Effect of polarization orthogonalization in wavelength division multiplexing soliton transmission system

15 September 1994 OPTICS COMMUNICATIONS Optics Communications 111 (1994) 39-42

EI£EVIER

Effect of polarization orthogonalization in wavelength division multiplexing soliton transmission system R. Ohhira, M. Matsumoto, A. H a s e g a w a Department of Communication Engineering, Faculty of Engineering, Osaka University, 2-1, Yamada-oka, Suita, Osaka 565, Japan Received 9 March 1994; revised manuscript received 14 June 1994

Abstract We show that the polarization orthogonalization between the adjacent WDM channels, combined with guiding filters, reduces significantly the deviation in the pulse arrival time caused by initial overlap.

1. Introduction

2. Theory

A major concern in wavelength division multiplexing (WDM) soliton transmission systems, is the biterror mainly caused by the frequency shift (velocity change) which originates from initial overlap [ 1 ]. Interactions between two solitons in one channel have been studied extensively and they are found to depend sensitively on phase, amplitude and polarization of the interacting solitons [2]. Practically noninteraction property of orthogonally polarized solitons in one channel, in fact, has been suggested to construct a polarization multiplex soliton transmission system [ 3 ]. In this manuscript, we study effectiveness of the reduction of soliton interactions between two different channels having orthogonal polarizations on the initial overlap by solving the coupled nonlinear Schr6dinger (NLS) equations and by the use of perturbation theory, and show that the deviation of center positions of two solitons having orthogonal polarizations at different channel are reduced approximately to a third compared with the case of identical polarizations. Insertion of frequency filters [4-6], however, is essential.

The coupled NLS equations for solitons having different polarizations with appropriate frequency filters are given by

. Ou

(

2

)

lul2 + -~lvl 2

. ~ 02u u=i~lU+l]Jl~--~,

(1) . Ov

102v

1~--Z + ~

+

(

2

)

. . 02v

Ivl2 + 71uf v = iaRv +l/JRb-~ , (2)

where, p = 1 for an identical polarization and p = 3 for an orthogonal polarization, u and v represent a faster and slower soliton, respectively. We consider here the guiding center variables [7] under the condition of the amplifier spacing Za being much smaller than the collision length Lcou [8] as well as the dispersion distance. 6 represents the excess gain and fl the curvature of the guiding frequency filter. Here, the four-wave mixing terms are ignored in the coupled NLS, Eqs. (1) and (2) because the phase matching condition is generally difficult to be met in a single

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102u

IO--Z+~T~+

R. Ohhira et al. / Optics Communications 111 (1994) 39-42

40

mode fiber. In order to obtain analytical estimates o f the effectiveness o f the polarization orthogonalization and guiding filters, we use the energy and m o m e n t u m conservation relations derived from Eq. ( 1 ),

aj

d--Z"

dr~2 _ 232r/2 - 2f12( ½r/~ + x2r/2), dZ d Z --

5f12K2r/2+

[Ul 2

~

(7)

dT,

(8)

where we put here

lul2dT

--cx)

v ( T , Z ) = r/2 sech r/2(T +

K2Z)

x e x p [ - i K 2 T + i(r/~ - K22)Z/2] --OO

and

--c~

oc

dZ --

u ( T , Z ) = r/, sech rh[T + (Ax + K I ) Z ]

- u-fiT) aT

u*

× exp{--i(AK + K1 ) T + i[r/~ - (AK + K, )2 ] Z / 2 } .

0o

~ -u-g-~)dT --

oo

--

o c

\ OT 2 0 T

+231

-

i4p f

I~

OT z OT

'2°1ul2 -0~ dT,

(4)

to construct perturbation results. The adiabatic change o f the amplitude and velocity parameters r/l and K| for u is obtained from substituting one soliton solution for u, dr/1 = 26,r/1 - 2fll(lr/~ dZ

dXldZ --

4 2+ ~]~lKlnl

+ K~r/l),

~1 a?

(5)

201u12"~

Ivl --ff-T-~a~,

(6)

The coupling terms (second terms in the right-hand sides of Eqs. (6) and (8)) are effective only when u is close to v in time. Therefore, (r/l, Xl ) and (r/2, K2) converge respectively to (1,0), for a choice o f bl = 3ill and b2 = 3fl2 i f u and v are separated [5]. We solve Eqs. (1) and (2) numerically to obtain the frequency variation Ox and the center position deviation b T. In the numerical simulations, instead o f a parabolic filter as shown in the right hand side of Eqs. (1) and (2), we use the frequency response o f a Fabry-POrot filter in order to simulate a realistic situation. To calculate 0K and ST, we used the following relations.

bT = W -l f

TlqlZ dT,

(9)

t / --oc

OK = W - 1 1 m f

~--~qdT. Oq*

(10)

--O~3

r)12dr

where we put

u ( T , Z ) = 711 sech

r/l(T +

Here, W = ff_~ Iq(Z, and q = u or v. Eqs. ( 5 ) - ( 8 ) are used to compare with the simulation results. Results of the numerical simulation and the perturbation theory are shown in Fig. 1. Here, the initial conditions are chosen to be r/1 = r/2 = 1,xl = x2 =

KIZ)

x e x p [ - i K i T + i(r/~ - K ~ ) Z / 2 ] and

0andAx

v ( T , Z ) = r/2 sech r/2[T + ( - A K +

K2)Z]

x e x p { - i ( - A K + x2 ) T + i [r/~ - ( - A K + K2 )2 ] Z/2}. Here, AK designates deviation from the normalized center frequencies of u and v. In a similar manner, one finds for parameters 1/2 and K2 for v

=

10.

