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 âRepresentation of signals in a combined domain: Bilinear signal dependenceâ 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].
We conclude this Introduction with some remarks about the signals with which we are dealing, and with some remarks about notation conventions. We consider scalar optical signals, which can be described by, say, (x, y, z, t), where x, y, z denote space variables and t represents the time variable. Very often we consider signals in a plane z = constant, in which case we can omit the longitudinal space variable z from the formulas. Furthermore, the transverse space variables x and y are combined into a two-dimensional column vector r. The signals with which we are dealing are thus described by a funct (r, t).
We will throughout denote column vectors by boldface, lowercase symbols, while matrices will be denoted by boldface, uppercase symbols; transposition of vectors and matrices is denoted by the superscript t. Hence, for instance, the two-dimensional column vectors r and q represent the space and spatial-frequency variables [x, y]t and [u, v]t, respectively, and qtr represents the inner product ux + vy. Moreover, in integral expressions, dr and dq are shorthand notations for dx dy and du dv, respectively.