I've been reading up on laser cutting and it seems that 808nm at around 2 watts is typical to cut paper. How would one calculate the wavelength and power required to cut an arbitrary material?
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...
Aside from the "it's complicated answer," there's also the "check for prior experimental research" answer. Without digging into it very far I used one trick. I found this Wikipedia article on Laser cutting and clicked ctrl+f to search the document for 'paper' and came up with this hit:
CO2 lasers are used for industrial cutting of many materials including mild steel, aluminium, stainless steel, titanium, Taskboard, paper, wax, plastics, wood, and fabrics.
So if I were building a paper laser cutting apparatus of some kind I'd search further to see what co2 lasers were capable of, and perhaps experiment with building one.