Page 64 - Fiber Optic Communications Fund

Optical Fiber Transmission 45

Refractive index n 1 Core Cladding

n 2

a r

Figure 2.13 Refractive-index profile of a step-index fiber.

Substituting Eq. (2.32) in Eq. (2.28) and using Eqs. (2.29) and (2.31), we obtain
2
2
2
f 1 f 1 f f 2 2
+ + + + k n (r)f = 0, (2.33)
0
2
r 2 r r r 2 z 2
where k = ∕c = 2∕ is the free-space wavenumber. The above equation is known as the Helmholtz
0
0
equation. We solve Eq. (2.33) by separation of variables:
f(r,, z)= R(r)Φ()Z(z). (2.34)
This technique may not work for all types of partial differential equations. Especially, if the partial differential
equation is nonlinear, the method of separation of variables fails. Substituting Eq. (2.34) in Eq. (2.33), we
obtain ( )
2
2
2
d R 1 dR 1 d Φ d Z 2 2
+ ΦZ + RZ + RΦ+ k n (r)RΦZ = 0. (2.35)
2
dr 2 r dr r d 2 dz 2 0
Dividing Eq. (2.35) by RΦZ, we obtain
( )
2
2
2
1 d R 1 dR 2 2 1 d Φ d Z 1
+ + k n (r)+ =− . (2.36)
2
2
R dr 2 r dr 0 Φr d 2 dz Z
In Eq. (2.34), we assumed that f can be decomposed into three parts R, Φ, and Z which are functions of r,
, and z, respectively. Since the right-hand side of Eq. (2.36) depends only on z while the left-hand side of
Eq. (2.36) depends only on R and Φ, they can be equated only if each of them is a constant independent of r,
2
, and z. Let this constant be :
2
1 d Z 2
− = , (2.37)
Z dz 2
Z(z)= A e iz + A e −iz . (2.38)
1 2
Using Eq. (2.34) and substituting Eq. (2.38) in Eq. (2.32), we obtain
[ −i(t−z) −i(t+z) ]
(r,, z, t)= R(r)Φ() A e + A e . (2.39)
2
1
The first and second terms represent forward- and backward-propagating waves, respectively. In this section,
let us consider only the forward-propagating modes by setting A = 0. For example, the laser output is
2
launched to the fiber from the left so that only forward-propagating modes are excited. If the fiber medium has