I'm a bit Gnostic myself, sometimes when someone says "Jesus" in front of me, I poke my head up and say "yes?". No one ever thinks that's funny, no other gnostics around when I need one.
I've had two members see my earlier post in this thread with questions about beam exanders, here's what I wrote back for advice (edited with more info):
All my expanders are ones I found on ebay I'd like to keep, all the way up to 6 inch diameter for 160X expansion but 10X is plenty, even 3 to 4X is enough expansion to take care of the the uber high divergence of just about any multimode diode to make the laser perform close to that of a DPSS 532nm laser. At 10X expansion a multimode diode will have a far tighter divergence than a 532nm DPSS laser without an expander on it.
You can make an expander with a G2 lens and a PCX or plano-convex lens by de-focusing the output of your lasers lens to put a spot on the PCX lens which covers about 80% of its diameter when it is at it's focal length away from the G2 lens. Make the tube the large PCX lens is in have some way of moving in and out closer or further away from your G2 lens and you can focus it to infinity that way.
I've been playing around with some PCX lenses for this purpose and had a host made for me to hold a 50.8 mm/2 inch diameter PCX lens in a tube which can slide in and out from the lasers G2 lens and it is in the mail to me now, haven't received it yet. This kind of beam expander is fairly easy to make compared to a normal Galilean beam expander, but it isn't at all typical. Typical Galiliean beam expanders use a concave-plano or double concave lens for its input coupled with a PCX or plano-convex lens on its output and the positive and negative focal lengths of each lens must be carefully chosen.
My host which uses a de-focused G2 lens which in effect acts as an expander lens coupled with a 2 inch diameter 4 inch focal length PCX lens to fully collimate the light the rest of the way to infinity focus is an experiment, I don't know of anyone else using this kind of beam expander so I will have to see how well it works for me. One of the possible problems is if the G2 lens is defocused too far (to produce a spot wide enough to cover most of my 2 inch diameter PCX lens), part of the beam or energy of the laser diode it is coupled with might be physically cut-off from being able to have all of the light enter the G2 lens, some of the light overshooting the edges of the small lens inside and going nowhere, if too far away from the laser diode. However, if you get a 1/2 to 1 inch diameter PCX lens with a focal length of four inches or more, the longer the FL the better, I believe the output beam from a de-focused G2 lens (three element, or what have you) can be expanded large enough to cover close to the entire aperture of a one inch or smaller diameter PCX lens at four inches away without the G2 input lens being de-focused too far to cut off or loose any of the light from the diode, perhaps a much larger diameter output lens too, but the further away or longer the focal length and or the smaller the PCX lens diameter, the less the G2 lens needs to be de-focused from infinity. When I write G2 lens any lens of that type will act the same way, a three element lens or any lens in a threaded holder for a standard M9x0.5 Module. Although a three element lens will probably clip faster than a G2 or single element lens, it very well may clip some when perfectly focused to infinity for some multimode diodes anyway.
I don't think clipping is a critical issue for most G2 type lenses, not unless the G2 lens is screwed more than half way out of the aixis module. Just take your laser and de-focus the lens and check how far out you can unscrew it before there is any observable drop off of output power and see how big a spot it can make at the focal length of the collimation lens you want to use, if it can make a spot big enough to cover 80 percent or more of the lens diameter and you don't see the output intensity dim when looking at the spot through laser safety goggles, you should be good to use that lens. Another idea to check for clipping, if you defocus the laser to where you have, for example, a 1 inch diameter spot at the focal length distance of the 1 inch diameter PCX lens you want to use (distance from the G2 lens), just make sure the spot is the right size at the FL and then shoot the beam at a far wall. The light will be so dilute as it continues to expand to make a large circle on the wall you can clearly see if the beam is being clipped without having to deal with the lasers intense glare and much safer to look at. I don't recommend looking at a 1 or 2 inch spot from a laser diode, it can be entirely too bright at some power levels to look at, not only from a glare standpoint, but from a safety standpoint too.
I believe this design is better than a regular Galilean beam expander for two reasons, one being you only need two lenses total instead of three and another is you do not need to find exactly matched lenses, the G2 lens can be adjusted as a variable spot size lens, just put your PCX lens at its focal length away from the G2 lens, adjust the G2 to put a spot about 80 percent the diameter of your PCX output lens and you are done! No math, no fuss, no calculating the negative focal length lens you need, just be sure the focal length away from your G2 lens is right, make some room for adjustment to and fro for your output lens so you can adjust the beam to its lowest divergence/infinity focus and you are done.
If you buy a beam expander, just about any visible wavelength beam expander will work, provided the input hole is large enough for the beam width of your lasers output and its AR coating, if it has one, is in the proper band or has a wide enough bandwidth for the color or wavelength of the diode you are using. The biggest problem with beam expanders is finding one which can attach to your host, I've even used black electrical tape to put them on some of my lasers. If you use a beam expander on a multimode laser diode without corrective optics first, you will see a far more pronounced ribbon-like output instead of a rounded beam, the beam was that way when smaller before expansion, just that it isn't so evident when the beam is only a couple of mm wide. For myself, if the divergence is low and the power is high enough that doesn't bother me at all. When I use my 6 inch diameter beam expander with my 800mw JetLasers which has a ribbon-like output on a dark night, the uber fat beam is so awesome going out like a cone where you can see the perspective of the beam appearing to become much smaller in the distance (of course, it actually isn't) to where it seems to have a sharp point on the end, but if I try the same thing with a 100mw laser the beam is too weak with so much expansion to be very impressive, but it will put a nice spot on a super high cloud layer you can't always get without expansion.
I should mention one other gotcha with beam expanders, if you have a 20X beam expander and its output lens is only 25mm wide and your input beamwidth is 2mm wide, the output lens isn't big enough to expand your laser beam that wide, you would need to use a smaller diameter input beam with a beam expander like that. i.e. 20X 2mm = 40mm, the output lens would need to be closer to 45mm to work for an input beam that wide for 20X expansion. I had a China ebay seller rip me off selling a 20X beam expander with what appeared to be a 50mm output lens, according to the photo shown in the listing, but a received one with a much smaller output lens, no way could it be used with a 2mm wide input beam (as his listing specified it could/Laserlands) to produce a 20X 40mm wide output beam. For my 160X beam expander, its input can't take a beamwidth much wider than about 1.0mm, sadly, because I have a 4 watt output 532nm DPSS laser I wanted to use it with which has an output beam which is about 2 mm wide. I can down size the output of my 4 watt laser using a beam expander backwards and then shoot into the big 160X expander, might do that someday, but will likely loose over 500mw of my output power doing so from the added losses of more lenses.
Forgive my repetition of some terms so much, thought I needed to do so to be clear on some points.
