Wigner distribution function and its application to first-order optics

Abstract

The Wigner distribution function of optical signals and systems has been introduced. The concept of such functions is not restricted to deterministic signals, but can be applied to partially coherent light as well. Although derived from Fourier optics, the description of signals and systems by means of Wigner distribution functions can be interpreted directly in terms of geometrical optics: (i) for quadratic-phase signals (and, if complex rays are allowed to appear, for Gaussian signals, too), it leads immediately to the curvature matrix of the signal; (ii) for Luneburg’s first-order system, it directly yields the ray transformation matrix of the system; (iii) for the propagation of quadratic-phase signals through first-order systems, it results in the well-known bilinear transformation of the signal’s curvature matrix. The zeroth-, first-, and second-order moments of the Wigner distribution function have been interpreted in terms of the energy, the center of gravity, and the effective width of the signal, respectively. The propagation of these moments through first-order systems has been derived. Since a Gaussian signal is completely described by its three lowest-order moments, the propagation of such a signal through first-order systems is known as well.