The coupling of two parallel dielectric fibers

Physica

47 (1970)

THE

501-514

COUPLING II.

0 North-Holland

OF TWO

CHARACTERISTICS R.

PARALLEL

OF THE

VANCLOOSTER

I.aboratorium

Publishing

DIELECTRIC

COUPLING

t and

~OOYKristallografie

IN TWO

FIBERS FIBERS

P. PHARISEAU

en Studie van de Vaste Stof,

Rijksuniversiteit

Received

Co.

Gent, Belgie

7 October

1969

Synopsis In a previous dielectric from could

fibers

a reciprocity be calculated

expressions the

paper r), it has been could

normal

be described

theorem. exactly. form.

how

a system

It was mentioned The

for the coefficients. mode

shown

with

Finally,

object

Further,

the coupling

between

of differential

how the coefficients

of the present the basic system

a numerical

study

for

work.

two

equations,

parallel derived

of these equations

is to obtain

will be discussed a particular

analytic passing

mode

will

to be

presented.

1. Introductiolt. First we evaluate by strictly theoretical methods all coupling coefficients and corrections, using the translational addition theorems for cylindrical vector wave functions. These calculations determine our basic system completely. This system represents a set of coupled first order linear equations. In order to solve that system most easily, we passed to the normal mode form, which decouples the system. Furthermore, it is shown how successive approximations lead to the determination of the beat-length. In the case of a HEii mode we have checked the theory numerically. The results are in agreement with the experimental curves of Bracey 2). Finally a comparison is made with the important work of Jonesa). 2. Computatio~z of coupling coefficients. 2.1. In a previous paper-l) we have shown that

Computation

of ~nfm,,~~s.

The electromagnetic fields are expressed as functions of m and n. In the case kn # 0, we have to evaluate the six following integrals:

1 Aspirant

of the “Nationaal

Fonds

voor

Wetenschappelijk

501

Onderzoek”

Belgium.

(2)

(6)

(71

Ai = 21,k12; 1; = A2, I.1 = 1.1,nrnl; The

modified

22 Hankel

kZ2,

il2, n?n 1 ;

h’

=

hkl2

h

=

12 ,,lr,l

; ;

1”

=y

l’kl2

I’

=

Pnrrr,

; ;

‘4 ;;; ”

=

A ;.j;

‘4 :;.”

7:

‘4 :;,::,

;

(8)

functions Z
~12 J/i

THE

COUPLING

&2Kk

,f,’ =

$

OF TWO

_

;i; T

$

2

K&(h’A;

The arguments

-

h&z)(hP

?I!?!

=

Ik(il2a2)

hh’PP’)

FIBERS.

II

503

-

-

(11)

h’P’).

Bessel

functions

are:

Kk = Kk(~;as);

;

K, =

Jn = J,(illal); Ik

DIELECTRIC

>

of the occurring

Jlc = Jk(%az)

(k; -

II,

2 -

PARALLEL

p&

;

K&24

=

;

[ as&ir2)

l.;..,;Kn*k

zc [ yy_

=

Kn*k(A2d);

(12)

From the expressions for the coupling coefficients in this case, we conclude that the EH and HE modes will always be coupled. One expects, however, the strongest coupling when lm > 0. This corresponds to coupling of modes of the same kind. As a trivial result, it was noticed that EH or HE modes do not couple with TE or TM modes, and conversely. Next we drew our attention to the case n = k = 0. It can be proven that there is no mutual coupling of TE(TM) modes with TM(TE) modes. In the other cases (coupling of TE modes, coupling of TM modes), we established, taking into account the integrals

==

12, aKoW4

2

Tt

au2

k2

lo(A24

(13)

Ko(A24,

J

--- -27~ 4the following

~2 --

+

:x

aIov2a2) Ko(M), aa2

(14)

results :

c*nLl.012 =

:<

Ko(liu2)

2

Jo(bl)

Jo(Ab2)

Ko(Aaa1)

Ko(G2)

~___

1

1

1

yi&

2%

n;3,

hh’PP’

~Ko Af -a;-- (A&z2) Io(lzu2)

2xK&d)

-

ng2 $

x

A&4;*

(12~2)

x

Ko(G2)]

2 m

>

0,

k > 0,

(15)

2.2. Computation of c,,,,,.,,+,. In this cast straightforward. The results of ow analysis at-(’ :

too,

all wlculations

;LI-(’

THE

COUPLING

OF TWO

PARALLEL

DIELECTRIC

FIBERS.

