Page 67 - Fiber Optic Communications Fund

48 Fiber Optic Communications

R(r) and dR∕dr at the core–cladding interface leads to the following equations:

C J ( a)= D K ( a), (2.50)
1 m
1 m
1
2
′ ′
C J ( a)= D K ( a), (2.51)
1 1 m 1 1 2 m 2
′
where denotes differentiation with respect to the argument. Dividing Eq. (2.51) by Eq. (2.50), we obtain the
following eigenvalue equation:
′
′
J ( a) K ( a)
m 1 2 m 2
= (2.52)
J ( a) K ( a)
m
2
1
m
1
where
√
2 2
= k n − 2 (2.53)
1
0 1
and
√
2 2
2
= − k n . (2.54)
2 0 2
Note that in Eq. (2.52), the only unknown parameter is the propagation constant . It is not possible to solve
Eq. (2.52) analytically. Eq. (2.52) may be solved numerically to obtain the possible values of .Itwouldbe
easier to solve Eq. (2.52) numerically if we avoid differentiations in Eq. (2.52). This can be done using the
following identities:
′
aJ ( a)=−mJ ( a)+ aJ m−1 ( a), (2.55)
1
1
m
1
m
1
1
′
aK ( a)=−mK ( a)− aK m−1 ( a). (2.56)
2
2
2
m
2
m
2
Using Eqs. (2.55) and (2.56) in Eq. (2.52), we obtain
J m−1 ( a) K m−1 ( a)
1
2
2
=− . (2.57)
J ( a) 1 K ( a)
1
2
m
m
The propagation constants obtained after solving Eq. (2.57) lie in the interval [k n , k n ]. It is convenient
0 2 0 1
to define the normalized propagation constant
2
2
∕k − n 2 2
0
b = 2 2 (2.58)
n − n
1 2
so that when = k n , b = 0 and when = k n , b = 1. For any guided mode of a step-index fiber, we have
0 2 0 1
0 < b < 1. Eq. (2.57) can be solved for various design parameters such as wavelength and core radius a, and
the numerically calculated propagation constant can be plotted as a function of a specific design parameter.
Instead, it is more convenient to define the normalized frequency
√ √
2
2
2
V = a + = k a n − n 2
0
1 2 1 2
2fa 2 2 1∕2
= (n − n ) , (2.59)
c 1 2
where f is the mean frequency of the light wave. Using Eq. (2.7), Eq. (2.59) may be rewritten as
2a
V = NA. (2.60)