Step-index fibres

3.3 Optical waveguides and fibres

3.3.1 Step-index fibres

The index of refraction in a step-index fibre (Fig. 3.7) is cylindrically symmetric and homogeneous within the core and the cladding, respectively. Its value declines from n 1 within the core at r = a step-like to the value n2 of the cladding. According to the geometry we look for solutions of the form E = E(r, 0) e —i(wt— Oz) . The wave equation for cylindrical (r, 0) components is complicated, since the er and e o unit vectors are not constant.

For the

Ez

and

Hz

components, a scalar wave equation still holds, where

V i (r,

0) stands for the transverse part of the nabla operator,

(v

2, k2 - ,32 ) ii

e

zz

=

O.

Fortunately, one gets a complete system of solutions if one first evaluates the compo- nents

{Ez , Hz }

and then constructs {E r ,

So, H r., 7-4,}

by means of Maxwell's equations,

V x H =

—iw€0 /4E and V x E -= iwp,o H, (3.14) the result of which is given in Eqs. (3.18) and (3.17).

The propagation constant

0

must still be determined, and the Helmholtz equation for {Ez ,

Hz }

in cylindrical coordinates with k1,2 = ni,2w/c is

7ô2

± IT a ± r 12

a

a:2

(k? /32)

)

^{ HEzz((rrl)

With the help of the trial solutions {Ez , H z } = {e(r), h(r)}e±i 4 , this is reduced to a Bessel equation for the radial distribution of the amplitudes,

( r ) 0 e

r: he (r) =

The curvature of the radial amplitudes {e(r),

h(r)}

depends on the sign of k

-

^02•

Within the core we can permit positive, convex curvatures corresponding to oscillating solutions; but within the cladding the amplitude must decline rapidly and therefore

{BJe(k i r)1Je(ki a) r--+0 oc

(k i r) 'e

core,

-+ OC

BI-(e(nr)IKe(Ka) oc

cladding.

must have a negative curvature

—

otherwise radiation results in an unwanted loss of energy (see Fig.

3.7):

within the core

0 <

k

= k? -

02 , within the cladding O>

—k 2 =

k

- [32 .

In other words, the propagation constant must have a value between the

wavenumbers ki = niw/c

of the homogeneous core and of the cladding material,

n i w/c <

₃

< n2 w/c,

and differ only a little from

k1 , 2

for small differences in the index of refraction A

= (n i — n2)/n i

(Eq.

(1.7)).

Such

waveguides

are called weakly guiding.

By definition it holds that

ki + o = kf - k3.

Since

ki k2,

the transverse

wavevectors

k1 and i are small compared with the propagation constant

0,

k

2 2.6,(nic0/0 2 < 02 .

The transverse solution must be finite, thereby keeping only the Bessel functions Jf (modified Bessel functions

Ke)

of the first kind within the core (cladding):

{ AMkir)IJe(kia)

^OC

(kir)

core,

Al(e(Kr)11(f(Ka)

^{T --+ 00 OC}

'e

e — kr ,\/vr

cladding, r->0

By defining the coefficients, we have already taken care that the components

{

^{ez , Hz }}

are continuous at

r =

a. For the

{E0 , 7-41

contributions we obtain conditions from

Eqs. (3.14),

=

H ( r (1)) =

—i0 kf — (32

— 0 2

[

^r

^{l e (r)}

e

(r)

^{- f - cobto}

a ^a

^{r h( r) e}

^{iN ,}

[r)

2

d

_r

h( WC°71i 0

0

Or

e(r) e.

(3.17)

Here we use appropriate

wavenumbers k i

for core

(i = 1)

and cladding

(i = 2).

In weakly guiding fibres we usually have

f3/(kF — (3 2 )r 1/2Aki a >> 1;

hence the

(r,

components are much stronger than the

z

components with which we started our solutions; these waves are nearly transverse.

Once the radial contributions are calculated, all six field components are known:

1 2

[h(r) +

,37-10(r, (IS)] ^,

wconi

1 2 [ e(r) 0E0 (r, 45)] • cutioni

To determine the propagation constant

(3,

we substitute

fe(r), h(r)}

from eq.

(3.16),

use boundary conditions

(3.1)

at

r =

a, and obtain after some algebra a system of

(3.15)

e(r) =

h(r)=

(3.16)

r(r4)

= Nr(r4) =

(3.18)

3.3 Waveguides 77

linear equations in

A

and

B.

