DSpace at VNU: Crosstalk effect in the case of three monomode plan wave guides

VNU . JOURNAL O F S C IE N C E , Mathematics - Physics, T.XXI, N02, 2006

C R O SS T A L K E F F E C T IN THE CASE
O F TH R EE MONOMODE PLAN WAVE GUIDES
D inh V an H oang, Mai H ong H anh
College o f sciences V ietnam N ational U niversity
A b stra c t. In th is paper, we exam ined the crosstalk effect in the case of th re e
monomode pro p ag atin g wave in th ree plan wave guides.
On th e basis of solving p ro pagatin g wave equations, we have received the
influence of s tru c tu re p a ra m e te r as the refractive index difference, the lengths
of p ro p a g a tin g waves, th e d iam eter of wave guides, th e sep arated distance
betw een two adjacen t wave guides etc... on the crosstalk effect.
Key words: wave guides optics, optical communication.

L I n tr o d u c tio n
Since th e n in e tie s of la s t cen tu ry , m ankind h as gone into th e period of infob rea k out. By th e tech n iq u e WDM, one can obtain a larg e gigabit a t far in te rv a l of
optical tra n s m iss io n line. H ow ever, one of th e defects in th is m u ltican al
com m unication is th e exhib itio n of c ro sstalk effect - th e power exchange betw een
th e two w aves p ro p a g a tin g in two ad jacen t canals. T his phenom enon re s u lts in the
noise of in fo rm a tio n which needed exclude.
T he c ro ssta lk effect h a s been studied in the case of two ad jacen t can a ls th a t
m ay be co nsidered as two p la n wave guides [1-4]
In th is p a p er, we h ave e n la rg ed th e resea rch to th e case th re e ad ja ce n t plan
w ave guides. On th e b a sis of resolving th e p ro p ag atin g wave equations p re se n te d in
section 2 , we h av e m ade a stu d y of th e influence of s tru c tu re p a ra m e te rs of wave
guides
as th e d iffe re n t of refractiv e index, th e len g th of p ropagating wave,
the
d iam ete r of w ave guide e tc... on th e c ro sstalk in te rv al - a ch arac teristic q u a n tity of
c ro sstalk effect. T hese re s e a rc h ’s re s u lts have been in d icated in section 3. A t last,
discussion a n d conclusions h ave given in section 4.

2. B a sic e q u a tio n s
We supposed th e re a re th re e p lan wave guides in which the p lan w aves
p ro p ag ate follow ing th e Oz directio n as seen in fig 1
T hese wa-ve guides h av e th e w idths of lị, l2ĩ / 3, th e refractive index n u n 2i n 3
an d s e p a ra tin g d ista n c e s of d u d 2 .
T he p ro p a g a tin g w aves h ave forms:
.E i (y,z) = a iul(y)e~jp'*

(1 )

E 2(y,z) = a2u2(y)e~J^

(2)

34


C ro s sta lk Effect in th e C ase o f Three M on om ode P la n W ave G u id e s
E 3{y,z) = azu A y ) e ilhz

35
(3)

H ere a u a 2, a 3 - c o n stan ts, /?!, jS2, /?3 - p ro p ag atio n c o n sta n ts, u x(y)f u 2(y), ỉ/3(y)
- am p litu d e functions of waves.
W hen th e pow er exchange betw een th e wave guides a p p eared , c o n sta n ts a x
become functions slowly changed by z.

The H elm holtz e q u atio n for each wave, in th is case, h a s th e follow ing form:

V2i£. +k?Eị = - S m

(i,m = 1,2,3)

(4)

H ere th e source Sm d e m o n stra te s th e field of one wave suffered th e influence
of th e a n o th e r wave. Follow ing [ 1 ] we can give
s m = K - n 2) K K = K - k 2) K

(5)

27Ĩ
W ith k0 = — - wave num ber, A - velocity of lig h t in vacuum .
A

From (4), (5), we have the system of equations for three plan wave guides, as follows.
v

2£ ,

+ k fE l = - ( k ị - k 2)E 2

(6 )

V 2E 2 + % E 2 = -[(* 32 - k 2)E3 + (kỉ - k 2)Ex]

(7)

V % +k%E3 = - ( k l - k 2)E 2

(8)

d2a
Solving th is system of eq u atio n s a fte r th e a p p ro x im atio n of neg lectin g —
dzÔCL■
before — - , we received a new system of equations:
dz


Dinh Van Hoangy Mai Hong H an h

36

(9)

^
= - j C 2la2(z)eJ
dz

( 10)

dz
%

= - j C l2aẠ z)e-j^

- jC 32a3(z)e-J^

( 11)

w ith
&p\ —P\

$2 >^03 ~ Pz

02
2

C 32

=

k ĩ —ị? \
0/?
I u3(y)u2(y)dy
p2 Ả

c 23 = o n

í

2

í

u3(y)u2(y)dy

2/*3

2

T his system is solved n u m erically for d ifferen t cases, depending on th e diverse
form s of function Ui(y).

