The Fourier Transform in Optics: Analogous Experiment and Digital Calculus

Abstract

Discrete optics and digital optics are fast becoming a classical chapter in optics and physics in general, despite their relative recent arrival on the scientific scene. In fact their spectacular blooming began precisely at the time of the computer revolution which made possible fast discrete numerical computation. Discrete mathematics in general and discrete optics in particular although predated digital optics, even by centuries, received a new impetus from the development of digital optics. Formalisms were designed to deal with the specific problems of discrete numerical calculation. Of course, these theoretical efforts were done not only for the benefit of optics but of all quantitative sciences. Diffractive optics in general and the newly formed scientific branch of digital holography, turned out to be especially suited to benefit from the development of discrete mathematics. One reason is that the optical diffraction in itself is a mathematical transform. An ordinary optical element such as the lens turned out to be a genuine natural optic computer, namely one that calculates the Fourier transform (Goodman, 1996, chapter 5). The problem is that the discrete mathematics is not at all the same thing as continuous mathematics. For instance for the most common theoretical tool in diffractive optics, the continuous (physical) Fourier transform (CFT) we have a discrete correspondent named discrete Fourier transform (DFT). We need DFT because the Fourier transform rarely yields closed form expressions and generally can be computed only numerically, not symbolically. Of course, no matter how accurate, by its very nature DFT can be only an approximation of CFT. But there is another advantage offered by DFT which inclines the balance in favour of discrete optics. The reason is somehow accidental and requires some explanation. It is the discovery of the Fast Fourier transform, for short FFT, (Cooley & Tukey, 1965), which was followed by a true revolution in the field of discrete optics because of the reduction with orders of magnitude of the computation time, especially for large loads of input data. FFT stirred also an avalanche of fast computation algorithms based on it. The property that allowed the creation of these fast algorithms is that, as it turns out, most diffraction formulae contain at their core one or more Fourier transforms which may be rapidly calculated using the FFT. The method of discovering a new fast algorithm is oftentimes to reformulate the diffraction formulae so that to identify and isolate the Fourier transforms it contains. We contributed ourselves to the development of the field with the creation of an