How to determine if a optical system is afocal?

Question

I have to come up with a method to measure the focal length of a concave lens that's not in my lab guidebook. So I decide that I will build an afocal system with a convex lens and a concave lens and measure the focal length of the convex lens instead. But how can I determine if the foci of the two lens coincide? I will be using reading microscope which can measure length down to an error of 0.001mm, so I guess I should not drag down the accuracy by too much.

P.S. we just have a "parallel generator" as the light source. Not sure what that is, but I think it's a point source fixed at a focus of a lens.

Answer 1

Use an auxiliary lens. First run your collimated source beam through the lens and mark the focus location. Then stick your afocal lens assembly in between the source and the auxiliary lens. If the focus location moves, you're not perfectly afocal.

Answer 2

Both Carl and Johannes (the latter in a comment) have given you good answers. I reproduce Johannes's comment:

Shine a laser beam through the system. If exit beam does not diverge, both foci coincide and you have realized an afocal system

and you say

@Johannes I'm a bit worried that the rail won't be long enough for me to determine if the exit beam diverges. I think should do some calculation to estimate the angle of divergence if the foci don't coincide.

If your method is going to work, then the rail has to be long enough! Otherwise, you can't make your method work! The distance between the two lens elements has to be $|f_1| - |f_2|$ where $f_1, f_2$ are the two focal lengths (converging and diverging, respectively). Morever, the longer your combined system, the more sensitive your alignment will be. So it can be self defeating to restrict yourself to a short rail.

Try using several converging lenses: the less optical power (i.e. the longer the focal length) of the converging lens, the greater the system magnification $|f_1/f_2|$ and the greater the sensitivity Johannes's and Carl's tests a greater sensitivity.

Here's also a rough method that will get you in the neighbourhood of the afocal working region before you fine tune with Carl's or Johannes's methods. The bigger the magnification, the better the method will work.

Your concave and convex lens are on a rail:

Your configuration is called a Galilean Telescope (a configuration with two converging lenses a distance $|f_1|+|f_2|$ apart, i.e. with a focus inside the system, is called a Keplerian telescope).

Hold your telescope up to your eye, with the diverging lens nearest to you and look at image coming through the converging lens at the far end:

You'll find, especially if the system has a high magnification, that the range of axial positions for the converging lens wherefor you get anything like an in-focus image is very small. Adjust the converging lens's axial position lens for clear magnified image by carefully sliding it along the rail. When you get the clearest image, load the assembly into the test jig you set up for Johannes's or Carl's test and refine it. You'll find it likely won't need much refinement.

Also, with Johannes's test, the "aligned" position is going to be a collimated laser beam at the output, so you'll find this most accurately by aiming at a screen several metres from the output. Be sure to take heed of laser safety: collimated laser beams are the most hazardous, so use a low power (class II, ISO60825) and you need to keep anyone not involved with the test away from the beam. If a test helper is checking for collimation on a far screen, be sure they wear alignment goggles (i.e. safety goggles with typically an OD2/OD3 rating at the test wavelength so they don't fully block the laser light but let you see enough for alignment).