The modified Bessel functions are shown in the figures 6 and 7. The functions are monotonically increasing functions of , and functions are monotonically decreasing functions of .
(c)
If is complex
Then the solutions are
Hankel function of kind
Hankel function of kind
In our analysis, will either be real or imaginary. Therefore we have to deal with Bessel, Neumann and Modified Bessel functions only. The Hankel functions are needed for analyzing propagation in a lossy medium.
(d)
Now , where is the propagation constant of the wave along the direction. If we assume the
situation is lossless i.e. when the wave travels in the direction, its amplitude does not change as a function of , then should be a real quantity.
If becomes imaginary, the function becomes an exponentially decaying function, and there is no wave propagation.
(e)
For wave propagation inside an optical fiber we assume that the material is lossless.
Then the dielectric constant is a real quantity. This makes
functions are monotonically increasing functions of 
, and 
functions are monotonically decreasing functions of 
. 
is complex 
Hankel function of 
kind 
Hankel function of 
kind 
will either be real or imaginary. Therefore we have to deal with Bessel, Neumann and Modified Bessel functions only. The Hankel functions are needed for analyzing propagation in a lossy medium. 
, where 
is the propagation constant of the wave along the 
direction. If we assume the
direction, its amplitude does not change as a function of 
, then 
should be a real quantity. 
becomes imaginary, the function 
becomes an exponentially decaying function, and there is no wave propagation. 
is a real quantity. This makes 
a real quantity. 
is a real quantity. 
