# Wigner distribution in optics

## Martin J. Bastiaans

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 Dolin and Walther in the sixties, 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.

While the mechanical phase space is connected to classical mechanics, where the movement of particles is studied, the phase space in optics is connected to geometrical optics, where the propagation of optical rays is considered. And where the position and momentum of a particle are the two important quantities in mechanics, in optics we are interested in the position and the direction of an optical ray. We will see that the Wigner distribution represents an optical field in terms of a ray picture, and that this representation is independent of whether the light is partially coherent or completely coherent.

We will observe that a Wigner distribution description is in particular useful when the optical signals and systems can be described by quadratic-phase functions, i.e., when we are in the realm of first-order optics: spherical waves, thin lenses, sections of free space in the paraxial approximation, etc. Although formulated in Fourier-optical terms, the Wigner distribution will form a link to such diverse fields as geometrical optics, ray optics, matrix optics, and radiometry.

Sections 1.2 through 1.7 will mainly deal with optical signals and systems. We treat the description of completely coherent and partially coherent light fields in Section 1.2. The Wigner distribution is introduced in Section 1.3 and elucidated with some optical examples. Properties of the Wigner distribution are considered in Section 1.4. In Section 1.5 we restrict ourselves to the one-dimensional case and observe the strong connection of the Wigner distribution to the fractional Fourier transformation and rotations in phase space. The propagation of the Wigner distribution through Luneburgĺs first-order optical systems is the topic of Section 1.6, while the propagation of its moments is discussed in Section 1.7. The final Section 1.8 is devoted to the broad class of bilinear signal representations, known as the Cohen class, of which the Wigner distribution is an important representative.

To: Papers by Martin J. Bastiaans