errruuugghhhh umm It's rather complex and there is no specific answer as a LOT of factors affect it
Wavelength, Photon Energy, Material Absorption Coefficient, Pulse Energy, Emitted Power, Material Band Gap Energy are just a few of them. It's not even possible to make a direct comparison between two different wavelengths unless you use the rather abstract figure of Watt/Seconds and that's hit and miss at best (and only tells about 5% of the story)
The effect you are thinking of PhotoThermal Vs PhotoChemical is a product of material band gap energy and photon energy, to get a photochemical effect in a CO2 cut you would need the material bandgap to be under 0.117eV (the photon energy of a normal CO2 laser beam) above that the effect is PhotoThermal and you cannot avoid carbonisation.
If you drop to say a 445nm laser that has a Photon Energy of 2.788eV so any material with a bandgap under that figure will undergo a PhotoChemical process (assuming no material defects exist) and not suffer from carbonisation.
NDFF (Neodymium-Doped Fluoride Fibre) will emit in the 380nm range but are both expensive and pretty much low power, if you want to get into the 157 - 351 range you would be looking at Excimer lasers to get anything like higher power (up to a few hundred watts) or Krypton - Ion or Argon - Ion if you want bench level test lasers. But before getting into all of that be advised you are looking at some pretty serious $$$$'s (quality labby lasers cost a LOT of money, figure on high end machine kinds of money just for the quality sources)
UV lasers are typically used for manufacturing Fiber Bragg Gratings and refractive laser eye surgery.
Also please be aware once you start hitting the UV wavelength the risks of skin cancer have been demonstrably proven over many years
get down into the 10 - 121 nm range and you are playing with really nasty toys, Photon Energies in the 10 eV up to 124 eV photon energies (although absorbed by air quite well) and you are in the entirely ionising radiation range so about as dangerous as lasers get.
Just the opposite, frequency/wavelength follows the inverse proportionality law, high frequency = shorter wavelength hence the reason to get a 532nm laser you frequency double (second harmonic generation) a 1,064 nm such as an Nd-YAG
Overall it's a really complex and exact science but with a LOT of possible variables that can change almost from second to second depending on so many factors
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Short version:
Find out the BandGap energy of what you want to cut, find an affordable laser with a Photon Energy above the figure you have for the material