II

505

The other results are

m > 0, +

I > 0,

(22)

Jo

Jo(nla2)

Ko(&a~)

Ko(A2a2)

~O,,LS,OZl = @l

m < 0,

3. Com@&tion

1 __-

1

1

ipuw 37 -jjF

X

I! < 0.

of th,e corrections

(23) anmi. The correction

terms Snmr, which

are given by

will now be computed exactly. As the surface integral is taken over the cross-sectional area of the second fiber, we’ll use the addition theorems once more. The integrands turn out to be double series, which after integration reduce to single summations. The last summations are rapidly convergent, Eight integrals remain to be evaluated:

s

(n:“,‘,,,,x rng’n:J-e, dS =

:= -

s

“x ci Kn+s(Kn+s k; s

(vz~,~~~ x r$&)

-e,

Km)

dS =

= x C Kn+s(Kn+s + Kn-,) Bs, s

B,,

(31)

(32)

\vith

The summation integrals :

index

s runs

from

--oo

t9

-1 00 in c‘ach cast.

.I,,

N, arc

(38)

THE

COUPLING

OF TWO

The first of these integrals

PARALLEL

DIELECTRIC

can be calculated

FIBERS.

using standard

II

507

results

from

the theory of Bessel functions: A, = v(1 -

s) 1,1,-r

The second integral

+ f&

73

-

stz

(

can be evaluated

>

immediately

I2 s .

and gives:

B, = sl;. Other

(40)

correction

terms

such as ~&s,

will give rise to the same results

iBsmr = finma = a,,,). 4. Discussion of the system of basic equations. differential equations can be written as:

The

system

of coupled

with the obvious notations: 1 lusrn -

1-

1

and

&,

We suppose that fiber propagating

vnm =

1 + I&,

(45)

there are Nr(N2) possible modes in the first (second) in the sense of the z axis. A same number of modes

can propagate in the opposite sense. Each of the 2Nr modes in the first fiber is expected to interact with all modes of the second one. We can assume Nr = Ns = N, without any restriction and we put: q

%Ll -- al, . ..) _ %Sl

== olN+l,

. . .,

a&2

== “2N+l>

.

_ %I&

== aBN+l>

Olkll -

+

_

aN>

a&

=

a2N,

.,

ak+/2 =

a3N>

. . .,

0Lkk2 =

a4N.

(46)

Herein, s is the lowest value of the index m for PZ= 0; k is the greatest value of the index n, while t represents the greatest value of m for n = k.

In matrix

form, we can write:

--Ii In shorthand diu dz

notations,

we

7~ Ca,

for the with adequate definitions matrix N. 1. Next eve discuss the system

4iV

x

41V matrix

(‘, and

M’e perform a transformation from thcs set of variables a new set (@I, /S,, . .., 04~)) by means of thy relation : (S is a constant

u: = sg in order to obtain

equations

matrix).

arc no longer coupkd,

or

and \YVol,taill

11c

, 4X,

is then readily integrated. The only remaining matrix S from the relation = A

., ,Y,

a system

j -1 1) 2,

S-lCS

(~1, 32,

4iV ‘= 4rVmatrix),

(/I is a diagonal The 4N differential modes. A system like

the, co1

CS = .S,,l.

probl~nn is to det~rntin

THE

This results T(

COUPLING

OF TWO

in a homogeneous

Cdk-

PARALLEL

DIELECTRIC

set of 4N linear

equations

FIBERS.

11

509

:

with 4N unknowns

A(j) &k) skj = 0.

(54)

k=l

In order

to obtain

det(cij

-

a nonzero

solution,

we should

have

16~) = 0.

(55)

4N values of 1 satisfy this equation; introducing these values (54) all sii can be determined. The outlined method must be worked out numerically. 2. Introducing the simplification G,Ul,kl.J = 0;

C1&2,kZl

=

into equations

01

(56)

we get two systems, one for forward travelling waves, another one for backward travelling waves. The systems are still too involved in order to obtain solutions in an analytic form. 3. At last, we determine the amplitudes of the coupled modes in the case

CL,kU

+ %,,,2,kZl

C=Z0;

This means that only coupling For forward travelling modes,

m

0

for

nfk;

mfl.

between equal modes is taken the system reduces to:

(57)

into

account.

(58) (59) with q1,,,2 = a2;

c&1 == “1; Integration

of this system

wnC,t,,l,~L,,L2 = Cl

gives:

id cos bz + b sin bz + 012(O)$.

