The total energy integral functions for apodization T1 = (1 - r2)n and T2 = (2r - r2)n are compared for unobstructed apertures. The energy values increase linearity for T2 and with the (2n + 3) power for T1. For obstructed apertures, these values for T1 increase with higher powers at small image radii and assume linear characteristics only at larger radii. This transition occurs as the obstruction approaches zero. Partial energy functions for a region centered some distance from the central region show the same general trend as the image irradiance functions.
Hubert F. A. Tschunko, "Imaging performance of annular apertures. 5: Total and partial energy integral functions," Appl. Opt. 22, 137-141 (1983)