When choosing compressor you have to consider if it will be able to constantly provide desired pressure and volume of air. Small airbrush compressors little compressors boast 
amazing pressure figures but really poor flow rates Air Compressor For Laser Cutting adequate only for airbrush pistols. Actually if small compressor would be able to push enough air it will likely overheat in short while.
Our laser cutter came with small membrane pump (Smallest on the picture), I really do not know very well what specifications it has but stream of air was extremely weak. It is enough if all you want to accomplish is engrave, although even then we got considerable smoke traces around engraved picture.
it had been rated for level of ~2.4cfm and had a 1 gallon container which can be filled to 58 PSI. It were able to give us constant 1 bar (14 PSI) of pressure, nonetheless it had to run constantly 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 concern with overheating since compressor engine isn’t on on a regular basis. Using reserve atmosphere in tank we can also drive up to 3.5bar (50psi) for short 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.