When choosing compressor you have to consider if it can constantly provide desired pressure and volume of air. Little airbrush compressors small compressors boast amazing pressure figures but really poor flow rates enough limited to airbrush pistols. Also if small compressor would be able to push enough air it’ll Air Compressor For Laser Cutting likely overheat in couple of minutes.
Our laser cutter was included with small membrane pump (Smallest on the picture), I do not know very well what specifications it has but blast of air was extremely weak. It is sufficient if all you want to accomplish is engrave, although also then we got considerable smoke traces around engraved picture.
it was rated for volume 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, 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 unit with 13 gallon 
tank which is filled to ~100psi. With this unit we are able to run at continuous ~2bar (31psi) without concern with overheating since compressor engine is not on on a regular basis. Using reserve atmosphere in tank we are able to also push up to 3.5bar (50psi) for short jobs.
The best setup for me would be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.