Page 186 - Introduction to Information Optics

3.2. Light Propagation in Optical Fibers 1 .1

modified Bessel equation. The solutions of these two equations are the Bessel
functions. Thus, F(p) can be expressed as

F(o) - m m P ^ a (3 ->?) >
{
\C-K m (yp)+D'I m (yp) , p > «, "'^
where J m is the mth order first kind Bessel function, Y m is the mth order second
kind Bessel function, K m is the mth order modified second Bessel function, I M
is the mth order modified first kind Bessel function, and A, B, C, and D are
constants.
When p -»0, Y m(Kp) -»• oo. Since light energy cannot be infinite in the real
world, B must be zero (i.e., B — 0). Similarly, when p -» oo, / m(yp) -> GO. Again,
since light energy cannot be infinitely large, D must be zero (i.e., D — 0). Thus,
Eq. (3.22) can be simplified into

The Bessel functions J m(Kp] and K m(yp) can be found by looking at Bessel
function tables or calculated by computers from series expressions, as given by

,n /-v\ 2n + m
(3.24a)

2n m
l"'^(m -n - 1)! (x\ ~

( _i r + i^ (3,24,5)

£7 , (3.24c)
./=! J

Substituting Eqs. (3.16), (3.17), and (3.23) into Eq. (3.14), we can get the final
solution of light field £.,

(3 25)
'

Then, by using the Maxwell equation, we can get H z, E f>, E tj), H f>, and H, lt.