Analysis and optimization of convergent Fourier-transform setups for optical correlators

Fraunhofer diffraction at finite distances can be produced by a variety of optical setups. Convergent arrangements, in which the diffracting object is illuminated by a converging spherical wave-front, are of special interest since they enable an easy control over the size of the resulting diffraction pattern. In particular, such arrangements are frequently used in optical correlators as a means to obtain the Fourier transform of the input scene. We analyze the effect of the wave aberration introduced by the convergent optical system in the resulting Fourier transform, both experimentally and by a computer simulation, as well as its impact over the correlation plane. Also, we introduce a simple optimization method that gives good results in alleviating the loss of space invariance, which is the most; important side effect found in the study.