When Air Compressor For Laser Cutting choosing compressor you should consider if it will be able to constantly provide desired pressure and level of air. Small airbrush compressors little compressors boast impressive pressure figures but really poor flow rates enough limited to airbrush pistols. Even if small compressor can push enough air it'll likely overheat in couple of minutes.
Our laser cutter came with little membrane pump (Smallest on the picture), I really do not know very well what specifications it has but stream of air was very weak. It is adequate if all you have to to accomplish 
is engrave, although also then we got significant smoke traces around engraved picture.
it had been rated for volume of ~2.4cfm and had a 1 gallon container which may be filled to 58 PSI. It were able to give us constant 1 bar (14 PSI) of pressure, but it had to operate all the time and overheat badly within five 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 continuous ~2bar (31psi) without fear of overheating since compressor engine isn't on at all times. Using reserve air flow in tank we are able to 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.