Multipath dispersion in fibre optic cable

[MalachiK]

Homework Statement

I saw this question on this forum

Multipath dispersion of a pulse of light in an optical fiber.
How is this problem solved?

and the answer given

Multipath dispersion is can be solved by:

1) Making the fibre (core) very narrow.

2) By making the cladding which surrounds the core very close to the core refractive index. The closer the two refractive indexes the better as any light that is less than a certain angle will be lost therefore only light that is at the right angle, preferably straight to the optical fibre, reaches the receiver.

At first I thought that it was obvious why reducing the width of the fibre would reduce the multipath dispersion. But having tried the problem with maths, I'm not so sure.

Can anyone tell me where my maths has gone wrong?

The Attempt at a Solution

I've tried to calculate the maximum possible path difference along a fibre of length l and width d that has a maximim transmission angle of to the normal of the core cladding boundary $$\theta$$. I've come up with an expression for this path difference in terms of the length of the fibre... <Attached are my scribbles> (L is the longest path and $$\Delta$$L is the greatest possible path difference.)


The problem with this is that the maximum path difference is independant of the width of the fibre and depends only on the angle..

 

[Nate810]

which means that reducing the width of the fibre does not reduce the maximum path difference. This makes me think that the solution given in the answer is wrong. Any help would be really appreciated. Thank you!