011(z) := eia

cos bz 121 +

c2.

h2

b=

id b sin bz

+ q(0)

$

&-clcz;

f = hl = h2 = h,

guides

1, 1,

(6’)

sin bz

d=---p.

hl

-

(62) h2

2

a--2;

In the case of identical

sin bz

(60)

(63)

we have b = lc11 =

1~21 =

ICI,

d = 0.

(64)

(65)

(66)

(671

for pnl,, and c,; ,,,, ,,,,,2 ;u-c’ nicntionvtl

Expressions

iir tlicl lost.

5. N~wmical calculations of coufiling corfficients allri cowcxti0~L.s. 5. 1 Espression for @‘(A). \I.e firstly determine the &~I-i\xti\;e of thv left hand side of the eigenvaluc~ equation @(/l) 1 0. It can he shmvn that in the case IZ :F 0, @‘(/L) has formally thv same \Ve can b:rite @(A) as follo\vs: @(IL)

=

k’;‘[A

(21,

v)

-

U(21,

7))

eslxession

/.-I (If,

71) +

for the t\vo kinds

H(,LL,

7’)

)

of modes.

(71)

\\-ith (721

(73)

THE

COUPLING

OF TWO

PARALLEL

DIELECTRIC

FIBEKS.

11

511

and 6 = k; -

c = k; + k;;

k;.

(74)

We notice that A(u, v) - B(u, v) = 0 in the case of EH modes, while A (24, v) + B(Zc, ZJ) = 0 in the case of HE modes. Using these relations, and the derivations of A and B with respect to h:

we obtain @‘(4

== k;lA (u, v) & B(sL, ~)][A’(zl, V) f ~‘(56, v)] =

(77) In the above expression modes. 5.2.

Expression

x

[k‘i(w

the upper (lower) sign is used for the EH

for ~~,,~i,,~,,~~.We found

+

“172)(Ko

rk

K2n)

+

(k’fql

+

kiq,)(Ko

for even modes (upper signs) and odd modes Introducing the notation N = K~vW’(h)(ak$l, ac n+ml,nrrL2=

(HE)

{K0[(4

-i-4)

VI+

24721

f

K2[(4

T

K2rL)l

(78)

(lower signs), respectively. we can write -4)

wl}(- l)n(-iW1. (79)

The indices of refraction

in the core and the cladding are given by ni and ns.

512

Again, the upper signs are foI- the evc’~I Illocl~~s, \Vllil~C the’ llJ\vc’~ SigllS for the odd modes. The summations illwlved run from - 00 to -t-co. L\.(, (‘al1 traIl~fo1-Ill exprcxssions to : 00

arc

tl1e

5.4. Numerical work. As already mentioned, WC‘ studied the HE311 mode. In order to compare with the results of Bracey, the case n/A”x 0.2146

THE

COUPLING

OF TWO

PARALLEL

0

DIELECTRIC

I

FIBERS.

II

513

I L

2

2

Fig. 2. Comparison

of theoretical

d/a

: theoretical _._.-.:

theoretical -

with experimental

results without

results.

correction;

results with correction;

: experimental

was examined. In fig. 2 the results agreement is very good.

results of Braceyz).

of this comparison

are shown.

The

5.5. Comparison in the work of Jones. Using a Green-function method, Jones obtained a system of differential equations. Our work led to a system of the same kind, using a different method. The results of A. L. Jones can be found in putting a,,

= 0

and

K nfk

v(

e-Mr’U) = -.--- -. a 1 JXViqii d

(83)

Finally, the expressions for the coupling coefficients seem to be less involved in our case. All square roots have been eliminated; the same expression was obtained for EH and HE modes. 6. Co?tclusio~z. Summarizing, in these articles a theory about the coupling of dielectric fibers has been presented, using a reciprocity theorem. Although the results are very satisfying, we believe that this theory could be extended, starting from a better knowledge of the electromagnetic field between the two fibers. Acknowledgments. This work is part of a research program on fiber optics, sponsored by NFWO (Belgium). One of us (R.V.) thanks this organ-

izatioil for a grant. \\‘e also wish to thank \Y. Dekinder for their continuous inter& \vork.

3) Jonc~s, .\. I,.. J. Opt. Sot. .\mc~r. 55 (1965) 261.

Prof. l)r. \\.. L>di~~rS~l- aiid Jr. during the prcpration of thih