Simple methods for measuring spatial coherence and their relation to the Wigner function

Measurements of the state of coherence of an optical field are important, for example, in the characterization of sources used for illumination in imaging systems. However, within the second order theory of coherence, this characterization amounts to the determination of a four-dimensional complex distribution. Therefore, the use of traditional techniques such as Young´s two-pinhole setup would require a very large set of measurements. In this presentation several alternative approaches are described, which were inspired by the description of the field in terms of what is known as phase-space distributions, such as the Wigner and ambiguity functions. In particular, the Wigner function provides an intuitive description of partially coherent fields, which resembles the radiance from the theory of radiometry, i.e. it provides a weight to each ray in the system. One technique that can be used for the characterization of the field´s coherence, known as phase-space tomography, is based on the recovery of the Wigner (or ambiguity) function from measurements of the optical intensity over a region of space. We will discuss the generalization of this technique beyond the paraxial regime through the use of generalizations of the Wigner and ambiguity functions. We will also present an alternative technique where the weight distribution of the subset of rays passing through a given point is recovered through the introduction of a transparent binary mask with a phase discontinuity. This technique provides simultaneously measurements of the coherence of the field at all pairs of points centered at this discontinuity.