A number of flattened irradiance distributions have been proposed
and analyzed in the literature, including the super-Gaussian, flattened-Gaussian, Fermi-Dirac, and super-Lorentzian as well as
generalizations of these functions to include multiple shape parameters. Previous work has made comparisons between these different families of functions and examined the effects of propagation (diffraction) on the shape of the beam profile. In this paper, we examine the normalization of different functions, comparisons of profile shapes using different parameters within each family, the slope of the profiles at the half-height point of the irradiance, and two conditions that permit matching shapes of profiles from different families. Then, we summarize the results of diffraction for variation of the profile shape parameters, beam propagation, and diameter of the exit aperture on the shape of a beam as it leaves the optics. Results are also presented which identify the regions and amounts of aspheric surface sag that is required to produce a flattened beam profile as compared to a top-hat profile profile.