chaos, 10 , pp .515-528. Hasegawa, A. and Y . Kodama (1995). Solitons in Optical Communications , Oxford University Press, Oxford . Hasegawa, A., Y . Kodama and A. Marut.a (1997). Recent progr~ in dispersion managed soli ton transmission technologies. Opt. Fiber Technol., 3, pp .197-213. Inoue, T., H. Sugahara, A. Maruta and Y. Kodama (2000) . Interactions bet.ween dispersion managed soli tons in optical-timedivision-mulLiplexed system. IEEE Photon. Technol. Lelt., 12, pp .299-301. Kodama, Y ., A. Maruta and A. Hasegawa (1994) . Long distance communications with soli tons. Q7Iantum Op!., 6, pp.463516. Maruta, A., T . Inolle , Y. Nonaka and Y. Yoshika (2002) . Bisoliton propagating in dispersion-managed syst.em and its application to high-speed and long-haul opt.ical transmission. IEEE J. Selected Topics in Quantum Electron. , 8, pp.640-650 . Sugahara H., H. Kato, T . Tnoue, A. Maruta and Y. Kodama (1999). Optimal dispersion management for a wavelength division mult.iplexed optical soliton transmission syst.em. IEEE/OSA .T. Lightwave Technol. , 11, pp . I547-1559 .

s

Fig. 2. Interaction distance Z/ for various 51 and R with 8 = -0 .1, T.. = 3.

Ill' 8=-0.1

'"

P

10"

cl ;:J

UT'

0

.~

S=2.2 R=O.9

ler ler

Cl)

1:: ......

10-" 10.10 8

0

·8

Time T Fig. 3. Waveform of in-phase bi-soliton for (8,8, R) = (-0.1,2 .2,0.9).

Ur

--z-o

8=-0.1

'"...-..

· ···· Z-0.45 - - Z-O.5

S=2.2 R=O.9

10"

E-<

ft

l N IO'-" ;:J

10'"

0

.~

ler

.s

10-"

Cl)

10·)0

o

-8

4

8

Time T Fig. 4. Waveform of anti-phase bi-soliton for (8 , S, R) (-0 .1,2.2,0.9).

=

8. CONCLUSIONS \Ve have summarized a systemat.ic procedure to synthesize a transmission control system based on the variational met.hod and compared the ability to reduce the timing jitter for some typical control systems. The similar manner is applicable to the other harmful perturb at ions such as the interactions between neighboring solitons, the interactions between neigh boring channels in WDM system, and tiber's higher order effects. We have

"

634