We have developed a computer model calculating bare cavity transverse eigenmodes for super-gaussian unstable resonators, including aperture diffraction in the gain medium. This generalized simulation, based on the Fox and Li Power Method, reduces the input parameters to five: rod longitudinal position, cavity magnification, super-gaussian order of the output coupler reflectivity, and Fresnel numbers for the cavity and rod apertures. Using two-dimensional FFT's to discretize the Huygen-Fresnel numbers, the output fields at the plane of the rod aperture and exiting the output coupler were subjected to beam quality (M2) and extraction efficiency (Xeff) analysis. Beam quality was found to be the most sensitive to cavity magnification, with M2 values varying as much as 30% or more with 3% shifts in magnification, which can occur during rod lensing. Avoiding peaking M2 values is demonstrated with design curves for two different cavity Fresnel numbers, and super-gaussian orders. The cavity Fresnel number and the super-gaussian order are shown to only weakly affect beam quality, although extraction efficiency varies strongly with the latter. Finally, optimized rod longitudinal position was explored for promising combinations of the other four parameters, and it was found to be near the high reflector (HR) end of the cavity, in terms of M2 analysis.