3. T he in flu e n c e o f str u c tu r e p a ra m eter o f w ave g u id e s on th e c r o ssta lk
in terv a l
3.1 D e fin itio n : C ro ssta lk in te rv a l L 0 is th e in te rv al d eterm in ed since the
tran sm issio n of lig h t in one w ave guide begins u n til th e power exchange a p p ears.
3.2 E xpression o f fu n c tio n Uị(y) a n d va lu es o f p a r a m e te r . We ta k e for
function Ui(y) th e follow ing expressions
( 12 )

Uj(y) = Ae s'y-,u2(y) = Be Sỉy;u3(y) = Ce s*y
w here A = c =1, B =1, <51=<53=1,<52=2.
N um erical v a lu e s of p a ra m e te rs are chosen
Ằị = Ă2 = y?3 = 133ịxm.a^ - a2 = a 3 = 1Ịum.cl^ = d 2 = d3 = 1CT4m
nx = n 2 = n 3 = 1.5 \n = 1.4999

3.3 The influence o f refra ctive index difference on L0
U sing M atlab lan g u ag e and s ta rtin g from (9) - (12), we plotted th e curves
|a-(2)|2v ersu s

z.

In

fig.2,

a re

p resen te d

th e

c u rv e s|a t(z)| w hen

An = (nx - n) = (n 2 - rt) = (n3 - n) varies.
From figure 2 a n d ta b le 1, one can see th a t th e dim inution of An re s u lts in th e
in crease of L 0.


C ro s sta lk E ffect in th e Case o f T hree M on om ode P la n W ave G u id e s

37

T a b le 1
An

0.01

0.001

0.0001

L 0(m)

23.585

235.136

2350.8

F ig.2. The diag ram allow s to determ in e th e dependence of L 0 on An

3.4 The depen den ce o f L0 on s e p a r a tin g d ista n c e d l9 d 2
By th e sam e m ethod of calculation w ith all o th e r p a ra m e te rs rem ainin g
anchanged b u t dị varies, we obtained fig .3 and tab le 2
T a b le 2
ư,=rf 2(m)

10-5

10 “

10-3

10-2

L q(m)

2351.2

2350.8

2353

2369.6

Fig. 3. T he d iag ra m allow s to determ in e th e dependence of 0 on d lt d 2

T he received re s u lts show th a t L 0 is alm ost u n ch an g ed w hen dj varies.


Dinh Van Hoang, Mai Hong H anh

38

3.5 The influence o f w ave length on L0
In th is case, we v aried only th e len g th of Aj. The re s u lts from fig.4 and tab le 3
in d icate th a t th e in cre ase of Aj will lead to in th e au g m e n ta tio n of L 0 i.e. th e
c ro sstalk effect will dim in ish a t longer w ave lengths.

T able 3
A (nm)
L 0(m)

1.08

1.33

1.55

1909.1

2350.8

2739.4

Fig. 4. The d iag ra m allow s to d ete rm in e th e dependence of L 0 on A.

3.6 The charge o f L0 when a m p litu d e o f wave varies
W hen th e a m p litu d e B of wave p ro p ag atin g in th e second wave guide is
v aried, we obtained fig 5 and tab le 4.

Fig. 5. The d iag ram allow s to d e te rm in e the dependence of L 0 on B


C ro s sta lk Effect in the Case o f Three M onom ode P l a n W ave G u ides

39

T a b le 4
B

1

2

3

L 0(m)

4707.5

2350.8

1567.2

From fig 5 and tab le 4, L 0 decreases by in c re a sin g B. T his show s th a t the
stro n g in te ra c tio n betw een th e w aves p ro p ag atin g in 3 w ave guides re s u lts in the

in crease of c ro sstalk effect.

4. D isc u s sio n and c o n c lu s io n s
From o b tain ed re s u lts p resen ted above, we could reveal some following
conclusions:
- The
p a ra m e te rs
influence of
p ro p ag atin g

c ro sstalk effect depends clearly on th e change in th e stru c tu re
of wave guides. The m ost sensitive p a ra m e te rs which dim in ish the
c ro sstalk effect a re refractive index difference An and wave length A
in wave guides.

■ The distribution of am plitude functions Uj can create the transformation of
cro sstalk effect. T he obtain ed re s u lts also in d icate t h a t d ifferen t wave functions will
give d iverse crosstalkv effect and th is point needs to be fu rth e r exam ined.
- The m ethod of calculátion used in th is p a p e r m ay be applied to th e case of
m ore th a n 3 w ave guides or th e case of m ultim ode w ave guides.

Reference:
1.

A .Y ariv, Q u a n tu m E lectronics T h ird E d itio n , J o h n W iley & Sons, N.Y. 1988
as

applied

to

Microwave

2.

H .H u an g , Couple M ode Theory
T ra n sm issio n , N e th e rla n d s 1984.

and

O ptical

3.

T. T am ir, G uided Wave O ptoelectronics, S p rin g er-V erlag N.Y. 1990

4.

B.E.A S aleh , M .c Teich, F u n d a m en ta ls o f p h o to n ics, Jo h n W illey & Sons N Y
1991.