IR Laser focusing question This is for a laser cutter.
10.4k IR laser outputs a beam at approximately 1/4" diameter, 400 watts. The beam is then focused using a single ZnSe lens with a 2.5" or 5" focal length. The final cutting kerf ends up being about a 0.009" wide gap. We are interested in a tighter focus.
We have been told that it can't be done by a couple of suppliers.
Is it possible to focus a 1/4" IR beam to a spot size tighter that 0.009"?
 A: The minimum spot size can be estimated by calculating the beam waist using Gaussian beam optics (wikipedia):
$$
\text{smallest spot size}= 2\omega_0=\frac{2\lambda}{\pi NA}
$$
Where $\omega_0$ is defined as the beam radius at the focus point, $\lambda = 10.4$ µm is the wavelength, and NA is the numerical aperture of your optical system. We can calculate NA as follows:
$$
NA=\frac{D}{2f}
$$
Where $D$ = 0.25" is the beam diameter, and the focus length of the lens $f$ = 2.5" or 5". So $NA$ = 0.05 or 0.025 respectively.
This means the smallest spot size $2w_0$ = 132 µm = 0.0052" for the 2.5" lens and 265 µm = 0.0104" for the 5" lens. I would say your supplier is ballpark right, the 0.009" you achieve is close to the physical limit. A few remarks:


*

*you should use the 2.5" lens, this should give you a smaller focus (but typically smaller focal lengths give higher aberrations, which brings us to the next point)

*the exact laser beam profile is quite important. Is your beam gaussian?
Aberrations in the beam distort the shape of the focus point and can make the spot size bigger.

*you did not mention your definition of your beam diameter, this is a science on itself: wikipedia. If you want the last drop out of your optical system, these things get important.

A: Yes, it is possible. 0.009" = 228 $\mu$m is about 20x larger than the wavelength of your  (I'm assuming) CO2 laser. So. In theory, you can focus much tighter. 
There are some practical issues however. I am not at all familiar with the specifics of laser-cutting tools, but here are some general things to think about:
To focus tighter, you need a higher numerical aperture (NA) with a relatively clean, well-behaved laser beam. If you have laser light coming into your lens from all directions, it won't focus nicely. If your beam is well collimated, even if the profile looks nasty, then you can focus well. If you have a lot of diffuse scattering, etc, you can clean up the collimation by using a spatial filter at the expense of laser power. 
The next step is to improve your NA. This basically means getting a lens with a shorter focal length. If you aren't utilizing the entire diameter of your lens, you can also improve your focus by expanding the beam size with a telescope (made from two ZnSe lenses) before focusing it (counterintuitively, this will make your focus spot smaller). This is probably more trouble than it's worth unless you already have big optics and a small beam diameter, although since your beam is relatively small, you might consider this. 
Issues with using a lens with a smaller focal length include: 
(A) Shorter working distance (distance between lens and target surface). To dramatically reduce spot size, you'll have to go with working distances in the millimeter range. If all you're looking for is a factor of 2, then basically your spot size will scale with focal length. I'm not sure if this will work in your tool, but if you can bring the surface in close, then definitely try it. 
(B) Lens damage. With a really tight focusing lens, the beam begins to significantly focus before it leaves the lens material. If the intensity of the light increases beyond the lens damage threshold, you'll obviously have problems. Whether this will be an issue depends on the details of your setup.  You might mitigate this problem by using a plano-convex lens "backward" (with curved side toward the focus), but it may not be necessary, and this will add aberration. 
(C) Thickness of lens. The shorter the focal length, the thicker the lens. This may be a (minor) problem if the lens mount of your tool is incompatible.
Additionally, you want to make sure that there are no other aberrations in the focus due to misalignment or spherical aberration. I recommend going with an aspherical lens if you're trying for a tight focus.
Finally, depending on how nicely your material cuts, a tighter focus might be a moot point. If the cut edge roughness is on the order of the laser focus width, then reducing your focus won't give you a narrower line anyhow. 
Hope this helps! 
