When choosing compressor you have to consider if it can constantly provide desired pressure and 
volume of air. Little airbrush compressors little compressors boast impressive pressure figures but really poor flow rates sufficient limited to airbrush pistols. Also if small compressor can push enough air it’ll Air Compressor For Laser Cutting likely overheat in short while.
Our laser cutter was included with small membrane pump (Smallest upon the picture), I do not know what specifications it has but stream of air was extremely weak. It is enough if all you have to to accomplish is engrave, although even then we got considerable smoke traces around engraved picture.
it was rated for volume of ~2.4cfm and had a 1 gallon container which may be filled to 58 PSI. It managed to give us continuous 1 bar (14 PSI) of pressure, nonetheless it had to run all the time and overheat badly within five minutes.
Third and current compressor we got was 6cfm rated belt driven device with 13 gallon container which is filled to ~100psi. With this unit we can run at continuous ~2bar (31psi) without fear of overheating since compressor engine is not on continuously. Using reserve air flow in tank we are able to also press up to 3.5bar (50psi) for brief jobs.
The best setup in my opinion will be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.