Setting a laser cutter focus in the meters range

Disclaimer: I know a bit about optics and laser beams. I have no specialization when it comes to laser cutters but I can make educated guesses.

The formulas for a Gaussian Beam are of critical relevance here. The laser cutter operates by focusing a laser beam to a tight focus with a certain waist size $w_0$. You could tell me how large the waist is but I would guess it is sub-mm to ensure high intensity for cutting and a small size to allow for machining small internal radii where necessary. I'll suppose the waist is 100 microns. I'll also guess that the cutter uses light with a wavelength of 10 micron.

The spatial profile (that is the radius of the beam as a function of distance) of any Gaussian beam (and the beam coming out of your cutter is certainly a guassian beam) is given by

$$ w(z) = w_0 sqrt{1 + left(frac{z}{z_R}right)^2} $$

With $z_R = frac{pi w_0^2}{lambda}$. For the beam I described we have

begin{align} w_0 =& 100 mu m z_Rapprox& 3mm end{align}

If $zgg z_R$ we can approximate $1 + left(frac{z}{z_R}right)^2approx left(frac{z}{z_R}right)^2$ which leads to:

$$ w(z) approx frac{lambda}{pi w_0} z $$

We see that away from the focus of size $w_0$ at $z=0$ the beam expands linearly as it propagates with a slope of $frac{lambda}{pi w_0}$. Importantly the smaller the focus is the faster the beam diverges. This is a general feature in optics and explains why you typically must focus at short distances to get a small focus.

In your case lets ask the question, if there is a waist of $w_0=100mu m$ at $z=0$ then how big must the beam be 1 meter away?

$$ w(z=1 m) approx 3 cm $$

The beam will have a radius of approximately 3 cm, or a diameter of a little more than 2 inches. This is a fairly reasonable size for a lens though probably much bigger than the OEM lens used in the laser cutting rig.

Without getting into too many details we can try to assess the feasibility of this. I'll assume the following optics scheme. It sounds like the rig natively creates a focused beam 8mm from the output of the cutting head. If you let this propagate some further distance the beam will begin diverging. Eventually it will reach a waist of 3 cm. At this point you can place a lens which will collimate the beam. The focal length will depend on the rate of divergence. After this if you place a lens with a focal length of 1 m you will get your 100 micron spot 1 meter away as desired.

Now what are the practical concerns. There are many. These aren't listed in any particular order

1. The radius of curvature for a 1 meter focal lenght lens is quite large. Typical optics I've worked with have much smaller focal lengths typically. Getting the right form for this lens may require very precise machining/polishing which may result in an expensive lens. It may be quite difficult to avoid optical aberrations this will affect the beam quality. This would require optics simulations to assess.
2. Beam intensity. All optics which you use will need to be able to sustain 200W of laser power passing through them without major thermal deformation. This is a materials challenge. Apparently it has already been overcome for some of the laser rig optics so this is presumably surmountable but may be costly as well.
3. Beam intensity again. Will it be ok to have this high of power passing through water? Will water significantly absorb at the CO2 wavelength? Just saw the important comment by @Andreas H. This is the most important point. Your beam is going to be STRONGLY absorbed by water at the 200 W is going create a steam explosion when it hits the water. If nothing breaks I doubt you'll be able to cut through that.
4. Fluid aberrations. Any flow of water during the cutting (either natural flow or flow due to heating by the optical beam) will cause the beam to develop aberrations and misalignment which could ruin the cut.
5. Finally this isn't a problem, just something that needs to be taken into account. Water has a different index of refraction than air. This will need to be taken into account when picking the focal length. However, this may cause another issue. At the passage between air and water the light rays all bend a little bit. In a ray optics picture this is no problem. However, it may cause some optical aberrations that will again make decrease the quality of the focus and thsu the cutting power/precision.

Finally, the more you can handle a larger waist the easier things will be. All of these problems will get better. However, increasing the waist decrease the intensity (intensity is inversely proportional to waist squared) so I'm sure there is a limit to how large you can make it given the laser power until it stops cutting well.

Good luck with your application. Consider consulting with the laser cutter manufacturer before attempting this mod.