When choosing compressor you should consider if it will be able to constantly provide desired pressure and volume of air. Small airbrush compressors small compressors boast amazing pressure figures but actually poor flow rates adequate only for airbrush pistols. Even if small compressor would be able to push enough air it will likely overheat in couple of minutes.
Our laser cutter was Air Compressor For Laser Cutting included with small membrane pump (Smallest on the picture), I do not know very well what specifications it has but blast of air was very weak. It is sufficient if all you want to accomplish is engrave, although actually then we got substantial smoke traces around engraved picture.
it had been rated for level of ~2.4cfm and had a 1 gallon tank which may be filled to 58 PSI. It managed to give us constant 1 bar (14 PSI) of pressure, but 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 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 isn’t on at all times. Using reserve surroundings in tank we can also force up to 3.5bar (50psi) for short jobs.
The best setup in my opinion would be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.