Abstract

The premise of this submission is to identify the role of the Wigner function [1] in paraxial wave optics, where it can be naturally introduced on the basis of the analogy between the paraxial wave equation and the Schrödinger equation of nonrelativistic quantum mechanics. Also known as the Wolf function, it describes the cross section of a paraxial beam using a real-valued function of displacement from axis and transverse components of the wave vector. This can be perceived as a quasi-probability distribution over the system's phase space, following the dynamics of a state space density of an ensemble of free noninteracting particles. In a (2+1)–dimensional picture this corresponds to the propagation of straight linear rays where the Wolf function describes the local phase space intensity. There is no notion of relative phase but negative values can be present, providing an alternative basis for description of interference phenomena. This corresponds exactly to the concept of pencils of positive and negative rays, developed by E. C. G. Sudarshan in the late 70’s and 80’s [2, 3].

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