# Applications of the Wigner distribution function to partially coherent light beams

## Martin J. Bastiaans

The paper presents a review of the Wigner distribution function and of some of its applications to optical problems, in particular to the description of partially coherent light beams and their propagation through first-order optical ABCD-systems. The Wigner distribution function describes an optical signal in space and spatial frequency (i.e., direction) simultaneously and can thus be considered the local spatial-frequency spectrum (or ray pattern) of the optical signal. Although derived in terms of Fourier optics, the description of an optical signal by means of its Wigner distribution function closely resembles the ray concept in geometrical optics; the Wigner distribution function thus presents a link between partial coherence and radiometry.

The description of partially coherent light by means of its mutual coherence function and its cross-spectral density forms the basis for the definition of the Wigner distribution function. Some examples of the Wigner distribution function for partially coherent light are given to illustrate its concept, and some of its most important properties are presented.

The modern theory of modal expansions is introduced, both to the cross-spectral density and to the Wigner distribution function. Inequalities for the Wigner distribution function are derived with the help of these modal expansions.

The propagation of the Wigner distribution function through linear optical systems is considered. For 'black-box' type systems, with an input and an output plane, an input-output relationship in terms of the so-called 'ray-spread function' is introduced, on the analogy of the well-known coherent point-spread function; special attention is paid to a circular aperture and to the important class of first-order optical ABCD-systems. For systems that are described by a differential equation, such as the Helmholtz equation and the wave equation, a corresponding differential equation - the 'transport equation' - is presented to describe the propagation in terms of the Wigner distribution function.

Some applications of the Wigner distribution function to specific optical problems are considered.
(i) An uncertainty principle for partially coherent light is described, similar to the one for completely coherent light, but taking into account the overall degree of coherence of the light. The uncertainty principle describes a relation between the width and the divergence of a partially coherent light beam.
(ii) A matrix of second-order moments of the Wigner distribution function of general partially coherent light beams - containing both the beam width and the beam divergence, and mixed moments - is considered. Special attention is given to the propagation of these moments through ABCD-systems.
(iii) Another matrix of second-order moments is presented, from which invariants for these moments can be derived when the partially coherent light beam propagates through ABCD-systems.
(iv) Special attention is given to the description of partially coherent Gaussian beams and ABCD-systems. It is shown that the well-known bilinear ABCD-law arises in a very natural way, when we describe such beams by their Wigner distribution function. The description includes the case of 'twisted' Gaussian beams.

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