Page 184 - Introduction to Information Optics

3.2. Light Propagation in Optical Fibers

Fig. 3.3. Description of optical fiber under cylindrical coordinate.

Under the cylindrical coordinate as illustrated by Fig. 3.3, Eq. (3.12) can be
written as
1 1 a2
^L( ^\ ^L , z)
2
p dp \ dp) p d(f) 2 dz 2
2 2 2
I d d Id d
p dp dp 2 2 p , cp, z)
2
d E z(p,(l>, Z) , 1

(3.13)

Equation (3.13) is a partial differential equation, which contains three variables
(p, 4>, z). Since this is a linear differential equation, it can be solved by the
method of variable separation; that is,

(3.14)

Each function of F(p), <P(<|!>), and Z(z) satisfies one ordinary differential
equation. Substituting Eq. (3.14) into Eq. (3.13), the following three equations
can be derived:
2
d Z(z) 2
+ /i Z(z) = 0, (3.l5a)
dz

= 0, (3.l5b)
2
d F(p) I m
^L + __ (3.1
dp 2 P dp