Apodization, Point Spread Variance, and Wave Front Variance

Apodization influences the intensity of rays passing through various portions of the pupil aperture. In a ray-tracing computation of the point spread, each ray at the image plane is weighted by the apodizing factor of the portion of the pupil through which it passed. In computing the variance, or second moment, of the point spread, for example, the squares of the distances of the rays from the center of the point spread are weighted by the apodizing factor associated with the portion of the pupil through which they passed, e.g, Stiles-Crawford apodization (1,2): Here we seek to find an attenuation of the wave aberration at the pupil, using a function of the apodizing factor that would result in a point spread of variance equivalent to that of the apodized unattenuated aberrated wave.