=1, <51=<53=1,<52=2.

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

C R O S S T A L K E F F E C T IN THE CASE

O F TH R EE MONOMODE PLAN WAVE GUIDES

D in h V an H oang, Mai H ong H anh College o f sciences V ietnam N a tio n a l U niversity

A b s tra c t. In th is p ap er, we exam ined th e crosstalk effect in the case of th re e monomode p ro p a g a tin g wave in th re e plan wave guides.

On th e b a sis of solving p ro p a g a tin g wave equations, we have received th e influence of s tr u c tu r e p a r a m e te r as the refractive index difference, the le n g th s of p r o p a g a tin g waves, th e d ia m e te r of wave guides, th e s e p a ra te d d istance betw een two a d ja c e n t wave guides etc... on the crosstalk effect.

Key words: w ave 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 an k in d h a s gone into th e period of info- b re a k out. By th e te c h n iq u e WDM, one can o b tain a larg e gigabit a t far in te rv a l of optical tr a n s m is s io n line. H ow ever, one of th e d efects in th is m u ltic a n a l co m m u n icatio n is th e ex h ib itio n of c ro sstalk effect - th e pow er exchange b etw een th e two w av es p ro p a g a tin g in two a d ja ce n t can als. T h is phenom enon re s u lts in th e noise of in fo rm a tio n w hich n eed ed exclude.

T he c ro s s ta lk effect h a s b een stu d ied in th e case of two a d ja ce n t c a n a ls th a t m ay be c o n sid ere d as two p la n w ave guides [1-4]

In th is p a p e r, we h av e e n la rg e d th e re se a rc h to th e case th re e a d ja c e n t p lan w ave guides. O n th e b a sis of reso lv in g th e p ro p a g a tin g w ave eq u atio n s p re s e n te d in section 2, we h a v e m ad e 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 re fra c tiv e index, th e le n g th of p ro p ag a tin g w ave, th e d ia m e te r of w ave g uide e tc ... on th e c ro sstalk in te rv a l - a c h a ra c te ristic q u a n tity of c ro ssta lk effect. T h ese r e s e a rc h ’s re s u lts h av e been in d ic ate d in section 3. A t la st, d iscu ssio n a n d conclusions h av e given in section 4.

2. B a s ic e q u a tio n s

We su p p o sed th e re a re th re e p la n w ave guides in w hich th e p la n w aves p ro p ag a te follow ing th e Oz d irec tio n as seen in fig 1

T h ese wa-ve g u id es h a v e th e w id th s of lị, l 2ĩ /3, th e refractiv e index n u n 2i n 3 an d s e p a r a 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 av e form s:

.E i (y,z) = a iul (y)e~jp'* (1)

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

34

C r o s s ta lk E ffect in th e C a se o f T hree M o n o m o d e P l a n W ave G u id e s 35

E 3{y,z) = azu A y ) e ilhz (3)

H ere a u a 2, a3 - c o n sta n ts, /?!, jS2, /?3 - p ro p ag a tio n c o n sta n ts, u x(y)f u 2(y), ỉ/3(y) - am p litu d e fu n ctio n s of waves.

W hen th e pow er exchange b etw een th e w ave g u id es a p p e a re d , c o n sta n ts a x become functions slowly ch an g ed by z.

T he 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 s tra te s th e field of one w ave su ffered th e influence of th e a n o th e r w ave. Follow ing [1] we can give

s m = K - n 2) K K = K - k 2) K (5)

2 7Ĩ

W ith k0 = — - w ave n u m b er, A - velocity of lig h t in vacu u m .

A

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

(6)

v 2 £ , + k fE l = - ( k ị - k 2)E 2

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

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

(7)

(

)

dz- ÔCL ■

before — - , we received a new sy stem of eq u atio n s:

dz

36 D inh Van Hoangy Mai Hong H a n h

^ = - j C 2la2(z)eJ

dz (9)

dz (10)

% = - j C l2a Ạ z)e-j ^ - jC 32a3(z)e-J^ (11)

w ith

&p\ — P\ $2 > ^03 ~ Pz 02

í

2 2

k ĩ — ị? \

C 32 = 0/? I u3(y)u 2(y)dy p2 Ả

2

c 23 = o n í u3(y)u2(y)dy 2/*3

T his system is solved n u m erica lly for d ifferen t cases, d ep en d in g on th e d iv erse form s of function _Ui(y).

3. T he in flu e n c e o f s tr u c tu r e p a r a m e te r o f w a v e g u id e s o n th e c r o s s ta lk

3.1 D e f in i ti o n: C ro ssta lk in te rv a l L0 is th e in te rv a l d e te rm in e d since th e tra n sm issio n of lig h t in one w ave guide begins u n til th e pow er exchange a p p e a rs.

