# Optics issue with focusing beam

1. Aug 19, 2013

### Degeneration

Hello all,
I've got an optics problem that I'm having some difficulty with, where I want to send a beam through a converging lens to obtain a certain image size. My incident profile is that of a ring with a certain thickness and use I use this as my object. I want an image of a certain size, so I've been using the thin lens equation to determine the lens' properties (magnification, focal length) - but I don't know if this is valid. The object has a significant ratio of the ring's (full width half max)/(ring radius), and when it is magnified through the lens, I want this ratio to remain the same.

Here is the issue: this ring-shaped laser beam (using an axicon to produce this shape) is collimated like so, and when I run simulations it turns out that this ratio of FWHM/Rad is NOT retained in the image. Is this use of the thin lens equation incorrect? I suspect it is because different parts of the ring are treated differently, i.e. the ring itself is treated as an uncollimated point source when passing through the lens, while the width of the ring is a collimated beam. Could anyone offer any insight? Perhaps a better way to compute the FWHM and radius of the ring passing through a converging lens with given parameters?

Many thanks

P.S. Here is the image linked to in case the hyperlink doesn't show: http://www.wavelength-tech.com/images/axicon01.gif [Broken]

Last edited by a moderator: May 6, 2017
2. Aug 19, 2013

### Simon Bridge

The thin lens equation does have a lot of approximations in it - so, sure, if your lens is fat and the ring of light is incedent well-off the axis, then the thin lens equation will give bad results.

Note: If I understand you - your modelling of the beam source as an object probably needs to be looked at.
You realize if the incident light is parallel, you should be able to put a screen anywhere and get a ring?
i.e. I think you need to remodel the geometry of your setup.

3. Aug 19, 2013

### Redbelly98

Staff Emeritus
If your lens is plano-convex as suggested in your diagram, you should orient it so the curved side is turned toward the collimated beam, and the flat side is turned toward the converging/diverging beam. This will minimize aberrations in focusing or collimating the beam.

4. Aug 20, 2013

### Degeneration

Thanks, this is helpful. Would a double convex lens produce less aberrations than a plano convex (assuming they're identical in their focusing power). Would the general lensmakers equation be a better choice?

5. Aug 20, 2013

### Andy Resnick

I'm confused- are you projecting a ring onto a screen and want to image that? Are you trying to increase/decrease the diameter of the propagating beam? What/where is your object plane?

6. Aug 20, 2013

### Redbelly98

Staff Emeritus
No/yes.

If you get a lens with the same curvature on each face, then no, the aberrations would be worse.
If you get a so-called "best form" lens, which has a slight curvature on one face and greater curvature on the other face, then aberrations will be slightly better than a plano-convex lens. But in all three optics/laser labs I worked in over the years, we always used plano-convex lenses and it was good enough. Of course, we oriented the lenses properly for minimal aberrations. Doing that, if you aren't already, is a fast and easy thing to try out next.

What is the diameter of your beam, and what are the diameter and focal length of the lens in your current setup? Getting a lens with a longer focal length (which would then be thinner) should improve on any aberrations as well.

Also, is your lens AR-coated for a wavelength range that includes your laser?

I don't understand this question. To me, it reads "would this equation be a better choice than using a double convex lens?", which doesn't make sense, so I am missing what it is you are asking.

From the figure linked to earlier, it looks like the OP wants to make a magnified image of the beam profile, perhaps to see it more easily.

7. Aug 22, 2013

### Degeneration

Right. I'm using the lens to magnify the image so that I get the desired size within a given length, mainly because I don't have too much room on my bench. The beam initially starts out as a gaussian with a 3mm diameter, send it through one lens, have a gap, and then send it through an axicon to produce a ring. I want my final image to be a ring with a certain FWHM to radius ratio on a detector some distance away, and I use this lens in the image to get the desired magnification.

I haven't been very clear here and I apologize. In that image, let's consider a cross section of the ring as shown, so that it looks like a beam has been split into an upper and lower one. Furthermore, let's consider that the width of this ring isn't constant as it appears to be in the diagram. As the image (the ring) gets larger, the radius and the width both grow. However, what I seem to see is that the FWHM increases at a rate different than the growth of the radius. So if I find this perfect ratio, let's call it R*, at a distance S1 from the lens, I want to be able to use it as an object and set the lens and its focal length so that at a distance S2 from the lens, I get a magnified image with ratio R*.

But according to my earlier statement, this ratio R* is only attained at one point as the beam propagates since FWHM and radius increase at different rates. So when I the ring is incident on the lens, it doesn't actually have ratio R*, but rather something different R'. Then, it appears that the image has ratio R', not R*. Simple geometric optics isn't working for me, and I was wondering if anyone had experience working around it. Thanks!

8. Aug 23, 2013

### Andy Resnick

Maybe I still misunderstand, but AFAIK with your setup you do not have independent control over the ring diameter and the 'thickness'. Is your beam (prior to the axicon) collimated? What happens if you move that first lens back and forth along the optical axis?

In any case, if you want to 'magnify the ring', what I think you mean is to increase the divergence angle β? I guess I still don't understand, but Thorlabs has a potential solution:

http://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=4277
(select the 'beamshape' tab)

One possible approach would be to place a lens between the axicon and ground glass screen (as discussed on the Thorlabs page). By imaging the ground glass screen with (for example) a zoom lens you will vary the size of the ring diameter, and by moving the intermediate lens you can (I assume) vary the ring thickness.