How can beam divergence be corrected for in a $\rm CO_2$ laser cutter?

First, some questions about your setup:

1. Does the laser have to be stationary? Though, if your diagram is at all to scale, it may be too large to move. In that case,
2. Do the optics have to move? Could you design this system like a CNC end-milling machine and have the table move beneath the laser? Though, 15-foot movement might be too much for a motion table.
3. Are fiber optics not workable? There are optical fibers that can transport high power CO$_2$ laser light. A quick google search brings up these:
   • http://www.jtingram.com/id10.html with a link to a PDF catalog for fibers than can handle 12kW/cm$^2$.
   • http://www.fibrephotonics.com/page/143/High-power-CO2-laser-delivery.htm, though only up to 50W.

Second, here's an experiment you can try to see if the beam divergence actually is a problem. Repeat your spot size measurements at various distance, but instead of melting a brick directly, put your focusing lens in front of the beam. Then, move the brick back and forth until you find the distance from the lens with the smallest melted spot. Measure the size of this spot and the distance from the lens versus the distance between the laser and the lens. If the variation in spot size and focus distance is too large, read on for a possible solution.

If you want to transport a beam a long distance, the trick is to start with a larger beam diameter. Figure 5.5 in the PDF linked to in @akhmeteli's comment to his answer shows that large beams don't diverge as fast as small beams. The equation for the diameter of a beam as a function of distance from its narrowest point is $$\omega(z) = \omega_0 \sqrt{1+\left(\frac{z\lambda}{\pi \omega_0^2 n}\right)^2}$$ where $z$ is the distance from the narrowest spot (which should be the laser aperture), $\omega_0$ is the diameter of the beam at its narrowest, $n$ is the index of refraction of the material the laser travels through (which is 1 for air), and $\lambda$ is the wavelength of the laser (10.6 microns for a CO$_2$ laser). Far from the laser, the waist is approximately $$\omega(z) \approx \frac{z\lambda}{\pi \omega_0 n}.$$ So, if you double the diameter of the beam ($\omega_0$), your halve the divergence angle (the asymptotic cone in your plot).

The setup I envision is shown in the diagram below:

I chose to use off-axis paraboloid mirrors instead of lenses since they have a higher damage threshold than lenses (at least the ones from ThorLabs do). But, you can also substitute lenses and flat mirrors to achieve the same effect. In any case, use two mirrors or two lenses in a confocal arrangement (distance between elements equals the sum of their focal lengths) to substantially increase the diameter of the beam and collimate it so that you can transport it any distance without worrying about divergence. Then, use a third mirror or lens mounted on your motion system to direct and focus the beam on the cutting surface.

Some advantages to this setup:

• The laser light does not change substantially during its trip down the long arm of the system, so the final focus does not change location with respect to the final focusing element. The larger the beam diameter, the better this approximation is. The relevant equation is 36.7 on page 36.6 in your PDF. $R(z)$ should not change substantially for a wide beam, so it will focus to the same spot after the final focusing lens.
• The larger beam is easier on the optical system in terms of damage from absorbed laser power since the power is diffused over a larger area.
• For the same reason, the larger beam is safer (I've gotten my hand in front of a 1-mm-diameter, 1-W CO$_2$ laser a couple of times and it hurts!)

There are a lot of instruments to measure CO2 laser beam characteristics, for example, our group uses this one, but it is not cheap, and you can only use it in the intermittent mode for 90W.

It does not seem easy to be able to focus the CO2 laser beam at distances from 2 to 17 feet. In principle, you could use a set of ZnSe lenses with different focal length. Our group uses lenses with f=1700mm and f=400mm (appr.) Lenses for CO2 lasers are not cheap either. Maybe there are adjustable lenses, but I don't know much about that.

Divergence typically does not depend on the beam power.

"If beam angle is what is affecting the focus, the difference in beam angle would be greater between zero and the mid point distance of the graph than the midpoint and the end point of the graph. This would mean that a most of the divergence error would occur nearest the laser. Is this true? What am I missing? " I am afraid I don't understand what you are talking about.

The quality of the beam does depend on the laser. What you need is mode quality $M^2$, but it should be decent for an industrial laser. So a decent laser should have a beam close to a Gaussian beam. Typically, the exit mirror of a laser is flat, so the minimum radius of the beam is at the exit of the laser. You can get a much smaller spot using a lens though. However, for a small spot, its depth is also small (the length of a Gaussian beam waist is the length of a region near the focus where the light intensity is at least one half of that in the focus). The beam waist length is proportional to an inverse square of the beam's width in the focus.

I think the best is to solve the problem at its root: your laser is divergent, You are right, your lens will focus at different positions depending on its distance from the laser, and that's quite tricky to correct dynamically on a moving system.

So: collimate the laser imposing the right beam diameter right at its output, before sending it to the head. Build a galilean telescope (also known as beam expander). You need 2 high damage threshold lenses (probably the same type as your last focusing lens) with the right focal lengths. The ratio of their focal lengths will be the magnifying ratio for the beam diameter. Choose them to slightly underfill your subsequent optics. Put them at a distance equal to the sum of their focal length and that's it. Adjusting finely their distance you can correct for the divergence, while having the beam size that you want and constant for all the rest of the path.

I would opt for one negative and one positive lens, so you magnify the beam diameter avoiding a laser focus in air, which with 90W can ionize air or at least create a bad pointing stability. Even if air is not broken at the focus, small air movements can deflect the beam (and temperature raises at the focus, so air moves even more).

And please... don't forget your safety goggles, and think about a protection cage around the setup. 90W do not forgive...