Generalized Aperture and the Three-Dimensional Diffraction Image

It is shown that when a lens produces an image of a point source the three-dimensional diffraction pattern which results is the three-dimensional Fourier transform of a generalization of the lens aperture. This implies similar Fourier relations in one and two dimensions. An explicit form of the former is derived which demonstrates that the amplitude distribution on an arbitrarily directed line through the focus is the Fourier transform of the projection of the generalized aperture onto that line. These relations hold in aberrant as well as in ideal systems. Some examples are worked out by using the one-dimensional relation.