Me, me, me, me..... arm up.... I love expanders, I have large lenses but unable to machine the fixtures. Also, pretty broke for a few weeks
I can send you to edmund optics for some basic info:
http://www.edmundoptics.com/technical-resources-center/lasers/beam-expanders
Also, pick just about any telescope you like, they all function as a beam expander in reverse.
There are some folk here who know expanders and I've learned a few things, here's an old thread on what I did learn. I wish some of the senior laser techies would chime in on the subject in this thread, as it is of intense interest to me:
http://laserpointerforums.com/f49/diy-beam-expander-42814-4.html
One of the problems with using a telescope as a beam expander is the curve for the eye piece is rather shallow compared to the diameter of the laser beam hitting it and because of that, they don't always make very good beam expanders. The size of the eye piece, or concave negative focal length lens should be so small so that a significant portion of the lens is filled by the diameter of the laser beam hitting it, close to the entire diameter but even half or a quarter and even less would work to expand the beam. The deeper and more bowl shaped the more negative its focal point will be and when coupled with a positive focal point lens of the right diameter and distance from the expanding concave lens, you have a beam expander.
The correct distance between a negative focal point concave lens and a positive focal point PCX or plano convex lens depends on the focal points of each, the positive or PCX lens sets the majority of the distance. Add the focal lengths of the negative and positive lenses together and you have the approximate distance they need to be separated, if I understand correctly. If you had a 100mm FL PCX lens (flat side towards the laser) and a -10mm FL concave lens, then the beam should be expanded by 10X and the distance between the lenses 110mm center of lens to center of lens, if I understand the principle correctly. Maybe it would just be 100mm between the two? I'm not sure.
My understanding is that the expansion factor or amount of expansion is the negative FL lens divided by the positive FL lens and thus in this example, a 10X beam expander. I believe you could improperly pair negative and positive FL lenses together and still make them work in this example, just that you could either overshoot or undershoot the diameter of the output beam to match the size of the PCX lens on the end of the tube. If you expand too much you waste some light by hitting the side walls of the tube, if too little you have not expanded the beam enough for the available diameter of the PCX lens on the end of the tube, producing less expansion than might otherwise be had. Anyone know if I am correct on this or not? I am thinking this is what happens when you don't have your telescope focused correctly, same with a beam expander except in reverse.