Fig. 1 shows the variation of the deviation of the center frequency (a) and center position (b) of a soliton induced by initial overlap as a function of the propagation distance Z normalized to the dispersion distance. In these figures, solid and dotted curves show the results o f the perturbation theory described

R. Ohhira et aL / Optics Communications I11 (1994) 39-42 0

.~.---'~

0

x

41

l

i

x. -o-

X

-0.02

-0.02

-0.04 ,,-1 ~r' ¢)

-0.06

/×

/

-0.08

-0.1

e-

i

-0.12

:;

-0.14 0

- -

Orthogonal

........

Identical

-0.) i

r..)

Polarization

-0.12 i

Polarization

I

I

I

I

I

5

10

15

20

25

-0.14

30

o

?

l

I

I

I

I

- -

Orthogonal

........

Identical

0.8 x

x

x

0.6

x

?

o

I0

?

without

x

filtering

o

?

15

i

I

b

i

o

20

i

filtering

?

o

25

i

o x

b

Polarization

30

i without filtering filtering

4

x

o

x x x ..................................

..........

r" 0

o o

3

o o

Q.

2 0.4

o

Normalized Distance Z

Polarization

×

o

5

Normalized Distance Z

~

X

-0.06

f:'

O

"~

X

X

-0.04

-0.08

o.)

i X

X

o

..

o o

~

o

o

o

o

o

0 0

o

I

,."

0.2

o x

i

l

I

I

I

5

10

15

20

25

30

Normalized Distance Z Fig. I. Variation of the central frequency ~c (a) and the central position To (b) along the normalized distance of propagation Z due to the initial overlap (Ax = 10, ~ = 0.05). The solid and the dotted curves show the results of the perturbation theory. The results of the numerical simulations are plotted as crosses and circles. The discrepancy between the simulation adn the perturbation theory originates from the amplitude oscillation in the simulation produced by the filter. Significant reduction of the center frequency deviation as well as the central position deviation are seen for the orthogonal polarization. by Eqs. (9) and (10), while circles and crosses are the simulation results. Either results show clear reduction of the deviation both in the center frequency and center position for the case of orthogonal polarization as expected from Eqs. ( 1 ) and (2). The discrepancy between the perturbation theory and simulation seen in Fig. 1 is the consequence of the amplitude variations caused by filters in the simulation results. In order to check the effectiveness of the guiding center filters,

x

x

x

x

x

x

x

x

I

I

f

I

I

5

10

15

20

25

x

30

Normalized Distance Z Fig. 2. Numerical simulations for variation of the central frequency x (a) and the central position TO (b) as a function of the normalized distance of propagation Z due to the initial overlap (Ax = 10) with and without the guiding filters. It is clear that the insertion of filter is essential. we run simulations without filters. Fig. 2 shows the comparison between simulation results with (shown already in Fig. 1 ) and without filters. From those resuits, it is clear that the help of the guiding filters is essential in achieving the desired results. In summary, based both on theory and simulation, we have shown that providing orthogonal polarizations to solitons at different wavelength channels at the input of a guiding filter, significantly reduces the deviation of the central position induced by the initial overlap. When more than two channels are used in W D M , the effect of initial overlaps can be reduced by providing orthogonal polarizations only to adjacent channels because two channels with sufficiently large frequency separation suffers insignificant devi-

42

R. Ohhira et al. /Optics Communications 111 (1994) 39-42

ations. In addition to the effect of polarizations, we have also studied effects o f difference in amplitudes between different channels. However the results are mixed; although the smaller amplitude deviates less, the large amplitude deviates more. This research is in part supported by G r a n t in Aid for Specially P r o m o t e d Research of The Ministry o f Education, Science and Culture.

References [1 ] Y. Kodama and A. Hasegawa, Optics Lett. 16 (1991) 2O8

[2] C. Desem and P.L. Chu, Optical Solitons-Theory and Experiment (Cambridge University Press, 1992). [3] S.G. Evangelides, L.F. Mollenauer, J.P. Gordon and N.S. Bergano, Lightwave Techn. 10 (1992) 28. [4] A. Mecozzi, J.D. Moores, H.A. Haus and Y. Lai, Optics Lett. 16 (1991) 1841. [5] Y. Kodama and A. Hasegawa, Optics Lett. 17 (1992) 31. [6] A. Mecozzi and H.A. Haus, Optics Lett. 17 (1992) 988. [7] A. Hasegawa and Y. Kodama, Phys. Rev. Lett. 66 (1991) 161. [8] L.F. Mollenauer, S.G. Evangelides and J.P. Gordon, Lightwave Yechn. 9 ( 1991 ) 362.