Page 187 - Introduction to Information Optics

172 3. Communication with Optics

The constants K and y can be found by using the following boundary
condition between the core and the cladding surfaces. Mathematically, it can
be expressed as

(3.26)
p op

Substituting Eq. (3.25) into Eq. (3.26), we obtain
A-J m(Ka)=C-K m(ya\ (3.27«)
A-K-J m(Ka] =C-yKUy fl ), (3.27J»)

where symbol prime indicates differentiation with respect to the argument.
Dividing Eq. (3.27a) by Eq. (3.Tib), we get following formular (so-called
dispersion relationship):

J m(Kd) K m(ya)
(.5.25)
K • J' m(Ka] y • K' m(ya)
To understand Eq. (3.28), let us substitute Eq. (3.20) into Eq. (3.28). Then, Eq.
(3.28) becomes

2
• J' m(Jn\kl - ft a)
Equation (3.29) determines the possible values of propagation constant ft, as
discussed in the following:
lftz
1. Since the term e represents the light propagation in z direction, the ft
is called the propagation constant. For nonattenuation propagation, ft
must be a real number. For simplicity, assume that the light only
propagates in one direction and ft must be larger than zero for the
selected propagation directions (i.e., ft > 0).
2 2
2. The conditions K = n^k^ — ft > 0 and /? > 0 result in ft < n^. The
2
2
conditions of y — ft — n\k^ > 0 and /i > 0 result in /? > n 2k 0. Thus, the
overall constraint on ft is
n 2k 0<ft<n lk 0. (3.30)

3. Multiplying l/k Q on both sides of Eq. (3.30), we get

n 2<-<n l. (3,3 J)