3.2 E x p re ss io n o f f u n c t io n Uị(y) a n d v a lu e s o f p a r a m e t e r . We ta k e for function Ui(y) th e follow ing ex p ressio n s

w here A =

c =1,

=1, <51=<53=1,<52=2.

N u m erical v a lu e s of p a ra m e te rs a re chosen

Ằị = Ă

2

3.3 The influence o f r e fra c tiv e index difference on L0

U sing M atlab la n g u a g e a n d s ta r tin g from (9) - (12), we p lo tted th e curves

|a-(2)|2v e rsu s z. In fig.2, a re p re s e n te d th e c u rv e s|a t(z)| w hen An = (nx - n) = (n 2 - rt) = (n3 - n) v aries.

From figure 2 a n d ta b le 1, one can see t h a t th e d im in u tio n of An r e s u lts in th e in crease of L 0.

in te r v a l

Uj(y) = Ae s'y-,u2(y) = Be Sỉy;u3(y) = Ce s*y (12)

C r o s s ta lk E ffect in th e C ase o f T h re e M o n o m o d e P l a n W ave G u id e s 37 T a b le 1

An 0.01 0.001 0.0001

L0(m) 23.585 235.136 2350.8

F ig.2. T he d iag ra m allow s to d e te rm in e th e d ep en d en ce of L0 on An 3.4 The d e p e n d e n c e o f L0 on s e p a r a tin g d is ta n c e d l9 d 2

By th e sam e m eth o d of calcu latio n w ith all o th e r p a ra m e te rs rem ain in g anchanged b u t dị v a rie s, we o b tain ed fig.3 an d ta b le 2

T a b le 2

ư,= rf2(m) 10-5 10“ 10-3 10-2

Lq( m) 2351.2 2350.8 2353 2369.6

Fig. 3. T he d ia g ra m allow s to d e te rm in e th e d ep en d en ce of 0 on d lt d 2 T he receiv ed re s u lts show t h a t L0 is alm o st u n c h an g e d w hen dj v aries.

38 D inh Van Hoang, M ai Hong H anh

3.5 The influence o f w a ve length on L0

In th is case, we v a rie d only th e le n g th of Aj. T he re s u lts from fig.4 an d ta b le 3 in d ic ate th a t th e in c re a se of Aj will lead to in th e a u g m e n ta tio n of L0 i.e. th e c ro ssta lk effect will d im in ish a t longer w ave len g th s.

T ab le 3

A (nm) 1.08 1.33 1.55

L0(m) 1909.1 2350.8 2739.4

Fig. 4. The d ia g ra m allow s to d e te rm in e th e d ependence of L0 on A.

3.6 The ch a rg e o f L0 w hen a m p litu d e o f wave va ries

W hen th e a m p litu d e B of w ave p ro p a g a tin g in th e second wave guide is v aried , we o b tain ed fig 5 a n d ta b le 4.

Fig. 5. T he d iag ra m allow s to d e te rm in e th e d ependence of L0 on B

C r o s s t a lk E ffect in th e C ase o f T hree M o n o m o d e P l a n W ave G u id e s 39 T a b le 4

B _{1 2} 3

L0(m) 4707.5 2350.8 1567.2

From fig 5 an d tab le 4, L0 d ecreases by in c re a s in g B . T h is show s t h a t the stro n g in te ra c tio n b etw een th e w aves p ro p ag a tin g in 3 w ave guides re s u lts in the in crease of c ro sstalk effect.

4. D is c u s s io n an d c o n c lu s io n s

From o b tain ed re s u lts p re se n te d above, we could rev eal som e following conclusions:

- T he c ro sstalk effect d ep en d s clearly on th e ch an g e in th e s tru c tu re p a ra m e te rs of wave guides. T he m ost sen sitiv e p a ra m e te r s w hich d im in ish the influence of c ro sstalk effect a re refractiv e index d ifferen ce An a n d w ave len g th A p ro p a g a tin g in w ave guides.

■ The distribution of am plitude functions Uj can create the transform ation of c ro sstalk effect. T he o b tain ed re s u lts also in d ic ate t h a t d iffe re n t wave fu n ctio n s will give d iv erse crosstalkv effect a n d th is p o in t n eed s to be f u r th e r exam ined.

- T he m eth o d of c alcu látio n used in th is p a p e r m ay be ap p lied 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 2. H .H u an g , Couple M ode Theory as a p p lied to Microwave a n d O ptical

T ra n sm issio n , N e th e rla n d s 1984.

3. T. T a m ir, G u id ed Wave O ptoelectronics, S p rin g e r-V e rla g N.Y. 1990

4. B.E.A S aleh , M .c Teich, F u n d a m e n ta ls o f p h o to n ic s, J o h n W illey & Sons N Y 1991.