When choosing compressor you need to consider if it can constantly provide 
desired Air Compressor For Laser Cutting pressure and volume of air. Small airbrush compressors small compressors boast amazing pressure figures but really poor flow rates sufficient only for airbrush pistols. Even if small compressor can push enough air it’ll likely overheat in short while.
Our laser cutter was included 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 adequate if all you have to to accomplish is engrave, although also then we got significant smoke traces around engraved picture.
it was 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 continuous 1 bar (14 PSI) of pressure, but it had to operate on a regular basis 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 fear of overheating since compressor engine is not on all the time. Using reserve air flow in tank we are able to also drive 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.