Towards digital Fourier optics - From the diffraction to the Fourier's space

Fourier optics, which requires the coherence of light sources, is based on a scalar representation of the electromagnetic field, justified by the independence of its spatio-temporal components. The aim of this article is to find the light amplitude at a point in space in order to deduce the intensity collected by an optical sensor. To achieve this goal, a wavelet decomposition is first performed. Then, its microscopic formulation is presented with the wave equation, leading to a macroscopic representation to study its applications.

Secondly, the study of experimental constraints makes it necessary to define several domains in order to interpret light phenomena. In particular, the notion of near-field and far-field is developed for the study of diffraction. Several classic examples are treated, such as diffraction by an edge and by different apertures. The Fourier transform is then used to obtain light intensity. Then, thanks to the introduction of a simple converging lens, Fourier optics is used in a range of applications compatible with the reduced dimensions of optical systems.

This article concludes with the technical implementation of the Fourier transform in optics. The classical notion of frequency is then introduced in its spatial form. The definition and some properties of the Fourier transform are briefly recalled, before tackling well-known examples such as the diffraction grating. These examples are treated in analog form, then updated in digital form to introduce sampling.

A second article [E 4 151] looks at analog and digital examples applied to different industrial fields.

At the end of the article, readers will find a glossary and a table of symbols used.