Laser Cutter Power/Wavelength to Cut Paper

Here are some of the relevant bits of physics/questions to ask:

• To cut a material, it needs to absorb heat faster than it can lose it.
• heat is conducted away: this is typically linear with temperature gradient
• heat can be radiated away: this is more important at higher temperatures (follows $T^4$ relationship)
• laser power may be reflected or absorbed: the right wavelength will be the one that is "mostly absorbed" and will depend on the material.
• the mechanism for "cutting" matters: is it melting, burning, or enhancing a chemical reaction (etching, oxidation)
• does the cut have to be very small (narrow)?
• does the surrounding material deform if it gets hot?
• how many meters of material do you want to cut per unit time?

Based on the above considerations it is hard to give a general "formula" for the power and wavelength. I would say that higher power means you can cut faster - which in turn means that you can make a narrower cut without heating up the surrounding material (it didn't have time to heat up). This is also a cleaner cut. As for the wavelength - typically you will use a laser that can efficiently generate a lot of power (CO2 laser can have efficiency up to 20% but a longer wavelength, around 10 µm), and that is well absorbed by the material. You also want to be able to focus it to a small spot - both to increase the power density and to make a fine cut. The longer the wavelength, the harder it is to focus something because of diffraction...