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How is that sometimes laser cutting by burning material or "flame cutting" causes ignition of the whole material like here where the whole paper starts to burn and sometimes doesn't allowing precise cutting like here?

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If I wanted to laser cut combustible materials, I would do so in an inert atmosphere - no oxygen, no burning. Of course you can never prevent local charring - chemical decomposition due to heat - but you can prevent a chemical reaction with oxygen by removing the oxygen.

Other than that, careful control of the laser intensity and time profile can help: a small intense pulse will vaporize material locally without heating the bulk, and this means that insufficient material will heat up to start a fire. In essence there is no "hot fuel" if you tweak the parameters just so.

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