When choosing compressor you should consider if it will be able to constantly provide desired pressure and volume of air. Small Air Compressor For Laser Cutting airbrush compressors small compressors boast impressive pressure figures but really poor flow rates adequate limited to airbrush pistols. Also if small compressor can push enough air it will likely overheat in couple of minutes.
Our laser cutter came with little membrane pump (Smallest upon the picture), I really do not know very well what specifications it has but blast of air was extremely weak. It is enough if all you want to do is engrave, although also then we got considerable 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 constant 1 bar (14 PSI) of pressure, nonetheless it had to run at all times and overheat badly within five 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 continuous ~2bar (31psi) without concern with overheating since compressor engine is not on on a regular basis. Using reserve air in tank we can also force up to 3.5bar (50psi) for short jobs.
The best setup for me will be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.