Convolution relation within the three-dimensional diffraction image

The amplitude in the image of a point source is (a constant times) the three-dimensional Fourier transform of the lens pupil, i.e., of the amplitude distribution of the converging waves in angle space. If the three-dimensional pupil has axial symmetry, it can be factored into a spherical shell and a one-dimensional function of distance along the axis. The amplitude in the image is given by the convolution of the Fourier transforms of the factors. On the axis, the amplitude is the Fourier transform of the one-dimensional function. Therefore the amplitude anywhere in the image is the convolution of the amplitude on the axis and the Fourier transform of the spherical shell.