Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Optica Acta International Journal of Optics

Transport Equations for the Wigner Distribution Function

Martin J. Bastiaans

Published Oct 1, 1979• 64 citations
Publisher

Abstract

Equations have been derived which describe the transport of the Wigner distribution function in homogeneous and inhomogeneous media. In a weakly inhomogeneous medium, the transport equation can be formulated in geometrical optical terms as follows: along a geometrical optical light ray, the Wigner distribution function has a constant value.

DOI
10.1080/713819904

Related papers

Transport Equations for the Wigner Distribution Function in an Inhomogeneous and Dispersive Medium

Martin J. Bastiaans

The Wigner distribution function of an optical signal, which can be considered as the momentary temporal and local spatial spectrum of the signal, is defined. Equations are derived which describe the transport of the Wigner distribution function in a medium, e.g. a plasma, that is weakly inhomogeneous in space and time, and exhibits weak dispersion for the temporal as well as the spatial frequency variable. The transport equations are compared with the eikonal equation in geometrical optics giving, as a geometrical-optical formulation of the transport equations, that along the path of a geometrical-optical light ray the Wigner distribution function has a constant value.

Optica Acta International Journal of Optics•Nov 1, 1979•42 citations

The Wigner distribution function and its applications to optics

Martin J. Bastiaans

The paper presents a review of some applications of the Wigner distribution function to optics. The Wigner distribution function F(x,u) of a signal describes the signal in space x and spatial frequency u, simultaneously, and can be considered as a local spatial frequency spectrum of the signal. Although derived in terms of Fourier optics, the description of a signal by means of its Wigner distribution function closely resembles the ray concept in geometrical optics. Some examples are given to show this resemblance. Properties of the Wigner distribution function are discussed, showing, for instance, its relation to Heisenberg’s uncertainty principle and to some well‐known radiometric quantities. The concept of the Wigner distribution function is not restricted to deterministic signals; it can easily be extended to stochastic signals. Some examples of stochastic signals are considered. The propagation of signals through linear systems can readily be expressed in terms of Wigner distribution functions. Again, the description of systems by Wigner distribution functions can be interpreted directly in terms of geometrical optics: for Luneburg’s first‐order system, for instance, such a description immediately yields the ray transformation matrix of the system, and the relation between the input and the output Wigner distributioin function reads Fo(x,u) = Fi(Ax+Bu,Cx+Du). Transport equations for the Wigner distribution function in a general linear medium are derived. In the Liouville approximation they take the form of a first‐order partial differential equation, which allows, once more, a geometric‐optical interpretation: along the path of a geometric‐optical light ray the Wigner distribution function has a constant value.

AIP conference proceedings•Jan 1, 1980•18 citations

The Wigner distribution function applied to optical signals and systems

MJ Martin Bastiaans

In this paper the Wigner distribution function has been introduced for optical signals and systems. The Wigner distribution function of an optical signal appears to be in close resemblance to the ray concept in geometrical optics. This resemblance reaches even farther: although derived from Fourier optics, the Wigner distribution functions of some elementary optical systems can directly be interpreted in terms of geometrical optics.

SPIE milestone series•Jan 1, 2006•0 citations

Analog of the Wigner-Moyal equation for the electromagnetic field

José Tito Mendonça, N. L. Tsintsadze

The evolution equation for the Wigner distribution of the classical electromagnetic field is derived for a nonstationary and inhomogeneous optical medium, which is formally similar to the Wigner-Moyal equation for a quantum system. The geometric optics approximation is discussed in detail, and the conservation equation for the number of photons is justified. The influence of dispersion is also considered.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Sep 1, 2000•30 citations

Wigner distribution function in nonlinear optics

Daniela Dragoman

Transformation laws for the Wigner distribution function, the radiant intensity, the radiant emittance, and the first- and second-order moments of the Wigner distribution function through an inhomogeneous, Kerr-type medium have been derived as well as for the beam quality factor and the kurtosis parameter. It is shown that the inhomogeneous Kerr-type medium can be approximated from the Wigner-distribution-function transformation-law point of view with a symplectic ABCD matrix with elements depending on the field distribution.

