When choosing Air Compressor For Laser Cutting compressor you should consider if it will be able to constantly provide desired pressure and level of air. Little airbrush compressors small compressors boast amazing pressure figures but actually poor flow rates enough limited to airbrush pistols. Also if small compressor can push enough air it will likely overheat in short while.
Our laser cutter came with small 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 accomplish is engrave, although also then we got considerable smoke traces around engraved picture.
it had been rated for volume 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 
on a regular basis and overheat badly within 5 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 are able to run at continuous ~2bar (31psi) without concern with overheating since compressor engine is not on all the time. Using reserve air in tank we can also drive 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.