What is the relationship between tube size and power output for a $\rm CO_2$ Laser?

Question

I am in the process of buying a replacement tube for my $\rm CO_2$ Laser cutter and something I've always wondered is, what is the relationship between the tube size and power output for a sealed tube $\rm CO_2$ Laser? In general it seems higher power tubes are longer and usually have a larger diameter, however, I'd like to understand the physics that determine what size is required for a desired power output.

Answer 1

Important Carbon dioxide laser characteristics are high unsaturated gain, high-power output and good efficiency. It has been experimentally determined that gain is a strong function of tube diameter and that output power is essentially independent of diameter. The measured gain for three tube diameters is plotted in the curve below as a function of the tube current at optimum gas pressures for maximum output power. The gain curves for larger diameter tubes can be substantially changed simply by adding some turbulence to the gas flow. The gain reaches its maximum at low currents, falling off as current increases. This fall-off is apparently caused in part by the heating of the gas on the tube axis. Note that the current for maximum gain is smaller than that for maximum output power.

These tubes can be used as amplifiers as well as oscillators by simply directing a laser beam down the length of the tube. Once the power density in the beam exceeds 25 to 50 W per square cm, the gain becomes saturated and decreases to a low value. Thus, for power densities much less than this figure, power amplification is achieved by using small bore tubes at low currents. For power densities much larger than 25 to 50 W per square cm, the power amplification if almost independent of the tube bore; thus, the tube should be operated at currents which produce maximum power.

Answer 2

If you have a longer tube, you can put in more gas and increase the amount of energy you pump into it, thus increasing the laser power.