How to determine the focal length of a lens that is equivalent to two lenses?

Question

For context on this question, I've included a whole diagram explaining the situation:

I have a light bulb with a 10° spread that I want to narrow down into a beam and send through a diffraction grating.

The calculation I've done so far requires 3 lenses: the first lens makes the light parallel, the second lens focuses the light, and the third lens makes the light parallel again in a narrow beam.

However, it's certainly possible to consolidate the first two lenses into one lens.

At the end of the day, I'm going to need to shop for these lenses and the exact sizes I want will not be available so I'll need to figure out how to do this anyways.

So my question is: given the beam spread of 10 degrees (represented by f_bulb), and a convex lens (f_lens), where will the beam converge on the other side of the lens (f_focus)?

Answer 1

In the thin lens approximation the following relationship is satisfied $$ \frac{1}{f} = \frac{1}{object} + \frac{1}{image} $$ where $f$ is the focal length, $object$ is the distance of the lens to the object (your lamp), and $image$ is the distance from the lens to the image. So, if you place the object at the distance $object=2f$, you expect to obtain a focus at $image = 2f$. Nevertheless, since your light source is not a point source, you will encounter several issues.

If I would have to setup the experiment, I would not bother too much about the divergence of the light rays. Instead I would try using an iris to cut out the center part of the light and try to observe something using only this part. If the light intensity is too low, I would buy the cheapest lens with a focal lens of about $f=100mm$ and diameter 1''.