High quality, monochromatic laser beams are governed by diffraction instead of geometrical optics. Talking about rays doesn't really tell the full story. The parameter of a laser beam which expresses how well collimated it is is called the Rayleigh range, $z_R$. The units of $z_R$ are units of distance, and you can think of it roughly as 'the beam will not start to diverge significantly within $z_R$ of the smallest spot.' Interestingly, it turns out that this parameter is directly related to the smallest spot size of the laser beam, $\omega_0$. This parameter is known as the beam waist. The relationship between them is
$$
z_r=\frac{\pi\omega_0^2}{\lambda}.
$$
The point is; **the smaller you want to make the beam, the less collimated it will be.** This only gets more true if the beam is of poor quality. Just to put some numbers to it, a red laser pointer focused to a spot size of $1\,\text{mm}$ is collimated over a range of $5\,\text{m}$, but if you focus it down to a spot size of $10\,\mu\text{m}$ it will only be collimated over a range of $0.5\,\text{mm}$.

First, some questions about your setup:

- 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,
- 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.
- 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:

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!*)

## Best Answer

On point 1, you are correct. The uncertainty principle leads to the finite spot size when focusing a laser beam and the spread of the beam thereafter. In general, in order to produce a smaller spot, you need to focus the beam in at a high angle (short focal length lens), meaning a larger spread in transverse momentum of the photons. The same thing can be said for imaging optics. In order for a microscope to look at a small object, you need a short focal length microscope objective that can collect light at a high angular spread.

As for point 2, "focusing" of a laser beam without a lens can happen due to some non-linear effects if you are at very-high intensities in some type of medium (air or otherwise), but that is a different subject all together. However, spreading of a laser beam is an inherent property of the beam due to the uncertainty relationship as well. Because the laser has a finite spot size, it must also have a spread in the transverse momentum. This means that as the laser propagates, the beam will in general expand. The far-field angular divergence (full angle of the "cone" of the beam) is given by (for $TEM_{0,0}$):

$\theta = \frac{2\lambda_0}{\pi n w_0}$

where $\lambda_0$ is the wavelength, $n$ is the index of refraction and $w_0$ is the waist size of the beam. You can see here that the divergence angle is inversely proportional to the waist size of the beam, $w_0$. For larger beams, there is less divergence.