Collimating light from Optical fibre problem

[XCBRA]

Homework Statement

A thin lens of focal length f is used to collimate the light emerging from two different optical fibres.
When used with a first fibre of small core diameter and numerical aperture NA it acheives a diffraction limited divergence of θ1 for the collumated beam. Find the separation u between the lens and the fibre that provides the best collimation. Find θ1 and the diameter D of the collumated beam immediately behind the lens.

When used with a second fibre of a large core diameter d but the same numerical aperture, the beam diverges much faster than it did using the first fibre. Calculate this larger divergence angle 2 as a function of f and d

Homework Equations

NA = sin(θ1)

1/u+1/v=1/f

The Attempt at a Solution

I am not entirely sure how to approach this problem as it is not a point source. I would like to say that:

for best collimation u = f -- from lens equation
and then

D=f tan(θ1) ≈ fθ1 ≈ f NA -- from trigonometry

however I am not sure if this is correct. Any hints on how to approach this question would be greatly appreciated. Many thanks

 

[aimilvping]

wow, so difficult to me!