The problem of image evaluation for small aberrations by means of the Maréchal method of phase-error balancing is well known. For the case of large phase-errors, a number of authors have suggested spot diagram averaging. This method implies an association of ray density with energy flux, which is well justified by experience for large aberrations, but certainly subject to question for small aberrations. Using the radius of gyration as a measure, the ray density method of averaging over the image plane is shown to be equivalent to the simpler and more physical process of averaging over the exit pupil using the radius as a weighting factor. In addition, a proof for the justification of the ray density method is given along with the geometrical intensity law. This proof stems from an application of Liouville’s theorem. Finally, the two limits are compared with results published in the literature for various combinations of aberrations.
JOHN B. DEVELIS, "Comparison of Methods for Image Evaluation," J. Opt. Soc. Am. 55, 165-173 (1965)