In 1932, Wigner introduced a distribution function in mechanics that permitted a description of mechanical phenomena in a phase space. Such a Wigner distribution was introduced in optics by Walther in 1968, to relate partial coherence to radiometry. A few years later, the Wigner distribution was introduced in optics again (especially in the area of Fourier optics), and since then, a great number of applications of the Wigner distribution have been reported. It is the aim of this chapter to review the Wigner distribution and some of its applications to optical problems, especially with respect to partial coherence and first-order optical systems. The chapter is roughly an extension to two dimensions of a previous review paper on the application of the Wigner distribution to partially coherent light, with additional material taken from some more recent papers on the twist of partially coherent Gaussian light beams and on second- and higher-order moments of the Wigner distribution. Some parts of this chapter have already been presented before and have also been used as the basis for a lecture on "Representation of signals in a combined domain: Bilinear signal dependence" at the Winter College on Quantum and Classical Aspects of Information Optics, The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, January 2006. In order to avoid repeating a long list of ancient references, we will often simply refer to the references in Ref. 4; these references and the names of the first authors are put between square brackets like [Wigner, Walther, 1,2].
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