Definition, properties and applications of the generalized temporal-spatial Wigner distribution function

The Wigner function is a powerful tool for analyzing signals and synthesizing systems. For one dimensional function, the Wigner domain is a two dimensional function that has two axes: the space/ time axis and the spatial temporal frequency axis, respectively. Basic optical elements as lenses, free space propagation, prisms and gratings, are expressed as known simple geometrical operations over the Wigner domain. In this paper we demonstrate the use of a generalized Wigner function definition that simultaneously handle both space and lime degrees of freedom, together with spatial and temporal frequencies. The generalized representation provides the ability to express simply also devices that effect both temporal and spatial parameters of the beam. Thus, it could be used for various applications that combine spatial and temporal handling such as temporal signal processing and super-resolution. The abilities of the new definition are demonstrated by several examples.