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