Abstract

The optical Fourier transform need not be restricted to the praxial case; it maybe generalized to describe large-angle diffraction from weakly scattering space-filling objects. The three-dimensional optical transform is based on the geometry of the Ewald–Laue construction: (1) The object is illuminated by a plane wave k; (2) each plane-wave component k' of scattered light is focused at the point Mk' with respect to the object-space origin (M is a dimensional factor scaling the wave vector into a displacement vector); (3) the image at the focal sphere is translated by–Mk; and (4) steps 1–3 are continued while k is scanned through all directions. The resulting image contains the product of the object function's Fourier transform and fixed geometric factors. A simple optomechanical apparatus produces visible transforms by means of limited k scanning above the ocular flicker rate; the demonstration is pedagogically potent. Applications include mapping of diffraction data into reciprocal space without electronic processing of data. Current research invokes Fourier-plane holography to form tomographic images from diffractometric data entirely through optical processing.

© 1990 Optical Society of America