When choosing compressor you should consider if it can constantly provide desired pressure and volume of air. Small airbrush compressors small compressors boast amazing pressure figures but really poor flow rates adequate limited to airbrush pistols. Even if small compressor would be able to 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 really do not know what specifications it has but stream of air was very weak. It is enough if all you want to do is engrave, although actually then we got considerable smoke traces around engraved picture.
it had been rated for volume of ~2.4cfm and had a 1 Air Compressor For Laser Cutting china gallon container which can be filled to 58 PSI. It managed to give us continuous 1 bar (14 PSI) of pressure, but it had to run on a regular basis 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 concern with overheating since compressor engine is not on all the time. Using reserve atmosphere in tank we can also push up to 3.5bar (50psi) for brief jobs.
The best setup in my opinion will be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.
