When choosing compressor you should consider if it can constantly provide desired pressure and volume of air. Small airbrush compressors little compressors boast impressive pressure figures but actually poor flow rates sufficient limited to airbrush pistols. Even if small compressor would be able to push enough air it’ll likely overheat in short while.
Our laser cutter was included with small membrane pump (Smallest on the picture), I do not know what specifications it has but stream of air was extremely weak. It is enough if all you have to to accomplish is engrave, although even then we got significant smoke traces around engraved picture.
it was rated for volume of ~2.4cfm and had a 1 gallon tank which can be filled to 58 PSI. It managed to give us constant 1 bar (14 PSI) of pressure, nonetheless it had to run at all times and overheat badly within 5 minutes.
Third and current compressor we got was 6cfm rated belt driven unit with 13 gallon tank which is filled to ~100psi. With this unit we can run at constant ~2bar (31psi) without concern with overheating since compressor engine is not on all the time. Using reserve surroundings in tank we are able to also push up to 3.5bar (50psi) for brief jobs.
The best setup for me would be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and Air Compressor For Laser Cutting software air control via solenoid valve.