Page 65 - Fiber Optic Communications Fund

46 Fiber Optic Communications

no defect, there would be no reflection occurring within the fiber and the assumption of a forward-propagating
mode is valid. From the left-hand side of Eq. (2.36), we obtain
( )
2
2
r 2 d R 1 dR 2 [ 2 2 2 ] 1 d Φ
+ + r k n (r)− =− . (2.40)
R dr 2 r dr 0 Φ d 2
The left-hand side of Eq. (2.40) is a function of r only and the right-hand side is a function of only. As
2
before, each of these terms should be a constant. Let this constant be m :
2
1 d Φ 2
− = m , (2.41)
Φ d 2
Φ()= B e im + B e −im . (2.42)
1
2
The first and second terms represent the modes propagating in counter-clockwise and clockwise directions,
respectively, when m is positive. Let us consider only one set of modes, say counter-clockwise modes, and set
B = 0. If the initial conditions at the input end of the fiber are such that both types of modes are excited, we
2
can not ignore the second term in Eq. (2.42). In Section 2.4.6, we will study how to combine various modes
to satisfy the given initial conditions. Using Eq. (2.41) in Eq. (2.40), we obtain
[ ]
2
d R 1 dR 2 2 2 m 2
+ + k n (r)− − R = 0. (2.43)
dr 2 r dr 0 r 2
2
Using Eq. (2.30) for n (r), we obtain
2
d R 1 dR m 2 2 2 2
+ − R +(k n − )R = 0 r < a, (2.44)
dr 2 r dr r 2 0 1
2
d R 1 dR m 2 2 2 2
+ − R +(k n − )R = 0 r ≥ a. (2.45)
0 2
dr 2 r dr r 2
2 2
2 2
2
Fiber modes can be classified into two types: (i) k n < < k n –these modes are called guided modes
0 2 0 1
2 2
2
and (ii) < k n –these modes are called radiation modes. It can be shown that there exists no mode when
0 2
2
2 2
> k n .
0 1
2.4.1 Guided Modes
2 2
2
Since < k n , the last term in Eq. (2.44) is positive and the solution of Eq. (2.44) for this case is given by
0 1
the Bessel functions
R(r)= C J ( r)+ C Y ( r), r ≤ a, (2.46)
1 m
2 m
1
1
√
2 2
2
where = k n − , J ( r) and Y ( r) are the Bessel functions of first kind and second kind, respec-
m
m
1
1
1
0 1
tively, and are plotted in Fig. 2.14. The solution Y ( r) has to be rejected, since it becomes −∞ as r → 0.
1
m
Therefore,
R(r)= C J ( r), r ≤ a. (2.47)
1
1 m
2
2 2
Since > k n , the last term in Eq. (2.45) is negative. The solution of Eq. (2.45) is given by the modified
0 2
Bessel function
R(r)= D K ( r)+ D I ( r), r ≥ a, (2.48)
2
2
1 m
2 m