When choosing compressor you need to consider if it can constantly provide desired pressure and Air Compressor For Laser Cutting volume of air. Little airbrush compressors little compressors boast amazing pressure figures but actually poor flow rates adequate only for airbrush pistols. Actually if small compressor would be able to push enough air it will 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 have to to accomplish is engrave, although also then we got considerable smoke traces around engraved picture.
it had been rated for level of ~2.4cfm and had a 1 gallon tank which can 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 5 minutes.
Third and current compressor we got was 6cfm rated belt driven device with 13 gallon tank which is filled to ~100psi. With this unit we are able to run at constant ~2bar (31psi) without fear of overheating since compressor engine isn’t on constantly. Using reserve atmosphere in tank we are able to also force 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.