On improving self-Fourier function theory for application to optical information processing

It is known that generally the space spectrum of an object is unlike the object itself. However, in Fourier optics, there are many important functions, which are their own Fourier transforms, defined as self-Fourier functions (SFFs). Caola has discovered a set of SFF's and proposed how to construct a SFF from an arbitrary and transformable function, i.e., Caola's SFF. Lohmann et al discussed Caola's SFF and showed that Caola discovered all SFFs. Proceeding from the cyclic property of a transform, Lohmann et al furthermore proposed that, for an arbitrary and cyclic transform, there is a self-transform function (STF), which can be constructed from an arbitrary and transformable function. Caola's SFF is just a special case of Lohmann's STF. In this paper, it is shown that the set of SFFs as discovered by Caola does not include all SFFs, and that the Caola's SFF is not applicable to odd functions. An equation can be used to construct a SFF from an arbitrary and transformable odd function.