It is cast in transparent form with symbols

X = k i a,

Y =

Ka

and

.11 (x) = dJ (x) I dx,

Bwl-to 4(X) ± ^K(Y) 7tA ( 1

- (3 X Je(X) Y K e(Y) \

nT4(X) , nW(Y) i

^{opp ( 1}

X Je (X) Y ( e (Y) )

and yields a characteristic eigenvalue equation

(4(X) K(Y) k?J(X) k3K /f (Y)

X Je(X) Y Ke(Y) X 'MX) Y Ke(Y)

£202

( 1 2 + ) 2

X

Y

Numerical treatment of this transcendental equation, for

f =

0, 1, 2, ..., gives solu- tions

(Xe m ,

Ve rn ) and a propagation constant Oem ; this treatment is elaborate and is covered extensively in the literature [89J. As we did in section 1.7 on ray optics, we restrict ourselves to the case of weakly guiding waves at small differences of the indices of refraction n 1 r= n2 or k1 k2 (3 and end up with

J(X) OY) ±,e

⁽ 1 1 )

X MX) Y K e(Y) x2

y2

The derivatives can be replaced by the identities

4(X) = ±,hTi(X) ^fMX)

^{and K(Y) =}

Kw_

(17)

eKe(Y)

and substitution delivers the conditions for each sign in

±

1, which may be associated with two classes of modes,

0,

(3.19)

HEem modes:

ate, modes:

'it —1 (Xern

Xe rn Je(X,e m ) Je+i(X.ern) X em Je(Xem )

Ke-1(Yern)

Ye7TI Ke Y£m,) '

Ke ±i

(Km)

YemKe(Yem.)

^•

Since

ki + K 2 k? - k,

a further condition is

)(Ira

^{ye2rn v2 ,}

(3.20)

(3.21) where the V parameter is a measure for the number of modes. It increases with

= w 2 (7 4 ^{_ 7} 4) a

^{2/2 .}

frequency w because of V Graphical solutions for the propagation constant can now be obtained from Fig. 3.8.

Eq. (3.19) gives one more condition for the coefficients

(A, B)

from eq. (3.16), which fix the amplitudes. The `+' sign holds for the HE modes, and the `-' sign for the EH modes:

[A ±

i(w p, o I 1 3)B]e =

0. (3.22)

From this it can be seen that the electric and magnetic fields are 90° out of phase.

12 •

Y

YK0/K

i

I

E02

• • •

TE03

8

78

3 Light propagation in matter

Fig.

3.8:

Graphical solution of the values for j3 in a step-index fibre for V =

10.

Left: TE and TM modes. Right: HE and EH modes.

Let us now have a look at some special cases.

(1) f = 0: TE and TM modes

For

=

0 we have A = 0

(B =

0), i.e. either the

E

or the H field is purely transverse.

Hence for

=

0 the TE/TM denominations are sensibly used. We have indicated the graphical conditions for the (degenerate) TE0,, and TM0,, modes in Fig. 3.8. TE/TM modes will not be guided for V <X01 = ki a = 2.405 because J0 (2.405) = 0. There is no guiding at all below the corresponding cut-off frequency w _-cut - 2.405c/a(nT-n3) 1 / 2 , which is directly obtained from eq. (3.15). At J1 (5.520) = 0 the next mode appears, and the fibre is no longer monomode.

(2) t > 0: HE and EH modes

The lowest mode is the HEll mode, which exists down to

X =

0. Thus the core fixes the mode for arbitrarily weakly curved transverse amplitudes - the part of the energy propagating within the cladding increases more and more. On doing the mathemat- ical treatment we assume the cladding to have an infinite extension - which involves technical limits. Owing to eqs. (3.20) and (3.22), the HE and EH modes differ. This difference not only is revealed by the unequal propagation constants, but also mani- fests itself in the domination of the z components in the corresponding H (HE), resp.

E

(EH), parts.

(3) t > 0: LP modes

By means of a recursion formula, one can show that the modes with indices HE.em and Elie +2 ,,, are degenerate. Their linear superposition results in linearly polarized transverse

E

and H fields, which are orthogonal and have a weak longitudinal field.

Modes constructed in that way are so-called

linearly polarized

modes

(LP),

Elie+2,rn

3.3 Waveguides 79