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Apologies if this is the wrong Stackexchange. All the laser-cutting searches led to graphicdesign.stackexchange.com. ?!?!?!?

I have access to a laser cutting rig, a 200W CO2 beast, that needs to do a bit of an unusual job - cut something underwater. We're not looking to have the hardware submerged, but we'd like to change the focus distance from ~8mm to (wait for it) about 1m. (That's meters, not millimeters.) If we prefocus to about 1m, traversing the boundary between air and water will bring the focus distance in to about .75m, which is where we need it. It's just that none of us have enough optics in our blood to know how to go about finding a lens to fit our focus distance.

I'm assuming that doing this will affect our focus quality considerably. Is the patient dead, Doc? If the water itself is a deal-breaker, MAYBE we can work something else out, but we're never going to get a cutter head into the space where we need to do the cut - so we're stuck doing a sniper job.

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    $\begingroup$ Do I understand correctly you want to penetrate water with the CO2 laser beam? Water is strongly absorbing at CO2 laser wavelength. The penetration depth will be in the order of 10µm (yes micrometers!) --> en.wikipedia.org/wiki/Electromagnetic_absorption_by_water $\endgroup$
    – Andreas H.
    Sep 11 '20 at 6:33
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    $\begingroup$ The absorption depth here, as @AndreasH. provided, pretty much puts an end to this concept. You're going to have to submerge the rig and evacuate the volume between the laser output and the cutting surface. $\endgroup$ Sep 11 '20 at 13:26
  • $\begingroup$ Well, it looks like we're going to be draining the swamp - as it were - so we're looking at a focus in the 1m range... $\endgroup$ Sep 12 '20 at 2:27
  • $\begingroup$ There are laser cutters that use a fiber optic to carry the laser beam to the target. Look into getting one of those? $\endgroup$
    – Jon Custer
    Sep 16 '20 at 14:32
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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_R\approx& 3mm \end{align}

If $z\gg z_R$ we can approximate $1 + \left(\frac{z}{z_R}\right)^2\approx \left(\frac{z}{z_R}\right)^2$ 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 $w_0=100\mu m$ 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.

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  • $\begingroup$ Oh I see you're using CO2, I'll adjust the wavelengths $\endgroup$
    – Jagerber48
    Sep 11 '20 at 4:10
  • $\begingroup$ Ok, I got numbers wrong on my first pass this doesn't seem as improbable as I first thought. I don't have time to clean up the answer right now. In short: it looks like you need a lens with a diameter of a couple inches (this isn't too bad) and a focal length of 1m. I think very long focal length lenses are hard to make and hard to get accurate, there will still be issues with aberrations I think but something may be available. $\endgroup$
    – Jagerber48
    Sep 11 '20 at 4:15
  • $\begingroup$ Also check out gaussianbeam.sourceforge.net $\endgroup$
    – Jagerber48
    Sep 11 '20 at 4:15
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    $\begingroup$ A 30cm lens does sound unlikely to be practical at sufficient quality - but a pair of parabolic mirrors from amateur telescopes might get you closer. $\endgroup$ Sep 11 '20 at 9:00

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