Use of the Wigner distribution function in optical problems

The paper presents a review of the Wigner distribution function (WDF) and of some of its applications to optical problems. The WDF describes a signal in space and spatial frequency simultaneously, and can be considered as the local spatial-frequency spectrum of the signal. Although derived in terms of Fourier optics, the description of a signal by means of its WDF closely resembles the ray concept in geometrical optics. The concept of the WDF is not restricted to deterministic signals; it can be applied to stochastic signals as well, thus presenting a link between partial coherence and radiometry. Properties of the WDF and its propagation through linear systems are considered; again, the description of systems by WDFs can be interpreted directly in geometric-optical terms. Three main categories of optical problems are considered, in which the concept of the WDF can be applied usefully. First, the application to geometric-optical systems, i.e., systems where a single ray remains a single ray. Second, the application to problems in which the signal appears quadratically, like in the case of partial coherence. Third, the application to problems where properties of the signal are discussed in space and spatial frequency simultaneously; the uncertainty principle in Fourier optics might be an example. The WDF approach is extremely useful when two or more categories are combined: for instance, the propagation of partially coherent light through geometric-optical systems, or the formulation of uncertainty relations for partially coherent light.