Page 185 - Introduction to Information Optics

170 3. Communication with Optics
where m is an integer and ft is a constant. The solution for Eq. (3.\5a) is

Z(z)=e iftz , (3.1.6)

which represents how light propagates in the z direction. Thus, the constant /?
is often called the propagation constant. The solution for Eq. (3.156) is

0(0) =e irn<j \ (3.17)

which represents how the light field changes along the angular direction. Due
to the periodic nature of the angular function, i.e., 0(0) = 0(0 + 2n), m must
be an integer. Equation (3.15c) is a little bit complicated. For the step index
fiber, an analytical solution can be obtained. As shown in Fig. 3.3, the refractive
index distribution for a step index fiber can be expressed as

n 1, p < a
n(p) = (3.18)
p > «,
n 2
where a is the radius of the fiber core. Substituting Eq. (3.18) into Eq. (3.15c),
the following equations are obtained:

2
d F(p) | ldF(p) in
2 nf fcg — F(p) - 0, (3.19«)
dp p dp P'
2
d F(p) , IdF(p) m'
F(p) - 0, p > a. (3.196)
dp* p dp
Equations (3.19a) and (3.l9b) can be further simplified by defining two new
constants:
2
2 2
K = n k - (^ (3.20a)
2 2
y = /i - nlki (3.206)
Substituting Eqs. (3.20a) and (3.206) into Eqs. (3.19a) and (3.196), we get

2
d F(p) , 1 dF(p) nr
—]F(p ) =0, (3.2 Ifl)
dp* P dp
2
d F(p) IdF(p)
dp 2 H p dp + -y ) F(p) = 0, p > a. (3.216)

Equation (3.2la) is the well-known Bessel equation and Eq. (3.216) is the