Applied Optics•Jul 20, 1996•22 citations

Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles

Miguel Angel Alonso

This tutorial gives an overview of the use of the Wigner function as a tool for modeling optical field propagation. Particular emphasis is placed on the spatial propagation of stationary fields, as well as on the propagation of pulses through dispersive media. In the first case, the Wigner function gives a representation of the field that is similar to a radiance or weight distribution for all the rays in the system, since its arguments are both position and direction. In cases in which the field is paraxial and where the system is described by a simple linear relation in the ray regime, the Wigner function is constant under propagation along rays. An equivalent property holds for optical pulse propagation in dispersive media under analogous assumptions. Several properties and applications of the Wigner function in these contexts are discussed, as is its connection with other common phase-space distributions like the ambiguity function, the spectrogram, and the Husimi, P, Q, and Kirkwood–Rihaczek functions. Also discussed are modifications to the definition of the Wigner function that allow extending the property of conservation along paths to a wider range of problems, including nonparaxial field propagation and pulse propagation within general transparent dispersive media.

Advances in Optics and Photonics•Nov 17, 2011•175 citations
Equations have been derived which describe the transport of the Wigner distribution function in homogeneous and inhomogeneous media. In a weakly inhomogeneous medium, the transport equation can be formulated in geometrical optical terms as follows: along a geometrical optical light ray, the Wigner distribution function has a constant value.
The Wigner distribution function of an optical signal, which can be considered as the momentary temporal and local spatial spectrum of the signal, is defined. Equations are derived which describe the transport of the Wigner distribution function in a medium, e.g. a plasma, that is weakly inhomogeneous in space and time, and exhibits weak dispersion for the temporal as well as the spatial frequency variable. The transport equations are compared with the eikonal equation in geometrical optics giving, as a geometrical-optical formulation of the transport equations, that along the path of a geometrical-optical light ray the Wigner distribution function has a constant value.
The paper presents a review of some applications of the Wigner distribution function to optics. The Wigner distribution function F(x,u) of a signal describes the signal in space x and spatial frequency u, simultaneously, and can be considered as a local spatial frequency spectrum of the signal. Although derived in terms of Fourier optics, the description of a signal by means of its Wigner distribution function closely resembles the ray concept in geometrical optics. Some examples are given to show this resemblance. Properties of the Wigner distribution function are discussed, showing, for instance, its relation to Heisenberg’s uncertainty principle and to some well‐known radiometric quantities. The concept of the Wigner distribution function is not restricted to deterministic signals; it can easily be extended to stochastic signals. Some examples of stochastic signals are considered. The propagation of signals through linear systems can readily be expressed in terms of Wigner distribution functions. Again, the description of systems by Wigner distribution functions can be interpreted directly in terms of geometrical optics: for Luneburg’s first‐order system, for instance, such a description immediately yields the ray transformation matrix of the system, and the relation between the input and the output Wigner distributioin function reads Fo(x,u) = Fi(Ax+Bu,Cx+Du). Transport equations for the Wigner distribution function in a general linear medium are derived. In the Liouville approximation they take the form of a first‐order partial differential equation, which allows, once more, a geometric‐optical interpretation: along the path of a geometric‐optical light ray the Wigner distribution function has a constant value.
Transformation laws for the Wigner distribution function, the radiant intensity, the radiant emittance, and the first- and second-order moments of the Wigner distribution function through an inhomogeneous, Kerr-type medium have been derived as well as for the beam quality factor and the kurtosis parameter. It is shown that the inhomogeneous Kerr-type medium can be approximated from the Wigner-distribution-function transformation-law point of view with a symplectic ABCD matrix with elements depending on the field distribution.
This tutorial gives an overview of the use of the Wigner function as a tool for modeling optical field propagation. Particular emphasis is placed on the spatial propagation of stationary fields, as well as on the propagation of pulses through dispersive media. In the first case, the Wigner function gives a representation of the field that is similar to a radiance or weight distribution for all the rays in the system, since its arguments are both position and direction. In cases in which the field is paraxial and where the system is described by a simple linear relation in the ray regime, the Wigner function is constant under propagation along rays. An equivalent property holds for optical pulse propagation in dispersive media under analogous assumptions. Several properties and applications of the Wigner function in these contexts are discussed, as is its connection with other common phase-space distributions like the ambiguity function, the spectrogram, and the Husimi, P, Q, and Kirkwood–Rihaczek functions. Also discussed are modifications to the definition of the Wigner function that allow extending the property of conservation along paths to a wider range of problems, including nonparaxial field propagation and pulse propagation within general transparent dispersive media.